Advance Elements of Laser Circuits and Systems: Nonlinear Applications in Engineering [1st ed. 2021] 3030641023, 9783030641023

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Ofer Aluf

Advance Elements of Laser Circuits and Systems Nonlinear Applications in Engineering

Advance Elements of Laser Circuits and Systems

Ofer Aluf

Advance Elements of Laser Circuits and Systems Nonlinear Applications in Engineering

Ofer Aluf Netanya, Israel

ISBN 978-3-030-64102-3 ISBN 978-3-030-64103-0 (eBook) https://doi.org/10.1007/978-3-030-64103-0 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book on laser circuits and systems: Nonlinearity applications in engineering covers and deals with two separate engineering and scientific areas and what between. It gives advance analysis methods for laser circuits and systems which represent many applications in engineering and science. Lasers circuits and systems come in many topological structures and represent many specific implementations which stand the target scientific and engineering features. The main types of lasers are categories by the lasing medium and are classified as either laser diode (semiconductor laser), Ti: Sapphire laser, ion-channel and log wavelength laser, solid-state laser, Nd: YAG, mid-infrared Q-switched microchip laser, gas laser, dual-wavelength laser, helium–neon laser, carbon dioxide laser, metal vapor laser, rare gas ion laser, excimer laser, chemical laser, and gas dynamic carbon dioxide laser. Additionally, there are dye laser, free-electron laser (FEL), X-ray laser, quantum cascade laser, lead salt laser, antimonide laser, and femtosecond laser. Ultrafast laser micromachining using picosecond or femtosecond laser can be used for the most challenging applications. Picosecond laser micromachining utilized a very short pulse laser (typically 8–12 picosecond duration), to prevent the formation of a heat-affected zone. Femtosecond laser micromachining system is used in ultrafast machining. Sometimes 20-fold reduction in pulse duration relative to a picosecond system gives advantages in terms of material interaction. By using picosecond or femtosecond laser, we can get a very high peak power density. These types of laser can be used to machine otherwise transparent materials such as silica or sapphire. Nonlinear absorption enables the light to interact with transparent materials, causing very strong, extremely localized heating and ablation. The strength of the laser ablation process results in a very low debris process. High-power (Peta watt, 1015 w) laser involves real-time controls and femtosecond precision timing systems. We can synchronize operation of lasers to specific beamline clock and get a broadband laser pulse stretchers with pulse compress. Laser diodes consist of a p–n diode with an active region where electrons and holes recombine resulting in light emission. In addition, a laser diode contains an optical cavity where stimulated emission takes place. The laser cavity consists of a waveguide terminated on each end by a mirror. There are some longitudinal modes to laser diode. Rate equations for each longitudinal mode (λ), with photon density Sλ and carrier density Nλ , are couple into specific mode. In the basic electrical equivalent v

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circuit of a laser diode, the effects of spontaneous emission and self-pulsations are included. The self-pulsations are represented by a negative resistance in the model. Application of this model suggests purely electronic methods of suppressing relaxation oscillations in laser diodes. When dealing with MRI technology, needs are addressed by fiber optics MRI system. The laser diode circuitry is the component responsible for the conversion of MR signals from electrical to the optical state. The vertical-cavity surface-emitting laser (VCSEL) employs current-blocking layers to funnel the current to the active region located between the mirrors of the laser. Ti: Sapphire lasers are tunable lasers which emit red and near-infrared light in the range from 650 to 1100 nm. These lasers are mainly used in scientific research because of their tenability and their ability to generate ultrashort pulses. The ion-channel laser (ICL) is an ultracompact version of the free-electron laser (FEL), with the undulator replaced by an ion channel. Analysis of the resulting scaled equations shows a realistic prospect of building a compact ICL source for fundamental wavelength down to UV and harmonics potentially extending to X-rays. A laser is capable of working in the terahertz range (long-wavelength light from the far infrared to one millimeter). It enables to examine better than can be done using chemical analysis. Solid-state laser is a laser that uses a gain medium that is a solid, rather than a liquid such as in dye lasers or a gas as in gas lasers. Semiconductor-based lasers are also in the solid state but are generally considered as a separate class from solid-state lasers. Quantum composers laser features three series of diode pumped, solid state, Nd: YAG lasers. Gas laser is a laser in which an electric current is discharged through a gas to produce coherent light. Lasers emitting simultaneously at multiple wavelengths and can fit for wide applications in many fields such as environmental monitoring, laser radar, spectral analysis, THz research, etc. It is a good solution if you need a special measure compare to traditional laser systems like one wavelength laser. The laser circuits and systems analyze as linear and nonlinear dynamical systems and their dynamics under parameters variations. This book is aimed at newcomers to linear and nonlinear dynamics and chaos laser system and circuits. The presentation stresses analytical and numerical methods, concrete examples, and geometric intuition. The laser systems and circuits analysis is developed systematically, starting with first-order differential equations and their bifurcation, followed by phase plane analysis, limit cycles and their bifurcations, chaos, iterated maps, period doubling, renormalization and strange attractors. Additionally, the book dealing with delayed laser circuits which characterized by overall variables delayed with time. Each variable has specific delay parameter and can be inspect for dynamics. More realistic laser circuit’s models should include some of the past states of laser circuits systems, that is, ideally, a real laser circuits should be modeled by differential equations with time delays. The use of delay differential equations (DDEs) in the modeling of laser type’s circuit’s dynamics is currently very active, largely due to progress achieved in the understanding of the dynamics of several classes of delayed differential equations and laser circuits and systems. This book is designed for advanced undergraduate or graduate students in electronics, physics, RF, and electronic engineering, mathematics who will interest in laser circuits dynamics and innovative analysis methods. It is also addressed to electrical and optic engineers, physics experts and researchers

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in physics, electronics, engineering, and mathematics who use dynamical systems as modeling tools in their studies. Therefore, only a moderate mathematical and electronic semiconductor background in geometry, linear algebra, analysis, and differential equations is required. Each book chapter includes various laser systems and circuits drawing and their equivalent analyses circuits. Laser circuits fixed points and stability analysis done by using much estimation. Various bifurcations laser systems and circuits are discussed. In this book, we try to provide the reader with explicit procedures for application of laser systems and circuits mathematical representations to particular research problems. Special attention is given to numerical implementation of the developed techniques. Let us briefly characterize the content of each chapter. Chapter 1. Dynamical and Nonlinearity of Laser Diode Circuits. In this chapter, laser diode coupled delay rate equations, laser diode intrinsic and package electrical equivalents circuits, MRI system, and electron photon exchanges into VCSEL are discussed. The laser diode is a p–n diode with an active region where electrons and holes recombine resulting in light emission. The elements of the intrinsic laser equivalent circuit are derived from the coupled rate equations which describe the interplay between the injected carrier and photon densities in the active region of the laser diode. The parameter τ is the latent period which is related to parasitic effects of the stimulated emission process. We draw the laser diode intrinsic and package electrical equivalent circuits and analyze the stability under parameters variation. MRI system laser diode is one element of fiber optic MRI system. One of these technologies is phased array coils with high number of elements. These are multichannel coils with each channel. A miniature fiber optic transmission (FOT) system is used and it contains a laser diode connected to a photodetector with fiber optic cabling. We inspect the diode laser circuitry under variation of delay parameters which are related to circuit’s internal circuitry microstrip lines. Vertical-cavity surface-emitting lasers (VCSELs) are a semiconductor lasers that have many advantages. Due to VCSEL active region parasitic effects, there is a time delay τ of the carrier–photon interaction time. The VCSEL rate equations include this time delay parameter, and under different values, there is a stability switching concerning the operation of VCSEL laser element. Chapter 2. Ti: Sapphire Laser Systems with Delay Parameters in Time Stability Analysis. In this chapter, sapphire laser systems are discussed. Ti: Sapphire lase (Ti: Al2 O3 laser, titanium-sapphire laser, or Ti: Sapphires) is a tunable laser which emit red and near-infrared light in the range from 650 nm to 1100 nm. In many scientific experiments, the Ti: Sapphire laser is used. It is very tenability and can generate ultrashort pulses. We use linearly polarized femtosecond laser pulses from a Ti: Sapphire laser to analyze the multiphoton ionization of D2 (D2 is a molecule of deuterium). There is a predictive model for the ionization rate of a diatomic molecule in the field of a short pulse with wavelength in the near-IR region. There are two equivalent models which give a good overlap with experimental results on the ionization of rate-gas atoms in the tunneling regime, interacting with femtosecond laser pulses. Shaping ultrashort pulses with a resolution approach 10 fs is done by

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a liquid-crystal spatial light modulator within reflective optics pulse-shaping apparatus. The principle is to use a spatial light modulator as a phase modulator. A spatial light modulator (SLM) is an object that imposes some form of spatially varying modulation on a beam of light. There is phase dispersion at higher orders, and especially cubic- and quartic-phase dispersions, stability analysis is done. Additionally, Ti: Sapphire laser can operate with mode of multiple pulse. It is mode locked by a semiconductor saturable absorber mirrors (SESAMs). The system (multipulse operation of Ti: Sapphire laser mode locked by an ion-implanted semiconductor saturable absorber mirror) differential equations are inspected for stability. Chapter 3. Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering. In this chapter, ion-channel and long-wavelength laser’s analysis is discussed. Ion-channel system includes relativistic electron beam propagating through plasma in the ion-focused regime. Growth is enhanced by optical guiding in the ion channel, which acts as a dielectric waveguide. It is a transport of relativistic electron beams (REBs) in plasmas and mechanism of ion focusing. The basic mechanism is laser-driven acceleration in ion channel. A long laser beam propagating through un-dense plasma produces a positively charged ion channel by expelling plasma electrons in the transverse direction. The perturbed equations of motion, average over betatron period, are inspected for stability and stability switching. The ion-channel laser perturbed system equations of motion are inspected for behavior analysis. A semiconductor laser (Laser diode) is a device that causes laser oscillation by the flowing an electric current to semiconductor. Practically, the mechanism of light emission is the same as a lighting-emitting diode (LED). The optical gain is produced in a semiconductor material. The nonlinear dynamics of the lasing nodes are described mathematically by multimode rate equations of the photon number Sp of the lasing modes and the injected electron number N and analysis is done. Chapter 4. Solid State Laser Nonlinearity Applications in Engineering. In this chapter, solid-state lasers are analyzed for best performances. A solid-state laser is a laser that uses a gain medium that is a solid. It is very different from liquid laser such as in dye lasers or a gas lasers. Lasers are commonly designed by the type of lasing material employed: Solid-state lasers have lasing material distributed in a solid matrix such as the ruby or neodymium “YAG” lasers. Solid-state lasers use optical pumping and such pump sources are relatively cheap and can provide very high powers. High-power diode lasers array can be utilized to pump solid-state laser materials efficiently. Laser materials with high intrinsic quantum efficiencies and an improved scalable cavity design make high output powers. The powerful and compact laser sources have fundamental research application such as X-ray, plasma, and higher-harmonic generation. The simplest laser model of a single-mode laser with homogeneously laser medium and saturate absorber is simulated and analyzed for stability behavior. In many applications, we use laser diode-pumped microchip solidstate laser (DPSSLs). One application is a self-aligned optical feedback vibrometry technique. The laser diode-pumped microchip solid-state laser is useful for compact, durable, coherent light sources. In laser, we have the phenomena of light injectioninduced. It is injection locking and return light-induced instabilities. Important is the basic properties of laser dynamics as well as their practical importance. One group of

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lasers is Nd stoichiometric lasers. Stoichiometric laser materials are pure chemical compounds capable of emitting coherent light in the undiluted state as opposed to conventional laser materials where the active ions or molecules are dispersed in a host. We characterize the light injection model for rotating-wave approximation fields and inspect stability. A combination of gain switching with passive Q-switching of a miniature diode-pumped solid-state laser is done. A composite pumping pulse, consisting of a long, low-intensity pulse and a following short, high-intensity pulse, is used to reduce the timing jitter. The simple model of a passively Q-switched laser is presented and inspected for stability. Chapter 5. Nd: YAG, Mid-Infrared and Q-Switched Microchip Lasers Stability Analysis. In this chapter, Nd: YAG, mid-infrared, and Q-switched lasers are analyzed for best performances and stability is discussed. A diode-pumped Nd: YAG laser is Q-switched by a GaAs saturable absorber and intra-cavity frequency doubled by a KTP crystal. At pump power, the device produces high quality pulses at specific pulse repetition rate. We characterize the dynamics of the pulse formation by rate equations and an energy-level model which accounts for the various energy-transfer process in GaAs. A diode-dumped Nd: YAG laser is Q-switched by a GaAs saturable absorber and intra-cavity frequency doubled by a KTP crystal. The dynamics of a pulse formation can be described by rate equations and an energy-level model which accounts for the various energy-transfer processes in GaAs and stability is discussed. Semiconductor saturable absorber mirror (SESAM) is a key component of ultrafast passive mode-locked laser sources. The key SESAM parameters are saturation fluence, modulation depth, and non-saturable losses are measurable with a high accuracy. We control these parameters to obtain stable pulse generation for a given laser. The model function for the nonlinear reflectivity is based on a simple two-level travelling wave system and the behavior is analyzed. Ion-doped crystalline laser mainly operates in the mid-IR spectral range between 2 and 5 μm. Types are rare-earth and transition-metal-based ionic crystals, and color-center lasers. They are compact all-solid-state room-temperature tunable sources, belonging to class of vibrionic lasers. The four-level scheme which characterized by balance equation for the population of the upper laser level n and dynamic is inspected. Chapter 6. Gas Laser Systems Stability Analysis Under Parameters Variation. In this chapter, gas laser systems are discussed and there stability behavior is analyzed. Lasing from molecular nitrogen is used in many scientific and industrial applications. The discharge pumped nitrogen laser, operating in a broad range of gas pressures, from several mill bars to the atmospheric pressure, and repetition rates from several hertz to several kilo hertz. It is robust source of high-power near-UV radiation. Achieving nitrogen lasing via remote excitation would pave the way to many potential applications. It is remotely initiated lasing from molecular gases by femtosecond filaments. The nitrogen laser emits radiation at 337 nm, and its active medium is gaseous nitrogen. The gas is confined within a pressure vessel, usually at a total pressure between 101 and 2700 Pa. The nitrogen laser generates pulses of duration between 300 ps (at atmospheric pressure) and 10 ns (at 2700 Pa). The repetition rate changes from 1 to 100 Hz. The dynamical behavior of nitrogen laser is discussed. Metal vapor lasers are devices in which the lasing medium is a vapor

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of metal atoms or ions, sometimes mixed with another gas. Metal vapor lasers use a variety of metal types to generate a variety of laser lines for applications throughout the electromagnetic spectrum. A self-consistent model copper vapor laser (CVL) is very helpful for characterization of the copper laser and its implementation. It uses vapors of copper as the lasing medium in a three-level laser. It can produce green laser light (510.6 nm) and yellow laser light (578.2 nm). The model for a high repetition rate copper vapor laser is described. Equations for the discharge pulse, laser pulse, and inter-pulse afterglow are discussed and analyzed for stability. Chapter 7. Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (I). In this chapter, filters systems in many circuits are inspected for dynamical behavior and stability analysis. Dual-wavelength lasers are not common in scientific facilities but some applications need to use them. Dual-wave lasers demonstrate simultaneous dual-wavelength lasing and outcome of operating in multiple wavelength is possible. A four-energy-level system of dual-wavelength Ti: Sapphire laser is characterized by rate equation. Two sub-resonators structure gives dualwavelength laser. The Ti: Sapphire laser system simultaneously generates sequence of femtosecond pulses at two independent wavelength regions. It shows a high degree of synchronization and reduced relative timing jitter. The analytic model is analyzed for stability. A vertical external-cavity surface-emitting laser (VECSEL) is a semiconductor laser based on a surface-emitting semiconductor gain chip and a laser resonator. VECSELs generate high optical power in diffraction limited beams and have wavelength versatility. VECSEL unit is constructed from a semiconductor gain chip and an external laser resonator, and additionally, there are arrangements for pumping and cooling. Er-doped fiber lasers (EDFLs) are a erbium-doped fiber amplifier (EDFA) which operate in the particular regime where coherent oscillation of ASE occurs due to feedbacks mean. EDFLs are used as sources for coherent light signal generation, while EDFAs are used as wave-wave amplifiers for coherent light signal generation. EDFLs can be pumped with compact, efficient laser diodes. A compact high-power yellow pulsed laser uses an intra-cavity sum-frequency mixing in a diodeend-pumped Q-switched Nd: YVO4 dual-wavelength laser. A three-mirror configuration forming two separate laser cavities is used to optimize the gain match for simultaneous dual-wavelength emission in Q-switched operation. A diode-pumped Nd-doped laser is a compact all-solid-state sources, in the blue, green, and red spectral regions by use of intra-cavity frequency doubling. These sources cover the region from 550 to 650 nm. The dynamic of yellow pulsed laser is discussed for stability. Chapter 8. Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (II). In this, we discuss a dual-wavelength laser system and stability analysis under parameters variation. The dynamic responses of the photon and carrier densities in an asymmetric dual quantum well laser exhibit wavelength switching. The laser consists of two different quantum wells isolated by a high and/or thick barrier layer that results in an inhomogeneous carrier injection into the two wells. The self-pulsing of the Tm3+ -doped silica fiber laser is operated near 2 μm. There are various self-pulsing regimes for a range of pumping rates where the fiber is endpumped with a high-power Nd: YAG laser which operated at 1.319 μm in a linear bidirectional cavity. There are a rich variety of nonlinear phenomena, ranging from

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self-pulsing to self-Q-switching and to a modulated quasi-cw wave. Quantum cascade (QC) lasers are semiconductor devices in which radiation is generated by electronic transitions within an artificial crystal, heterostructure crystal. This heterostructure consists of alternating layers of two semiconductor materials, with the layer thickness determining the electronic states inside the crystal, and thereby, the frequency of the emitted light as well as the electrical transport. The operation of QC lasers in the range of frequencies 1–10 THz is demonstrated by employing super lattice active material and developing a waveguide concept based on interface modes called surface Plasmon. Chapter 9. Laser Circuits and Systems Bifurcation Behaviors—Investigation, Comparison, and Conclusions. In this chapter, we summary the main topics regarding laser circuits and systems, inspect behavior, dynamics, stability, comparison, and conclusion. Laser systems and circuits are an integrated part of every industrial and scientific system. There are many laser source types which characterized by different parameters. Plasma physics is also an area which is integrated with laser systems and circuits. Laser systems and circuits can be represented by set of nonlinear differential equations with delays (DDEs) or without delays. The dynamic of laser systems and circuits is analyzed by nonlinear dynamic, fixed points, bifurcations; BOA, stability, and stability switching are inspected. Plasma diagnostic is also a very important in every laser system which has characterized by overall parameters and different diagnostic methods. Some laser systems and circuits models may involve some delay-dependent parameters. The presence of laser systems and circuits parameters often greatly complicates the task of an analytical study of such models. The laser systems and circuits stability of a given steady state is simply determined by the graphs of some functions of τ which can be expressed explicitly. Most application problems, we look at one such function and locate its zeros. This function has number of zeros, providing thresholds for stability switches. As time delay increases, stability changes from stable to unstable to stable, implying that a large delay can be stabilizing. Netanya, Israel

Ofer Aluf

Introduction

The laser stands for light amplifications by stimulated emission of radiation. We get laser by some excited atoms which emit photons then in turn stimulated other atoms to emit photons. To make a powerful laser, we trap atoms between two mirrors. This bounces the photons back and forth, increasing the stimulation of other atoms. The light from a laser is made up of one color (one wavelength). All the waves in light from a laser travel in the same direction, making a concentrated beam. The lasers have implementation like DVD players, medicine, eye surgery, space exploration, to drill holes in diamonds, communication by fiber optics, Internet and TV, etc. The electrons in atoms are special glasses, crystals, or gases absorb energy from an electrical current or another laser and become “excited.” The excited electrons move from a lower energy orbit to a higher-energy orbit around the atom’s nucleus. When they return to their normal or “ground” state, the electrons emit photons (particles of light). All photons are at the same wavelength and are “coherent,” meaning the crests and troughs of the light waves are all in lockstep. In contrast, ordinary visible light comprises multiple wavelengths and is not coherent. The particular wavelength of light is determined by the amount of energy released when the excited electron drops to a lower orbit and the laser light is directional. Whereas a laser generates a very tight beam, a flash light product light that is diffuse. Because laser light is coherent, it stays focused for vast distances, even to the moon and back. Modern lasers can produce pulses that are billions of times more powerful. Some lasers, such as ruby lasers, emit short pulses of light. Others, like helium–neon gas lasers or liquid dye lasers, emit light that is continuous. The ruby laser emits pulses of light lasting only billionths of a second. Laser light does not need to be visible and can be invisible infrared light and then pass through special optics that convert them to visible green light and then to visible, high-energy, ultraviolet light for optimum interaction with the target. There are much kind of lasers we can analyze and simulate the dynamics and stability, laser diodes, Ti: Sapphire laser systems, ionchannel and long-wavelength lasers, solid-state laser, Nd: YAG, mid-infrared, and Q-switched microchip lasers, gas lasers, and dual-wavelength laser systems. In laser diode circuits, we can characterize the laser diode intrinsic and package electrical equivalent circuit and analyze the stability under parameters variations. Each laser system or circuit can be representing by set of differential equations which depend xiii

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on laser system’s variable parameters. The laser system differential equations are related to the rate equations most of the times, or other process that happened in a particular laser type. The investigation of laser systems or circuits differential equation bifurcation theory, includes the study of possible changes in the structure of the orbits of a differential equation depending on variable parameters. The book illustrates certain observations and analyzes local bifurcations of an appropriate arbitrary scalar differential equation. Since the implicit function theorem is the main ingredient used in these generalizations, then it includes a precise statement of the theorem. Additional analyze the bifurcations of a laser system’s differential equation on the circle. Bifurcation behavior of specific differential equations can be encapsulated in certain pictures called bifurcation diagrams. This is done for optimization of lasers systems parameters and to get the best performance. Dynamics (chaos, fractals) change with laser systems that evolution in time. There are two types of dynamical systems: differential equations and iterated map (difference equations). Differential equation is described the evolution of a systems in continues time. Iterated map is arising in problems where the time is discrete. Differential equations can be divided to two main groups, ordinary differential equations and partial differential equations. Differential equation system can be represented as the below set. • dξ2 dξn dξi dξ1 = f 1 (ξ1 , . . . , ξn ); = f 2 (ξ1 , . . . , ξn ); . . . ; = f n (ξ1 , . . . , ξn ); ξi = dt dt dt dt

Some of laser systems or circuits can be represented as equation in dimensions “one.” Basic notions of laser system and circuit stability and bifurcations of vector fields are easily explained for scalar autonomous equations dimension one—because their flows are determined from the equilibrium points. Numerical solutions of each equations lead to scalar maps and show some of the “anomalies” albeit profound and exciting that may arise when numerical approximation is poor period doubling bifurcation, chaos, etc. Laser system rate equations can be turned to the dynamics and bifurcations of periodic solutions of non-autonomous equations with periodic coefficients dimension one and one half, where scalar maps reappear naturally as Poincare maps. Laser system schematic’s investigation the dynamics of planar autonomous equations—dimension two—where, in addition to equilibria, new dynamical behavior, such as periodic and homoclinic orbits, appears. Laser systems and circuit schematics stability of an equilibrium point is inspected. The subtle topological aspects of linear systems as well as the standard theory of Liapunov functions are discussed. Center manifolds and the method of Liapunov Schmidt are done to make a reduction to a scalar autonomous equation. Periodic orbit—Poincare—Andronov—Hopf bifurcation—and its analysis can be reduced to that of a non-autonomous periodic equation. Some of laser system’s rate differential equations or other laser phenomenon differential equations are delayed in time. The laser system rate differential equations variables delay in time are due to parasitic phenomenon inside the laser. The laser system rate delay differential equations in

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population dynamics incorporate a time delays and are the result of the existence of some stage structure. The laser system through stage survival rate is often a function of time delays, and the models may involve some delay-dependent parameters. The laser system analysis combines graphical information with analytical work to effectively analyze the local stability of some models involving delay-dependent parameters. The stability of a given steady state is simply determined by the graphs of some functions of τ which can be expressed explicitly and thus can be easily depicted by MATLAB software. We look at function and locate its zeros. This function often has only two zeros, providing thresholds for stability switches. The common scenario in laser system rate equation is that as time delay increases, stability changes from stable to unstable to stable, implying that a large delay can be stabilizing. We incorporate of a time delay which is the result of the existence of some stage structure. There are practical guidelines that combine graphical information with analytical to effectively analyze the local stability of laser models involving delay-dependent parameters. Generally, we analyze the occurrence of any possible stability switching resulting from the increase of value of the time delay τ for the general laser system characteristic equation D(λ, τ ) = 0. D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; Pn (λ, τ ) =

n  k=0

pk (τ ) · λk ; Q m (λ, τ ) =

m 

qk (τ ) · λk

k=0

n, m ∈ N0 ; n > m and pk (·), qk (·) : R+0 → R are continuous and differentiable function of τ .

Contents

1 Dynamical and Nonlinearity of Laser Diode Circuits . . . . . . . . . . . . . 1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Laser Diode Intrinsic and Package Electrical Equivalent Circuits Stability Analysis for Parameters Variation . . . . . . . . . . . . 1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dynamic of Electron-Photon Exchanges into VCSEL, Stability Optimization Under Delayed Carrier-Photon Interaction in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ti: Sapphire Laser Systems with Delay Parameters in Time Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire Laser Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire Laser Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . 2.3 Multipulse Operation of a Ti: Sapphire Laser Mode Semiconductor Saturable-Absorbed Mirror Pulse Energies and Gain Delayed in Time Stability Analysis . . . . . . . . . . . . . . . . . . 2.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Ion Channel Laser Perturbed Differential Equations of Motion Stability Optimization Under Delayed Variables in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Solid State Laser Nonlinearity Applications in Engineering . . . . . . . 4.1 Solid State Laser Controlled by Semiconductor Devices Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nanometer-Vibration Measurement with Microchip Solid-State Laser Instability Under Parameters Variation . . . . . . . . 4.3 Doppler-Shift with a Microchip Solid State Laser Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nd:YAG, Mid-Infrared and Q-Switched Microchip Lasers Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nd:YAG Laser Passively Q-Switched with GaAs Stability Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers Rate Equations Stability Analysis . . . . . . . . . . . . . . . . . . 5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations Instability Under Delayed Variables in Time . . . . . . . . . . . . . . . . . . . 5.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Gas Laser Systems Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Nitrogen Gas Laser Filament Plasma Kinetic Equations Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . 6.2 A Quasi-Two Level Analytic Model for Metal Vapor Laser System Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Self-consistent Model Copper Vapor Laser (CVL) Circuity Stability Analysis Under Parameters Variation . . . . . . . . .

227 228

275 314 330 338 339 341 373 414 432 456 468 469 471 499 521 540 552 553 554 638 655

Contents

6.4 A Self-consistent Model Copper Vapor Laser (CVL) Electron Density Upper and Lower Laser Levels Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser Stability Analysis Under Delay Variables in Time . . . . . . . . . . . . . . 7.2 Dual-Wavelength Emission from Vertical External-Cavity Surface-Emitting Laser Stability Analysis Under Delay Parameters in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Quasi-periodic in Erbium-Doped Fiber Laser Nonlinearity and Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser Analysis Under Delay Variables in Time . . . . . . . . . . . . . . . . . . . . . . 7.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength Lasing Stability Analysis Under Parameters Variation and Delay Variables in Time . . . . . . . . 8.2 Tm3+ -Doped Silica Fibre Lasers Nonlinearity and Stability Analysis Under Parameters Variation . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Terahertz Dual-Wavelength Quantum Cascade Laser Stability Analysis Under Parameters Variations and Delay Variables in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

676 696 705 707 708

763 784 798 821 830 831

832 867

890 923 935

9 Laser Circuits and Systems Bifurcation Behaviors Investigation, Comparison and Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

937 968

Appendix A: Laser X-ray Guiding System . . . . . . . . . . . . . . . . . . . . . . . . . .

969

Appendix B: Plasma Diagnostics Fundamental . . . . . . . . . . . . . . . . . . . . . .

997

Appendix C: Laser Beam Shaping, Jitter and Crosstalk . . . . . . . . . . . . . . 1025 Appendix D: Plasma Mirror System—Modeling, Characterization, Optic Design, Operation and Working Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

xx

Contents

Appendix E: High Power Laser/Target Diagnostic System Optical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

Chapter 1

Dynamical and Nonlinearity of Laser Diode Circuits

Laser diode consists of a p-n diode with an active region where electrons and holes recombine resulting in light emission. The laser diode contains an optical cavity where stimulated emission takes place. The laser cavity consists of a waveguide terminated on each end by a mirror. Other names to Laser Diode (LD) are injection laser diode or ILD. The laser diode is the type of laser produced with a wide range of uses that include fiber optic communications, barcode readers, laser pointers, etc. The laser diode complete equivalent circuit is separated into two parts. The first part represents the intrinsic electrical equivalent circuit of the laser chip. The second part is the electrical equivalent circuit of the package including the major parasitic elements. The elements of the intrinsic laser equivalent circuit are derived from the coupled rate equations which describe the interplay between the injected carrier and photon densities in the active region of the laser diode. Coupled rate equations parasitic effects of the stimulated emission process is talent period τ . The investigation of our laser diode coupled delay rate equations, differential equations is based on bifurcation theory, a study of possible changes in the structure of the orbits of a delay differential equations (DDEs) depending on τ parameter. Additionally, we check the laser diode intrinsic and package electrical equivalent circuit’s stability switching for parameters variation. An optical system which capable of transmitting MRI signals consist laser diode circuitry, responsible for the conversion of MR signal from the electrical to the optical state. We investigate the dynamic and stability of MRI system laser diode circuit under parameters variation. Laser diode transmitter RF section circuit contains 5 O microstrip, DC bias circuit, RFC, and additional discrete components, input signal is RF. The stability of the circuit is analyzed under parameters variation. The vertical-cavity surface-emitting laser (VCSEL) is a type of semiconductor laser diode with laser beam emission perpendicular from the top surface. VCSEL applications include fiber optic communications, precision sensing, etc. There are two main VCSEL’s variables, Carrier density in the active region N(t) and Photon density in the active region S(t). Due to VCSEL active region parasitic effects, there is a time delay (τ ) of the carrier-photon interaction time. The VCSEL rate equations with delay parameter (τ ) of the carrier-photon interaction time are © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_1

1

2

1 Dynamical and Nonlinearity of Laser Diode Circuits

analysed for stability switching under variation of (τ ) parameter. We review the VCSEL structure, characteristics and inspect deeply the main variables and parameters. First is to write the VCSEL rate equations with delay parameter (τ ) of the carrier-photon interaction time. We combine graphical information with analytical work to effectively study the local stability of the VCSEL model involving delay dependent parameters [1–7].

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis The laser diode consists of a p-n diode with an active region where electrons and holes recombine resulting in light emission. A laser diode contains an optical cavity where stimulated emission takes place. The laser cavity consists of a waveguide terminated on each end by a mirror. Photons, which are emitted into the waveguide, can travel back and forth in this waveguide provided they are reflected at the mirrors. The distance between the two mirrors is the cavity length, labeled L. The light in the waveguide is amplified by stimulated emission. Stimulated emission is a process where a photon triggers the radiative recombination of an electron and hole creating an additional photon with the same energy and phase as the incident photon. This “cloning” of photons results in a coherent beam. The stimulated emission process yields an increase in photons as they travel along the waveguide. The laser diode lasing condition is as follow, when combined with the waveguide losses, stimulated emission yields a net gain per unit length g. The number of photons can therefore be maintained if the roundtrip amplification in a cavity of length, L, including the partial reflection at the mirrors with reflectivity R1 and R2 equals unity (R1 · R2 · exp(2 · g · L) = 1). In case roundtrip amplification R1 · R2 · exp(2 · g · L) < 1 is less than one the number of photons steadily decreases. If the roundtrip amplification (R1 · R2 · exp(2 · g · L) > 1), the number of photons increases as the photons travel back and forth in the cavity and no steady  statevalue 1 would be obtained. The gain required for lasing equals: g = 2·L · ln R11·R2 . The process is as follow: initially, the gain is negative if no current is applied to the laser diode as absorption dominates in the waveguide. As the laser current is increased, the absorption first decreases and gain increases. The current for which the gain satisfies the lasing condition is the threshold current of the laser Ith . When below the threshold current very little light is emitted by the laser. If we applied current larger than the threshold current, the output power, Pout increases linearity with the applied ·(I − Ith ) where h ·ν is the energy per current. The output power equals: Pout = η· h·ν q photon. The factor η indicates that only a fraction of the generated photons contribute to the output power of the laser as photons are partially lost through the other mirror and through the waveguide. The laser diode consists of a cavity, which is the region between two mirrors with reflectivity R1 and R2 , and a gain medium (quantum wall). The optical mode originates in spontaneous emission, which is confined to the cavity

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

3

by the waveguide. The optical mode is amplified by the gain medium and partially reflected by the mirrors. The modal gain depends on the gain of the medium, multiplied with the overlap between the gain medium and the optical mode (confinement factor ). The modal gain is g(N ) · . Lasing occurs when the round trip optical gain equals the losses. Laser with modal gain g(N ) ·  and waveguide lossα, this condition implies: R1 · R2 · exp[2 · (g(N ) − α) · L] = 1 where  L is the length of the 1 1 cavity. The distributed loss of the mirrors is L · ln √ R ·R . 1 2 We need to analyze the dynamical of a double heterojunction laser diode circuits using scattering parameters [1]. The laser diode is highly coherent light output that can be modulated to carry coded information at a high density and speed along a fiber optic cable. The modulation response characteristics of laser diode is studied to find the optimum operation conditions. The modulation response of the laser diode is determined by solving the rate equations numerically. The disadvantage that it does not take into account the influence of the package parasitic and device circuit interactions on the modulation response. The solution is to use a circuit analysis based on the complete electrical circuit model of the conventional double heterostructure lasers or quantum well lasers. The complete equivalent circuit of a laser diode can be separated into two parts. The first part is the intrinsic electrical equivalent circuit of the laser chip and the second part is the electrical equivalent circuit of the package including the major parasitic elements. The elements of the intrinsic laser equivalent circuit are derived from the coupled rate equations which describe the interplay between the injected carrier and photon densities in the active region of the laser diode. We define the parasitic effects of the stimulated emission process as a latent period τ . The laser diode coupled single-mode delayed differential rate equations are given by the following DDEs (Delay Differential Equations), Ne = Ne (t); N ph =N ph (t); Ni = Ni (t): 1 d Ne (t) Ne (t) = − A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) − dt q ·a·d τs d N ph (t) N ph (t) Ne (t) = A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) − +β · dt τ ph τs where variable N ph is the photon density. Variable Ne is the electron density, q is the electronic charge, d is the thickness of the active region, a is the area of the diode contact stripe, I is the injected current, Nom is the minimum electron density to obtain a positive gain, A is a constant related to the stimulated emission process, τs is the spontaneous emission lifetime, τ ph is the photon lifetime and β is the fraction of the spontaneous emission that is coupled to the lasing mode. The model differential equations assume that the inversion is homogenous and the gain is linear in the difference between Ne and Nom . The carrier density Ne is a function of the junction voltage, where Ni is the intrinsic carrier  and V is the junction voltage.   density  The q·V q·V carrier density equation: Ne = Ni · exp 2·k·T ; Ne (t − τ ) = Ni (t − τ ) · exp 2·k·T

4

1 Dynamical and Nonlinearity of Laser Diode Circuits

   d Ne (t) d Ni (t) q·V q·V ; = · exp 2·k·T dt dt 2·k·T     1 q·V d Ni (t) q·V = · exp − − A · exp − · dt q ·a·d 2·k·T 2·k·T     q·V Ni (t) − Nom · N ph (t − τ ) − Ni (t − τ ) · exp 2·k·T τs     d N ph (t) q·V = A · Ni (t − τ ) · exp − Nom · N ph (t − τ ) dt 2·k·T   N ph (t) β q·V − + · Ni (t) · exp τ ph τs 2·k·T 

Ne (t) = Ni (t) · exp

We get two possible sets of differential equations: 1 No.1: d Ndte (t) = q·a·d − A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) −

Ne (t) τs

d N ph (t) N ph (t) Ne (t) = A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) − +β · dt τ ph τs To find the equilibrium points (fixed points) of the Laser diode coupled delay rate equation: limt→∞ Ne (t − τ ) = limt→∞ Ne (t); limt→∞ N ph (t − τ ) = limt→∞ N ph (t); t  τ ; t − τ ≈ t ∀ t → ∞.   d N ph (t) 1 d Ne (t) N∗ ∗ = 0; = 0; − A · Ne∗ − Nom · N ph − e =0 dt dt q ·a·d τs ∗ N ph     N∗ ∗ A · Ne∗ − Nom · N ph − + β · e = 0 ⇒ A · Ne∗ − Nom · τ ph τs ∗ N ph N∗ ∗ = −β · e N ph τ ph τs

    1 N∗ ∗ − A · Ne∗ − Nom · N ph − e = 0 ⇒ A · Ne∗ − Nom · q ·a·d τs ∗ N 1 ∗ − e N ph = q ·a·d τs We can write the equation:

∗ N ph τ ph

−β ·

Ne∗ τs

=

1 q·a·d

−

Ne∗ τs

 ∗ ∗ N ph N ph Ne∗ 1 τs 1 ∗ ⇒ Ne = · − − (β − 1) = τs τ ph q ·a·d τ ph q ·a·d (β − 1)   ∗  N ph 1 1 1 τs ∗ − Nom · N ph · − − · − A· q ·a·d τ ph q ·a·d τs (β − 1)

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

 ∗ N ph τs 1 =0 · − τ ph q ·a·d (β − 1) 2  ∗  A · τs · N ph 1 1 τs ∗ − + Nom · N ph · + A· q ·a·d (β − 1) · τ ph (β − 1) q · a · d ∗ N ph 1 1 1 =0 · · − + (β − 1) τ ph (β − 1) q · a · d 2  ∗ 

 A · τs · N ph 1 1 1 τs ∗ − · N ph + Nom − · · + A· (β − 1) · τ ph (β − 1) q · a · d (β − 1) τ ph 1 1 1 + =0 · + q ·a·d (β − 1) q · a · d 2  ∗ 

 A · τs · N ph 1 1 1 τs ∗ · N ph + Nom · · + A· (1 − β) · τ ph (β − 1) q · a · d (β − 1) τ ph  1 1 · 1− =0 + q ·a·d (1 − β) 2  ∗

  A · τs · N ph 1 1 τs 1 ∗ + Nom + · · + A· − · N ph (1 − β) · τ ph (1 − β) q · a · d (1 − β) τ ph  1 1 · 1− =0 + q ·a·d (1 − β) 2  ∗ 

 A · τs · N ph 1 1 1 τs ∗ − Nom · N ph · · + − A· (1 − β) · τ ph (1 − β) τ ph (1 − β) q · a · d  1 1 · 1− =0 + q ·a·d (1 − β) −2 ± 22 − 4 · 1 · 3  ∗ 2 ∗ ∗ ∗ (1) N ph · 1 + N ph · 2 + 3 = 0; N ph = ; N ph 2 · 1 −2 + 22 − 4 · 1 · 3 = 2 · 1 −2 − 22 − 4 · 1 · 3 ∗ (2) N ph = ; (Ne∗ )(1) 2 · 1 ⎡ ⎛ ⎞ ⎤ − + 22 − 4 · 1 · 3 2 1 τs 1 ⎠− ⎦ ·⎣ = ·⎝ (β − 1) τ ph 2 · 1 q ·a·d

5

6

1 Dynamical and Nonlinearity of Laser Diode Circuits



Ne∗

(2)

⎡ =

⎛

1 ⎝ τs ·⎣ · τ ph (β − 1)

−2 −

⎞ ⎤ 22 − 4 · 1 · 3 1 ⎠− ⎦ 2 · 1 q ·a·d

The first fixed point:   ∗ (1) ,  N ∗ (1) E (1) N ph e ⎡ ⎛ ⎛ ⎞ ⎤⎞ −2 + 22 − 4 · 1 · 3 −2 + 22 − 4 · 1 · 3 1 τ 1 ⎦⎠ s ⎣ ⎝ ⎝ ⎠ · = , · − 2 · 1 τ ph 2 · 1 q ·a·d (β − 1)

The second fixed point:   ∗ (2) ,  N ∗ (2) E (2) N ph e ⎡ ⎛ ⎛ ⎞ ⎤⎞ −2 − 22 − 4 · 1 · 3 − 22 − 4 · 1 · 3 − 2 1 τ 1 ⎦⎠ s ⎣ ⎝ ⎝ ⎠ · = , · − 2 · 1 τ ph 2 · 1 q ·a·d (β − 1)

Stability analysis: The standard local stability analysis about any one of the equilibrium point  of Laser diode delayed rate equations consists in  adding  to its coordinates  N ph Ne arbitrarily small increments of exponential form n ph n e ·eλ·t , and retaining the first order terms in n ph , n e . The system of two homogeneous equations leads to polynomial characteristics equation in the eigenvalue λ. The polynomial characteristic equations accept by set the below photon density and electron density equations. The photon density and  density fixed values with arbitrarily small increments  electron of exponential form n ph n e ·eλ·t are: i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc. (i) + n ph · eλ·t ; Ne (t) = Ne(i) + n e · eλ·t ; N ph (t − τ ) N ph (t) = N ph (i) + n ph · eλ·(t−τ ) = N ph

Ne (t − τ ) = Ne(i) + n e · eλ·(t−τ ) ;

d N ph (t) d Ne (t) = n ph · λ · eλ·t ; = n e · λ · eλ·t dt dt

d N ph (t − τ ) d Ne (t − τ ) = n ph · λ · eλ·t · e−λ·τ ; = n e · λ · eλ·t · e−λ·τ dt dt We choose the above expressions for our N ph (t), Ne (t) as small displacement   (i) + n ph n e from the system fixed points at time t = 0: N ph (t = 0) = N ph n ph ; Ne (t = 0) = Ne(i) + n e for t > 0, λ < 0 the selected fixed point is stable otherwise t > 0, λ > 0 is unstable. Our system tends to the selected fixed point exponentially for t > 0, λ < 0 otherwise go away from the selected fixed point exponentially. The eigenvalue λ parameter establishes if the fixed point is stable or

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

7

Table 1.1 N ph , Ne expressions for eigenvalue λ and time parameter λ0

N ph (t = 0) =

t >0

(i) N ph

N ph (t = 0) = N ph + n ph

Ne (t = 0) = Ne(i) + n e

Ne (t = 0) = Ne(i) + n e

(i)

t →∞

(i)

+ n ph

(i)

N ph (t) = N ph + n ph · e−|λ|·t

N ph (t) = N ph + n ph · e|λ|·t

Ne (t) = Ne(i) + n e · e−|λ|·t

Ne (t) = Ne(i) + n e · e|λ|·t

(i) N ph (t → ∞) = N ph

N ph (t → ∞) n ph · e|λ|·t

Ne (t → ∞) = Ne(i)

Ne (t → ∞) n e · e|λ|·t

unstable, additionally his absolute value |λ| establishes the speed of flow toward or away from the selected fixed point [8, 9] (Table 1.1). The speeds of flow toward or away from the selected fixed point for Laser diode delayed rate equations: 1 d Ne (t) Ne (t) = − A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) − dt q ·a·d τs

    1 (i) + n ph · eλ·t · e−λ·τ − A · Ne(i) + n e · eλ·t · e−λ·τ − Nom · N ph q ·a·d  1  (i) − · Ne + n e · eλ·t τs      1 (i) = − A · Ne(i) − Nom + n e · eλ·t · e−λ·τ · N ph + n ph · eλ·t · e−λ·τ q ·a·d 1 1 − · Ne(i) − · n e · eλ·t τs τs

n e · λ · eλ·t =

n e · λ · eλ·t

1 n e · λ · eλ·t = q ·a·d     (i) (i) − A · Ne(i) − Nom · n ph · eλ·t · e−λ·τ − A · Ne − Nom · N ph (i) − A · N ph · n e · eλ·t · e−λ·τ − A · n e · n ph · eλ·t · e−λ·τ · eλ·t · e−λ·τ 1 1 − · Ne(i) − · n e · eλ·t τs τs

  1 1 (i) − A · Ne(i) − Nom · N ph − · Ne(i) q ·a·d τs   − A · Ne(i) − Nom · n ph · eλ·t · e−λ·τ

n e · λ · eλ·t =

(i) − A · N ph · n e · eλ·t · e−λ·τ − A · n e · n ph · e2·λ·t · e−2·λ·τ 1 − · n e · eλ·t τs

8

1 Dynamical and Nonlinearity of Laser Diode Circuits

n e · n ph multiplication is very small n e · n ph → ε   1 1 (i) n e · λ · eλ·t = − A · Ne(i) − Nom · N ph − · Ne(i) q ·a·d τs   − A · Ne(i) − Nom · n ph · eλ·t · e−λ·τ 1 (i) − A · N ph · n e · eλ·t · e−λ·τ − · n e · eλ·t τs At fixed point:

1 q·a·d

  (i) − A · Ne(i) − Nom · N ph −

1 τs

· Ne(i) = 0

  n e · λ · eλ·t = −A · Ne(i) − Nom · n ph · eλ·t · e−λ·τ 1 (i) − A · N ph · n e · eλ·t · e−λ·τ − · n e · eλ·t τs Dividing the two sides of the above equation by eλ·t term −A ·



Ne(i)



− Nom · n ph · e

−λ·τ

 1 (i) −λ·τ + n e −A · N ph · e − −λ =0 τs

d N ph (t) N ph (t) Ne (t) = A · (Ne (t − τ ) − Nom ) · N ph (t − τ ) − +β · dt τ ph τs     (i) n ph · λ · eλ·t = A · Ne(i) + n e · eλ·(t−τ ) − Nom · N ph + n ph · eλ·(t−τ )   1  (i) 1  − · N ph + n ph · eλ·t + β · · Ne(i) + n e · eλ·t τ ph τs     (i)  n ph · λ · eλ·t = A · Ne(i) − Nom + n e · eλ·(t−τ ) · N ph + n ph · eλ·(t−τ ) 1 1 1 1 (i) · N ph − · n ph · eλ·t + β · · Ne(i) + β · · n e · eλ·t τ ph τ ph τs τs   (i) n ph · λ · eλ·t = A · Ne(i) − Nom · N ph   (i) + A · Ne(i) − Nom · n ph · eλ·(t−τ ) + A · N ph · n e · eλ·(t−τ ) 1 1 (i) + A · n e · n ph · e2·λ·(t−τ ) − · N ph − · n ph · eλ·t τ ph τ ph 1 1 + β · · Ne(i) + β · · n e · eλ·t τs τs −

n e · n ph Multiplication is very small n e · n ph → ε   (i) n ph · λ · eλ·t = A · Ne(i) − Nom · N ph   (i) (i) + A · Ne − Nom · n ph · eλ·(t−τ ) + A · N ph · n e · eλ·(t−τ )

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

9

1 1 1 1 (i) · N ph − · n ph · eλ·t + β · · Ne(i) + β · · n e · eλ·t τ ph τ ph τs τs   (i) n ph · λ · eλ·t = A · Ne(i) − Nom · N ph 1 1 (i) − · N ph + β · · Ne(i) τ ph τs   (i) (i) + A · Ne − Nom · n ph · eλ·(t−τ ) + A · N ph · n e · eλ·(t−τ ) 1 1 − · n ph · eλ·t + β · · n e · eλ·t τ ph τs

−

  (i) At fixed point: A · Ne(i) − Nom · N ph −

1 τ ph

(i) · N ph +β ·

1 τs

· Ne(i) = 0

  n ph · λ · eλ·t = A · Ne(i) − Nom · n ph · eλ·t · e−λ·τ 1 1 (i) + A · N ph · n e · eλ·t · e−λ·τ − · n ph · eλ·t + β · · n e · eλ·t τ ph τs Dividing the two sides of the above equation by eλ·t term   1 1 (i) n ph · λ = A · Ne(i) − Nom · n ph · e−λ·τ + A · N ph · n e · e−λ·τ − · n ph + β · · ne τ ph τs

      1 1 (i) =0 n ph · A · Ne(i) − Nom · e−λ·τ − − λ + n e · A · N ph · e−λ·τ + β · τ ph τs We can summary our system arbitrarily small increments equation: 

 1 n ph · A · − Nom · e − −λ τ ph   1 (i) =0 · e−λ·τ + β · + n e · A · N ph τs

   1 (i) −A · Ne(i) − Nom · e−λ·τ · n ph + n e −A · N ph · e−λ·τ − − λ = 0 τs 

Ne(i)



−λ·τ

We get the Laser diode delayed rate equations eigenvalue matrix: 

       (i) A · Ne(i) − Nom · e−λ·τ − τ1ph − λ A · N ph · e−λ·τ + β · τ1s 0 n ph  −λ·τ  (i) = · (i) 1 −λ·τ n 0 −A · Ne − Nom · e −A · N ph · e − τs − λ e     (i) · e−λ·τ + β · τ1s A · Ne(i) − Nom · e−λ·τ − τ1ph − λ A · N ph   (A − λ · I ) = (i) −A · Ne(i) − Nom · e−λ·τ −A · N ph · e−λ·τ − τ1s − λ

10

1 Dynamical and Nonlinearity of Laser Diode Circuits

det(A − λ · I ) 

   (i) A · Ne(i) − Nom · e−λ·τ − τ1ph − λ A · N ph · e−λ·τ + β · τ1s   = 0 ⇒ det =0 (i) −A · Ne(i) − Nom · e−λ·τ −A · N ph · e−λ·τ − τ1s − λ       1 1 (i) A · Ne(i) − Nom · e−λ·τ − − λ · −A · N ph · e−λ·τ − − λ τ ph τs    −λ·τ  (i) 1 (i) =0 · A · N ph · e−λ·τ + β · + A · Ne − Nom · e τs    (i)  (i) −A2 · N ph · Ne − Nom · e−2·λ·τ − τ1s · A · Ne(i) − Nom · e−λ·τ   −λ · A · Ne(i) − Nom · e−λ·τ (i) (i) + τ1ph · A · N ph · e−λ·τ + τ ph1·τs + τ1ph · λ + λ · A · N ph · e−λ·τ +λ · τ1s + λ2   (i) +A · Ne(i) − Nom · A · N ph · e−2·λ·τ  (i)  1 +A · Ne − Nom · β · τs · e−λ·τ = 0

    (i) (i) −A2 · N ph · Ne(i) − Nom · e−2·λ·τ + A2 · Ne(i) − Nom · N ph · e−2·λ·τ      (i) (i) + A · Ne(i) − Nom · τ1s · (β − 1) + τ 1ph · A · N ph + λ · A · N ph − Ne(i) + Nom · e−λ·τ   + τ ph1·τs + τ 1ph + τ1s · λ + λ2 = 0

   (i)  (i) (i) −A2 · N ph · Ne − Nom · e−2·λ·τ + A2 · Ne(i) − Nom · N ph · e−2·λ·τ = 0 1 + τ ph · τs



1 1 + τ ph τs





 1 1 (i) · λ + λ2 + A · [Ne(i) − Nom ] · · (β − 1) + · N ph τs τ ph

(i) + λ · A · [N ph − Ne(i) + Nom ]} · e−λ·τ = 0

D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ n = 2; m = 1; n < m; Pn (λ, τ ) =  

1 + τ ph · τs



1 1 + τ ph τs



 1 1 (i) · N ph Ne(i) − Nom · · (β − 1) + τs τ ph   (i) − Ne(i) + Nom + λ · A · N ph

Q m (λ, τ ) = A ·

The expression for Pn (λ, τ ): Pn (λ, τ ) =

n=2 

λk · pk (τ )

k=0

p0 (τ ) =

1 1 1 ; p1 (τ ) = + ; p2 (τ ) = 1 τ ph · τs r ph τs

· λ + λ2 

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

The expression for Q m (λ, τ ): Q m (λ, τ ) =

m=1 

11

λk · qk (τ )

k=0



 1 1 (i) − Nom · · (β − 1) + · N ph q0 (τ ) = A · τs τ ph   (i) − Ne(i) + Nom q1 (τ ) = A · N ph 

Nc(i)



The coefficients pi , q j are not depend on τ parameter; i = 0,1, n ; j = 0, m The homogeneous system for N ph , Ne leads to a characteristic equation for the 2  eigenvalue λ having the form P(λ, τ ) + Q(λ, τ ) · e−λt = 0; P(λ) = aj · λj j=0    Q(λ) = 1j=0 c j · λ j . The coefficients a j (qi , q1 , τ ), c j (qi , qk , τ ) ∈  depend on qi , qk and delay τ. The parameters qi , qk are any Laser diode delayed rate equation’s parameters, other parameters kept as a constant. 1 1 1 ; a1 = + ; a2 = 1 τ ph · τs τ ph τs    1  (i) 1 (i) · N ph c0 = A · Nc − Nom · · (β − 1) + τs τ ph   (i) c1 = A · N ph − Ne(n) + Nom

a0 =

Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, aj , and cj . The coefficients aj , cj are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for q1 , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0 Furthermore P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [2, 4] can also be verified in the present case: (a) If λ = i · ω; ω ∈  then P(i · ω) + Q(i · ω) = 0. (b) If |Q(λ)/P(λ)| is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0. Pn (λ = i · ω, τ ) =  Q m (λ = i · ω, τ ) = A ·

1 − ω2 + τ ph · τs 

Ne(i)



1 1 + τ ph τs

 ·i ·ω

 1 1 (i) − Nom · · (β − 1) + · N ph τs τ ph



12

1 Dynamical and Nonlinearity of Laser Diode Circuits

  (i) + i · ω · A · N ph − Ne(i) + Nom Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )    1  (i) 1 1 (i) = + A · Ne − Nom · · (β − 1) + · N ph − ω2 τ ph · τs τs τ ph 

   1 1 (i) (i) + A · N ph − Ne + Nom · i · ω = 0 + + τ ph τs 2    1 1 1 2 |P(i · ω, τ )|2 = − ω2 + + · τ ph · τs τ ph τs    1 1 1 2 1 2 + −2· ω = · ω2 + ω4 2 + τ τ τ · τ ph s ph s τ ph · τs  

 1 1 (i) |Q(i · ω, τ )| = A · − Nom · · (β − 1) + · N ph τs τ ph 2  (i) + ω2 · A2 · N ph − Ne(i) + Nom 2

2

Ne(i)

2

1 F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 =  2 τ ph · τs 2   1  (i) 1 (i) 2 − A · Ne − Nom · · (β − 1) + · N ph τs τ ph   !  2  1 1 2 1 (i) 2 (i) + + −2· − A · N ph − Ne + Nom · ω2 + ω4 τ ph τs τ ph · τs We define the following parameters for simplicity:

" " " 0, 2, 4

2   (i)  1 1 1 (i) 2 = − A · N − N · − 1) + · N · (β  om e ph 2 0 τs τ ph τ ph · τs    2 #  # 1 1 2 1 (i) = + −2· − Ne(i) + Nom ; =1 − A2 · N ph 2 4 τ ph τs τ ph · τs #

Hence F(ω, τ ) = 0 implies solving the polynomial.

2 k=0

" 2·k

· ω2·k = 0 and its roots are given by

1 − ω2 ; PI (i · ω, τ ) = PR (i · ω, τ ) = τ ph · τs



1 1 + τ ph τs

 ·ω

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

 

 1 1 (i) · N ph Ne(i) − Nom · · (β − 1) + τs τ ph   (i) Q I (i · ω, τ ) = ω · A · N ph − Ne(i) + Nom

Q R (i · ω, τ ) = A ·

sin θ (τ ) =

13



−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2

cos θ (τ ) = −

PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) |Q(i · ω, τ )|2

We use different parameters terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 2 m = 1; n >; m; Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

2 $

a j · λ j ; Q(λ) =

j=0

1 $

c j · λ j ; P(λ)

j=0

= a0 + a1 · λ + a2 · λ ; Q(λ) = c0 + c1 · λ 2

n, m ∈ N0 ; n > m and a j , c j : R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “-” denotes complex and conjugate. functions of λ and differentiable in τ. The coefficients  P(λ), Q(λ) are analytic    a j Nom , τs , τ ph , β, . . . ∈ R and c j Nom , τs , τ ph , β, . . . ∈ R depend on Laser diode delayed rate equations systems Nom , τs , τ ph , β, . . . values. Unless strictly necessary, the designation of the varied arguments: (Nom , τs , τ ph , β, τ, . . .) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments and direct substitution show that a0 + c0 = 0 [2, 3].    1  (i) 1 1 (i) ; c0 = A · Ne − Nom · · (β − 1) + · N ph a0 = τ ph · τs τs τ ph    1  1 1 (i) = 0 + A · Ne(i) − Nom · · (β − 1) + · N ph τ ph · τs τs τ ph ∀ Nom , τs , τ ph , β, τ, . . . ∈ R+ i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see [2], Sect. 1.3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire Nom , τs , τ ph , β, τ, . . . domain of interest.

14

1 Dynamical and Nonlinearity of Laser Diode Circuits

P(λ) (b) | Q(λ) | is bounded for |λ| → ∞; Re(λ) ≥ 0. No roots bifurcation from ∞.

1 ·λ Indeed, in the limit | Q(λ) | = | a0 +ac01+c |. P(λ) ·λ+a2 ·λ2 (c) The following expression exists: F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 = 2 "  2·k has at most a finite number of zeros. Indeed, this is a polynomial 2·k · ω

k=0

in ω (degree in ω4 ).   (d) Each positive root ω Nom , τs , τ ph , β, τ, . . . of F(ω) = 0 is continuous and differentiable with respect to Nom , τs , τ ph , β, τ, . . . The condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i · ω) Q(−i · ω) = Q(i · ω) thus, ω > 0 maybe on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ - plane, whereby increasing the change its sign from parameters: Nom , τs , τ ph , β, τ, . . . Re(λ)  may, atthe crossing,    (2)  ∗ (1) ∗ (2) ∗ (1) (1) (2) N ph , Ne or E N ph , Ne∗ (−) to (+), i.e. from a stable focus E to an unstable one or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to Nom , τs , τ ph , β, τ, . . . and system parameters.  ∂Reλ ; a, d, q, A, τs , τ ph , β, τ ∂ Nom λ=iω   ∂Reλ Λ−1 (a) = ; Nom , d, q, A, τs , τ ph , β, τ ∂a λ=iω   ∂Reλ −1 Λ (d) = ; a, Nom , q, A, τs , τ ph , β, τ ∂d λ=iω   ∂Reλ Λ−1 (q) = ; a, d, Nom , A, τs , τ ph , β, τ ∂q λ=iω   ∂Reλ Λ−1 (A) = ; a, d, q, Nom , τs , τ ph , β, τ ∂ A λ=iω   ∂Reλ −1 Λ (τs ) = ; a, d, q, A, Nom , τ ph , β, τ ∂τs λ=iω     ∂Reλ Λ−1 τ ph = ; a, d, q, A, τs , Nom , β, τ ∂τ ph λ=iω   ∂Reλ Λ−1 (β) = ; a, d, q, A, τs , τ ph , Nom , τ ∂β λ=iω   ∂Reλ −1 Λ (τ ) = ; a, d, q, A, τs , τ ph , β, Nom ∂τ λ=iω Λ−1 (Nom ) =



= const = const = const = const = const = const = const = const = const

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

15

P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ) When writing and inserting λ = i · ω into Laser diode delayed rate equations characteristic equation ω must satisfy the following equations. sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω)|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equations that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note:F(ω) is dependent on τ. Now it is important to notice that if τ ∈ / I , assume / I ,ω(τ ) that I ⊆ R0+ is the set where ω(τ ) is a positive root of F(ω) and for τ ∈ is not defined. Then for all τ in I , ω(τ ) is satisfied that F(ω) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we can not have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of as the solution of sin θ(τ ) = . . . ; cos θ (τ ) = . . .. sin θ (τ ) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos θ (τ ) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

And the relation between the argument θ(τ ) and τ · ω(τ ) for τ ∈ I must be described ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ N0 . Hence we can define the maps ; n ∈ N0 ; τ ∈ I . Let us introduce the τn : I → R0+ given by τn (τ ) = θ(τ )+2·n·π ω(τ ) functions: I → R Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ. In the following, the subscripts λ, ω, a, d, q, A, τs , τ ph , β, . . . indicate the   Lut us first concentrate on Λ(x), remember in  corresponding partial derivatives. λ a, d, q, A, τs , τ ph , β, . . . and ω a, d, q, A, τs , τ ph , β, . . . and keeping all parameters except one (x) and τ. The derivation closely follows that in the reference [BK]. Differentiating Laser diode delayed rate equations characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = a, d, q, A, τs , τ ph , β, . . . 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω)

16

1 Dynamical and Nonlinearity of Laser Diode Circuits

i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains: 

∂λ ∂x

=

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = QR + i · QI Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 When (x) can be any Laser diode delayed rate equations parameters a, d, q, A, τs , τ ph , β, . . . and time delay τ etc. where for convenience, we have dropped the arguments (i · ω, x) and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] x . We Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = −F Fω define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ   1 1 1 2 PR = ·ω − ω ; PI = + τ ph · τs τ ph τs    1  (i) 1 (i) Q R = A · Ne − Nom · · (β − 1) + · N ph τs τ ph   (i) − Ne(i) + Nom ; PRω = −2 · ω Q I = ω · A · N ph     1 1 (i) PI ω = ; Q Rω = 0; Q I ω = A · N ph + − Ne(i) + Nom τ ph τs PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; PRω ·   1 2 − ω ; Q Rω · Q R = 0 PR = −2 · ω · τ ph · τs   Fτ = 2 · (PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I z · Q I ) = 0

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

17

   1 1 1 ; PI · − ω2 · + τ ph · τs τ ph τs   1 1 + PRω = −2 · ω2 · τ ph τs      1  (i) 1 (i) (i) 2 = A · Ne − Nom · · (β − 1) + · N ph · N ph − Ne(i) + Nom τs τ ph 

PR · PI ω =

QR · QIω

V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 F(ω, τ ) = 0. Differentiating with respect to τ and we get ∂ω ∂ω + Fτ = 0; τ ∈ I ⇒ ∂τ ∂τ  ∂ω Fz ∂Reλ Fτ = ωτ = − ; = − ; Λ−1 (τ ) = Fω ∂τ λ−iω ∂τ Fω !   −2 · U + τ · |P|2 + i · Fω   Λ−1 (τ ) = Re Fτ + i · 2 · V + ω · |P|2  

  −1  ∂Reλ sign Λ (τ ) = sign ∂τ λ=iω

Fω ·

  V + ∂ω ·U ∂ω ∂τ ·τ sign Λ−1 (τ ) = sign{Fω } · sign +ω+ |P|2 ∂τ

!

We shall presently examine the possibility of stability transitions (bifurcations) diode delayed rate equations system, about the equilibrium point  Laser   ∗ E (∗) N ph , Ne∗ as a result of a variation of delay parameter τ . The analysis consists in identifying the roots of our system characteristics equation situated on the imaginary axis of the complex λ-plane where by increasing the delay parameter τ , Re λ may at the crossing, changes its sign from − to +, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [4]. Λ−1 (τ ) =



∂Reλ ∂τ

 λ=i·ω

; a, d, q, A, τs , τ ph , β, Nom = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω )

18

1 Dynamical and Nonlinearity of Laser Diode Circuits



      1 1 1 1 1 2 2 U= −ω · + + + ·2·ω τ ph · τs τ ph τs τ ph τs 

    (i)  1 1 (i) (i) (i) − A · Ne − Nom · · (β − 1) + · N ph · A · N ph − Ne + Nom τs τ ph We get the expression for F(ω, τ ) Laser diode delayed rate equations parameter values. We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. Parameter τ has possible values [0.001…10] second (τ ∈ [0.001 . . . 10]). It can be express by 3D function F(ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Laser diode delayed rate equations system. 

−1

Λ (τ ) =  =

∂Reλ ∂τ ∂Reλ ∂τ

 λ=i·ω



λ=i·ω

!   −2 · U + τ · |P|2 + i · Fω  ; Λ−1 (τ )  = Re Fτ + 2 · i · V + ω · |P|2      2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 =  2 Fτ2 + 4 · V + ω · P 2

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . ; cos(τ ) = . . . when ω = ω+ (τ ) if only ω+ is feasible. Additionally, when all Laser diode delayed rate equations parameters are known the stability switch due to various time delay values τ is described in the following expression:   sign Λ−1 (τ ) = sign{Fω (ω(τ ), τ )}· 

U · (ω(τ )) · ωτ (ω(τ )) + V · (ω(τ )) sign τ · ωτ (ω(τ )) + ω(τ ) + |P(ω(τ ))|2 Remark We know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyse stability. F function is independent on τ parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ(τ ) = . . . ; cos(τ ) = . . . then −PR · Q I + PI · Q R ; cos(ω · τ ) |Q|2 (PR · Q R + PI · Q I =− )|Q|2 = Q 2R + Q 2I ; |P|2 = PR2 + PI2 |Q|2

sin(ω · τ ) =

We can plot the stability switch diagram based on g(τ ) function behavior.

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

19

  ∂Reλ g(τ ) = Λ−1 (τ ) = ; g(τ ) ∂τ λ=i·ω      2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 −1 = Λ (τ ) = 2  Fτ2 + 4 · V + ω · P 2   ∂Reλ −1 sign[g(τ )] = sign[Λ (τ )] = sign ∂τ λ=i·ω ⎧ ⎫ 2 · [Fω · (V + ω · P 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −F · (U + τ · P 2 )] ⎪ ⎬ τ sign[g(τ )] = sign ⎪ F 2 + 4 · (V + ω · P 2 )2 ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎩ ⎭ 2    Fτ2 + 4 · V + ω · P 2 > 0; sign Λ−1 (τ )      = sign Fω · V + ω · P 2 − Fτ · U + τ · P 2  

 −1   Fτ    2 2 sign Λ (τ ) = sign Fω · V + ω · P − ; ωτ · U +τ · P Fω  −1 Fτ ∂ω ∂ F/∂ω = − ; ωτ = =− Fω ∂τ ∂ F/∂τ       sign Λ−1 (τ ) = sign Fω · V + ω · P 2 + ωτ · U + τ · P 2      sign Λ−1 (τ ) = sign Fω · V + ωτ · U + ω · P 2 + ωτ · τ · P 2  

 −1    (V + ωτ · U ) 2 sign Λ (τ ) = sign Fω · P · + ω + ωτ · τ ; sign P 2 > 0 P2  

 −1  (V + ωτ · U ) sign Λ (τ ) = sign [Fω ] · + ω + ωτ · τ P2    (V + ωτ · U ) + ω + ω · τ sign Λ−1 (τ ) = sign[Fω ] · sign τ P2   −1  V + ωτ · U + ω + ωτ · τ sign Λ (τ ) = sign[Fω ] · sign P2    ωτ · U + ω + ω · τ V = 0 ⇒ sign Λ−1 (τ ) = sign[Fω ] · sign τ P2 Fτ = 0; Fω = 0 ⇒ ωτ = −

  Fτ = 0 ⇒ sign Λ−1 (τ ) = sign[Fω ] · sign[ω] Fω

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]

20

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.2 Sign of Λ−1 (τ ) Vs sign of sign[Fω ] and sign[ω] sign[Fω ]

sign[ω]

sign[Λ−1 (τ )]

±

±

+

±

∓

–

Table 1.3 System cases behavior for sign[Λ−1 (τ )]sign Case

Crossing behavior

sign[Λ−1 (τ )]

>0

Crossing proceeds from (−) to (+) respectively (stable to unstable)

sign[Λ−1 (τ )] < 0

Crossing proceeds from (+) to (−) respectively (unstable to stable)

   1 1 1 2 − ω2 ; PI ω · PI = + · ω; Q Rω · Q R = 0 τ ph · τs τ ph τs 2  (i) Q I ω · Q I = A2 · N ph − Ne(i) + Nom · ω 

PRω · PR = −2 · ω ·

Fω = 2 · [PRω · PR + PI ω · PI − Q I ω · Q I ] !     2  1 1 2 1 (i) 2 2 (i) −ω + + + A · N ph − Ne + Nom = −2 · ω · 2 · τ ph · τs τ ph τs

       1 1 1 1 1 2 2 U= + ·2·ω −ω · + + τ ph · τs τ ph τs τ ph τs 

    (i)  1 1 (i) (i) (i) − A · Ne − Nom · · (β − 1) + · N ph · A · N ph − Ne + Nom τs τ ph We check the sign of Λ−1 (τ ) according the following rule (Table 1.2). We can summary our system cases behavior in the following table (Table 1.3). No. 2:     d Ni (t) 1 q·V q·V = · exp − − A · exp − · dt q ·a·d 2·k·T 2·k·T     q·V Ni (t) Ni (t − τ ) · exp − Nom · N ph (t − τ ) − 2·k·T τs     d N ph (t) q·V = A · Ni (t − τ ) · exp − Nom · dt 2·k·T   N ph (t) β q·V N ph (t − τ ) − + · Ni (t) · exp τ ph τs 2·k·T

1.1 Laser Diode Coupled Delay Rate Equations Stability Analysis

21

Remark It is reader exercise to analyze the second possible set of differential equations. To find the equilibrium points (fixed points), stability analysis, Laser diode delayed rate equations eigenvalue matrix, and stability switching inspection.

1.2 Laser Diode Intrinsic and Package Electrical Equivalent Circuits Stability Analysis for Parameters Variation The complete equivalent circuit of a laser diode can be separated into two parts. The first part represents the intrinsic electrical equivalent circuit of the laser chip itself. The second part is the electrical equivalent circuit of the package including the major parasitic elements. The elements of the intrinsic laser equivalent circuit are derived from the coupled rate equations which describe the interplay between the injected carrier and photon densities in the active region of the laser diode. We describe the laser diode coupled single-mode rate equation parameters. Nph is the photon density, Ne is the electron density, q is the electronic charge, d is the thickness of the active region, a is the area of the diode contact stripe, I is the injected current, Nom is the minimum electron density required to obtain a positive gain, A is a constant related to the stimulated emission process, τS is the spontaneous emission lifetime, τph is the photon lifetime and β is the fraction of the spontaneous emission that is coupled to the lasing mode. We describe the carrier density as a function of the junction voltage q·V (Ne = Ni · e 2·k·T ; Ni is the intrinsic carrier density and V is the junction voltage). We define Rd as the differential resistance of the laser diode and n 0ph , n 0e and n om are the 0 , Ne0 and Nom , respectively [1]. normalized steady-state values of N ph Rd =

2·k·T 1 N0 · a · q · d 2·k·T τs · ; Id = e ; Rd = · 0 2 q Id τs q Ne · a · d

Id is the normalized current. 0 ; n 0e = A · τ ph · Ne0 ; n om = A · τ ph · Nom n 0ph = A · τ S · N ph

The basic intrinsic electrical equivalent circuit of the laser diode is describe below (Fig. 1.1). Vsup is the power source that drive the laser diode with its parasitic resistance Rp . Rse is the series resistance with the inductor (Li ). Ri =

1 Rd 2·k·T τs  · ; Ri = · 0 0 q2 Ne · a · d n 0ph + 1 A · τ S · N ph +1

22

1 Dynamical and Nonlinearity of Laser Diode Circuits

Fig. 1.1 Basic intrinsic equivalent circuit of laser diode

Rd · τ ph 2 · k · T · τ ph Rd · τ ph   L i =  ; Li ≈  ≈ n 0 0 2 0 0 A · N ph ph · q · Ne · a · d n ph + β · n 0e − n om τs N 0 · a · q2 · d ; Ci = e Rd 2·k·T n 0e n0    ≈ β · Rd ·  e 2 Rse = β · Rd ·  n 0ph · n 0ph + β · n 0e − n om n 0ph Ci =

Rse ≈ β ·

2·k·T · q2

τ ph

2  0 a · d · A · τs · N ph

The resistance Ri including the differential resistance (Rd ) of the laser diode models and is damping due to the spontaneous and stimulated recombination terms in the rate equations. The resistance Rse models damping due to spontaneous emission coupled into the lasing mode. Resistances Ri and Rse are responsible for the damping of the electro-optical resonance. The capacitance Ci represents the active layer diffusion capacitance of the laser diode. The inductance L i arises from the small signal analysis of the rate equations and represents the resonance phenomenon of the laser diode with the capacitance Ci . The resonant frequency ( fr ) of the circuit is , 0 A · N ph τ ph 1 1 1 ·√ · ; fr = ; L i · Ci = fr = 0 2·π 2·π τ ph A · N ph L i · Ci We can increase the value of the resonant frequency by decreasing the values of L i and Ci since the values of L i and Ci are depend on the laser diode parameters, the resonant frequency can be increased by decreasing the photon lifetime (τ ph ),

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

23

0 increasing the photon density (n 0ph ) (or increasing the N ph ), or by increasing the 0 gain coefficient A, that is, increasing electron density n e . There is a close relationship between the circuit components of the equivalent circuit and the laser diode parameters. We derive circuit differential equations.

I R p = I Ri + ICi + I L i ; I L i = I Rse ; V A2 = VRi = VCi = VL i + VRse ; V A3 = VRse d V A2 d ILi ; VL i = L i · dt dt 1 d V A2 · Rse ; ICi · = Ci dt

V A 2 = I Ri · R i ; I C i = C i · VRse = I Rse · Rse = I L i 1 · Ci

-

d ILi ICi · dt; V A2 = L i · dt 1 d ILi + I L i · Rse + I L i · Rse ; · ICi · dt = L i · Ci dt 

1 d 1 d ILi d 2 ILi d ILi + I L i · Rse ⇒ · Rse · ICi · dt = L i · · IC i = L i · + 2 dt Ci dt Ci dt dt V A2 =

Vsup − V A2 Vsup − V A2 ; I R p = I Ri + I C i + I L i ⇒ Rp Rp V A2 d V A2 1 + ILi ; ILi = = + Ci · · VL i ·dt Ri dt Li

IRp =

VL i = V A2 − VRse = V A2 − I L i · Rse ; I L i   1 1 Rse = · V A2 − I L i · Rse · dt; I L i = · V A2 ·dt − · I L i ·dt Li Li Li Vsup − V A2 1 V A2 d V A2 Rse + = + Ci · · V A2 ·dt − · I L i ·dt Rp Ri dt Li Li

 1 d Vsup − V A2 V A2 d V A2 Rse + = + Ci · · V A2 ·dt − · I L i ·dt dt Rp Ri dt Li Li d Vsup → ε(= 0) dt −

1 d V A2 1 d V A2 d 2 V A2 1 Rse = + Ci · · · + · V A2 − · ILi 2 Rp dt Ri dt dt Li Li

1 d 2 ILi d ILi d V A2 d 2 ILi d ILi · R = L · Rse · IC i = L i · + ⇒ · + se i Ci dt 2 dt dt dt 2 dt We define new variables for simplicity: Y1 =

d V A2 dt

;

d ILi dt

= Y2

24

1 Dynamical and Nonlinearity of Laser Diode Circuits

Y1 = L i ·

1 dY2 1 1 dY1 Rse + Y2 · Rse ; − + · Y1 = · Y1 + Ci · · V A2 − · ILi dt Rp Ri dt Li Li

We can summary our system differential equation: dY1 = −Y1 · dt



1 1 + Rp Ri

 ·

1 Rse dY2 1 Rse 1 − · V A2 + · ILi ; = Y1 − Y2 · Ci L i · Ci L i · Ci dt Li Li

d V A2 d ILi =Y1 ; = Y2 ; V A2 = V A2 (t); I L i = I L i (t); Y1 = Y1 (t); Y2 = Y2 (t) dt dt   We define four functions: f i = f i V A2 , I L i , Y1 , Y2 ∀ i = 1, 2, 3, 4   dY2   dY1 = f 1 V A2 , I L i , Y1 , Y2 ; = f 2 V A2 , I L i , Y1 , Y2 dt dt   d ILi   d V A2 = f 3 V A2 , I L i , Y1 , Y2 ; = f 4 V A2 , I L i , Y1 , Y2 dt dt     1 1 1 1 Rse · f 1 V A2 , I L i , Y1 , Y2 = −Y1 · + − · V A2 + · ILi Rp Ri Ci L i · Ci L i · Ci     1 Rse f 2 V A2 , I L i , Y1 , Y2 = Y1 − Y2 · ; f 3 V A2 , I L i , Y1 , Y2 Li Li   = Y1 ; f 4 V A2 , I L i , Y1 , Y2 = Y2 To find system fixed points (equilibrium points), we set dY2 d V A2 d ILi d V A2 dY1 d ILi = 0; = 0; =0; = 0; = 0 ⇒ Y1∗ = 0; = 0 ⇒ Y2∗ = 0 dt dt dt dt dt dt   1 dY1 1 1 1 · = 0 ⇒ −Y1∗ · + − · V A∗2 dt Rp Ri Ci L i · Ci Rse + · I ∗ = 0 −→ Y1∗ = 0Rse · I L∗i = V A∗2 L i · Ci L i     System fixed point: E ∗ V A∗2 , I L∗i , Y1∗ , Y2∗ = Rse · I L∗i , I L∗i , 0, 0 Stability analysis: The standard local stability analysis about one of the equilibrium points of laser diode basic intrinsic electrical equivalent circuit consists in adding to coordinate [V A2 I L i Y1 Y2 ] arbitrarily small increment of exponential form v A2 · eλ·t , i L i ·eλ·t , y1 ·eλ·t , y2 ·eλ·t respectively and retaining the first order terms in [v A2 , i L i , y1 , y2 ]. The system of one homogeneous equation leads to a polynomial characteristic equation in the eigenvalue λ. The polynomial characteristic equation accept by set the below laser diode intrinsic electrical equivalent circuit respect to time into one system equation. Our system fixed points/values with arbitrarily small increment of the exponential form v A2 · eλ·t , i L i · eλ·t , y1 · eλ·t , y2 · eλ·t are i = 0 (first fixed point),

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

25

i = 1 (second fixed point), and i = 2 (third fixed point) [8, 9]. V A2 (t) = V A(i)2 + v A2 · eλ·t ; I L i (t) = I L(i)i + i L i · eλ·t ; Y1 (t) = Y1(i) + y1 · eλ·t ; Y2 (t) = Y2(i) + y2 · eλ·t d V A2 (t) d I L i (t) dY1 (t) = v A2 · λ · eλ·t ; = i L i · λ · eλ·t ; dt dt dt dY2 (t) λ·t λ·t = y2 · λ · e = y1 · λ · e ; dt We choose the above expressions for our V A2 (t), I L i (t), Y1 (t), and Y2 (t) as small displacement [v A2 , i L i , y1 , y2 ] from our laser diode intrinsic equation circuit fixed points at time t = 0. For λ < 0, t > 0 the selected fixed point is stable otherwise λ > 0, t > 0 is unstable. Our laser diode intrinsic equation circuit tends to the selected fixed point exponentially for λ < 0, t > 0, otherwise go away from the selected fixed point exponentially. Eigenvalue λ is the parameter which establish if the fixed point is stable or unstable, additionally his absolute value |λ| establish the speed of the flow toward or away from the selected fixed point. The laser diode intrinsic electrical equivalent circuit variables for negative eigenvalue (λ < 0) are as follow (Table 1.4). The laser diode intrinsic electrical equivalent circuit variables for positive eigenvalue (λ > 0) are as follow (Table 1.5). d V A2 =Y1 ; v A2 · λ · eλ·t =Y1(i) + y1 · eλ·t . At fixed point Y1(i) = 0; −v A2 · λ+y1 = 0 dt d ILi dt

= Y2 ; i L i ·λ·eλ·t = Y2(i) + y2 ·eλ·t . At fixed point Y2(i) = 0; −i L i ·λ+ y2 = 0

 1 1 1 1 Rse · + − · V A2 + · ILi Rp Ri Ci L i · Ci L i · Ci    1    1 1 1 (i) λ·t · · = − Y1 + y1 · e + − · V A(i)2 + v A2 · eλ·t Rp Ri Ci L i · Ci

dY1 = −Y1 · dt y1 · λ · eλ·t



Table 1.4 System variables for λ < 0 and time parameter Time (t)

λ0

(i)

(i)

V A2 (t) = V A2 + v A2 · e−|λ|·t ; I L i (t) = I L i + i L i · e−|λ|·t (i)

(i)

Y1 (t) = Y1 + y1 · e−|λ|·t ; Y2 (t) = Y2 + y2 · e−|λ|·t t > 0; t → ∞

(i)

(i)

V A2 (t → ∞) = V A2 ; I L i (t → ∞) = I L i (i)

(i)

Y1 (t → ∞) = Y1 ; Y2 (t → ∞) = Y2

26

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.5 System variables for λ > 0 and time parameter Time (t)

λ>0

t=0

V A2 (t = 0) = V A2 + v A2 ; I L i (t = 0) = I L i + i L i

(i)

(i)

(i)

(i)

Y1 (t = 0) = Y1 + y1 ; Y2 (t = 0) = Y2 + y2 (i)

(i)

V A2 (t) = V A2 + v A2 · e|λ|·t ; I L i (t) = I L i + i L i · e|λ|·t

t>0

(i)

(i)

Y1 (t) = Y1 + y1 · e|λ|·t ; Y2 (t) = Y2 + y2 · e|λ|·t V A2 (t → ∞) = v A2 · e|λ|·t ; I L i (t → ∞) = i L i · e|λ|·t

t > 0; t → ∞

Y1 (t → ∞) = y1 · e|λ|·t ; Y2 (t → ∞) = y2 · e|λ|·t

+

  Rse · I L(i)i + i L i · eλ·t L i · Ci

 1 1 1 1 · + − · V A(i)2 Rp Ri Ci L i · Ci   1 Rse 1 1 · + · I L(i)i − y1 · + · eλ·t L i · Ci Rp Ri Ci 1 Rse − v A2 · · eλ·t + i L i · · eλ·t L i · Ci L i · Ci

y1 · λ · eλ·t = −Y1(i) ·

At fixed point −Y1(i) ·  −y1 ·

1 1 + Rp Ri



+

1 Rp

 ·

1 Ri





·

1 Ci

−

1 L i ·Ci

· V A(i)2 +

Rse L i ·Ci

· I L(i)i = 0

1 1 Rse − y1 · λ − v A2 · + i Li · =0 Ci L i · Ci L i · Ci

   R dY2 1 Rse 1  (i) se (i) = Y1 − Y 2 · ; y2 · λ · eλ·t = · Y1 + y1 · eλ·t − Y2 + y2 · eλ·t · dt Li Li Li Li

y2 · λ · eλ·t =

1 Rse 1 Rse λ·t · Y1(i) − Y2(i) · + y1 · · eλ·t − y2 · ·e Li Li Li Li

At fixed point L1i · Y1(i) − Y2(i) · RLsei = 0; y1 · L1i − y2 · RLsei − y2 · λ = 0. We can summary our laser diode intrinsic electrical equivalent circuit arbitrarily small increment equations:  − v A2 · λ+y1 = 0; −i L i · λ + y2 = 0; −y1 · − y1 · λ − v A2 ·

1 Rse + i Li · =0 L i · Ci L i · Ci

y1 ·

1 Rse − y2 · − y2 · λ = 0 Li Li

1 1 + Rp Ri

 ·

1 Ci

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

27

The arbitrarily small increments Jacobian of our laser diode intrinsic electrical equivalent circuit is as follow: ⎛

11 . . . ⎜ .. . . ⎝ . . 41 · · ·

⎞ ⎛ ⎞ v A2 11 . . . 14 14 ⎜ i ⎟ .. ⎟ · ⎜ L i ⎟ = 0; A − λ · I = ⎜ .. . . .. ⎟ ⎟ ⎝ . . . ⎠ . ⎠ ⎜ ⎝ y1 ⎠ 44 41 · · · 44 y2 ⎞

⎛

11 = −λ; 12 = 0; 13 = 1; 14 = 0; 21 = 0; 22 = −λ; 23 = 0; 24 = 1 31 = −

  1 1 Rse 1 1 · ; 32 = ; 33 = − + − λ; 34 = 0; 41 = 0 L i · Ci L i · Ci Rp Ri Ci

1 Rse ; 44 = − −λ Li Li ⎞ ⎛ 11 . . . 14 ⎟ ⎜ det(A − λ · I ) = 0; det ⎝ ... . . . ... ⎠ = 0 42 = 0; 43 =

41 · · · 44

⎛ ⎜ det(A − λ · I ) = −λ · det⎝ ⎛

−λ Rse L i ·Ci

0  1 1 − R p + Ri · 1 Li

0

⎞

1 1 Ci

−λ

0

⎞ ⎟ ⎠

− RLsei − λ

0 −λ 1 ⎟ ⎜ se 0 + det ⎝ − L i1·Ci LRi ·C ⎠ i Rse 0 0 − Li − λ

!       1 1 Rse Rse 1 det(A − λ · I ) = −λ · −λ · − + −λ · − −λ + 2 · Rp Ri Ci Li L i · Ci 

   1 Rse −λ +λ· − · − L i · Ci Li

    

 1 1 Rse Rse 1 · + +λ · +λ + 2 det(A − λ · I ) = −λ · −λ · Rp Ri Ci Li L i · Ci

    1 Rse +λ· · +λ L i · Ci Li  

  1 Rse 1 1 1 1 1 Rse · · det(A − λ · I ) = −λ · −λ · + · + + + Rp Ri Ci L i Rp Ri Ci Li !      Rse Rse 1 1 · · λ2 + ·λ + λ2 + 2 +λ· L · C L L L i · Ci i i i i · Ci



det(A − λ · I ) = λ2 ·

1 1 + Rp Ri



·

1 Rse · + Ci L i



1 1 + Rp Ri



·

1 Rse · + Ci Li

28

1 Dynamical and Nonlinearity of Laser Diode Circuits

Rse λ3 + λ4 − λ · 2 L i · Ci     Rse 1 1 · · λ2 +λ· + L i · Ci Li L i · Ci   Rse Rse 1 · −λ· 2 +λ· =0 L i · Ci Li L i · Ci   1 1 1 Rse 4 3 · det(A − λ · I ) = λ + λ · + + Rp Ri Ci Li   1 Rse 1 1 1 · + λ2 · + · + Rp Ri Ci L i L i · Ci   1 1 1 Rse · + + det(A − λ · I ) = 0 ⇒ λ4 + λ3 · Rp Ri Ci Li   1 1 1 R 1 se · =0 + λ2 · + · + R R C L L i · Ci p i i   i 

λ2 · λ2 + λ · 4 $

λk ·

k=0

=



1 1 + Rp Ri

# k

= 0;

1 1 + Rp Ri

·

# 4

 ·

1 1 + Rp Ri

Rse 1 + + Ci Li

= 1;



# 3

=

·

1 1 + Rp Ri

1 1 Rse · + Ci L i L i · Ci

 ·

=0

1 Rse # + ; 2 Ci Li

# # 1 Rse 1 · + ; = 0; =0 1 0 Ci L i L i · Ci

We have three possible cases: 

 1 1 1 Rse · + + λ2 = 0; λ2 + λ · Rp Ri Ci Li   . Case A: 1 Rse 1 1 1 · = 0 + + · + Rp Ri Ci L i  L i · Ci 1 1 1 Rse · + + λ2 = 0; λ2 + λ · Rp Ri Ci Li   . Case B: 1 Rse 1 1 1 · =0 + + · + Rp Ri Ci L i L i · Ci 

  1 + C: λ2 = 0; λ2 + λ · + R1i · C1i + RLsei Rp    1 + R1i · C1i · RLsei + L i1·Ci = 0. Rp We analyze cases B, C since the eigenvale’s quadratic equation is equal to zero. Totally we get four (4) eigenvalues for our laser diode basic intrinsic electrical equivalent circuit’s characteristic equation.    Rse 1 1 1 · + Case B: λ1 = 0; λ2 = 0; λ2 + λ · + + Rp Ri Ci Li    1 + R1i · C1i · RLsei + L i1·Ci = 0. Rp Case

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

29

0 or λ1 =0; λ2 = 0 or λ1 = 0; λ2 = 0 and λ2 + λ ·  Case C:λ1 = 0; λ2 =  1 + R1i · C1i + RLsei + R1p + R1i · C1i · RLsei + L i1·Ci = 0. Rp Eigenvalues stability discussion: Our laser diode intrinsic equation circuit system involving (N > 2, N = 4), the characteristic equation is of degree N "  N variables = 4 ( 4k=0 λk · k = 0) and must often be solved numerically. Expect in some particular cases, such an equation has (N = 4) distinct roots that can be real or complex. These values are the eigenvalues of the 4 × 4 Jacobian matrix A. The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 4-variables (V A2 I L i Y1 Y2 ) laser diode intrinsic equation circuit system has four eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the four eigenvalues are real and negative (stable steady state), (2) the four eigenvalues are real, three of them are negative (unstable steady state), (3) and (4) two eigenvalues are complex conjugates with a negative real part and the other eigenvalues are real and negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) two eigenvalues are complex conjugates with a negative real part and at least one eigenvalue is positive (unstable steady state). For each of the five non-degenerated cases, if at least one of the eigenvalues is equal to zero then system’s fixed points are on a line [3, 4]. The complete intrinsic electrical equivalent circuit of the laser diode includes terms which are added to the rate equations for explaining the different phenomena appearing in the light output of the laser diodes and the different geometry lasers. The complete intrinsic electrical equivalent circuit includes another capacitor parallel to Ci for explaining the space-charge effects in the active region of the laser diode. The name of the capacitance is as the space-charge capacitance Csc , of the active region. Additionally there is a non-uniform electron density which arrives at an equivalent circuit containing a series resistor Rse1 , with the inductor. The equation of Csc is given − 21  where Csc(0) is the zero-bias space-charge capacitance. by Csc = Csc(0) · 1 − VVD VD is the heterojunction built-in potential (typical value 1.65 v). The resistance Rse1 (·A)2 ·τ · N 0 −N is given by R =  s ( e om ) · N 0 · L where  is the optical confinement se1

 2·π·L 2 ef f 2· 1+ w

ph

i

factor normal to the junction plane, L e f f is the effective carrier diffusion length and w is the width of the active layer. Resistor Rse1 models the lateral carrier diffusion and the optical confinement in the active region. The complete intrinsic electrical equivalent circuit of the laser diode (Fig. 1.2) which takes into account all the major factors affecting the dynamic response of the laser diode is shown below [1]. V A2 = VCsc = VCi = VRi ; V A4 = VRse ; V A3 − V A4 Vsup − V A2 = VL i ; V A2 − V A3 = VRse1 ; = IRp Rp

30

1 Dynamical and Nonlinearity of Laser Diode Circuits

Fig. 1.2 Complete intrinsic electrical equivalent circuit of the laser diode

ICsc = Csc · I Ri =

d V A2 d V A2 ; IC i = C i · dt dt

V A2 V A2 − V A3 ; I Rse1 = I L i = I Rse ; I Rse1 = Ri Rse1

VL i = V A 3 − V A 4 = L i · Vsup −V A2 Rp

= Csc ·

d ILi V A4 ; I Rse = ; I R p = ICsc + ICi + I Ri + I Rse1 dt Rse

d V A2 d V A2 V A − V A3 VA ; V A2 = VRse1 + VL i + VRse + Ci · + 2 + 2 dt dt Ri Rse1

d ILi V A2 − V A3 + I L i · Rse ; V A2 = I L i · Rse1 + L i · dt Rse1 V A4 1 = = · (V A3 − V A4 ) · dt Rse Li   1 1 V A3 d V A2 Vsup −V A2 · V A2 − + = + C + · (C ) sc i Rp dt  Ri Rse1 Rse1 

1 V A2 − V A3 = · Rse1 Li

V A4 1 = · Rse Li

-

(V A3 − V A4 ) · dt ⇒

d V A2 d V A3 − dt dt

(V A3 − V A4 ) · dt ⇒

·

1 1 = · (V A3 − V A4 ) Rse1 Li

1 1 d V A4 = · · (V A3 − V A4 ) Rse dt Li

1 d V A4 1 1 d V A2 d V A3 1 1 1 1 · · V A3 − · V A4 ; · · · V A3 − · V A4 = − = Rse dt Li Li Rse1 dt Rse1 dt Li Li

1 1 1 d V A3 d V A2 1 = − · · · V A3 + · Rse1 dt Rse1 dt Li Li d V A2 Rse1 d V A3 Rse1 = − · V A3 + · V A4 V A4 ⇒ dt dt Li Li

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

31

  Vsup 1 V A2 1 V A3 d V A2 + − = (Csc + Ci ) · + · V A2 − Rp Rp dt Ri Rse1 Rse1  

 d Vsup 1 d Vsup V A2 1 V A3 d V A2 · V A2 − ; + →ε − = (Csc + Ci ) · + dt R p Rp dt Ri Rse1 Rse1 dt   Vsup 1 1 1 V A3 d V A2 · V A2 − + = (Csc + Ci ) · + + Rp dt Ri Rse1 Rp Rse1   1 d V A3 d V A2 d 1 1 1 d 2 V A2 {. . .} ⇒ (Csc + Ci ) · − =0 · + + + 2 dt dt Ri Rse1 Rp dt Rse1 dt   d V A2 1 1 1 d 2 V A2 · + + + (Csc + Ci ) · dt 2 Ri Rse1 Rp dt   1 d V A2 Rse1 Rse1 − · · V A3 + · V A4 = 0 − Rse1 dt Li Li   d V A2 1 1 1 d 2 V A2 · + + + (Csc + Ci ) · 2 dt Ri Rse1 Rp dt 1 1 d V A2 1 + − · · V A3 − · V A4 = 0 Rse1 dt Li Li   d V A2 1 1 1 1 d 2 V A2 · + + + · V A3 − · V A4 = 0 (Csc + Ci ) · dt 2 Ri Rp dt Li Li We define new variable: Y =

d V A2 dt

;

dY dt

=

d 2 V A2 dt 2

Rse1 Rse d V A3 Rse1 d V A4 Rse =Y − = · V A3 + · V A4 ; · V A3 − · V A4 dt Li Li dt Li Li   1 1 1 1 dY ·Y + + + · V A3 − · (Csc + Ci ) · dt Ri Rp Li Li   1 1 dY ·Y =− + V A4 = 0 ⇒ (Csc + Ci ) · dt Ri Rp 1 1 − · V A3 + · V A4 Li L i

dY 1 · =− dt (Csc + Ci )

1 1 + Ri Rp

·Y −

1 1 · V A3 + · V A4 L i · (Csc + Ci ) L i · (Csc + Ci )

We can summary our system differential equations: Y = Y (t); V A3 = V A3 (t); V A4 = V A4 (t) 1 dY =− · dt (Csc + Ci )



1 1 + Ri Rp

 ·Y

32

1 Dynamical and Nonlinearity of Laser Diode Circuits

−

1 L i · (Csc + Ci )

· V A3 +

1 L i · (Csc + Ci )

· V A4

Rse1 d V A3 Rse1 d V A4 =Y− · V A3 + · V A4 ; dt Li Li dt Rse Rse = · V A3 − · V A4 Li Li   We define three functions: gi = gi Y, V A3 , V A4 ∀ i = 1, 2, 3   d V A3 dY = g1 Y, V A3 , V A4 ; dt dt   d V A4   = g3 Y, V A3 , V A4 = g2 Y, V A3 , V A4 ; dt     1 1 1 · g1 Y, V A3 , V A4 = − · + Ri Rp (Csc + Ci ) 1 1 Y− · V A3 + · V A4 L i · (Csc + Ci ) L i · (Csc + Ci )     Rse1 Rse1 g2 Y, V A3 , V A4 = Y − · V A3 + · V A4 ; g3 Y, V A3 , V A4 Li Li Rse Rse · V A3 − · V A4 = Li Li At fixed points:

dY dt

= 0;

d V A3 dt

= 0;

d V A4 dt

=0

d V A4 d V A3 Rse1 = 0 ⇒ V A∗3 = V A∗4 ; = 0 ⇒ Y∗ − · V A∗3 dt dt Li Rse1 + · V A∗3 = 0 ⇒ Y ∗ = 0 Li       E ∗ Y ∗ , V A∗3 , V A∗4 = 0, V A∗3 , V A∗3 = 0, V A∗4 , V A∗4 Stability analysis: The standard local stability analysis about one of the equilibrium points of laser diode complete intrinsic electrical equivalent circuit consists in adding to coordinate [Y V A3 V A4 ] arbitrarily small increment of exponential form y · eλ·t , v A3 ·eλ·t , v A4 ·eλ·t Respectively and retaining the first order terms in [y, v A3 , v A4 ]. The system of one homogeneous equation leads to a polynomial characteristic equation in the eigenvalue λ. The polynomial characteristic equation accept by set the below laser diode complete intrinsic electrical equivalent circuit respect to time into one system equation. Our system fixed points/values with arbitrarily small increment of the exponential form y · eλ·t , v A3 · eλ·t , v A4 · eλ·t are i = 0 (first fixed point ), i = 1 (second fixed point), and i = 2 (third fixed point).

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

33

Y (t) = Y (i) + y · eλ·t ; V A3 (t) = V A(i)3 + v A3 · eλ·t ; V A4 (t) dY (t) = y · λ · eλ·t = V A(i)4 + v A4 · eλ·t ; dt d V A3 (t) d V A4 (t) = v A3 · λ · eλ·t ; = v A4 · λ · eλ·t dt dt We choose the above expressions for our Y (t), V A3 (t), and V A4 (t) as small displacement [y, v A3 , v A4 ] from our laser diode complete intrinsic equation circuit fixed points at time t = 0. For λ < 0, t > 0 the selected fixed point is stable otherwise λ > 0, t > 0 is unstable. Our laser diode complete intrinsic equation circuit tends to the selected fixed point exponentially for λ < 0, t > 0, otherwise go away from the selected fixed point exponentially. Eigenvalue λ is the parameter which establish if the fixed point is stable or unstable, additionally his absolute value |λ| establish the speed of the flow toward or away from the selected fixed point. The laser diode complete intrinsic electrical equivalent circuit variables for negative eigenvalue (λ < 0) are as follow (Table 1.6). The laser diode complete intrinsic electrical equivalent circuit variables for positive eigenvalue (λ > 0) are as follow (Table 1.7). dY 1 · =− dt (Csc + Ci )



1 1 + Ri Rp

 ·Y −

1 1 · V A3 + · V A4 L i · (Csc + Ci ) L i · (Csc + Ci )

Table 1.6 System variables for λ < 0 and time parameter Time (t)

λ0

V A4 (t) = V A(i)4 + v A4 · e−|λ|·t (i)

(i)

Y (t → ∞) = Y (i) ; V A3 (t → ∞) = V A3 ; V A4 (t → ∞) = V A4

t > 0; t → ∞

Table 1.7 System variables for λ > 0 and time parameter Time (t)

λ>0

t=0

Y (t = 0) = Y (i) + y; V A3 (t = 0) = V A(i)3 + v A3 (i)

V A4 (t = 0) = V A4 + v A4 t>0

(i)

Y (t) = Y (i) + y · e|λ|·t ; V A3 (t) = V A3 + v A3 · e|λ|·t (i)

V A4 (t) = V A4 + v A4 · e|λ|·t t > 0; t → ∞ Y (t → ∞) = y · e|λ|·t ; V A3 (t → ∞) = v A3 · e|λ|·t ; V A4 (t → ∞) = v A4 · e|λ|·t

34

1 Dynamical and Nonlinearity of Laser Diode Circuits

    1 1 1 · y·λ·e =− + · Y (i) + y · eλ·t Ri Rp (Csc + Ci )   1 − · V A(i)3 + v A3 · eλ·t L i · (Csc + Ci )   1 + · V A(i)4 + v A4 · eλ·t L i · (Csc + Ci )   1 1 1 · Y (i) y · λ · eλ·t = − · + Ri Rp (Csc + Ci ) 1 1 · V A(i)3 + · V A(i)4 − L i · (Csc + Ci ) L i · (Csc + Ci )   1 1 1 1 · y · eλ·t − · · v A3 · eλ·t − + Ri Rp L i · (Csc + Ci ) (Csc + Ci ) 1 · v A4 · eλ·t + L i · (Csc + Ci ) λ·t

  1 1 1 1 · Y (i) − · · V A(i)3 + Ri Rp L i · (Csc + Ci ) (Csc + Ci ) 1 · V A(i)4 = 0 + L i · (Csc + Ci )

− At fixed point:

−

1 · (Csc + Ci )

v A3 · λ · eλ·t



1 1 + Ri Rp



·y−y·λ−

1 1 · v A3 + · v A4 = 0 L i · (Csc + Ci ) L i · (Csc + Ci )

Rse1 d V A3 Rse1 =Y − · V A3 + · V A4 dt Li Li  R   Rse1  (i) se1 = Y (i) + y · eλ·t − · V A3 + v A3 · eλ·t + · V A(i)4 + v A4 · eλ·t Li Li Rse1 Rse1 · V A(i)3 + · V A(i)4 Li Li Rse1 · v A3 · eλ·t + · v A4 · eλ·t Li

v A3 · λ · eλ·t = Y (i) − + y · eλ·t − At fixed point: Y (i) −

· V A(i)3 +

Rse1 Li

· V A(i)4 = 0

Rse1 Rse1 · v A3 − v A3 · λ + · v A4 = 0 Li Li  Rse Rse  (i) Rse · V A3 − · V A4 ; v A4 · λ · eλ·t = · V A3 + v A3 · eλ·t = Li Li Li  Rse  (i) λ·t − · V A4 + v A4 · e Li y−

d V A4 dt

Rse1 Li

Rse1 Li

1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

v A4 · λ · eλ·t = At fixed point:

35

Rse Rse Rse Rse · V A(i)3 − · V A(i)4 + · v A3 · eλ·t − · v A4 · eλ·t Li Li Li Li · V A(i)3 −

Rse Li

Rse Li

· V A(i)4 = 0

Rse Rse · v A3 − · v A4 − v A4 · λ = 0 Li Li We can summary our laser diode complete intrinsic electrical equivalent circuit arbitrarily small increment equations:   1 1 1 1 ·y−y·λ− · · v A3 + Ri Rp L i · (Csc + Ci ) (Csc + Ci ) 1 · v A4 = 0 + L i · (Csc + Ci ) −

y−

Rse1 Rse1 Rse Rse · v A3 − v A3 · λ + · v A4 = 0; · v A3 − · v A4 − v A4 · λ = 0 Li Li Li Li

The arbitrarily small increments Jacobian of our laser diode complete intrinsic electrical equivalent circuit is as follow: ⎛ ⎜ ⎝

− (Csc1+Ci ) ·



1 Ri

+

1 Rp



− λ − L i ·(Csc1 +Ci ) − RLse1i − λ

1 0 ⎛

⎜ A−λ· I =⎝

− (Csc1+Ci ) ·



Rse Li

1 Ri

1 0

+

1 Rp



1 L i ·(Csc +Ci ) Rse1 Li − RLsei − λ

⎞ ⎛

⎞

⎟ ⎜ ⎟ ⎠ · ⎝ v A3 ⎠ = 0 v A4

− λ − L i ·(Csc1 +Ci ) − RLse1i − λ Rse Li

y

1

L i ·(Csc +Ci ) Rse1 Li Rse − Li − λ

⎞ ⎟ ⎠

det(A − λ · I ) = 0     Rse1 − RLse1i − λ 1 1 1 Li − λ · det · + Rse − RLsei − λ Ri Rp (Csc + Ci ) Li   Rse1 1 1 Li · det + 0 − RLsei − λ L i · (Csc + Ci )   1 − RLse1i − λ 1 · det + Rse 0 L i · (Csc + Ci ) Li    1 1 1 det(A − λ · I ) = − −λ · · + Ri Rp (Csc + Ci )

 det(A − λ · I ) = −

36

1 Dynamical and Nonlinearity of Laser Diode Circuits

     Rse1 Rse Rse Rse1 − −λ · − −λ − · Li Li Li Li   1 Rse Rse 1 + · − · −λ + L i · (Csc + Ci ) Li L i · (Csc + Ci ) L i    1 1 1 +λ · · det(A − λ · I ) = − + Ri Rp (Csc + Ci )

     Rse1 Rse Rse · Rse1 +λ · +λ − Li Li L i2   1 Rse Rse 1 − · · +λ + L i · (Csc + Ci ) Li L i · (Csc + Ci ) L i    1 1 1 · det(A − λ · I ) = − + +λ · Ri Rp (Csc + Ci )    1 Rse1 Rse − ·λ λ2 + λ · + L L L · +C ) (C i  i  sc  i   i

1 Rse Rse1 1 1 + + + · (Csc + Ci ) Ri Rp Li Li     1 1 Rse1 Rse 1 1 · · −λ· + + + Li Li Ri Rp (Csc + Ci ) L i · (Csc + Ci )

 1 R Rse 1 1 se1 det(A − λ · I ) = −λ · λ2 + λ · + )+( + ) ·( (Csc + Ci ) Ri Rp Li Li      1 Rse 1 1 1 Rse1 · · + + + + Li Li Ri Rp (Csc + Ci ) L i · (Csc + Ci )

det(A − λ · I ) = −λ3 − λ2 ·

    rse1 rse 1 1 1 + · λ · = 0; = −1; + + k 3 2 ri rp li li (csc + ci ) k=0    #  # 1 1 1 rse1 rse 1 · · + ; =− + + =0 1 0 li li ri rp (csc + ci ) li · (csc + ci ) 3 $

k

#

#

#

 =−

We have three possible cases: Case A: λ1 = 0      Rse1 1 1 1 Rse + λ2 + λ · · + + (Csc + Ci ) Ri Rp Li Li   1 1 Rse1 Rse 1 1 ·( + + = 0 + + )· Li Li Ri R p (Csc + Ci ) L i · (Csc + Ci ) Case B: λ1 = 0  λ2 + λ ·

1 · (Csc + Ci )



1 1 + Ri Rp



 +

Rse1 Rse + Li Li



1.2 Laser Diode Intrinsic and Package Electrical Equivalent …

 +

Rse1 Rse + Li Li

37

   1 1 1 1 · · + =0 + Ri Rp (Csc + Ci ) L i · (Csc + Ci )

Case C: λ1 = 0      Rse1 1 1 1 Rse + · λ2 + λ · + + (Csc + Ci ) Ri Rp Li Li     1 1 1 Rse1 Rse 1 · · + =0 + + + Li Li Ri Rp (Csc + Ci ) L i · (Csc + Ci ) We analyze case B, C since the eigenvale’s quadratic equation is equal to zero. Totally we get three (3) eigenvalues for our complete intrinsic electrical equivalent circuit of the laser diode characteristic equation. Eigenvalues stability discussion: Our complete intrinsic electrical equivalent circuit of the laser diode system involving N "variables (N > 2, N = 3), the characteristic  equation is of degree N = 3 ( 3k=0 λk · k = 0) and must often be solved numerically. Expect in some particular cases, such an equation has (N = 3) distinct roots that can be real or complex. These values are the eigenvalues of the 3 × 3 Jacobian matrix A. The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 3-variables (Y V A3 V A4 ) complete intrinsic electrical equivalent circuit of the laser diode system has three eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the three eigenvalues are real and negative (stable steady state), (2) the three eigenvalues are real, two of them are negative (unstable steady state), (3) and (4) one eigenvalue is complex conjugates with a negative real part and the other eigenvalues are real and negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) one eigenvalue is complex conjugates with a negative real part and at least one eigenvalue is positive (unstable steady state). For each of the five non-degenerated cases, if at least one of the eigenvalues is zero then system’s fixed points are on a line [10, 11].

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis Fiber optics MRI system addresses all the needs of MRI technology. One of these technologies is phased array coils with high number of elements. The phase array is a type of MRI coil that allows parallel imaging. These are multi-channel coils with each channel having its own receiver when channel system increases the number of receiver channels over traditional MRI coils. Fast scan time is the most obvious benefit of phase array. The fact is by using multiple receiver coils in parallel scan

38

1 Dynamical and Nonlinearity of Laser Diode Circuits

time in Fourier imaging can be considerably reduced. In order to minimize the total size of system interconnections and improve the system safety we use a miniature fiber optic transmission (FOT) system. The TOF system consists of a receiver coil with active detuning, a low-noise preamplifier, and a laser diode connected to a photodetector with fiber optic cabling. The laser diode circuitry is the component responsible for the conversion of MR signals from electrical to the optical state. The circuitry is placed after the preamplifiers in the RF signal path of the fiber-optic MRI probe. The circuit is preceded by preamplifiers from the fact of purely noise performance. Diode laser circuitry is integrated electrical-optical-electrical (E-O-E) conversion process and is associated with a loss in the transmitted power. Adding preamplifiers provide that the gain is sufficient enough to move the signal away from the noise floor even after attenuation. The laser diode circuitry consists of an L-match section, a Bias-T, and the laser diode itself (Fig. 1.3). The L-match is responsible for the matching of the low impedance laser diode to the high impedance output of the preamplifier. The Bias-T combines the RF signal with the appropriate DC current injection which required biasing the laser. The laser diode holds the micro-lens to couple the generated optical energy to the fiber. Due to laser diode circuit microstrip parasitic effects there are instabilities on the circuit operation. The circuit microstrip parasitic effects are represented by delay lines elements which delay the current in time that flow through them [6]. We consider ideal delay lines then the voltages that fall are neglected (Vτi → ε ∀ i = 1, 2). The voltage on the laser diode is V and the current is I (V = f (I )). We consider ideal delay lines Vτi → ε ∀ i = 1, 2 and the delay is in the currents. We write circuit equations:

Fig. 1.3 MRI system laser diode circuit with delay lines

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

39

X (t) − V A1 Rp d V A1 d IL1 ; V A 1 − V A 2 = VL 1 = L 1 · IC 1 = C 1 · dt dt   d V A3 − V A4 IC2 (t) = I L 1 (t − τ1 ); IC2 = C2 · dt d IL2 ; I R1 = I L 2 ; I L 1 + IC 1 = I R p VL 2 = V A 7 − V A 4 = L 2 · dt V A5 = V A4 ; V A2 = V A3 ; V A5 = V A6 ; I R p =

Vsup − V A7 ; V A6 = V = f (I ); I (t) = I L 2 (t − τ2 ) + IC2 (t − τ2 ) R1   2 $ I (t) = I L 2 (t − τ2 ) + I L 1 t − τk I R1 =

k=1

X (t) − V A1 ⇒ I R p · R p = X (t) − V A1 ; V A1 = X (t) − I R p · R p Rp  d V A1 d = C1 · X (t) − I R p · R p = C1 · dt dt

IRp = IC 1

d IRp d X (t) − C1 · R p · ; V A1 dt dt d IL1 d IL1 ⇒ V A2 = V A1 − L 1 · = L1 · dt dt

I R p − I L 1 = C1 · − V A 2 = VL 1

d IL1 V A2 = V A3 = X (t) − I R p · R p − L 1 · dt   d V A3 − V A4 1 ⇒ V A3 − V A4 = · IC2 · dt IC 2 = C 2 · dt C2 1 V A4 = V A3 − · IC2 · dt; V A4 C2 1 d IL1 − = X (t) − I R p · R p − L 1 · · IC2 · dt dt C2 d IL2 d IL2 ⇒ V A7 = V A4 + L 2 · dt -dt 1 d IL1 d IL2 − = X (t) − I R p · R p − L 1 · · IC2 · dt + L 2 · dt C2 dt

V A7 − V A4 = L 2 · V A7

Vsup − V A7 Vsup 1 = − · V A7 R1 R R1  1 Vsup 1 1 d IL1 d IL2 − = − · X (t) − I R p · R p − L 1 · · IC2 · dt + L 2 · R1 R1 dt C2 dt

I R1 = I R1

40

1 Dynamical and Nonlinearity of Laser Diode Circuits

  Vsup d 1 1 d IL1 d IL2 I R1 = − − · X (t) − I R p · R p − L 1 · · IC2 · dt + L 2 · dt R1 R1 dt C2 dt d I R1 dt

=

  d IRp d2 IL1 d2 IL2 d Vsup 1 1 d X (t) 1 · − · R p − L1 · − · − · I + L · 2 C2 dt R1 R1 dt dt C2 dt 2 dt 2

 d IRp d Vsup d X (t) d 2 IL1 1 d 2 IL2 d I R1 1 · − · I + L · → ε; =− − · Rp − L1 · C2 2 dt dt R1 dt dt dt 2 C2 dt 2 d IL2 IC2 (t) = I L 1 (t − τ1 )&I R1 = I L 2 ; dt  d IRp d 2 IL1 1 d 2 IL2 1 d X (t) − τ + L · − · I · =− − · Rp − L1 · (t ) L 1 2 1 R1 dt dt dt 2 C2 dt 2

V A5 = V A6 = V = f (I ); V A5 = V A4 = V = f (I ); I (t) = I L 2 (t − τ2 ) + IC2 (t − τ2 )   V A4 = V ⇒ V = f I L 2 (t − τ2 ) + IC2 (t − τ2 )    2 $ τk V = f I L 2 (t − τ2 ) + I L 1 t − k=1

V A4 = X (t) − I R p · R p − L 1 ·

1 d IL1 − · dt C2

-

IC2 · dt; V A4 = f (I )

1 d IL1 X (t) − I R p · R p − L 1 · − · IC 2 · dt C2    2 $ τk dt = f I L 2 (t − τ2 ) + I L 1 t − k=1

-

1 d IL1 − X (t) − I R p · R p − L 1 · · I L 1 (t − τ1 )· dt C2    2 $ τk dt = f I L 2 (t − τ2 ) + I L 1 t − k=1

We can summary our system differential equations: I R p − I L 1 = C1 ·

d IRp d X (t) − C1 · R p · dt dt

 d IRp d IL2 d X (t) d 2 IL1 1 d 2 IL2 1 − τ + L · − · I · =− − · Rp − L1 · (t ) L 1 2 1 dt R1 dt dt dt 2 C2 dt 2

1 d IL1 − X (t) − I R p · R p − L 1 · · I L 1 (t − τ1 )· dt C2    2 $ τk dt = f I L 2 (t − τ2 ) + I L 1 t − k=1

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

41

By differentiation of the above equation we get the following differential equation: d 2 IL1 1 d X (t) d I R p − · I L 1 (t − τ1 ) − · Rp − L1 · 2 dt dt dt C2    2 $ d = τk f I L 2 (t − τ2 ) + I L 1 t − dt k=1 The above system differential equations are sophisticated to solve analytically and it is recommended to solve numerically by using MATLAB software or other software. We use for our analysis the intrinsic electrical equivalent circuit of the laser diode (Fig. 1.4). It is reader exercise to use the complete intrinsic electrical equivalent circuit of the laser diode which takes into accounts all the major factors affecting the laser diode’s dynamic response. There is a close relationship between the intrinsic electrical equivalent circuit of the laser diode components and the laser diode parameters. X (t) − V A1 d VC1 d V A1 ; IC 1 = C 1 · = C1 · Rp dt dt d IL1 ; V A2 = V A3 = V A1 ; V A1 − V A2 = L 1 · dt

IRp = VC1

IC2 (t) = I L 1 (t − τ1 ); VC2 = V A3 − V A4 ; VL 1 = V A1 − V A2 ; IC2 = C2 ·

 d  V A3 − V A4 dt

Fig. 1.4 MRI system laser diode, intrinsic electrical equivalent circuit of the laser diode

42

1 Dynamical and Nonlinearity of Laser Diode Circuits

I R1 =

Vsup − V A7 d IL2 ; VL 2 = V A 7 − V A 4 = L 2 · ; I R1 = I L 2 ; V A4 = V A5 ; I L i = I Rse R1 dt I L 2 (t − τ2 ) + IC2 (t − τ2 ) = I Ri + ICi + I L i ; I Ri = IC i = C i ·

V A5 Ri

d V A5 ; VRi = V A5 ; VCi = V A5 dt

d ILi V A6 ; I Rse = V A5 − V A6 = L i · ; I R p = I L 1 + IC1 ; I L 2 (t − τ2 ) dt Rse   2 $ τk = I Ri + ICi + I L i + IL1 t − k=1

X (t) − V A1 ⇒ I R p · R p = X (t) − V A1 ; V A1 = X (t) − I R p · R p Rp  d V A1 d = C1 · X (t) − I R p · R p = C1 · dt dt

IRp = IC 1

IC 1 = C 1 · = V A1

d IRp d X (t) d IL1 − R p · C1 · ; V A1 − V A2 = L 1 · ⇒ V A2 dt dt dt d IL1 ; V A2 = V A3 − L1 · dt d IL1 ; IC 2 dt 1 − V A4 = · IC2 · dt C2

V A3 = V A2 = X (t) − I R p · R p − L 1 ·

 d V A3 − V A4 ⇒ V A3 dt 1 d IL1 = V A3 − · IC2 · dt; V A4 = X (t) − I R p · R p − L 1 · C2 dt 1 · IC2 · dt − C2 = C2 ·

V A4

Vsup − V A7 ⇒ I R1 · R1 = Vsup − V A7 R1 d IL2 = Vsup − I R1 · R1 ; V A7 − V A4 = L 2 · dt

I R1 = V A7

Vsup − I R1 · R1  1 d IL1 d IL2 − − X (t) − I R p · R p − L 1 · · IC2 · dt = L 2 · dt C2 dt



d d IL1 1 Vsup − I R1 · R1 − X (t) − I R p · R p − L 1 · · − dt dt C2

-

IC2 · dt = L 2 ·

d IL2 dt



1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

d Vsup d X (t) d I R p d I R1 − · R1 − + · Rp dt dt dt dt 2 2 d IL1 1 d I L 2 d Vsup →ε + L1 · + · IC 2 = L 2 · ; 2 dt C2 dt 2 dt −

d I R1 d X (t) d I R p d 2 IL1 1 · R1 − + · Rp + L1 · + · IC 2 dt dt dt dt 2 C2 d 2 IL2 = L2 · ; IC2 (t) = I L 1 (t − τ1 ) dt 2 d X (t) d I R p d 2 IL1 d I R1 · R1 − + · Rp + L1 · dt dt dt dt 2 2 1 d IL2 + · I L 1 (t − τ1 ) = L 2 · C2 dt 2

−

d IL1 I Ri · Ri = V A5 ; V A5 = V A4 = X (t) − I R p · R p − L 1 · dt d V A5 1 · IC2 · dt; ICi = Ci · − C2 dt  1 d d IL1 X (t) − I R p · R p − L 1 · − · IC2 · dt IC i = C i · dt dt C2  2 d X (t) d I R p d IL1 1 − · IC 2 IC i = C i · − · Rp − L1 · dt dt dt 2 C2  d X (t) d I R p d 2 IL1 1 − · Rp − L1 · − · I − τ IC i = C i · (t ) L1 1 dt dt dt 2 C2 d ILi V A6 = I Rse · Rse ; V A5 − V A6 = L i · dt 1 d IL1 d ILi − · IC2 · dt − I Rse · Rse = L i · X (t) − I R p · R p − L 1 · dt C2 dt

 d 1 d IL1 d ILi X (t) − I R p · R p − L 1 · − · IC2 · dt − I Rse · Rse = L i · dt dt C2 dt d X (t) d I R p d 2 IL1 1 d I Rse − · Rp − L1 · · − · IC 2 − 2 dt dt dt C2 dt d 2 ILi Rse = L i · ; IC2 (t) = I L 1 (t − τ1 ) dt 2 d 2 IL1 1 d X (t) d I R p − · Rp − L1 · − · I L 1 (t − τ1 ) dt dt dt 2 C2

43

44

1 Dynamical and Nonlinearity of Laser Diode Circuits

−

d I Rse d 2 ILi · Rse = L i · ; I Rse = I L i dt dt 2

d X (t) d I R p d 2 IL1 1 d 2 ILi d ILi − · I = L · − τ − − · Rp − L1 · · R (t ) L 1 se i 1 dt dt dt 2 C2 dt dt 2 I Ri · Ri = V A5 ; I Ri · Ri = X (t) − I R p · 1 d IL1 − · IC2 · dt Rp − L1 · dt C2

 1 d d IL1 I Ri · Ri = X (t) − I R p · R p − L 1 · − · IC2 · dt dt dt C2 d I Ri d X (t) d I R p d 2 IL1 1 · Ri = − · Rp − L1 · − · IC 2 dt dt dt dt 2 C2 d I Ri d X (t) d I R p d 2 IL1 1 · Ri = − · Rp − L1 · − · I L 1 (t − τ1 ) dt dt dt dt 2 C2 1 · ICi · dt; I Ri · I Ri · R i = V A 5 ; V A 5 = Ci

 1 d 1 I Ri · R i = · ICi · dt; · ICi · dt Ri = Ci dt Ci 1 d I Ri 1 d I Ri · Ri = = · IC i ⇒ · ICi ; I Rse = I L i dt Ci dt C i · Ri We can summary our MRI system laser diode circuitry differential equations: 1. −

d IRp d I R1 d X (t) d 2 IL1 1 d 2 IL2 + · I L 1 (t − τ1 ) = L 2 · · R1 − + · Rp + L1 · 2 dt dt dt dt C2 dt 2



2. ICi 3.

4.

d X (t) d I R p d 2 IL1 1 − · Rp − L1 · = Ci · − · I L 1 (t − τ1 ) 2 dt dt dt C2



d X (t) d I R p d 2 IL1 1 d 2 ILi d ILi − · Rp − L1 · · R − · I = L · − τ − (t ) L 1 se i 1 dt dt dt 2 C2 dt dt 2 d X (t) d I R p d 2 IL1 d I Ri · Ri = − · Rp − L1 · dt dt dt dt 2 1 1 d I Ri = − · I L 1 (t − τ1 ); · IC i C2 dt C i · Ri

Additionally KCL at nodes A1 and A5 : I R p = I L 1 + IC1 ; I L 2 (t − τ2 ) +    I L 1 t − 2k=1 τk = I Ri + ICi + I L i For simplicity we define the following function in time:

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

ξ (t, τ1 ) = −

45

d 2 IL1 1 d X (t) d I R p + · Rp + L1 · + · I L 1 (t − τ1 ) dt dt dt 2 C2

We can summary again our system differential equations: −

d I R1 d 2 IL2 · R1 + ξ (t, τ1 ) = L 2 · ; IC i dt dt 2 d 2 ILi d ILi · Rse = L i · = −Ci · ξ (t, τ1 ); −ξ (t, τ1 ) − dt dt 2

1 d I Ri d I Ri · Ri = −ξ (t, τ1 ); = · IC i dt dt C i · Ri  I R p = I L 1 + IC1 ; I L 2 (t − τ2 ) + I L 1 t −

2 $

 τk

= I Ri + I C i + I L i

k=1

We can write differential equation (4) as follow: L1 · 4. →1. ; −

d 2 IL1 d I Ri d X (t) d I R p 1 · Ri + − · Rp − =− · I L 1 (t − τ1 ) 2 dt dt dt dt C2 d I R1 dt

· R1 −

d I Ri dt

d 2 IL2 dt 2

· Ri = L 2 · 

I L 2 (t − τ2 ) + I L 1 t − 

2 $

 τk

= I Ri + I C i + I L i ⇒ I C i

k=1

= I L 2 (t − τ2 ) + I L 1 t −

2 $

 τk

− I Ri − I L i

k=1

2. ICi = Ci ·



d X (t) dt

−

d IRp dt

· Rp − L1 ·

 I L 2 (t − τ2 ) + I L 1 t −

2 $

d 2 IL1 dt 2

−

1 C2

 · I L 1 (t − τ1 )

 τk

− I Ri − I L i

k=1



d X (t) d I R p d 2 IL1 1 − · Rp − L1 · = Ci · − · I L 1 (t − τ1 ) dt dt dt 2 C2 3. →

d X (t) dt

−

d IRp dt

· Rp − L1 ·

d 2 IL1 dt 2

−

1 C2

· I L 1 (t − τ1 ) −

d ILi dt



· Rse = L i ·

d 2 ILi dt 2

46

1 Dynamical and Nonlinearity of Laser Diode Circuits

 d X (t) d I R p d I Ri d X (t) d I R p 1 − · Rp − − · Ri + − · Rp − · I L 1 (t − τ1 ) dt dt dt dt dt C2 2 1 d ILi d ILi · Rse = L i · − · I L 1 (t − τ1 ) − C2 dt dt 2 d I Ri d ILi d 2 ILi · Ri − · Rse = L i · dt dt dt 2 1 d I Ri d I Ri = · IC i ⇒ dt C i · Ri dt     2 $ 1 · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i = C i · Ri k=1 We can summary our system differential equations: L2 ·

d 2 IL2 d I R1 d I Ri · R1 − · Ri =− dt 2 dt dt

d X (t) d I R p d 2 IL1 1 − · Rp − L1 · − · I L 1 (t − τ1 ) 2 dt dt dt C2   2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci Ci Ci k=1 d 2 ILi d I Ri d ILi d 2 ILi = − ⇒ L · · R · R i se i dt 2 dt dt dt 2     2 $ 1 d ILi · Rse = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i − Ci dt k=1     2 $ d I Ri 1 · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i = dt C i · Ri k=1

Li ·

&& d 2 IL2 d IL2 d I Ri · R1 − · Ri =− dt 2 dt dt     2 $ d IL2 1 · R1 − =− · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt Ci k=1 I R1 = I L 2 ⇒ L 2 ·

L2 ·

d 2 IL2 dt 2

d 2 IL1 1 d X (t) d I R p − · Rp − L1 · − · I L 1 (t − τ1 ) 2 dt dt dt C2

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

47

  2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci C C i i k=1

    2 $ d 2 ILi 1 d ILi · Rse Li · = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i − 2 dt Ci dt k=1     2 $ 1 d I Ri = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt C i · Ri k=1

We can summary our system differential equation in this stage:     2 $ d 2 IL2 d IL2 1 L2 · · R1 − =− · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt 2 dt Ci k=1 d X (t) d I R p d 2 IL1 1 − · Rp − L1 · − · I L 1 (t − τ1 ) dt dt dt 2 C2   2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci C C i i k=1

    2 $ d 2 ILi 1 d ILi · Rse Li · = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i − 2 dt Ci dt k=1     2 $ 1 d I Ri = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt C i · Ri k=1

&&& d IRp d IRp d X (t) − R p · C1 · ⇒ dt dt dt 1 1 d X (t) − = · · IC 1 ; IC 1 = I R p − I L 1 Rp dt R p · C1

IC 1 = C 1 ·

  d IRp 1 d X (t) 1 = − · · I Rp − IL1 dt Rp dt R p · C1 d X (t) d I R p d 2 IL1 1 − · Rp − L1 · − · I L 1 (t − τ1 ) 2 dt dt dt C2   2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci Ci Ci k=1

  d2 IL1 d X (t) 1  d X (t) 1 1 − − · I R p − IL1 − L 1 · − · I L 1 (t − τ1 ) = · I L 2 (t − τ2 ) dt dt C1 C2 Ci dt 2

48

1 Dynamical and Nonlinearity of Laser Diode Circuits ⎛ ⎞ 2 $ 1 1 1 ⎝ + · IL1 t − τk ⎠ − · I Ri − · ILi Ci Ci Ci k=1

1 1 d 2 IL1 1 · IRp − · IL1 − L 1 · − · I L 1 (t − τ1 ) C1 C1 dt 2 C2   2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci C C i i k=1 We can summary our system differential equation at this stage:     2 $ d 2 IL2 d IL2 1 · R1 − 1. L 2 · =− · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt 2 dt Ci k=1 2.

1 1 d 2 IL1 1 · IRp − · IL1 − L 1 · − · I L 1 (t − τ1 ) C1 C1 dt 2 C2   2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci Ci Ci k=1

    2 $ d 2 ILi 1 d ILi · Rse 3. L i · = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i − 2 dt Ci dt k=1     2 $ 1 d I Ri = 4. · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt C i · Ri k=1 5.

  d IRp 1 d X (t) 1 · · I Rp − IL1 = − dt Rp dt R p · C1 Our above system variables are I L 2 , I Ri , I L i , I R p , I L 1 We define the following new variables:

dI dY2 ; dtL 2 dt

d 2 IL1 dt 2

=

dI dY1 ; dtL 1 dt

= Y1 ;

d 2 IL2 dt 2

= Y2 d 2 ILi dY3 d I L i ; = Y3 = 2 dt dt dt

We can write our system differential equation with the new variables:     2 $ dY2 1 = −Y2 · R1 − L2 · · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt Ci k=1 1 1 1 dY1 − · IRp − · IL1 − L 1 · · I L 1 (t − τ1 ) C1 C1 dt C2

=

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

49

  2 $ 1 1 1 1 = · I L 2 (t − τ2 ) + · IL1 t − τk − · I Ri − · ILi Ci Ci C C i i k=1     2 $ 1 dY3 = Li · · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt Ci k=1

d IL1 d IL2 = Y1 ; = Y2 dt dt     2 $ 1 d I Ri d ILi = = Y3 · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i ; dt C i · Ri dt k=1 − Y3 · Rse ;

  d IRp 1 d X (t) 1 = − · · I Rp − IL1 dt Rp dt R p · C1 Finally, we summary our system differential equations:     2 $ dY2 R1 1 = −Y2 · − · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt L2 Ci · L 2 k=1 1 dY1 1 1 = · IRp − · IL1 − · I L 1 (t − τ1 ) dt C1 · L 1 C1 · L 1 C2 · L 1   2 $ 1 1 − · IL1 t − τk − · I L 2 (t − τ2 ) Ci · L 1 Ci · L 1 k=1 1 1 · I Ri + · ILi Ci · L 1 Ci · L 1     2 $ 1 dY3 = · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i dt L i · Ci k=1 +

Rse d I L 1 d IL2 = Y1 ; = Y2 ; Li dt dt     2 $ 1 d I Ri d ILi = = Y3 · I L 2 (t − τ2 ) + I L 1 t − τk − I Ri − I L i ; dt C i · Ri dt k=1 − Y3 ·

  d IRp 1 d X (t) 1 = − · · I Rp − IL1 dt Rp dt R p · C1 Our above system variables are I L 2 , I Ri , I L i , I R p , I L 1 , Y1 , Y2 , Y3 To find the equilibrium point (fixed points) of our MRI system Laser diode circuitry by using intrinsic electrical equivalent circuit of laser diode.

50

1 Dynamical and Nonlinearity of Laser Diode Circuits

lim I L 1 (t − τ1 ) = I L 1 (t); lim I L 2 (t − τ2 ) t→∞   2 $ = I L 2 (t); lim I L 1 t − τk = I L 1 (t); t  τ1 ; t  τ2

t→∞

t→∞

And setting 1, 2, 3 t

2 $ k=1

τk ;

d IL1 dt

= 0;

d IL2 dt

k=1

= 0;

d ILi dt

= 0;

d I Ri dt

= 0;

d IRp dt

= 0;

dYi dt

= 0∀i =

d IL1 d IL2 d ILi = 0 ⇒ Y1∗ = 0; = 0 ⇒ Y2∗ = 0; = 0 ⇒ Y3∗ = 0 dt dt dt

  1 · I L∗2 + I L∗1 − I R∗i − I L∗i = 0 ⇒ I L∗2 + I L∗1 − I R∗i − I L∗i = 0 Ci · L 2  1 1 1 1 1 1 1 ∗ · I L∗1 − · IRp − + + · I L∗2 + · I R∗i + · I∗ = 0 C1 C1 C2 Ci Ci Ci Ci L i  1  ∗ d X (t) − · I R p − I L∗1 = 0 I L∗2 + I L∗1 − I R∗i − I L∗i = 0; dt C1 −

We consider our RF feeding source as X (t) =  + ξ (t);   ξ (t) d X (t) = dt 



 d dξ (t) dξ (t) d X (t) →ε + ; → ε; →ε dt dt dt dt

  1  ∗ d X (t) →ε − · I R p − I L∗1 = 0 ⇒ I R∗ p dt C1  1 1 1 1 1 ∗ · I L∗1 − = IL1 ; − + · I L∗2 + · I R∗i + · I∗ = 0 C2 Ci Ci Ci Ci L i I L∗2 + I L∗1 − I R∗i − I L∗i = 0 ⇒ I L∗i = I L∗2 + I L∗1 − I R∗i 1 1 1 − · I∗ + · I∗ − · I∗ = 0 C 2 L 1 C i Ri C i Ri 1 · I ∗ = 0 ⇒ −I L∗1 = 0 ⇒ I L∗1 = 0; I R∗ p = 0; I L∗2 C2 L 1 − I R∗i − I L∗i = 0 ⇒ I L∗2 = I R∗i + I L∗i

−

We can define our fixed points (equilibrium points):     E ∗ I L∗1 , I L∗2 , I R∗i , I R∗ p , I L∗i , Y1∗ , Y2∗ , Y3∗ = 0, I R∗i + I L∗i , I R∗i , 0, I L∗i , 0, 0, 0 ε

First we analyze our system stability for no delays: τk → ε ∀ k = 1, 2; τ1 +τ2 →

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

51

We get the following set of system differential equations:   R1 1 dY2 = −Y2 · − · I L 2 + I L 1 − I Ri − I L i dt L2 Ci · L 2  dY1 1 1 1 1 1 1 · IL1 − · IRp − · + + · IL2 = dt C1 · L 1 L1 C1 C2 Ci Ci · L 1 1 1 + · I Ri + · ILi Ci · L 1 Ci · L 1   1 dY3 = · I L 2 + I L 1 − I Ri − I L i dt L i · Ci Rse d I L 1 d IL2 = Y1 ; = Y2 ; − Y3 · Li dt dt  d ILi  1 d I Ri = · I L 2 + I L 1 − I Ri − I L i ; dt C i · Ri dt  d X (t)  d IRp 1 =− →ε · I Rp − IL1 ; = Y3 ; dt R p · C1 dt Stability analysis: The standard local stability analysis about any one of the equilibrium points of MRI system Laser diode circuitry consists in adding to its coordinates [I L 1 I L 2 I Ri I R p I L i Y1 Y2 Y3 ] arbitrarily small increments of exponen  tial form i L 1 i L 2 i Ri i R p i L i y1 y2 y3 · eλ·t , and retaining the first order terms in i L 1 i L 2 i Ri i R p i L i y1 y2 y3 . The system of eight homogeneous equations leads to a polynomial characteristics equation in the eigenvalue λ. The polynomial characteristics equation accept by set the below equations [8]. The MRI system Laser diode circuitry fixed values  with arbitrarily small increments of exponential form  i L 1 i L 2 i Ri i R p i L i y1 y2 y3 · eλ·t are i = 0 (first fixed point) i = 1 (second fixed point), i = 2 (third fixed point), etc., I L 1 (t) = I L(i)1 + i L 1 · eλ·t ; I L 2 (t) = I L(i)2 + i L 2 · eλ·t ; I Ri (t) = I R(i)i + i Ri · eλ·t ; I R p (t) = I R(i)p + i R p · eλ·t Y1 (t) = Y1(i) + y1 · eλ·t ; Y2 (t) = Y2(i) + y2 · eλ·t ; Y3 (t) d I L 1 (t) = i L 1 · λ · eλ·t = Y3(i) + y3 · eλ·t ; dt d I R p (t) d I L 2 (t) d I Ri (t) = i L 2 · λ · eλ·t ; = i Ri · λ · eλ·t ; dt dt dt dY (t) 1 = y1 · λ · eλ·t = i R p · λ · eλ·t ; dt

52

1 Dynamical and Nonlinearity of Laser Diode Circuits

dY2 (t) dY3 (t) = y2 · λ · eλ·t ; = y3 · λ · eλ·t ; I L i (t) dt dt d I L i (t) = I L(i)i + i L i · eλ·t ; = i L i · λ · eλ·t dt We choose the above expressions for our I L 1 (t), I L 2 (t), I Ri (t), I R p (t), I L i (t), Y1 (t),   Y2 (t), Y3 (t) as small displacement i L 1 i L 2 i Ri i R p i L i y1 y2 y3 from the system fixed points at time t = 0 I L 1 (t = 0) = I L(i)1 + i L 1 ; I L 2 (t = 0) = I L(i)2 + i L 2 ; I Ri (t = 0) = I R(i)i + i Ri ; I R p (t = 0) = I R(i)p + i R p Y1 (t = 0) = Y1(i) + y1 ; Y2 (t = 0) = Y2(i) + y2 ; Y3 (t = 0) = Y3(i) + y3 For t > 0, λ < 0 the selected fixed point is stable otherwise t > 0, λ > 0 is unstable. Our system tends to the selected fixed point exponentially for t > 0, λ < 0 otherwise go away from the selected fixed point exponentially. The eigenvalue λ is the parameter which if the selected fixed point is stable or unstable, additionally his absolute value |λ| establish the speed of flow toward or away from the selected fixed point.   R1 1 dY2 = −Y2 · − · I L 2 + I L 1 − I Ri − I L i dt L2 Ci · L 2  R  1 y2 · λ · eλ·t = − Y2(i) + y2 · eλ·t · L2   1 − · I L(i)2 + i L 2 · eλ·t + I L(i)1 + i L 1 · eλ·t − I R(i)i − i Ri · eλ·t − I L(i)i − i L i · eλ·t Ci · L 2 R1 R1 λ·t y2 · λ · eλ·t = −Y2(i) · − y2 · ·e L2 L2  1 1 1 1 − · I L(i)2 + · I L(i)1 − · I R(i)i − · I (i) Ci · L 2 Ci · L 2 Ci · L 2 Ci · L 2 L i 1 1 + · i L 2 · eλ·t + · i L 1 · eλ·t Ci · L 2 Ci · L 2 1 1 λ·t λ·t − · i Ri · e − · i Li · e Ci · L 2 Ci · L 2 R1 y2 · λ · eλ·t = −Y2(i) · L2  1 1 1 1 R1 λ·t (i) (i) (i) (i) − · IL2 + · IL1 − · I Ri − · I L i − y2 · ·e Ci · L 2 Ci · L 2 Ci · L 2 Ci · L 2 L2  1 1 1 1 − · i L 2 · eλ·t + · i L 1 · eλ·t − · i Ri · eλ·t − · i L i · eλ·t Ci · L 2 Ci · L 2 Ci · L 2 Ci · L 2

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

At fixed point −Y2(i) ·



53 

(i) (i) (i) (i) R1 1 1 1 1 L 2 − Ci ·L 2 · I L 2 + Ci ·L 2 · I L 1 − Ci ·L 2 · I Ri − Ci ·L 2 · I L i = 0

R1 1 1 1 1 − y2 · λ − · i L2 − · i L1 + · i Ri + · i Li = 0 L2 Ci · L 2 Ci · L 2 Ci · L 2 Ci · L 2  dY1 1 1 1 1 1 · IL1 = · IRp − · + + dt C1 · L 1 L1 C1 C2 Ci 1 1 1 − · IL2 + · I Ri + · ILi Ci · L 1 Ci · L 1 Ci · L 1   1 y1 · λ · eλ·t = · I R(i)p + i R p · eλ·t C1 · L 1      1 1 1 1 1 − · I L(i)1 + i L 1 · eλ·t − · + + · I L(i)2 + i L 2 · eλ·t L1 C1 C2 Ci Ci · L 1     1 1 · I R(i)i + i Ri · eλ·t + · I L(i)i + i L i · eλ·t + Ci · L 1 Ci · L 1  1 1 1 1 1 y1 · λ · eλ·t = · I L(i)1 · I R(i)p − · + + C1 · L 1 L1 C1 C2 Ci 1 1 1 − · I L(i)2 + · I R(i)i + · I (i) Ci · L 1 Ci · L 1 Ci · L 1 L i  1 1 1 1 1 λ·t · eλ·t + i Rp · · e − i L1 · · + + C1 · L 1 L1 C1 C2 Ci 1 1 1 − i L2 · · eλ·t + i Ri · · eλ·t + i L i · · eλ·t Ci · L 1 Ci · L 1 Ci · L 1

−y2 ·

 1 1 1 1 1 (i) · I L(i)1 ·I − · + + C1 · L 1 R p L1 C1 C2 Ci At fixed point 1 1 1 − · I (i) + · I (i) + · I (i) = 0 Ci · L 1 L 2 C i · L 1 Ri Ci · L 1 L i  1 1 1 1 1 − i L1 · · + + C1 · L 1 L1 C1 C2 Ci 1 1 1 · + i Ri · + i Li · =0 Ci · L 1 Ci · L 1 Ci · L 1

− y1 · λ + i R p · − i L2

  1 dY3 Rse = · I L 2 + I L 1 − I Ri − I L i − Y3 · dt L i · Ci Li  1 y3 · λ · eλ·t = · I L(i)2 + i L 2 · eλ·t + I L(i)1 + i L 1 · eλ·t L i · Ci      R  se − I R(i)i + i Ri · eλ·t − I L(i)i + i L i · eλ·t − Y3(i) + y3 · eλ·t · Li

54

1 Dynamical and Nonlinearity of Laser Diode Circuits

 1 · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i + i L 2 · eλ·t L i · Ci  Rse Rse λ·t − i Ri · eλ·t − i L i · eλ·t − Y3(i) · − y3 · ·e Li Li

y3 · λ · eλ·t = +i L 1 · eλ·t

1 Rse · (I L(i)2 + I L(i)1 − I R(i)i − I L(i)i ) − Y3(i) · L i · Ci Li 1 + · (i L 2 · eλ·t + i L 1 · eλ·t − i Ri · eλ·t L i · Ci Rse λ·t − i L i · eλ·t ) − y3 · ·e Li

y3 · λ · eλ·t =

At fixed point

1 L i ·Ci

  · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i − Y3(i) ·

Rse Li

=0

  1 Rse · i L 2 + i L 1 − i Ri − i L i − y3 · − y3 · λ = 0 L i · Ci Li d IL1 = Y1 ; i L 1 · λ · eλ·t = Y1(i) + y1 · eλ·t ; Y1(i) = 0; −i L 1 · λ + y1 = 0 dt d IL2 = Y2 ; i L 2 · λ · eλ·t = Y2(i) + y2 · eλ·t ; Y2(i) = 0; −i L 2 · λ + y2 = 0 dt

  2 $   d I Ri d I Ri 1 1 · I L 2 + I L 1 − I Ri − I L i ⇒ · I L k − I Ri − I L i = = dt C i · Ri dt C i · Ri k=1

1 i Ri · λ · eλ·t = · C i · Ri      I L(i)1 + i L 1 · eλ·t + I L(i)2 + i L 2 · eλ·t − I R(i)i + i Ri · eλ·t − I L(i)i + i L i · eλ·t   1 · I L(i)1 + I L(i)2 − I R(i)i − I L(i)i C i · Ri   · i L 1 · eλ·t + i L 2 · eλ·t − i Ri · eλ·t − i L i · eλ·t

i Ri · λ · eλ·t = +

1 C i · Ri

  1 · I L(i)1 + I L(i)2 − I R(i)i − I L(i)i = 0; −i Ri · λ C · Ri At fixed point i   1 + · i L 1 + i L 2 − i Ri − i L i = 0 C i · Ri d ILi = Y3 ; i L i · λ · eλ·t = Y3(i) + y3 · eλ·t ; Y3(i) = 0; −i L i · λ + y3 = 0 dt

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

i Rp

55

  d IRp 1 =− · I R p − I L 1 ; i R p · λ · eλ·t dt R p · C1    1 =− · I R(i)p + i R p · eλ·t − I L(i)1 + i L 1 · eλ·t R p · C1     1 1 · λ · eλ·t = − · I R(i)p − I L(i)1 − · i R p · eλ·t − i L 1 · eλ·t R p · C1 R p · C1

    At fixed point − R p1·C1 · I R(i)p − I L(i)1 = 0; −i R p · λ − R p1·C1 · i R p − i L 1 = 0 We can summary our system arbitrarily small increments equations:  1 1 1 1 1 − i L1 · · + + − y1 · λ + i R p · C1 · L 1 L1 C1 C2 Ci 1 1 1 + i Ri · + i Li · =0 − i L2 · Ci · L 1 Ci · L 1 Ci · L 1 R1 1 1 − y2 · λ − · i L2 − · i L1 L2 Ci · L 2 Ci · L 2 1 1 + · i Ri + · i Li = 0 Ci · L 2 Ci · L 2 − y2 ·

1 1 1 1 Rse · i L2 + · i L1 − · i Ri − · i L i − y3 · L i · Ci L i · Ci L i · Ci L i · Ci Li − y3 · λ = 0; −i L 1 · λ + y1 = 0; −i L 2 · λ + y2 = 0 1 1 · i L1 + · i L2 C i · Ri C i · Ri 1 1 − · i Ri − · i Li = 0 C i · Ri C i · Ri − i Ri · λ +

−i L i · λ + y3 = 0; −i R p · λ −

1 1 · i Rp + · i L1 = 0 R p · C1 R p · C1

The small increments Jacobian of our MRI system Laser diode circuitry system is as below:

56

1 Dynamical and Nonlinearity of Laser Diode Circuits

⎛

y1

⎞

⎜ y⎟ ⎜ 2⎟ ⎜ ⎟ ⎛ ⎞ ⎜ y3 ⎟ ⎜ ⎟ 11 . . . 18 ⎜ ⎟ ⎜ .. . . .. ⎟ ⎜ i L 1 ⎟ ⎟ = 0; 11 = −λ; 12 = 13 = 0; 14 ⎝ . . . ⎠·⎜ ⎜ i L2 ⎟ ⎜ ⎟ 81 · · · 88 ⎜i ⎟ ⎜ Ri ⎟ ⎜ ⎟ ⎝ i Li ⎠ i Rp

1 1 1 1 1 ; 15 = − · + + L1 C1 C2 Ci Ci · L 1 

=−

1 1 1 ; 17 = ; 18 = ; 21 = 0; 22 Ci · L 1 Ci · L 1 C1 · L 1 R1 1 1 = −λ − ; 23 = 0; 24 = − ; 25 = − L2 Ci · L 2 Ci · L 2

16 =

1 1 ; 27 = ; 28 = 0; 31 = 32 = 0; 33 Ci · L 2 Ci · L 2 Rse 1 =− − λ; 34 = 35 = Li Ci · L i

26 =

1 ; 38 = 0; 41 = 1; 42 = 43 = 0; 44 Ci · L i = −λ; 45 = . . . = 48 = 0

36 = 37 = −

51 = 0; 52 = 1; 53 = 54 = 0; 55 = −λ; 56 = 57 = 58 = 0; 61 = 62 = 63 = 0 64 = 65 = =−

1 1 ; 66 = −λ − ; 67 C i · Ri C i · Ri

1 ; 68 = 0; 71 = 72 = 0; 73 = 1 C i · Ri

74 = 75 = 76 = 0; 77 = −λ; 78 = 0; 81 1 = 82 = 83 = 0; 84 = ; 85 = 86 = 87 = 0 R p · C1 88 = −λ −

1 R p · C1

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

⎛

11 . . . ⎜ .. . . A−λ· I =⎝ . .

57

⎞ 18 .. ⎟; det(A − λ · I ) . ⎠

81 · · · 88

=0⇒

8 $

  λk · f k L 1 , L 2 , Ci , Ri , R p , C1 , Rse , L i , . . . = 0

k=0

   We get zeros of the polynomial 8k=0 λk · f k L 1 , L 2 , Ci , Ri , R p , C1 , Rse , L i , . . . = 0 with real coefficients f 1 , f 2 , . . . which are functions of our system parametersL 1 , L 2 ,8 Ci , .k . .. The characteristic equation is a polynomial of degree N = 8; k=0 λ · f k (L 1 , L 2 , . . .) = 0. Eigenvalue stability discussion: Our MRI system Laser diode circuitry system involving N variables (N > 2, N = 8), the characteristic equation is of degree N = 8 and must often be solved numerically. Expect in some particular cases, such an equation has (N = 8) distinct roots that can be real or complex. These values are the eigenvalues of the 8x8 Jacobian matrix (A). The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 8-variables I L 1 I L 2 I Ri I R p I L i Y1 Y2 Y3 system has eight eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the eight eigenvalues are real and negative (stable steady state), (2) the eight eigenvalues are real, seven of them are negative (unstable steady state), (3) and (4) at least two eigenvalues are complex conjugates with a negative real part and the other are negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) at least two eigenvalues are complex conjugates with a negative real part and the other, one of them real is positive (unstable steady state) [12]. Second we analyze our system stability for τ1 = τ, τ2 = 0; τ1 = 0, τ2 = τ ; τ1 = τ2 = τ . We choose the first case τ21 = τ, τ2 = 0. We summary out system differential equations (τ1 = τ, τ2 = 0; k=1 τk = τ ):   R1 1 dY2 = −Y2 · − · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i dt L2 Ci · L 2  1 dY1 1 1 1 1 · I L 1 (t − τ ) = · IRp − · IL1 − · + dt C1 · L 1 C1 · L 1 L1 C2 Ci 1 1 1 − · I L 2 (t) + · I Ri + · ILi Ci · L 1 Ci · L 1 Ci · L 1   1 dY3 = · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i dt L i · Ci Rse d I L 1 d IL2 ; − Y3 · = Y1 ; = Y2 Li dt dt

58

1 Dynamical and Nonlinearity of Laser Diode Circuits

 d ILi  d I Ri 1 · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i ; = = Y3 dt C i · Ri dt   d IRp 1 d X (t) 1 = − · · I Rp − IL1 dt Rp dt R p · C1 I L 1 = I L 1 (t); I L 2 = I L 2 (t); I Ri = I Ri (t); I L i = I L i (t); I R p = I R p (t); Yk = Yk (t) ∀ k = 1, 2, 3 We already define the MRI system Laser diode circuitry fixed values with arbi trarily small increments of exponential form i L 1 i L 2 i Ri i R p i L i y1 y2 y3 · eλ·t (i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc.) and complete it. I L 1 (t − τ ) = I L(i)1 + i L 1 · eλ·(t−τ ) ⇒ I L 1 (t − τ ) d I L 1 (t − τ ) = I L(i)1 + i L 1 · eλ·t · e−λ·τ ; = i L 1 · λ · eλ·t · e−λ·τ dt  1 dY1 1 1 1 1 · I L 1 (t − τ ) = · IRp − · IL1 − · + dt C1 · L 1 C1 · L 1 L1 C2 Ci 1 1 1 − · I L 2 (t) + · I Ri + · ILi Ci · L 1 Ci · L 1 Ci · L 1     1 1 y1 · λ · eλ·t = · I R(i)p + i R p · eλ·t − · I L(i)1 + i L 1 · eλ·t C1 · L 1 C1 · L 1    1 1 1 − · I L(i)1 + i L 1 · eλ·(t−τ ) · + L1 C2 Ci     1 1 · I L(i)2 + i L 2 · eλ·t + · I R(i)i + i Ri · eλ·t − Ci · L 1 Ci · L 1   1 (i) λ·t · ILi + i Li · e + Ci · L 1  1 1 1 1 1 (i) (i) λ·t y1 · λ · e = · I L(i)1 ·I − ·I − · + C1 · L 1 R p C1 · L 1 L 1 L1 C2 Ci 1 1 1 − · I (i) + · I (i) + · I (i) Ci · L 1 L 2 C i · L 1 Ri Ci · L 1 L i 1 1 + · i R p · eλ·t − · i L 1 · eλ·t C1 · L 1 C1 · L 1  1 1 1 1 · i L 1 · eλ·(t−τ ) − − · + · i L 2 · eλ·t L1 C2 Ci Ci · L 1 1 1 + · i Ri · eλ·t + · i L i · eλ·t Ci · L 1 Ci · L 1

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

 1 1 1 1 1 · I L(i)1 · I R(i)p − · I L(i)1 − · + C1 · L 1 C1 · L 1 L1 C2 Ci At fixed point 1 1 1 − · I L(i)2 + · I R(i)i + · I (i) = 0 Ci · L 1 Ci · L 1 Ci · L 1 L i − y1 · λ − −

  1 1 1 · e−λ·τ · i L 1 + + C1 C2 Ci 1 1 1 + · i Rp + · i Ri + · i Li = 0 C1 · L 1 Ci · L 1 Ci · L 1

1 · L1

1 · i L2 Ci · L 1



  dY2 R1 1 = −Y2 · − · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i dt L2 Ci · L 2

 R  1 y2 · λ · eλ·t = − Y2(i) + y2 · eλ·t · L2   1 (i) (i) (i) (i) − · I L 2 + i L 2 · eλ·t + I L 1 + i L 1 · eλ·(t−τ ) − I Ri − i Ri · eλ·t − I L i − i L i · eλ·t Ci · L 2

y2 · λ · eλ·t = −Y2(i) · 1 Ci · L 2 1 − Ci · L 2 −

R1 L2

  R1 λ·t · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i − y2 · ·e L2   · i L 2 · eλ·t + i L 1 · eλ·(t−τ ) − i Ri · eλ·t − i L i · eλ·t

At fixed point −Y2(i) ·

R1 L2

−

1 Ci ·L 2

  · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i = 0

R1 1 1 − y2 · λ − · i L2 − i L1 · · e−λ·τ L2 Ci · L 2 Ci · L 2 1 1 + · i Ri + · i Li = 0 Ci · L 2 Ci · L 2 − y2 ·

  1 dY3 Rse = · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i − Y3 · dt L i · Ci Li 1 y3 · λ · eλ·t = · L i · Ci   I L(i)2 + i L 2 · eλ·t + I L(i)1 + i L 1 · eλ·(t−τ ) − I R(i)i − i Ri · eλ·t − I L(i)i − i L i · eλ·t Rse Rse λ·t − y3 · ·e Li Li   1 y3 · λ · eλ·t = · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i L i · Ci − Y3(i) ·

59

60

1 Dynamical and Nonlinearity of Laser Diode Circuits

− Y3(i) ·

  Rse 1 + · i L 2 · eλ·t + i L 1 · eλ·(t−τ ) − i Ri · eλ·t − i L i · eλ·t Li L i · Ci

Rse λ·t ·e Li

− y3 ·

At fixed point

  · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i − Y3(i) ·

1 L i ·Ci

− y3 · λ − y3 · −

1 · i Ri L i · Ci

Rse Li

=0

Rse 1 1 + · i L2 + i L1 · · e−λ·τ Li L i · Ci L i · Ci 1 − · i Li = 0 L i · Ci

d IL1 = Y1 ; i L 1 · λ · eλ·t = Y1(i) + y1 · eλ·t ; Y1(i) dt = 0; −i L 1 · λ + y1 = 0 d IL2 = Y2 ; i L 2 · λ · eλ·t = Y2(i) + y2 · eλ·t ; Y2(i) dt = 0; −i L 2 · λ + y2 = 0   1 d I Ri = · I L 2 (t) + I L 1 (t − τ ) − I Ri − I L i dt C i · Ri 1 i Ri · λ · eλ·t = · C i · Ri  I L(i)2 + i L 2 · eλ·t + I L(i)1 + i L 1 · eλ·t · e−λ·τ − I R(i)i  −i Ri · eλ·t − I L(i)i − i L i · eλ·t   1 · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i C i · Ri   · i L 2 · eλ·t + i L 1 · eλ·t · e−λ·τ − i Ri · eλ·t − i L i · eλ·t

i Ri · λ · eλ·t = +

1 C i · Ri

At fixed point

  · I L(i)2 + I L(i)1 − I R(i)i − I L(i)i = 0

1 Ci ·Ri

1 1 · i Ri − i Ri · λ + · i L2 C i · Ri C i · Ri 1 1 + i L1 · · e−λ·τ − · i Li = 0 C i · Ri C i · Ri

−

d ILi = Y3 ; i L i · λ · eλ·t = Y3(i) + y3 · eλ·t ; Y3(i) dt

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

61

= 0; −i L i · λ + y3 = 0

i Rp

  d IRp 1 =− · I R p − I L 1 ; i R p · λ · eλ·t dt R p · C1    1 =− · I R(i)p + i R p · eλ·t − I L(i)1 + i L 1 · eλ·t R p · C1     1 1 · λ · eλ·t = − · I R(i)p − I L(i)1 − · i R p · eλ·t − i L 1 · eλ·t R p · C1 R p · C1

    At fixed point − R p1·C1 · I R(i)p − I L(i)1 = 0; −i R p · λ − R p1·C1 · i R p − i L 1 = 0 our system arbitrarily small increments equations with delay  We can summary 2 τ1 = τ, τ2 = 0; k=1 τk = τ :    1 1 1 1 −λ·τ ·e · i L1 − y1 · λ − · + + L1 C1 C2 Ci 1 1 1 1 − · i L2 + · i Rp + · i Ri + · i Li = 0 Ci · L 1 C1 · L 1 Ci · L 1 Ci · L 1 R1 1 − y2 · λ − · i L2 L2 Ci · L 2 1 1 1 − i L1 · · e−λ·τ + · i Ri + · i Li = 0 Ci · L 2 Ci · L 2 Ci · L 2

− y2 ·

Rse 1 1 + · i L2 + i L1 · · e−λ·τ Li L i · Ci L i · Ci 1 1 − · i Ri − · i L i = 0; −i L 1 · λ + y1 = 0; −i L 2 · λ + y2 = 0 L i · Ci L i · Ci

− y3 · λ − y3 ·

1 1 1 · i Ri − i Ri · λ + · i L2 + i L1 · · e−λ·τ C i · Ri C i · Ri C i · Ri 1 − · i L i = 0; −i L i · λ + y3 = 0 C i · Ri

−

−i R p · λ −

1 1 · i Rp + · i L1 = 0 R p · C1 R p · C1

The small increments system  Jacobian of our MRI  Laser diode circuitry system 2  with delay is as below τ1 = τ, τ2 = 0; τk = τ : k=1

62

1 Dynamical and Nonlinearity of Laser Diode Circuits

⎛

y1

⎞

⎜ y⎟ ⎜ 2⎟ ⎜ ⎟ ⎛ ⎞ ⎜ y3 ⎟ ⎜ ⎟ ϒ11 . . . ϒ18 ⎜ ⎟ ⎜ .. . . .. ⎟ ⎜ i L 1 ⎟ ⎟ = 0; ϒ11 = −λ; ϒ12 = ϒ13 = 0; ϒ14 ⎝ . . . ⎠·⎜ ⎜ i L2 ⎟ ⎜ ⎟ ϒ81 · · · ϒ88 ⎜i ⎟ ⎜ Ri ⎟ ⎜ ⎟ ⎝ i Li ⎠ =−

1 · L1





i Rp

1 1 1 + + C1 C2 Ci



 · e−λ·τ ; ϒ15 = −

1 Ci · L 1

1 1 1 ; ϒ17 = ; ϒ18 = ; ϒ21 = 0; ϒ22 Ci · L 1 Ci · L 1 C1 · L 1 R1 1 =− − λ; ϒ23 = 0; ϒ24 = − · e−λ·τ L2 Ci · L 2

ϒ16 =

1 1 1 ; ϒ26 = ; ϒ27 = ; ϒ28 Ci · L 2 Ci · L 2 Ci · L 2 Rse = 0; ϒ31 = ϒ32 = 0; ϒ33 = −λ − Li

ϒ25 = −

1 1 1 · e−λ·τ ; ϒ35 = ; ϒ36 = − ; ϒ37 L i · Ci L i · Ci L i · Ci 1 =− ; ϒ38 = 0; ϒ41 = 1; ϒ42 = ϒ43 = 0 L i · Ci

ϒ34 =

ϒ44 = −λ; ϒ45 = ϒ46 = ϒ47 = ϒ48 = 0; ϒ51 = 0; ϒ52 = 1; ϒ53 = ϒ54 = 0; ϒ55 = −λ; ϒ56 = ϒ57 = ϒ58 = 0 1 1 · e−λ·τ ; ϒ65 = ; ϒ66 C i · Ri C i · Ri 1 =− C i · Ri

ϒ61 = ϒ62 = ϒ63 = 0; ϒ64 = =−

1 − λ; ϒ67 C i · Ri

ϒ68 = 0; ϒ71 = ϒ72 = 0; ϒ73 = 1; ϒ74 = ϒ75 = ϒ76 = 0; ϒ77 = −λ; ϒ78 = 0 1 ; ϒ85 R p · C1 1 = −λ − R p · C1

ϒ81 = ϒ82 = ϒ83 = 0; ϒ84 = = ϒ86 = ϒ87 = 0; ϒ88

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

63

⎞ ϒ11 . . . ϒ18 ⎟ ⎜ A − λ · I = ⎝ ... . . . ... ⎠; det(A − λ · I ) = 0 ϒ81 · · · ϒ88 1 1 1 1 σ1 = − ; σ2 = ; σ3 = + L1 C1 C2 Ci ⎛

1 1 1 R1 ; σ5 = ; σ6 = ; σ7 = − Ci · L 1 Ci · L 1 C1 · L 1 L2 1 1 σ8 = − ; σ9 = Ci · L 2 Ci · L 2

σ4 = −

1 1 1 ; σ11 = − ; σ12 = L i · Ci L i · Ci C i · Ri 1 1 Rse =− ; σ14 = ; σ15 = C i · Ri R p · C1 Li

σ10 = σ13

2 We need to get the characteristic equation for the case τ1 = τ, τ2 = 0; k=1 τk = τ and discuss stability. We study the occurrence of any possible stability switching, resulting from the increase of the value of the time delay τ for the general characteristic equation D(λ, τ ). The expression for the characteristic equation is in the form D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n > m.   ϒ11 = −λ; ϒ12 = ϒ13 = 0; ϒ14 = σ1 · σ2 + σ3 · e−λ·τ ; ϒ15 = σ4 ϒ16 = σ5 ; ϒ17 = σ5 ; ϒ18 = σ6 ; ϒ21 = 0; ϒ22 = σ7 − λ; ϒ23 = 0; ϒ24 = σ8 · e−λ·τ ϒ25 = σ8 ; ϒ26 = σ9 ; ϒ27 = σ9 ; ϒ28 = 0; ϒ31 = ϒ32 = 0; ϒ33 = −λ − σ15 ϒ34 = σ10 · e−λ·τ ; ϒ35 = σ10 ; ϒ36 = σ11 ; ϒ37 = σ11 ; ϒ38 = 0; ϒ41 = 1; ϒ42 = ϒ43 = 0

ϒ44 = −λ; ϒ45 = ϒ46 = ϒ47 = ϒ48 = 0; ϒ51 = 0; ϒ52 = 1; ϒ53 = ϒ54 = 0; ϒ55 = −λ; ϒ56 = ϒ57 = ϒ58 = 0 ϒ61 = ϒ62 = ϒ63 = 0; ϒ64 = σ12 · e−λ·τ ; ϒ65 = σ12 ; ϒ66 = σ13 − λ; ϒ67 = σ13 ϒ68 = 0; ϒ71 = ϒ72 = 0; ϒ73 = 1; ϒ74 = ϒ75 = ϒ76 = 0; ϒ77 = −λ; ϒ78 = 0 ϒ81 = ϒ82 = ϒ83 = 0; ϒ84 = σ14 ; ϒ85 = ϒ86 = ϒ87 = 0; ϒ88 = −λ − σ14

64

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.8 Matrix A − λ · I (8x8), matrix elements by rows and columns −λ

0

0

σ1 · (σ2 + σ3 · e−λ·τ )

σ4

σ5

σ5

σ6

0

σ7 − λ

0

σ8 · e−λ·τ

σ8

σ9

σ9

0

0

0

−λ − σ15

σ10

1

0

0

−λ

0

1

0

0

· e−λ·τ

· e−λ·τ

σ10

σ11

σ11

0

0

0

0

0

−λ

0

0

0

0

0

0

σ12

σ12

σ13 − λ

σ13

0

0

0

1

0

0

0

−λ

0

0

0

0

σ14

0

0

0

−λ − σ14

  det(A − λ · I ) = (−λ) · det(A1 ) − σ1 · σ2 + σ3 · e−λ·τ · det(A2 ) + σ4 · det(A3 ) − σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 ) Matrix A − λ · I is 8x8 dimensions. Matrix A1 , A2 ,…, A6 are 7 × 7 dimension. Matrix A − λ · I (8x8) determinant: matrix elements by rows and columns (Table 1.8)   det(A − λ · I ) = −λ · det(A1 ) − σ1 · σ2 + σ3 · e−λ·τ · det(A2 ) + σ4 · det(A3 ) − σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 ) Matrix A1 (7 × 7) determinant: matrix elements by rows and columns (Table 1.9) det(A1 ) = (σ7 − λ) · det(A11 ) + σ8 · e−λ·τ · det(A12 ) − σ8 · det(A13 ) + σ9 · det(A14 ) − σ9 · det(A15 ) Matrix A11 (6 × 6) determinant: matrix elements by rows and columns (Table 1.10). det(A11 ) = (−λ − σ15 ) · det(A111 ) − σ10 · e−λ·τ · det(A112 ) + σ10 · det(A113 ) − σ11 · det(A114 ) + σ11 · det(A115 ) Table 1.9 Matrix A1 (7 × 7) elements by rows and columns σ7 − λ

0

σ8 · e−λ·τ

0

−λ − σ15

σ10

0

0

−λ

1

0

0

· e−λ·τ

· e−λ·τ

σ8

σ9

σ9

0

σ10

σ11

σ11

0

0

0

0

0

−λ

0

0

0

0

0

σ12

σ12

σ13 − λ

σ13

0

0

1

0

0

0

−λ

0

0

0

σ14

0

0

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

65

Table 1.10 Matrix A11 (6 × 6) elements by rows and columns −λ − σ15

σ10 · e−λ·τ

σ10

σ11

σ11

0

0

−λ

0

0

0

0

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

σ13

0

1

0

0

0

−λ

0

0

σ14

0

0

0

−λ − σ14

Table 1.11 Matrix A111 (5 × 5) elements by rows and columns −λ

0

0

0

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

σ13

0

0

0

0

−λ

0

σ14

0

0

0

−λ − σ14

Matrix A111 (5 × 5) determinant: matrix elements by rows and columns (Table 1.11). det(A111 ) = λ4 · (σ13 − λ) · (λ + σ14 ) Matrix A112 (5 × 5) determinant: matrix elements by rows and columns (Table 1.12) det(A112 ) = 0 Matrix A113 (5 × 5) determinant: matrix elements by rows and columns (Table 1.13). det(A113 ) = 0 Matrix A114 (5 × 5) determinant: matrix elements by rows and columns (Table 1.14). Table 1.12 Matrix A112 (5 × 5) elements by rows and columns 0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13 − λ

σ13

0

1

0

0

−λ

0

0

0

0

0

−λ − σ14

66

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.13 Matrix A113 (5 × 5) elements by rows and columns 0

−λ

0

0

0

0

0

0

0

0

· e−λ·τ

0

σ12

σ13 − λ

σ13

0

1

0

0

−λ

0

0

σ14

0

0

−λ − σ14

Table 1.14 Matrix A114 (5 × 5) elements by rows and columns 0

−λ

0

0

0

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

1

0

0

−λ

0

0

σ14

0

0

−λ − σ14

det(A114 ) = λ2 · σ13 · (λ + σ14 ) Matrix A115 (5 × 5) determinant: matrix elements by rows and columns (Table 1.15). det(A115 ) = λ2 · (σ13 − λ) · (λ + σ14 ) Matrix A12 (6 × 6) determinant: matrix elements by rows and columns (Table 1.16). Table 1.15 Matrix A115 (5 × 5) elements by rows and columns 0

−λ

0

0

0

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

0

1

0

0

0

0

0

σ14

0

0

−λ − σ14

Table 1.16 Matrix A12 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10

σ11

σ11

0

0

0

1

0

0

0

0

0

−λ

0

0

0

0

0

σ12

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

0

0

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

67

Table 1.17 Matrix A13 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ11

σ11

0

0

0

−λ

0

0

0

1

0

0

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

−λ − σ14

det(A12 ) = 0 Matrix A13 (6 × 6) determinant: matrix elements by rows and columns (Table 1.17). det(A13 ) = −(−λ − σ15 ) · det(A131 ) + σ10 · e−λ·τ · det(A132 ) − σ11 · det(A133 ) + σ11 · det(A134 ) Matrix A131 (5 × 5) determinant: matrix elements by rows and columns (Table 1.18). det(A131 ) = λ2 · (σ13 − λ) · (λ + σ14 ) Matrix A132 (5 × 5) determinant: matrix elements by rows and columns (Table 1.19). det(A132 ) = 0 Table 1.18 Matrix A131 (5 × 5) elements by rows and columns 0

−λ

0

0

0

1

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

σ13

0

0

0

0

−λ

0

0

σ14

0

0

−λ − σ14

Table 1.19 Matrix A132 (5 × 5) elements by rows and columns 0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

σ13

0

0

1

0

−λ

0

0

0

0

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.20 Matrix A133 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

0

0

· e−λ·τ

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

σ14

0

−λ − σ14

Matrix A133 (5 × 5) determinant: matrix elements by rows and columns (Table 1.20). det(A133 ) = −λ · σ13 · (λ + σ14 ) Matrix A134 (5 × 5) determinant: matrix elements by rows and columns (Table 1.21). det(A134 ) = −λ · (σ13 − λ) · (λ + σ14 ) Matrix A14 (6 × 6) determinant: matrix elements by rows and columns (Table 1.22). det(A14 ) = −(−λ − σ15 ) · det(A141 ) + σ10 · e−λ·τ · det(A142 ) − σ10 · det(A143 ) + σ11 · det(A144 ) Table 1.21 Matrix A134 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.22 Matrix A14 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

0

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

−λ − σ14

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Table 1.23 Matrix A141 (5 × 5) elements by rows and columns 0

−λ

0

0

0

1

0

−λ

0

0

· e−λ·τ

0

σ12

σ12

σ13

0

0

0

0

−λ

0

0

σ14

0

0

−λ − σ14

Matrix A141 (5 × 5) determinant: matrix elements by rows and columns (Table 1.23). det(A141 ) = λ2 · σ12 · (λ + σ14 ) Matrix A142 (5 × 5) determinant: matrix elements by rows and columns (Table 1.24). det(A142 ) = 0 Matrix A143 (5 × 5) determinant: matrix elements by rows and columns (Table 1.25). det(A143 ) = −λ · σ13 · (λ + σ14 ) Matrix A144 (5 × 5) determinant: matrix elements by rows and columns (Table 1.26). det(A144 ) = −λ · σ12 · (λ + σ14 ) Table 1.24 Matrix A142 (5 × 5) elements by rows and columns 0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

0

0

−λ − σ14

Table 1.25 Matrix A143 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

0

0

· e−λ·τ

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

σ14

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.26 Matrix A144 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.27 Matrix A15 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

0

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

σ14

0

0

−λ − σ14

Matrix A15 (6 × 6) determinant: matrix elements by rows and columns (Table 1.27) det(A15 ) = −(−λ − σ15 ) · det(A151 ) + σ10 · e−λ·τ · det(A152 ) − σ10 · det(A153 ) + σ11 · det(A154 ) Matrix A151 (5 × 5) determinant: matrix elements by rows and columns (Table 1.28) det(A151 ) = 0 Matrix A152 (5 × 5) determinant: matrix elements by rows and columns (Table 1.29). det(A152 ) = 0 Matrix A153 (5 × 5) determinant: matrix elements by rows and columns (Table 1.30). Table 1.28 Matrix A151 (5 × 5) elements by rows and columns 0

−λ

0

0

0

1

0

−λ

0

0

· e−λ·τ

0

σ12

σ12

σ13 − λ

0

0

0

0

0

0

0

σ14

0

0

−λ − σ14

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Table 1.29 Matrix A152 (5 × 5) elements by rows and columns 0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

0

−λ − σ14

Table 1.30 Matrix A153 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

det(A153 ) = −λ · (σ13 − λ) · (λ + σ14 ) Matrix A154 (5 × 5) determinant: matrix elements by rows and columns (Table 1.31). det(A154 ) = −λ · σ12 · (λ + σ14 ) Matrix A2 (7 × 7) determinant: matrix elements by rows and columns (Table 1.32). Table 1.31 Matrix A154 (5 × 5) elements by rows and columns 0

0

−λ

0

0

1

0

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.32 Matrix A2 (7 × 7) elements by rows and columns 0

σ7 − λ

0

σ8

σ9

σ9

0

0

−λ − σ15

σ10

σ11

σ11

0

1

0

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

0

σ12

σ13 − λ

σ13

0

0

0

1

0

0

−λ

0

0

0

0

0

0

0

−λ − σ14

0

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.33 Matrix A21 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10

σ11

σ11

0

1

0

0

0

0

0

0

0

−λ

0

0

0

0

0

σ12

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

0

0

0

−λ − σ14

det(A2 ) = −(σ7 − λ) · det(A21 ) − σ8 · det(A22 ) + σ9 · det(A23 ) − σ9 · det(A24 ) Matrix A21 (6 × 6) determinant: matrix elements by rows and columns (Table 1.33). det(A21 ) = (λ + σ15 ) · det(A211 ) + σ10 · det(A212 ) − σ11 · det(A213 ) + σ11 · det(A214 )

Matrix A211 (5 × 5) determinant: matrix elements by rows and columns (Table 1.34). det(A211 ) = −λ2 · (σ13 − λ) · (λ + σ14 ) Matrix A212 (5 × 5) determinant: matrix elements by rows and columns (Table 1.35). det(A212 ) = 0 Table 1.34 Matrix A211 (5 × 5) elements by rows and columns 1

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13 − λ

σ13

0

0

0

0

−λ

0

0

0

0

0

−λ − σ14

Table 1.35 Matrix A212 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

0

0

0

0

0

σ13 − λ

σ13

0

0

1

0

−λ

0

0

0

0

0

−λ − σ14

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Table 1.36 Matrix A213 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

0

0

−λ − σ14

Matrix A213 (5 × 5) determinant: matrix elements by rows and columns (Table 1.36). det(A213 ) = λ · σ13 · (λ + σ14 ) Matrix A214 (5 × 5) determinant: matrix elements by rows and columns (Table 1.37). det(A214 ) = λ · (σ13 − λ) · (λ + σ14 ) Matrix A22 (6 × 6) determinant: matrix elements by rows and columns (Table 1.38). det(A22 ) = (−λ − σ15 ) · det(A221 ) − σ11 · det(A222 ) + σ11 · det(A223 ) Matrix A221 (5 × 5) determinant: matrix elements by rows and columns (Table 1.39). Table 1.37 Matrix A214 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

0

−λ − σ14

Table 1.38 Matrix A22 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ11

σ11

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

σ13 − λ

σ13

0

0

0

1

0

−λ

0

0

0

0

0

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.39 Matrix A221 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

σ13 − λ

σ13

0

0

0

0

−λ

0

0

0

0

0

−λ − σ14

Table 1.40 Matrix A222 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

det(A221 ) = λ · (σ13 − λ) · (λ + σ14 ) Matrix A222 (5 × 5) determinant: matrix elements by rows and columns (Table 1.40). det(A222 ) = σ13 · (λ + σ14 ) Matrix A223 (5 × 5) determinant: matrix elements by rows and columns (Table 1.41). det(A223 ) = (σ13 − λ) · (λ + σ14 ) Matrix A23 (6 × 6) determinant: matrix elements by rows and columns (Table 1.42). det(A23 ) = (−λ − σ15 ) · det(A231 ) − σ10 · det(A232 ) + σ11 · det(A233 ) Matrix A231 (5 × 5) determinant: matrix elements by rows and columns (Table 1.43). Table 1.41 Matrix A223 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

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Table 1.42 Matrix A23 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10

σ11

0

1

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

σ12

σ13

0

0

0

1

0

−λ

0

0

0

0

0

0

−λ − σ14

Table 1.43 Matrix A231 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

−λ

0

0

0

0

σ12

σ13

0

0

0

0

−λ

0

0

0

0

0

−λ − σ14

det(A231 ) = λ · σ12 · (λ + σ14 ) Matrix A232 (5 × 5) determinant: matrix elements by rows and columns (Table 1.44). det(A232 ) = σ13 · (λ + σ14 ) Matrix A233 (5 × 5) determinant: matrix elements by rows and columns (Table 1.45). Table 1.44 Matrix A232 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

Table 1.45 Matrix A233 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.46 Matrix A24 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10

σ11

0

1

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

σ12

σ13 − λ

0

0

0

1

0

0

0

0

0

0

0

0

−λ − σ14

det(A233 ) = σ12 · (λ + σ14 ) Matrix A24 (6 × 6) determinant: matrix elements by rows and columns (Table 1.46). det(A24 ) = (−λ − σ15 ) · det(A241 ) − σ10 · det(A242 ) + σ11 · det(A243 ) Matrix A241 (5 × 5) determinant: matrix elements by rows and columns (Table 1.47). det(A241 ) = −λ · (σ13 − λ) · (λ + σ14 ) Matrix A242 (5 × 5) determinant: matrix elements by rows and columns (Table 1.48). det(A242 ) = (σ13 − λ) · (λ + σ14 ) Table 1.47 Matrix A241 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

−λ

0

0

0

0

σ12

σ13 − λ

0

0

0

0

0

0

0

0

0

0

−λ − σ14

Table 1.48 Matrix A242 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

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Table 1.49 Matrix A243 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

Matrix A243 (5 × 5) determinant: matrix elements by rows and columns (Table 1.49). det(A243 ) = σ12 · (λ + σ14 ) Matrix A3 (7 × 7) determinant: matrix elements by rows and columns (Table 1.50). det(A3 ) = −(σ7 − λ) · det(A31 ) − σ8 · e−λ·τ · det(A32 ) + σ9 · det(A33 ) − σ9 · det(A34 )

Matrix A31 (6 × 6) determinant: matrix elements by rows and columns (Table 1.51). det(A31 ) = 0 Table 1.50 Matrix A3 (7 × 7) elements by rows and columns 0

σ7 − λ

σ8 · e−λ·τ

0

0

0

−λ − σ15

σ10

1

0

0

−λ

0

1

0

0

σ9

· e−λ·τ

· e−λ·τ

σ9

0

σ11

σ11

0

0

0

0

0

0

0

0

0

0

σ12

σ13 − λ

σ13

0

0

0

1

0

0

−λ

0

0

0

0

σ14

0

0

−λ − σ14

Table 1.51 Matrix A31 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ11

σ11

0

1

0

−λ

0

0

0

0

0

0

0

0

0

· e−λ·τ

0

0

σ12

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.52 Matrix A32 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ11

σ11

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

σ13 − λ

σ13

0

0

0

1

0

−λ

0

0

0

0

0

0

−λ − σ14

Matrix A32 (6 × 6) determinant: matrix elements by rows and columns (Table 1.52). det(A32 ) = (−λ − σ15 ) · det(A321 ) − σ11 · det(A322 ) + σ11 · det(A323 ) Matrix A321 (5 × 5) determinant: matrix elements by rows and columns (Table 1.53). det(A321 ) = λ · (σ13 − λ) · (λ + σ14 ) Matrix A322 (5 × 5) determinant: matrix elements by rows and columns (Table 1.54). det(A322 ) = σ13 · (λ + σ14 ) Matrix A323 (5 × 5) determinant: matrix elements by rows and columns (Table 1.55). Table 1.53 Matrix A321 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

σ13 − λ

σ13

0

0

0

0

−λ

0

0

0

0

0

−λ − σ14

Table 1.54 Matrix A322 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

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Table 1.55 Matrix A323 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

Table 1.56 Matrix A33 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ11

0

1

0

0

−λ

0

0

0

1

0

0

0

0

0

0

0

σ12 · e−λ·τ

σ13

0

0

0

1

0

−λ

0

0

0

0

σ14

0

−λ − σ14

det(A323 ) = (σ13 − λ) · (λ + σ14 ) Matrix A33 (6 × 6) determinant: matrix elements by rows and columns (Table 1.56). det(A33 ) = (−λ − σ15 ) · det(A331 ) − σ10 · e−λ·τ · det(A332 ) + σ11 · det(A333 ) Matrix A331 (5 × 5) determinant: matrix elements by rows and columns (Table 1.57). det(A331 ) = σ12 · e−λ·τ · λ · (λ + σ14 ) Matrix A332 (5 × 5) determinant: matrix elements by rows and columns (Table 1.58). det(A332 ) = σ13 · (λ + σ14 ) Table 1.57 Matrix A331 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

0

0

0

0

σ12 · e−λ·τ

σ13

0

0

0

0

−λ

0

0

0

σ14

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.58 Matrix A332 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

Table 1.59 Matrix A333 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

0

0

0

1

0

0

0

0

0

σ14

−λ − σ14

Matrix A333 (5 × 5) determinant: matrix elements by rows and columns (Table 1.59). det(A333 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A34 (6 × 6) determinant: matrix elements by rows and columns (Table 1.60). det(A34 ) = (−λ − σ15 ) · det(A341 ) − σ10 · e−λ·τ · det(A342 ) + σ11 · det(A343 ) Matrix A341 (5 × 5) determinant: matrix elements by rows and columns (Table 1.61). det(A341 ) = 0 Matrix A342 (5 × 5) determinant: matrix elements by rows and columns (Table 1.62). Table 1.60 Matrix A34 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ11

0

1

0

0

−λ

0

0

0

1

0

0

0

0

· e−λ·τ

0

0

0

σ12

σ13 − λ

0

0

0

1

0

0

0

0

0

0

σ14

0

−λ − σ14

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Table 1.61 Matrix A341 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

0

0

· e−λ·τ

0

0

σ12

σ13 − λ

0

0

0

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.62 Matrix A342 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

det(A342 ) = (σ13 − λ) · (λ + σ14 ) Matrix A343 (5 × 5) determinant: matrix elements by rows and columns (Table 1.63). det(A343 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A4 (7 × 7) determinant: matrix elements by rows and columns (Table 1.64). Table 1.63 Matrix A343 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

0

0

0

1

0

0

0

0

0

σ14

−λ − σ14

Table 1.64 Matrix A4 (7 × 7) elements by rows and columns 0

σ7 − λ

0

σ8 · e−λ·τ

σ8

σ9

0

0

0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

0

1

0

0

−λ

0

0

0

0

1

0

0

−λ

0

0

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

0

0

1

0

0

−λ

0

0

0

0

σ14

0

0

−λ − σ14

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.65 Matrix A41 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

0

1

0

−λ

0

0

0

0

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

−λ − σ14

det(A4 ) = −(σ7 − λ) · det(A41 ) − σ8 · e−λ·τ · det(A42 ) + σ8 · det(A43 ) − σ9 · det(A44 )

Matrix A41 (6 × 6) determinant: matrix elements by rows and columns (Table 1.65). det(A41 ) = −(−λ − σ15 ) · det(A411 ) + σ10 · e−λ·τ · det(A412 ) − σ10 · det(A413 ) + σ11 · det(A414 ) Matrix A411 (5 × 5) determinant: matrix elements by rows and columns (Table 1.66). det(A411 ) = λ2 · σ12 · e−λ·τ · (λ + σ14 ) Matrix A412 (5 × 5) determinant: matrix elements by rows and columns (Table 1.67). det(A412 ) = λ · σ13 · (λ + σ14 ) Table 1.66 Matrix A411 (5 × 5) elements by rows and columns 1

−λ

0

0

0

0

0

−λ

0

0

· e−λ·τ

0

σ12

σ12

σ13

0

0

0

0

−λ

0

0

σ14

0

0

−λ − σ14

Table 1.67 Matrix A412 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

0

0

−λ − σ14

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Table 1.68 Matrix A413 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

0

0

· e−λ·τ

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

σ14

0

−λ − σ14

Matrix A413 (5 × 5) determinant: matrix elements by rows and columns (Table 1.68). det(A413 ) = 0 Matrix A414 (5 × 5) determinant: matrix elements by rows and columns (Table 1.69). det(A414 ) = −λ · σ12 · e−λ·τ · (λ + σ14 ) Matrix A42 (6 × 6) determinant: matrix elements by rows and columns (Table 1.70). det(A42 ) = (−λ − σ15 ) · det(A421 ) − σ10 · det(A422 ) + σ11 · det(A423 ) Matrix A421 (5 × 5) determinant: matrix elements by rows and columns (Table 1.71). Table 1.69 Elements by rows and columns 1

0

−λ

0

0

0

0

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.70 Matrix A42 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10

σ11

1

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

σ12

σ13

0

0

0

1

0

−λ

0

0

0

0

0

0

−λ − σ14

0

84

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.71 Matrix A421 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

−λ

0

0

0

0

σ12

σ13

0

0

0

0

−λ

0

0

0

0

0

−λ − σ14

Table 1.72 Matrix A422 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

det(A421 ) = λ · σ12 · (λ + σ14 ) Matrix A422 (5 × 5) determinant: matrix elements by rows and columns (Table 1.72). det(A422 ) = σ13 · (λ + σ14 ) Matrix A423 (5 × 5) determinant: matrix elements by rows and columns (Table 1.73). det(A423 ) = σ12 · (λ + σ14 ) Matrix A43 (6 × 6) determinant: matrix elements by rows and columns (Table 1.74). det(A43 ) = (−λ − σ15 ) · det(A431 ) − σ10 · e−λ·τ · det(A432 ) + σ11 · det(A433 ) Matrix A431 (5 × 5) determinant: matrix elements by rows and columns (Table 1.75). Table 1.73 Matrix A423 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

85

Table 1.74 Matrix A43 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ11

0

1

0

0

−λ

0

0

0

1

0

0

0

0

0

0

0

σ12 · e−λ·τ

σ13

0

0

0

1

0

−λ

0

0

0

0

σ14

0

−λ − σ14

Table 1.75 Matrix A431 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

0

0

0

0

σ12 · e−λ·τ

σ13

0

0

0

0

−λ

0

0

0

σ14

0

−λ − σ14

det(A431 ) = σ12 · e−λ·τ · λ · (λ + σ14 ) Matrix A432 (5 × 5) determinant: matrix elements by rows and columns (Table 1.76). det(A432 ) = σ13 · (λ + σ14 ) Matrix A433 (5 × 5) determinant: matrix elements by rows and columns (Table 1.77). Table 1.76 Matrix A432 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13

0

0

0

1

−λ

0

0

0

0

0

−λ − σ14

Table 1.77 Matrix A433 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

0

0

0

1

0

0

0

0

0

σ14

−λ − σ14

86

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.78 Matrix A44 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ10

0

1

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

0

0

0

1

0

0

0

0

0

0

σ14

0

−λ − σ14

det(A433 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A44 (6 × 6) determinant: matrix elements by rows and columns (Table 1.78). det(A44 ) = (−λ − σ15 ) · det(A441 ) − σ10 · e−λ·τ · det(A442 ) + σ10 · det(A443 ) Matrix A441 (5 × 5) determinant: matrix elements by rows and columns (Table 1.79). det(A441 ) = 0 Matrix A442 (5 × 5) determinant: matrix elements by rows and columns (Table 1.80). det(A442 ) = σ12 · (λ + σ14 ) Table 1.79 Matrix A441 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

0

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.80 Matrix A442 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

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Table 1.81 Matrix A443 (5 × 5) elements by rows and columns 1

0

0

−λ

0

1

0

0

0 0 · e−λ·τ

0

0

0

σ12

0

0

1

0

0

0

0

0

σ14

−λ − σ14

0

Matrix A443 (5 × 5) determinant: matrix elements by rows and columns (Table 1.81). det(A443 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A5 (7 × 7) determinant: matrix elements by rows and columns (Table 1.82). det(A5 ) = −(σ7 − λ) · det(A51 ) − σ8 · e−λ·τ · det(A52 ) + σ8 · det(A53 ) − σ9 · det(A54 )

Matrix A51 (6 × 6) determinant: matrix elements by rows and columns (Table 1.83). det(A51 ) = −(−λ − σ15 ) · det(A511 ) + σ10 · e−λ·τ · det(A512 ) − σ10 · det(A513 ) + σ11 · det(A514 ) Table 1.82 Matrix A5 (7 × 7) elements by rows and columns 0

σ7 − λ

σ8 · e−λ·τ

0

0

0

−λ − σ15

σ10

1

0

0

−λ

0

1

0

0

σ8

· e−λ·τ

· e−λ·τ

σ9

0

σ10

σ11

0

0

0

0

−λ

0

0

0

0

0

σ12

σ12

σ13 − λ

0

0

0

1

0

0

0

0

0

0

0

σ14

0

0

−λ − σ14

Table 1.83 Matrix A51 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

0

1

0

−λ

0

0

0

0

0

0

−λ

0

0

· e−λ·τ

0

0

σ12

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

σ14

0

0

−λ − σ14

88

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.84 Matrix A511 (5 × 5) elements by rows and columns 1

−λ

0

0

0

0

0

−λ

0

0

· e−λ·τ

0

σ12

σ12

σ13 − λ

0

0

0

0

0

0

0

σ14

0

0

−λ − σ14

Matrix A511 (5 × 5) determinant: matrix elements by rows and columns (Table 1.84). det(A511 ) = 0 Matrix A512 (5 × 5) determinant: matrix elements by rows and columns (Table 1.85). det(A512 ) = λ · (σ13 − λ) · (λ + σ14 ) Matrix A513 (5 × 5) determinant: matrix elements by rows and columns (Table 1.86). det(A513 ) = 0 Matrix A514 (5 × 5) determinant: matrix elements by rows and columns (Table 1.87). det(A514 ) = −λ · σ12 · e−λ·τ · (λ + σ14 ) Table 1.85 Matrix A512 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

0

−λ − σ14

Table 1.86 Matrix A513 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

0

0

· e−λ·τ

0

0

σ12

σ13 − λ

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

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Table 1.87 Matrix A514 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

1

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.88 Matrix A52 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10

σ11

1

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

σ12

σ13 − λ

0

0

0

1

0

0

0

0

0

0

0

0

−λ − σ14

0

Matrix A52 (6 × 6) determinant: matrix elements by rows and columns (Table 1.88). det(A52 ) = (−λ − σ15 ) · det(A521 ) − σ10 · det(A522 ) + σ11 · det(A523 ) Matrix A521 (5 × 5) determinant: matrix elements by rows and columns (Table 1.89). det(A521 ) = 0 Matrix A522 (5 × 5) determinant: matrix elements by rows and columns (Table 1.90). det(A522 ) = (σ13 − λ) · (λ + σ14 ) Matrix A523 (5 × 5) determinant: matrix elements by rows and columns (Table 1.91). Table 1.89 Matrix A521 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

−λ

0

0

0

0

σ12

σ13 − λ

0

0

0

0

0

0

0

0

0

0

−λ − σ14

90

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.90 Matrix A522 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

Table 1.91 Matrix A523 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

det(A523 ) = σ12 · (λ + σ14 ) Matrix A53 (6 × 6) determinant: matrix elements by rows and columns (Table 1.92). det(A53 ) = (−λ − σ15 ) · det(A531 ) − σ10 · e−λ·τ · det(A532 ) + σ11 · det(A533 ) Matrix A531 (5 × 5) determinant: matrix elements by rows and columns (Table 1.93). Table 1.92 Matrix A53 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ11

0

1

0

0

−λ

0

0

0

1

0

0

0

0

· e−λ·τ

0

0

0

σ12

σ13 − λ

0

0

0

1

0

0

0

0

0

0

σ14

0

−λ − σ14

Table 1.93 Matrix A531 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

0

0

0

0

0

0

0

0

σ14

0

−λ − σ14

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

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Table 1.94 Matrix A532 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

0

0

0

1

0

0

0

0

0

0

−λ − σ14

det(A531 ) = 0 Matrix A532 (5 × 5) determinant: matrix elements by rows and columns (Table 1.94). det(A532 ) = (σ13 − λ) · (λ + σ14 ) Matrix A533 (5 × 5) determinant: matrix elements by rows and columns (Table 1.95). det(A533 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A54 (6 × 6) determinant: matrix elements by rows and columns (Table 1.96). det(A54 ) = −(λ + σ15 ) · det(A541 ) − σ10 · e−λ·τ · det(A542 ) + σ10 · det(A543 ) Table 1.95 Matrix A533 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

0

0

0

1

0

0

0

0

0

σ14

−λ − σ14

Table 1.96 Matrix A54 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ10

0

1

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

0

0

0

1

0

0

0

0

0

0

σ14

0

−λ − σ14

92

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.97 Matrix A541 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

0

0

0

0

0

0

0

0

σ14

0

−λ − σ14

Matrix A541 (5 × 5) determinant: matrix elements by rows and columns (Table 1.97). det(A541 ) = 0 Matrix A542 (5 × 5) determinant: matrix elements by rows and columns (Table 1.98). det(A542 ) = σ12 · (λ + σ14 ) Matrix A543 (5 × 5) determinant: matrix elements by rows and columns (Table 1.99). det(A543 ) = σ12 · e−λ·τ · (λ + σ14 ) Matrix A6 (7 × 7) determinant: matrix elements by rows and columns (Table 1.100). det(A6 ) = −(σ7 − λ) · det(A61 ) − σ8 · e−λ·τ · det(A62 ) Table 1.98 Matrix A542 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

0

0

0

1

0

0

0

0

0

0

−λ − σ14

Table 1.99 Matrix A543 (5 × 5) elements by rows and columns 1

0

0

−λ

0

1

0

0

0 0 · e−λ·τ

0

0

0

σ12

0

0

1

0

0

0

0

0

σ14

−λ − σ14

0

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

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Table 1.100 Matrix A6 (7 × 7) elements by rows and columns 0

σ7 − λ

0

σ8 · e−λ·τ

σ8

σ9

σ9

0

0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

σ11

1

0

0

−λ

0

0

0

0

1

0

0

−λ

0

0

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

σ13

0

0

1

0

0

0

−λ

0

0

0

σ14

0

0

0

+ σ8 · det(A63 ) − σ9 · det(A64 ) + σ9 · det(A65 ) Matrix A61 (6 × 6) determinant: matrix elements by rows and columns (Table 1.101). det(A61 ) = −(−λ − σ15 ) · det(A611 ) + σ10 · e−λ·τ · det(A612 ) − σ10 · det(A613 ) + σ11 · det(A614 ) − σ11 · det(A615 ) Matrix A611 (5 × 5) determinant: matrix elements by rows and columns (Table 1.102). det(A611 ) = −λ2 · σ14 · (σ13 − λ) Matrix A612 (5 × 5) determinant: matrix elements by rows and columns (Table 1.103). Table 1.101 Matrix A61 (6 × 6) elements by rows and columns 0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

σ11

1

0

−λ

0

0

0

0

0

0

−λ

0

0

· e−λ·τ

0

0

σ12

σ12

σ13 − λ

σ13

0

1

0

0

0

−λ

0

0

σ14

0

0

0

Table 1.102 Matrix A611 (5 × 5) elements by rows and columns 1

−λ

0

0

0

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

σ13

0

0

0

0

−λ

0

σ14

0

0

0

94

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.103 Matrix A612 (5 × 5) elements by rows and columns 1

0

0

0

0

0

0

−λ

0

0

0

0

σ12

σ13 − λ

σ13

0

1

0

0

−λ

0

0

0

0

0

Table 1.104 Matrix A613 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

σ13

0

1

0

0

−λ

0

0

σ14

0

0

det(A612 ) = 0 Matrix A613 (5 × 5) determinant: matrix elements by rows and columns (Table 1.104). det(A613 ) = 0 Matrix A614 (5 × 5) determinant: matrix elements by rows and columns (Table 1.105). det(A614 ) = −λ · σ14 · σ13 Matrix A615 (5 × 5) determinant: matrix elements by rows and columns (Table 1.106). det(A615 ) = −λ · σ14 · (σ13 − λ) Matrix A62 (6 × 6) determinant: matrix elements by rows and columns (Table 1.107). Table 1.105 Matrix A614 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

1

0

0

−λ

0

0

σ14

0

0

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

95

Table 1.106 Matrix A615 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

0

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

σ13 − λ

0

1

0

0

0

0

0

σ14

0

0

Table 1.107 Matrix A62 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10

σ11

σ11

1

0

0

0

0

0

0

1

0

−λ

0

0

0

0

0

σ12

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

0

0

0

det(A62 ) = 0 Matrix A63 (6 × 6) determinant: matrix elements by rows and columns (Table 1.108). det(A63 ) = (−λ − σ15 ) · det(A631 ) − σ10 · e−λ·τ · det(A632 ) + σ11 · det(A633 ) − σ11 · det(A634 ) Matrix A631 (5 × 5) determinant: matrix elements by rows and columns (Table 1.109). det(A631 ) = −λ · σ14 · (σ13 − λ) Matrix A632 (5 × 5) determinant: matrix elements by rows and columns (Table 1.110) Table 1.108 Matrix A63 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ11

σ11

1

0

0

−λ

0

0

0

1

0

0

0

0

· e−λ·τ

0

0

0

σ12

σ13 − λ

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

96

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.109 Matrix A631 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

0

0

· e−λ·τ

0

0

σ12

σ13 − λ

σ13

0

0

0

0

−λ

0

0

σ14

0

0

Table 1.110 Matrix A632 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

0

0

0

0

0

σ13 − λ

σ13

0

0

1

0

−λ

0

0

0

0

0

det(A632 ) = 0 Matrix A633 (5 × 5) determinant: matrix elements by rows and columns (Table 1.111). det(A633 ) = σ13 · σ14 Matrix A634 (5 × 5) determinant: matrix elements by rows and columns (Table 1.112) Table 1.111 Matrix A633 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

σ13

0

0

1

0

−λ

0

0

0

σ14

0

Table 1.112 Matrix A634 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

0

0

0

0

σ12 · e−λ·τ

σ13 − λ

0

0

1

0

0

0

0

0

σ14

0

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

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Table 1.113 Matrix A64 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

1

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

0

1

0

0

−λ

0

0

0

σ14

0

0

det(A634 ) = σ14 · (σ13 − λ) Matrix A64 (6 × 6) determinant: matrix elements by rows and columns (Table 1.113). det(A64 ) = (−λ − σ15 ) · det(A641 ) − σ10 · e−λ·τ · det(A642 ) + σ10 · det(A643 ) − σ11 · det(A644 ) Matrix A641 (5 × 5) determinant: matrix elements by rows and columns (Table 1.114) det(A641 ) = −λ · σ14 · σ12 Matrix A642 (5 × 5) determinant: matrix elements by rows and columns (Table 1.115). det(A642 ) = 0 Table 1.114 Matrix A641 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

σ13

0

0

0

0

−λ

0

0

σ14

0

0

Table 1.115 Matrix A642 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

σ13

0

0

1

0

−λ

0

0

0

0

0

98

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.116 Matrix A643 (5 × 5) elements by rows and columns 1

0

0

−λ

0

1

0

0

0 0 · e−λ·τ

0

0

0

σ12

0

0

1

0

−λ

σ13

0

0

0

σ14

0

Matrix A643 (5 × 5) determinant: matrix elements by rows and columns (Table 1.116). det(A643 ) = σ13 · σ14 Matrix A644 (5 × 5) determinant: matrix elements by rows and columns (Table 1.117). det(A644 ) = σ12 · σ14 Matrix A65 (6 × 6) determinant: matrix elements by rows and columns (Table 1.118) det(A65 ) = (−λ − σ15 ) · det(A651 ) − σ10 · e−λ·τ · det(A652 ) + σ10 · det(A653 ) − σ11 · det(A654 ) Table 1.117 Matrix A644 (5 × 5) elements by rows and columns 1

0

0

−λ

0

0

1

0

0

−λ

0

0

0

σ12 · e−λ·τ

σ12

0

0

1

0

0

0

0

0

σ14

0

Table 1.118 Matrix A65 (6 × 6) elements by rows and columns 0

0

−λ − σ15

σ10 · e−λ·τ

σ10

σ11

1

0

0

−λ

0

0

0

1

0

0

−λ

0

0

0

0

σ12 · e−λ·τ

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

σ14

0

0

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

99

Table 1.119 Matrix A651 (5 × 5) elements by rows and columns 1

0

−λ

0

0

0

1

0

−λ

0

· e−λ·τ

0

0

σ12

σ12

σ13 − λ

0

0

0

0

0

0

0

σ14

0

0

Table 1.120 Matrix A652 (5 × 5) elements by rows and columns 1

0

0

0

0

0

1

0

−λ

0

0

0

0

σ12

σ13 − λ

0

0

1

0

0

0

0

0

0

0

Matrix A651 (5 × 5) determinant: matrix elements by rows and columns (Table 1.119). det(A651 ) = 0 Matrix A652 (5 × 5) determinant: matrix elements by rows and columns (Table 1.120). det(A652 ) = 0 Matrix A653 (5 × 5) determinant: matrix elements by rows and columns (Table 1.121). det(A653 ) = σ44 · (σ13 − λ) Matrix A654 (5 × 5) determinant: matrix elements by rows and columns (Table 1.122). det(A654 ) = σ12 · σ14 Table 1.121 Matrix A653 (5 × 5) elements by rows and columns 1

0

0

−λ

0

1

0

0

0 0 · e−λ·τ

0

0

0

σ12

0

0

1

0

0

σ13 − λ

0

0

0

σ14

0

100

1 Dynamical and Nonlinearity of Laser Diode Circuits

Table 1.122 Matrix A654 (5 × 5) elements by rows and columns 1

0

0

−λ

0

1

0

0

0 −λ · e−λ·τ

0

0

0

σ12

0

0

1

0

0

σ12

0

0

0

σ14

0

The determinants and sub determinants structure is as follow (Table 1.123). det(A11 ) = λ5 − λ4 · (σ13 − σ15 + σ11 − σ14 ) − λ3 · [σ15 · σ13 + (σ13 − σ15 + σ11 ) · σ14 ] − λ2 · σ14 · σ15 · σ13 0 = 1; 1 = −(σ13 − σ15 + σ11 − σ14 ); 2 = −[σ15 · σ13 + (σ13 − σ15 + σ11 ) · σ14 ]; 3 = −σ14 · σ15 · σ13 det(A11 ) =

5 $

λk · (5−k) = λ5 · 0 + λ4 · 1 + λ3 · 2 + λ2 · 3

k=2

det(A13 ) = −λ5 + λ4 · (σ13 − σ15 − σ14 ) + λ3 · [σ15 · σ13 + σ11 + (σ13 − σ15 ) · σ14 ] + λ2 · (σ15 · σ13 + σ11 ) · σ14 4 = −1; 5 = σ13 − σ15 − σ14 ; 6 = σ15 · σ13 + σ11 + (σ13 − σ15 ) · σ14 ; 7 = (σ15 · σ13 + σ11 ) · σ14 det(A13 ) =

5 $

λk · (9−k) = λ2 · 7 + λ3 · 6 + λ4 · 5 + λ5 · 4

k=2

det(A14 ) = λ4 · σ12 + λ3 · (σ12 · σ15 + σ12 · σ14 ) + λ2 · (σ10 · σ13 − σ11 · σ12 + σ12 · σ15 · σ14 ) + λ · (σ10 · σ13 − σ11 · σ12 ) · σ14 8 = σ12 ; 9 = σ12 · σ15 + σ12 · σ14 ; 10 = σ10 · σ13 − σ11 · σ12 + σ12 · σ15 · σ14 11 = (σ10 · σ13 − σ11 · σ12 ) · σ14 det(A14 ) =

4 $ k=1

λk · (12−k) = λ4 · 8 + λ3 · 9 + λ2 · 10 + λ · 11

2

− λ) · (λ + σ14 )

det(A115 ) = λ2 · (σ13

· σ13 · (λ + σ14 )

det(A114 ) = λ

det(A112 ) = 0 det(A113 ) = 0

− λ) · (λ + σ14 )

det(A111 ) = λ4 · (σ13

+ σ11 · det(A115 ) det(A132 ) =

− λ) · (λ + σ14 )

det(A134 ) = −λ · (σ13

· σ13 · (λ + σ14 )

0 det(A133 ) = −λ

− λ) · (λ + σ14 )

det(A131 ) = λ2 · (σ13

+ σ11 · det(A134 )

· det(A132 )

− σ11 · det(A133 )

+ σ10 · e

−λ·τ

− σ15 ) · det(A131 )

det(A13 ) = −(−λ

− σ11 · det(A114 )

· det(A112 )

det(A12 ) = 0

+ σ10 · det(A113 )

− σ10 · e

−λ·τ

= (−λ − σ15 ) · det(A111 )

det(A11 ) · det(A142 )

· σ12 · (λ + σ14 )

det(A144 ) = −λ

· σ13 · (λ + σ14 )

det(A143 ) = −λ

det(A142 ) = 0

· σ12 · (λ + σ14 )

det(A141 ) = λ2

+ σ11 · det(A144 )

− σ10 · det(A143 )

+ σ10 · e

−λ·τ

· det(A141 )

det(A14 ) = −(−λ − σ15 )

det(A1 ) = (σ7 − λ) · det(A11 ) + σ8 · e−λ·τ · det(A12 ) − σ8 · det(A13 ) + σ9 · det(A14 ) − σ9 · det(A15 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 )

det(A − λ · I ) = −λ · det(A1 ) − σ1 · (σ2 + σ3 · e−λ·τ ) · det(A2 ) + σ4 · det(A3 )

Table 1.123 Determinants and sub determinants structure

· σ12 · (λ + σ14 )

det(A154 ) = −λ

− λ) · (λ + σ14 )

det(A153 ) = −λ · (σ13

det(A151 ) = 0 det(A152 ) = 0

+ σ11 · det(A154 )

− σ10 · det(A153 )

+ σ10 · e−λ·τ · det(A152 )

− σ15 ) · det(A151 )

det(A15 ) = −(−λ

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis 101

102

1 Dynamical and Nonlinearity of Laser Diode Circuits

det(A15 ) = −λ3 · σ10 + λ2 · [σ10 · (σ13 − σ14 ) − σ12 · σ11 ] + λ · (σ10 · σ13 · σ14 − σ11 · σ12 · σ14 ) 12 = −σ10 ; 13 = σ10 · (σ13 − σ14 ) − σ12 · σ11 ; 14 = σ10 · σ13 · σ14 − σ11 · σ12 · σ14 det(A15 ) =

3 $

λk · (15−k) = λ · 14 + λ2 · 13 + λ3 · 12

k=1

det(A1 ) = (σ7 − λ) · det(A11 ) + σ8 · e−λ·τ · det(A12 ) − σ8 · det(A13 ) + σ9 · det(A14 ) − σ9 · det(A15 ) det(A1 )|det(A12 )=0 = (σ7 − λ) · det(A11 ) − σ8 · det(A13 ) + σ9 · det(A14 ) − σ9 · det(A15 )   det(A1 )|det(A12 )=0 = (σ7 − λ) · λ5 · 0 + λ4 · 1 + λ3 · 2 + λ2 · 3   − σ8 · λ2 · 7 + λ3 · 6 + λ4 · 5 + λ5 · 4   + σ9 · λ4 · 8 + λ3 · 9 + λ2 · 10 + λ · 11   − σ9 · λ · 14 + λ2 · 13 + λ3 · 12 We define for simplicity the following sub determinants products: det(A1 ) = .det(A1 )|det(A12 )=0 = det(A1 )(1) − det(A1 )(2) + det(A1 )(3) − det(A1 )(4) det(A1 )(1) = −λ6 · 0 + λ5 · (σ7 · 0 − 1 ) + λ4 · (σ7 · 1 − 2 ) + λ3 · (σ7 · 2 − 3 ) + λ2 · σ7 · 3 det(A1 )(2) = λ2 · σ8 · 7 + λ3 · σ8 · 6 + λ4 · σ8 · 5 + λ5 · σ8 · 4 det(A1 )(3) = λ4 · σ9 · 8 + λ3 · σ9 · 9 + λ2 · σ9 · 10 + λ · σ9 · 11 det(A1 )(4) = λ · σ9 · 14 + λ2 · σ9 · 13 + λ3 · σ9 · 12 det(A1 ) = det(A1 )|det(A2 )=0 = −λ6 · 0 + λ5 · (σ7 · 0 − 1 − σ8 · 4 ) + λ4 · (σ7 · 1 − 2 + σ9 · 8 − σ8 · 5 ) + λ3 · (σ7 · 2 − 3 − σ8 · 6 + σ9 · 9 − σ9 · 12 )

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

103

+ λ2 · (σ7 · 3 − σ8 · 7 + σ9 · 10 − σ9 · 13 ) + λ · (σ9 · 11 − σ9 · 14 ) 6 = −0 ; 5 = σ7 · 0 − 1 − σ8 · 4 ; 4 = σ7 · 1 − 2 + σ9 · 8 − σ8 · 5 3 = σ7 · 2 − 3 − σ8 · 6 + σ9 · 9 − σ9 · 12 ; 2 = σ7 · 3 − σ8 · 7 + σ9 · 10 − σ9 · 13 1 = σ9 · 11 − σ9 · 14 det(A1 ) = det(A1 )|der(A2 )=0 =

6 $

λk · Ωt = λ6 · 6 + λ5 · 5

k=1 2

+ λ · 4 + λ · 3 + λ · 2 + λ · 1 4

3

Table 1.124. det(A21 ) = λ5 + λ4 · (σ14 − σ13 + σ15 ) − λ3 · (σ15 · σ13 + σ11 + σ14 · (σ13 − σ15 )) − λ2 · σ14 · (σ15 · σ13 + σ11 ) det(A21 ) =

5 $

λk · Tk = λ2 · T2 + λ3 · T3 + λ4 · T4 + λ5 · T5 ; T2

k=2

= −σ14 · (σ15 · σ13 + σ11 ) T3 = −(σ15 · σ13 + σ11 + σ14 · (σ13 − σ15 )) T4 = σ14 − σ13 + σ15 ; T5 = 1 det(A22 ) = λ4 + λ3 · (σ15 − σ13 + σ14 ) + λ2 · (σ14 · (σ15 − σ13 ) − σ15 · σ13 − σ11 ) − λ · σ14 · (σ15 · σ13 + σ11 ) det(A22 ) =

4 $

λk · Hk = λ4 · H4 + λ3 · H3 + λ2 · H2 + λ · H1

k=1

H4 = 1; H3 = σ15 − σ13 + σ14 ; H2 = σ14 · (σ15 − σ13 ) − σ15 · σ13 − σ11 ; H1 = −σ14 · (σ15 · σ13 + σ11 ) det(A23 ) = −λ3 · σ12 − λ2 · σ12 · (σ15 + σ14 ) + λ · (σ11 · σ12 − σ10 · σ13 − σ12 · σ15 · σ14 ) + (σ11 · σ12 − σ10 · σ13 ) · σ14

det(A214 ) = λ · (σ13 − λ) · (λ + σ14 )

· (λ + σ14 )

det(A213 ) = λ · σ13

− λ) · (λ + σ14 )

det(A211 ) = −λ2 · (σ13

det(A223 ) = (σ13 − λ) · (λ + σ14 )

+ σ14 )

det(A222 ) = σ13 · (λ

− λ) · (λ + σ14 )

det(A221 ) = λ · (σ13

+ σ11 · det(A223 )

− σ11 · det(A213 )

det(A212 ) = 0

− σ11 · det(A222 )

+ σ10 · det(A212 )

+ σ11 · det(A214 )

− σ15 ) · det(A221 )

det(A22 ) = (−λ

+ σ15 ) · det(A211 )

det(A21 ) = (λ

det(A2 ) = −(σ7 − λ) · det(A21 ) − σ8 · det(A22 ) + σ9 · det(A23 ) − σ9 · det(A24 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 )

det(A − λ · I ) = −λ · det(A1 ) − σ1 · (σ2 + σ3 · e−λ·τ ) · det(A2 ) + σ4 · det(A3 )

Table 1.124 Determinants and sub determinants structure

det(A233 ) = σ12 · (λ + σ14 )

det(A232 ) = σ13 · (λ + σ14 )

· (λ + σ14 )

det(A231 ) = λ · σ12

+ σ11 · det(A233 )

− σ10 · det(A232 )

− σ15 ) · det(A231 )

det(A23 ) = (−λ

det(A243 ) = σ12 · (λ + σ14 )

− λ) · (λ + σ14 )

det(A242 ) = (σ13

− λ) · (λ + σ14 )

det(A241 ) = −λ · (σ13

+ σ11 · det(A243 )

− σ10 · det(A242 )

− σ15 ) · det(A241 )

det(A24 ) = (−λ

104 1 Dynamical and Nonlinearity of Laser Diode Circuits

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

det(A23 ) =

3 $

105

λk · Bk = λ3 · B3 + λ2 · B2 + λ · B1 + B0 ; B3 = −σ12

k=0

B2 = −σ12 · (σ15 + σ14 ) B1 = σ11 · σ12 − σ10 · σ13 − σ12 · σ15 · σ14 ; B0 = (σ11 · σ12 − σ10 · σ13 ) · σ14 det(A24 ) = −λ4 + λ3 · (σ13 − σ15 − σ14 ) + λ2 · (σ15 · σ13 + σ10 + σ14 · (σ13 − σ15 )) + λ · (σ11 · σ12 − σ10 · σ13 + σ14 · (σ15 · σ13 + σ10 )) + σ14 · (σ11 · σ12 − σ10 · σ13 )

det(A24 ) =

4 $

λk · Ak = λ4 · A4 + λ3 · A3 + λ2 · A2 + λ · A1 + A0 ; A4 = −1

k=0

A3 = σ13 − σ15 − σ14 ; A2 = σ15 · σ13 + σ10 + σ44 · (σ13 − σ15 ) A1 = σ11 · σ12 − σ10 · σ13 + σ14 · (σ15 · σ13 + σ10 ); A0 = σ14 · (σ11 · σ12 − σ10 · σ13 ) det(A2 ) = −(σ7 − λ) · det(A21 ) − σ8 · det(A2 ) + σ9 · det(A23 ) − σ9 · det(A24 )   det(A2 ) = −(σ7 − λ) · λ2 · T2 + λ3 · T3 + λ4 · T4 + λ5 · T5   − σ8 · λ4 · H4 + λ3 · H3 + λ2 · H2 + λ · H1   + σ9 · λ3 · B3 + λ2 · B2 + λ · B1 + B0   − σ9 · λ4 · A4 + λ3 · A3 + λ2 · A2 + λ · A1 + A0 det(A2 ) =

6 $

λk · k = λ6 · 6 + λ5 · 5 + λ4 · 4

k=0

+ λ3 · 3 + λ2 · 2 + λ · 1 + 0 6 = T5 ; 5 = T4 − T5 · σ7 ; 4 = T3 − T4 · σ7 − σ8 · H4 − A4 · σ9 3 = T2 − T3 · σ7 − σ8 · H3 − A3 · σ9 + B3 · σ9 ; 2 = −T2 · σ7 − σ8 · H2 + B2 · σ9 − A2 · σ9 1 = −H1 · σ8 + B1 · σ9 − A1 · σ9 ; 0 = σ9 · (B0 − A0 ) (Table 1.125)

det(A31 ) = 0

− λ) · (λ + σ14 )

det(A323 ) = (σ13

− λ) · (λ + σ14 )

det(A321 ) = λ · (σ13 · (λ + σ14 )

det(A322 ) = σ13

· (λ + σ14 )

det(A333 ) = σ12 · e−λ·τ

+ σ14 )

det(A332 ) = σ13 · (λ

· λ · (λ + σ14 )

det(A331 ) = σ12 · e−λ·τ

· det(A332 ) + σ11 · det(A333 )

− σ11 · det(A322 )

+ σ11 · det(A323 )

· det(A331 ) − σ10 · e−λ·τ

det(A33 ) = (−λ − σ15 )

− σ15 ) · det(A321 )

det(A32 ) = (−λ

det(A3 ) = −(σ7 − λ) · det(A31 ) − σ8 · e−λ·τ · det(A32 ) + σ9 · det(A33 ) − σ9 · det(A34 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 )

det(A − λ · I ) = −λ · det(A1 ) − σ1 · (σ2 + σ3 · e−λ·τ ) · det(A2 ) + σ4 · det(A3 )

Table 1.125 Determinants and sub determinants structure

·(λ + σ14 )

det(A343 ) = σ12 · e−λ·τ

− λ) · (λ + σ14 )

det(A342 ) = (σ13

det(A341 ) = 0

+ σ11 · det(A343 )

− σ10 · e−λ·τ · det(A342 )

− σ15 ) · det(A341 )

det(A34 ) = (−λ

106 1 Dynamical and Nonlinearity of Laser Diode Circuits

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

107

det(A31 ) = 0 det(A32 ) = (−λ − σ15 ) · det(A321 ) − σ11 · det(A322 ) + σ11 · det(A323 ) det(A32 ) = (−λ − σ15 ) · λ · (σ13 − λ) · (λ + σ14 ) − σ11 · σ13 · (λ + σ14 ) + σ11 · (σ13 − λ) · (λ + σ14 ) det(A32 ) = λ4 + λ3 · (σ14 − σ13 + σ15 ) − λ2 · (σ15 · σ13 + σ11 + σ14 · (σ13 − σ15 )) − λ · σ14 · (σ15 · σ13 + σ11 ) det(A32 ) =

4 $

λk · Pk = λ · P1 + λ2 · P2 + λ3 · P3 + λ4 · P4 ; P1

k=1

= −σ14 · (σ15 · σ13 + σ11 ) P2 = −(σ15 · σ13 + σ11 + σ14 · (σ13 − σ15 )); P3 = σ14 − σ13 + σ15 ; P4 = 1 det(A33 ) = (−λ − σ15 ) · det(A331 ) − σ10 · e−λ·τ · det(A332 ) + σ11 · det(A333 ) det(A33 ) = (−λ − σ15 ) · σ12 · e−λ·τ · λ · (λ + σ14 ) − σ10 · e−λ·τ · σ13 · (λ + σ14 ) + σ11 · σ12 · e−λ·τ · (λ + σ14 )  det(A33 ) = e−λ·τ · −λ3 · σ12 − λ2 · σ12 · (σ15 + σ14 ) + λ · (σ11 · σ12 − σ10 · σ13 − σ15 · σ12 · σ14 ) +(σ11 · σ12 − σ10 · σ13 ) · σ14 ] det(A33 ) = e−λ·τ ·

3 $

  λk · Yk = e−λ·τ · Y0 + λ · Y1 + λ2 · Y2 + λ3 · Y3

k=0

Y0 = (σ11 · σ12 − σ10 · σ13 ) · σ14 ; Y1 = σ11 · σ2 − σ10 · σ3 − σ15 · σ12 · σ14 Y2 = −σ12 · (σ5 + σ4 ) Y3 = −σ12 det(A34 ) = (−λ − σ15 ) · det(A341 ) − σ10 · e−λ·τ · det(A34 ) + σ11 · det(A343 ) det(A4 ) = −σ10 · e−λ·τ · (σ13 − λ) · (λ + σ14 ) + σ11 · σ1 · e−λ·τ · (λ + σ14 )

108

1 Dynamical and Nonlinearity of Laser Diode Circuits

det(A34 ) = e−λ·τ ·  2  λ · σ10 + λ · (σ11 · σ12 − σ10 · σ13 + σ14 · σ10 ) + σ14 · (σ11 · σ12 − σ10 · σ13 ) det(A34 ) = e−λ·τ ·

2 $

  λk · Ik = e−λ·τ · I0 + I1 · λ + I2 · λ2

k=0

I0 = σ14 · (σ11 · σ12 − σ10 · σ13 ) I1 = σ11 · σ12 − σ10 · σ13 + σ14 · σ10 ; I2 = σ10 det(A3 ) = −(σ7 − λ) · det(A31 ) − σ8 · e−λ·τ · det(A32 ) + σ9 · det(A33 ) − σ9 · det(A34 ) det(A3 )|det(A31 )=0 = −σ8 · e + σ9 · e

−λ·τ

·

 3 $

−λ·τ

·

 4 $

λ · Yk

λ · Pk

k=1

 k

 k

− σ9 · e

−λ·τ

·

 2 $

k=0

 λ · Ik k

k=0

det(A3 ) = det(A3 )|der (A31 )=0  4  3  2   ! $ $ $ −λ·τ k k k =e · −σ8 · λ · Pk + σ9 · λ · Yk − σ9 · λ · Ik k=1

k=0

k=0



det(A3 ) = e−λ·τ · −λ4 · P4 · σ8 + λ3 · (Y3 · σ9 − P3 · σ8 ) + λ2 · (Y2 · σ9 − P2 · σ8 − σ9 · I2 ) + λ · (Y1 · σ9 − P1 · σ8 − σ9 · I1 ) + σ9 · (Y0 − I0 ) det(A3 ) = e

−λ·τ

·

 4 $

 λ · ϒk ; ϒ0 = σ9 · (Y0 − I0 ); ϒ1 k

k=0

= Y1 · σ9 − P1 · σ8 − σ9 · I1 ϒ2 = Y2 · σ9 − P2 · σ8 − σ9 · I2 ; ϒ3 = Y3 · σ9 − P3 · σ8 ; ϒ4 = −P4 · σ8 Table 1.126.  det(A41 ) = e−λ·τ · λ4 · σ12 + λ3 · σ12 (·σ15 + σ14 ) + λ2 · (σ10 · σ13 − σ11 · σ12 + σ12 · σ15 · σ14 ) + λ · (σ10 · σ13 − σ11 − σ12 ) · σ14 ]

· det(A412 )

· e−λ·τ · (λ + σ14 )

det(A414 ) = −λ · σ12

det(A413 ) = 0

· (λ + σ14 )

det(A412 ) = λ · σ13

· e−λ·τ · (λ + σ14 )

det(A411 ) = λ2 · σ12

+ σ11 · det(A414 )

− σ10 · det(A413 )

+ σ10 · e

−λ·τ

− σ15 ) · det(A411 )

det(A41 ) = −(−λ

· (λ + σ14 )

det(A423 ) = σ12

· (λ + σ14 )

det(A421 ) = λ · σ12 · (λ + σ14 )

det(A422 ) = σ13 det(A433 ) = σ12 · e−λ·τ · (λ + σ14 )

det(A432 ) = σ13

· λ · (λ + σ14 ) · (λ + σ14 )

det(A431 ) = σ12 · e−λ·τ

+ σ11 · det(A433 )

· det(A432 )

− σ10 · e

+ σ11 · det(A423 )

−λ·τ

− σ15 ) · det(A431 )

det(A43 ) = (−λ

− σ10 · det(A422 )

− σ15 ) · det(A421 )

det(A42 ) = (−λ

det(A4 ) = −(σ7 − λ) · det(A41 ) − σ8 · e−λ·τ · det(A42 ) + σ8 · det(A43 ) − σ9 · det(A44 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 )

det(A − λ · I ) = −λ · det(A1 ) − σ1 · (σ2 + σ3 · e−λ·τ ) · det(A2 ) + σ4 · det(A3 )

Table 1.126 Determinants and sub determinants structure

· (λ + σ14 )

det(A443 ) = σ12 · e−λ·τ

+ σ14 )

det(A442 ) = σ12 · (λ

det(A441 ) = 0

+ σ10 · det(A443 )

− σ10 · e−λ·τ · det(A442 )

− σ15 ) · det(A441 )

det(A44 ) = (−λ

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis 109

110

1 Dynamical and Nonlinearity of Laser Diode Circuits

det(A41 ) = e

−λ·τ

·

 4 $

 λ · Ek k

  = e−λ·τ · λ4 · E 4 + λ3 · E 3 + λ2 · E 2 + λ · E 1

k=1

E 1 = (σ10 · σ13 − σ11 · σ12 ) · σ14 ; E 2 = σ10 · σ13 − σ11 · σ12 + σ12 · σ15 · σ14 E 3 = σ12 (·σ15 + σ14 ); E 4 = σ12 det(A42 ) = −λ3 · σ12 − λ2 · σ12 − (σ15 + σ14 ) + λ · (σ11 · σ12 − σ10 · σ13 − σ15 · σ12 · σ14 ) + (σ11 · σ12 − σ10 · σ13 ) · σ14 det(A42 ) =

3 $

λk · Tk = λ3 · T3 + λ2 · T2 + λ · T1 + T0 ; T3

i=0

= −σ12 ; T2 = −σ12 · (σ15 + σ14 ) T1 = σ11 · σ12 − σ10 · σ13 − σ15 · σ12 · σ14 ; T0 = (σ11 · σ12 − σ10 · σ13 ) · σ14  det(A43 ) = e−λz · −λ3 · σ12 − λ2 · σ12 · (σ15 + σ14 ) +λ · (σ11 · σ12 − σ10 · σ13 − σ12 · σ13 · σ14 ) + σ14 · (σ11 · σ12 − σ10 · σ13 )]  det(A43 ) = e−λ·τ ·

3 $

 λk · L k

k=0

  = e−λ·z · λ3 · L 3 + λ2 · L 2 + λ · L 1 + L 0 L 3 = −σ12 ; L 2 = −σ12 · (σ15 + σ14 ); L 1 = σ11 · σ12 − σ10 · σ13 − σ12 · σ15 − σ14 L 0 = σ14 · (σ11 · σ12 − σ10 · σ13 ) det(A44 ) = 0 det(A4 ) = −(σ7 − λ) · det(A41 ) − σ8 · e−λ·τ · det(A42 ) + σ8 · det(A43 ) − σ9 · det(A44 ) det(A4 ) = −(σ7 − λ) · e

−λ·τ

·

 4 $ i=1

 λ · Ek k

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

− σ8 · e

−λ·τ

·

 3 $

 −λ·τ

 3 $

111



λ · Tk + σ8 · e · λ · Lk k=0 k=0 ⎧ ⎞ ⎞ ⎛ ⎞⎤⎫ ⎛ ⎡⎛ 4 3 3 ⎬ ⎨ $ $ $ −λ·τ k k k det(A4 ) = e · (−σ7 + λ) · ⎝ λ · E k ⎠ + σ8 · ⎣ ⎝ λ · Lk ⎠ − ⎝ λ · Tk˙ ⎠⎦ ⎭ ⎩ k=1 k=0 k=0 k

k

 det(A4 ) = e−λ·τ · λ5 · E 4 + λ4 · [E 3 + σ8 · (L 3 − T3 ) − E 4 · σ7 ] + λ3 · [E 2 + σ8 · (L 2 − T2 ) − σ7 · (E 3 + σ8 · (L 3 − T3 ))] + λ2 · [E 1 + σ8 · (L 1 − T1 ) − σ7 · (E 2 + σ8 · (L 2 − T2 ))] +λ · [σ8 · (L 0 − T0 ) − σ7 · (E 1 + σ8 · (L 1 − T1 ))] − σ7 · σ8 · (L 0 − T0 )} det(A4 ) = e

−λ·τ

·

 5 $

 λ · χk ; χ5 = E 4 ; χ4 = E 3 + σ8 · (L 3 − T3 ) − E 4 · σ7 k

k=0

χ3 = E 2 + σ8 · (L 2 − T2 ) − σ7 · (E 3 + σ8 − (L 3 − T3 )); χ2 = E 1 + σ8 − (L 1 − T1 ) − σ7 · (E 2 + σ8 · (L 2 − T2 )) χ1 = σ8 · (L 0 − T0 ) − σ7 · (E 1 + σ8 − (L 1 − T1 )); χ0 = −σ7 · σ8 · (L 0 − T0 ) Table 1.127.  det(A51 ) = e−λ·τ · −σ10 · λ3 + λ2 · (σ10 · σ13 − σ12 · σ11 − σ10 · σ14 ) +λ · σ14 · (σ10 · σ13 − σ12 · σ11 )}  det(A51 ) = e

−λ·τ

·

3 $

 λ · Gk k

k=1

  = e−λ·τ · λ · G 1 + λ2 · G 2 + λ3 · G 3 ; G 3 = −σ10 G 2 = σ10 · σ13 − σ12 · σ11 − σ10 · σ14 ; G 1 = σ14 · (σ10 · σ13 − σ12 · σ11 ) det(A52 ) = σ10 · λ2 + λ · (σ11 · σ12 − σ10 · σ13 + σ14 · σ10 ) + σ14 · (σ11 · σ12 − σ10 · σ13 ) det(A52 ) =

2 $

λk · Ik = λ2 · I2 + λ · I1 + I0 ; I0 = σ14 · (σ11 · σ12 − σ10 · σ13 )

i=0

I1 = σ11 · σ12 − σ10 · σ13 + σ14 · σ10 ; I2 = σ10

· e−λ·τ · (λ + σ14 )

det(A514 ) = −λ · σ12

det(A513 ) = 0 (λ + σ14 )

det(A523 ) = σ12

− λ) · (λ + σ14 )

det(A522 ) = (σ13

− λ) · (λ + σ14 )

det(A512 ) = λ · (σ13

· e−λ·τ · (λ + σ14 )

det(A533 ) = σ12

− λ) · (λ + σ14 )

det(A332 ) = (σ13

det(A531 ) = 0

+ σ11 · det(A33 )

+ σ11 · det(A523 )

· det(A532 )

− σ10 · e

det(A521 ) = 0

− σ10 · det(A513 )

· det(A512 )

−λ·τ

−σ15 ) · det(A531 )

det(A53 ) = (−λ

− σ10 · det(A522 )

−σ15 ) · det(A521 )

det(A52 ) = (−λ

det(A511 ) = 0

+σ11 · det(A514 )

+ σ10 · e

−λ·τ

−σ15 ) · det(A511 )

det(A51 ) = −(−λ

det(A5 ) = −(σ7 − λ) · det(A51 ) − σ8 · e−λ·τ · det(A52 ) + σ8 · det(A53 ) − σ9 · det(A54 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 )

Table 1.127 Determinants and sub determinants structure   det(A − λ · I ) = −λ · det(A1 ) − σ1 · σ2 + σ3 · e−λ·τ · det(A2 ) + σ4 · det(A3 )

(λ + σ14 )

det(A543 ) = σ12 · e−λ·τ

+σ14 )

det(A542 ) = σ12 · (λ

det(A541 ) = 0

+ σ10 · det(A543 )

− σ10 · e−λ·τ · det(A542 )

+σ15 ) · det(A541 )

det(A54 ) = −(λ

112 1 Dynamical and Nonlinearity of Laser Diode Circuits

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

113

 det(A53 ) = e−λ·τ · σ10 · λ2 + λ · (σ11 · σ12 − σ10 · σ13 + σ10 · σ14 ) +σ14 · (σ11 · σ12 − σ10 · σ13 )} det(A33 ) = e−λ·τ ·

 2 $

 λk · Jk

  = e−λ·τ · λ2 · J2 + λ · J1 + J0

k=0

J0 = σ14 · (σ11 · σ12 − σ10 · σ13 ) J1 = σ11 · σ12 − σ10 · σ13 + σ10 · σ14 ; J2 = σ10 ; det(A54 ) = 0 det(A5 ) = −(σ7 − λ) · det(A51 ) − σ8 · e−λ·τ · det(A52 ) + σ8 · det(A53 ) − σ9 · det(A54 )

  det(A5 ) = −(σ7 − λ) · e−λ·τ · λ · G 1 + λ2 · G 2 + λ3 · G 3   − σ8 · e−λ·τ · λ2 · I2 + λ · I1 + I0   + σ8 · e−λ·τ · λ2 · J2 + λ · J1 + J0 −λ·τ ·  Since J22 = I2 ; 3J1 = I1 ; J0 = I0 then det(A5 ) = −(σ7 − λ) · e λ · G1 + λ · G2 + λ · G3

  det(A5 ) = e−λ·τ · λ4 · G 3 + λ3 · (G 2 − σ7 · G 3 ) + λ2 · (G 1 − G 2 · σ7 ) − λ · G 1 · σ7  4  $ λk · t k det(A5 ) = e−λ·τ · k=1

t4 = G 3 ; t3 = G 2 − σ7 · G 3 ; t2 = G 1 − G 2 · σ7 ; t1 = −G 1 · σ7 Table 1.128. det(A61 ) = −(−λ − σ15 ) · det(A611 ) + σ10 · e−λ·τ · det(A612 ) − σ10 · det(A613 ) + σ11 · det(A614 ) − σ11 · det(A615 ) det(A61 ) = λ4 · σ14 + λ3 · σ14 · (σ15 − σ13 ) − λ2 · σ14 · (σ15 · σ13 + σ11 ) det(A61 ) =

4 $

λk · Vk = λ2 · V2 + λ3 · V3 + λ4 · V4 ; V2 = −σ14 · (σ15 · σ13 + σ11 )

k=2

V3 = σ14 · (σ15 − σ13 ); V4 = σ14 det(A63 ) = (−λ − σ15 ) · det(A631 ) − σ10 · e−λ·τ · det(A632 ) + σ11 · det(A633 ) − σ11 · det(A634 )

det(A612 ) = 0 det(A613 ) = 0 det(A614 ) = −λ · σ14 · σ13 det(A615 ) = −λ · σ14 · (σ13 − λ)

· σ14 · (σ13 − λ)

det(A611 ) = −λ2

− σ11 · det(A615 )

det(A631 ) = −λ · σ14 · (σ13 − λ) det(A632 ) = 0 det(A633 ) = σ13 · σ14 det(A634 ) = σ14 · (σ13 − λ)

+ σ11 · det(A633 )

· det(A632 )

− σ11 · det(A634 )

− σ10 · e

+ σ11 · det(A614 )

· det(A612 )

−λ·τ

−σ15 ) · det(A631 )

det(A63 ) = (−λ

− σ10 · det(A613 )

+ σ10 · e

−λ·τ

−σ15 ) · det(A611 )

det(A61 ) = −(−λ · det(A642 )

det(A641 ) = −λ · σ14 · σ12 det(A642 ) = 0 det(A643 ) = σ13 · σ14 det(A644 ) = σ12 · σ14

− σ11 · det(A644 )

+ σ10 · det(A643 )

− σ10 · e

−λ·τ

−σ15 ) · det(A641 )

det(A64 ) = (−λ

det(A6 ) = −(σ7 − λ) · det(A61 ) − σ8 · e−λ·τ · det(A62 ) + σ8 · det(A63 ) − σ9 · det(A64 ) + σ9 · det(A65 )

− σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 ); det(A62 ) = 0

Table 1.128 Determinants and sub determinants structure   det(A − λ · I ) = −λ · det(A1 ) − σ1 · σ2 + σ3 · e−λ·τ · det(A2 ) + σ4 · det(A3 )

det(A651 ) = 0 det(A652 ) = 0 det(A653 ) = σ14 · (σ13 − λ) det(A654 ) = σ12 · σ14

− σ11 · det(A654 )

+ σ10 · det(A653 )

− σ10 · e−λ·τ · det(A652 )

−σ15 ) · det(A651 )

det(A65 ) = (−λ

114 1 Dynamical and Nonlinearity of Laser Diode Circuits

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

115

det(A63 ) = −λ3 · σ14 + λ2 · σ14 · (σ13 − σ15 ) + λ · σ14 · (σ15 · σ13 + σ11 ) det(A63 ) =

3 $

λk · Wk = λ · W1 + λ2 · W2 + λ3 · W3 ; W1 = σ14 · (σ15 · σ13 + σ11 )

k=1

W2 = σ14 · (σ13 − σ15 ); W3 = −σ14 det(A64 ) = (−λ − σ15 ) · det(A641 ) − σ10 · e−λ·τ · det(A642 ) + σ10 · det(A643 ) − σ11 · det(A644 ) det(A64 ) = λ2 · σ14 · σ12 + λ · σ13 · σ14 · σ12 + σ14 · (σ10 · σ13 − σ11 · σ12 ) det(A64 ) =

2 $

λk · X i = X 0 + λ · X 1 + λ2 · X 2 ; X 0 = σ14 · (σ10 · σ13 − σ11 · σ12 )

k=0

X 1 = σ15 · σ14 · σ12 ; X 2 = σ14 · σ12 det(A65 ) = (−λ − σ15 ) · det(A651 ) − σ10 · e−λ·τ · det(A652 ) + σ10 · det(A653 ) − σ11 · det(A654 ) det(A65 ) = −λ · σ10 · σ14 + σ14 · (σ10 · σ13 − σ11 · σ12 ); det(A65 ) =

1 $

λk · Z k = Z 0 + λ · Z 1

k=0

Z 0 = σ14 · (σ10 · σ13 − σ11 · σ12 ); Z 1 = −σ10 · σ14 det(A6 ) = −(σ7 − λ) · det(A61 ) + σ8 · det(A63 ) − σ9 · det(A64 ) + σ9 · det(A65 ) det(A6 ) = λ5 · V4 + λ4 · (V3 − σ7 · V4 ) + λ3 · (V2 − σ7 · V3 + σ8 · W3 ) + λ2 · (−σ7 · V2 + σ8 · W2 − σ9 · X 2 ) + λ · (W1 · σ8 − X 1 · σ9 + σ9 · Z 1 ) + (σ9 · Z 0 − X 0 · σ9 ) det(A6 ) =

5 $

λk · Ok = O0 + λ · O1 + λ2 · O2 + λ3 · O3 + λ4 · O4 + λ5 · O5

k=0

Oo = σ9 · Z 0 − X 0 · σ9 ; O1 = W1 · σ8 − X 1 · σ9 + σ9 · Z 1

116

1 Dynamical and Nonlinearity of Laser Diode Circuits

O2 = −σ7 · V2 + σ8 − W2 − σ9 · X 2 O3 = V2 − σ7 · V3 + σ8 · W3 ; O4 = V3 − σ7 · V4 ; O5 = V4 We can summary our intermediate results: det(A1 ) = det(A1 )|det(A1 )=0 =

6 $

λk · k

k=1

= λ6 · 6 + λ5 · 5 + λ4 · 4 + λ3 · 3 + λ2 · 2 + λ · 1 det(A2 ) =

6 $

λk · k = λ6 · 6 + λ5 · 5 + λ4 · 4 + λ3 · 3

k=0

+ λ2 · 2 + λ · 1 + 0 det(A3 ) = e−λ·τ · 

det(A4 ) = e

−λ·τ

·

 4 $

 λk · ϒk

= e−λ·τ ·

k=0

ϒ0 + λ · Y1 + λ2 · ϒ2 + λ3 · ϒ3 + λ4 · ϒ4

 5 $



 λ · χk k

k=0

  = e−λ·τ · χ0 + λ · χ1 + λ2 · χ3 + λ3 · χ3 + λ4 · χ4 + λ5 · χ5 det(A5 ) = e

−λ·τ

·

 4 $

 λ · tk k

= e−λ·τ ·

k=1

  4 λ · G 3 + λ3 · (G 2 − σ7 · G 3 ) + λ2 · (G 1 − G 2 · σ7 ) − λ · G 1 · σ7 det(A6 ) =

5 $

λk · Ok = O0 + λ · O1 + λ2 · O2 + λ3 · O3 + λ4 · O4 + λ5 · O5

k=0

Finally we get the expression for det(A − λ · I ).   det(A − λ · I ) = (−λ) · det(A1 ) − σ1 · σ2 + σ3 · e−λ·τ · det(A2 ) + σ4 · det(A3 ) − σ5 · det(A4 ) + σ5 · det(A5 ) − σ6 · det(A6 ) det(A − λ · I ) = (−λ) ·

 6 $ k=1

 λ − k k

 6   $  −λ·τ i · λ · k − σ1 · σ2 + σ3 · e k=0

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

+ σ4 · e

−λ·τ

 4 $

·

 − σ5 · e

−λ·τ

· 

+ σ5 · e

−λ·τ

·

 λ · ϒk k

k=0 5 $

 λ · χk k

k=0 4 $

 λ − tk

− σ6 ·

k

k=1

det(A − λ · I ) = (−λ) ·

 6 $

 5 $

λ · Ωk

−σ5 ·

− σ1 · σ2 ·

λ · Ok  6 $

 λ · k k

k=0



λ · Ok

 4 $

 5 $

k=0

 k

k

k=0

+ σ4 ·

 5 $

 k

k=1

− σ6 ·

117

 λk · ϒk

k=0

− σ1 · σ3 ·



λ · χi

+ σ5 ·

k

 λk · k

k=0

 4 $

k=0

 6 $

! · e−λ·τ

λ · tk k

k=1

We get our MRI system laser diode equivalent circuit characteristic equation: D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e − σ1 · σ2 ·

 6 $

−λ·τ

 λ · k k

− σ6 ·

k=0

Q m (λ, τ ) = σ4 ·

 4 $

 λ · ϒk

 5 $

 5 $

λk − χi

+ σ5 ·

λ · k

k=1

 λ · Ok

k=0



 k

k

− σ1 · σ3 ·

k

k=0

− σ5 ·

; Pn (λ, τ ) = (−λ) ·

 6 $

 6 $

 4 $

k=0

 λ · k k

k=0



λ k · tk

k=1

Pn (λ, τ ) = −λ7 · 6 − λ6 · (3 + σ1 · σ2 · 6 ) − λ5 · (4 + 3 · σ1 · σ2 + O5 · σ6 ) − λ4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) − λ3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − λ2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − λ · (1 · σ1 · σ2 + O1 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 )

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Q m (λ, τ ) = −σ1 · σ3 · 6 − λ6 − λ5 − (5 · σ1 · σ2 + χ5 · σ5 ) + λ4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) + λ3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) + λ2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ3 + t2 · σ5 ) + λ · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 ) We get the characteristics equation for stability analysis. We study the occurrence of any possible stability switching, resulting from the increase of the value of the time delay τ or any other circuit parameter for the general characteristic equation D(λ,τ). D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n > m, n = 7, m = 6 Q m (λ, τ ) =

m $

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + . . . + q6 (τ ) · λ6 ; q0 (τ )

k=0

= σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 q1 (τ ) = ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ; q2 (τ ) = ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 q3 (τ ) = ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ; q4 (τ ) = ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 q5 (τ ) = −(5 · σ1 · σ2 + χ5 · σ5 ); q6 (τ ) = −σ1 · σ3 · 6 Pn (λ, τ ) =

n $

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + . . . + p7 (τ ) · λ7 ; p0 (τ )

k=0

= −(σ1 · σ2 · 0 + O0 · σ6 ) p1 (τ ) = −(1 · σ1 · σ2 + O1 · σ6 ); p2 (τ ) = −(1 + σ1 · σ2 · 2 + O2 · σ6 ) p3 (τ ) = −(2 + 3 · σ1 · σ2 + σ6 · O3 ); p4 (τ ) = −(3 + σ1 · σ2 · 4 + σ6 · O4 ) p5 (τ ) = −(4 + 5 · σ1 · σ2 + O5 · σ6 ); p6 (τ ) = −(5 + σ1 · σ2 · 6 ); p7 (τ ) = −6

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

119

The homogeneous system for Y1 , Y2 , Y3 , I L 1 , I L 2 , I Ri , I L i , I R p leads to characteristics equation  for the eigenvalue λ having the form P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; Q(λ, τ ) = 6j=0 c j · λ j 7    P(λ, τ ) = a j · λ j . The coefficients a j (qi , qk , τ ), c j (qi , qk , τ ) ∈ R depend j=0

on qi , qk and delay τ. qi , qk are any MRI system laser diode equivalent circuit parameters, other parameters kept as constant. a0 = −(σ1 · σ2 · 0 + O0 · σ6 ); a1 = −(1 · σ1 · σ2 + O1 · σ6 ); a2 = −(1 + σ1 · σ2 · 2 + O2 · σ6 ) a2 = −(1 + σ1 · σ2 · 2 + O2 · σ6 ); a3 = −(2 + 3 · σ1 · σ2 + σ6 · O3 ) a4 = −(3 + σ1 · σ2 · 4 + σ6 · O4 ); a5 = −(4 + 5 · σ1 · σ2 + O5 · σ6 ) a6 = −(5 + σ1 · σ2 · 6 ); a7 = −6 ; c0 = σ4 · ϒ0 − σ1 · σ3 − σ5 · γ0 c1 = ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ; c2 = ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 c3 = ϒ3 · σ4 − σ1 · σ3 · 3 − γ3 · σ5 + σ5 · t3 ; c4 = ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 c5 = −(5 · σ1 · σ2 + χ5 · σ5 ); c6 = −σ1 · σ3 · 6 Unless strictly necessary, the designation of varied arguments (qi , qk ) will subsequently be omitted from P, Q, aj , and cj . The coefficients aj , cj are continuous, and differentiable functions of their arguments, and direct substitution shows that a0 +c0 = 0;−(σ1 · σ2 · 0 + O0 · σ6 )+σ4 ·ϒ0 −σ1 ·σ3 −σ5 ·χ0 = 0 for qi , qk ∈ R+ ; that is λ = 0 is not of P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; P(λ) = P(λ, τ ); Q(λ) = Q(λ, τ ) [2, 3]. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [2, 4] can also be verified in the present case: 1. If λ = i · ω, ω ∈ R then P(i · ω) + Q(i · ω) = 0. 2. If |Q(λ)/P(λ)| is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk .

120

1 Dynamical and Nonlinearity of Laser Diode Circuits

We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω. Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0  Pn (λ = i · ω, τ ) = i · ω7 · 8 − ω5 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) + ω3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) −ω · (1 · σ1 · σ2 + O1 · σ6 )} + ω6 · (5 + σ1 · σ2 · 6 ) − ω4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + ω2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 ) Q m (λ = i · ω, τ ) = ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) − ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 ) 0 + i · −ω5 · (5 · σ1 · σ2 + χ5 · σ5 ) − ω3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) +ω · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 )}

⎧ i ∀k =4·n−3 ⎪ ⎪ ⎨ −1 ∀ k = 4 · n − 2 λk = i k · ω k ; i k = ; n ∈ R0 ; n = 1, 2, 3, . . . ⎪ −i ∀ k = 4 · n − 1 ⎪ ⎩ 1 ∀k =4·n Q m (λ = i · ω, τ ) = Q m (i · ω, τ ) = Q(i · ω, τ ) Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = ω6 · [5 + σ1 · σ2 · 6 + σ1 · σ3 · 6 ] + ω4 · [ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 − 3 − σ1 · σ2 · 4 − σ6 · O4 ] + ω2 · [1 + σ1 · σ2 · 2 + O2 · σ6 − (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 )] + (σ4 · ϒ0 − σ1 · σ3 − σ3 · χ0 ) − σ1 · σ2 · 0 − O0 · σ6 0 + i · ω7 · 6 − ω5 · [(4 + 5 · σ1 · σ2 + O5 · σ6 ) + (5 · σ1 · σ2 + χ5 · σ5 )] + ω3 · [(2 + 3 · σ1 · σ2 + σ6 · O3 ) − (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 )] +ω · [(ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ) − (1 · σ1 · σ2 + O1 · σ6 )]} = 0

Pn (λ = i · ω, τ ) = Pn (i · ω, τ ) = P(i · ω, τ )  |P(i · ω, τ )|2 = ω6 · (5 + σ1 · σ2 · 6 ) − ω4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) 2 +ω2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 )  + ω7 · 6 − ω5 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) + ω3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − ω · (1 · σ1 · σ2 + O1 · σ6 )}2  |Q(i · ω, τ )|2 = ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) 2 −ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 )

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

121

 + −ω5 · (5 · σ1 · σ2 + χ5 · σ5 ) − ω3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) +ω · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 )}2 F(ω, τ ) = P(i · ω, τ )| − |Q(i · ω, τ )| = 2

2

7 $

2.k · ω2·k

k=0

Hence F(ω, τ ) = 0 implies

7 

2.k · ω2.k = 0 and its roots are given by solving

k=0

the above polynomial. Remark It is the reader exercise to find the 2·k ; k = 0, 1, . . . , 7 expressions.

PR (i · ω, τ ) = ω6 · (5 + σ1 · σ2 · 6 ) − ω4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + ω2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 ) PI (i · ω, τ ) = ω7 · 6 − ω5 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) + ω3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − ω · (1 · σ1 · σ2 + O1 · σ6 ) Q R (i · ω, τ ) = ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) − ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 ) Q I (i · ω, τ ) = −ω5 · (5 · σ1 · σ2 + χ5 · σ5 ) − ω3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) + ω · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ) sin θ (τ ) =

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2

cos θ (τ ) = −

PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) |Q(i · ω, τ )|2

We use different parameters terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 7; m = 6; n > m. Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ); P(λ) =

7 $ j=0

a j · λ j ; Q(λ) =

6 $ j=0

cj · λj

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1 Dynamical and Nonlinearity of Laser Diode Circuits

P(λ) = −λ7 · 6 − λ6 · (5 + σ1 · σ2 · 6 ) − λ5 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) − λ4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) − λ3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − λ2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − λ · (1 · σ1 · σ2 + O1 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 ) Q(λ) = −σ1 · σ3 · 6 · λ6 − λ5 · (5 · σ1 · σ2 + χ5 · σ5 ) + λ4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) + λ3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) + λ2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) + λ · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 ) n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “–” denotes complex and conjugate. function in λ and differentiable  P(λ), Q(λ) are analytic   in τ. The coefficients a j Y1 , Y2 , Y3 , I L 1 , τ, . . . ∈ R and c j Y1 , Y2 , Y3 , I L 1 , τ, . . . ∈ R depend on MRI system laser diode equivalent circuit’s Y1 , Y2 , Y3 , I L 1 , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: will subsequently be omitted P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0. − (σ1 · σ2 · 0 + O0 · σ6 ) + σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 = 0 ∀ Y1 , Y2 , Y3 , I L 1 , τ, . . . ∈ R+ i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang [2], Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire   Y1 , Y2 , Y3 , I L 1 , τ, . . . domain of interest. 1 1 1 P(λ) 1 (b) 1 Q(λ) 1 is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. 1 1 1 1 1 1 1 q0 (τ )+q1 (τ )·λ+...+q6 (τ )·λ6 1 Indeed, in the limit: 1 Q(λ) = 1 p0 (τ )+ p1 (τ )·λ+...+ p7 (τ )·λ7 1. P(λ) 1 (c) The following expressions exist: F(ω) = P(i · ω)|2 − |Q(i · ω)|2

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 =

7 # $ i=0

2·k

· ω2·k

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

123

Has at most a finite number of zeros. Indeed, this is a polynomial in ω (Degree in ω14 ).   (d) Each positive root ω Y1 , Y2 , Y3 , I L 1 , τ, . . . of F(ω) = 0 is continuous and differentiable with respect to Y1 , Y2 , Y3 , I L 1 , τ, . . .. The condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i · ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 maybe on eigenvalue of characteristic equation. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plan, whereby increasing the parameters:Y1 , Y2 , Y3 , I L 1 , τ, . .., Reλ may at the crossing change its

sign from (−) to (+), i.e. from a stable focus E ∗ I L∗1 , I L∗2 , I R∗i , I R∗ p , I L∗i , Y1∗ , Y2∗ , Y3∗ =   0, I R∗i + I L∗i , I R∗i , 0, I L∗i , 0, 0, 0 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to Y1 , Y2 , Y3 , I L 1 , τ, . . . and system parameters.  ∂Reλ Λ (τ ) = ; R p , C1 , L 1 , C2 , R1 , L 2 , Ri , Ci , L i , Rse = const ∂τ λ=i·ω   ∂Reλ −1 Λ (τ ) = ; τ, C1 , L 1 , C2 , R1 , L 2 , Ri , Ci , L i , Rse = const ∂ R p λ=i·ω   ∂Reλ Λ−1 (τ ) = ; R p , τ, L 1 , C2 , R1 , L 2 , Ri , Ci , L i , Rse = const ∂C1 λ=i·ω   ∂Reλ −1 Λ (τ ) = ; R p , C1 , τ, C2 , R1 , L 2 , Ri , Ci , L i , Rse = const ∂ L 1 λ=i·ω   ∂Reλ Λ−1 (τ ) = ; R p , C1 , L 1 , τ, R1 , L 2 , Ri , Ci , L i , Rse = const ∂C2 λ=i·ω   ∂Reλ Λ−1 (τ ) = ; R p , C1 , L 1 , C2 , τ, L 2 , Ri , Ci , L i , Rse = const ∂ R1 λ=i·ω   ∂Reλ −1 Λ (τ ) = ; R p , C1 , L 1 , C2 , R1 , τ, Ri , Ci , L i , Rse = const ∂ L 2 λ=i·ω   ∂Reλ Λ−1 (τ ) = ; R p , C1 , L 1 , C2 , R1 , L 2 , τ, Ci , L i , Rse = const ∂ Ri λ=i·ω   ∂Reλ Λ−1 (τ ) = ; R p , C1 , L 1 , C2 , R1 , L 2 , Ri , τ, L i , Rse = const ∂Ci λ=i·ω   ∂Reλ −1 Λ (τ ) = ; R p , C1 , L 1 , C2 , R1 , L 2 , Ri , Ci , τ, Rse = const ∂ L i λ=i·ω −1



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1 Dynamical and Nonlinearity of Laser Diode Circuits

Λ−1 (τ ) =



∂Reλ ∂ Rse

 λ=i·ω

; R p , C1 , L 1 , C2 , R1 , L 2 , Ri , Ci , L i , τ = const

P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ) When writing and inserting λ = i · ω into MRI system laser diode circuit’s characteristic equation, ω must satisfy the following equations: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω, τ )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equations that, by squaring and adding the sides, ω must be a positive root for F(ω) = P(i · ω)|2 − |Q(i · ω)|2 . Note: F(ω) is dependent onτ. Now it is important to notice that if τ ∈ / I (assume / I , ω(τ ) that I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for τ ∈ is not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . .; cos θ (τ ) = . . . and the relation between the arguments θ (τ ) and τ · ω(τ ) for τ ∈ I must be describing below [10]. sin θ (τ ) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos θ (τ ) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ N0 Hence we can define the maps τn : I → R+0 given by τn (τ ) = θ(τ )+2·n·π ; n∈ ω(τ ) N0 , τ ∈ I . Let us introduce the functions: I → R; Sn (τ ) = τ − τn (τ ), τ ∈ I, n ∈ N0 that is continuous and differentiable in τ. In the following, the subscripts Let us first λ, ω, R p , C1 , L 1 , . . . indicate the corresponding partial  derivatives.   concentrate on Λ(x), remember in λ ω, R p , C1 , L 1 , . . . and ω R p , C1 , L 1 , C2 , . . . , keeping all parameters except one (x) andτ. The derivation closely follows that in reference [BK]. Differentiating MRI system Laser diode circuit characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = R p , C1 , L 1 , C2 , . . .

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis



∂λ ∂x

−1

=

125

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP ; Q λ = ∂∂λQ ; Px = ∂∂ Px ; Q x = ∂∂Qx , substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω); i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) And that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains:  −1 11 ∂λ 1 1 1 ∂x λ=i·ω

=

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = QR + i · QI Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 When (x) can be any MRI system laser diode circuit’s parameter R p , C1 , L 1 , C2 , . . . and time delayτ, etc., Where for convenience, = 2 · we have dropped the arguments(i, ω, x), and where Fω [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; ωx = − FFωx Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]. We define U and V functions: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ. PR = ω6 · (5 + σ1 · σ2 · 6 ) − ω4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + ω2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 ) PI = ω7 · 6 − ω5 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) + ω3 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − ω · (1 · σ1 · σ2 + O1 · σ6 ) Q R = ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · l4 ) − ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + c2 · σ5 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 )

126

1 Dynamical and Nonlinearity of Laser Diode Circuits Q I = −ω5 · (5 · σ1 · σ2 + χ5 · σ5 ) − ω3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) + ω · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 )

PRω = 6 · ω5 · (5 + σ1 · σ2 · 6 ) − 4 · ω3 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + 2 · ω · (1 + σ1 · σ2 · 2 + O2 · σ6 ) PI ω = 7 · ω6 · 6 − 5 · ω4 · (4 + 5 · σ1 · σ2 + O5 · σ6 ) + 3 · ω2 · (2 + 3 · σ1 · σ2 + σ6 · O3 ) − (1 · σ1 · σ2 + O1 · σ6 ) Q Rω = 6 · ω5 · σ1 · σ3 · 6 + 4 · ω3 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) − 2 · ω · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) Q I ω = −5 · ω4 · (5 · σ1 · σ2 + χ5 · σ5 ) − 3 · ω2 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · l3 ) + (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · c1 ) PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; ωτ = −

Fτ Fω

PRω · PR = {6 · ω5 · (5 + σ1 · σ2 · 6 ) − 4 · ω3 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + 2 · ω · (1 + σ1 · σ2 · 2 + O2 · σ6 )}  · ω6 · (5 + σ1 · σ2 · 6 ) − ω4 · (3 + σ1 · σ2 · 4 + σ6 · O4 ) + ω2 · (1 + σ1 · σ2 · 2 + O2 · σ6 ) − (σ1 · σ2 · 0 + O0 · σ6 )}

0 Q Rω · Q R = 6 · ω5 · σ1 · σ3 · 6 + 4 · ω3 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) − 2 · ω · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 )}· 0 ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 −σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 ) − ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 )

QR · QIω

+(σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 )} 0 = ω6 · σ1 · σ3 · 6 + ω4 · (ϒ4 · σ4 − σ1 · σ3 · 4 − χ4 · σ5 + σ5 · t4 )

2 −ω2 · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 ) + (σ4 · ϒ0 − σ1 · σ3 − σ5 · χ0 ) 0 · −5 · ω4 · (5 · σ1 · σ2 + χ5 · σ5 ) − 3 · ω2 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 )

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis

127

+(ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 )} 0 Q I · Q Rω = −ω5 · (5 · σ1 · σ2 + χ5 · σ5 ) − ω3 · (ϒ3 · σ4 − σ1 · σ3 · 3 − χ3 · σ5 + σ5 · t3 ) 0 +ω · (ϒ1 · σ4 − σ1 · σ3 · 1 − χ1 · σ5 + σ5 · t1 )} · 6 · ω5 · σ1 · σ3 · 6 + 4 · ω3 · (ϒ4 · σ4 −σ1 · σ3 · 4 − χ4 · σ5 + σ5 · l4 ) − 2 · ω · (ϒ2 · σ4 − σ1 · σ3 · 2 − χ2 · σ5 + t2 · σ5 )}

V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0; F(ω, τ ) = 0 +Fτ = 0; τ ∈ I ⇒ Differentiating with respect to τ and we get Fω · ∂ω ∂τ 

∂ω ∂τ

= − FFωτ



∂ω Fτ = ωτ = − ; Λ−1 (τ ) ∂τ F ω λ=i·ω !   2 −2 · U + τ · |P| + i · Fω   = Re Fτ + i · 2 · V + ω · |P|2  

     ∂Reλ ; sign Λ−1 (τ ) sign Λ−1 (τ ) = sign ∂τ λ=i·ω ! ∂ω V + ∂τ · U ∂ω ·τ +ω+ = sign{Fω } · sign |P|2 ∂τ −1

Λ (τ ) =

∂Reλ ∂τ

;

We shall presently examine the possibility of stability transitions (bifurcations) MRI system laser diode circuit, about the equilibrium   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ pointE I L , I L 2 , I Ri , I R p , I L i , Y1 , Y2 , Y3 .     E ∗ I L∗1 , I L∗2 , I R∗i , I R∗ p , I L∗i , Y1∗ , Y2∗ , Y3∗ = 0, I R∗i + I L∗i , I R∗i , 0, I L∗i , 0, 0, 0 as a result of a variation of delay parameter τ. The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ plane, Where by increasing the delay parameter τ, Reλ may at the crossing, changes its sign from “−” to “ + ”, i.e. from a stable focus E ∗ to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect toτ. Λ−1 (τ ) =



∂Reλ ∂C1

 λ=i·ω

; R p , τ, L 1 , C2 , R1 , L 2 , Ri , Ci , L i , Rse = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) After getting the expression for F(ω, τ ) MRI system laser diode circuit’s parameter values, we find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values τ ∈ [0.001 . . . 10], it can be express by 3D function F(ω, τ ) = 0. The next step is to plot the stability switch diagram based on different delay values of our system.

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1 Dynamical and Nonlinearity of Laser Diode Circuits

!    −2 · U + τ · |P|2 + i · Fω ∂Reλ  ; Λ−1 (τ )  Λ (τ ) = = Re ∂τ λ=i·ω Fτ + i · 2 · V + ω · |P|2      2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 = 2  Fτ2 + 4 · V + ω · P 2 −1



The stability switch occurs only on those delay values (τ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . .; cos θ (τ ) = . . . when ω = ω+ (τ ) if only ω+ is feasible. Additionally, when all MRI system laser diode circuit parameters are known and the stability switch due to various time delay valuesτ is describe in the following expression:   sign Λ−1 (τ ) = sign{Fω (ω(τ ), τ )}· 

U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) sign τ · ωτ (ω(τ )) + ω(τ ) + |P(ω(τ ))|2 Remark We know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F function is independent on τ parameter F(ω, τ ) = 0. The results: we find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values ofω. F(ω) = 0 ⇒ ω1 , ω2 , . . . and next is to find those ω, τ values which fulfill sin θ (τ ) = . . . ; cos θ (τ ) = . . .. Finally, we plot the stability which diagram. −1



∂Reλ ∂τ



; g(τ ) g(τ ) = Λ (τ ) = λ=i·ω      2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 = 2  Fτ2 + 4 · V + ω · P 2    −1  ∂Reλ sign[g(τ )] = sign Λ (τ ) = sign ∂τ λ=i·ω        2 2 · Fω · V + ω · P − Fτ · U + τ · P 2 = sign  2 Fτ2 + 4 · V + ω · P 2 2  Fτ2 + 4 · V + ω · P 2 > 0        sign Λ−1 (τ ) = sign Fω · V + ω · P 2 − Fτ · U + τ · P 2

  F      Fτ τ sign Λ−1 (τ ) = sign [Fω ] · V + ω · P 2 − · U + τ · P2 ; ωτ = − ; ωτ Fω Fω  −1 ∂ F/∂ω ∂ω = =− ∂τ ∂ F/∂τ

1.3 MRI System Laser Diode Circuitry Dynamic and Stability Analysis Table 1.129 Sign of Λ−1 (τ ) for stability switching   sign[Fω ] τ ·U sign V +ω + ω + ω · τ τ 2 P

129

sign[Λ−1 (τ )]

±

±

+

±

∓

−

     sign Λ−1 (τ ) = sign [Fω ] · V + ωτ · U + ω · P 2 + ωτ · τ · P 2   

  V + ωτ · U 1 · sign Λ−1 (τ ) = sign [Fω ] · + ω + ω · τ τ P2 P2  

   V + ωτ · U 1 + ω + ω · τ sign 2 > 0 ⇒ sign Λ−1 (τ ) = sign [Fω ] · τ P P2   −1  V + ωτ · U + ω + ωτ · τ sign Λ (τ ) = sign[Fω ] · sign P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] −1 We check the sign  of Λ (τ ) according the following rule (Table 1.129).  −1 proceeds from (−) to (+) respectively If sign Λ (τ ) > 0 then   the crossing (stable to unstable). If sign Λ−1 (τ ) < 0 then the crossing proceeds from (+) to (−) respectively (unstable to stable). The stability switching can occur for specific ω values.

1.4 Dynamic of Electron-Photon Exchanges into VCSEL, Stability Optimization Under Delayed Carrier-Photon Interaction in Time In this sub chapter, we discuss the dynamic of electron-photon exchanges into VCSEL, stability optimization under delayed carrier-photon interaction in time. Vertical-cavity surface-emitting lasers (VCSELs) employ current-blocking layers to funnel the current to the active region located between the mirrors of the laser. The implementation depth is limited so that the lateral resistance can become substantial for devices with large-area contacts. First we describe the VCSEL DC characteristics. Above the diode threshold potential (typically 1.5–1.6 v), the VCSEL current has an almost linear dependence on the applied forward-bias voltage, specified by a resistance Rd . Lasing occurs when the injected current exceeds the VCSEL’s threshold current Ith , and the optical output power is linearly proportional to the input current by the slope efficiency factor η. VCSELs typically have thresholds currents below 1 mA and slope efficiencies of 0.3–0.5 mW/mA. At higher current densities (above

130

1 Dynamical and Nonlinearity of Laser Diode Circuits

10 mA), the slope efficiency decreases due to self-heating effects in the VCSEL. VCSELs can be designed to lase light of a variety of Datacom wavelengths. GaAsbased VCSELs operating at 850 nm designed for optical interconnects and short distances links are the most mature and common VCSEL technology. Longer wavelength GaAs and InP-based VCSELs lasing at 1.3 and 1.55 μm have developed for use in long-distance links due to lower loss and fiber dispersion at those wavelengths. The VCSEL structure is formed by two horizontal electrically conductive distributed Bragg reflector (DBR) surrounding an optical gain region. The highly reflective DBR mirrors provide the necessary optical feedback for lasing to occur. The gain region of most high speed VCSELs is constructed of several thin semiconductor layers that form multiple quanta well (MQW). The VCSEL’s optical characteristics and the emitted wavelength are thus determined by the gain profile of the quantum well material and the geometry of the DBR. VCSEL optical response is regulated by two coupled differential equations governing the carrier (N) and photon (S) density in the active region. When the laser is turned on, a current Ij flows into the laser’s gain region volume Vqw , increasing the carrier density. Photon generation begin as spontaneous emission (β), until the carrier density exceeds the threshold level, stimulated emission occurs, determined by the gain coefficient G. The time difference between the current being applied and the light emission is the laser’s turn on delay. N(t) and S(t) are the densities in the VCSEL active region of the carrier and photon respectively. Due to VCSEL active region parasitic effects, there is a time delay (τ) of the carrier-photon interaction time. We introduce the VCSEL carrier-photon interaction active region volume (Fig. 1.5) [7, 13]. Terminology: N—Carrier density in the active region. S—Photon density in the active region. N0 —Carrier density in the active region at t = 0. η—Injection efficiency. τn —Carrier recombination lifetime. τp —Photon lifetime. G—Gain coefficient. Ntr —Carrier’s transparency number. β—Spontaneous emission coupling coefficient. ε—Gain compression factor. I—VCSEL’s injection current. Ioff —VCSEL’s offset current. Fig. 1.5 VCSEL carrier-photon interaction active region volume

N0

S0 N, S N, S

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

131

τ—Time delay due to VCSEL active region parasitic effects. The VCSEL rate equation with delay parameter (τ) of the carrier-photon interaction time:   η · I − Io f f N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) d S(t) d N (t) = − ; − dt q τn 1 + ε · S(t − τ ) dt S(t) β · N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) =− + + τp τn 1 + ε · S(t − τ ) N = N (t); S = S(t) Optical output powerP0 = k·S; k-scaling factor accounting for the output coupling efficiency of the VCSEL. T = T0 + (I · V − P0 ) · Rth − τth ·

dT dt

Rth —VCSEL’s thermal impedance (which related the change in device temperature to the power dissipated as heat). τth —Thermal time constant (which is necessary to account for the nonzero response time of the device temperature, observed to be on the order of 1 μs). T0 —Ambient temperature. V —Laser voltage. term disappears. Expression (I · V − P0 ) models the Under DC condition dT dt power dissipated in the VCSEL, where we assume that any power not carried in the optical output is dissipated as heat in the device. We describe the key VCSEL parameters as functions of temperature, in particular the laser gain and the thermal physics of the device. We partition the thermal threshold current into a constant value of threshold current Itho plus an empirical offset current Io f f (T ).   Po = η · I − Itho − Io f f (T ) All static thermal effects are accounted for via the offset current model. This offset current using a polynomial function of temperature. Io f f (T ) =

∞ $

ak · T k = a0 + a1 · T + a2 · T 2 + a3 · T 3 + a4 · T 4 + . . .

k=0

It fit of the threshold current Ith (T ) to the expression Itho + Io f f (T ). The equilibrium points (fixed points) of our system ddtN = 0; ddtS = 0; limt→∞ S(t − τ ) = S; limt→∞ N (t − τ ) = N . We define as VCSEL fixed points: N ( j) , S ( j) ; j = 0, 1, 2, . . . index of VCSEL fixed point is (j).

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1 Dynamical and Nonlinearity of Laser Diode Circuits

    G · N ( j) − Ntr · S ( j) η · I − Io f f S ( j) N ( j) − = 0; − − q τn 1 + ε · S ( j) τp  ( j)  ( j) G · N − Nn · S β · N ( j) + + =0 τn 1 + ε · S ( j)   τn · η · I − Io f f τn τn ( j) ( j) ·S − N = ; A= τ p · (β − 1) τ p · (β − 1) (β − 1) · q   τn · η · I − Io f f B= (β − 1) · q     A = A τn , τ p , β ; B = B τn , η, β, I, Io f f ; N ( j) = A · S ( j) − B 

      2  A · ε η · I − Io f f A B ( j) S ·ε · +G· A +S · − G · B − G · Ntr − + τn τn q τn     η · I − Io f f B − =0 + q τn ( j)

  η · I − Io f f A B A·ε ε 1 ; ; = = = τn τ p · (β − 1) τn τ p · (β − 1) τn (β − 1) · q

2  ε + G · τ n · τ p · (β − 1)       τn · η · I − Io f f η · I − Io f f β 1 − G · Ntr − ·ε −G· · + S ( j) · τ p · (β − 1) q (β − 1) · q (β − 1)   η · I − Io f f β =0 · − q (β − 1)



S ( j)

  η · I − Io f f β ε + G · τn ; 3 = − · ; 2 1 = τ p · (β − 1) q (β − 1)     τn · η · I − Io f f η · I − Io f f β 1 −G· · G · Ntr − ·ε = τ p · (β − 1) q (β − 1) · q (β − 1)  ( j) 2 S · 1 + S ( j) · 2 + 3 = 0; N ( j) ≥ 0; S ( j) ≥ 0

S

( j)

=

−2 ±



22 − 4 · 1 · 3

( j)

−2 = ± 2 · 1

; S 2 · 1   A · 2 − 4 ·  ·  1 3 2 2 · A =− +B ± 2 · 1 2 · 1

22 − 4 · 1 · 3 2 · 1

; N ( j)

We can check our Electron-photon exchanges into VCSEL system for τ = 0; τ > 0.The standard local stability analysis about any one of the equilibrium points of the Electron-photon exchanges into VCSEL system consists in adding to coordinate

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

133

[S, N] arbitrarily small increments of exponential form [s, n] · eλt and retaining the first order terms in S, N. The system of two homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below carrier density in the active region (N) and photon density in the active region (S) and derivatives with respect to time into VCSEL rate equation with delay parameter (τ) of the carrier-photon interaction time. Electronphoton exchanges into VCSEL system fixed values with arbitrarily small increaments of exponential form [s, n] · eλt are   j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc., S(t) = S ( j) + s · eλ·t ; N (t) = N ( j) + n · eλ·t . We choose these expressions for ourselves S(t), N(t) as a small displacement [s, n] from Electron-photon exchanges into VCSEL system fixed points in time t = 0. S(t = 0) = S ( j) + s; N (t = 0) = N ( j) + n For t > 0; λ < 0 the selected fixed point is stable otherwise t > 0; λ > 0 is unstable. Our system tends to the selected fixed point exponentially for t > 0; λ < 0 otherwise go away from the selected fixed point (equilibrium point) exponentially. Eigenvalue λ parameter is established if the fixed point is stable or unstable; additionally, his absolute value |λ| establishes the speed of flow toward or away from the selected fixed point. The speed of flow toward or away from the selected fixed point for VCSEL rate equations with delay parameter (τ) system with respect to time are = λ · s · eλ·t ; d Ndt(t) = λ · n · eλ·t . as follow: d S(t) dt Additionally S(t − τ ) = S ( j) + s · eλ(t−τ ) = S ( j) + s · eλ·t · e−λ·τ ; N (t − τ ) = N ( j) + n · eλ·(t−τ ) = N ( j) + n · eλ·t · e−λ·τ d S(t − τ ) d N (t − τ ) = λ · s · ei·t · e−λ·τ ; = λ · n · eλ·t · e−λ·τ . dt dt   η · I − Io f f N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) d N (t) = − − dt q τn 1 + ε · S(t − τ )   ( j)   N + n · eλ·t η · I − Io f f λ·t λ·n·e = − q τ  n ( j)   ( j) λ·t −λ·τ − Ntr · S + s · eλ·t · e−λ·τ G· N +n·e ·e   − 1 + ε · S ( j) + s · eλ·t · e−λ·τ   η · I − Io f f N ( j) n · eλ·t λ·t − − λ·n·e = q τn τ   ( j)n   ( j) λ·t −λ·τ − Ntr · S + s · eλ·t · e−λ·τ G· N +n·e ·e   − 1 + ε · S ( j) + s · eλ·t · e−λ·τ

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1 Dynamical and Nonlinearity of Laser Diode Circuits



   N ( j) + n · eλ·t · e−λ·τ − Ntr · S ( j) + s · eλ·t · e−λ·τ = N ( j) · S ( j) + N ( j) · s · eλ·t · e−λ·τ + n · eλ·t · e−λ·τ · S ( j) + n · s · e2·λ·t · e−2·λ·τ − Ntr · S ( j) − Ntr · s · eλ·t · e−λ·τ

λ·n·e

λ·t

  η · I − Io f f n · eλ·t N ( j) = − − q τn τn  ( j) ( j) ( j) G · N · S + N · s · eλ·t · e−λ·τ + n · eλ·t · e−λ·τ · S ( j) +n · s · e2·λ·t · e−2·λ·τ − Ntr · S ( j) − Ntr · s · eλ·t · e−i·τ   − 1 + ε · S ( j) + s · eλ·t · e−λ·τ

→ → 0 ⇒ n · s · e2·λ·t · e−2·λ·τ  ( j) η · I − Io f f N n · eλ·t − = − q τn τn  ( j) ( j) ( j) G · N · S + N · s · eλ·t · e−λ·τ + n · eλ·t · e−λ·τ · S ( j) −Ntr · S ( j) − Ntr · s · eλ·t · e−λ·τ   − 1 + ε · S ( j) + s · eλ·t · e−λ·τ

n · s λ·n·e

λ·t

  η · I − Io f f N ( j) n · eλ·t − λ·n·e = − q τn τn   ( j) ( j)  ( j) G · N · S + N · s + n · S ( j) · eλ·t · e−λ·τ −Ntr · S ( j) − Ntr · s · eλ·t · e−λ·τ   − 1 + ε · S ( j) + ε · s · eλ·t · e−λ·τ !   1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ  ·  1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ   η · I − Io f f N ( j) n · eλ·t λ·t λ·n·e = − − q τn τn   ( j) ( j)  ( j) ( j) · s + n · S  · eλ·t · e−λ·τ − Ntr · S ( j) G · N · S + N  −Ntr · s · ei·t · e−λ·τ · 1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j) − ε2 · s 2 · e2·λ·t · e−2·λ·τ λ·t

s2 → 0 λ·n·e

λ·t

  η · I − Io f f N ( j) n · eλ·t − = − q τn τn

0

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

135

   · s + n · S ( j)  · eλ·t · e−λ·τ − Ntr ·S ( j) G · N ( j) · S ( j) + N( j) −Ntr · s · eλ·t · e−λ·τ · 1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j)  ( j) ( j)  ( j)   N · S + N · s + n · S ( j) · eλ·t · e−λ·τ − Ntr · S ( j) − Ntr · s · eλ·t · e−λ·τ ·   1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ   = N ( j) · S ( j) · 1 + ε · S ( j)       + N ( j) · s + n · S ( j) · 1 + ε · S ( j) · eλ·t · e−λ·τ − Ntr · S ( j) · 1 + ε · S ( j)   − Ntr · 1 + ε · S ( j) · s · eλ·t · e−λ·τ − N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ   − N ( j) · s 2 + n · s · S ( j) · eλ·t · e−λ·τ · ε · eλ·t · e−λ·τ + Ntr · S ( j) · ε · s · eλ·t · e−λ·τ + Ntr · s 2 · e2·λ·t · e−2·λ·τ · ε s 2 → 0; n · s → 0 

   N ( j) · S ( j) + N ( j) · s + n · S ( j) · eλ·t · e−λ·τ − N p · S ( j) − Ntr · s · eλ·t · e−λ·τ ·    1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ       = N ( j) · S ( j) · 1 + ε · S ( j) + N ( j) · s + n · S ( j) · 1 + ε · S ( j) · eλ·t · e−λ·τ   − Ntr · S ( j) · 1 + ε · S ( j)   − Nn · 1 + ε · S ( j) · s · eλ·t · e−λ·τ − N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ + N p · S ( j) · ε · s · eλ·t · e−λ·τ

  η · I − Io f f N ( j) n · eλ·t − λ·n·e = − q τn τn   ( j)   ( j) ( j)  ( j) + N ·s + n · S ( j)  G · N · S  · 1 + ε · S · 1 + ε · S ( j) · eλ·t · e−λ·τ − Ntr · S ( j) · 1 + ε · S ( j) −Ntr · 1 + ε · S ( j) · s · eλ·t ·e−λ·τ − N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ +Ntr · S ( j) · ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j)   η · I − Io f f N ( j) n · eλ·t λ·t − λ·n·e = − q τn τn   ( j) ( j)  ( j) G · N · S + N  · s + n · S( j) · eλ·t · e−λ·τ − Ntr · S ( j) −Ntr · s · eλt · e−λ·τ · 1 + ε· S ( j) − N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ +Ntr · S ( j) · ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j) λ·t

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1 Dynamical and Nonlinearity of Laser Diode Circuits

  η · I − Io f f n · eλ·t N ( j) λ·n·e = − − q τn τn   ( j) ( j)  ( j) G · N · S + N · s + n · S ( j) · eλ·t · e−λ·τ − N p · S ( j) −Ntr · s · eλ·t · e−λ·τ · 1 + ε · S ( j) −  2 1 + ε · S ( j)   G · −N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ + No · S ( j) · ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j)   η · I − Io f f N ( j) n · ei·t λ·t − λ·n·e = − q τn τn  ( j)  ( j)   ( j) G · N − Ntr · S + N · s + n · S ( j) · eλ·t · e−λ·τ −Nn · s · eλ·t · e−λ·τ   − 1 + ε · S ( j)   G · −N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ + Ntr · S ( j) · ε · s · eλ·t · e−λ·τ −  2 1 + ε · S ( j)     η · I − Io f f G · N ( j) − Ntr · S ( j) N ( j) n · eλ·t λ·t   − λ·n·e = − − q τn τn 1 + ε · S ( j)    ( j) ( j) − Ntr · s G· N ·s+n·S   − · eλ·t · e−λ·τ 1 + ε · S ( j)   G · Ntr − N ( j) λ·t −λ·τ ( j) −  2 · S · ε · s · e · e j) ( 1+ε·S λ·t

At fixed points: limt→∞ N (t − τ ) = N (t); limt→∞ S(t − τ ) = S(t); t  τ ⇒ t −τ ≈t     η · I − Io f f G · N ( j) − Nr · S ( j) N ( j)   − − =0 q τn 1 + ε · S ( j)    G · N ( j) · s + n · S ( j) − Ntr · s n   λ·n =− − · e−λ·τ τn 1 + ε · S ( j)   G · Ntr − N ( j) −λ·τ ( j) −  2 · S · ε · s · e j) ( 1+ε·S    G · N ( j) · s + n · S ( j) − Ntr · s n   −λ·n− − · e−λ·τ τn 1 + ε · S ( j)   G · Ntr − N ( j) −λ·τ ( j) −  =0 2 · S · ε · s · e j) ( 1+ε·S

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …





G · S ( j) 1  · e−λ·τ + −λ − τn 1 + ε · S ( j)   S ( j) · ε  · s · e−λ·τ = 0 1−  1 + ε · S ( j)



  G · N ( j) − Ntr  · ·n−  1 + ε · S ( j)

S(t) β · N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) d S(t) =− + + dt τp τn 1 + ε · S(t − τ )   ( j) β · N + n · eλt S ( j) + s · eλ·t λ·t λ·s·e =− + τp τn   ( j)   ( j) λ·t −λ·τ − Ntr · S + s · eλ·t · e−λ·τ G· N +n·e ·e   + 1 + ε · S ( j) + s · eλ·t · e−λ·τ S ( j) β · N ( j) s · eλ·t β · n · eλ·t λ · s · eλ·t = − + − + τp τn τp τn   ( j)   ( j) λ·t −λ·τ − Ntr · S + s · eλ·t · e−λ·τ G· N +n·e ·e   + 1 + ε · S ( j) + s · eλ·t · e−λ·τ We already got the expression for

G·( N ( j) +n·eλ·t ·e−λ·τ −Nn )·( S ( j) +s·eλ·t ·e−λ·τ ) 1+ε·( S ( j) +s·eλ·t ·e−λ·τ )

    G · N ( j) + n · eλ·t · e−λ·τ − Ntr · S ( j) + s · eλ·t · e−λ·τ   1 + ε · S ( j) + s · eλ·t · e−λ·τ   ( j) ( j)  ( j) · s + n · S ( j)  · eλ·t · e−λ·τ − Ntr ·S ( j) G · N · S + N  −Ntr · s · eλ·t · e−λ·τ · 1 + ε · S ( j) − ε · s · eλ·t · e−λ·τ =  2 1 + ε · S ( j) s 2 → 0; n · s → 0     G · N ( j) + n · ei·t · e−λ·τ − Ntr · S ( j) + s · eλ·t · e−λ·τ   1 + ε · S ( j) + s · eλ·t · e−λ·τ     ( j) ( j)  ε · S ( j) + N ( j) · s + n · S ( j) G · N (·j)S ·λ·t1 +−λ·τ · e ·e − Ntr · S ( j) · 1 + ε · S ( j) 1 + ε · S ( j) · s · ei·t ·e−λ·τ − N ( j) · S ( j) · ε · s · eλ·t · e−λ·t −Ntr · 1 + ε · S λ·t ( j) +No · S · ε · s · e · e−λ·τ =  2 1 + ε · S ( j)     G · N ( j) + n · eλ·t · e−λ·τ − Nr · S ( j) + s · eλ·t · e−λ·τ   1 + ε · S ( j) + s · eλ·t · e−λ·τ

137

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1 Dynamical and Nonlinearity of Laser Diode Circuits

    G · N ( j) − Ntr · S( j) + N ( j) · s + n · S ( j) · eλ·t · e−λ·t −Ntr · s · eλ·t · e−λ·τ   = 1 + ε · S ( j)   G · −N ( j) · S ( j) · ε · s · eλ·t · e−λ·τ + Ntr · S ( j) · ε · s · eλ·t · e−λ·τ +  2 1 + ε · S ( j)       G · N ( j) + n · eλt · e−λ·τ − Nn · S ( j) + s · eλ·t · e−λ·τ G · N ( j) − Ntr · S ( j)     = 1 + ε · S ( j) + s · eλ·t · e−λ·τ 1 + ε · S ( j)    G · N ( j) · s + n · S ( j) − Ntr · s   + · eλ·t · e−λ·τ 1 + ε · S ( j)   G · Ntr − N ( j) λ·t −λ·τ ( j) +  2 · S · ε · s · e · e 1 + ε · S ( j)   G · N ( j) − Ntr · S ( j) S ( j) β · N ( j) s · eλ·t β · n · eλ·t λ·t   λ·s·e =− + − + + τp τn τp τn 1 + ε · S ( j)    G · N ( j) · s + n · S ( j) − Ntr · s   + · eλ·t · e−λ·τ 1 + ε · S ( j)   G · Ntr − N ( j) λ·t −λ·τ ( j) +  2 · S · ε · s · e · e 1 + ε · S ( j)   G · N ( j) − Ntr · S ( j) S ( j) β · N ( j) s · eλ·t β · n · eλ·t λ·t   λ·s·e =− + + − + τp τn τn τp 1 + ε · S ( j)    ( j) ( j) − Ntr · s G· N ·s+n·S   + · eλ·t · e−λ·τ j) ( 1+ε·S   G · Ntr − N ( j) λ·t −λ·τ ( j) +  2 · S · ε · s · e · e j) ( 1+ε·S At fixed points: limt→∞ N (t − τ ) = N (t); limt→∞ S(t − τ ) = S(t); t  τ ⇒ t −τ ≈t   G · N ( j) − Ntr · S ( j) β · N ( j) S ( j)   + + − =0 τp τn 1 + ε · S ( j)    G · N ( j) · s + n · S ( j) − Ntr · s β · n · eλ·t s · eλ·t λ·t   λ·s·e = − + · eλ·t · e−λ·τ τn τp 1 + ε · S ( j)   G · Ntr − N ( j) λ·t −λ·τ ( j) +  2 · S · ε · s · e · e 1 + ε · S ( j)

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

139

   G · N ( j) · s + n · S ( j) − Ntr · s β ·n s   λ·s = − + · e−λ·τ τn τp 1 + ε · S ( j)   G · Ntr − N ( j) −λ·τ ( j) +  2 · S · ε · s · e j) ( 1+ε·S   G · S ( j) β −λ·τ  ·e + ·n τn 1 + ε · S ( j)       G · N ( j) − Ntr S ( j) · ε 1  ·   · e−λ·τ · s = 0 + 1−  + −λ − τp 1 + ε · S ( j) 1 + ε · S ( j) We can summarize our last results:    G · S ( j) 1 −λ·τ  ·e + −λ − · τn 1 + ε · S ( j)     G · N ( j) − Ntr S ( j) · ε  · 1−   · s · e−λ·τ = 0 n−  1 + ε · S ( j) 1 + ε · S ( j)   β G · S ( j) −λ·τ  ·e + ·n τn 1 + ε · S ( j)       G · N ( j) − Ntr S ( j) · ε 1  ·   · e−λ·τ · s = 0 + 1−  + −λ − τp 1 + ε · S ( j) 1 + ε · S ( j) The small increments Jacobian of our Electron-photon exchanges into VCSEL system is as follow: ⎛

     G· N ( j) −Ntr ( j) 1 +  G·S ( j)  · e−λ·τ   −λ − − · 1 −  S ·ε( j)  · e−λ·t ⎜ j) j) ( ( τ n 1+ε·S 1+ε·S 1+ε·S ⎜   ⎜   ⎝ G· N ( j) −Ntr ( j) ( j) ·ε β G·S 1 S −λ·τ    ·   · e−λ·τ −λ − τ p + 1 −  ( j) · e ( j) ( j) τn + 1+ε·S

1+ε·S

⎞ ⎟ ⎟ ⎟· ⎠

  n =0 s

1+ε·S

(A − λ · I )       ⎞ ⎛ G· N ( j) −N G·S ( j) S ( j) ·ε −λ·τ −λ·t − 1+ε·S ( j) tr · 1 − 1+ε·S −λ − τ1n + 1+ε·S ( j) ) · e ( j) ) · e ( ( ) ( ⎠     =⎝ G· N ( j) −N β G·S ( j) S ( j) ·ε −λ·τ · 1+ε·S ( j) tr · e−λ·τ −λ − τ1p + 1 − 1+ε·S ( j) ) τn + (1+ε·S ( j) ) · e ( ( )

(A − λ · I ) = 0 ⎛ ⎜ −λ − ⎜ ⇒ det ⎜ ⎜ ⎝ β

τn



1 τn

+

+

G·S ( j) (1+ε·S ( j) )

G·S ( j) (1+ε·S ( j) )

·e

−λ·τ

· e−λ·τ



( j)

−Ntr ) − G·(N · [1 (1+ε·S ( j) ) ( j) S ·ε .− (1+ε·S ( j) ) ] · e−λ·τ S ( j) ·ε −λ − τ1p + (1 − (1+ε·S ( j) ) ) ( j)

−Ntr ) · e−λ·τ · G·(N (1+ε·S ( j) )

⎞ ⎟ ⎟ ⎟=0 ⎟ ⎠

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1 Dynamical and Nonlinearity of Laser Diode Circuits

    S ( j) ·ε · For simplicity we define the function: g1 S ( j) , N ( j) = 1 − 1+ε·S ( j) ( ) G·( N ( j) −Ntr ) (1+ε·S ( j) )   G · S ( j)  g2 S ( j) , N ( j) =  1 + ε · S ( j)



det(A − λ · I ) = 0 ⎞ ⎛       −g1 S ( j) , N ( j) · e−λ·τ −λ − τ1n + g2 S ( j) , N ( j) · e−λ·τ ⎠=0 ⇒ det ⎝  ( j) ( j)  −λ·τ   β −λ − τ1p + g1 S ( j) , N ( j) · e−λ·τ ·e τn + g2 S , N





  ( j) ( j)  −λ·τ  ( j) ( j)  −λ·τ 1 1 −λ − · −λ − ·e ·e + g2 S , N + g1 S , N τn τp 

      β − + g2 S ( j) , N ( j) · e−λ·τ · −g1 S ( j) , N ( j) · e−λ·τ = 0 τn   1 1 − λ · g1 S ( j) , N ( j) · e−λ·τ + ·λ τp τn   1 1 + − · g1 S ( j) , N ( j) · e−λ·τ τn · τ p τn  ( j) ( j)  −λ·τ   ·e + g2 S , N · λ + g2 S ( j) , N ( j) · e−λ·τ ·     1 − g2 S ( j) , N ( j) · g1 S ( j) , N ( j) · e−2·λ·τ τp     β + · g1 S ( j) , N ( j) · e−λ·τ + g2 S ( j) , N ( j) · τ  n  g1 S ( j) , N ( j) · e−2·λ·τ = 0

λ2 + λ ·

      1 1 1 + λ +λ· + + g2 S ( j) , N ( j) − g1 S ( j) , N ( j) · λ τp τn τn · τ p  ( j) ( j)  1  ( j) ( j)   ( j) ( j)  1 β · − · e−λ·τ = 0 + · g1 S , N · g1 S , N +g2 S , N τp τn τn 

2

The characteristic equation D(λ, τ ) and we study the occurrence of any possible stability switching, resulting from the increase of the value of the time delay τ for general characteristic equationD(λ, τ ) [3, 4]. D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−τ ·λ ; n, m ∈ N0 ; n > m; n = 2; m = 1  1 1 1 + + τp τn τn · τ p      Q m (λ, τ ) = g2 S ( j) , N ( j) − g1 S ( j) , N ( j) · λ 

Pn (λ, τ ) = λ2 + λ ·

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

141

  1     1 β + g2 S ( j) , N ( j) · + · g1 S ( j) , N ( j) − · g1 S ( j) , N ( j) τp τn τn Pn (λ, τ ) =

n=2 $

Pk (τ ) · λk = P0 (τ ) + P1 (τ ) · λ + P2 (τ ) · λ2 ; P0 (τ )

k=0

=

1 1 1 ; P1 (τ ) = + τn · τ p τp τn

P2 (τ ) = 1; Q m (λ, τ ) =

m=1 $

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ

k=0

  1   β + · g1 S ( j) , N ( j) q0 (τ ) = g2 S ( j) , N ( j) · τp τn  ( j) ( j)      1 − · g1 S , N ; q1 (τ ) = g2 S ( j) , N ( j) − g1 S ( j) , N ( j) τn The homogeneous system for S, N leads to a characteristic equation for the eigenvalue λ having the form P(λ, τ ) + Q(λ, τ ) · e−τ ·λ = 0; P(λ) =

2 $

a j · λ j ; Q(λ) =

j=0

1 $

cj · λj

j=0

  The coefficients a j (qi , qk , τ ), c j (qi , qk , τ ) ∈ R depend on qi , qk and delay τ . Parameters qi , qk are any electron-photon exchanges into VCSEL system’s global parameters, other parameters kept as a constant. a0 =

1 1 1 ; a1 = + ; a2 = 1 τn · τ p τp τn

  1   β c0 = g2 S ( j) , N ( j) · + · g1 S ( j) , N ( j) τp τn       1 − · g1 S ( j) , N ( j) ; c1 = g2 S ( j) , N ( j) − g1 S ( j) , N ( j) τn Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, aj , and cj . The coefficients aj , cj are continuous, and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−τ ·λ = 0. a0 + c0 =

  1     1 1 β + g2 S ( j) , N ( j) · + · g1 S ( j) , N ( j) − · g1 S ( j) , N ( j) τn · τ p τp τn τn

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis can be verified in the present case. 1. If 1λ = i1 · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0. 1 1 is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. 2. If 1 Q(λ) P(λ) 1 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real numberω :Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0. Pn (λ = i · ω, τ ) = −ω2 +

1 +i ·ω· τn · τ p



1 1 + τp τn



     1   Q m (λ = i · ω, τ ) = i · ω · g2 S ( j) , N ( j) − g1 S ( j) , N ( j) + g2 S ( j) , N ( j) · τp  ( j) ( j)   ( j) ( j)  1 β − + · g1 S , N · g1 S , N τn τn Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )   1   1 1 = −ω2 + + g2 S ( j) , N ( j) · + [β − 1] · · g1 S ( j) , N ( j) τn · τ p τp τn    ( j) ( j)   ( j) ( j)  1 1 + g2 S , N + − g1 S , N +i ·ω· τp τn 2    1 1 1 2 2 2 2 +ω · + = ω4 |P(i · ω, τ )| = −ω + τn · τ p τp τn   1 1 1 1 2 − 2 · ω2 · + 2 2 + ω2 · + τn · τ p τn · τ p τp τn    1 1 2 1 1 2 4 2 |P(i · ω, τ )| = ω + ω · + −2· + 2 2 τp τn τn · τ p τn · τ p   1 1 1 2 4 2 |P(i · ω, τ )| = ω + ω · + 2 + 2 2 2 τp τn τn · τ p   2   |Q(i · ω, τ )|2 = ω2 · g2 S ( j) , N ( j) − g1 S ( j) , N ( j)   ( j) ( j)  1  ( j) ( j)  2 1 + g2 S , N + (β − 1) · · g1 S , N · τp τn F(ω, τ ) = P(i · ω, τ )|2 − |Q(i · ω, τ )|2

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

 =ω +ω · 4

+

2

1 1 + 2 2 τp τn



143

  2   − g2 S ( j) , N ( j) − g1 S ( j) , N ( j)

!

  ( j) ( j)  1  ( j) ( j)  2 1 1 S · S − g , N + − 1) · · g , N (β 2 1 τn2 · τ p2 τp τn

We define the following parameters for simplicity:0 , 2 , 4 .   ( j) ( j)  1  ( j) ( j)  2 1 1 0 = 2 2 − g2 S , N · + (β − 1) · · g1 S , N τn · τ p τp τn     2   1 1 2 = + 2 − g2 S ( j) , N ( j) − g1 S ( j) , N ( j) ; 4 = 1 2 τp τn Hence F(ω, τ ) = 0 implies

2 

2,k · ω2·k = 0 and its roots are given by solving

k=0

the above polynomial. 1 ; PI (i · ω, τ ) = ω · PR (i · ω, τ ) = −ω + τn · τ p



2

1 1 + τp τn



  1   1 Q R (i · ω, τ ) = g2 S ( j) , N ( j) · + (β − 1) · · g1 S ( j) , N ( j) τp τn Q I (i · ω, τ ) = ω · [g2 (S ( j) , N ( j) ) − g1 (S ( j) , N ( j) )] sin θ (τ ) =

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2

cos θ (τ ) = −

PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) |Q(i · ω, τ )|2

We use different parameters terminology from our last characteristics parameter definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 2; m = 1; n > m; Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

2 $ j=0

a j · λ j = ao + a1 · λ + a2 · λ2 ; Q(λ) =

1 $

c j · λ j = c0 + c1 · λ

j=0

n, m ∈ N0 ; n > m; a j , c j : R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “-” denotes complex

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1 Dynamical and Nonlinearity of Laser Diode Circuits

and conjugate. P(λ),  Q(λ) are analytic functions in λ and differentiable in τ . The coefficients a j η, τn , τ p , G, β, ε, . . . ∈ R and c j η, τn , τ p , G, β, ε, . . . ∈ R depend on electron-photon exchanges into VCSEL system’s η, τn , τ p , G, β, ε, . . . values. Unless strictly necessary, the designation of the varied arguments: η, τn , τ p , G, β, ε, . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous, and differentiable functions of their argument, and direct substitution show that a0 + c0 = 0 [2, 3]. a0 =

  1     1 1 β ; c0 = g2 S ( j) , N ( j) · + · g1 S ( j) , N ( j) − · g1 S ( j) , N ( j) τn · τ p τp τn τn   1   1 β + g2 S ( j) , N ( j) · + · g1 S ( j) , N ( j) τn · τ p τp τn   1 − · g1 S ( j) , N ( j) = 0 ∀ η, τn , τ p , G, β, ε, . . . τn

i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis scan also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire η, τn , τ p , G, β, ε, . . . domain of interest. P(λ) is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (b) Q(λ) in the limit:   ( j) ( j)      g2 S , N − g1 S ( j) , N ( j) · λ + g2 S ( j) , N ( j) · 1 11 + β ·g S ( j) ,N ( j) − 1 ·g S ( j) ,N ( j) 11 1 )1 1 1 τn 1 (  ) τn 1 ( 1 =1 in the limit : 1 Q(λ) 1 1 1 1 P(λ) 1 2 λ +λ·

τp

1 τp

+ τn + τn ·τ p

(c) The following expressions exist: F(ω, τ ) =P(i · ω, τ )|2 − |Q(i · ω, τ )|2 F(ω, τ ) = P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = 2k=0 2·k · ω2·k Has at most a finite number of zeros. in ω (degree in ω4 ).   Indeed, this is a polynomial (d) Each positive root ω η, τn , τ p , G, β, ε, τ, . . . of F(ω) = 0 is continuous and differentiable with respect to η, τn , τ p , G, β, ε, τ, . . .. The condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i · ω), and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ - plane, whereby increasing the parameters: η, τn , τ p , G, β, ε, τ, . . . Re λ may, at the crossing change its sign from (−) to (+), i.e. from a stable focus:

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

145

  E ( j) N ( j) , S ( j) ⎛ ⎞   A · 2 − 4 ·  ·  22 − 4 · 1 · 3 1 3 −2 2  · A 2 ⎠ = ⎝− +B ± , ± 2 · 1 2 · 1 2 · 1 2 · 1 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to η, τn , τ p , G, β, ε, τ, . . . and system parameters. −1



Λ (τ ) =

∂Reλ ∂τ

 λ=i·ω

; η, τn , τ p , G, β, ε, . . .

 ∂Reλ ; τn , τ p , G, β, ε, τ, . . . ∂η λ=i·ω   ∂Reλ Λ−1 (τn ) = ; η, τ p , G, β, ε, τ, . . . ∂τn λ=i·ω     ∂Reλ −1 Λ τp = ; η, τn , G, β, ε, τ, . . . ∂τ p λ=i·ω   ∂Reλ Λ−1 (G) = ; η, τn , τ p , β, ε, τ, . . . ∂G λ=i·ω   ∂Reλ −1 Λ (β) = ; η, τn , τ p , G, ε, τ, . . . ∂β λ=i·ω   ∂Reλ −1 Λ (ε) = ; η, τn , τ p , G, β, τ, . . . ∂ε λ=i·ω Λ−1 (η) =



P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ) When writing and inserting λ = i ·ω; ω ∈ R into electron-photon exchanges into VCSEL system’s characteristic equation, ω must satisfy the following equations: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω)|2 = 0 in the view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equations that by squaring and adding the sides, ω must be a positive root of F(ω, τ ) = P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = 0. Note: In case that F(ω) is dependent on τ . It is important to notice that if τ ∈ / I

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1 Dynamical and Nonlinearity of Laser Diode Circuits

(assume that I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ / I, ω(τ ) is not defined). Then for all τ ∈ I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches [5]. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angel θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . . ; cos θ (τ ) = . . . sin θ (τ ) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos θ (τ ) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

And the relation between the argument θ(τ ) and τ · ω(τ ) for τ ∈ I must be ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ R0 . Hence we can define the maps τn : I → R+0 given by τn (τ ) = θ(τ )+2·n·π ; n ∈ R0 ; τ ∈ I . Let us introduce the functions: I → R ω(τ ) Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ R0 that is continuous and differentiable in τ . In the following, the subscripts λ, ω, η, τn , τ p , G, β, ε, . . . indicate the corresponding partial derivatives. on Λ(x), remember  Let us first concentrate   in λ η, τn , τ p , G, β, ε, . . . and ω η, τn , τ p , G, β, ε, . . . , and keeping all parameters except one (x) and τ . Differentiating electron-photon exchanges into VCSEL system’s characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = η, τn , τ p , G, β, ε, . . . 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

∂ P(λ,x) ∂ Q(λ,x) ; Q λ (λ, x) ; Px (λ, x) Where Pλ (λ, x) = = = ∂λ ∂λ ∂ P(λ,x) ∂ Q(λ,x) ; Q x) = (λ, x ∂x ∂x Substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω); i · Pλ (i · ω) = Pω (i · ω) i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains:  −1 11 ∂λ 1 1 1 ∂x   λ=i·ω i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 = Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = QR + i · QI Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

147

Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 When (x) can be any electron-photon exchanges into VCSEL system’s parameter η, τn , τ p , G, β, ε, . . . and time delayτ , etc. Where for conve= nient, we have dropped the arguments (i · ω, x) and where Fω = 2 · 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] Fx [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ .   1 1 1 ; PI = ω · + τn · τ p τp τn  ( j) ( j)  1   1 · Q R = g2 S , N + (β − 1) · · g1 S ( j) , N ( j) τp τn PR = −ω2 +

     1 1 Q I = ω · g2 S ( j) , N ( j) − g1 S ( j) , N ( j) ; PRω = −2 · ω; PI ω = + ; Q Rω = 0 τp τn

    Q I ω = g2 S ( j) , N ( j) − g1 S ( j) , N ( j) ; PRτ = 0 Fτ PI τ = 0; Q Rτ = 0; Q I τ = 0; ωτ = − Fω   1 2 ; Q Rω · Q R = 0; Fτ PRω · PR = −2 · ω · −ω + τn · τ p = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0  1 1 2 + ; Q I ω· τp τn   2   Q I = ω · g2 S ( j) , N ( j) − g1 S ( j) , N ( j) 

PI ω · PI = ω ·

      1 2 · g2 S ( j) , N ( j) − g1 S ( j) , N ( j) ; PI · PRω PR · PI ω = −ω + τn · τ p   1 1 2 = −2 · ω · + τp τn    1   1 + (β − 1) · · g1 S ( j) , N ( j) · Q R · Q I ω = g2 S ( j) , N ( j) · τp τn  ( j) ( j)    ( j) ( j)  − g1 S , N g2 S , N

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1 Dynamical and Nonlinearity of Laser Diode Circuits

Q I · Q Rω = 0; V = 0. F(ω, τ ) = 0, differentiating with respect to τ and we get Fτ ∂ω ∂ω + Fτ = 0; τ ∈ I ⇒ = − ; Λ−1 (τ ) Fω · ∂τ ∂τ Fω   ∂Reλ ∂ω Fτ = ωτ = − = ; ∂τ λ=i·ω ∂τ Fω !

Λ

−1

 −2 · U + τ · |P|2 + i · Fω   (τ ) = Re Fτ + i · 2 · V + ω · |P|2

 

 2 0 ∂Reλ ; sign Λ−1 (τ ) = sign ∂τ λ=i·ω

V + ∂ω ·U ∂ω ∂τ ·τ sign Λ (τ ) = sign{Fω } · sign +ω+ 2 |P| ∂τ 

−1



!

We shall presently examine the possibility of stability transitions (bifurcations) electron-photon exchanges into VCSEL system, about the equilibrium points   E ( j) N ( j) , S ( j) ⎛ ⎞   A · 2 − 4 ·  ·  2  − 4 ·  ·  1 3 1 3 2 2 2 · A −2 ⎠ = ⎝− +B ± , ± 2 · 1 2 · 1 2 · 1 2 · 1 j = 0, 1, 2, . . . as a result of a variation of delay parameterτ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ—plane, where by increasing the delay parameterτ , Re λ may at the crossing, changes its sign from – to +, i.e. from a stable focus E(∗) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ . Λ−1 (τ ) =



∂Reλ ∂τ

 λ=i·ω

; η, τn , τ p , G, β, ε, . . .

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω )      ( j) ( j)   ( j) ( j)  1 1 1 2 2 · g2 S , N U = −ω + − g1 S , N +2·ω · + τn · τ p τp τn   ( j) ( j)  1  ( j) ( j)  1 − g2 S , N · · + (β − 1) · · g1 S , N τp τn     ( j) ( j)  − g1 S ( j) , N ( j) g2 S , N Then we get the expression for F(ω, τ ) electron-photon exchanges into VCSEL system’s parameter values. We find those ω, τ values which fullfill F(ω, τ ) = 0.

1.4 Dynamic of Electron-Photon Exchanges into VCSEL …

149

We ignore negative, complex, and imaginary values of ω for specific τ values. In our case F(ω, τ ) is independent on τ values. We can express by 3D or 2D function F(ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our electron-photon exchanges into VCSEL system. !   −2 · U + τ · |P|2 + i · Fω   = Re Λ (τ ) = Fτ + 2 · i · V + ω · |P|2 λ=i·ω        2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 ∂Reλ Λ−1 (τ ) = = 2  ∂τ λ=i·ω Fτ2 + 4 · V + ω · P 2 −1



∂Reλ ∂τ



The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . ; cos θ (τ ) = . . . when ω = ω+ (τ ) if only ω+ is feasible. Additionally, when all electron-photon exchanges into VCSEL system’s parameters are known and the stability switch due to various time delay values τ is described in the following expression:   sign Λ−1 (τ ) = sign{Fω (ω(τ ), τ )}·

U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) sign τ · ωτ (ω(τ )) + ω(τ ) + |P(ω(τ ))|2 Remark We know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delay values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyse stability. F function is independent on τ parameter values F(ω, τ ) = 0. The results: We find those ω, τ values which fullfil F(ω, τ ) = 0. We ignore negative, complex and imaginary values of ω. Next is to find those ω, τ values which fullfil sin θ (τ ) = . . . ; cos θ (τ ) = . . . −PR · Q I + PI · Q R ; cos(ω · τ ) |Q|2 (PR · Q R + PI · Q I ) =− ; |Q|2 = Q 2R + Q 2I |Q|2

sin(ω · τ ) =

Finally, we plot the stability switch diagramg(τ ) = Λ−1 (τ ) =

∂τ

λ=i·ω

  2 · Fω · V + ω · P − Fτ · U + τ · P 2 −1 g(τ ) = Λ (τ ) = 2  Fτ2 + 4 · V + ω · P 2         −1  2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 sign[g(τ )] = sign Λ (τ ) = sign 2  Fτ2 + 4 · V + ω · P 2 



 2

 ∂Reλ 

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1 Dynamical and Nonlinearity of Laser Diode Circuits

2    Fτ2 + 4 · V + ω · P 2 > 0; sign Λ−1 (τ )      = sign Fω · V + ω · P 2 − Fτ · U + τ · P 2  

 Fτ    2 2 sign Λ (τ ) = sign [Fω ] · V + ω · P − · U +τ · P Fω  −1 Fτ ∂ω ∂ F/∂ω =− ωτ = − ; ωτ = Fω ∂τ ∂ F/∂τ      sign Λ−1 (τ ) = sign [Fω ] · V + ωτ · U + ω · P 2 + ωτ · τ · P 2 



−1

 

   V + ωτ · U  sign Λ−1 (τ ) = sign [Fω ] · P 2 · + + ω · τ (ω ) τ P2  

 2  −1  V + ωτ · U sign P > 0 ⇒ sign Λ (τ ) = sign [Fω ] · + (ω + ωτ · τ ) P2    V + ωτ · U sign Λ−1 (τ ) = sign[Fω ] · sign + ω + ω · τ τ P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] 

    1 1 1 2 +ω· Fω = 2 · + τn · τ p τp τn       2 − ω · g2 S ( j) , N ( j) − g1 S ( j) , N ( j)  −2 · ω · −ω2 +



   1 1 1 2 +ω· + τn · τ p τp τn  ( j) ( j) 2    ( j) ( j)  − g1 S , N −ω · g2 S , N

 Fω = 2 · 2 · ω · ω2 −

  We check the sign of Λ−1 (τ ), if sign Λ−1 (τ ) > 0 then proceeds   the crossing from (−) to (+) respectively (stable to unstable). If sign Λ−1 (τ ) < 0 then the crossing proceeds from (+) to (−) respectively (unstable to stable).

1.5 Questions 1.

We have Laser diode coupled delay rate equations which characterize our system. variable N ph is the photon density. Variable Ne is the electron density, q is the electronic charge, d is the thickness of the active region, a is the area of

1.5 Questions

151

the diode contact stripe, I is the injected current, Nom is the minimum electron density to obtain a positive gain, A is a constant related to the stimulated emission process, τs is the spontaneous emission lifetime, τ ph is the photon lifetime and β is the fraction of the spontaneous emission that is coupled to the lasing mode. The carrier density Ne is a function of the junction voltage, where Ni is the intrinsic carrier density and V is the junction voltage.   1 q·V d Ni (t) = · exp − dt q ·a·d 2·k·T     q·V q·V · (Ni (t − τ ) · exp − A · exp − 2·k·T 2·k·T Ni (t) − Nom ) · N ph (t − τ ) − τs     d N ph (t) q·V = A · Ni (t − τ ) · exp − Nom · N ph (t − τ ) dt 2·k·T   N ph (t) β q·V + · Ni (t) · exp − τ ph τs 2·k·T (second possible sets of differential equations) The model differential equations assume that the inversion is homogenous and the gain is linear in the difference between Ne and Nom . The carrier density Ne is a function of the junction voltage, where Ni is the intrinsic carrierdensity  and V is q·V the junction voltage. The carrier density equation: Ne = Ni ·exp 2·k·T ; Ne (t −   q·V . We define the parasitic effects of the stimulated τ ) = Ni (t − τ ) · exp 2·k·T emission process as a latent period τ . 1.1 Find system equilibrium points (fixed points) and explain the meaning. 1.2 Discuss stability and stability switching of our system (τ variation). 1.3 We define our τs (spontaneous emission lifetime) andτ ph (photon lifetime) 2 parameters 3 as a function of our latent period parameter τ ; τs = τ − 1 · 2 τ, τ ph = τ − 2 · τ ; 1 , 2 ∈ R . Find system fixed points and discuss stability/stability switching for different values of τ parameter (τ > 0). 1.4 What are the restrictions on 1 , 2 ; 1 , 2 ∈ R parameters which are related to τs , τ ph expressions (see 1.3)? (τs > 0, τ ph > 0). 1.5 Discuss our system stability and stability switching for the following cases: (1) 1 > 0; 2 = 0 (2) 1 = 0; 2 > 0 (3) 1 = 0; 2 = 0, 1 , 2 ∈ R. 2.

The dynamic behavior of a laser is determined by the interaction of the intracavity light field with the gain medium. Essentially, the intracavity laser power can grow or decay exponentially according to the difference between gain and resonator losses, whereas the rate of change in the gain is determined by stimulated and spontaneous emission (and possibly by other effects such as quenching and energy transfer). With certain approximations, including a not too high laser

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1 Dynamical and Nonlinearity of Laser Diode Circuits

gain, the dynamics of the intracavity optical power P and the gain coefficient g in a continuous-wave laser can be described with the following coupled delay differential equations. Due to the delay of the optical power P which amplified by gain coefficient g, the multiplication element is P(t − τ ) · g(t − τ ). I · P(t) dg g(t − τ ) · P(t − τ ) dP g(t − τ ) · P(t − τ ) (g − gss ) − ; − = =− dt TR TR dt τg E sat

where T R is the cavity round-trip time, I the cavity loss, gss the small-signal gain (for a given pump intensity), τg the gain relaxation time (often close to the upper-state lifetime), and E sat the saturation energy of the gain medium. 2.1. Find system equilibrium points (fixed points) and explain the Meaning. 2.2 Discuss stability and stability switching of our system for different values of τ parameter. 2.3 We define τg (gain3relaxation time) as a function of τ and ;  ∈ R √ parameters, τg = τ −  · τ . Find system fixed points and discuss stability/stability switching for different values of τ parameter (τ > 0). 2.4 What are the restrictions on ;  ∈ R parameter which is related to τg expression (See 2.3)? (τg > 0). 2.5 We define τg (gain relaxation time) as a function of τ and ;  ∈ R 3 √ parameters, τg = τ −  · τ . Find system fixed points and discuss stability/stability switching for different values of τ parameter (τ > 0). 3.

We have system of back to back connection laser diodes (D1, D2) which are attached to fiber optic. The supply signal (square wave alternates between Vsup1 and Vsup2 , Vsup 1 > 0; Vsup 2 < 0; |Vsup 1 | > |Vsup 2 |) is connected through resistor Rp to the set of two laser diodes.

The durations of the positive voltage pulse and the negative voltage pulse are Tb and Ta respectively (Ta = Tb ; Ta > 0; Tb > 0). At t=0 switch S1 changes his position from OFF to ON state. Investigate the operation of the system when you use the basic intrinsic equivalent circuit of the laser diode. Some of the laser diodes (D1 and D2) parameters are not the same (see below list):

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153

Ri−D1 =

1 2·k·T τs−D1 · · 0 0 2 q Ne · a · d (A · τ S−D1 · N ph + 1)

Ri−D2 =

1 2·k·T τs−D2 · · 0 0 q2 Ne · a · d (A · τ S−D2 · N ph + 1)

L i−D1 ≈

2 · k · T · τ ph−D1 2 · k · T · τ ph−D2 ; L i−D2 ≈ 0 0 A · N ph · q 2 · Ne0 · a · d A · N ph · q 2 · Ne0 · a · d Ne0 · a · q 2 · d 2·k·T n 0e 0 · (n ph + β) · (n 0e − n om−D1 )

Ci−D1 = Ci−D2 = Ci = Rse−D1 = β · Rd · Rse−D1 = β · Rd ·

n 0ph

n 0ph · (n 0ph

n 0e + β) · (n 0e − n om−D2 )

3.1 Represent our system by set of differential equations (separate between the operation of D1 and D2 diodes). 3.2 Find system fixed points (equilibrium points). Discuss how your results are affected by the diodes different basic intrinsic equivalent circuit’s parameters? 3.3 Discuss our system stability and bifurcation for different diode’s basic intrinsic equivalent circuit’s parameters. Is there any stability switching for different diode’s parameter values? 3.4 We change the direction of D1, how it influences our system dynamic? Find fixed points and discuss stability. 3.5 We change the direction of D2, how it influences our system dynamic? Find fixed points and discuss stability. 4.

A semiconductor laser is biased with current I above threshold. The steady state values of carrier and photon densities are n and np , respectively. Since I > Ith , n ≈ n th . Suppose at time t = 0, the carrier density is suddenly increased from n to n + n. The new value of the carrier density is not the steady state value and we need to improve it, if at all, the steady state value is recovered. The carrier density n + n is greater than n th , and, consequently, the gain 4 g is greater than the threshold gain g5 th . In steady state, can never be greater than g5 th , but this restriction does not hold in non-steady state situations. Another way is to perturbe the photon density at time t = 0 to n p + n p . If the carrier or photon densities in a laser are distributed from their steady state values than it is important to know if these quantities return to their steady state values. If they do the laser is stable. If they don’t, the laser is unstable. Inspecting the recovery dynamics associated with such carrier Density or photon density perturbations help for understanding the laser dynamic. The coupled differential equations for the perturbations in the carrier and photons densities constitute a second order linear system. Due to laser operation parasitic effects there is

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1 Dynamical and Nonlinearity of Laser Diode Circuits

a delay in n p (t) and n(t) variables in time (n p (t) → n p (t − τ1 ) and n(t) → n(t − τ2 ); τ1 > 0; τ2 > 0; τ1 = τ2 ). The coupled equations give the following identical second order delay differential equations for the perturbations (relaxation oscillations). d 2 n p (t) dn p (t) +γ · + ω2R · n p (t − τ1 ) = 0; ω R = dt 2 dt

,

1 1 1 ; γ = + τst · τ p τr τst

d 2 n(t) dn(t) + ω2R · n(t − τ2 ) = 0 +γ · 2 dt dt τst τr τp n sp 4 g g5 th Np np n p n

differential stimulated emission time. differential recombination time. cavity photon lifetime. spontaneous emission factor. gain which is related to the net stimulate rate. threshold gain. total photon number. average photon density. photon density perturbed additive term in time. carrier density perturbed additive term in time.

g = g5 The carrier density at which the gain 4 g equals the threshold gain g5 th (4 th ) is called the threshold carrier density. 4.1 Find system fixed points (equilibrium points). 4.2 Discuss system stability and stability switching for the following cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 4.3 What happened if the differential stimulated emission time (τst ) and differential recombination time (τr ) are very big (τst → ∞; τr → ∞)? How the system’s dynamic changes? Discuss stability and stability switching in that case. 4.4 What happened if the cavity photon lifetime τ p is very big (τ p → ∞)? How the system’s dynamic changes? Discuss stability and stability switching in that case. d 2 n (t) 4.5 The second derivative in time of n p (t) variable is neglected ( dt 2p → ε). How the system dynamic is changed? Find fixed points and discuss stability and stability switching for the following cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 5.

We have laser diode which an electric current input is converted to an output of photons. The time-dependent relation between the input electric current and output photon is described by a pair of equations describing the time evolution of photon and carrier densities inside the laser medium. This pair of equations are the laser rate equations, we analyze the “kinetics” by the coupled rate equations.

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155

The differential equations are local photon and injected carrier conservation equations. We define some parameters: z: spatial dimension along the length of the laser (distance along the active medium with z=0 at the center of the laser). R: laser reflectors of (power) reflectivities placed at z = ± L2 . X+ , X- : forward (X+ ) and backward (X- ) propagating photon densities (which are proportional to the light intensities). N: local carrier densities. Ntr : electron density where the semiconductor medium becomes transparent. C:group velocity of the waveguide mode. A: gain constant in s −1 /(unit carrier density), differential optical gain. β: fraction of spontaneous emission entering the lasing mode. τs : spontaneous recombination lifetime of the carrier. J: uniform pump current density. τ : delay parameter (related toN ∗ (t − τ ) · P ∗ (t − τ )). e: electronic charge. d:thickness of the active region in which the carriers are confined. P: total local photon density and the boundary conditions have been used (P = X + + X − ). 6 +L/2 “*”: denotes the spatial average −L/2 ddzL . ∗

P·( L )

·P) 2 f 1 , f 2 : factor functions ( f 1 = (N , f 2 = P ∗ ·(1+R) ). If the following conditions N ∗ ·P ∗ 1 ln R are satisfied f 1 = 1; f 2 = − 2 · (1−R) , we can recognize the following rate equations: N ∗ = N ∗ (t); P ∗ = P ∗ (t)

d P∗ N∗ P∗ = A · f 1 · N ∗ (t − τ ) · P ∗ (t − τ ) − 2 · c · (1 − R) · f 2 · +2·β · dt L τs J N∗ dN∗ = − − A · f 1 · N ∗ (t − τ ) · P ∗ (t − τ ) dt e·d τs The first of these conditions ( f 1 = 1) requires, for the quantities N and P, that the spatial average of the product equals the product of the spatial averages. This condition is not satisfied in general, but it will be true if the electron density N is uniform, as in the case when R approaches unity. The second condition ln R ) requires the photon loss rate to be inversely proportional to ( f 2 = − 21 · (1−R) the conventional photon lifetime. It will also be satisfied if R is very close to unity. Due to laser’s parasitic effects there is a delay in time of τ which is related to the multiplication element (N ∗ (t) · P ∗ (t) → N ∗ (t − τ ) · P ∗ (t − τ )). 5.1 Find system fixed points (equilibrium points). 5.2 Discuss stability and stability switching for different values of τ parameter. 5.3 Discuss system stability and stability switching if the group velocity of the waveguide mode c is a function of parameter τ and fundamental group √ velocity of the waveguide mode c0 ; c0 ∈ R+ c = c0 · τ 2 + τ .

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1 Dynamical and Nonlinearity of Laser Diode Circuits

5.4 Discuss system stability and stability switching if the fraction of spontaneous3emission √ entering the lasing modeβ is a function of parameter τ , β = 0 · τ + τ 3 ; 0 ∈ R. What are the restrictions on0 parameter if we know thatβ > 0. 5.5 If we can neglect our uniform pump current density (J), J → ε; How the system dynamic change? Find fixed points and discuss stability and stability switching for different values of τ parameter. 6.

We have a system of two laser diodes (D1, D2) which are attached to fiber optic. There are two power sources in our system; one is fix DC source (Vsup ) and the other is a square wave which is alternated between Vsup 1 andVsup 2 (Vsup > Vsup 1 ; Vsup 1 > 0; Vsup 2 < 0; |Vsup | > |Vsup 2 |, |Vsup 1 | = |Vsup 2 |). At t = 0 switch S1 changes his position from OFF to ON state. Investigate the operation of the system when you use the complete intrinsic electrical equivalent circuit of the laser diode. Some of the laser diodes (D1 and D2) parameters are not the same (see below list):

Ri−D1 =

1 2·k·T τs−D1 · · 0 0 q2 Ne · a · d (A · τ S−D1 · N ph + 1)

Ri−D2 =

1 2·k·T τs−D2 · · 0 0 q2 Ne · a · d (A · τ S−D2 · N ph + 1)

L i−D1 ≈

2 · k · T · τ ph−D1 2 · k · T · τ ph−D2 ; L i−D2 ≈ 0 0 A · N ph · q 2 · Ne0 · a · d A · N ph · q 2 · Ne0 · a · d Ne0 · a · q 2 · d 2·k·T n 0e · (n 0ph + β) · (n 0e − n om−D1 )

Ci−D1 = Ci−D2 = Ci = Rse−D1 = β · Rd ·

n 0ph

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157

Rse−D1 = β · Rd ·

n 0ph

·

(n 0ph

n 0e + β) · (n 0e − n om−D2 )

Csc and Rse1 elements have the same value for D1 and D2. 6.1 Represent our system by set of differential equations (saperate between the operation of D1 and D2 diodes). 6.2 Find system fixed points (equilibrium points). Discuss how your results are affected by the diodes different complete intrinsic electrical equivalent circuit’s parameters? 6.3 Discuss our system stability and bifurcation for different diode’s complete intrinsic electrical equivalent circuit’s parameters. Is there any stability switching for different diode’s parameter values? 6.4 We change the direction of D1, how it influences our system dynamic? Find fixed points and discuss stability. 6.5 We change the direction of D2, how it influences our system dynamic? Find fixed points and discuss stability. 7.

We have MRI system laser diode circuit with delay lines. The laser diode circuitry consists of T—matching network (Ca , Cb , L 1 ), a Bias—T (C1 , L 2 ) match network, and laser diode itself. The T—matching network is responsible for the matching of the low impedance laser diode to the high impedance output of the pre amplifier. The Bias—T combines the RF signal with the appropriate DC current injection which required biasing the laser. The laser diode holds the micro-lens to couple the generated optical energy to the fiber. Due to laser diode circuit microstrip parasitic effects there are instabilities on the circuit operation. The circuit microstrip parasitic effects are represented by delay lines elements which delay the current in time that flow through them. We consider ideal delay lines then the voltages that fall are neglected (Vτi → ε ∀ i = 1, 2, 3).

7.1 Find system fixed points (equilibrium points). 7.2 Find system delay differential equations (DDEs) and Jacobian.

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1 Dynamical and Nonlinearity of Laser Diode Circuits

7.3 Discuss stability and stability switching for different value of τ parameter, cases: (1) τ1 = τ ; τ2 = 0; τ3 = 0 (2) τ1 = 0; τ2 = τ ; τ3 = 0 (3) τ1 = τ ; τ2 = τ ; τ3 = 0 (4) τ1 = 0; τ2 = τ ; τ3 = τ (5) τ1 = τ ; τ2 = 0; τ3 = τ (6) τ1 = τ ; τ2 = τ ; τ3 = τ . 7.4 We replace the elements in T matching network to L a , L b , C2 , find fixed points and discuss stability and stability switching for the cases in (7.3). 7.5 We replace system T—matching network with PI—matching network (Ca , Cb , L 1 ). Find system fixed points and discuss stability and stability switching for different value of τ parameter, cases: (1) τ1 = τ ; τ2 = 0; τ3 = 0 (2) τ1 = 0; τ2 = τ ; τ3 = 0 (3) τ1 = τ ; τ2 = τ ; τ3 = 0 (4) τ1 = 0; τ2 = τ ; τ3 = τ (5) τ1 = τ ; τ2 = 0; τ3 = τ (6) τ1 = τ ; τ2 = τ ; τ3 = τ .

8.

The MRI signals, after being coupled to the optical beam in the fiber, are then transmitted to the receiver. The main responsibility of the receiver is to convert optical signals back to electricity, and it uses a special photo detector (photo diode). An analysis of the laser beam that falls on the Photo Diode (PD) reveals that the envelope of the beam consists of two components, the one generated by MRI signals and the one generated by the DC biasing current of the laser diode. The current that the photo detector generates is the sum of these components converted. As the demodulation of the optical carrier is performed internally by the photo detector, the only circuitry needed is a Bias—T to separate these two components. The schematics of the receiver is as follow.

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159

Circuit parasitic effects are represented by two delay lines Tau1, Tau2, which Vτ1 , Vτ2 → ε (delay on the current which flow through τ1 , τ2 ). The Bias—T circuit (C1 , L 1 ), separates DC component of the signal from radio frequency component. The biasing circuit for photo diode is V_bias, R1 . The photo diode used is high speed diode which is a 100 um diameter InGaAs photodetector in a plastic housing with an SC receptacle. MRI scanner is represented by R_scanner. The photo diode equivalent circuit is shown below.

IL—current generated by the incident light and its proportional to the amount of light, ID —diode current, Cj —junction capacitance, Rsh —shunt resistance, Rs —series resistance, I’ = IRSH —shunt resistance current, VD1 —voltage across the diode, Iout —output current, Vo —output voltage. Using the above equivalent circuit by neglecting the current through junction capaciatance, the output current I0 is given as follow: Iout = I L − I0 · (e I0 q k T

V D1 Vt

− 1) − C j ·

VD1 d VD1 k·T − , ; Vt = dt Rsh q

photodiode reverse saturation current, electron charge, Boltzmann’s constant, absolute temperature of the photodiode

Cj ·

V D1 d VD1 VD1 → ε ⇒ Iout = I L − I0 · (e Vt − 1) − . dt Rsh

8.1 Find system Delay Differential Equations (DDEs) and fixed points. 8.2 Discuss stability and stability switching for different values of τ parameter (Cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = √ τ ; τ2 = τ ).   8.3 We change the value of resistor R1 , R1 → R1 ; R1 =  · R1 ;  ∈ R+ . Discuss stability and stability switching for different values of  parameter (τ is constant).

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1 Dynamical and Nonlinearity of Laser Diode Circuits

8.4 How system dynamic is changed if we disconnect photo diode equivalent circuit’s Rsh resistor? Find fixed points and discuss stability and stability switching for the following cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 8.5 We change the polarity of the photo diode (D1 ), How the dynamic of the circuit is change? Is it possible to get the right outcome? Explain deeply your answer. 9.

We have a system of Vertical-Cavity Surface-emitting laser (VCSELs) which employ current-blocking layers to funnel the current to the active region located between the mirrors of the laser. The VCSEL carrier-photon interaction active region volume is describe in the next figure.

The interaction active region volume schematic includes additionally mechanism which takes some carriers (carriers (α factor, 0 < α < 1) in the active region volume) are amplified by squaring and reinject into the active region volume. The time takes that mechanism to happened is second. The VCSEL rate equations with delay parameter (τ) of the carrier-photon interaction time and delay parameter  of the additional mechanism: η · (I − Io f f ) N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) d N (t) = − − dt q τn 1 + ε · S(t − τ ) 2 2 − α · N (t) + α · N (t − ) S(t) β · N (t) G · (N (t − τ ) − Ntr ) · S(t − τ ) d S(t) =− + + dt τp τn 1 + ε · S(t − τ ) N S S0 N0 η τn τp G Ntr β

Carrier density in the active region. Photon density in the active region. Photon density in the active region at t = 0. Carrier density in the active region at t = 0. Injection efficiency. Carrier recombination lifetime. Photon lifetime. Gain coefficient. Carrier’s transparency number. Spontaneous emission coupling coefficient.

1.5 Questions

ε I Ioff τ 

161

Gain compression factor. VCSEL’s injection current. VCSEL’s offset current. Time delay due to VCSEL active region parasitic effects. Time delay which related to the additional mechanism.

9.1 Find system fixed points and draw functions of each fixed point for different values of αparameter (0 < α < 1). 9.2 Discuss stability and stability switching for different values of  parameter (other parameters are constant). 9.3 Discuss stability and stability switching for different values of τ parameter (other parameters are constant). 9.4 We change the additional mechanism so some carriers (carriers (αfactor, 0 < α < 1) in the active region volume) are amplified by tripling ([]3 (t − )) and reinject into the active region volume. Find fixed points and discuss stability and stability switching for different values of τ and parameters. 9.5 We change the additional mechanism so some carriers (carriers (αfactor, 0√< α < 1) in the active region volume) are attenuated by square root ( [](t − )) and reinject into the active region volume. Find fixed points and discuss stability and stability switching for different values of τ and parameters. √√ 10. Vertical cavity surface-emitting laser (VCSEL’s) are the 4 4 transmitters in short distance optical communication system such as LAN’s, where 1-2Gb/s signal must be carried 0–3 km over multimode fiber. Laser diode rate-equation parameters are extracted for simulation of on-off keyed digital communication links. The extraction procedure uses measurements of the current voltage light characteristic, the AC small-signal response above threshold, and the turn-on delay due to an isolated pulse. Many characteristics of laser diode behavior can be described by a set of rate equations, which may be written as ηi · I dN = − R(N ) −  · vgr · G(N , S) · S dt q · Nω · Vact dS S = Nω · Rβ (N ) + Nω ·  · vgr · G(N , S) · S − dt τp where N is the carrier density, I is the injection current, and S is the normaltot , with Stot being the photon number. This definition ized photon density VSact provides symmetry in the rate-equation stimulated emission terms. ηi is the current injection efficiency, q is the electron charge, Nω the number of QW’s (Quantum Well’s), Vact is the volume of a single QW, R(N ) is the carrier recombination rate,  is the carrier confinement factor, vgr is the velocity of the light in the cavity, G(N , S) is the gain function, including gain compression, Rβ (N ) is the coupled spontaneous emission, and τ p is the photon lifetime. The act output power is thus ηc ·h·c··S·V , where ηc is the output coupling efficiency, λ·τ p

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1 Dynamical and Nonlinearity of Laser Diode Circuits

and h·c is the photon energy. The recombination rate R(N ) may be written λ A · N + B · N 2 + C · N 3 . When B and C are set to zero, linearized form τNn often used when simple rate equations are desired. The coupled spontaneous emission Rβ (N ) may be written as β B · B · N 2 or β A · A · N when the linearized recomG(N ) bination rate is used. The gain function may be written as G(N , S) = 1+ε··S where εis the gain suppression factor. The gain confinement factor appears tot is equal to  · S in the gain suppression term because the photon density VSmod e where V mod e is the volume of the optical mode. The gain carrier dependence be R(N ) }. The gain function may be linearized about written as G(N ) = G 0 · ln{ R(N th ) the transparency carrier density, yielding G(N ) = m · G 0 · ( NN0 − 1), where m is a linearization constant. There are three model variations which define R(N ) and G(N , S), as shown in the below table. Model 1

Model 2

R(N )

A·N +B·

Rβ (N )

βB · B · N 2 2 0 R(N ) G 0 · ln R(N th )

G(N )

N2

+C ·

N3

A·N +B·

Model 3 N2

+C ·

βB · B · N 2   m · G 0 · NN0 − 1

N3

A·N βA · A · N 2 0 R(N ) G 0 · ln R(N th )

10.1 Find system fixed points and jacobian for all models (1, 2, 3). 10.2 Discuss stability for each system differential equations models (1, 2, 3). Classify fixed points for different parameter’s values. 10.3 The carrier confinement factor  = 0, How the system dynamic is change? Find fixed points and discuss stability. 10.4 Due to parasitic effects there is a delay τ in the multiplication Elements G(N (t), S(t))· S(t) → G(N (t −τ ), S(t −τ ))· S(t −τ ), discuss stability and stability switching for different values of τ parameter. 10.5 The number of QW’s (Quantum Well’s) Nω = 1, how the system dynamic is changed? Discuss stability for that case.

References 1. M.S. Ozyazici, The complete electrical equivalent circuit of a double heterojunction laser diode using scattering parameters. J. Optoelectron Adv Mater 6(4), 1243–1253 (2004) 2. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993) 3. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002) 4. J. Kuang, Y. Cong. Stability of Numerical methods for Delay Differential Equations. Elsevier Science (2007) 5. B. Balachandran, T. Kalmár-Nagy D.E. Gilsinn, Delay Differential Equations: Recent Advances and New Directions (Hardcover). 1st edn. (Springer; 2009)

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6. O. Gokalp Memis, Y. Eryaman, O. Aytur, E. Atalar, Miniaturized fiber-optic transmission system for MRI signals. Magn. Reson. Med. 59,165–173 (2008) 7. P.V. Mena, J.J. Morikuni, S.-M. Kang, A.V. Harton, K.W. Wyatt, A simple rate-equation-based thermal VCSEL model. J. Lightwave Technol. 17(5) (1999) 8. S.H. Strogatz, Nonlinear Dynamics and Chaos. Westview Press 9. J.C. Sprott, Simplifications of the lorenz attractor. Nonlinear Dynam. Psychol. Life Sci. 13(3), 271–278 10. G. Stepan, T. Insperger, R. Szalai, Delay, Parametric excitation, and the nonlinear dynamics of cutting process. Int. J. Bifurcat. Chaos, 15(9), 2783-2798 (2005) 11. R. Rand, A. Barcilon, T. Morrison, Parametric resonance of hopf bifurcation. Nonlinear Dynam. 39, 411–421 (2005) 12. H.D.I. Abarbanel, M. I. Rabinovich, M. M. Sushchik, Introduction to Nonlinear Dynamics for Physicists (World Scientific, 1993) 13. X. Shi, C. Qi, G. Wang, J. Hu, Rate-equation-based VCSEL model and simulation. ComputAided Des Comput Graph (2009)

Chapter 2

Ti: Sapphire Laser Systems with Delay Parameters in Time Stability Analysis

Ti: Sapphire lase (Ti: Al2 O3 laser, titanium-sapphire laser, or Ti: Sapphires) is a tunable laser which emit red and near infrared light in the range from 650 to 1100 nm. In many scientific experiments the Ti: Sapphire laser is used. It is very tenability and can generate ultrashort pulses. The lasing medium is Titanium-sapphire and the crystal of sapphire (Al2 O3 ) that is doped with titanium ions. In many systems the Ti: Sapphire laser is pumped with another laser with a wavelength of 514–532 nm. Another laser can be argon-ion laser (514.5 nm) and frequency doubled Nd:YAG, Nd:YLF and Nd:YVO lasers (527–532 nm). The most efficient operation of Ti: Sapphire laser is at wavelengths near 800 nm. Additionally, the other widely use of Titanium-doped sapphire is transition-metal-doped gain medium for tunable lasers and femto second solid-state lasers. Sapphire has an excellent thermal conductivity, and alleviating thermal effects for high laser powers and intensities. The Ti3+ ion has a very large gain bandwidth which allowing the generation of a very short pulses and wide wavelength tenability. The maximum gain and laser efficiency are obtained around 800 nm. The Ti3+ doping concentration has to be kept fairly low because otherwise no good crystal is possible. The upper-state lifetime of Ti: Sapphire is short (3.2 us), and the saturation power is very high. The Ti: Sapphire pump intensity needs to be high and we get a pump source with high beam quality. The Ti: Sapphire has relatively high laser cross sections, which reduces the tendency of Ti: Sapphire laser for Q-switching instabilities. Ultrashort pulses from Ti: Sapphire laser can be generated with passive mode locking, in the form of kerr lens mode locking (KLM). A pulse duration around 100 fsec is easily achieved and is typical for commercial devices. Typical output powers of mode-locked Ti: Sapphire lasers are of the order 0.3–1 w, whereas continuous-wave versions sometimes generate several watts. A typical pulse repetition rate is 80 MHz, and devices with multi-gigahertz repetition rates are available and use as frequency comb sources. Ti: Sapphire is also used for multi-pass amplifiers and regenerative amplifiers. Nonlinear frequency conversion is used to extend further the range of emission wavelengths of Ti: Sapphire laser system. Linearly polarized femtosecond laser pulses from Ti: sapphire laser is related to multiphoton ionization of D2 molecule. The rate of ionization of the molecule © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_2

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2 Ti: Sapphire Laser Systems with Delay Parameters …

is much lower than that of the atom. The interaction of strong laser fields with diatomic molecules characterized by the following processes: non-sequential ionization of molecules, high-order harmonic generation and fragmentation of diatomic molecules. They are closely related to the tunneling ionization (TI) of the molecules to a singly charged ion. There are two aspects which are related to finding a predictive model for the ionization rate of a diatomic molecule in the field of a short pulse with wavelength in the near-IR region. The first one is related to the fact that there was no experimentally proven model capable of providing a quantitative fit with the data on rare-gas atoms. The second is related to the difficulty in modeling the TI rate of molecules (orientation of the neutral molecule in the laser field). The second order differential equation for the rotation of the molecule with reduced mass, polarizability and inter-nuclear distance in the field of laser with frequency (ω) and field strength (F) has a stability switching under parameter variation and it is inspected. Ultra-broad bandwidth pulses from a Ti: sapphire laser is characterized by effects of higher-order phase dispersion on the fidelity of the shaped pulse. The phase dispersion at higher orders cubic phase and quartic phase dispersions. The phase dispersion differential equations are inspected for stability under parameters variation. Semiconductor saturable-absorber mirror mode-locked Ti: sapphire soliton laser is operated in the multiple-pulse regime. The stability of the pair of solitons is bounded by two limits. A stability criterion from the evolution of pulse energies as well as the gain is inspected in a set of differential equations. We analyze energies and gain delayed in time stability analysis due to parasitic effects [1–4].

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire Laser Stability Analysis Under Parameters Variation We use linearly polarized femtosecond laser pulses from a Ti: Sapphire laser to analyze the multiphoton ionization of D2 (D2 is a molecule of Deuterium). Deuterium (Hydrogen-2, Symbol D or 2 H) is one of two stable isotopes of hydrogen. The nucleus of deuterium, called a deuteron, contains one proton and one neutron, where the far more common protium has no neutron in the nucleus. We can compare between two mechanisms, one is the ion signal versus intensity curve of the singly charged molecular ion, second is the ion signal versus the intensity curve of the Ar atom with similar ionization potential. The ionization energy or ionization potential is the energy necessary to remove an electron from the neutral atom. It is a minimum for the alkali metals which have a single electron outside a closed shell. It generally increases across a row on the periodic maximum for the noble gases which have closed shells. Comparing between two mechanisms gives the outcome that the rate of ionization of the molecule is much lower than that of atom. There are two-centre nature of the potential which produces the barrier for tunneling, and the orientation of the molecular axis with respect to the laser polarization. A molecule can have

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire … Fig. 2.1 Structure of strong laser field pulse and diatomic molecules experiment

Strong Laser Field (pulse)

167

Diatomic Molecules

more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is aligned with the z-axis in a Cartesian coordinate system. Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is generated. The effective barrier is equivalent to that produced by an atom with an effective charge of less than unity [1]. {Rate of ionization of the molecule}  Rate of ionization of an atom}. We investigate the interaction of strong laser fields with diatomic molecules (Fig. 2.1). There are some mechanism’s processes, first the non-sequential ionization of molecules, second the high-order harmonic generation, and third the fragmentation of diatomic molecules. Diatomic molecules are molecules composed of only two atoms, of the same or different chemical elements (The prefix di- is of Greek origin, meaning “two”). If a diatomic molecule consists of two atoms of the same element, such as hydrogen (H2 ) or oxygen (O2 ), then it is said to be homo nuclear. All processes are related to the Tunneling Ionization (TI) of the molecule to a singly charged ion. Tunnel ionization is a process in which electrons in an atom (or a molecule) pass through the potential barrier and escape from the atom (or molecule). In an intense electric field, the potential barrier of an atom (molecule) is distorted drastically. Therefore, the length of the barrier that electrons have to pass decreases and electrons can escape from the atom (molecule) easily. We can consider that the ionization rate of the diatomic molecules aligned in the direction of the field. It should be enhanced compared to an atom with a similar ionization potential. There is an impact on fragmentation of molecular ions which involve the ionization of aligned species. The total rate of tunneling ionization (TI) is affected by the fact that at the first stage of ionization the neutral molecules are randomly oriented with respect to the polarization of the laser. The rate of multiphoton ionization of an N2 molecule is suppressed compared to that of an Ar atom with the same ionization potential. This suppression is more pronounced in the case of ionization of D2 compared to Ar . The random orientation of the molecules gives the result of suppression. There is an analytic expression for the single-electron wave function of the D2 molecule and we predict the rate of tunneling ionization (TI) of this molecule. There is a predictive model for the ionization rate of a diatomic molecule in the field of a short pulse with wavelength in the near-IR region. There are two equivalent models which give a good overlap with experimental results on the ionization of rate-gas atoms in the tunneling regime, interacting with femtosecond laser pulses. There is a difficult in modeling

168

2 Ti: Sapphire Laser Systems with Delay Parameters …

the tunneling ionization (TI) rate of molecules which is related to the orientation of the neutral molecule in the laser field. There is a second order differential equation, gives the rotation of a molecule with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F). α · F2 d 2 θ  ω0 2 2 2 + · cos (τ ) · sin(2 · θ ) = 0; τ = ω · t; ω = ; dτ 2 0 dτ 2 ω μ · R2 = d(ω · t)2 = ω2 · dt 2 θ Angle between the polarization of the laser and inter nuclear axis. ω0 Establish the rotation dynamic of the molecule. ω0 Quantity determines the rotation dynamics of the molecules. ω θ0 Molecule initially angle with the direction of laser polarization. μ reduced mass of the molecule. α Polarization factor (polarizability). R Inter nuclear distance in the field of laser. ω Laser frequency (we consider it as a constantin our  system).  α·F 2 F α 2 ω0 Calculated parameter ω0 = μ·R 2 ; ω0 = R · μ . (ω0 = 7.9 × 10−4 au (Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations) for D2 molecule at a peak intensity of 1014 W/cm2 ). 2·π·ω ω0

The division between the period of oscillation for the molecule around the   . · T polarization axis and the laser period Tmolecule = 2·π·ω Laser ω0

Tmolecule Period of oscillation for the molecule around the polarization axis. Laser periodLaser period. TLaser We can write our system second order differential equation: dθ 1 d 2 θ  ω0 2 · 2 + · cos2 (ω · t) · sin(2 · θ ) = 0; y = 2 ω dt ω dt dθ 1 dy  ω0 2 · · cos2 (ω · t) · sin(2 · θ ) = 0; + =y ω2 dt ω dt We can write our system differential equations:  ω 2 dθ dy 0 = −ω2 · =y · cos2 (ω · t) · sin(2 · θ ); dt ω dt dy dθ = −ω02 · cos2 (ω · t) · sin(2 · θ ); =y dt dt

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

169

The rotation dynamics of the molecule is determined by the quantity ωω0 . The molecule initially making an angel θ0 with the direction of laser polarization will times the laser period. The oscillate around the polarization axis with a period of 2·π·ω ω0 molecule in the Ti: Sapphire laser field should be considered as randomly oriented. TI is defined as the tunneling ionization of the molecules to a charged ion. The interaction of strong laser fields with diatomic molecules involves the following processes: 1. Non sequential ionization of molecules. 2. High-order harmonic generation. 3. Fragmentation of diatomic molecules. Predictive mode for ionization of diatomic molecules, short laser pulse with wavelength in the infra-red (IR) region is constructed from 1. No experimental model which fit with data on rare-gas atom. 2. Orientation of the neutral molecule in the laser field. Typical experiment is to use Ti: Sapphire femtosecond laser at 800 nm with pulse length of 200 fsec which hit Ar and D2 and cause to ionization. These two particles have similar ionization potential (D2 15.46 eV and Ar 15.76 eV). We investigate the dynamic and stability of our system for two cases: (1) Regular differential equations (2) Delay Differential Equations (DDEs)—parasitic effects which are related to laser diatomic molecules interaction. Case (1): Our femtosecond laser pulses from a Ti: Sapphire interact with diatomic molecule—differential equations for the rotation of a molecule (with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F)).

dy dθ α · F2 = −ω02 · cos2 (ω · t) · sin(2 · θ ); = y; ω02 = dt dt μ · R2 α · F2 dy dθ =− =y · cos2 (ω · t) · sin(2 · θ ); dt μ · R2 dt At fixed points

dθ dt

α·F 2 ∗ = 0; dy = 0 then y ∗ = 0; − μ·R 2 · cos (ω · t) · sin(2 · θ ) = 0. dt 2

α · F2 = 0; cos2 (ω · t) = 0 ⇒ sin(2 · θ ∗ ) = 0; 2 · θ ∗ = k · π ⇒ θ ∗ μ · R2 π = k · ∀ k = . . . , −2, −1, 0, +1, +2, . . . 2   System fixed points: E ∗ (θ ∗ , y ∗ ) = k · π2 , 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . Stability analysis: The standard local stability analysis about any one of the equilibrium points (fixed points) of diatomic molecule—differential equations for the

170

2 Ti: Sapphire Laser Systems with Delay Parameters …

rotation of a molecule system consists in adding to coordinates [θ y] arbitrary small increments of exponential form [θ y] · eλ·t , and retaining the first order terms in θ y. The system of two homogeneous equations leads to a polynomial characteristics equation in the eigenvalue λ. The polynomial characteristics equations accept by set of below Angle between the polarization of the laser and inter nuclear axis and Angle between the polarization of the laser and inter nuclear axis derivative respect to the time into diatomic molecule—system differential equations for the rotation of a molecule. Diatomic molecule – differential equations for the rotation of a molecule system fixed values with arbitrarily small increments of exponential form [θ y] · eλ·t are: i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc. θ (t) = θ (i) + θ · eλ·t ; y(t) = y (i) + y · eλ·t We choose the above expressions for our θ (t), y(t) as small displacement [θ y] from the system fixed points at time t = 0 (θ (t = 0) = θ (i) +θ ; y(t = 0) = y (i) + y). For t > 0; λ < 0 the selected fixed point is stable otherwise t > 0; λ > 0 is unstable. Our system tends to the selected fixed point exponentially for t > 0; λ < 0 otherwise go away from the selected fixed point exponentially. Eigenvalue λ is the parameter which establish if the fixed point is stable or unstable, additionally his absolute value |λ| establish the speed of flow toward or away from the selected fixed point. The next table describes diatomic molecule—differential equations for the rotation of a molecule system variables for different eigenvalue λ and t values (Table 2.1). The speeds of flow toward or away from the selected fixed point diatomic molecule— differential equations for the rotation of a molecule system angle (θ ) between the polarization of the laser and inter nuclear axis and angle (θ ) derivative respect to time are as follow: dθ (t) dy(t) = λ · θ · eλ·t ; = λ · y · eλ·t dt dt α · F2 dy =− · cos2 (ω · t) · sin(2 · θ ) ⇒ λ · y · eλ·t dt μ · R2 Table 2.1 Molecular system’s angle (θ) and angle derivative for positive (λ > 0) and negative (λ < 0) eigenvalues λ0

θ(t = 0) = θ (i) + θ y(t = 0) = y

t>0

+y

θ(t) = θ (i) + θ · e−|λ|·t y(t) = y

t →∞

(i)

(i)

+y·e

−|λ|·t

θ(t = 0) = θ (i) + θ y(t = 0) = y (i) + y θ(t) = θ (i) + θ · e|λ|·t y(t) = y (i) + y · e|λ|·t

θ(t → ∞) = θ (i)

θ(t → ∞) ≈ θ · eλ·t

y(t → ∞) = y (i)

y(t → ∞) ≈ y · eλ·t

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

171

α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·t ]) μ · R2 sin(2 · [θ (i) + θ · eλ·t ]) = 2 · sin(θ (i) + θ · eλ·t ) · cos(θ (i) + θ · eλ·t ) =−

sin(θ (i) + θ · eλ·t ) = sin(θ (i) ) · cos(θ · eλ·t ) + cos(θ (i) ) · sin(θ · eλ·t ) cos(θ (i) + θ · eλ·t ) = cos(θ (i) ) · cos(θ · eλ·t ) − sin(θ (i) ) · sin(θ · eλ·t ) sin(2 · [θ (i) + θ · eλ·t ]) = 2 · sin(θ (i) + θ · eλ·t ) · cos(θ (i) + θ · eλ·t ) = 2 · {sin(θ (i) ) · cos(θ · eλ·t ) + cos(θ (i) ) · sin(θ · eλ·t )} · {cos(θ (i) ) · cos(θ · eλ·t ) − sin(θ (i) ) · sin(θ · eλ·t )} = 2 · sin(θ (i) ) · cos(θ · eλ·t ) · cos(θ (i) ) · cos(θ · eλ·t ) − 2 · sin(θ (i) ) · cos(θ · eλ·t ) · sin(θ (i) ) · sin(θ · eλ·t ) + 2 · cos(θ (i) ) · sin(θ · eλ·t ) · cos(θ (i) ) · cos(θ · eλ·t ) − 2 · cos(θ (i) ) · sin(θ · eλ·t ) · sin(θ (i) ) · sin(θ · eλ·t ) sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ (i) ) · cos2 (θ · eλ·t ) − sin(2 · θ · eλ·t ) · sin2 (θ (i) ) + sin(2 · θ · eλ·t ) · cos2 (θ (i) ) − sin(2 · θ (i) ) · sin2 (θ · eλ·t ) sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ (i) ) · [cos2 (θ · eλ·t ) − sin2 (θ · eλ·t )] + sin(2 · θ · eλ·t ) · [cos2 (θ (i) ) − sin2 (θ (i) )] cos2 (θ · eλ·t ) − sin2 (θ · eλ·t ) = cos(2 · θ · eλ·t ); cos2 (θ (i) ) − sin2 (θ (i) ) = cos(2 · θ (i) ) sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ (i) ) · cos(2 · θ · eλ·t ) + sin(2 · θ · eλ·t ) · cos(2 · θ (i) ) π θ ∗ = θ (i) = k · ∀ k = . . . , −2, −1, 0, +1, +2, . . . ; 2 · θ (i) 2 = k · π ∀ k = . . . , −2, −1, 0, +1, +2, . . . sin(2 · θ (i) ) = sin(k · π ) = 0; cos(2 · θ (i) ) = cos(k · π ) = (−1)k sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ · eλ·t ) · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . ξ(θ, λ) = 2 · θ · eλ·t ; sin(2 · θ · eλ·t ) = sin(ξ(θ, λ)) =

∞  n=0

(−1)n · [ξ(θ, λ)]2·n+1 (2 · n + 1)!

sin(ξ(θ, λ)) =

∞  n=0

(−1)n · [ξ(θ, λ)]2·n+1 = ξ(θ, λ) (2 · n + 1)!

[ξ(θ, λ)]3 [ξ(θ, λ)]5 [ξ(θ, λ)]7 − + − + ··· 3! 5! 7! ∞  (−1)n sin(ξ(θ, λ)) = · [ξ(θ, λ)]2·n+1 = 2 · θ · eλ·t (2 · n + 1)! −

[2 · θ

n=0 λ·t · e ]3

3!

+

[2 · θ · eλ·t ]5 [2 · θ · eλ·t ]7 − + ··· 5! 7!

172

2 Ti: Sapphire Laser Systems with Delay Parameters … sin(ξ(θ, λ)) =

∞  n=0

(−1)n · [ξ(θ, λ)]2·n+1 = 2 · θ · eλ·t (2 · n + 1)!

23 · θ 3 · e3·λ·t 25 · θ 5 · e5·λ·t 27 · θ 7 · e7·λ·t − + − + ··· 3! 5! 7!

We consider θ m → ε ∀ m ≥ 2 then sin(ξ(θ, λ)) =

∞  n=0

sin(2 · [θ

(i)

23 ·θ 3 ·e3·λ·t 3!

+

25 ·θ 5 ·e5·λ·t 5!

−

27 ·θ 7 ·e7·λ·t 7!

+ ··· → ε

(−1)n · [ξ(θ, λ)]2·n+1 ≈ 2 · θ · eλ·t (2 · n + 1)!

+ θ · eλ·t ]) = 2 · θ · eλ·t · (−1)k ∀ k = · · · , −2, −1, 0, +1, +2, . . .

λ · y · eλ·t = −

α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·t ]) μ · R2

α · F2 · cos2 (ω · t) · 2 · θ · eλ·t · (−1)k μ · R2 α · F2 −λ·y− · cos2 (ω · t) · 2 · θ · (−1)k = 0 μ · R2 dθ = y ⇒ λ · θ · eλ·t = y (i) + y · eλ·t ; y∗ = y (i) = 0; −λ · θ + y = 0 dt =−

We can summary our diatomic molecule—differential equations for the rotation of a molecule system’s arbitrarily small increments equations: α · F2 · cos2 (ω · t) · 2 · θ · (−1)k = 0; −λ · θ + y = 0 ∀ k μ · R2 = . . . , −2, −1, 0, +1, +2, . . .   α·F 2 2 k y −λ − μ·R 2 · cos (ω · t) · 2 · (−1) · =0 θ 1 −λ  α·F 2 2 k −λ − μ·R 2 · cos (ω · t) · 2 · (−1) A−λ· I = ; det(A − λ · I ) = 0 1 −λ −λ·y−

α · F2 · cos2 (ω · t) · 2 · (−1)k = 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2 α · F2 α · F2 2 2 k λ =− · cos (ω · t) · 2 · (−1) ⇒ λ1,2 = ± −2 · · cos2 (ω · t) · (−1)k 2 μ· R μ · R2 λ2 +

μ > 0 (Reduced mass of the molecule), α > 0 (Polarization factor (polarizability)).

F α  α · F2 2 · · cos (ω · t) > 0; λ = ± 2 · · − cos2 (ω · t) · (−1)k 1,2 2 μ· R R μ

  F α α F · 2 · > 0 λ1,2 = ± · 2 · · cos2 (ω · t) · −(−1)k R μ R μ

2·

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

We define u = cos(ω · t); u 2 = cos2 (ω · t). u is positive. 

173

√ u 2 = |u|, if we cannot be sure that



[cos(ω ·

t)]2

cos(ω · t) ∀ 0 ≤ ω · t ≤ π2 − cos(ω · t) ∀ π2 ≤ ω · t ≤ π

 α · 2 · · |cos(ω · t)| · −(−1)k ; λ1,2 μ

 α · 2 · · |cos(ω · t)| · j · (−1)k μ

= |cos(ω · t)| = F R F =± R

λ1,2 = ±

We consider in our analysis k ≥ 0 (k is natural number k ∈ N). 

λ1,2

λ1,2

 (−1)k =

1 ∀ k = 0, 2, 4, . . . j ∀ k = 1, 3, 5, . . .

⎧ F α ⎪ ⎪ ± · 2 · · |cos(ω · t)| · j ∀ k = 0, 2, 4, . . . ⎨ R μ =

⎪ F α ⎪ ⎩ ± · 2 · · |cos(ω · t)| · j 2 ∀ k = 1, 3, 5, . . . R μ

⎧ F α ⎪ ⎪ ⎨ ± j · R · 2 · μ · |cos(ω · t)| ∀ k = 0, 2, 4, . . . =

⎪ α F ⎪ ⎩ ∓ · 2 · · |cos(ω · t)| ∀ k = 1, 3, 5, . . . R μ

If the eigenvalues are pure imaginary λ1,2 = ± j ·

F R

·



2·

0, 2, 4, . . . then all the solutions are periodic with period T =

α μ

· |cos(ω · t)| ∀ k = F R

√

·

2·π . 2· μα ·|cos(ω·t)|

The

oscillations have  fixed amplitudes and fixed point is a center. If λ1 ·λ2 < 0, in our case: F λ1,2 = ∓ R · 2 · μα · |cos(ω · t)| ∀ k = 1, 3, 5, . . . then eigenvalues are real and have opposite signs; hence the fixed point is a saddle point. The next figure illustrates the system fixed points (equilibrium points) phase portrait and classification (Fig. 2.2) [5, 6]. Case (2): Our femtosecond laser pulses from a Ti: Sapphire interact with diatomic molecule—delay differential equations for the rotation of a molecule (with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F)). Due to parasitic effects there is a delay on angle between the polarization of the laser and inter nuclear axis (θ (t) → θ (t − τ2 )) and y parameter (y(t) → y(t − τ1 )), there is no effect of delay on θ, y variables derivative in time. dy(t) dθ (t) α · F2 = −ω02 · cos2 (ω · t) · sin(2 · θ (t − τ2 )); = y(t − τ1 ); ω02 = dt dt μ · R2

174

2 Ti: Sapphire Laser Systems with Delay Parameters …

Fig. 2.2 System fixed points (equilibrium points) phase portrait and classification

α · F2 dy(t) dθ (t) =− = y(t − τ1 ) · cos2 (ω · t) · sin(2 · θ (t − τ2 )); 2 dt μ· R dt At fixed points dθ = 0; dy = 0; limt→∞ y(t − τ1 ) = y(t); limt→∞ θ (t − τ2 ) = dt dt α·F 2 2 ∗ θ (t); t  τ1 , τ2 then y ∗ = 0; − μ·R 2 · cos (ω · t) · sin(2 · θ ) = 0 α · F2 = 0; cos2 (ω · t) = 0 ⇒ sin(2 · θ ∗ ) = 0; 2 · θ ∗ μ · R2 π = k · π ⇒ θ ∗ = k · ∀ k = . . . , −2, −1, 0, +1, +2, . . . 2   System fixed points: E ∗ (θ ∗ , y ∗ ) = k · π2 , 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . Diatomic molecule—differential equations for the rotation of a molecule system fixed values with arbitrarily small increments of exponential form [θ y] · eλ·t are: i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc. θ (t − τ2 ) = θ (i) + θ · eλ·(t−τ2 ) ; y(t − τ1 ) = y (i) dθ (t) dy(t) = λ · θ · eλ·t ; = λ · y · eλ·t + y · eλ·(t−τ1 ) ; dt dt We get two delayed differential equations respect to coordinates [θ y] arbitrarily small increments of exponential [θ y] · eλ·t . α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·(t−τ2 ) ]); λ · θ · eλ·t μ · R2 = y (i) + y · eλ·(t−τ1 )

λ · y · eλ·t = −

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

175

α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·t · e−λ·τ2 ]); λ · θ · eλ·t μ · R2 = y (i) + y · eλ·t · e−λ·τ1

λ · y · eλ·t = −

We have three possible cases: (1) τ1 = τ A ; τ2 = 0 (2) τ1 = 0; τ2 = τ A (3) τ1 = τ A ; τ2 = τ A . We analyze the stability of our system for the second case: τ1 = 0; τ2 = τ A . λ · y · eλ·t = −

α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·t · e−λ·τ A ]); λ · θ · eλ·t μ · R2

= y (i) + y · eλ·t sin(2 · [θ (i) + θ · eλ·t ]) = 2 · sin(θ (i) + θ · eλ·t · e−λ·τ A ) · cos(θ (i) + θ · eλ·t · e−λ·τ A ) sin(θ (i) + θ · eλ·t ) = sin(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) + cos(θ (i) ) · sin(θ · eλ·t · e−λ·τ A ) cos(θ (i) + θ · eλ·t ) = cos(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) − sin(θ (i) ) · sin(θ · eλ·t · e−λ·τ A )

sin(2 · [θ (i) + θ · eλ·t ]) = 2 · sin(θ (i) + θ · eλ·t · e−λ·τ A ) · cos(θ (i) + θ · eλ·t · e−λ·τ A ) = 2 · {sin(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) + cos(θ (i) )· sin(θ · eλ·t · e−λ·τ A )} · {cos(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) − sin(θ (i) ) · sin(θ · eλ·t · e−λ·τ A )} = 2 · sin(θ (i) )· cos(θ · eλ·t · e−λ·τ A ) · cos(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) − 2 · sin(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) · sin(θ (i) ) · sin(θ · eλ·t · e−λ·τ A ) + 2 · cos(θ (i) ) · sin(θ · eλ·t · e−λ·τ A ) · cos(θ (i) ) · cos(θ · eλ·t · e−λ·τ A ) − 2 · cos(θ (i) ) · sin(θ · eλ·t · e−λ·τ A ) · sin(θ (i) ) · sin(θ · eλ·t · e−λ·τ A ) sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ (i) ) · cos2 (θ · eλ·t · e−λ·τ A ) − sin(2 · θ · eλ·t · e−λ·τ A ) · sin2 (θ (i) ) + sin(2 · θ · eλ·t · e−λ·τ A ) · cos2 (θ (i) ) − sin(2 · θ (i) ) · sin2 (θ · eλ·t · e−λ·τ A ) sin(2 · [θ (i) + θ · eλ·t ]) = sin(2 · θ (i) ) · [cos2 (θ · eλ·t · e−λ·τ A ) − sin2 (θ · eλ·t · e−λ·τ A )] + sin(2 · θ · eλ·t · e−λ·τ A ) · [cos2 (θ (i) ) − sin2 (θ (i) )] cos2 (θ · eλ·t · e−λ·τ A ) − sin2 (θ · eλ·t · e−λ·τ A ) = cos(2 · θ · eλ·t · e−λ·τ A ); cos2 (θ (i) ) − sin2 (θ (i) ) = cos(2 · θ (i) ) sin(2 · [θ (i) + θ · eλ·t · e−λ·τ A ]) = sin(2 · θ (i) ) · cos(2 · θ · eλ·t · e−λ·τ A ) + sin(2 · θ · eλ·t · e−λ·τ A ) · cos(2 · θ (i) ) π θ ∗ = θ (i) = k · ∀ k = . . . , −2, −1, 0, +1, +2, . . . ; 2 · θ (i) 2 = k · π ∀ k = . . . , −2, −1, 0, +1, +2, . . . sin(2 · θ (i) ) = sin(k · π ) = 0; cos(2 · θ (i) ) = cos(k · π ) = (−1)k sin(2 · [θ (i) + θ · eλ·t · e−λ·τ A ]) = sin(2 · θ · eλ·t · e−λ·τ A ) · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . .

176

2 Ti: Sapphire Laser Systems with Delay Parameters … ξ(θ, λ, τ A ) = 2 · θ · eλ·t · e−λ·τ A ; sin(2 · θ · eλ·t e−λ·τ A ) = sin(ξ(θ, λ, τ A )) =

∞  n=0

(−1)n · [ξ(θ, λ, τ A )]2·n+1 (2 · n + 1)!

sin(ξ(θ, λ, τ A )) =

∞  n=0

(−1)n · [ξ(θ, λ, τ A )]2·n+1 (2 · n + 1)!

[ξ(θ, λ, τ A )]3 3! [ξ(θ, λ, τ A )]7 [ξ(θ, λ, τ A )]5 − + ··· + 5! 7! ∞  (−1)n sin(ξ(θ, λ, τ A )) = · [ξ(θ, λ, τ A )]2·n+1 (2 · n + 1)! n=0 = ξ(θ, λ, τ A ) −

[2 · θ · eλ·t · e−λ·τ A ]3 3! [2 · θ · eλ·t · e−λ·τ A ]7 [2 · θ · eλ·t · e−λ·τ A ]5 − + ··· + 5! 7! ∞  (−1)n sin(ξ(θ, λ, τ A )) = · [ξ(θ, λ, τ A )]2·n+1 (2 · n + 1)! n=0 = 2 · θ · eλ·t · e−λ·τ A −

23 · θ 3 · e3·λ·t · e−3·λ·τ A 3! 27 · θ 7 · e7·λ·t · e−7·λ·τ A 25 · θ 5 · e5·λ·t · e−5·λ·τ A − + ··· + 5! 7! = 2 · θ · eλ·t · e−λ·τ A −

We consider θ m → ε ∀ m ≥ 2 then 23 · θ 3 · e3·λ·t · e−3·λ·τ A 25 · θ 5 · e5·λ·t · e−5·λ·τ A 27 · θ 7 · e7·λ·t · e−7·λ·τ A + − + ··· → ε 3! 5! 7! ∞  (−1)n sin(ξ(θ, λ, τ A )) = · [ξ(θ, λ, τ A )]2·n+1 ≈ 2 · θ · eλ·t · e−λ·τ A (2 · n + 1)! n=0

sin(2 · [θ (i) + θ · eλ·t · e−λ·τ A ]) = 2 · θ · eλ·t · e−λ·τ A · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . λ · y · eλ·t = − =−

α · F2 · cos2 (ω · t) · sin(2 · [θ (i) + θ · eλ·t · e−λ·τ A ]) μ · R2

α · F2 · cos2 (ω · t) · 2 · θ · eλ·t · e−λ·τ A · (−1)k μ · R2

−λ·y−

α · F2 · cos2 (ω · t) · 2 · θ · e−λ·τ A · (−1)k = 0 μ · R2

dθ = y ⇒ λ · θ · eλ·t = y (i) + y · eλ·t ; y∗ = y (i) = 0; −λ · θ + y = 0 dt

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

177

 α·F 2 2 −λ·τ A y −λ − μ·R · (−1)k 2 · cos (ω · t) · 2 · e · =0 θ 1 −λ  α·F 2 2 −λ·τ A −λ − μ·R · (−1)k 2 · cos (ω · t) · 2 · e A−λ· I = ; det(A − λ · I ) = 0 1 −λ



α · F2 · cos2 (ω · t) · 2 · e−λ·τ A · (−1)k = 0 ∀ k μ · R2 = . . . , −2, −1, 0, +1, +2, . . .

λ2 +

α · F2 · cos2 (ω · t) · 2 · e−λ·τ A · (−1)k = 0 ∀ k μ · R2 = . . . , −2, −1, 0, +1, +2, . . .

D(λ, τ A ) = λ2 +

We study the occurrence of any possible stability switching resulting from the increase the value of time delay τ A for the general characteristic equation D(λ, τ A ). If we choose τ A parameter then, D(λ, τ A ) = Pn (λ, τ A ) + Q m (λ, τ A ) · e−λ·τ A . The expression for Pn (λ, τ A ) is Pn (λ, τ A ) =

n 

pk (τ A ) · λk = p0 (τ A ) + p1 (τ A ) · λ + p2 (τ A ) · λ2 + · · ·

k=0

The expression for Q m (λ, τ A ) is Q m (λ, τ A ) =

m 

qk (τ A ) · λk = q0 (τ A ) + q1 (τ A ) · λ + q2 (τ A ) · λ2 + · · ·

k=0

The general characteristic equation D(λ, τ A ) is as follow: α · F2 · cos2 (ω · t) · 2 · (−1)k · e−λ·τ A μ · R2 = 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . .

D(λ, τ A ) = λ2 +

D(λ, τ A ) = Pn (λ, τ A ) + Q m (λ, τ A ) · e−λ·τ A n = 2; m = 0; n > m The expression for Pn (λ, τ A ): Pn=2 (λ, τ A ) = p1 (τ A ) · λ + p2 (τ A ) · λ2 .

n=2 k=0

pk (τ A ) · λk = p0 (τ A ) +

p0 (τ A ) = 0; p1 (τ A ) = 0; p2 (τ A ) = 1 The expression for Q m (λ, τ A ): Q m=0 (λ, τ A ) =

m=0 k=0

qk (τ A ) · λk = q0 (τ A )

178

2 Ti: Sapphire Laser Systems with Delay Parameters …

q0 (τ A ) =

α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2

The homogeneous system for θ y leads to a characteristic equation for the eigen2 j value λ having the form P(λ) + Q(λ) · e−λ·τ A ; P(λ) = j=0 a j · λ ; Q(λ) = 0 j j=0 c j ·λ and the coefficients {a j (qi , qk ), c j (qi , qk )} ∈ R depend on qi , qk but not on τ A , qi , qk are any molecule—differential equations for the rotation of a molecule system’s parameters, other parameters keep as a constant. α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k μ · R2 = . . . , −2, −1, 0, +1, +2, . . .

a0 = 0; a1 = 0; a2 = 1; c0 =

Unless strictly necessary, the designation of the variation arguments (qi , qk ) will subsequently be omitted from P, Q, a j ,c j . The coefficients a j and c j are continuous, and differential functions of their arguments, and direct substitution shows that α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2 

1 + l ; l = 0, 1, 2, . . . ; l ∈ N0 a0 + c0 = 0 ∀ qi , qk ∈ R+ ; ω · t = π · 2

a0 + c0 =

λ = 0 is not a of P(λ) + Q(λ) · e−λ·τ A = 0. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (BK] can also be verified in the present case: 1. 2.

If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0.  Q(λ)   P(λ)  is bounded for |λ| → ∞, Re λ ≥ 0. No roots bifurcation from ∞.

3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable respect to qi , qk . Remark Please do not confuse ω which is related to our stability analysis and ω which is Laser frequency (we consider it as a constant in our system) We assume that Pn (λ, τ A ) = Pn (λ) and Q m (λ, τ A ) = Q m (λ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i ·ω, τ A )+ Q m (λ = i · ω, τ A ) = 0. Pn (λ = i · ω, τ A ) = −ω2 ; Q m (λ = i · ω, τ A ) =

α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

179

α · F2 · cos2 (ω · t)· μ · R2 2 · (−1)k = 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . .

Pn (λ = i · ω, τ A ) + Q m (λ = i · ω, τ A ) = −ω2 +

|Pn (λ = i · ω, τ A )|2 = ω4 ; |Q m (λ = i · ω, τ A )|2 α2 · F 4 · cos4 (ω · t) · 4 · (−1)2·k μ2 · R 4 F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 =

= ω4 −

α2 · F 4 · cos4 (ω · t) · 4 · (−1)2·k μ2 · R 4

Since (−1)2·k = 1; ∀ k = . . . , −2, −1, 0, +1, +2, . . . then F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = ω4 −

α2 · F 4 · cos4 (ω · t) · 4 μ2 · R 4

Remark Please note that F(ω) is a function and F is one of system’s parameter. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 =

2 

2·k · ω2·k

k=0

= 4 · ω + 2 · ω + 0 4

2

α2 · F 4 · cos4 (ω · t) · 4 · (−1)2·k ; ∀ k μ2 · R 4 = . . . , −2, −1, 0, +1, +2, . . .

4 = 1; 2 = 0; 0 = −

Since (−1)2·k = 1; ∀ k = . . . , −2, −1, 0, +1, +2, . . . then 0 = − μα 2·F · ·R 4 cos4 (ω · t) · 4.  Hence F(ω) = 0 implies 2k=0 2·k · ω2·k = 0 and its roots are given by solving the above polynomial. Furthermore PR (i · ω, τ A ) = −ω2 ; PI (i · ω, τ A ) = 0; Q I (i · ω, τ A ) = 0. 2

Q R (i · ω, τ A ) =

4

α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2 −PR (i·ω,τ A )·Q I (i·ω,τ A )+PI (i·ω,τ A )·Q R (i·ω,τ A ) |Q(i·ω,τ A )|2 A )+PI (i·ω,τ A )·Q I (i·ω,τ A ) − PR (i·ω,τ A )·Q R (i·ω,τ |Q(i·ω,τ A )|2

Hence sin θ (τ A ) = And cos θ (τ A ) =

sin θ (τ A ) = 0; cos θ (τ A ) = −

−ω2 ·

α·F 2 μ·R 2

α 2 ·F 4 μ2 ·R 4

· cos2 (ω · t) · 2 · (−1)k

· cos4 (ω · t) · 4 · (−1)2·k

180

2 Ti: Sapphire Laser Systems with Delay Parameters …

Since (−1)2·k = 1; ∀ k = . . . , −2, −1, 0, +1, +2, . . . then cos θ (τ A ) = − cos θ (τ A ) =

α·F 2 · cos2 (ω · t) · 2 · μ·R 2 α 2 ·F 4 · cos4 (ω · t) · 4 μ2 ·R 4 2 2

−ω2 ·

(−1)k

μ· R ω · (−1)k · 2 · α · F 2 cos2 (ω · t)

We can summary our results for sin θ (τ A ), cos θ (τ A ): μ · R2 ω2 · · (−1)k ∀ k 2 2 2 · α · F cos (ω · t) = . . . , −2, −1, 0, +1, +2, . . .

sin θ (τ A ) = 0; cos θ (τ A ) =

 Which jointly with F(ω) = 0 ⇒ 2k=0 Ξ2·k · ω2·k = 0 that are continuous and differentiable in τ A based on Lemma 1.1. Hence we use Theorem 1.2 and this prove the Theorem 1.3. We use different parameters terminology from our last characteristic parameters definition k → j; pk (τ A ) → a j ; qk (τ A ) → c j ; n = 2; m = 0; n > m additionally Pn (λ, τ A ) → P(λ, τ A ); Q m (λ, τ A ) → Q(λ, τ A ) then P(λ, τ A ) =

2 

a j · λ ; Q(λ, τ A ) = j

j=0

2 

cj · λj.

j=0

α · F2 P(λ) = λ2 ; Q(λ) = · cos2 (ω · t) · 2 · (−1)k μ · R2 = 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable function of τ A such that a0 + c0 = 0. In the following “—” denotes complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ A . The coefficients: {a j (α, F, μ, R, . . .) & c j (α, F, μ, R, . . .)} ∈ R; depend on differential equations for the rotation of a molecule system’s parameters α, F, μ, R, . . .. Unless strictly necessary, the designation of the variation arguments. These parameters α, F, μ, R, . . . will subsequently be omitted from P, Q, aj , cj . The coefficients aj , cj are continuous (for c0 coefficient you need to choose k as an odd or even numbers to keep continuity), and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 ∀ α, F, μ, R, . . . ∈ R+ [7, 8, 9]. α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k μ · R2 = . . . , −2, −1, 0, +1, +2, . . .

a0 = 0; a1 = 0; a2 = 1; c0 =

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

181

α·F 2 μ·R 2

· cos2 (ω · t) · 2 · (−1)k = 0 ∀ k = ..., 1  −2, −1, 0, +1, +2, . . . ; α, F, μ, R, . . . ∈ R+ For ω · t = π · 2 + l ; l = 0, 1, 2, . . . ; l ∈ N0 i.e. λ = 0 is not a root of characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see [7], Sect. 3.4) can also verified in the present case. 1. If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. i.e. P and Q have no common imaginary roots. This condition can be verified numerically in the entire (differential equations for the rotation of a molecule system’s parameters α, F, μ, R, . . .)  of interest. domain   is bounded for |λ| → ∞, Re λ ≥ 0. No roots bifurcation from ∞. 2.  Q(λ) P(λ)      α·F 2 2 k  Q(λ)   μ·R 2 · cos (ω · t) · 2 · (−1)     ∀ k = . . . , −2, −1, 0, +1, +2, . . .  P(λ)  =   λ2 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω.

F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = ω4 −

α2 · F 4 · cos4 (ω · t) · 4 μ2 · R 4

4. Each positive root ω(α, F, μ, R, . . . ∈ R+ ) of F(ω) = 0 is continuous and differentiable respect to α, F, μ, R, . . . ∈ R+ . This condition can be assessed numerically. In addition, since the coefficients in P and Q are real, we have, P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus λ = i · ω; ω > 0 may be on eigenvalue of characteristic equation. The analysis consists in identifying the roots of characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the parameters α, F, μ, R, τ A . . ., Re λ may, at the  crossing,  change its sign from (−) to (+), i.e. from a stable focus E ∗ (θ ∗ , y ∗ ) = k · π2 , 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . to an unstable one, or vice versa. This feature may be further assessed by examining the sign  partial derivatives with respect to α, F, μ, R, τ A . . . parameters,  of the when other parameters are constant (α, μ, R, τ A . . .). −1 (F) = ∂Reλ ∂ F λ=i·ω  ∂Reλ ; α, μ, F, τ A . . . = const ∂ R λ=i·ω 

∂Reλ −1 (α) = ; R, μ, F, τ A . . . = const ∂α λ=i·ω −1 (R) =



182

2 Ti: Sapphire Laser Systems with Delay Parameters …

 ∂Reλ ; R, α, F, τ A . . . = const ∂μ λ=i·ω

 ∂Reλ −1 (τ A ) = ; R, α, F, μ . . . = const ∂τ A λ=i·ω

−1 (μ) =



When writing P(λ) = PR (λ) + i · PI (λ) and Q(λ) = Q R (λ) + i · Q I (λ), and inserting λ = i · ω; ω > 0 into differential equations for the rotation of a molecule system’s characteristic equation ω must satisfy the following: sin(ω · τ A ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ A ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω)|2 = 0 in the view of requirement (a) above, (g, h) ∈ R. Furthermore, it follows above equations sin(ω · τ A ) and cos(ω · τ A ) that, by squaring and adding sides, ω must be positive root of F(ω) = |P(i · ω)|2 −|Q(i · ω)|2 = 0. Note F(ω) is / I (assume that I ⊆ R+0 independent of τ A . Now it is important to notice that if τ A ∈ is the set where ω(τ A ) is a positive root of F(ω) and for τ A ∈ / I ; ω(τ A ) is not define. Then for all τ A in I, ω(τ A ) is satisfies that F(ω, τ A ) = 0). Then there are no positive ω(τ A ) solutions for F(ω, τ A ) = 0, and we cannot have stability switches. For any τ A ∈ I , where ω(τ A ) is a positive solution of F(ω, τ A ) = 0, we can define the angle θ (τ A ) ∈ [0, 2 · π ] as the solution of sin θ (τ A ) = . . . ; cos θ (τ A ) = . . . and the relation between the argument θ (τ A ) and ω(τ A ) · τ A for τ A ∈ I must be ω(τ A ) · τ A = θ (τ A ) + n · 2 · π ∀ n ∈ N0 . Hence we can define the maps τ An : )+n·2·π ; n ∈ N0 ; τ A ∈ I . Let us introduce the I → R+0 given by τ An (τ A ) = θ(τ Aω(τ A) functions I → R; Sn (τ ) = τ A − τ An (τ A ); τ A ∈ I ; n ∈ N0 that are continuous and differentiable in τ. In the following, the subscripts λ, ω, α, μ, R, τ A . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(α, μ, R, τ A . . . ∈ R+ ); ω(α, μ, R, τ A . . . ∈ R+ ) and keeping all parameters except one (x) and τ A . The deviation closely follows that in reference [BK]. Differentiating differential equations for the rotation of a molecule system’s characteristic equation P(λ) + Q(λ) · e−λ·τ A = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: Remark x = α, μ, R, F, . . . ∈ R+

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ A · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP ; Q λ = ∂∂λQ , … etc. Substituting λ = i · ω and bearing P(−i · ω) = P(i ·ω) ; Q(−i · ω) = Q(i ·ω) then i · Pλ (i ·ω) = Pω (i ·ω) ; i · Q λ (i ·ω) = Q ω (i ·ω) that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …



∂λ ∂x =

−1

183

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ A · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = QR + i · QI . Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 When (x) can be any system parameters α, μ, R, F . . . and delay parameter τ A , etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = −

Fx Fω

We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay τ A . Furthermore PR (i · ω, τ A ) = −ω2 ; PI (i · ω, τ A ) = 0 ; Q I (i · ω, τ A ) = 0 Q R (i · ω, τ A ) =

α · F2 · cos2 (ω · t) · 2 · (−1)k ∀ k = . . . , −2, −1, 0, +1, +2, . . . μ · R2

Remark Please do not confuse ω which is related to our stability analysis and ω which is Laser frequency (we consider it as a constant in our system).

PRω (i · ω, τ A ) = −2 · ω; PI ω (i · ω, τ A ) = 0; Q Rω (i · ω, τ A ) = 0; Q I ω (i · ω, τ A ) = 0 PRτ A (i · ω, τ A ) = 0; PI τ A (i · ω, τ A ) = 0; Q Rτ A (i · ω, τ A ) = 0; Q I τ A (i · ω, τ A ) = 0 V = (PR · PI τ A − PI · PRτ A ) − (Q R · Q I τ A − Q I · Q Rτ A ) = 0 PR · PI ω = 0; PI · PRω = 0; Q R · Q I ω = 0; Q I · Q Rω = 0 ⇒ U = 0; Fω = 4 · ω3 ; Fω = 0

184

2 Ti: Sapphire Laser Systems with Delay Parameters …

Fτ A = 2 · [(PRτ A · PR + PI τ A · PI ) − (Q Rτ A · Q R + Q I τ A · Q I )] Fτ = 0; ωτ A = − A = 0 Fω ∂ω ∂ω Fτ + Fτ A = 0; τ A ∈ I ; =− A ∂τ A ∂τ A Fω 

∂Reλ −1 (τ A ) = ; −1 (τ A ) ∂τ A λ=i·ω   ∂ω −2 · [U + τ A · |P|2 ] + i · Fω Fτ ; = Re = ωτ A = − A 2 ∂τ A Fω Fτ A + i · 2 · [V + ω · |P|   

∂Reλ sign{ −1 (τ A )} = sign ∂τ A λ=i·ω   ∂ω U · ∂τ +V ∂ω −1 A sign{ (τ A )} = sign{Fω } · sign τ A · +ω+ ∂τ A |P|2 Fω ·

sign{ −1 (τ A )} = sign{4 · ω3 } · sign{ω} We shall presently examine the possibility of stability transitions (bifurcations) in a differential equations for the rotation of a molecule system, about the equilibrium points E ∗ (θ ∗ , y ∗ ) = k · π2 , 0 ∀ k = . . . , −2, −1, 0, +1, +2, . . . as a result of a variation of delay parameter τ A . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λplane where by increasing the delay parameter τ A , Re λ may at the crossing, change its sign from − to +, i.e. from a stable focus E ∗ (θ ∗ , y ∗ ) to unstable one, or vice versa. This feature may be further assessed by examining  thesign of the partial −1 when other derivatives with respect to τ A , sign{ (τ A )} = sign ∂Reλ ∂τ A λ=i·ω parameters are constant where ω ∈ R+ . We need to plot the stability switch diagram based on different delay values of our system. Since it is very complex function we recommended to solve it numerically rather than analytic.    ∂Reλ −2 · [U + τ A · |P|2 ] + i · Fω = Re ∂τ A λ=i·ω Fτ A + i · 2 · [V + ω · |P|2 

∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ A · (U + τ A · P 2 )} −1 (τ A ) = = ∂τ A λ=i·ω Fτ2A + 4 · (V + ω · P 2 )2 −1 (τ A ) =



The stability switch occurs only on those delay values τ A which fit the equation: τ A = ωθ++(τ(τAA)) and θ+ (τ A ) is the solution of sin θ (τ A ) = . . . ; cos θ (τ A ) = . . . when ω = ω+ (τ A ) if only ω+ is feasible [10, 11]. Additionally when all differential equations for the rotation of a molecule system parameters are known and the stability switch due to various time delay values τ A is describe in the following expression: sign{ −1 (τ A )} = sign{Fω (ω(τ A ), τ A )} · sign{τ A · ωτ A (ω(τ A )) + ω(τ A )

2.1 Linearly Polarized Femtosecond Laser Pulses from Ti: Sapphire …

+

U (ω(τ A )) · ωτ A (ω(τ A )) + V (ω(τ A )) |P(ω(τ A ))|2

185



Remark We know F(ω, τ A ) = 0 implies it roots ωi (τ A ) and finding those delays values τ A which ωi is feasible. There are τ A values which ωi are complex or imaginary numbers, then unable to analyze stability. Lemma 1.1 Assume that ω(τ ) is a positive real root F(ω, τ ) = 0 defined for τ ∈ I , which is continuous and differentiable. Assume further that if λ = i · ω; ω ∈ R, then Pn (i · ω, τ ) + Q n (i · ω, τ ) = 0; τ ∈ R hold true. The functions Sn (τ ); n ∈ N0 , are continuous and differentiable on I. Theorem 1.2 Assume that ω(τ ) is a positive real root F(ω, τ ) = 0 defined for τ ∈ I ; I ⊆ R+0 , and some τ ∗ ∈ I ; Sn (τ ∗ ) = 0 for some n ∈ N0 then a pair of simple conjugate pure imaginary roots λ+ (τ ∗ ) = i · ω(τ ∗ ); λ− (τ ∗ ) = −i · ω(τ ∗ ) of D(λ, τ ) = 0 exist at τ = τ ∗ which crosses the imaginary axis from left to right if δ(τ ∗ ) > 0 and cross the imaginary axis from right to left if δ(τ ∗ ) < 0 where δ(τ ∗ ) = sign



   d Re λ d Sn (τ ) |λ=i·ω = sign{Fω (ω(τ ∗ ), τ ∗ )} · sign |τ =τ ∗ dτ dτ

Theorem 1.3 The characteristic equation has a pair of simple and conjugate pure imaginary roots λ = ±ω(τ ∗ ), ω(τ ∗ ) real at τ ∗ ∈ I if Sn (τ ∗ ) = τ ∗ − τn (τ ∗ ) = 0 for some n ∈ N0 . If ω(τ ∗ ) = ω+ (τ ∗ ), this pair of simple conjugate pure imaginary roots crosses crosses the imaginary axis the imaginary axis from left to right if δ+ (τ ∗ ) > 0 and  Re  λ |λ=i·ω+ (τ ∗ ) . from right to left if δ+ (τ ∗ ) < 0 where δ+ (τ ∗ ) = sign d dτ    d Sn (τ ) d Re λ δ+ (τ ) = sign |λ=i·ω+ (τ ∗ ) = sign |τ =τ ∗ dτ dτ ∗



If ω(τ ∗ ) = ω− (τ ∗ ), this pair of simple conjugate pure imaginary roots cross the imaginary axis from left to right, if δ− (τ ∗ ) > 0 and crosses   Re λthe imaginary axis from right to left. If δ− (τ ∗ ) < 0 where δ− (τ ∗ ) = sign d dτ |λ=i·ω− (τ ∗ ) =   n (τ ) −sign d Sdτ |τ =τ ∗ .   Re λ ∗ |λ=i·ω(τ ∗ ) = 0, If ω+ (τ ) = ω− (τ ∗ ) = ω(τ ∗ ) then (τ ∗ ) = 0 and sign d dτ the same is true when Sn (τ ∗ ) = 0 the following result can be useful in identifying values of τ where stability switches happened. Remark Lemma 1.1 and Theorems 1.2 and 1.3, In our analysis we discuss delay parameter τ A .

186

2 Ti: Sapphire Laser Systems with Delay Parameters …

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire Laser Stability Analysis Under Parameters Variation We can shape ultrashort pulses with a resolution approach 10 fsec by a liquid-crystal spatial light modulator within reflective optics pulse-shaping apparats. The principle is to use a spatial light modulator as a phase modulator. A Spatial Light Modulator (SLM) is an object that imposes some form of spatially varying modulation on a beam of light. Usually, and SLM modulates the intensity of the light beam. It is possible to produce devices that modulate the phase of the beam or both the intensity and the phase simultaneously. SLMs are used extensively in holographic data storage setups to encode information into a laser beam. Liquid crystal SLMs can help to solve problems related to laser micro particle manipulation. In this case spiral beam parameters can be changed dynamically. The outcome is a variety of complex ultrafast waveforms, including odd pulses, high repetition rate pulse trains, and asymmetric pulse trains. There is a limitation of shaping ultra-broad-bandwidth pulses. The synthesis of general waveforms may require compensating for nonlinear spatial dispersion of frequency in the masking plane. The implementation of cubic and quartic-phase modulations of the pulse helps to compensate for large amounts of high-order phase dispersion. One of the powerful methods for synthesizing complex femtosecond optical waveforms is pulse shaping within a dispersion-free gratinglens apparatus. The optical pulse is first transformed from the time domain into the frequency domain. We modify the spectrum of the pulse by a frequency dependent complex linear filter. When a filter is a linear filter (but not necessarily time-invariant), and its input is a complex signal, then, by linearity, If the filter is real, then filtering of complex signals can be carried out by simply performing real filtering on the real and imaginary parts separately (thereby avoiding complex arithmetic) (Fig. 2.3). Finally, the pulse is transformed back into the time domain. We get output pulse which as a temporal field profile that is essentially the Fourier transform of the frequency filter imposed on the electric-field spectrum of the pulse [2, 12]. Optical pulse (Time domain) Conversion unit (from time domain to frequency domain) Frequency dependent complex linear filter

Conversion unit (from frequency domain to time domain)

Optical pulse (Time domain) with Modifies spectrum

Fig. 2.3 Spectrum modify procedure of the pulse by a frequency dependent complex linear filter

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

187

There are many applications which use shape pulses, observation of the fundamental dark soliton in optical fiber and mode-selective excitation of coherent photons. By femtosecond pulse shaping- we dynamic control of molecular and chemical system. The result is system which driven into user-specific final quantum state by use of tailored optical fields to interact coherently with and to manipulate the dynamic evolution of the system. Other areas which can be implemented are solid state electronics and opto electronics. Mainly the pulses can be shaped by dynamically reconfigurable filters (masks), which is a good method. The method includes the use of a single liquid-crystal modulator within a pulse shaper. It can alter either pulse or amplitude independently, the use of two liquid-crystal modulators within a modified pulse shaper, which provide independent control of both phase and amplitude. We use acousto-optic modulator within a pulse shaping apparatus, which simultaneously modulate phase and amplitude. Spectral holographic techniques are complex optical processing and waveform synthesis by convolution and time-to-space conversion. Shaped pulses are influenced by quality of the input pulses which were generated by the use of spatial soliton pulse compression techniques. The mode locked Ti: Sapphire lasers extend these methods to pulses that approach 10 fsec in duration and, to explore the limits imposed by short-duration, ultra-broad bandwidth optical pulses. It is possible to produce pulses less than 10 fsec in duration directly from a laser oscillator. It is a challenge to extend spectral-domain pulse-shaping techniques to ultra-broad bandwidth pulses and investigate the limitations inherent in tailoring ultra-short wave form. The schematic of the ultra-broad bandwidth pulse-shaping apparatus is described below (Fig. 2.4). The diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams (red and blue in our case) travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light and the grating acts as the dispersive element. The spherical mirror is a mirror which has the shape of a piece cut out of a spherical surface. There are two types of spherical mirrors: concave and convex. The concave mirror is a spherical mirror who’s reflecting surface faces towards the center of the sphere and convex mirror is a spherical mirror with reflecting surface curved outwards. In our system we use a concave mirror as the first reflector of the light comes from the first grating and direct to Liquid Crystal SLM and as the second reflector from the light comes from Liquid Crystal SLM to the second grating. Concave mirrors have a curved surface with a center of curvature equidistant from every point on the mirror’s surface. An object beyond the center of curvature forms a real and inverted image between the focal point and the center of curvature. Input Optical Pulse

Grating

red blue

red

Spherical mirror blue (concave)

red

Liquid crystal blue SLM

red

Spherical mirror Grating (concave) blue

Fig. 2.4 Ultra-broad bandwidth pulse-shaping apparatus schematic

Output Optical Pulse

188

2 Ti: Sapphire Laser Systems with Delay Parameters …

Our pulse-shaping system is performed with one-dimensional liquid-crystal SLM. The system consists of a pair of gratings placed at the focal planes of a unit magnification confocal pair of concave focal-length gold spherical mirrors. Mid way through the apparatus the optical frequencies are spatially separated, with a linear spatial dispersion given by ddλx ≈ d·cosf θd ( ddλx —linear spatial dispersion, f —focal length, d—grating period, θd —diffraction angle, x—position of each frequency component in the masking plane). The compensation of cubic and quartic phase dispersions is very important in our analysis. It is a compensation of high-order phase dispersion in pulse compression and pulse-amplification. The propagation of ultra-board bandwidth pulses through dispersive media results in severe pulse broadening as frequency dependent phase shifts accumulate. For optical pulses with spectra not  1 , the phase of an optical pulse propagating through dispersive too broad ω ω0 media can be expressed by Taylor series: (ω) = (ω0 ) + (1) (ω) · (ω − ω0 ) + +

1 · (2) (ω) · (ω − ω0 )2 2!

1 1 · (3) (ω) · (ω − ω0 )3 + · (4) (ω) · (ω − ω0 )4 + · · · 3! 4!

 n  where ω0 is the center frequency of the pulse and (n) (ω) = ddωn ω=ω is the 0 nth-order derivative of the phase evaluated at ω0 . The group delay and the linear d (1) dispersion of an optical pulse are given by the linear term  (ω) = dω |ω=ω0 and 2 the quadratic term (2) (ω) = ddω2 |ω=ω0 respectively. In optics the group delay is the rate of change of the total phase shift with respect to angular frequency. We can get from our system variety of temporal waveforms by using ultra-board bandwidth pulses coupled with programmable pulse shaping techniques. There are limitations which ultra-board bandwidths impose on pulse shaping. Large bandwidths affect the quality of shaped pulses: 1. Higher order phase dispersion of the pulse during propagation through the shaper, which can lead to pulse broadening and loss of temporal resolution. 2. Higher order dispersion of the frequency components in the Fourier plane of the shaper, which can lead to deviations from the ideal pulse shape. 3. Amplitude and phase filtering of the pulse from other optical elements (gratings, mirrors) in the shaper, which lead to pulse broadening. There are effects of higher-order phase dispersion on the fidelity of the shaped pulse. Any residual phase dispersion can be automatically compensated for the adjustment of the phases of individual modulator pixels. Any material (SLM) contributions to positive second order group velocity dispersion (GVD) within the shaper can be canceled if the position of the final grating is adjusted. Dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. Group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. There is phase dispersion at higher orders, specially cubic and quartic phase dispersions.

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

189

Both the material and the grating contribution can be analyze and we can estimate the material contributions of quadratic cubic and quartic phase distortions by the , where L is the propagation distance then fiber propagation parameter β(ω) = (ω) L  n  d β (n) (ω) (n) β (ω) = L = dωn . The derivatives β (n) (ω) are given in terms of the index of refraction.

ω=ω0



λ3 d 2 n (3) λ4 d 2n d 3n · ; β (ω) = − · 3 · + λ · 2 · π · c2 dλ2 4π 2 c3 dλ2 dλ3 

λ5 3 d 2n d 3n λ2 d 4 n β (4) (ω) = 3 4 · · 2 +λ· 3 + · π ·c 4 dλ dλ 4 dλ4

β (2) (ω) =

The refraction index (n) is dependent on the wavelength (λ), n = n(λ). The refraction index (n) is equal to the speed of light divided by the wave’s speed v. The wave speed v is related to both the frequency f and the wavelength λ (v = λ · f ). The wavelength in a medium is λ = vf and the wavelength in vacuum is λ0 = cf . The frequency (f) does not change as light moves from one medium to another. c/ f c c/ f λ0 ;n = = ; n(λ) = λ v/ f v λ

2  (2) d (ω) β  = β (2) (ω) = ; β (3) (ω) L dω2 ω=ω0

3  (3) (ω) d β = = ; β (4) (ω) L dω3 ω=ω0

4  (4) (ω) d β = = L dω4 ω=ω0 n(λ) =

We take the second, third, and fourth derivative of propagation parameter (β(ω)) in ω as a constant at ω = ω0 . We define new variables: Y1 , Y2 , Y3 , Y4 . d 2n d 3n d 4n dn dY1 dY2 dY3 , Y2 = = = = , Y = , Y = 3 4 dλ dλ dλ2 dλ dλ3 dλ dλ4 

3 4 λ dY1 (3) λ dY2 ; β (ω) = − 2 3 · 3 · Y2 + λ · β (2) (ω) = · 2 · π · c2 dλ 4π c dλ 

5 2 λ 3 λ dY3 · Y2 + λ · Y3 + · β (4) (ω) = 3 4 · π ·c 4 4 dλ dY1 2 · π · c2 λ3 dY1 ⇒ = β (2) (ω) = · · β (2) (ω) 2 · π · c2 dλ dλ λ3 

λ4 dY2 dY2 (3) ⇒ 3 · Y2 + λ · β (ω) = − 2 3 · 3 · Y2 + λ · 4π c dλ dλ 2 3 4π c = − 4 · β (3) (ω) λ Y1 =

190

2 Ti: Sapphire Laser Systems with Delay Parameters …

4π 2 c3 dY2 3 = − 5 · β (3) (ω) − · Y2 dλ λ λ 

3 λ5 3 λ2 dY3 ⇒ · Y2 β (4) (ω) = 3 4 · · Y2 + λ · Y3 + · π ·c 4 4 dλ 4 2 3 4 λ dY3 π ·c + λ · Y3 + · β (4) (ω) · = 4 dλ λ5 π 3 · c4 3 λ2 dY3 dY3 · Y2 + λ · Y3 + · = · β (4) (ω) ⇒ 5 4 4 dλ λ dλ 3 4 4 π 3 · c4 (4) · β (ω) − 2 · Y2 − · Y3 = 2· λ λ5 λ λ We can summary our system set of differential equations with the new variables: dY2 3 dY1 2 · π · c2 4 · π 2 · c3 (2) · β (ω); · β (3) (ω) − · Y2 = = − 3 5 dλ λ dλ λ λ 4 · π 3 · c4 dY3 3 4 (4) = · β (ω) − 2 · Y2 − · Y3 dλ λ7 λ λ 2 · π · c2 dY1 2 · π · c2 (2) = · β (ω) ⇒ Y = · β (2) (ω); λ 2 3 3 dλ λ λ  3 2 · π · c2 · β (2) (ω) = √ 3 Y2 √ √ 3 1 Y2 ( 3 Y2 )5 1  = = ; 3 λ 2 · π · c2 · β (2) (ω) λ5 ( 3 2 · π · c2 · β (2) (ω))5 √ 1 ( 3 Y2 )2  = λ2 ( 3 2 · π · c2 · β (2) (ω))2 √ 1 ( 3 Y2 )7 =  λ7 ( 3 2 · π · c2 · β (2) (ω))7 We get the following two system differential equations: √ 4 · π 2 · c3 · ( 3 Y2 )5 dY2 =−  · β (3) (ω) dλ ( 3 2 · π · c2 · β (2) (ω))5 √ 3 · 3 Y2 − · Y2 3 2 · π · c2 · β (2) (ω) √ 4 · π 3 · c4 · ( 3 Y2 )7 dY3 =  · β (4) (ω) dλ ( 3 2 · π · c2 · β (2) (ω))7 √ √ 3 · ( 3 Y2 )2 4 · 3 Y2  −  · Y − · Y3 2 3 2 · π · c2 · β (2) (ω) ( 3 2 · π · c2 · β (2) (ω))2

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

191

We can write our system’s two differential equations: dY2 3 4 · π 2 · c3 · β (3) (ω) 5 4 · (Y2 ) 3 · (Y2 ) 3 −  =−  3 3 2 (2) 5 2 (2) dλ ( 2 · π · c · β (ω)) 2 · π · c · β (ω) 4 · π 3 · c4 · β (4) (ω) dY3 7 =  · (Y2 ) 3 3 2 (2) 7 dλ ( 2 · π · c · β (ω)) 3 4 5 1 −  · (Y2 ) 3 −  · (Y2 ) 3 · Y3 3 3 2 (2) 2 2 (2) ( 2 · π · c · β (ω)) 2 · π · c · β (ω) At fixed points:

dY2 dλ

3 = 0; dY = 0. dλ

4 · π 2 · c3 · β (3) (ω) dY2 3 5 4 =0⇒−  · (Y2∗ ) 3 −  · (Y2∗ ) 3 = 0 3 3 dλ ( 2 · π · c2 · β (2) (ω))5 2 · π · c2 · β (2) (ω)   4 · π 2 · c3 · β (3) (ω) 3 ∗ 43 ∗ 13 − (Y2 ) ·  · (Y2 ) +  =0 3 ( 3 2 · π · c2 · β (2) (ω))5 2 · π · c2 · β (2) (ω) Option 1: Y2∗ = 0 4·π 2 ·c3 ·β (3) (ω) Option 2: √ 3 2 (2) (

2·π·c ·β

· (Y2∗ ) 3 + √ 3 1

(ω))5

(Y2∗ ) 3 = − 1

 3

3

(Y2∗ ) = −  3 1 3

3 2·π·c2 ·β (2) (ω)

= 0.

2 · π · c2 · β (2) (ω)

5

· 4 · π 2 · c3 · β (3) (ω) 2 · π · c2 · β (2) (ω)  4 3 · 3 2 · π · c2 · β (2) (ω) 4 · π 2 · c3 · β (3) (ω)

(Y2∗ ) 3 = −

3 · (2 · π · c2 · β (2) (ω)) 3 4 · π 2 · c3 · β (3) (ω)

(Y2∗ ) 3 = −

3 · 2 3 · π 3 · c 3 · [β (2) (ω)] 3 4 · π 2 · c3 · β (3) (ω)

(Y2∗ ) 3 = −

3 · 23 [β (2) (ω)] 3 4 8 · π ( 3 −2) · c( 3 −3) · 4 β (3) (ω)

4

1

4

1

4

8

4

4

1

4

[β (2) (ω)] 3 3 · 23 4 8 · π ( 3 −2) · c( 3 −3) · =− 4 β (3) (ω) 4

1 (Y2∗ ) 3

4

3 · 23 [β (2) (ω)] 3 2 1 · π − 3 · c− 3 · 4 β (3) (ω) 4

(Y2∗ ) 3 = − 1



Y2∗

4

4

4

2 1 3 · 23 [β (2) (ω)] 3 = − · π − 3 · c− 3 · 4 β (3) (ω)

3

; Y2∗ = −

33 · 24 [β (2) (ω)]4 · π −2 · c−1 · (3) 3 4 [β (ω)]3

192

2 Ti: Sapphire Laser Systems with Delay Parameters … dY3 3 4 · π 3 · c4 · β (4) (ω) ∗ 7 ∗ 5 = 0 ⇒  7 · (Y2 ) 3 −  2 · (Y2 ) 3 dλ 3 3 2 (2) 2 (2) 2 · π · c · β (ω) 2 · π · c · β (ω) 1 4 · (Y2∗ ) 3 · Y3∗ = 0 − 3 2 (2) 2 · π · c · β (ω) 1 7 4 · π 3 · c4 · β (4) (ω) 4  · (Y2∗ ) 3 · (Y2∗ ) 3 · Y3∗ =  3 3 2 (2) 2 (2) 7 2 · π · c · β (ω) ( 2 · π · c · β (ω)) 3 ∗ 35 −  · (Y2 ) ( 3 2 · π · c2 · β (2) (ω))2

Y3∗ =

4·π 3 ·c4 ·β (4) (ω) 7 √ 3 2·π ·c2 ·β (2) (ω)

7

· (Y2∗ ) 3 −  √ 3

√ 3

4 2·π ·c2 ·β (2) (ω)

 4·π 3 ·c4 ·β (4) (ω)

√ 3 2·π ·c2 ·β (2) (ω)

Y3∗ =

3

2·π ·c2 ·β (2) (ω)

∗ 6 7 · (Y2 ) 3

1

− √ 3

Y3∗ =

 3

2·π ·c2 ·β (2) (ω)

4 2·π ·c2 ·β (2) (ω)

4·π 3 ·c4 ·β (4) (ω) 7 √ 3 2·π ·c2 ·β (2) (ω)

5

· (Y2∗ ) 3

· (Y2∗ ) 3

√ 3 

2

∗ 4 2 · (Y2 ) 3

1

· (Y2∗ ) 3

1

· (Y2∗ ) 3

6

· (Y2∗ ) 3 −  √ 3

3

2·π ·c2 ·β (2) (ω)

 2

4

· (Y2∗ ) 3

4 = 0 ; 3 2 · π · c2 · β (2) (ω)

4 √ 3 2·π ·c2 ·β (2) (ω)

Option 1: Y2∗ = 0 ⇒ Y3∗ = 0 3 4 Option 2: Y2∗ = − 3 4·23 · π −2 · c−1 ·

[β (2) (ω)]4 ; Y2∗ [β (3) (ω)]3

= − 232 · π −2 · c−1 · 3

[β (2) (ω)]4 [β (3) (ω)]3

⎧ ⎪ 1 ⎨ Y3∗ = ·  4 ⎪ ⎩ 3

4 · π 3 · c4 · β (4) (ω) ∗ 6 7 · (Y2 ) 3  2 · π · c2 · β (2) (ω) 3 2 · π · c2 · β (2) (ω) ⎫ ⎪ ⎬  3 3 ∗ 43 −  2 · (Y2 ) ⎪ · 2 · π · c2 · β (2) (ω) 3 ⎭ 2 (2) 2 · π · c · β (ω)

 3

2 · π · c2 · β (2) (ω)

6

 = 22 · π 2 · c4 · [β (2) (ω)]2 ; 3 2 · π · c2 · β (2) (ω) = 2 3 · π 3 · c 3 · [β (2) (ω)] 3 1



1

2

1

4 · π 3 · c4 · β (4) (ω) 3 6 ∗ 43 · (Y2∗ ) 3 − 1 1 2 1 · (Y2 ) 2 2 4 (2) 2 2 · π · c · [β (ω)] 2 3 · π 3 · c 3 · [β (2) (ω)] 3   (4) π · β (ω) 1 3 4 ∗ 3 · (Y2∗ )2 − 1 Y3∗ = · 1 2 1 · (Y2 ) 4 [β (2) (ω)]2 2 3 · π 3 · c 3 · [β (2) (ω)] 3

Y3∗

1 = · 4



2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

193

3 2 (2) 8 3 [β (2) (ω)]4 36 −4 −2 [β (ω)] (Y2∗ )2 = − 2 · π −2 · c−1 · (3) = · π · c · 2 [β (ω)]3 26 [β (3) (ω)]6 4

3 3 [β (2) (ω)]4 3 4 (Y2∗ ) 3 = − 2 · π −2 · c−1 · (3) 2 [β (ω)]3 (Y2∗ ) 3 =

[β (2) (ω)] 3 [β (3) (ω)]4 36 [β (2) (ω)]8 · 6 · π −4 · c−2 · (3) 2 [β (ω)]6 16

34

−3 · c− 3 · 8 · π 23  π · β (4) (ω) 1 Y3∗ = · 4 [β (2) (ω)]2 4

8

4

−

[β (2) (ω)] 3 · [β (3) (ω)]4

16

34

3

− 83

− 43



·c 1 1 2 1 · 8 · π 2 3 · π 3 · c 3 · [β (2) (ω)] 3 2 3   1 35 1 3 [β (2) (ω)]6 [β (2) (ω)]5 ∗ (4) Y3 = · 3 · 3 2 · 3 · (3) · β (ω) − (3) 4 2 π ·c 2 [β (ω)]6 [β (ω)]4 We can summary our system fixed points in Table 2.2. (k) (ω) β (ω) = = L (k)



dkβ dωk

 ω=ω0

; k = 2, 3, 4

Stability analysis: Our ultra-board bandwidth pulse from a Ti: Sapphire laser (n) system, derivatives of propagation  k parameter β  (ω) in terms of the index of d n are presented by the following refraction n = n(λ) derivatives dλ k ; k = 2, 3, 4   2 dY2 d n d3n 1 : differential equations Y2 = dY = , Y = = 3 2 3 dλ dλ dλ dλ dY2 4 · π 2 · c3 · β (3) (ω) 5 =−  · (Y2 ) 3 3 2 (2) 5 dλ ( 2 · π · c · β (ω)) 3 4 · (Y2 ) 3 − 3 2 (2) 2 · π · c · β (ω) 4 · π 3 · c4 · β (4) (ω) dY3 7 =  · (Y2 ) 3 3 dλ ( 2 · π · c2 · β (2) (ω))7 Table 2.2 System (derivatives β (n) (ω) which given in terms of the index of refraction) fixed points Fixed point Y2∗

Y3∗

First fixed point

0

0

Second fixed point

− 232 · π −2 · c−1 ·

3

[β (2) (ω)]4 [β (3) (ω)]3

1 4

·

35 23

·

1 π 3 ·c2

·



3 23

·

[β (2) (ω)]6 [β (3) (ω)]6

· β (4) (ω) −

[β (2) (ω)]5 [β (3) (ω)]4



194

2 Ti: Sapphire Laser Systems with Delay Parameters …

3 4 5 1 · (Y2 ) 3 · Y3 −  · (Y2 ) 3 −  3 3 ( 2 · π · c2 · β (2) (ω))2 2 · π · c2 · β (2) (ω) We use linearization technique and we can approximate the phase portrait near a fixed point by that of a corresponding linear system. We consider our system dY2 3 = f 1 (Y2 , Y3 ); dY = f 2 (Y2 , Y3 ). dλ dλ 4 · π 2 · c3 · β (3) (ω) 5 f 1 (Y2 , Y3 ) = −  · (Y2 ) 3 3 2 (2) 5 ( 2 · π · c · β (ω)) 3 4 · (Y2 ) 3 − 3 2 · π · c2 · β (2) (ω) 4 · π 3 · c4 · β (4) (ω) 7 f 2 (Y2 , Y3 ) =  7 · (Y2 ) 3 3 2 · π · c2 · β (2) (ω) 3 4 5 1 −  · (Y2 ) 3 −  · (Y2 ) 3 · Y3 3 3 2 (2) 2 2 (2) 2 · π · c · β (ω) ( 2 · π · c · β (ω)) And we suppose that (Y2∗ , Y3∗ ) is a fixed point, f 1 (Y2∗ , Y3∗ ) = 0; f 2 (Y2∗ , Y3∗ ) = 0. Let u = Y2 −Y2∗ ; v = Y3 −Y3∗ denote the components of a small disturbance from the fixed point [5, 6, 13]. We need to inspect whether the disturbance grows or decays, 2 = dY since Y2∗ is a constant we need to derive differential equations for u and v. du dλ dλ then dY2 du = = f 1 (u + Y2∗ , v + Y3∗ ) dλ dλ du ∂ f1 ∂ f1 = f 1 (u + Y2∗ , v + Y3∗ ) = f 1 (Y2∗ , Y3∗ ) + u · +v· + O(u 2 , v2 , u · v) dλ dY2 dY3 Y2 = u + Y2∗ ; Y3 = v + Y3∗ ;

du ∂ f1 ∂ f1 +v· + O(u 2 , v2 , u · v), since f 1 (Y2∗ , Y3∗ ) = 0 = f 1 (u + Y2∗ , v + Y3∗ ) = u · dλ dY2 dY3

The partial derivatives are to be evaluated at the fixed point (Y2∗ , Y3∗ ); thus they are numbers, not functions. The O(u 2 , v2 , u · v) denotes quadratic terms in u and v. Since u and v are small, these quadratic terms are extremely small. Similarly we find Y2 = u + Y2∗ ; Y3 =v + Y3∗ ;

dY3 dv = = f 2 (u + Y2∗ , v + Y3∗ ) dλ dλ

dv = f 2 (u + Y2∗ , v + Y3∗ ) = f 2 (Y2∗ , Y3∗ ) dλ ∂ f2 ∂ f2 +u· +v· + O(u 2 , v2 , u · v) dY2 dY3 dv dλ

∂ f2 ∂ f2 = f 2 (u +Y2∗ , v +Y3∗ ) = u · dY +v · dY + O(u 2 , v2 , u ·v), since f 2 (Y2∗ , Y3∗ ) = 0 2 3

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

⎛ du ⎞ ⎜ dλ ⎟ ⎝ ⎠= dv dλ



∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3

195

 u · + quadratic terms v 

The Jacobian matrix at fixed point (Y2∗ , Y3∗ ): A =

∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3

. (Y2∗ ,Y3∗ )

The quadratic terms are tiny and we can neglect them. We get the linearized system. ⎛ du ⎞

  ∂ f1 ∂ f1 u ⎜ dλ ⎟ 2 ∂Y3 · ⎠ = ∂Y ⎝ ∂ f2 ∂ f2 dv v ∂Y2 ∂Y3 dλ 4 · π 2 · c3 · β (3) (ω) 5 ∂ f 1 (Y2 , Y3 ) 2 = −  5 · · (Y2 ) 3 ∂Y2 3 3 2 · π · c2 · β (2) (ω) − 3

3 2·π ·

c2

·

β (2) (ω)

·

∂ f 1 (Y2 , Y3 ) 4 1 · (Y2 ) 3 ; =0 3 ∂Y3

∂ f 2 (Y2 , Y3 ) 4 · π 3 · c4 · β (4) (ω) 7 4 =  7 · (Y2 ) 3 ∂Y2 3 3 2 · π · c2 · β (2) (ω) 3 5 2 −  2 · · (Y2 ) 3 3 3 2 · π · c2 · β (2) (ω) 4

− 3

β (2) (ω)

·

1 2 · (Y2 )− 3 · Y3 3

2·π · · ∂ f 2 (Y2 , Y3 ) 4 1 = − · (Y2 ) 3 3 2 (2) ∂Y3 2 · π · c · β (ω) c2

Case A: First fixed point E (0) (Y2(0) , Y3(0) ) = (0, 0).  A=

∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3

= (Y2∗ ,Y3∗ )=(0,0)

Then we have a plane of fixed points. Case B: Second fixed point



00 00



196

2 Ti: Sapphire Laser Systems with Delay Parameters …

E

 A=  =

(1)

(Y2(1) , Y3(1) )

∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3

∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3



3 3 [β (2) (ω)]4 1 35 1 = − 2 · π −2 · c−1 · (3) , · · · 2 [β (ω)]3 4 23 π 3 · c2   3 [β (2) (ω)]6 [β (2) (ω)]5 (4) · · β (ω) − 23 [β (3) (ω)]6 [β (3) (ω)]4

(Y2(1) ,Y3(1) )



 3   (2) 4 (2) 6 (2) 5 5 (Y2(1) ,Y3(1) )= − 232 ·π −2 ·c−1 · [β (3) (ω)]3 , 41 · 233 · π 31·c2 · 233 · [β (3) (ω)]6 ·β (4) (ω)− [β (3) (ω)]4 [β

(ω)]

[β

(ω)]

[β

(ω)]

(3)

∂ f 1 (Y2 , Y3 ) 4 · π · c · β (ω) |(Y (1) ,Y (1) ) = −  5 · 2 3 ∂Y2 3 2 · π · c2 · β (2) (ω) 2

3

2

3 (2) 4 3 3 5 −2 −1 [β (ω)] · − 2 · π · c · (3) 3 2 [β (ω)]3 1

3 (2) 4 3 3 3 4 −2 −1 [β (ω)] − · · − 2 · π · c · (3) 3 2 [β (ω)]3 2 · π · c2 · β (2) (ω) 3

∂ f 1 (Y2 , Y3 ) |(Y (1) ,Y (1) ) = 0 2 3 ∂Y3

4  (2) 4 3 4 · π 3 · c4 · β (4) (ω) 7 ∂ f 2 (Y2 , Y3 ) 33 −2 −1 [β (ω)] |(Y (1) ,Y (1) ) =  − 2 · π · c · (3) 7 · 2 3 ∂Y2 3 2 [β (ω)]3 3 2 · π · c2 · β (2) (ω) 2  (2) 4 3 5 33 −2 −1 [β (ω)] −  · · − 2 · π · c · (3) 2 [β (ω)]3 ( 3 2 · π · c2 · β (2) (ω))2 3 − 2  3 (2) 4 4 1 33 −2 −1 [β (ω)] − · π · c · · · · − 3 22 [β (3) (ω)]3 2 · π · c2 · β (2) (ω) 3    1 3 [β (2) (ω)]6 [β (2) (ω)]5 1 35 (4) · · β (ω) − · 3 · 3 2 · 4 2 π ·c 23 [β (3) (ω)]6 [β (3) (ω)]4 3

1

3 (2) 4 3 ∂ f 2 (Y2 , Y3 ) 3 4 −2 −1 [β (ω)] · − 2 · π · c · (3) = − 3 ∂Y3 2 [β (ω)]3 2 · π · c2 · β (2) (ω) (Y2(1) ,Y3(1) )

We define the following parameters: ∂ f 1 (Y2 , Y3 ) ∂ f 1 (Y2 , Y3 ) |(Y (1) ,Y (1) ) ; 2 = |(Y (1) ,Y (1) ) 2 3 2 3 ∂Y2 ∂Y3 ∂ f 2 (Y2 , Y3 )

3 = |(Y (1) ,Y (1) ) 2 3 ∂Y2

1 =

2.2 Ultraboard-Bandwidth Pulse from a Ti: Sapphire …

4 =  A=  A−λ· I =

=

∂ f1 ∂Y2 ∂ f2 ∂Y2 ∂ f1 ∂Y2 ∂ f2 ∂Y2

∂ f1 ∂Y3 ∂ f2 ∂Y3 ∂ f1 ∂Y3 ∂ f2 ∂Y3



∂ f 2 (Y2 , Y3 ) ∂Y3 (Y2(1) ,Y3(1) )

=



197

(Y2(1) ,Y3(1) )

−λ·

(Y2(1) ,Y3(1) )

0

1 − λ

3 4 − λ

1 2

3 4 10 01



=



=

1 0

3 4



1 − λ 2

3 4 − λ





det(A − λ · I ) = ( 1 − λ) · ( 4 − λ) = 1 · 4 − 1 · λ − 4 · λ + λ2 = λ2 − λ · ( 1 + 4 ) + 1 · 4 det(A − λ · I ) = 0 ⇒ λ2 − λ · ( 1 + 4 ) + 1 · 4 = 0; τ = trace(A) = 1 + 4 ;  = det(A) = 1 · 4 

1 + 4 + ( 1 + 4 )2 − 4 · 1 · 4 τ2 − 4 ·  = λ1 = 2 2  √

1 + 4 − ( 1 + 4 )2 − 4 · 1 · 4 τ − τ2 − 4 ·  = λ2 = 2 2 τ+

√

2

3 (2) 4 3 3 4 · π 2 · c3 · β (3) (ω) 5 −2 −1 [β (ω)] τ = trace(A) = −  5 · · − 2 · π · c · (3) 3 2 [β (ω)]3 3 2 · π · c2 · β (2) (ω) 1

3 (2) 4 3 3 4 −2 −1 [β (ω)] · · − 2 · π · c · (3) − 3 2 [β (ω)]3 2 · π · c2 · β (2) (ω) 3 1

3 (2) 4 3 3 4 −2 −1 [β (ω)] · − 2 · π · c · (3) − 3 2 [β (ω)]3 2 · π · c2 · β (2) (ω)  2

3 (2) 4 3 3 4 · π 2 · c3 · β (3) (ω) 5 −2 −1 [β (ω)]  = det(A) = −  · · − 2 · π · c · (3) 2 [β (ω)]3 ( 3 2 · π · c2 · β (2) (ω))5 3  1

3 (2) 4 3 3 3 4 −2 −1 [β (ω)] · − − · π · c · · 3 22 [β (3) (ω)]3 2 · π · c2 · β (2) (ω) 3  1

3 (2) 4 3 3 4 −2 −1 [β (ω)] · − 2 · π · c · (3) · − 3 2 [β (ω)]3 2 · π · c2 · β (2) (ω)

3

198

2 Ti: Sapphire Laser Systems with Delay Parameters …

Table 2.3 System (derivatives β (n) (ω) which given in terms of the index of refraction) classification of the second fixed point Condition

Outcome

0

Eigenvalues are either real with same sign (nodes—satisfy τ 2 − 4 ·  > 0), or complex conjugate (spiral and centers). Spirals satisfy τ 2 − 4 ·  < 0

=0

At least one of the eigenvalues is zero. There is either a whole line of fixed points or a plane of fixed points if A = 0

τ 2 − 4 ·  = 0 Star nodes and degenerate nodes live on this parabola; it is the borderline between nodes and spirals τ 0

Unstable spiral and nodes

τ =0

Neutrally stable centers live on the borderline, eigenvalues are purely imaginary

Classification of the second fixed point (Table 2.3).

2.3 Multipulse Operation of a Ti: Sapphire Laser Mode Semiconductor Saturable-Absorbed Mirror Pulse Energies and Gain Delayed in Time Stability Analysis Ti: Sapphire laser can be operates with mode of multiple-pulse. It is mode locked by a semiconductor saturable absorber mirrors (SESAM’s). First, saturable absorption is a property of materials where the absorption of the light decreases with increasing light intensity. Saturable absorbers are useful in laser cavities and the key parameters for a saturable absorber are its wavelength range where it absorbs and how fast it recovers which it dynamic response, saturation intensity which is the intensity level or pulse energy it saturates. We can describe it by the absorption rate (A) equation: A = 1+α I , A—absorption rate, I—light intensity,α—linear absorption, I0

I0 —saturation intensity. A semiconductor saturable absorber mirror (SESAM) is a mirror structure with an incorporated saturable absorber in semiconductor technology. It is used for the generation of ultrashort pulses by passive mode locking of various types of lasers. The SESAM contains a semiconductor Bragg mirror and near the surface a single quantum well absorber layer. The Bragg mirror material has larger hand gap energy and no absorption occurs in that region. It is a saturable Bragg reflector (SBRs). We can calculate the penetration of the optical field into a SESAM by matrix method which is applied to other types of dielectric mirrors. The design of the structure influences the bandwidth and the chromatic dispersion. The absorber layer is placed in an anti-node of the electric field and the results are

2.3 Multipulse Operation of a Ti: Sapphire Laser …

199

maximum saturable absorption and the smallest possible saturation fluency. Another option is to use multiple absorber layers which give us high modulation depth. The multiple absorber layers are placed in separate anti-nodes or near one anti-node. Figure 2.5 describes the structure of typical SESAM at specific wavelength. On a GaAs substrate, a GaAs/AlGaAs Bragg mirror is grown within the top layers, there is an InGaAs quantum well. The most important characteristics of a SESAM are modulation depth, saturation fluence, recovery time and non saturable losses. The modulation depth is the maximum nonlinear change in reflectivity and it is depends on the thickness and material of the absorber, optical wavelength, and the degree of optical field penetration into the absorber structure. The saturation fluence is the fluence of an incident short pulse which is required for causing significant absorption saturation. The recovery time is the exponential time constant of absorption recovery after a saturating pulse. In every SESAM operation there are some non saturable losses, which are unwanted. They lead to device heating while not contributing to the pulse shaping. The non saturable losses tend to be higher for SESAMs with a larger modulation depth and faster recovery [3, 14]. Basically, soliton laser is a mode-locked laser using pulse compression and solitons in a single mode fiber to force the laser itself to produce pulses of a well-defined shape and width. Pulse compression and solitons in single mode fibers result from the interaction of nonlinearity with “negative” group velocity dispersion. Operation of soliton lasers can be in multiple pulse modes. A higher order soliton is a soliton pulse the energy of which is higher than that of a fundamental soliton by a factor which is the square of an integer number. The temporal shape of such a pulse is not constant, but rather varies periodically during propagation. The soliton period is the period of this evolution. We can breakup of single pulses into multiple pulses by Kerr lens mode-locked Ti:Sapphire laser or by semiconductor saturable absorber mirrors (SESAM’s). We get pulses with larger spacing compare to single-pulse width. Ti: Sapphire laser includes titanium-doped sapphire crystal which characterized by the broad gain profile and excellent physical properties. The typical mode-locking mechanism for most Ti: Sapphire lasers is soft-aperture Kerrlens mode-locking (KLM). The Kerr effect of the gain crystal results in intensity

Laser Pulses GaAs Substrate

GaAs/AlAs Bragg mirror

Fig. 2.5 Structure of semiconductor saturable absorber mirror

InGaAs Quantum well

200

2 Ti: Sapphire Laser Systems with Delay Parameters …

dependent gain distribution and beam overlapping of the laser and pump beams. The short pulse experiences a higher gain than the multimode continuous-wave (CW) signals, resulting in stable mode-locking. The Ti:Sapphire crystal works as not only a gain crystal to supply energy but also a Kerr medium to start mode-locking. There is a limited controllability of the cavity features, such as the pump threshold to start mode-locking which depends on the nonlinearities inside the cavity, the stable regions to maintain mode-locking whose gap depends on the asymmetry of the cavity. The laser cavity experienced multiple pulses, and production of high-repetition rate soliton pulse sources for optical fiber communication systems. Erbium-doped fiber lasers are characterized by harmonic mode locking. C4+ r Yag Bulk laser incorporated the same characteristic of harmonic mode locking. Harmonic mode locking is a technique for achieving higher pulse repetition rates and is mainly applied to actively mode locked lasers. The principle is that multiple pulses circulate in the laser resonator with equal spacing. We achieve it by driving the modulator of an actively mode-locked laser with a harmonic of the resonator’s round-trip frequency. There is no proving that the pulses have equal pulse energies. The pulses are not always mutually phase coherent. The nonlinear polarization rotation or a semiconductor Bragg reflector is used as the passive mode-locking mechanisms. The passive mode locking is a technique of mode locking, based on a saturable absorber inside the laser resonator. The passive mode locking can be achieved by incorporating a saturable absorber with suitable properties into the laser resonator. The incorporation of the absorber may lead to passive Q switching, to Q—switching mode locking, or some noisy mode of operation, if the absorber properties are not appropriate. The steady state is considered, where a short pulse is already circulating in the laser resonator, it is assumed that there is a single circulating pulse, and a fast absorber. We analyze the dynamic and stability of the laser by using nonlinear dynamic and possible numerical simulations. It is interested to analyze the stability of a soliton in the laser when subject to perturbations from a filter and an absorber. We can define some system parameters: T TR g q Tg Ta Pg · Tg ; E α g0 ; q0 l0 β = 12  κ Ae f f lL

Time parameter. Round trip time. Saturable gain. Saturable absorption. Saturable gain recovery time. Saturable absorption recovery time (absorber recovery time). Saturation energies. Unsaturated values of gain and absorption. Linear loss (output coupling). Effect of the birefringent filter. HWHM of the filter. Self-phase-modulation parameter (calculated from the nonlinear refractive index Nonlinear refractive index n2 of sapphire. Effective mode area inside the crystal.   Total length of the Ti:Sapphire crystal per round trip κ =

2×π·n 2 ·l L λ0 ·Ae f f

.

2.3 Multipulse Operation of a Ti: Sapphire Laser …

ψ Q Qa , Qb r ls δ α lc

201

Electric field envelope that the pulses on two time scales. &   describes Total intra cavity energy Q = |ψ|2 · dt . Pulse energies. Ratio of round-trip time to soliton period as a measure of the strength   Q 2 ·κ 2 Q·κ of solitonic pulse formation r = 2×π·|β2 | = π·t0 . Total energy loss of soliton (ls = δ + α). Filter loss (energy loss due to filter). Absorber loss (energy loss due to absorber). Loss experienced by the continuum.

The state which the growth of dispersive continuum radiation is effectively suppressed happened when ls < lc ; ls = δ + α. We can consider that apart from filter and absorber perturbations, the discreteness of all effects like self-phase modulation, dispersion, gain, etc., causes shedding of continuum by the soliton. The discreteness is not covered by the current model but will become important for cases when r ≈ 1 or larger (r > 1). We can write the laser in-equilibrium condition of energy balance as g0 − l0 − δ − α = 0. The laser adjust it saturated gain (g) according the values of δ, α. The frame defined by the two lines separated by q0 therefore moves with respect to the zero-gain line. If the line g −l0 − q0 crosses the zero line, because of an increase in δ, continuum growth is possible. It is true also for soliton mode-locking schemes that use ultrafast saturable absorbers such as the Kerr lens nonlinear polarization rotation. The onset of continuum growth of (δ (filter loss) for instance) is the critical condition that destabilizes the soliton pulse in the laser cavity. We define (ε) parameter which characterize the absorber, ε = 0 for the ideally fast absorber, ε = 0 for slow absorber (growth occurs slightly earlier, indicated by ε = 0, but physical process is the same). The stability of the pair of solitons is bounded by two limits. The first limit, toward smaller |β2 |, is that of renewed shedding and growth of the continuum if (α2 + δ2 + ε) Q2 > q0 , and similarly to the single-soliton case. The process drives the transition from double to triple pulse. The second limit, toward larger |β2 |, is not absolute since the pulse pair always appears to experience lower loss than a single pulse with the same total energy, because of the dominant influence of the filter loss. We derive a stability criterion from the evolution of pulse energies Q a and Q b respectively as well as the gain (g)—soliton perturbation theory. Mainly we can present the system generalized complex Ginzburg-Landau equation (GCGLE) which is the master equation to describe the dynamical effects of laser by means of numerical simulations [15, 16]. i · TR · ψT −

β2 ∂ 2 ψ ∂ 2ψ · 2 + κ · |ψ|2 · ψ = i · (g − q − l0 ) · ψ + i · β · 2 2 ∂t ∂t

And the solution we get by neglecting the perturbations on the above equation’s right hand side, it is known as the Non Linear Schrodinger Equation (NLSE), which has the first-order soliton solution.

202

2 Ti: Sapphire Laser Systems with Delay Parameters …

ψ=



 κ·Q κ · Q2 Q2 · κ 2 · sech · t · exp i · ·T 4 · |β2 | 2 × |β2 | 8 · |β2 | · TR

We can consider our system differential equations when the perturbations on the right hand side are not neglected. d Qa = 2 · [g · (Q a + Q b ) − l0 − δa − αa ] · Q a dT d Qb TR · = 2 · [g · (Q a + Q b ) − l0 − δb − αb ] · Q b dT (g − g0 ) (Q a + Q b ) · g dg =− − dT Tg Pg · Tg · TR

TR ·

δa , δb Pulse filter losses (a, b). αa , αb Pulse absorber losses (a, b). The filter loss and absorber loss are given by the following expression: ls = δ + α. β 1 + · ls = 2 2 · t0 3 · t0

 '+∞ t 2 · |β2 | τF W H M · q(t) · dt; t0 = sech2 = t0 κ · Q a,b 1.763

−∞

We analyze the dynamic and stability of our system by using nonlinear dynamic. Our system variables are pulse energies: Q a , Q b . Our system (Multi pulse operation of Ti: Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror) differential equations when the perturbations on the right hand side are not neglected (Q a = Q a (T ); Q b = Q b (T ); g = g(T )). 1 d Qa =2· · [g · (Q a + Q b ) − l0 − δa − αa ] · Q a dT TR 1 d Qb dg =2· · [g · (Q a + Q b ) − l0 − δb − αb ] · Q b ; dT TR dT (g − g0 ) (Q a + Q b ) · g =− − Tg Pg · Tg · TR At fixed points:

d Qa dT

dg Qb = 0; ddT = 0; dT = 0.

2·

1 · [g ∗ · (Q a∗ + Q ∗b ) − l0 − δa − αa ] · Q a∗ = 0 TR

2·

1 · [g ∗ · (Q a∗ + Q ∗b ) − l0 − δb − αb ] · Q ∗b = 0 TR

2.3 Multipulse Operation of a Ti: Sapphire Laser …

−

203

(g ∗ g0 ) (Q a∗ + Q ∗b ) · g ∗ − =0 Tg Pg · Tg · TR

d Qa 1 =0⇒2· · [g ∗ · (Q a∗ + Q ∗b ) − l0 − δa − αa ] · Q a∗ = 0 dT TR Case 1: Q a∗ = 0; Case 2: g ∗ · (Q a∗ + Q ∗b ) − l0 − δa − αa = 0 ⇒ Q a∗ = g1∗ · (l0 + δa + αa ) − Q ∗b Qb Case 1: Q a∗ = 0; ddT = 0 ⇒ 2 · T1R · [g ∗ · Q ∗b − l0 − δb − αb ] · Q ∗b = 0 ∗ Case 1.1: Q a = 0; Q ∗b = 0.; Case 1.2: Q a∗ = 0; g ∗ · Q ∗b − l0 − δb − αb = 0 ⇒ Q ∗b = g1∗ (l0 + δb + αb ) ∗ dg 0) = 0 ⇒ − (g T−g =0 dT g (0) (0) (0) (0) First fixed point: E (Q a , Q b , g ) = (0, 0, g0 ). Case 1.2: Q a∗ = 0; Q ∗b = g1∗ · (l0 + δb + αb )

Case 1.1: Q a∗ = 0; Q ∗b = 0;

⇒ g ∗ = g0

1 · (l0 + δb + αb ) · g ∗ dg (g ∗ − g0 ) g∗ =0⇒− − =0 dT Tg Pg · Tg · TR (l0 + δb + αb ) − (g ∗ − g0 ) − =0 Pg · TR

(l0 + δb + αb ) = 0 ⇒ g ∗ − g0 Pg · TR (l0 + δb + αb ) ∗ (l0 + δb + αb ) =− ; g = g0 − Pg · TR Pg · TR

− (g ∗ − g0 ) −

Q ∗b =

1 1 ) · (l0 + δb + αb ) · (l0 + δb + αb ) = ( ∗ g b +αb ) g0 − (l0 +δ Pg ·TR

Second fixed point: ⎛

⎞ (l + δ + α ) 0 b b ⎠ (1) ⎝ ) · (l0 + δb + αb ), g0 − E (1) (Q a(1) , Q (1) b , g ) = 0, ( Pg · TR b +αb ) g0 − (l0 +δ Pg ·TR 1

Case 2: Q a∗ =

1 g∗

· (l0 + δa + αa ) − Q ∗b

1 d Qb =0⇒2· · [g ∗ · (Q a∗ + Q ∗b ) − l0 − δb − αb ] · Q ∗b = 0 dT TR d Qb 1 1 · [g ∗ · ( ∗ · (l0 + δa + αa ) − Q ∗b + Q ∗b ) − l0 − δb − αb ] · Q ∗b = 0 =0⇒2· dT TR g 1 1 2· · [δa + αa − δb − αb ] · Q ∗b = 0; 2 · · [δa + αa − δb − αb ] = 0 ⇒ Q ∗b = 0 TR TR

204

2 Ti: Sapphire Laser Systems with Delay Parameters …

(g ∗ − g0 ) (Q a∗ + Q ∗b ) · g ∗ dg =0⇒− − =0 dT Tg Pg · Tg · TR   1 ∗ ∗ ∗ · g∗ · (l + δ + α ) − Q + Q 0 a a ∗ b b g (g − g0 ) − − Tg Pg · Tg · TR (l0 + δa + αa ) = 0 ⇒ −(g ∗ − g0 ) − =0 Pg · TR g ∗ − g0 = − Q a∗ =

(l0 + δa + αa ) (l0 + δa + αa ) ⇒ g ∗ = g0 − Pg · TR Pg · TR

1 1 ) · (l0 + δa + αa ) · (l0 + δa + αa ) − Q ∗b = ( (l0 +δa +αa ) g∗ g0 − Pg ·TR

Third fixed point: ⎞ + δ + α ) (l 0 a a ⎠ ) · (l0 + δa + αa ), 0, g0 − = ⎝( Pg · TR a +αa ) g0 − (l0 +δ Pg ·TR ⎛

E

(2)

(Q a(2) ,

(2) Q b , g (2) )

1

We can summary our system fixed points: (0) E (0) (Q a(0) , Q (0) b , g ) = (0, 0, g0 )

⎞ + δ + α ) (l 0 b b ⎠ (1) ) · (l0 + δb + αb ), g0 − E (1) (Q a(1) , Q b , g (1) ) = ⎝0, ( Pg · TR b +αb ) g0 − (l0 +δ Pg ·TR ⎞ ⎛ + δ + α ) 1 (l 0 a a (2) (2) (2) (2) ⎠ ) · (l0 + δa + αa ), 0, g0 − E (Q a , Q b , g ) = ⎝ ( Pg · TR g − (l0 +δa +αa ) ⎛

1

0

Pg ·TR

The system is not linear and we need to implement linearization technique and then to approximate the phase portrait near a fixed points by that of a corresponding linear system. 1 d Qa =2· · [g · (Q a + Q b ) − l0 − δa − αa ] · Q a dT TR 1 d Qb dg =2· · [g · (Q a + Q b ) − l0 − δb − αb ] · Q b ; dT TR dT (g − g0 ) (Q a + Q b ) · g =− − Tg Pg · Tg · TR We define three functions: f 1 (Q a , Q b , g) = 2· T1R ·[g·(Q a + Q b )−l0 −δa −αa ]· Q a

2.3 Multipulse Operation of a Ti: Sapphire Laser …

205

1 · [g · (Q a + Q b ) − l0 − δb − αb ] · Q b TR (g − g0 ) (Q a + Q b ) · g f 3 (Q a , Q b , g) = − − Tg Pg · Tg · TR

f 2 (Q a , Q b , g) = 2 ·

d Qa d Qb dg = f 1 (Q a , Q b , g); = f 2 (Q a , Q b , g); = f 3 (Q a , Q b , g) dT dT dT (∗) And suppose that (Q a(∗) , Q (∗) b , g ) is specific system fixed point. (∗) (∗) (∗) (∗) (∗) (∗) (∗) f 1 (Q a(∗) , Q (∗) b , g ) = 0; f 2 (Q a , Q b , g ) = 0; f 3 (Q a , Q b , g ) = 0

Let u = Q a − Q a∗ ; v = Q b − Q ∗b ; w = g − g ∗ . Denote the components of a small disturbance from the fixed point. To see whether the disturbance grows or decays, we need to derive differential equations for u, v, w. d Q a dv d Q b dw dg du = ; = ; = ; Q a∗ , Q ∗b , g ∗ − const dT dT dT dT dT dT And by substitution: du d Qa = = f 1 (Q a∗ + u, Q ∗b + v, g ∗ + w) dT dT d Qb dv = = f 2 (Q a∗ + u, Q ∗b + v, g ∗ + w) dT dT dg dw = = f 3 (Q a∗ + u, Q ∗b + v, g ∗ + w) dT dT And by Taylor series expansion: du d Qa ∂ f1 ∂ f1 = = f 1 (Q a∗ , Q ∗b , g ∗ ) + u · +v· dT dT ∂ Qa ∂ Qb ∂ f1 + O(u 2 , v2 , w2 , u · v · w) +w· ∂g d Qb dv ∂ f2 ∂ f2 = = f 2 (Q a∗ , Q ∗b , g ∗ ) + u · +v· dT dT ∂ Qa ∂ Qb ∂ f2 +w· + O(u 2 , v2 , w2 , u · v · w) ∂g dg dw ∂ f3 ∂ f3 = = f 3 (Q a∗ , Q ∗b , g ∗ ) + u · +v· dT dT ∂ Qa ∂ Qb ∂ f3 + O(u 2 , v2 , w2 , u · v · w) +w· ∂g Since f 1 (Q a∗ , Q ∗b , g ∗ ) = 0; f 2 (Q a∗ , Q ∗b , g ∗ ). = 0; f 3 (Q a∗ , Q ∗b , g ∗ ) = 0.

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2 Ti: Sapphire Laser Systems with Delay Parameters …

d Qa ∂ f1 du ∂ f1 ∂ f1 = =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT dT ∂ Qa ∂ Qb ∂g d Qb ∂ f2 dv ∂ f2 ∂ f2 = =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT dT ∂ Qa ∂ Qb ∂g dw dg ∂ f3 ∂ f3 ∂ f3 = =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT dT ∂ Qa ∂ Qb ∂g To simplify the notation, we have written

∂ f1 , ∂ f1 , ∂ f1 , ∂ f2 ∂ Q a ∂ Q b ∂g ∂ Q a

·

∂ f2 ∂ f2 , . ∂ Q b ∂g

And ∂∂Qf3a , ∂∂Qf3b , ∂∂gf3 . These partial derivatives are to be evaluated at the fixed point (Q a∗ , Q ∗b , g ∗ ); thus they are numbers, no functions. The shorthand O(u 2 , v2 , w2 , u · v · w) denotes quadratic terms in u, v, w. Since u, v, w are small, these quadratic terms are extremely small [5, 6]. ∂ f1 ∂ f1 ∂ f1 du =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT ∂ Qa ∂ Qb ∂g ∂ f2 dv ∂ f2 ∂ f2 =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT ∂ Qa ∂ Qb ∂g ∂ f3 dw ∂ f3 ∂ f3 =u· + O(u 2 , v2 , w2 , u · v · w) +v· +w· dT ∂ Qa ∂ Qb ∂g ⎛ du ⎞ ⎛ ∂ f1 ∂ f1 ∂ f1 ⎞ ⎛ ⎞ ⎜ dT ⎟ u ⎜ ⎟ ∂ Q ∂ Q ∂g ⎜ dv ⎟ ⎜ ∂ f2a ∂ f2b ∂ f2 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎝ ∂ Q a ∂ Q b ∂g ⎠ · ⎝ v ⎠ + quadratic terms ⎜ dT ⎟ ∂ f3 ∂ f3 ∂ f3 ⎝ ⎠ w ∂ Q a ∂ Q b ∂g dw dT ⎛ ∂ f1 ∂ f1 ∂ f1 ⎞ ∂Q

⎜ a A = ⎝ ∂∂Qf2a ∂ f3 ∂ Qa

∂ Q b ∂g ∂ f2 ∂ f2 ∂ Q b ∂g ∂ f3 ∂ f3 ∂ Q b ∂g

⎟ ⎠ (Q a∗ ,Q ∗b ,g ∗ )

It is called the Jacobian matrix A at the fixed points (Q a∗ , Q ∗b , g ∗ ). Since the quadratic terms are tiny, it is tempting to neglect them altogether. We get our linearized system whose dynamics can be analyzed by the methods of classification of linear systems. ⎛ du ⎞ ⎛ ∂ f1 ⎜ dT ⎟ ⎟ ⎜ ∂Q ⎜ dv ⎟ ⎜ ∂ f2a ⎟ = ⎝ ∂ Qa ⎜ ⎜ dT ⎟ ∂ f3 ⎠ ⎝ ∂ Qa dw dT

∂ f1 ∂ f1 ∂ Q b ∂g ∂ f2 ∂ f2 ∂ Q b ∂g ∂ f3 ∂ f3 ∂ Q b ∂g

⎞ ⎛ ⎞ u ⎟ ⎜ ⎟ ⎠ · ⎝ v⎠ w

2.3 Multipulse Operation of a Ti: Sapphire Laser …

f 1 (Q a , Q b , g) = 2 ·

207

1 · [g · (Q a2 + Q b · Q a ) − (l0 + δa + αa ) · Q a ] TR

∂ f1 1 =2· · [g · (2 · Q a + Q b ) − (l0 + δa + αa )] ∂ Qa TR ∂ f1 1 1 ∂ f1 =2· =2· · g · Qa ; · (Q a2 + Q b · Q a ) ∂ Qb TR ∂g TR f 2 (Q a , Q b , g) = 2 ·

1 · [g · (Q a · Q b + Q 2b ) − (l0 + δb + αb ) · Q b ] TR

∂ f2 1 ∂ f2 1 =2· · g · Qb; =2· · [g · (Q a + 2 · Q b ) ∂ Qa TR ∂ Qb TR 1 ∂ f2 =2· − (l0 + δb + αb )]; · (Q a · Q b + Q 2b ) ∂g TR (g − g0 ) (Q a + Q b ) · g − Tg Pg · Tg · TR ∂ f3 g ∂ f3 g =− ; =− ∂ Qa Pg · Tg · TR ∂ Q b Pg · Tg · TR + * 1 ∂ f3 (Q a + Q b ) 1 (Q a + Q b ) =− − =− · 1+ ∂g Tg Pg · Tg · TR Tg Pg · TR f 3 (Q a , Q b , g) = −

We can summary our Jacobian matrix A elements expressions: ∂ f1 ∂ Qa ∂ f1 ∂ Qb ∂ f2 ∂ Qa ∂ f2 ∂g ∂ f3 ∂ Qa ∂ f3 ∂g

1 TR 1 =2· TR 1 =2· TR 1 =2· TR =2·

· [g · (2 · Q a + Q b ) − (l0 + δa + αa )] 1 ∂ f1 =2· · (Q a2 + Q b · Q a ) ∂g TR ∂ f2 1 · g · Qb; =2· · [g · (Q a + 2 · Q b ) − (l0 + δb + αb )] ∂ Qb TR · g · Qa ;

· (Q a · Q b + Q 2b )

g ∂ f3 g ; =− Pg · Tg · TR ∂ Q b Pg · Tg · TR + * 1 (Q a + Q b ) =− · 1+ Tg Pg · TR

=−

Jacobian matrix A at the fixed points (Q a∗ , Q ∗b , g ∗ ): (0) First fixed point: E (0) (Q a(0) , Q (0) b , g ) = (0, 0, g0 ). 1 ∂ f1 | E (0) (Q a(0) ,Q (0) ,g(0) )=(0,0,g0 ) = −2 · · (l0 + δa + αa ) b ∂ Qa TR

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2 Ti: Sapphire Laser Systems with Delay Parameters …

∂ f1 | (0) (0) (0) (0) =0 ∂ Q b E (Q a ,Q b ,g )=(0,0,g0 ) ∂ f1 =0 | (0) (0) (0) (0) ∂g E (Q a ,Q b ,g )=(0,0,g0 ) ∂ f2 | (0) (0) (0) (0) =0 ∂ Q a E (Q a ,Q b ,g )=(0,0,g0 ) ∂ f2 1 | (0) (0) (0) (0) = −2 · · (l0 + δb + αb ) ∂ Q b E (Q a ,Q b ,g )=(0,0,g0 ) TR ∂ f2 =0 ∂g E (0) (Q a(0) ,Q (0) (0) b ,g )=(0,0,g0 ) ∂ f3 g0 | (0) (0) (0) (0) =− ∂ Q a E (Q a ,Q b ,g )=(0,0,g0 ) Pg · Tg · TR ∂ f3 g0 | (0) (0) (0) (0) =− ∂ Q b E (Q a ,Q b ,g )=(0,0,g0 ) Pg · Tg · TR

⎛ ⎜ A=⎝ ⎛ ⎜ =⎝

∂ f3 1 | (0) (0) (0) (0) =− ∂g E (Q a ,Q b ,g )=(0,0,g0 ) Tg ⎞ ∂ f1 ∂ f1

∂ f1 ∂ Q a ∂ Q b ∂g ∂ f2 ∂ f2 ∂ f2 ∂ Q a ∂ Q b ∂g ∂ f3 ∂ f3 ∂ f3 ∂ Q a ∂ Q b ∂g

−2 ·

1 TR

⎟ ⎠ (0) E (0) (Q a(0) ,Q (0) b ,g )=(0,0,g0 )

· (l0 + δa + αa ) 0 −2 · g0 − Pg ·Tg ·TR

⎞ 0 1 · (l0 + δb + αb ) 0 ⎟ ⎠ TR − Pg ·Tgg0 ·TR − T1g 0

Remark As long as the fixed point for the linearized system in not one of the borderline cases, we can neglect the quadratic terms. Then the linearized system gives a qualitatively correct picture of the phase portrait near specific fixed point (∗) (Q a(∗) , Q (∗) b , g ) [5, 6]. ⎛ ⎜ A−λ· I =⎝

−2 ·

1 TR

· (l0 + δa + αa ) − λ 0 −2 · − Pg ·Tgg0 ·TR

1 TR

⎞ 0 0 ⎟ · (l0 + δb + αb ) − λ 0 ⎠ g0 1 − Pg ·Tg ·TR − Tg − λ

1 det(A − λ · I ) = −[2 · · (l0 + δa + αa ) + λ]· TR ⎛ ( ) ⎞ − 2 · T1R · (l0 + δb + αb ) + λ 0 ( )⎠ det ⎝ − Pg ·Tgg0 ·TR − T1g + λ + * 1 · (l0 + δa + αa ) + λ · det(A − λ · I ) = − 2 · TR

2.3 Multipulse Operation of a Ti: Sapphire Laser …

209

* + * + 1 1 2· · (l0 + δb + αb ) + λ · +λ TR Tg + * 1 · (l0 + δa + αa ) + λ · det(A − λ · I ) = 0 ⇒ − 2 · TR * + * + 1 1 2· · (l0 + δb + αb ) + λ · +λ =0 TR Tg We get three eigenvalues: λ1 = −2 · T1R · (l0 + δa + αa ) ; λ2 = −2 · T1R · (l0 + δb + αb ); λ3 = − T1g . Our system’s typical modeling parameters are as follow: TR = 12.5ns (Round trip time), l0 = 0.08 (Total linear loss per round trip), Tg = 2.5us (Upper state lifetime of Ti: Sapphire), 4.5×10−3 < δa +αa < 5.2×10−3 and 4.5 × 10−3 < δb + αb < 5.2 × 10−3 for soliton operation time 0.6–2 ps. Three eigenvalues are real and negative numbers and the first fixed point is stable. Second fixed point: ⎛

⎡

⎤

⎞

(l0 + δb + αb ) ⎦ (l0 + δb + αb ) ⎠ ) , g0 − (l0 +δb +αb ) Pg · TR g0 − Pg ·TR ⎡ ⎡ ⎤ * + 1 ⎣ (l0 + δb + αb ) ⎣ (l0 + δb + αb ) ⎦ ) g0 − · ( =2· · TR Pg · TR g − (l0 +δb +αb )

(1) ⎝0, ⎣ ( E (1) (Q a(1) , Q (1) b ,g ) =

∂ f1 | (1) (1) (1) (1) ∂ Q a E (Q a ,Q b ,g )

+

*

0

Pg ·TR

−(l0 + δa + αa )] ∂ f1 ∂ f1 | (1) (1) (1) (1) = 0 | E (1) (Q a(1) ,Q (1) ,g(1) ) = 0; b ∂ Qb ∂g E (Q a ,Q b ,g ) ⎡ ⎤ + * ∂ f2 1 (l0 + δb + αb ) ⎣ (l0 + δb + αb ) ⎦ ) · ( | (1) (1) (1) (1) = 2 · · g0 − ∂ Q a E (Q a ,Q b ,g ) TR Pg · TR g − (l0 +δb +αb ) 0

⎡

+

*

Pg ·TR

⎡

⎤

∂ f2 1 ⎣ (l0 + δb + αb ) ⎣ (l0 + δb + αb ) ⎦ ) · ( | (1) (1) (1) (1) = 2 · · 2 · g0 − ∂ Q b E (Q a ,Q b ,g ) TR Pg · TR g − (l0 +δb +αb ) 0

−(l0 + δb + αb )] ⎡ ⎤2 1 ⎣ (l0 + δb + αb ) ⎦ ∂ f2 ) | (1) (1) (1) (1) = 2 · · ( ∂g E (Q a ,Q b ,g ) TR g − (l0 +δb +αb ) 0

Pg ·TR

Pg ·TR

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2 Ti: Sapphire Laser Systems with Delay Parameters …

)⎫ ⎧( b +αb ) ⎬ ⎨ g0 − (l0 +δ Pg ·TR − ⎩ Pg · Tg · TR ⎭ ( ) b +αb ) g0 − (l0 +δ Pg ·TR =− Pg · Tg · TR ⎡

⎤

∂ f3 | (1) (1) (1) (1) = ∂ Q a E (Q a ,Q b ,g ) ∂ f3 | (1) (1) (1) (1) ∂ Q b E (Q a ,Q b ,g )

∂ f3 1 | (1) (1) (1) (1) = − ∂g E (Q a ,Q b ,g ) Tg

⎢ ⎢ · ⎢1 + ⎣

( (l0 +δb +αb ) ) (l +δ +α ) g0 − 0 Pg b·T b R

Pg · TR

⎥ ⎥ ⎥ ⎦

We use the identification to simplify our above expressions. g0 · Pg · TR − (l0 + δb + αb ) (l0 + δb + αb ) = Pg · TR Pg · TR Pg · TR 1 = b +αb ) g0 · Pg · TR − (l0 + δb + αb ) g0 − (l0 +δ Pg ·TR

g0 −

We get the following Jacobian matrix A’s elements at the second fixed point: 1 ∂ f1 | (1) (1) (1) (1) = 2 · · (δb + αb − δa − αa ) ∂ Q a E (Q a ,Q b ,g ) TR ∂ f1 ∂ f1 | (1) (1) (1) (1) = 0 | E (1) (Q a(1) ,Q (1) ,g(1) ) = 0; b ∂ Qb ∂g E (Q a ,Q b ,g ) ∂ f2 1 | E (1) (Q a(1) ,Q (1) ,g(1) ) = 2 · · (l0 + δb + αb ) b ∂ Qa TR ∂ f2 1 | (1) (1) (1) (1) = 2 · · (l0 + δb + αb ) ∂ Q b E (Q a ,Q b ,g ) TR +2 * (l0 + δb + αb ) · Pg · TR ∂ f2 1 (1) (1) | (1) · (1) = 2 · ∂g E (Q a ,Q b ,g ) TR g0 · Pg · TR − (l0 + δb + αb )   g0 · Pg · TR − (l0 + δb + αb ) ∂ f3 | (1) (1) (1) (1) = − ∂ Q a E (Q a ,Q b ,g ) Tg · Pg2 · TR2 g0 · Pg · TR − (l0 + δb + αb ) ∂ f3 | E (1) (Q a(1) ,Q (1) ,g(1) ) = − b ∂ Qb Tg · TR2 · Pg2 + * ∂ f3 1 (l0 + δb + αb ) (1) (1) | (1) · 1+ (1) = − ∂g E (Q a ,Q b ,g ) Tg g0 · Pg · TR − (l0 + δb + αb )

2.3 Multipulse Operation of a Ti: Sapphire Laser …

211

det(A − λ · I ) = 0 * + * + 1 1 2· · (δb + αb − δa − αa ) − λ · 2 · · (l0 + δb + αb ) − λ · TR TR * + + * 1 (l0 + δb + αb ) − · 1+ −λ Tg g0 · Pg · TR − (l0 + δb + αb ) 2 3 g0 · Pg · TR − (l0 + δb + αb ) − − · Tg · TR2 · Pg2 2 +2 3 * (l0 + δb + αb ) · Pg · TR 1 · 2· =0 TR g0 · Pg · TR − (l0 + δb + αb ) * + * + 1 1 2· · (δb + αb − δa − αa ) − λ · 2 · · (l0 + δb + αb ) − λ · TR TR * + + * 1 (l0 + δb + αb ) − · 1+ −λ Tg g0 · Pg · TR − (l0 + δb + αb ) + * (l0 + δb + αb )2 1 =0 · +2· TR · Tg g0 · Pg · TR − (l0 + δb + αb ) We get eigenvalue characteristic equation

3 k=0

λk · ϒk = 0.

Remark it is reader task to find the exact expression for ϒk ; k = 0, 1, 2, 3. We get three eigenvalues λ1 , λ2 , λ3 . Eigenvalues stability discussion: Our system (Multi pulse operation of Ti: Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber   3 k mirror), the characteristic equation is of degree N = 3 k=0 λ · ϒk = 0 and must

212

2 Ti: Sapphire Laser Systems with Delay Parameters …

often be solved numerically. Expect in some particular cases, such an equation has (N = 3) distinct roots that can be real or complex. These values are the eigenvalues of the 3 × 3 Jacobian matrix (A). The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 3-variables (Q a , Q b , g) system has three eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the three eigenvalues are real and negative (stable steady state), (2) the three eigenvalues are real, two of them are negative (unstable steady state), (3) and (4) one eigenvalue is complex with a negative real part and the other eigenvalues are real and negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) one eigenvalue is complex with a negative real part and at least one eigenvalue is positive (unstable steady state) [5, 6]. Third fixed point: ⎛

⎞ (l + δ + α ) + δ + α ) (l a a 0 a a ⎠ (2) ⎝( 0 ) , 0, g0 − E (2) (Q a(2) , Q (2) b ,g ) = (l0 +δa +αa ) P · T g R g0 − Pg ·TR ⎡ ⎡ ⎤ + * ∂ f1 1 ⎣ (l0 + δa + αa ) ⎣ (l0 + δa + αa ) ⎦ ) · ( | (2) (2) (2) (2) = 2 · · 2 · g0 − ∂ Q a E (Q a ,Q b ,g ) TR Pg · TR g − (l0 +δa +αa ) 0

Pg ·TR

−(l0 + δa + αa )] ⎛ ⎞ + * ∂ f1 1 (l0 + δa + αa ) ⎝ (l0 + δa + αa ) ⎠ ) · ( | (2) (2) (2) (2) = 2 · · g0 − ∂ Q b E (Q a ,Q b ,g ) TR Pg · TR g − (l0 +δa +αa ) 0

⎡

Pg ·TR

⎤2

∂ f1 1 ⎣ (l0 + δa + αa ) ⎦ ∂ f 2 ) ; | (2) (2) (2) (2) = 2 · · ( | (1) (2) (2) (2) = 0 ∂g E (Q a ,Q b ,g ) TR ∂ Q a E (Q a ,Q b ,g ) g − (l0 +δa +αa ) 0

⎧ ⎨*

Pg ·TR

⎡ ⎤ + ∂ f2 1 (l0 + δa + αa ) ⎣ (l0 + δa + αa ) ⎦ ) g0 − · ( | (2) (2) (2) (2) = 2 · · ∂ Q b E (Q a ,Q b ,g ) TR ⎩ Pg · TR g − (l0 +δa +αa ) 0

Pg ·TR

−(l0 + δb + αb )} ( ∂ f3 ∂ f2 | (2) (2) (2) (2) = 0; | (2) (2) (2) (2) = − ∂g E (Q a ,Q b ,g ) ∂ Q a E (Q a ,Q b ,g )

g0 −

(l0 +δa +αa ) Pg ·TR

Pg · Tg · TR

)

2.3 Multipulse Operation of a Ti: Sapphire Laser …

213

( ) a +αa ) g0 − (l0 +δ Pg ·TR ∂ f3 | (2) (2) (2) (2) = − ∂ Q b E (Q a ,Q b ,g ) Pg · Tg · TR ⎡

⎤ ∂ f3 | (2) (2) (2) (2) ∂g E (Q a ,Q b ,g )

1 ⎢ ⎢ = − · ⎢1 + Tg ⎣

( (l0 +δa +αa ) ) (l +δ +α ) g0 − 0 Pg a·T a R

Pg · TR

⎥ ⎥ ⎥ ⎦

We use the identification to simplify our above expressions. g0 · Pg · TR − (l0 + δa + αa ) (l0 + δa + αa ) = Pg · TR Pg · TR Pg · TR 1 = a +αa ) g · P · T g0 − (l0 +δ 0 g R − (l0 + δa + αa ) Pg ·TR

g0 −

We get the following Jacobian matrix A’s elements at the third fixed point: 1 ∂ f1 | E (2) (Q a(2) ,Q (2) ,g(2) ) = 2 · · [l0 + δa + αa ] b ∂ Qa TR ∂ f1 1 | (2) (2) (2) (2) = 2 · · (l0 + δa + αa ) ∂ Q b E (Q a ,Q b ,g ) TR +2 * (l0 + δa + αa ) · Pg · TR ∂ f1 1 | E (2) (Q a(2) ,Q (2) ,g(2) ) = 2 · · b ∂g TR g0 · Pg · TR − (l0 + δa + αa ) ∂ f2 | (2) (2) (2) (2) = 0 ∂ Q a E (Q a ,Q b ,g ) ∂ f2 1 | (2) (2) (2) (2) = 2 · · (δa + αa − δb − αb ) ∂ Q b E (Q a ,Q b ,g ) TR ∂ f2 | (2) (2) (2) (2) = 0 ∂g E (Q a ,Q b ,g ) g0 · Pg · TR − (l0 + δa + αa ) ∂ f3 | (2) (2) (2) (2) = − ∂ Q a E (Q a ,Q b ,g ) Tg · Pg2 · TR2 g0 · Pg · TR − (l0 + δa + αa ) ∂ f3 | (2) (2) (2) (2) = − ∂ Q b E (Q a ,Q b ,g ) Tg · Pg2 · TR2 + * ∂ f3 1 (l0 + δa + αa ) | (2) (2) (2) (2) = − · 1 + ∂g E (Q a ,Q b ,g ) Tg g0 · Pg · TR − (l0 + δa + αa )

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2 Ti: Sapphire Laser Systems with Delay Parameters …

⎛

∂ f1 ∂ f1 ∂ f1 ∂ Q a ∂ Q b ∂g ∂ f2 ∂ f2 ∂ f2 ∂ Q a ∂ Q b ∂g ∂ f3 ∂ f3 ∂ f3 ∂ Q a ∂ Q b ∂g

⎜ A=⎝ ⎛ ⎜ =⎜ ⎝

2·

1 TR

−

2·

g0 ·Pg ·TR −(l0 +δa +αa ) Tg ·Pg2 ·TR2

∂ f1 ∂Q

∂ f3 ∂ Qa

2·

(2) E (2) (Q a(2) ,Q (2) b ,g )

· [l0 + δa + αa ]

⎜ a A − λ · I = ⎝ ∂∂Qf2a

⎜ =⎜ ⎝

⎟ ⎠

0

⎛

⎛

⎞

1 TR

∂ f1 ∂ f1 ∂ Q b ∂g ∂ f2 ∂ f2 ∂ Q b ∂g ∂ f3 ∂ f3 ∂ Q b ∂g

2·

1 TR

1 TR

· (δa + αa − δb − αb )

−

· (l0 + δa + αa )

g0 ·Pg ·TR −(l0 +δa +αa ) Tg ·Pg2 ·TR2

⎞

(2) E (2) (Q a(2) ,Q (2) b ,g )

· [l0 + δa + αa ] − λ 0

1 TR

·

(l0 +δa +αa )·Pg ·TR g0 ·Pg ·TR −(l0 +δa +αa )

2

0 − T1g · 1 +

(l0 +δa +αa ) g0 ·Pg ·TR −(l0 +δa +αa )

⎞ ⎟ ⎟ ⎠

⎛

⎟ ⎠

g ·P ·T −(l +δ +α ) − 0 g TR ·P 20·T 2 a a g g R

2·

2· 2·

1 TR

1 TR

⎞ 100 − λ · ⎝0 1 0⎠ 001

· (l0 + δa + αa )

2·

1 TR

·

· (δa + αa − δb − αb ) − λ −

g0 ·Pg ·TR −(l0 +δa +αa ) Tg ·Pg2 ·TR2

(l0 +δa +αa )·Pg ·TR g0 ·Pg ·TR −(l0 +δa +αa )

2

0 − T1g · 1 +

(l0 +δa +αa ) g0 ·Pg ·TR −(l0 +δa +αa )

−

det(A − λ · I ) = 0

 * + 1 1 2· · [l0 + δa + αa ] − λ · 2 · · (δa + αa − δb − αb ) − λ TR TR + + * * (l0 + δa + αa ) 1 −λ · − · 1+ Tg g0 · Pg · TR − (l0 + δa + αa ) 3 +2  2 * (l0 + δa + αa ) · Pg · TR g0 · Pg · TR − (l0 + δa + αa ) 1 +2· · · − − TR g0 · Pg · TR − (l0 + δa + αa ) Tg · Pg2 · TR2 * + 1 · 2· · (δa + αa − δb − αb ) − λ = 0 TR

 * + 1 1 2· · [l0 + δa + αa ] − λ · 2 · · (δa + αa − δb − αb ) − λ TR TR + + * * (l0 + δa + αa ) 1 −λ · − · 1+ Tg g0 · Pg · TR − (l0 + δa + αa ) (l0 + δa + αa )2 1 · +2· TR · Tg [g0 · Pg · TR − (l0 + δa + αa )] * + 1 · 2· · (δa + αa − δb − αb ) − λ = 0 TR We get eigenvalue characteristic equation

3 k=0

λk · ϒk = 0.

2.3 Multipulse Operation of a Ti: Sapphire Laser …

215

Remark it is reader task to find the exact expression for ϒk ; k = 0, 1, 2, 3. We get three eigenvalues λ1 , λ2 , λ3 . Eigenvalues stability discussion: Our system (Multi pulse operation of Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror),   3 k λ · ϒ = 0 and must often the characteristic equation is of degree N = 3 k k=0 be solved numerically. Expect in some particular cases, such an equation has (N = 3) distinct roots that can be real or complex. These values are the eigenvalues of the 3 × 3 Jacobian matrix (A). The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 3-variables (Q a , Q b , g) system has three eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the three eigenvalues are real and negative (stable steady state), (2) the three eigenvalues are real, two of them are negative (unstable steady state), (3) and (4) one eigenvalue is complex with a negative real part and the other eigenvalues are real and negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) one eigenvalue is complex with a negative real part and at least one eigenvalue is positive (unstable steady state) [5, 6].

2.4 Questions 1.

We have a system of stronger laser field pulse and diatomic molecules. The system is characterized by a second order differential equation, gives the rotation of a molecule with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F). The field strength (F) is dependent on the angle between the polarization of the laser and inter nuclear axis (θ ) according to the following function F = 0 + 1 · sin2 (θ ) 0 , 1 ∈ R+ . Our system second order differential equation: ω0 d 2θ + ( )2 · cos2 (τ ) · sin(2 · θ ) = 0 2 dτ ω α · F2 τ = ω · t; ω02 = ; dτ 2 = d(ω · t)2 = ω2 · dt 2 μ · R2 θ Angle between the polarization of the laser and inter nuclear axis. ω0 Establish the rotation dynamic of the molecule. ω0 Quantity determines the rotation dynamics of the molecules. ω θ0 Molecule initially angle with the direction of laser polarization. μ reduced mass of the molecule. α Polarization factor (polarizability). R Inter nuclear distance in the field of laser.

216

2 Ti: Sapphire Laser Systems with Delay Parameters …

ω Laser frequency (we consider it as a constantin our  system).  α·F 2 F α 2 ω0 Calculated parameter ω0 = μ·R 2 ; ω0 = R · μ (ω0 = 7.9 × 10−4 au (Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations) for D2 molecule at a peak intensity of 1014 W/cm2 ). 1.1 Find possible fixed points in our system. 1.2 Discuss stability of our system by using stability analysis about any one of the equilibrium points (fixed points) of diatomic molecule-differential equations for the rotation of a molecule system. 1.3 Plot our system fixed points on the phase portrait and classify them. 1.4 Discuss the case when there is a delay on angle between the polarization of the laser and inter nuclear axis θ (t) → θ (t − ). Find dynamics, stability behavior, and stability switching of our system. 2.

We have a system of stronger laser field pulse and diatomic molecules. The system is characterized by a second order differential equation, gives the rotation of a molecule with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F). There are interferences in our system which dependent on the angle between the polarization √ of the laser and inter nuclear axis (θ ) according to the function f (θ ) = | 3 θ + cos(θ )|. Our system second order differential equation (with interferences function): d 2 θ  ω0 2 + · f (θ ) · cos2 (τ ) · sin(2 · θ ) = 0 dτ 2 ω α · F2 τ = ω · t; ω02 = ; dτ 2 = d(ω · t)2 = ω2 · dt 2 μ · R2 θ Angle between the polarization of the laser and inter nuclear axis. ω0 Establish the rotation dynamic of the molecule. ω0 Quantity determines the rotation dynamics of the molecules. ω θ0 Molecule initially angle with the direction of laser polarization. μ reduced mass of the molecule. α Polarization factor (polarizability). R Inter nuclear distance in the field of laser. ω Laser frequency (we consider it as a constantin our  system).  α·F 2 F α 2 ω0 Calculated parameter ω0 = μ·R 2 ; ω0 = R · μ . (ω0 = 7.9 × 10−4 au (Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations) for D2 molecule at a peak intensity of 1014 W/cm2 ). 2.1 Find possible fixed points in our system.

2.4 Questions

217

2.2 Discuss stability of our system by using stability analysis about any one of the equilibrium points (fixed points) of diatomic molecule-differential equations for the rotation of a molecule system. 2.3 Plot our system fixed points on the phase portrait and classify them. 2.4 Discuss the case when there is a delay on angle between the polarization of the laser and inter nuclear axisθ (t) → θ (t − ). Find dynamics, stability behavior, and stability switching of our system. 3.

We have a system of stronger laser field pulse and diatomic molecules. The system is characterized by a second order differential equation, gives the rotation of a molecule with reduced mass (μ), polarization (α) and inter nuclear distance (R) in the field of a laser with frequency (ω) and field strength (F). The polarization parameter (α) is dependent on the angle between the polarization of√the laser and inter nuclear axis (θ ) according to the following function α = | θ + sin(θ )|. Additionally, there are interferences in our system which dependent on the angle between the polarization √ of the laser and inter nuclear axis (θ ) according to the function f (θ ) = | 5 θ + sin(θ )|. Our system second order differential equation (with interferences function): d 2 θ  ω0 2 + · f (θ ) · cos2 (τ ) · sin(2 · θ ) = 0 dτ 2 ω α · F2 τ = ω · t; ω02 = ; dτ 2 = d(ω · t)2 = ω2 · dt 2 μ · R2 θ Angle between the polarization of the laser and inter nuclear axis. ω0 Establish the rotation dynamic of the molecule. ω0 Quantity determines the rotation dynamics of the molecules. ω θ0 Molecule initially angle with the direction of laser polarization. μ reduced mass of the molecule. α Polarization factor (polarizability). R Inter nuclear distance in the field of laser. ω Laser frequency (we consider it as a constantin our  system).  α·F 2 F α 2 ω0 Calculated parameter ω0 = μ·R 2 ; ω0 = R · μ . (ω0 = 7.9 × 10−4 au (Atomic units (au or a.u.) form a system of natural units which is especially convenient for atomic physics calculations) for D2 molecule at a peak intensity of 1014 W/cm2 ). 3.1 Find possible fixed points in our system. 3.2 Discuss stability of our system by using stability analysis about any one of the equilibrium points (fixed points) of diatomic molecule-differential equations for the rotation of a molecule system. 3.3 Plot our system fixed points on the phase portrait and classify them.

218

2 Ti: Sapphire Laser Systems with Delay Parameters …

3.4 Discuss the case when there is a delay on angle between the polarization of the laser and inter nuclear axis θ (t) → θ (t − ). Find dynamics, stability behavior, and stability switching of our system. 4.

We have a system of Ultra-broad bandwidth pulse-shaping apparatus which includes two gratings, two spherical mirrors, and crystal SLM (Spatial Light Modulator). Any material (SLM) contributions to positive second order group velocity dispersion (GVD) within the shaper can be canceled if the position of the final grating is adjusted. There is phase dispersion at higher orders, specially cubic and quartic phase dispersions. Fiber propagation parameteris β(ω)  = dn β (ω) (n) (ω) (n) , where L is the propagation distance then β (ω) = L = dωn . L The derivatives β (n) (ω)are given in terms of the index of refraction.

ω=ω0

λ3 d 2 n (3) · ; β (ω) 2 · π · c2 dλ2 

λ4 d 2n d 3n =− 2 3 · 3· 2 +λ· 3 4π c dλ dλ 

5 2 λ 3 d n d 3n λ2 d 4 n β (4) (ω) = 3 4 · · 2 +λ· 3 + · 4 π ·c 4 dλ dλ 4 dλ

β (2) (ω) =

We take the second, third, and fourth derivative of propagation parameter (β(ω)) in ω as a constant at ω = ω0 . We define new variables: Y1 , Y2 , Y3 , Y4 . Y1 =

d 2n d 3n d 4n dn dY1 dY2 dY3 , Y2 = = = = , Y3 = , Y4 = 2 3 dλ dλ dλ dλ dλ dλ dλ4

Due to interferences in our system there are shifting in new variables Y2 , Y3 , Y2 (λ) → Y2 (λ − λ A ); Y3 (λ) → Y3 (λ − λ B ) and there is no effect of delay on 2 , dY3 . the derivatives of these variables dY dλ dλ 4.1 Find our system final two delay differential equations (DDEs). 4.2 Find our system fixed points and how there are dependent on system parameters? 4.3 Discuss the stability of our system by the standard local stability analysis about any one of the equilibrium points (fixed points) for the following cases: (a) λ A = 0; λ B = λ (b) λ A = λ ; λ B = 0. 4.4 Find the small increment Jacobian of our system. 4.5 Discuss stability switching for different values of λ parameter (cases (a), (b)). 5.

We have a system of Ultra-broad bandwidth pulse-shaping apparatus which includes two gratings, two spherical mirrors, and crystal SLM (Spatial Light Modulator). Any material (SLM) contributions to positive second order group velocity dispersion (GVD) within the shaper can be canceled if the position of the final grating is adjusted. There is phase dispersion at higher orders, specially

2.4 Questions

219

cubic and quartic phase dispersions. Fiber propagation parameteris β(ω)  = dn β (ω) (n) (ω) (n) , where L is the propagation distance then β (ω) = L = dωn . L ω=ω0

The derivatives β (n) (ω) are given in terms of the index of refraction. Due to other source of probe laser in our system there is a polynomial representation of the third, fourth, and fifth power of wavelength variable in our derivatives of β (n) (ω) in terms of index of refraction differential equations. 3  (2)

β (ω) =

k · λ k

k=0

·

d 2n dλ2

2·π · 4  k · λ k

 d 2n d 3n k=0 (3) β (ω) = − · 3 · + λ · 4π 2 c3 dλ2 dλ3 c2

5 

k · λ k

 3 d 2n d 3n λ2 d 4 n (4) · · β (ω) = · +λ· 3 + π 3 · c4 4 dλ2 dλ 4 dλ4 0 , 1 , . . . , 5 ∈ R+ k=0

We take the second, third, and fourth derivative of propagation parameter (β(ω)) in ω as a constant atω = ω0 . We define new variables: Y1 , Y2 , Y3 , Y4 . Y1 =

d 2n d 3n d 4n dn dY1 dY2 dY3 , Y2 = = = = , Y = , Y = 3 4 dλ dλ dλ2 dλ dλ3 dλ dλ4

5.1 Find system fixed point, How they dependent on system parameters? 5.2 Use linearization technique and try to approximate the phase portrait near a fixed point by that of a corresponding linear system. 5.3 Find the Jacobian matrix at each system’s fixed point. 5.4 Classify each fixed point according to eigenvalues and discuss stability. 5.5 How system’s parameters 0 , 1 , . . . , 5 ∈ R+ establish our system stability? Discuss each parameter in details. 6.

We have a system of Ultra-broad bandwidth pulse-shaping apparatus which includes two gratings, two spherical mirrors, and crystal SLM (Spatial Light Modulator). Any material (SLM) contributions to positive second order group velocity dispersion (GVD) within the shaper can be canceled if the position of the final grating is adjusted. There is phase dispersion at higher orders, specially cubic and quartic phase dispersions. Fiber propagation parameter is β(ω) = (n) dn β (ω) , where L is the propagation distance then β (n) (ω) =  L(ω) = ( dω n )ω=ω0 . L (n) The derivatives β (ω) are given in terms of the index of refraction. Due to other source of probe laser in our system there is a polynomial representation of the third, fourth, and fifth power of wavelength variable in our derivatives of

220

2 Ti: Sapphire Laser Systems with Delay Parameters …

β (n) (ω) in terms of index of refraction differential equations. 3  (2)

β (ω) =

k · λ k

k=0

2·π · 4 

=−

·

d 2 n (3) ; β (ω) dλ2

k · λ k

 d 2n d 3n · 3· 2 +λ· 3 4π 2 c3 dλ dλ

k=0,k=2 5 

k=0,k=3

β (4) (ω) =

3 · 4

c2

k · λ k

π 3 · c4  d 2n d 3n λ2 d 4 n ; 0 , 1 , . . . , 5 ∈ R+ · 2 +λ· 3 + · dλ dλ 4 dλ4

We take the second, third, and fourth derivative of propagation parameter (β(ω)) in ω as a constant atω = ω0 . We define new variables: Y1 , Y2 , Y3 , Y4 . Y1 =

d 2n d 3n d 4n dn dY1 dY2 dY3 , Y2 = = = = , Y3 = , Y4 = 2 3 dλ dλ dλ dλ dλ dλ dλ4

Additionally, due to interferences in our system there are shifting in new variables Y2 , Y3 , Y2 (λ) → Y2 (λ − λ A ); Y3 (λ) → Y3 (λ − λB )and there is no effect 2 , dY3 . of delay on the derivatives of these variables dY dλ dλ 6.1 Find our system final two delay differential equations (DDEs). 6.2 Find our system fixed points and how there are dependent on system parameters? 6.3 Discuss the stability of our system by the standard local stability analysis about any one of the equilibrium points (fixed points) for the following cases: (a) λ A = 0; λ B = λ (b) λ A = λ ; λ B = 0. 6.4 Find the small increment Jacobian of our system. 6.5 Discuss stability switching for different values of λ parameter (cases (a), (b)). 7.

We have a system of Ti: Sapphir laser mode semiconductor saturable absorber mirror (SESAM) that operate in a multipulse mode. We need to analyze the stability of a soliton in the laser when subject to perturbations from a filter and an absorber. The system generalized complex Ginzburg-Landau equation (GCGLE) which is the master equation to describe the dynamical effects of laser 2 by means of numerical simulations: i · TR · ψT − β22 · ∂∂tψ2 + κ · |ψ|2 · ψ = i · (g −

q − l0 ) · ψ + i · β · ∂∂tψ2 . We can consider our system delay differential equations (DDEs) when the perturbations on the right hand side are not neglected. 2

2.4 Questions

221

d Qa = 2 · [g(T ) · (Q a (T − τ1 ) + Q b (T − τ2 )) dT − l0 − δa − αa ] · Q a (T − τ1 ) d Qb = 2 · [g(T ) · (Q a (T − τ1 ) + Q b (T − τ2 )) TR · dT − l0 − δb − αb ] · Q b (T − τ2 ) dg (g(T ) − g0 ) (Q a (T − τ1 ) + Q b (T − τ2 )) · g(T ) − =− dT Tg Pg · Tg · TR

TR ·

τ1 , τ2 —Time delay parameters which are related to pulse energies, we get it for our system when the semiconductor saturable absorber mirror (SESAM) is not ideal in his functionality, δa , δb —Pulse filter losses (a, b), αa , αb — Pulse absorber losses (a, b), Q a , Q b —Pulse energies, l0 —Linear loss (output coupling), TR —Round trip time, Tg —Saturable gain recovery time, Pg · Tg —Saturation energies, g, Saturable gain, g0 —Unsaturated values of gain, T —Time parameter. 7.1 Find system fixed points and how they change for different values of system parameter (plot graphs of the behavior). 7.2 Discuss the stability of our system by the standard local stability analysis about any one of the equilibrium points (fixed points) for the following cases: (a) τ1 = 0; τ2 =  (b) τ1 = ; τ2 = 0. 7.3 Find the small increment Jacobian of our system and characteristic equation D(λ, ). 7.4 Discuss stability switching for different values of  parameter (cases (a), (b)). 8.

We have a system of Ti: Sapphir laser mode semiconductor saturable absorber mirror (SESAM) that operate in a multipulse mode. We need to analyze the stability of a soliton in the laser when subject to perturbations from a filter and an absorber. The system generalized complex Ginzburg-Landau equation (GCGLE) which is the master equation to describe the dynamical effects of laser 2 by means of numerical simulations: i · TR · ψT − β22 · ∂∂tψ2 + κ · |ψ|2 · ψ = i · (g −

q − l0 ) · ψ + i · β · ∂∂tψ2 . We can consider our system delay differential equations (DDEs) when the perturbations on the right hand side are not neglected. 2

d Qa = 2 · [g(T − τ ) · (Q a (T ) dT + Q b (T )) − l0 − δa − αa ] · Q a (T ) d Qb = 2 · [g(T − τ ) · (Q a (T ) TR · dT + Q b (T )) − l0 − δb − αb ] · Q b (T ) (g(T − τ ) − g0 ) (Q a (T ) + Q b (T )) · g(T − τ ) dg =− − dT Tg Pg · Tg · TR

TR ·

222

2 Ti: Sapphire Laser Systems with Delay Parameters …

τ —Time delay parameters which is related to Saturable gain (g): we get it for our system when the semiconductor saturable absorber mirror (SESAM) is not exactly the same as our original design, δa , δb —Pulse filter losses (a, b), αa , αb —Pulse absorber losses (a, b), Q a , Q b —Pulse energies, l0 —Linear loss (output coupling), TR —Round trip time, Tg —Saturable gain recovery time, Pg · Tg —Saturation energies, g—Saturable gain. g0 —Unsaturated values of gain, T —Time parameter. 8.1 Find system fixed points and how they change for different values of system parameter (plot graphs of the behavior). 8.2 Discuss the stability of our system by the standard local stability analysis about any one of the equilibrium points (fixed points) for different value of time delay parameter (τ ). 8.3 Find the small increment Jacobian of our system and characteristic equation D(λ, τ ). 8.4 Discuss stability switching for different values ofτ parameter. 9.

Semiconductor ring lasers (SRLs) are miniature ring laser devices with potential applications in optoelectronics, photonics and all-optical circuits. Semiconductor ring lasers are literally ring-shaped optical waveguides with a lasing medium. They have the ability to trap light in a ring, and recirculate it continuously as long as they remain powered. SRL have considerable gain and a low threshold pump power, making it a highly efficient laser system. The light was confined only to the film and did not reflect off of the outer surface of the optical fiber. There are nonlinear behaviors in semiconductor ring lasers (SRLs). A multimode model has been developed for linear and nonlinear interactions between modes in an SRL lasing uni-directionally. Heterodyne detection has been used to make high resolution measurements of the lasing spectra of an SRL in which the individual resonances associated with the coupled eigenvalues can be observed. There are number of key parameters characterizing the coupling mechanisms in the device and the semiconductor gain medium. There is an analytical model for mode interactions in an SRL. The analytic model helps us to extract key parameters characterizing the linear and nonlinear coupling mechanisms in the semiconductor laser medium. The analysis of nonlinear effects in a near single-mode laser can be simplified by considering only interactions pumped by the lasing mode. The analyses ignore the effect of the cavity resonance on the mode gain and analyze the interaction between a strong pump field at ω0 and a pair of low intensity fields at ω1 = ω0 +  and ω1 = ω0 − . The refractive index dispersion is usually specified in terms of the first and second order variation in refractive index n with wavelength or refractive index variation in terms of ω. We can then express the first and second dispersion of    χb in terms of n (χb = χb (n)). χb1 accounts for the linear gain and refractive index of the material. χb1 varies with frequency due to gain and refractive index dispersion.

2.4 Questions

223

ω dn  χb1 = n 2b = n 20 − 2 · n 0 · ·λ· ω0 dλ 2

3 2 2 2 n (ω) dn d + · λ· + 2 · n 0 · λ2 · 2 dλ dλ ω02 

 χb1

=

n 2b

=

 χ01



dχ 1 d 2χ 1 + ω · + (ω)2 · dω dω2 



The only variables in our system are n = n(λ); χ 1 = χ 1 (ω). We consider the   parameters of our system: χb1 , n b , n 0 , χ01 , ω, ω0 . 9.1 Find system fixed points and how they change for different values of system parameter (plot graphs of the behavior). 9.2 Find the small increment Jacobian of our system and characteristic equation. 9.3 Use linearization technique (if you need) and try to approximate the phase portrait near a fixed point by that of a corresponding linear system. 9.4 Classify each fixed point according to eigenvalues and discuss stability. 9.5 How system parameters influence our system stability. 10. There is an important directional mode switching in semiconductor ring lasers through optical injection co-propagating with the lasing mode. It based on the particular structure of a two dimensional asymptotic phase space. Semiconductor Ring Lasers (SRLs) have Bi-stability, possibility of operation in either of the two counter propagating directions. The switching of the direction of operation of a SRL consists of injecting an optical signal counter propagating to the lasing mode. Once the signal is removed, the SRL remains stable in this direction. There is a scheme based on injection from only one side of the SRL, which would remove the need to have an injection laser at both sides of the SRL in order to switch back and forth between the two directional sides. The analysis of directional switching within the existing models is complicated due to the large number of dynamical variables. The analysis is based on an asymptotic reduction to two dimensions of a general rate equation model for a SRL. The switching scheme relies on the special topology of the invariant manifolds of the system. The SRL is operated in a single longitudinal, single-transverse mode. The rate equations model for that device is as follow: d E 1,2 = κ · (1 + i · α) · [N · (1 − s · |E 1,2 |2 − c · |E 2,1 |2 ) − 1] · E 1,2 dt 1 − k · ei·φk · E 2,1 + · F1,2 (t) κ · τin Hint: The above differential equation is actually two differential equations: first, is with the first index and second is with the second index (Related to E i , Fi ; i = 1, 2).

224

2 Ti: Sapphire Laser Systems with Delay Parameters …

d E1 = κ · (1 + i · α) · [N · (1 − s · |E 1 |2 − c · |E 2 |2 ) − 1] · E 1 dt 1 − k · ei·φk · E 2 + · F1 (t) κ · τin d E2 = κ · (1 + i · α) · [N · (1 − s · |E 2 |2 − c · |E 1 |2 ) − 1] · E 2 dt 1 − k · ei·φk · E 1 + · F2 (t) κ · τin dN = γ · [μ − N − N · (1 − s · |E 1 |2 − c · |E 2 |2 ) · |E 1 |2 dt − N · (1 − s · |E 2 |2 − c · |E 1 |2 ) · |E 2 |2 ] E 1 , E 2 —Slowly varying envelopes of the counter-propagating fields, N — Carrier inversion density, κ—Field decay rate, γ —Carrier decay rate, α— Linewidth enhancement factor, μ—Renormalized injection current with μ ≈ 0 at transparency and μ ≈ 1 at lasing threshold. s, c: parameters which self and cross saturation effects are added phenomenologically and are molded by them, k—Amplitude that characterized the localized reflections result in a linear coupling between the two fields, φk —Phase shift that characterized the localized reflections result in a linear coupling between the two fields, τin —Flight time in the ring cavity, |E in j |2 , Injected power, —Detuning between both lasers. Remark Reflection of the counter-propagating modes occurs where ring cavity and Coupling waveguide meet and can also occur at the end facets of the coupling waveguide. Two counter-propagating modes are considered to saturate both their own and each other’s gain due to spectral hole burning effects. The term F1,2 (t) = E in j · ei··t represents the optically injected field in one of the two modes. The variables of our system are E 1 = E 1 (t); E 2 = E 2 (t); N = N (t). 10.1 Find system fixed points and how they change for different values of system parameter (plot graphs of the behavior). 10.2 Find the small increment Jacobian of our system and characteristic equation. 10.3 Use linearization technique (if you need) and try to approximate the phase portrait near a fixed point by that of a corresponding linear system. 10.4 Classify each fixed point according to eigenvalues and discuss stability. 10.5 How system parameters influence our system stability. Remark The rate equations model for that device include complex functions, so you need to analyze them differently than a regular real functions.

References

225

References 1. A. Talebpour, S. Larochelle, S.L. Chin, Suppressed tunneling ionization of the D2 molecule in an intense Ti: sapphire laser pulse. J. Phys. B: At. Mol. Opt. Phys. 31, L49–L58 (1998) 2. A. Efimov, C. Schaffer, D. H. Reitze, Programmable shaping of Ultrabroad-bandwidth pulses from a Ti: sapphire laser. J. Opt. Soc. Am. 12(10) (1995) 3. M.J. Lederer, B. Luther-Davies, H.H. Tan, C. Jagadish, N.N. Akhmediev, J.M. Soto-Crespo, Multipulse operation of a Ti: sapphire laser mode locked by an ion-implanted semiconductor saturable absorber mirror. J. Opt. Soc. Am. B 16(6) (1999) 4. F. Song, J.Q. Yao, D.W. Zhou, J.Y. Qiao, G.Y. Zhang, J.G. Tian, Rate-equation theory and experimental research on dual-wavelength operation of a Ti: sapphire laser. Appl. Phys. B 72, 605–610 (2001) 5. S.H. Strogatz, Nonlinear Dynamics and Chaos. Westview Press 6. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos. Text Appl. Math. (Hardcover) 7. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993) 8. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002) 9. J. Kuang, Y. Cong, Stability of numerical methods for delay differential equations. Elsev. Sci. (2007) 10. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33(5), 1144–1165 (Published electronically February 14, 2002) 11. Y. Kuang, Delay differential equations with applications in population dynamics. Math. Sci. Eng. 191 (1993 by Academic Press, Inc) 12. J.A. Davis, D.E. McNamara, D.M. Cottrell, T. Sonehara, Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator. Appl. Opt. 39(10) (2000) 13. J. Guckenheimer, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42 14. X. Han, H. Zeng, Kerr-lens mode-locked Ti: sapphire laser with an additional intra cavity nonlinear medium. Opt. Express, 16(23) (2008) 15. W. Van Saarloos, P.C. Hohenberg, Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Phys. Rev. Lett. 64(7) (1990) 16. N. N. Akhmediev, A. Ankiewicz, Multi soliton solutions of the complex Ginzburg-Landau equation. Phys. Rev. Lett. 79(21) (1997)

Chapter 3

Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Ion-channel system includes relativistic electron beam propagating through plasma in the ion-focused regime. Growth is enhanced by optical guiding in the ion channel, which acts as a dielectric waveguide. It is a transport of relativistic electron beams (REBs) in plasmas and mechanism of ion-focusing. The propagation in plasmas of short pulse, low emittance relativistic electron beams is related to plasma lens, the continuous plasma focus, the plasma wake field accelerator, and the beat wave accelerator. Beam injection and extraction from a plasma module is important in order to produce high quality electron beams with a plasma accelerator. There is a need for proper matching conditions to focus the incoming high brightness beam down to few microns size and to capture a high divergent beam at the exit without loss of beam quality. Plasma-based lenses provide focusing gradients of the order of kT/m with radially symmetric focusing. It gives compact and affordable alternative to permanent magnets in the design of transport lines. There is a coherent radiation from intense, relativistic electron beams in un-magnetized plasmas. The ion channel “free electron” laser (ICL), consisting of an intense, relativistic beam of electrons injected into a plasma less dense than the beam. The ion channel FEL (Free Electron Laser) makes use of ion-focusing to transport the beam, and a resonance, akin to that of the planer wiggler FEL, to produce coherent radiation. The ion channel acts as an optical fiber with step index of refraction. For high current, dielectric guiding eliminates the usual constraint that the Rayleigh length must be longer than the gain length. Rayleigh length or Rayleigh range is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. The three-dimensional free electron laser (FEL) power gain length’s increase due to the collective effects of an ultra-relativistic electron beam, geometric transverse wake-field, coherent synchrotron radiation, and micro bunching instability. The gain length is affected by an increase of the electron beam projected emittance. It is done even though the slice local emittance is preserved. The ion channel FEL consists of a tank of neutral gas, from centimeters to meters in length, through which a plasma column millimeters in width is produced by an ionizing laser pulse. The relativistic electron beam (REB) is injected within less than a recombination time, and with © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_3

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

proper matching, as in a continuous plasma focus. It is propagating in the axial (+z) direction. As the beam head propagates through the plasma, it continuously expels plasma electrons from the beam volume, leaving fixed the relatively immobile ions to provide focusing for the remainder of the beam. The transverse force on the beam due to self-fields is much less than the transverse electric field due to the ion charges. The expulsion of plasma electrons produces a cylindrical channel occupied by un-neutralized ions and this is the “ion channel”. We consider the motion in the ion-channel of a single electron, subject to an electromagnetic wave linearly polarized in the y direction. A Betatron is a type of cyclic particle accelerator which is a transformer with a torus-shaped vacuum tube as its secondary coil. An alternating current in the primary coils accelerates electrons in the vacuum around a circular path. The betatron period is a definition which is related to our ion-channel laser system. We get the perturbed equations of motion when averaging over the betatron period. Longitudinal modes in Long-wavelength lasers have mode-competition phenomena which we need to measure. The model behavior of an InGaAsP-InP Fabry Perot laser emitting in a long wavelength region is analyzed and stability is inspected. Modecompetition phenomena induce quasi-periodic hopping among several longitudinal modes, which reveal multimode like output spectra as the time-averaged spectra in long-wavelength lasers. There are variations of photon number and their frequency spectra in addition to the longitudinal mode spectra. The mechanism of the predicted multi-mode like oscillation is analyzed and theoretical model, variation of the photon number Sp , injected electron number N by set of differential equations. There is influence of instantaneous mode competition on the dynamics of semiconductor lasers. The nonlinear dynamics of the lasing modes are described mathematically by multimode rate equations of the photon number Sp of the lasing modes and the injected electron number N [1], 2, 3].

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis The basic mechanism is laser-driven acceleration in ion-channel. A long laser beam propagating through un-dense plasma produces a positively charged ion channel by expelling plasma electrons in the transverse direction. There is a dynamics of an electron in a resulting two-dimensional channel under the action of the laser field and the transverse electric field of the channel. We can enhance the axial momentum via amplification of betatron oscillations. Oscillations can be parametrically amplified when the betatron frequency, which increases with the wave amplitude, becomes comparable to the frequency of its modulations. There is a non-inertial (accelerated/decelerated) relativistic axial motion induced by the wave regardless of the angle between the laser electric field and the field of the channel and caused to modulation. For specific density, there is a well pronounced wave amplitude threshold above which the maximum electron energy is considerably enhanced. The energy

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

229

enhancement we get is accompanied by spectrum broadening. The regime of laser interaction with the target electrons is greatly influenced by the density of the target. The interaction takes place only at the surface of the target if the density is overcritical (solid density targets). The laser beam is able to go through the target if the density is sub-critical, as in the case of gas jets. The low-density targets have extended interaction length with the laser and a higher energy gain by the electrons. A sub-critical plasma layer occurs also with solid-density targets where a significant pre-pulse is present. A pre-pulse can create a transparent pre-plasma, extending many wavelengths from the target surface along the beam path. The main pulse interacts with low-density plasma before reaching the target. We consider laser interaction with low density sub-critical plasma in order to see if such interactions can generate hot electrons in addition to the ones produced at the critical surface. The electron acceleration mechanism in this regime depends on laser pulse duration. A laserproduced wake field, the wake field mechanism becomes less efficient for longer laser pulses. For long laser beam, its ponder motive pressure tends to expel plasma electrons from the beam in the transverse direction. The plasmas generate a counteracting electric field via charge separation. The laser can create a positively charged channel with a quasi-static transverse electric field evolving on an ion time scale. There are dynamics of an electron irradiated by a linearly polarized electromagnetic wave in a two-dimensional steady-state ion channel. The relativistic electron beam propagating through plasma in the ion-focused regime exhibits an electromagnetic instability at a resonant frequencyω ∼ 2 · γ 2 · ω p (γ —Lorentz factor, ω p —beam plasma radial frequency). There is an optical guiding inthe ion channel, which acts as a dielectric waveguide, with fiber parameters V ∼ 2 · IIA (I—Peak beam current, I A —Alfven current). The Alfven current I A is a useful scaling parameter for intense beam studies. The magnetic deflection of electrons in a charge-neutral beam defines an upper limit on the transportable current. When a high-current beam enters plasma, plasma particles move to cancel the beam generated electric field. There is a response of a plasma to a steady-state beam and characterized plasma Debye length. Plasmas are collections of ions and electrons governed by long-range electromagnetic interactions. Usually, the densities of ions and electrons, n e , and n i , are almost equal and the mixture is space-charge neutralized. On plasma transport of electron beams the beam density n b is much smaller than the plasma density n e …(n b  n e ). The beam drives out a few of the plasma electrons to achieve complete space-charge neutralization, where; n i = n e + n b . Relativistic electron beams generate strong magnetic focusing forces. The beams are self-contained with only partial space-charge neutralization. The transport of such beams in low-density plasmas (n b > n e ) is of considerable interest. The beam expels all the low-energy plasma electrons. The kinetic energy of plasma particles affects how they respond to beams. The velocity dispersion of plasma ions and electrons as Maxwell distributions with temperatures Ti and Te is presented. . Unconfined plasmas The plasma particles have average drift velocities: Vi and Ve e . The condition are stable against velocity space instabilities if |Vi , |Ve |  2·k·T me holds when moderate current beams propagate through dense plasmas. High-current pulsed electron beams can induce a large plasma electron drift velocity, leading to

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

a two stream instability and rapid plasma heating. Plasmas used for electron beam transport can have low temperature (k · Te < 100 eV), and large spatial extent. The methods for beam plasma generation include laser ionization, pulsed plasma guns, low-energy electron discharges or collisions of beam particles with a background gas. Beam-generated plasmas are often not fully ionized-neutral atoms are present in the beam volume. Atoms may influence the beam and plasma responses through collisions. There is a possible simple model to calculate the steady-state response of plasma to an injected beam. The beam enters unconfined plasma of infinite dimension with no included magnetic field. Low-energy electrons respond rapidly to the presence of the beam while the massive ions respond slowly. Assuming that beam pulse is long enough for plasma electrons to adjust to a modified equilibrium. The ion channel laser (ICL), consisting of an intense, relativistic beam of electrons injected into a plasma less dense than the beam. The ion channel laser (ICL) makes use of ion-focusing to transport the beam, and a resonance, akin to that of the planar Wiggler FEL, to produce coherent radiation. A uniform static axial magnetic field is induced in addition to the ion electro static field and Wiggler magnetic field, this is a possibility. The evolution of electron beam in the tapered planar Wiggler field is with self-electric and self-magnetic fields. Free-electron laser (FEL) operation often requires sufficiently large gain, which increase when the beam current is increased. In the high-current regime, the electron motion can be altered by the self-generated field effects. There is evolution of electron beam in the tapered planar Wiggler field with self-electric and self-magnetic fields. The ion channel laser (ICL) consists of a tank of neutral gas, through which a plasma column millimeters in width is produced by a laser pulse. A relativistic electron beams (REBs) is injected, propagating in the axial (+z) direction. It is done within less than a recombination time, and with proper matching. A relativistic electron beam, propagating through an under dense plasma (n b > n e ), expels plasma electron from the beam volume and beyond to produce an “ion-channel”, which then focusses the beam, and causes it to radiate. The structure of relativistic electron beam which injected and propagating in the axial (+z) direction is as follow (Fig. 3.1). The beam head is propagated through the plasma and expels plasma electrons from the beam volume. It is leaving fixed the relatively immobile ions to provide focusing for the remainder of the beam. Assumption: The transverse force on the beam due to self-fields is much less than the transverse electric fields due to the ion-charge. The plasma densities possibilities are as follow (Table 3.1). The  expulsion of plasma electron produces a cylindrical channel of radius b, b ≈ a · nn bp occupied by un-neutralized ions and this is the “ion-channel”. We define τr as the current rise time. The large radial plasma oscillations are not excited as plasma 4·π·n p ·e2 electrons are ejected from the channel and this requires ω p · τr  1; ω2p = . m E—Electron charge, m—Electron mass, ω p —Plasma frequency,ωb —beamplasma frequency, η—Overlap integral of the mode and beam radial profiles, τr — Current rise time, n p —Plasma density prior to channel formation, a—Relativistic electron beams (REBs) cylindrical channel radius, b—A cylindrical channel radius

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

y z b

231

a

x

Relativistic Electron beam Ion channel Quasi neutral Plasma

Figure 3.1 Structure of relativistic electron beam which injected and propagating in the axial (+z)

Table 3.1 System plasma densities possibilities, in term of n p and n b Condition Meaning (outcomes) np 

nb γ2

nb > n p

Transverse force on the beam due to self-fields is much less than the transverse electric field due to the ion charge (n p —Plasma density prior to channel formation, n b —Peak beam density, γ —Lorentz factor) The generalization to an over dense plasma

which expulsion of plasma electron happened, c —Speed of light in vacuum, I — Peak beam current, I A —Alfven current, τ —Beam length, τi —Time for the ions to collapse inward due to the radial electric field of the beam, ωβ —Betatron frequency which electrons oscillate in x and y, ω1 —Up shifted frequency which electrons are oscillating, ω—Frequency of radiation in the forward (+z) direction in the lab frame, A, ϕ - Eikonal amplitude and phase, respectively, A y —Vector potential at y direction, px , p y , pz —Canonical momenta in x, y and z, respectively, Vz —Betatron averaged drift velocity in z, εn —rms normalized emittance, ω—Detuning parameter, kβ —Betatron angular wave number, k z —Wave number in z direction, ψ—Phase variable, aβ —Betatron parameter which is analogous to the wiggler parameter in an Free Electron Laser (FEL), qx , q y , qz —System’s variables in x, y, and z directions, θx , θ y —System variables in x and y directions. The beam length (τ ) is short compare to the time for ions to collapse  inward due to the radial electric field of the beam (τi ),  τ < τi ; τi ∼ 41 · ( bc ) ·

mi m

·

IA I

> τ . Additionally I = π · a 2 · n b · e · c ; I A =

m·c3 , e

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we can write for the electrons which oscillate in x and y directions at the beta m·c tron frequencyωβ ∼ ω p · 2· pz . Electrons are oscillating with upshifted frequency ω1 ∼ γ ·ωβ in the center of momentum frame and radiate incoherently. The frequency of radiation in the forward (+z) direction, in the lab frame is ω ∼ 2·γ ·ω1 ∼ 2·γ 2 ·ωβ . The motion in the “ion-channel” of a single electron is dependent on the electromagnetic wave linearity polarized in the y direction. We define the vector potentialA y , 2 A y = ( mce ) · A · sin(ξ ); ξ = k z · z − ω · t + ϕ. We define A, ϕ as Eikonal amplitude and phase, respectively (the two parameters are related to Eikonal approximation method useful in wave scattering equation which occur  in our system). The 4·π·n b ·e2 2 b ωb = 2 · e · π·n . Assumption: The beam-plasma frequencyωb , ωb = m m an-harmonic in the transverse motion and second order terms in A are neglected, as well as derivatives of the slowly varying Eikonal quantities, and transverse gradients. We relate to the canonical momenta in x, y and z, respectively px , p y , pz , such that px = m · c · qx · sin(θx ) ; p y = m · c · q y · sin(θ y ); pz = m · c · qz . If the Eikonal dθ x = ωβ ; dty = ωβ . The phase amplitudeA, A = 0, qx , q y are constants then dθ dt variable ψ, ψ = θ y + ξ ; ψ = θ y + k z · z − ω · t + ϕ, average over the betatron period. The detuning parameter ω, ω = k z · Vz − ω + ωβ , where Vz is betatron (2+q 2 +q 2 )

q 2 +q 2

q 2 +q 2

x y average drift velocity in z, Vcz = 1 − 4·qx 2 y . The aβ , aβ2 = x 2 y ; aβ = 2 z is analogous to the wiggler parameter √ in an FEL. We can consider for the round 3 qz ·kβ ·a beam (Fig. 3.1) aβ , aβ = √2 aβ = 2a ·εn (initially the same for each particle).

q ·k ·a 2

Additionally, εn = z 4β ; kβ = (average over betatron period) [1]:

ωβ . c

The perturbed equations of motion derived

dθ y qy dψ dϕ 1 = k z · Vz − ω + + + · k z · c · 2 · A · cos(ψ) dt dt dt 2 qz dθ y qy 1 1 dqz = ωβ · [1 − = − · kz · c · · A · cos(ψ)] ; · A · sin(ψ) dt 2 · qy dt 2 qz q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2 is very small dϕ → 0. We consider that the derivative of Eikonal phase in time dϕ dt dt We get the following perturbed equations of motion (average over betatron period): 1 dϕ dψ · A · cos(ψ)] + ( = k z · Vz − ω + ωβ · [1 − → 0) dt 2 · qy dt qy 1 + · k z · c · 2 · A · cos(ψ) 2 qz

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

233

qy dqz 1 · A · sin(ψ); = − · kz · c · dt 2 qz q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2 Our system variables are ψ, qx , q y , qz .

Vz c

= 1−

(2+qx2 +q y2 ) ]. 4·qz2

(2+qx2 +q y2 ) 4·qz2

⇒ Vz = c · [1 −

We get the following perturbed equations of motion (average over betatron period): (2 + qx2 + q y2 ) dψ 1 = k z · c · [1 − ] − ω + ωβ · [1 − · A · cos(ψ)] dt 4 · qz2 2 · qy qy 1 + · k z · c · 2 · A · cos(ψ) 2 qz q y2 qy dq y dqz 1 1 1 = − · kz · c · = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) · A · sin(ψ) ; dt 2 qz dt 2 4 qz q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2 At fixed points:

dψ dt

=0;

dqz dt

=0;

dq y dt

=0;

dqx dt

=0

q y∗ · qx∗ 1 dqx = 0 ⇒ − · kz · c · · A · sin(ψ ∗ ) = 0 dt 8 (qz∗ )2 Case 1 q y∗ = 0 q y∗ = 0;

dq y 1 = 0 ⇒ − · ωβ · A · sin(ψ ∗ ) = 0; ωβ · A = 0 dt 2 ⇒ sin(ψ ∗ ) = 0 ⇒ ψ ∗ = n · π ∀ n = 0, 1, 2, ...; n ∈ N0

dψ (2 + (qx∗ )2 ) = 0 ⇒ k z · c · [1 − ] − ω + ωβ dt 4 · (qz∗ )2 1 · A · cos(ψ ∗ )] · [1 − 2 · (q y∗ = 0) =0

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

 cos(ψ ∗ ) = ± 1 − sin2 (ψ ∗ ); sin(ψ ∗ ) = 0 ⇒ cos(ψ ∗ ) = ±1 dψ (2 + (qx∗ )2 ) = 0 ⇒ k z · c · [1 − ] − ω + ωβ dt 4 · (qz∗ )2 1 · [1 − · A · (±1)] 2 · (q y∗ = 0) =0 ωβ > 0; ωβ · [1 − k z · c · [1 −

2 · (q y∗ = 0)

· A · (±1)] → ∓∞

(2 + (qx∗ )2 ) ] − ω → ±∞ 4 · (qz∗ )2

[k z · c − ω] − k z · c · ⇒ −k z · c ·

1

(2 + (qx∗ )2 ) → ±∞ 4 · (qz∗ )2

k z ·c−ω1 (2 + (qx∗ )2 ) → ±∞ − [k z · c − ω] ≈ ±∞ ∗ 2 4 · (qz )

kz · c ·

(2 + (qx∗ )2 ) → ∓∞ ⇒ qz∗ = 0 4 · (qz∗ )2

The first fixed point: E (0) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, qx∗ , 0, 0); n ∈ N0 ; qx∗ ∈ R. Remark The qx∗ values we choose establish first fixed point coordinate’s values. Case 2 qx∗ = 0 qx∗ = 0;

q y∗ 1 dqz = 0 ⇒ − · k z · c · ∗ · A · sin(ψ ∗ ) = 0 dt 2 qz

(q y∗ )2 dq y 1 1 = 0 ⇒ − · (ωβ + · k z · c · ∗ 2 ) · A · sin(ψ ∗ ) = 0 dt 2 4 (qz ) (2 + (q y∗ )2 ) dψ = 0 ⇒ k z · c · [1 − ] dt 4 · (qz∗ )2 1 · A · cos(ψ ∗ )] − ω + ωβ · [1 − 2 · q y∗ +

q y∗ 1 · k z · c · ∗ 2 · A · cos(ψ ∗ ) 2 (qz )

=0 z Case 2.1 qx∗ = 0; dq = 0 ⇒ − 21 · k z · c · dt

q y∗ qz∗

· A · sin(ψ ∗ ) = 0 ⇒ sin(ψ ∗ ) = 0

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

sin(ψ ∗ ) = 0 ⇒ ψ ∗ = n · π ∀ n = 0, 1, 2, ...; n ∈ N0  cos(ψ ∗ ) = ± 1 − sin2 (ψ ∗ ); sin(ψ ∗ ) = 0 ⇒ cos(ψ ∗ ) = ±1 k z · c · [1 − +

(2 + (q y∗ )2 ) 4

1 · kz · c · 2

· (qz∗ )2 q y∗ · (qz∗ )2

k z · c · [1 −

] − ω + ωβ · [1 −

1 · A · cos(ψ ∗ )] 2 · q y∗

A · cos(ψ ∗ ) = 0

(2 + (q y∗ )2 )

] + [ωβ − ω] 4 · (qz∗ )2 q y∗ 1 ωβ + { · kz · c · ∗ 2 − } · A · cos(ψ ∗ ) = 0 2 (qz ) 2 · q y∗ k z · c · [1 −

(2 + (q y∗ )2 )

] + [ωβ − ω] 4 · (qz∗ )2 q y∗ 1 ωβ + { · kz · c · ∗ 2 − } · (±1) = 0 2 (qz ) 2 · q y∗

k z · c · [1 − Case 2.1.1

kz · c −

(2 + (q y∗ )2 )

] + [ωβ − ω] 4 · (qz∗ )2 q y∗ 1 ωβ + { · kz · c · ∗ 2 − } · (+1) = 0 2 (qz ) 2 · q y∗

k z · c · (2 + (q y∗ )2 ) 4 · (qz∗ )2

+ [ωβ − ω] +

q y∗ 1 ωβ · kz · c · ∗ 2 − =0 2 (qz ) 2 · q y∗

q y∗ k z · c · (2 + (q y∗ )2 ) 1 ωβ · kz · c · ∗ 2 − − = ω − ωβ − k z · c 2 (qz ) 4 · (qz∗ )2 2 · q y∗ 2 · k z · c · (q y∗ )2 − k z · c · (2 + (q y∗ )2 ) · q y∗ − ωβ · 2 · (qz∗ )2 4 · (qz∗ )2 · q y∗ = (ω − ωβ − k z · c) 2 · k z · c · (q y∗ )2 − k z · c · (2 + (q y∗ )2 ) · q y∗ − ωβ · 2 · (qz∗ )2 = 4 · (qz∗ )2 · q y∗ · (ω − ωβ − k z · c) 2 · k z · c · (q y∗ )2 − k z · c · (2 + (q y∗ )2 ) · q y∗ = ωβ · 2 · (qz∗ )2 + 4 · (qz∗ )2 · q y∗ · (ω − ωβ − k z · c)

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

2 · k z · c · (q y∗ )2 − k z · c · (2 + (q y∗ )2 ) · q y∗ = 2 · (qz∗ )2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] 2 · k z · c · (q y∗ )2 − k z · c · 2 · q y∗ − k z · c · (q y∗ )3 = 2 · (qz∗ )2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] 2 · k z · c · (q y∗ )2 − k z · c · 2 · q y∗ − k z · c · (q y∗ )3 = 2 · (qz∗ )2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] 2 · (qz∗ )2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] = k z · c · q y∗ · [2 · q y∗ − 2 − (q y∗ )2 ] (qz∗ )2 = qz∗ = ±

k z · c · q y∗ · [2 · q y∗ − 2 − (q y∗ )2 ]

2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)]  k z · c · q y∗ · [2 · q y∗ − 2 − (q y∗ )2 ] 2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)]

The second fixed point: E (1) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, q y∗ ,  k z · c · q y∗ · [2 · q y∗ − 2 − (q y∗ )2 ] ± ) 2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] n ∈ N0 ; q y∗ ∈ R Remark The q y∗ values we choose establish qz∗ fixed point values.   (2 + (q y∗ )2 ) + [ωβ − ω] kz · c · 1 − 4 · (qz∗ )2   Case 2.1.2 q y∗ 1 ωβ · kz · c · ∗ 2 − + · (−1) = 0 2 (qz ) 2 · q y∗  kz · c · 1 −

(2 + (q y∗ )2 )

kz · c − kz · c ·



4 · (qz∗ )2 (2 + (q y∗ )2 ) 4 · (qz∗ )2

+ [ωβ − ω] −

q y∗ 1 ωβ · kz · c · ∗ 2 + =0 2 (qz ) 2 · q y∗

+ [ωβ − ω] −

q y∗ 1 ωβ =0 · kz · c · ∗ 2 + 2 (qz ) 2 · q y∗

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

[k z · c + ωβ − ω] − k z · c ·

(2 + (q y∗ )2 ) 4 · (qz∗ )2

q y∗ 1 ωβ =0 · kz · c · ∗ 2 + 2 (qz ) 2 · q y∗

−

[k z · c + ωβ − ω] +

−q y∗ · k z · c · (2 + (q y∗ )2 ) − 2 · q y∗ · k z · c · q y∗ + 2 · (qz∗ )2 · ωβ 4 · (qz∗ )2 · q y∗

=0 − q y∗ · k z · c · (2 + (q y∗ )2 ) − 2 · q y∗ · k z · c · q y∗ + 2 · (qz∗ )2 · ωβ = [−k z · c − ωβ + ω] · 4 · (qz∗ )2 · q y∗ − q y∗ · k z · c · (2 + (q y∗ )2 ) − 2 · q y∗ · k z · c · q y∗ = 2 · (qz∗ )2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ } k z · c · q y∗ · [−(2 + (q y∗ )2 ) − 2 · q y∗ ] = 2 · (qz∗ )2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ } (qz∗ )2 =

k z · c · q y∗ · [−(2 + (q y∗ )2 ) − 2 · q y∗ ]

2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ }  k z · c · q y∗ · [−(2 + (q y∗ )2 ) − 2 · q y∗ ] qz∗ = ± 2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ } The third fixed point: E (2) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, q y∗ ,  k z · c · q y∗ · [−(2 + (q y∗ )2 ) − 2 · q y∗ ] ± ) 2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ } n ∈ N0 ; q y∗ ∈ R Remark The q y∗ values we choose establish qz∗ fixed point values. z Case 2.2 qx∗ = 0; dq = 0 ⇒ − 21 · k z · c · dt

q y∗ qz∗

· A · sin(ψ ∗ ) = 0 ⇒ q y∗ = 0

(q y∗ )2 dq y 1 1 = 0 ⇒ − · (ωβ + · k z · c · ∗ 2 ) dt 2 4 (qz ) 1 · A · sin(ψ ∗ )|q y∗ =0 = − · ωβ · A · sin(ψ ∗ ) 2 =0

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

1 · ωβ · A · sin(ψ ∗ ) = 0; ωβ · A = 0 2 sin(ψ ∗ ) = 0 ⇒ ψ ∗ = n · π ∀ n = 0, 1, 2, ...; n ∈ N0 −

 cos(ψ ∗ ) = ± 1 − sin2 (ψ ∗ ); sin(ψ ∗ ) = 0 ⇒ cos(ψ ∗ ) = ±1   (2 + (q y∗ )2 ) dψ = 0 ⇒ kz · c · 1 − − ω + ωβ dt 4 · (qz∗ )2 · [1 −

q y∗ 1 1 ∗ · k · A · cos(ψ )] + · c · z 2 · q y∗ 2 (qz∗ )2

· A · cos(ψ ∗ ) = 0 1 ] 2 · (qz∗ )2 1 − ω + ωβ · [1 − · A · cos(ψ ∗ )] 2 · (q y∗ = 0)

qx∗ = 0; q y∗ = 0 ⇒ k z · c · [1 −

=0 kz · c − kz · c ·

1 ωβ · A · cos(ψ ∗ ) = 0 − ω + ωβ − 2 · (qz∗ )2 2 · (q y∗ = 0)

[k z · c − ω + ωβ ] − k z · c · kz · c ·

1 ωβ − · A · cos(ψ ∗ ) = 0 2 · (qz∗ )2 2 · (q y∗ = 0)

1 ωβ · A · (±1) = [k z · c − ω + ωβ ] − 2 · (qz∗ )2 2 · (q y∗ = 0)

ωβ · A · (±1) → ±∞ 2 · (q y∗ = 0)

ωβ · A · (±1)|  k z · c − ω + ωβ 2 · (q y∗ = 0) 1 ωβ · A · (±1) kz · c · =− ∗ 2 2 · (qz ) 2 · (q y∗ = 0) |

kz · c ·

1 → ∓∞ ⇒ qz∗ → 0 2 · (qz∗ )2

The fourth fixed point:E (3) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, 0, 0); n ∈ N0 . Case 3 sin(ψ ∗ ) = 0 ⇒ ψ ∗ = n · π ∀ n = 0, 1, 2, . . . ; n ∈ N0  cos(ψ ∗ ) = ± 1 − sin2 (ψ ∗ ); sin(ψ ∗ ) = 0 ⇒ cos(ψ ∗ ) = ±1

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

239

(2 + (q y∗ )2 + (qx∗ )2 ) dψ = 0 ⇒ k z · c · [1 − ] − ω + ωβ dt 4 · (qz∗ )2 q y∗ 1 1 ∗ · [1 − · A · cos(ψ )] + · c · · A · cos(ψ ∗ ) · k z 2 · q y∗ 2 (qz∗ )2 =0 [k z · c − ω + ωβ ] − +

1 · [k z · c · 2

q y∗ (qz∗ )2

[k z · c − ω + ωβ ] − +

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2 ωβ ] · A · cos(ψ ∗ ) = 0 q y∗

−

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2

q y∗ 1 ωβ · [k z · c · ∗ 2 − ∗ ] · A · (±1) = 0 2 (qz ) qy

k z · c · (2 + (q y∗ )2 + (qx∗ )2 )

−

+

4 · (qz∗ )2

q y∗ 1 ωβ · [k z · c · ∗ 2 − ∗ ] 2 (qz ) qy

· A · (±1) = [−k z · c + ω − ωβ ]

Case 3.1

−

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2

+

q y∗ 1 ωβ · [k z · c · ∗ 2 − ∗ ] · A 2 (qz ) qy

= [−k z · c + ω − ωβ ] −

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2

+ kz · c ·

q y∗ · A 2 · (qz∗ )2

−

ωβ · A 2 · q y∗

= [−k z · c + ω − ωβ ] −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ − ωβ · A · 2 · (qz∗ )2 4 · (qz∗ )2 · q y∗ = [−k z · c + ω − ωβ ] − k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ − ωβ · A · 2 · (qz∗ )2 = 4 · (qz∗ )2 · q y∗ · [−k z · c + ω − ωβ ] − k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ = 4 · (qz∗ )2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A · 2 · (qz∗ )2

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

2 · (qz∗ )2 · {2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A} = −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ (qz∗ )2 =

−k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗

 qz∗

=±

2 · {2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A} −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A}

The fifth fixed point: qx∗ ∈ R; q y∗ ∈ R E (4) (ψ ∗ , qx∗ , q y∗ , qz∗ )  = (n · π, qx∗ , q y∗ , ±

−k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A}

)

n ∈ N0 Remark The qx∗ , q y∗ values we choose establish qz∗ fixed point values. q y∗ k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 1 ωβ · [k + · c · − ∗ ] · A · (−1) − z 4 · (qz∗ )2 2 (qz∗ )2 qy Case 3.2 = [−k z · c + ω − ωβ ] −

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2

+

q y∗ 1 ωβ · [−k z · c · ∗ 2 + ∗ ] · A 2 (qz ) qy

= [−k z · c + ω − ωβ ] −

k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) 4 · (qz∗ )2

− kz · c ·

q y∗ · A 2 · (qz∗ )2

+

ωβ · A 2 · q y∗

= [−k z · c + ω − ωβ ] −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ + ωβ · A · 2 · (qz∗ )2 4 · (qz∗ )2 · q y∗ = [−k z · c + ω − ωβ ] − k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ + ωβ · A · 2 · (qz∗ )2 = 4 · (qz∗ )2 · q y∗ · [−k z · c + ω − ωβ ] − k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

241

= 4 · (qz∗ )2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A · 2 · (qz∗ )2 2 · (qz∗ )2 · {2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A} = −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ (qz∗ )2 =

−k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗

 qz∗

=±

2 · {2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A} −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A}

The six fixed point: qx∗ ∈ R; q y∗ ∈ R E (4) (ψ ∗ , qx∗ , q y∗ , qz∗ )  = (n · π, qx∗ , q y∗ , ±

−k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A}

)

n ∈ N0 Remark The qx∗ , q y∗ values we choose establish qz∗ fixed point values. We can summary the process to find system possible fixed points by the following flowchart. q ·q x = − 18 · k z · c · yq 2 x · A · sin(ψ) (*) dq dt z

dqz = − 21 · k z · c dt dq (***) dty = − 21 · (ωβ

(**)

(****)

dψ = kz · c · dt

qy · A · sin(ψ); qz q2 1 + 4 · k z · c · q y2 ) · A · sin(ψ) z (2 + qx2 + q y2 ) [1 − ] − ω + ωβ 4 · qz2

·

· [1 −

1 2 · qy

qy 1 · k z · c · 2 · A · cos(ψ) 2 qz dq y dqz x = 0; = 0; = 0; dψ = 0. At fixed points: dq dt dt dt dt We can summary our system fixed points (Fig. 3.2a–e): The first fixed point:E (0) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, qx∗ , 0, 0); n ∈ N0 ; qx∗ ∈ R. The second fixed point: · A · cos(ψ)] +

E (1) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, q y∗ ,  k z · c · q y∗ · [2 · q y∗ − 2 − (q y∗ )2 ] ) ± 2 · [ωβ + 2 · q y∗ · (ω − ωβ − k z · c)] n ∈ N0 ; q y∗ ∈ R

242

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

The third fixed point: E (2) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, q y∗ ,  k z · c · q y∗ · [−(2 + (q y∗ )2 ) − 2 · q y∗ ] ± ) 2 · {[−k z · c − ωβ + ω] · 2 · q y∗ − ωβ } n ∈ N0 ; q y∗ ∈ R

Figure 3.2 a Process finding system fixed points (flow chart), Case 1, b Process finding system fixed points (flow chart), Case 2, c Process finding system fixed points (flow chart), Case 2.1, d Process finding system fixed points (flow chart), Case 2.2, e Process finding system fixed points (flow chart), Case 3

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

Figure 3.2 (continued)

243

244

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Figure 3.2 (continued)

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

Figure 3.2 (continued)

245

246

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Figure 3.2 (continued)

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

247

The fourth fixed point: E (3) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, 0, 0, 0); n ∈ N0 . The fifth fixed point: qx∗ ∈ R; q y∗ ∈ R E (4) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, qx∗ , q y∗ ,  −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ + k z · c · q y∗ · A · 2 · q y∗ ± ) 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] + ωβ · A} n ∈ N0 The six fixed point: qx∗ ∈ R; q y∗ ∈ R E (4) (ψ ∗ , qx∗ , q y∗ , qz∗ ) = (n · π, qx∗ , q y∗ ,  −k z · c · (2 + (q y∗ )2 + (qx∗ )2 ) · q y∗ − k z · c · q y∗ · A · 2 · q y∗ ) ± 2 · {2 · q y∗ · [−k z · c + ω − ωβ ] − ωβ · A} n ∈ N0 Stability analysis: The standard local stability analysis about any one of the equilibrium points of Ion channel laser perturbed system consists in adding to its coordinated [ψ qx q y qz ] arbitrarily small increments of exponential terms [ψ, qx , q y , qz ] · eλ·t , and retaining the first order terms in ψ qx q y qz . The system of four homogeneous equations leads to a polynomial characteristic equation in the eigenvalueλ. The polynomial characteristic equation accepts by set the Ion channel laser perturbed system equations. The Ion channel laser perturbed system fixed values with arbitrarily small increments of exponential form [ψ, qx , q y , qz ] · eλ·t are; i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc., ψ(t) = ψ (i) + ψ · eλ·t ; qx (t) = qx(i) + qx · eλ·t q y (t) = q y(i) + q y · eλ·t ; qz (t) = qz(i) + qz · eλ·t dψ(t) dqx (t) = ψ · λ · eλ·t ; = qx · λ · eλ·t dt dt dq y (t) dqz (t) = q y · λ · eλ·t ; = qz · λ · eλ·t dt dt &&&& (2 + qx2 + q y2 ) dψ 1 = k z · c · [1 − ] − ω + ωβ · [1 − dt 4 · qz2 2 · qy qy 1 · A · cos(ψ)] + · k z · c · 2 · A · cos(ψ) 2 qz

248

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

 ψ ·λ·e

λ·t

= kz · c · 1 −

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2

 − ω + ωβ · 1 − +



1 2 · [q y(i) + q y · eλ·t ]

· A · cos(ψ

 (i)

λ·t

+ψ ·e )

[q y(i) + q y · eλ·t ] 1 · k z · c · (i) · A · cos(ψ (i) + ψ · eλ·t ) 2 [qz + qz · eλ·t ]2

We take each part from the above expression and develop it mathematically differently. (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 (2 + (qx(i) )2 + 2 · qx(i) · qx · eλ·t + (qx )2 · e2·λ·t

(1) =

+(q y(i) )2 + 2 · q y(i) · q y · eλ·t + (q y )2 · e2·λ·t ) 4 · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t ]

We consider (qx )2 , (q y )2 , and (qz )2 are very small and tend to zero, then:(qx )2 → ε, (q y )2 → ε, and (qz )2 → ε. (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 )

=

4 · [qz(i) + qz · eλ·t ]2 (2 + (qx(i) )2 + (q y(i) )2 + 2 · qx(i) · qx · eλ·t + 2 · q y(i) · q y · eλ·t )

(i)

4 · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t ]

(i)

(i)

(i)

(2 + (qx )2 + (q y )2 + 2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t ) (i) (i) 4 · [(qz )2 + 2 · qz · qz · eλ·t ] (i) (i) [(qz )2 − 2 · qz · qz · eλ·t ] · (i) 2 (i) [(qz ) − 2 · qz · qz · eλ·t ] (i) (i) (i) (i) (i) (i) [2 + (qx )2 + (q y )2 ] · (qz )2 − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i) (i) (i) +[2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t ] · (qz )2 (i) (i) (i) −[2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t ] · 2 · qz · qz · eλ·t = (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ] (i) 2 (i) 2 (i) 2 (i) (i) (i) [2 + (qx ) + (q y ) ] · (qz ) − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i) (i) (i) (i) (i) (i) +2 · [qx · qx + q y · q y ] · (qz )2 · eλ·t − 2 · [qx · qx + q y · q y ] · 2 · qz · qz · e2·λ·t = (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ] (i) (i) (i) (i) (i) (i) [2 + (qx )2 + (q y )2 ] · (qz )2 − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i)

=

(i)

(i)

(i)

(i)

(i)

+2 · [qx · qx + q y · q y ] · (qz )2 · eλ·t − 2 · [qx · qx · qz + q y · q y · qz ] · 2 · qz · e2·λ·t (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ]

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

249

We consider qx · qz ≈ 0; q y · qz ≈ 0; (qz )2 ≈ 0 then (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 {[2 + (qx(i) )2 + (q y(i) )2 ] + 2 · [qx(i) · qx + q y(i) · q y ] · eλ·t } · (qz(i) )2 =

−[2 + (qx(i) )2 + (q y(i) )2 ] · 2 · qz(i) · qz · eλ·t 4 · (qz(i) )4

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 [2 + (qx(i) )2 + (q y(i) )2 ] 2 · q y(i) 2 · qx(i) λ·t = + · q · e + · q y · eλ·t x 4 · (qz(i) )2 4 · (qz(i) )2 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 − · qz · eλ·t 4 · (qz(i) )3 1

=

2 · [q y(i) + q y · eλ·t ]

=

(2)

[q y(i) + q y · eλ·t ] (3)

[qz(i) + qz · eλ·t ]2

[q y(i) + q y · eλ·t ] [qz(i) + qz · eλ·t ]2

=

1

·

[q y(i) − q y · eλ·t ]

2 · [q y(i) + q y · eλ·t ] [q y(i) − q y · eλ·t ] q y(i) − q y · eλ·t

(q y )2 ≈0

=

q y(i) − q y · eλ·t

2 · [(q y(i) )2 − (q y )2 · e2·λ·t ] 1 1 = − · q y · eλ·t (i) 2 · qy 2 · (q y(i) )2 [q y(i) + q y · eλ·t ]

2 · (q y(i) )2

(qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t [q y(i) + q y · eλ·t ] (qz )2 ≈0 = (qz(i) )2 + 2 · qz(i) · qz · eλ·t (qz )2 ≈0

=

[q y(i) + q y · eλ·t ] (qz(i) )2 + 2 · qz(i) · qz · eλ·t

·

[(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

q y(i) · (qz(i) )2 − 2 · qz(i) · q y(i) · qz · eλ·t = [q y(i) + q y · eλ·t ]

+(qz(i) )2 · q y · eλ·t − 2 · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

(qz )2 ≈0

= [qz(i) + qz · eλ·t ]2 q y(i) · (qz(i) )2 − 2 · qz(i) · q y(i) · qz · eλ·t + (qz(i) )2 · q y · eλ·t − 2 · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

250

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Table 3.2 System’s first differential equation elements by adding variables their arbitrarily small increments of exponential term Expression 1

Final expression (i)

(i) (i) (2+[qx +qx ·eλ·t ]2 +[q y +q y ·eλ·t ]2 ) (i) 4·[qz +qz ·eλ·t ]2

+ 2 3

1 (i) 2·[q y +q y ·eλ·t ]

1 (i) 2·q y

(i)

(i) 4 · (qz )2 (i) 2 · qy · qy (i) 2 4 · (qz )

−

(i)

[q y +q y ·eλ·t ]

qy

[qz +qz ·eλ·t ]2

(qz )2

(i)

(i)

(i) · qx · eλ·t (i) 2 4 · (qz ) (i) (i) [2 + (qx )2 + (q y )2 ] · 2 · eλ·t − (i) 3 4 · (qz )

[2 + (qx )2 + (q y )2 ]

−

(i)

1 (i) 2·(q y )2 (i)

2·q y (i)

(qz )3

+

2 · qx

· qz · eλ·t

· q y · eλ·t · qz · eλ·t +

1 (i) (qz )2

· q y · eλ·t

We assume that qz · q y ≈ 0; (qz )2 ≈ 0 then 2 [q y(i) +q y ·eλ·t ] (qz ) ≈0 q y(i) ·(qz(i) )2 −2·qz(i) ·q y(i) ·qz ·eλ·t +(qz(i) )2 ·q y ·eλ·t = (i) λ·t 2 [qz +qz ·e ] (qz(i) )4 q y(i) 2·q y(i) 1 λ·t = (i) 2 − (i) 3 · qz · e + (i) 2 · q y · eλ·t (qz ) (qz ) (qz )

We can summary all algebraic manipulation in the next table (Table 3.2). cos(ψ (i) + ψ · eλ·t ) = cos(ψ (i) ) · cos(ψ · eλ·t ) − sin(ψ (i) ) · sin(ψ · eλ·t ) We use Taylor series to represent the functions cos(ψ · eλ·t ) and sin(ψ · eλ·t ) as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. The Taylor series are concentrated near to zero since (ψ → 0) · eλ·t which is called a Maclaurin series. The functions are approximated by using a finite number of terms of their Taylor series. It gives quantitative estimates on the error introduced by the use of such an approximations. cos(ψ · eλ·t ) =

∞  (−1)n · ψ (2·n) · e2·n·λ·t (2 · n)! n=0

ψ 4 · e4·λ·t ψ 2 · e2·λ·t + 2! 4! − . . . ∀ ψ · eλ·t

=1−

Since ψ 2 ≈ 0; ψ 4 ≈ 0; . . . . then cos(ψ · eλ·t ) ≈ 1 sin(ψ · eλ·t ) =

∞  n=0

(−1)n · ψ (2·n+1) · e(2·n+1)·λ·t (2 · n + 1)!

ψ 3 · e3·λ·t ψ 5 · e5·λ·t + 3! 5! − . . . ∀ ψ · eλ·t

= ψ · eλ·t −

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

251

Since ψ 3 ≈ 0; ψ 5 ≈ 0; . . . . then sin(ψ · eλ·t ) ≈ ψ · eλ·t cos(ψ (i) + ψ · eλ·t ) ≈ cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t sin(ψ (i) + ψ · eλ·t ) = sin(ψ (i) ) · cos(ψ · eλ·t ) + cos(ψ (i) ) · sin(ψ · eλ·t ) sin(ψ (i) + ψ · eλ·t ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t  ψ ·λ·e

λ·t

= kz · c · 1 −

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2

 − ω + ωβ · 1 − +



1 2 · [q y(i) + q y · eλ·t ]

· A · cos(ψ (i) + ψ · eλ·t )

[q y(i) + q y · eλ·t ] 1 · k z · c · (i) · A · cos(ψ (i) + ψ · eλ·t ) 2 [qz + qz · eλ·t ]2

ψ · λ · eλ·t

= k z · c · [1 − {

[2 + (qx(i) )2 + (q y(i) )2 ]

4 · (qz(i) )2 2 · q y(i) 2 · qx(i) λ·t + · q · e + · q y · eλ·t x 4 · (qz(i) )2 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 − · qz · eλ·t }] − ω (i) 3 4 · (qz ) 1 1 + ωβ · {1 − [ − · q y · eλ·t ] (i) 2 · qy 2 · (q y(i) )2 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ]} q y(i) 2 · q y(i) 1 · k z · c · { (i) − (i) · qz · eλ·t 2 (qz )2 (qz )3 1 + (i) · q y · eλ·t } · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ] (qz )2 +

ψ · λ · eλ·t = k z · c · [1 −

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz · eλ·t 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) λ·t − · q · e − · q y · eλ·t ] x 4 · (qz(i) )2 4 · (qz(i) )2 + kz · c · [



252

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

− ω + ωβ · {1 − ( − +

1 2·

(q y(i) )2

1 2 · q y(i)

· A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ]

· q y · eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ])}

q y(i) 1 · k z · c · { (i) } · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ] 2 (qz )2

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 (i) (i) λ·t · A · [cos(ψ ) − sin(ψ ) · ψ · e ] +

ψ · λ · eλ·t = k z · c · [1 − + kz · c · [ · eλ·t −

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · qx(i)

2 · q y(i)

· q y · eλ·t ] − ω 4 · (qz(i) )2 4 · (qz(i) )2 1 1 + ωβ · {1 − ( · A · cos(ψ (i) ) − · A · sin(ψ (i) ) (i) 2 · qy 2 · q y(i) 1 · ψ · eλ·t − · q y · eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ])} 2 · (q y(i) )2 +

· qx · eλ·t −

· qz

q y(i) 1 1 · k z · c · { (i) } · A · cos(ψ (i) ) − · k z 2 2 (qz )2

·c·{

q y(i) (qz(i) )2

} · A · sin(ψ (i) ) · ψ · eλ·t

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ]

+

ψ · λ · eλ·t = k z · c · [1 − + kz · c · [ −

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2

2 · qx(i) 4 · (qz(i) )2

4·

(qz(i) )3

· qx · eλ·t −

2 · q y(i) 4 · (qz(i) )2

· qz · eλ·t

· q y · eλ·t ] − ω

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

+ ωβ · {1 −

1 2·

· ψ · eλ·t +

q y(i)

· A · cos(ψ (i) ) +

1 2 · q y(i)

· A · sin(ψ (i) )

1

· q y · eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) (i) 2 2 · (q y ) q y(i) 1 1 + · k z · c · { (i) } · A · cos(ψ (i) ) − 2 2 2 (qz ) q y(i) · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·t (qz )2 2 · q y(i) 1 1 + · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 (i) (i) λ·t · A · [cos(ψ ) − sin(ψ ) · ψ · e ]  ψ ·λ·e

λ·t

= kz · c · 1 −

− ω + ωβ · {1 −

1

2 · q y(i)   q y(i)

[2 + (qx(i) )2 + (q y(i) )2 ]



4 · (qz(i) )2 · A · cos(ψ (i) )}

1 · kz · c · · A · cos(ψ (i) ) 2 (qz(i) )2  [2 + (qx(i) )2 + (q y(i) )2 ] · 2 + kz · c · · qz · eλ·t (i) 3 4 · (qz )  (i) 2 · q y(i) 2 · qx λ·t λ·t − · qx · e − · qy · e 4 · (qz(i) )2 4 · (qz(i) )2  1 1 · A · sin(ψ (i) ) · ψ · eλ·t + · qy + ωβ · (i) 2 · qy 2 · (q y(i) )2 +

· eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ]}   q y(i) 1 − · kz · c · · A · sin(ψ (i) ) · ψ · eλ·t 2 (qz(i) )2 2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 (i) (i) λ·t · A · [cos(ψ ) − sin(ψ ) · ψ · e ]

+

At fixed points:

dψ dt

=0

253

· ψ · eλ·t ])}

254

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering



[2 + (qx(i) )2 + (q y(i) )2 ]

kz · c · 1 −  · 1−

4 · (qz(i) )2 1

2 · q y(i)

ψ ·λ·e

λ·t

 − ω + ωβ

 (i)

· A · cos(ψ ) + 

= kz · c · −

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · qx(i) 4 · (qz(i) )2 

+ ωβ ·

q y(i) 1 · k z · c · { (i) } · A · cos(ψ (i) ) = 0 2 (qz )2

2·

· qx · e 1 q y(i)

λ·t

−

2 · q y(i) 4 · (qz(i) )2

· qz · eλ·t  · qy · e

λ·t

· A · sin(ψ (i) ) · ψ · eλ·t +

1 2 · (q y(i) )2

· q y · eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ]}   q y(i) 1 − · kz · c · · A · sin(ψ (i) ) · ψ · eλ·t (i) 2 2 (qz )   2 · q y(i) 1 1 λ·t λ·t + · kz · c · · q y · e − (i) · qz · e 2 (qz(i) )2 (qz )3 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·t ] ψ · λ · eλ·t = k z · c · [

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

2 · qx(i)

4 · (qz(i) )3 2 · q y(i)

· q y · eλ·t ] 4 · (qz(i) )2 4 · (qz(i) )2 1 1 + ωβ · { · A · sin(ψ (i) ) · ψ · eλ·t + (i) 2 · qy 2 · (q y(i) )2 1 · q y · eλ·t · A · cos(ψ (i) ) − · eλ·t (i) 2 2 · (q y ) −

· qx · eλ·t −

· qz · eλ·t

· A · sin(ψ (i) ) · q y · ψ · eλ·t } q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·t 2 (qz )2 1 1 + · k z · c · { (i) · q y · eλ·t · A · cos(ψ (i) ) 2 (qz )2 1 − (i) · eλ·t · A · sin(ψ (i) ) · ψ · q y · eλ·t (qz )2 2 · q y(i) 2 · q y(i) − (i) · qz · eλ·t · A · cos(ψ (i) ) + (i) (qz )3 (qz )3 −

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

· eλ·t · A · sin(ψ (i) ) · qz · ψ · eλ·t } Assumption: q y q · ψ ≈ 0 ; qz · ψ ≈ 0 ψ · λ · eλ·t = k z · c · [

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · qx(i)

2 · q y(i)

· q y · eλ·t ] 4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · { · A · sin(ψ (i) ) · ψ · eλ·t 2 · q y(i) 1 + · q y · eλ·t · A · cos(ψ (i) )} (i) 2 2 · (q y ) −

· qx · eλ·t −

· qz · eλ·t

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·t 2 (qz )2 1 1 + · k z · c · { (i) · q y · eλ·t · A · cos(ψ (i) ) 2 (qz )2 −

−

2 · q y(i) (qz(i) )3

ψ · λ · eλ·t = k z · c · [

· qz · eλ·t · A · cos(ψ (i) )}

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) − · q − · q y ] · eλ·t x 4 · (qz(i) )2 4 · (qz(i) )2 1 1 + ωβ · { · A · sin(ψ (i) ) · ψ + 2 · q y(i) 2 · (q y(i) )2 · q y · A · cos(ψ (i) )} · eλ·t q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·t 2 (qz )2 1 1 + · k z · c · { (i) · q y · A · cos(ψ (i) ) 2 (qz )2 −

−

2 · q y(i) (qz(i) )3

· qz · A · cos(ψ (i) )} · eλ·t

Divide the two sides of the above equation by eλ·t term.  ψ · λ = kz · c ·

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

· qz

255

256

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

2 · qx(i)

−

· qx −

2 · q y(i)

· qy ] 4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · { · A · sin(ψ (i) ) · ψ 2 · q y(i) 1 + · q y · A · cos(ψ (i) )} 2 · (q y(i) )2

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ 2 (qz )2 1 1 + · k z · c · { (i) · q y · A · cos(ψ (i) ) 2 (qz )2 −

−

2 · q y(i) (qz(i) )3

ψ · λ = kz · c ·

· qz · A · cos(ψ (i) )} [2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

− kz · c ·

2 · qx(i)

· qz 2 · q y(i)

· qx − k z · c ·

4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · · A · sin(ψ (i) ) · ψ 2 · q y(i) 1 + ωβ · · q y · A · cos(ψ (i) ) 2 · (q y(i) )2

· qy

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ 2 (qz )2 1 1 + · k z · c · (i) · q y · A · cos(ψ (i) ) 2 (qz )2 2 · q y(i) 1 − · k z · c · (i) · qz · A · cos(ψ (i) ) 2 (qz )3 −

ψ · λ = [ωβ ·

1 2 · q y(i)

· A · sin(ψ (i) ) −

· A · sin(ψ (i) )] · ψ − k z · c ·

q y(i) 1 · k z · c · { (i) } 2 (qz )2

2 · qx(i)

· qx 4 · (qz(i) )2 1 1 + [ωβ · · A · cos(ψ (i) ) + · k z · c (i) 2 2 2 · (q y ) ·

1 (qz(i) )2

· A · cos(ψ (i) ) − k z · c ·

2 · q y(i) 4 · (qz(i) )2

] · qy

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

+ [k z · c · −  ωβ ·

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · q y(i) 1 · k z · c · (i) · A · cos(ψ (i) )] · qz 2 (qz )3 1

· A · sin(ψ (i) ) −

2 · q y(i)

q y(i) 1 · k z · c · { (i) } 2 (qz )2 2 · qx(i)

· A · sin(ψ (i) )] · ψ − ψ · λ − k z · c ·

4 · (qz(i) )2 1 1 + [ωβ · · A · cos(ψ (i) ) + · k z (i) 2 2 2 · (q y ) ·c· 

1 (qz(i) )2

+ kz · c · −

· A · cos(ψ (i) ) − k z · c ·

2 · q y(i) 4 · (qz(i) )2

· qx

] · qy

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · q y(i) 1 · k z · c · (i) · A · cos(ψ (i) )] · qz = 0 2 (qz )3

We define the following global parameters for simplicity. 11 = ωβ ·

1

1 · A · sin(ψ ) − · k z · c · 2

2 · q y(i)

· A · sin(ψ (i) ); 12 = −k z · c · 13 = ωβ · ·c·

1 2·

1 (qz(i) )2

(q y(i) )2

q y(i) (qz(i) )2

2 · qx(i) 4 · (qz(i) )2

· A · cos(ψ (i) ) +

· A · cos(ψ (i) ) − k z · c ·

14 = k z · c · −



(i)

1 · kz 2 2 · q y(i)

4 · (qz(i) )2

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) 2 (qz )3

11 · ψ − ψ · λ + 12 · qx + 13 · q y + 14 · qz = 0 q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2



257

258

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

(q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) 1 qx · λ · eλ·t = − · k z · c · 8 (qz(i) + qz · eλ·t )2 · A · sin(ψ) sin(ψ) = sin(ψ (i) + ψ · eλ·t ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) 1 qx · λ · eλ·t = − · k z · c · 8 (qz(i) + qz · eλ·t )2 · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t )

=

(qz(i) + qz · eλ·t )2 q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t + q y · qx · e2·λ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t

Assumption q y · qx ≈ 0; (qz )2 ≈ 0 then (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t )

=

(qz(i) + qz · eλ·t )2 q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t

q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t

·

[(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

q y(i) · qx(i) · (qz(i) )2 − q y(i) · qx(i) · 2 · qz(i) · qz · eλ·t +(qz(i) )2 · q y(i) · qx · eλ·t − q y(i) · 2 · qz(i) · qz · qx · e2·λ·t =

+(qz(i) )2 · qx(i) · q y · eλ·t − qx(i) · 2 · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

Assumption qz · qx ≈ 0; qz · q y ≈ 0; (qz )2 then (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) (qz(i) + qz · eλ·t )2 q y(i) · qx(i) · (qz(i) )2 − q y(i) · qx(i) · 2 · qz(i) · qz · eλ·t =

+(qz(i) )2 · q y(i) · qx · eλ·t + (qz(i) )2 · qx(i) · q y · eλ·t (qz(i) )4

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

(q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) (qz(i) + qz · eλ·t )2

=

q y(i) · qx(i)

+

(qz(i) )2 q y(i) (qz(i) )2

−

q y(i) · qx(i) · 2 (qz(i) )3

· qx · eλ·t +

· qz · eλ·t

qx(i) (qz(i) )2

· q y · eλ·t

q y(i) · qx(i) q y(i) · qx(i) · 2 1 qx · λ · eλ·t = − · k z · c · [ − 8 (qz(i) )2 (qz(i) )3 (i) qy · qz · eλ·t + (i) · qx · eλ·t (qz )2 (i) qx + (i) · q y · eλ·t ] · A · [sin(ψ (i) ) (qz )2 + cos(ψ (i) ) · ψ · eλ·t ] q y(i) · qx(i) 1 1 qx · λ · eλ·t = [− · k z · c · A · + · kz · c (i) 2 8 8 (qz ) (i) (i) q y(i) q y · qx · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · A· z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t ] · [sin(ψ (i) ) 8 (qz )2 + cos(ψ (i) ) · ψ · eλ·t ] q y(i) · qx(i) 1 1 qx · λ · eλ·t = [− · k z · c · A · + · kz · c (i) 2 8 8 (qz ) (i) (i) q y(i) q y · qx · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · A· z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t ] · sin(ψ (i) ) 2 8 (qz ) q y(i) · qx(i) q y(i) · qx(i) · 2 1 1 · k + · c · A · + [− · k z · c · A · z 8 8 (qz(i) )2 (qz(i) )3 q y(i) 1 · qz · eλ·t − · k z · c · A · (i) · qx · eλ·t 8 (qz )2 (i) 1 qx − · k z · c · A · (i) · q y · eλ·t ] · cos(ψ (i) ) · ψ · eλ·t 8 (qz )2 q y(i) · qx(i) q y(i) · qx(i) · 2 1 1 qx · λ · eλ·t = [− · k z · c · A · + · kz · c · A · (i) 8 8 (qz )2 (qz(i) )3

259

260

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

q y(i) 1 · k z · c · A · (i) · qx · eλ·t 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t ] · sin(ψ (i) ) 2 8 (qz ) q y(i) · qx(i) 1 1 · ψ + · kz · c · A + [− · k z · c · A · (i) 8 8 (qz )2 · qz · eλ·t −

·

q y(i) · qx(i) · 2 (qz(i) )3

−

· qz · ψ · eλ·t −

q y(i) 1 · k z · c · A · (i) · qx · ψ · eλ·t 8 (qz )2

1 qx(i) · k z · c · A · (i) · q y · ψ · eλ·t ] · cos(ψ (i) ) · eλ·t 8 (qz )2

Assumption: qx · ψ ≈ 0; qz · ψ ≈ 0; q y · ψ ≈ 0 then q y(i) · qx(i) 1 1 + · kz · c · A qx · λ · eλ·t = [− · k z · c · A · (i) 2 8 8 (qz ) q y(i) q y(i) · qx(i) · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) · k z · c · A · (i) · q y · eλ·t ] · sin(ψ (i) ) 8 (qz )2 q y(i) · qx(i) 1 · ψ] · cos(ψ (i) ) · eλ·t + [− · k z · c · A · 8 (qz(i) )2

−

q y(i) · qx(i) 1 qx · λ · eλ·t = − · k z · c · A · · sin(ψ (i) ) (i) 2 8 (qz ) (i) (i) q 1 y · qx · 2 · qz · eλ·t · sin(ψ (i) ) + · kz · c · A · (i) 3 8 (qz ) q y(i) 1 − · k z · c · A · (i) · qx · eλ·t · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t · sin(ψ (i) ) 8 (qz )2 q y(i) · qx(i) 1 · ψ] · cos(ψ (i) ) · eλ·t + [− · k z · c · A · 8 (qz(i) )2 At fixed point

dqx dt

= 0; −

qx · λ · eλ·t =

1 8

· kz · c · A ·

q y(i) ·qx(i) (qz(i) )2

· sin(ψ (i) ) = 0 then

q y(i) · qx(i) · 2 1 · kz · c · A · · qz · eλ·t · sin(ψ (i) ) (i) 3 8 (qz )

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

q y(i) 1 · k z · c · A · (i) · qx · eλ·t · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t · sin(ψ (i) ) 2 8 (qz )

−

q y(i) · qx(i) 1 · ψ] · cos(ψ (i) ) · eλ·t + [− · k z · c · A · 8 (qz(i) )2 We divide the two sides of the above expression by eλ·t term. q y(i) · qx(i) · 2 1 · qz · sin(ψ (i) ) · kz · c · A · 8 (qz(i) )3 q y(i) 1 − · k z · c · A · (i) · qx · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · sin(ψ (i) ) 8 (qz )2

qx · λ =

−

q y(i) · qx(i) 1 · kz · c · A · · ψ · cos(ψ (i) ) (i) 2 8 (qz )

q y(i) · qx(i) 1 · kz · c · A · · cos(ψ (i) ) · ψ 8 (qz(i) )2 q y(i) 1 − · k z · c · A · (i) · sin(ψ (i) ) · qx − qx · λ 8 (qz )2 1 qx(i) − · k z · c · A · (i) · sin(ψ (i) ) · q y 8 (qz )2

−

+

q y(i) · qx(i) · 2 1 · kz · c · A · · sin(ψ (i) ) · qz = 0 (i) 3 8 (qz )

We define the following global parameters for simplicity q y(i) · qx(i) 1 · cos(ψ (i) ) 21 = − · k z · c · A · 8 (qz(i) )2 q y(i) 1 22 = − · k z · c · A · (i) · sin(ψ (i) ) 8 (qz )2 1 qx(i) · sin(ψ (i) ) 23 = − · k z · c · A · (i) 2 8 (qz ) (i) (i) q 1 y · qx · 2 · sin(ψ (i) ) 24 = · k z · c · A · 8 (qz(i) )3

261

262

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

21 · ψ + 22 · qx − qx · λ + 23 · q y + 24 · qz = 0 q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz (q y(i) + q y · eλ·t )2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) ) · A · sin(ψ) 2 4 (qz + qz · eλ·t )2 sin(ψ) = sin(ψ (i) + ψ · eλ·t ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t (q y(i) + q y · eλ·t )2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) ) 2 4 (qz + qz · eλ·t )2 · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2 (q y(i) )2 + 2 · q y(i) · q y · eλ·t + (q y )2 · e2·λ·t = (i) |(q y )2 ≈0; (qz )2 ≈0 (qz )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t (q y(i) )2 + 2 · q y(i) · q y · eλ·t ≈ (i) (qz )2 + 2 · qz(i) · qz · eλ·t (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2

≈

[(q y(i) )2 + 2 · q y(i) · q y · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

(q y(i) )2 · (qz(i) )2 − (q y(i) )2 · 2 · qz(i) · qz · eλ·t =

+2 · q y(i) · q y · eλ·t · (qz(i) )2 − 4 · q y(i) · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

Assumption: qz · q y ≈ 0; (qz )2 ≈ 0 then (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2 (q y(i) )2 · (qz(i) )2 − (q y(i) )2 · 2 · qz(i) · qz · eλ·t + 2 · q y(i) · q y · eλ·t · (qz(i) )2 ≈ (qz(i) )4 (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2

≈

(q y(i) )2 (qz(i) )2

−

(q y(i) )2 · 2 (qz(i) )3

· qz · eλ·t +

2 · q y(i) (qz(i) )2

· q y · eλ·t

(q y(i) )2 (q y(i) )2 · 2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · [ (i) − · qz · eλ·t 2 4 (qz )2 (qz(i) )3

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

+

2 · q y(i) (qz(i) )2

263

· q y · eλ·t ]) · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ]

(q y(i) )2 (q y(i) )2 · 2 1 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) − · k z · c · · qz · eλ·t 2 4 4 (qz )2 (qz(i) )3 2 · q y(i) 1 + · k z · c · (i) · q y · eλ·t ) · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] 4 (qz )2 (q y(i) )2 2 · q y(i) 1 1 1 q y · λ · eλ·t = − · ([ωβ + · k z · c · (i) ] + [ · k z · c · (i) · q y · eλ·t 2 4 4 (qz )2 (qz )2 (i) 2 (q y ) · 2 1 − · kz · c · · qz · eλ·t ]) · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] 4 (qz(i) )3   (q y(i) )2 1 1 λ·t q y · λ · e = − · ωβ + · k z · c · (i) · A · [sin(ψ (i) ) 2 2 4 (qz )  2 · q y(i) 1 1 (i) λ·t · k z · c · (i) · q y · eλ·t + cos(ψ ) · ψ · e ] − · 2 4 (qz )2  (q y(i) )2 · 2 1 − · kz · c · · qz · eλ·t · A · [sin(ψ (i) ) (i) 3 4 (qz ) + cos(ψ (i) ) · ψ · eλ·t ] (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · sin(ψ (i) ) 2 4 (qz )2   (q y(i) )2 1 1 − · ωβ + · k z · c · (i) · A · cos(ψ (i) ) 2 4 (qz )2  2 · q y(i) 1 1 λ·t · k z · c · (i) · q y · eλ·t ·ψ ·e − · 2 4 (qz )2  (q y(i) )2 · 2 1 − · kz · c · · qz · eλ·t · A 4 (qz(i) )3 · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] At fixed points

dq y dt

qy · λ · e

λ·t

= 0 ⇒ − 21 · ωβ +

1 4

· kz · c ·

(q y(i) )2 (qz(i) )2



· A · sin(ψ (i) ) = 0 then

  (q y(i) )2 1 1 = − · ωβ + · k z · c · (i) · A · cos(ψ (i) ) 2 4 (qz )2

264

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

 2 · q y(i) 1 1 · k z · c · (i) · q y · eλ·t ·ψ ·e − · 2 4 (qz )2  (q y(i) )2 · 2 1 λ·t − · kz · c · · qz · e · A · [sin(ψ (i) ) (i) 3 4 (qz ) λ·t

+ cos(ψ (i) ) · ψ · eλ·t ] (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ · eλ·t 2 4 (qz )2 2 · q y(i) 1 1 1 − · [A · · k z · c · (i) · q y · eλ·t − A · · k z 2 2 4 4 (qz ) (q y(i) )2 · 2 ·c· · qz · eλ·t ] · sin(ψ (i) ) (i) 3 (qz ) 2 · q y(i) 1 1 1 − · [A · · k z · c · (i) · q y · eλ·t − A · · k z 2 4 4 (qz )2 ·c·

(q y(i) )2 · 2 (qz(i) )3

· qz · eλ·t ] · cos(ψ (i) ) · ψ · eλ·t q y

(q y(i) )2 1 1 · λ · eλ·t = − · [ωβ + · k z · c · (i) ] 2 4 (qz )2 · A · cos(ψ (i) ) · ψ · eλ·t 2 · q y(i) 1 1 1 · [A · · k z · c · (i) · q y − A · · k z 2 2 4 4 (qz ) (i) 2 (q y ) · 2 ·c· · qz ] · sin(ψ (i) ) · eλ·t (qz(i) )3 2 · q y(i) 1 1 1 − · [A · · k z · c · (i) · q y · ψ · eλ·t − A · 2 2 4 4 (qz ) (q y(i) )2 · 2 · kz · c · · qz · ψ · eλ·t ] · cos(ψ (i) ) · eλ·t (i) 3 (qz ) −

Assumption q y · ψ ≈ 0; qz · ψ ≈ 0 then qy · λ · e

λ·t

  (q y(i) )2 1 1 = − · ωβ + · k z · c · (i) 2 4 (qz )2 · A · cos(ψ (i) ) · ψ · eλ·t  2 · q y(i) 1 1 − · A · · k z · c · (i) · q y 2 4 (qz )2

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

 (q y(i) )2 · 2 1 −A · · k z · c · · qz · sin(ψ (i) ) · eλ·t 4 (qz(i) )3 We divide the two sides of the above equation by eλ·t term. (q y(i) )2 1 1 q y · λ = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ 2 4 (qz )2 2 · q y(i) 1 1 1 − · [A · · k z · c · (i) · q y − A · 2 4 4 (qz )2 (i) 2 (q y ) · 2 · kz · c · · qz ] · sin(ψ (i) ) (qz(i) )3 (q y(i) )2 1 1 q y · λ = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) 2 4 (qz )2 2 · q y(i) 1 · ψ − · A · k z · c · (i) · sin(ψ (i) ) · q y 8 (qz )2 (q y(i) )2 · 2 1 + · A · kz · c · · sin(ψ (i) ) · qz 8 (qz(i) )3 (q y(i) )2 1 1 · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) 2 4 (qz )2 2 · q y(i) 1 · ψ − · A · k z · c · (i) · sin(ψ (i) ) · q y − q y · λ 8 (qz )2 (i) 2 (q y ) · 2 1 · sin(ψ (i) ) · qz = 0 + · A · kz · c · 8 (qz(i) )3

−

We define the following global parameters for simplicity 31

  (q y(i) )2 1 1 = − · ωβ + · k z · c · (i) · A · cos(ψ (i) ) 2 4 (qz )2

2 · q y(i) 1 32 = 0; 33 = − · A · k z · c · (i) · sin(ψ (i) 8 (qz )2 34 =

(q y(i) )2 · 2 1 · sin(ψ (i) ) · A · kz · c · 8 (qz(i) )3

31 · ψ + 32 · qx + 33 · q y − q y · λ + 34 · qz = 0

265

266

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

qy 1 dqz = − · kz · c · · A · sin(ψ) dt 2 qz q y(i) + q y · eλ·t 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ) 2 qz + qz · eλ·t sin(ψ) = sin(ψ (i) + ψ · eλ·t ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t q y(i) + q y · eλ·t 1 qz · λ · eλ·t = − · k z · c · [ (i) ]· A 2 qz + qz · eλ·t · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] q y(i) + q y · eλ·t qz(i) + qz · eλ·t

= =

[q y(i) + q y · eλ·t ] [qz(i) − qz · eλ·t ] · [qz(i) + qz · eλ·t ] [qz(i) − qz · eλ·t ] q y(i) · qz(i) − q y(i) · qz · eλ·t + qz(i) · q y · eλ·t − q y · qz · e2·λ·t (qz(i) )2 − (qz )2 · e2·λ·t

Assumption q y · qz ≈ 0; (qz )2 ≈ 0 then q y(i) + q y · eλ·t qz(i) + qz · eλ·t

≈ =

q y(i) · qz(i) − q y(i) · qz · eλ·t + qz(i) · q y · eλ·t (qz(i) )2 q y(i) qz(i)

−

q y(i) (qz(i) )2

· qz · eλ·t +

1 qz(i)

· q y · eλ·t

q y(i) q y(i) 1 qz · λ · eλ·t = − · k z · c · [ (i) − (i) · qz · eλ·t 2 qz (qz )2 1 + (i) · q y · eλ·t ] · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·t ] qz q y(i) q y(i) 1 qz · λ · eλ·t = − · k z · c · [ (i) − (i) · qz · eλ·t 2 qz (qz )2 1 + (i) · q y · eλ·t ] · A · sin(ψ (i) ) qz q y(i) q y(i) 1 − · k z · c · [ (i) − (i) · qz · eλ·t 2 qz (qz )2 1 + (i) · q y · eλ·t ] · A · cos(ψ (i) ) · ψ · eλ·t qz q y(i) 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ (i) ) 2 qz

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

q y(i) 1 · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz q y(i) q y(i) 1 − · k z · c · [ (i) · ψ − (i) · qz · ψ · eλ·t 2 qz (qz )2 1 + (i) · q y · ψ · eλ·t ] · A · cos(ψ (i) ) · eλ·t qz +

Assumption qz · ψ ≈ 0; q y · ψ ≈ 0 then q y(i) 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ (i) ) 2 qz q y(i) 1 + · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz −

At fixed point

dqz dt

q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) · eλ·t · ψ 2 qz

= 0 ⇒ − 21 · k z · c ·

q y(i) qz(i)

· sin(ψ (i) ) = 0 then

q y(i) 1 · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz

qz · λ · eλ·t =

−

q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) · eλ·t · ψ 2 qz

We divide the two sides of the above equation by eλ·t term. q y(i) 1 · k z · c · (i) · A · sin(ψ (i) ) · qz 2 (qz )2 1 1 − · k z · c · (i) · A · sin(ψ (i) ) · q y 2 qz

qz · λ =

−

q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) · ψ 2 qz

267

268

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) · ψ 2 qz 1 1 − · k z · c · (i) · A · sin(ψ (i) ) · q y 2 qz q y(i) 1 + · k z · c · (i) · A · sin(ψ (i) ) · qz − qz · λ = 0 2 (qz )2 −

We define the following global parameters for simplicity q y(i) 1 41 = − · k z · c · (i) · A · cos(ψ (i) ) 2 qz 1 42 = 0; 43 = − · k z · c · A · sin(ψ (i) ) (i) 2 · qz q y(i) 1 · k z · c · (i) · A · sin(ψ (i) ) 2 (qz )2 41 · ψ + 42 · qx + 43 · q y + 44 · qz − λ · qz = 0 44 =

We can summary our Ion channel laser perturbed system arbitrarily small increments equations: 11 · ψ − ψ · λ + 12 · qx + 13 · q y + 14 · qz = 0 21 · ψ + 22 · qx − qx · λ + 23 · q y + 24 · qz = 0 31 · ψ + 32 · qx + 33 · q y − q y · λ + 34 · qz = 0 41 · ψ + 42 · qx + 43 · q y + 44 · qz − λ · qz = 0 ⎞ ⎞ ⎛ 1 ... 0 11 . . . 14 ⎟ ⎜ ⎟ ⎜ I = ⎝ ... . . . ... ⎠; A = ⎝ ... . . . ... ⎠ 41 · · · 44 0 ··· 1 ⎞ ⎛ ⎛ 1 ... 11 . . . 14 ⎜ .. . . .. ⎟ ⎜ .. . . A−λ· I =⎝ . . . ⎠−⎝. . ⎛

⎛

11 . . . ⎜ .. . . {⎝ . . 41 · · ·

⎞ ⎛ 14 1 .. ⎟ − ⎜ .. . ⎠ ⎝.

44

0

⎞ 0 .. ⎟ · λ .⎠

41 · · · 44 0 ··· 1 ⎛ ⎞ ⎞ ψ ... 0 ⎜q ⎟ ⎜ x⎟ . . .. ⎟ ⎟ = 0 ; det(A − λ · I ) = 0 . . ⎠ · λ} · ⎜ ⎝ qy ⎠ ··· 1 qz

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

⎛

⎞ ⎛ ⎞ 14 1 ... 0 .. ⎟ − ⎜ .. . . .. ⎟ · λ} = 0;  = 0;  = 0 32 42 . ⎠ ⎝. . .⎠ 41 · · · 44 0 ··· 1 ⎞ ⎛ ⎞ ⎛ 1 ... 0 11 . . . 14 ⎜ .. . . .. ⎟ ⎜ .. . . .. ⎟ ⎝ . . . ⎠−⎝. . .⎠·λ

11 . . . ⎜ .. . . det{⎝ . .

0 ··· 1 41 · · · 44 ⎛ ⎞ (11 − λ) 12 13 14 ⎜ 21 (22 − λ) 23 24 ⎟ ⎟ =⎜ ⎝ 31 0 (33 − λ) 34 ⎠ 41 0 43 (44 − λ) ⎛ ⎞ (11 − λ) 12 13 14 ⎜ 21 (22 − λ) 23 24 ⎟ ⎟ det(A − λ · I ) = det⎜ ⎝ 31 0 (33 − λ) 34 ⎠ 41 0 43 (44 − λ) =0 ⎛

⎞ 23 24 (22 − λ) det( A − λ · I ) = (11 − λ) · det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) ⎛ ⎞ 23 24 21 − 12 · det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ) ⎛ ⎞ 24 21 (22 − λ) + 13 · det ⎝ 31 0 34 ⎠ 41 0 (44 − λ) ⎛ ⎞ 23 21 (22 − λ) − 14 · det⎝ 31 0 (33 − λ) ⎠ 41

0

43

No.1 ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ)   34 (33 − λ) = (22 − λ) · det 43 (44 − λ) = (22 − λ) · {(33 − λ) · (44 − λ) − 43 · 34 }

269

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

⎛

⎞ 23 24 (22 − λ) det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = (22 − λ) · {(33 − λ) · (44 − λ) − 43 · 34 } = (22 − λ) · {λ2 − (33 + 44 ) · λ + (33 · 44 − 43 · 34 )} ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = (22 − λ) · {λ2 − (33 + 44 ) · λ + (33 · 44 − 43 · 34 )} = λ2 · 22 − (33 + 44 ) · 22 · λ + (33 · 44 − 43 · 34 ) · 22 − λ3 + (33 + 44 ) · λ2 − (33 · 44 − 43 · 34 ) · λ ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = −λ3 + [22 + (33 + 44 )] · λ2 − [(33 + 44 ) · 22 + (33 · 44 − 43 · 34 )] · λ + (33 · 44 − 43 · 34 ) · 22 We define for simplicity the following system global parameters: ϒ1 = 22 + (33 + 44 ); ϒ2 = −[(33 + 44 ) · 22 + (33 · 44 − 43 · 34 )] ϒ3 = (33 · 44 − 43 · 34 ) · 22 ⎛

⎞ 23 24 (22 − λ) det ⎝ 0 (33 − λ) 34 ⎠ = −λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 0 43 (44 − λ) No.2 ⎞ 23 24 21 det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ)     34 34 (33 − λ) 31 = 21 · det − 23 · det 43 (44 − λ) 41 (44 − λ) ⎛

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

 31 (33 − λ) + 24 · det 43 41 ⎞ ⎛ 23 24 21 det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ) 

= 21 · [(33 − λ) · (44 − λ) − 43 · 34 ] − 23 · [31 · (44 − λ) − 41 · 34 ] + 24 · [31 · 43 − 41 · (33 − λ)] ⎛

⎞ 21 23 24 det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ) = 21 · [33 · 44 − 33 · λ − 44 · λ + λ2 − 43 · 34 ] − 23 · [31 · 44 − 31 · λ − 41 · 34 ] + 24 · [31 · 43 − 41 · 33 + 41 · λ] ⎛

⎞ 21 23 24 det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ) = 21 · 33 · 44 − 21 · 33 · λ − 21 · 44 · λ + 21 · λ2 − 21 · 43 · 34 − 23 · (31 · 44 − 41 · 34 ) + 23 · 31 · λ + 24 · (31 · 43 − 41 · 33 ) + 24 · 41 · λ ⎛

⎞ 21 23 24 det ⎝ 31 (33 − λ) 34 ⎠ 41 43 (44 − λ) = 21 · λ2 + [23 · 31 + 24 · 41 − 21 · 33 − 21 · 44 ] · λ + 21 · 33 · 44 − 21 · 43 · 34 − 23 · (31 · 44 − 41 · 34 ) + 24 · (31 · 43 − 41 · 33 ) ϒ4 = 21 ; ϒ5 = 23 · 31 + 24 · 41 − 21 · 33 − 21 · 44 ϒ6 = 21 · 33 · 44 − 21 · 43 · 34 − 23 · (31 · 44 − 41 · 34 ) + 24 · (31 · 43 − 41 · 33 )

271

272

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

⎛

⎞ 23 24 21 det ⎝ 31 (33 − λ) 34 ⎠ = ϒ4 · λ2 + ϒ5 · λ + ϒ6 41 43 (44 − λ) No.3 ⎞ 24 21 (22 − λ) det ⎝ 31 0 34 ⎠ 41 0 (44 − λ)     34 0 34 31 = 21 · det − (22 − λ) · det 0 (44 − λ) 41 (44 − λ)   31 0 + 24 · det 41 0 ⎛

= −(22 − λ) · [31 · (44 − λ) − 41 · 34 ] ⎛

⎞ 21 (22 − λ) 24 det ⎝ 31 0 34 ⎠ 41 0 (44 − λ)     34 0 34 31 = 21 · det − (22 − λ) · det 0 (44 − λ) 41 (44 − λ)   31 0 + 24 · det 41 0 = (−22 + λ) · [(31 · 44 − 41 · 34 ) − 31 · λ] ⎛

⎞ 21 (22 − λ) 24 det ⎝ 31 0 34 ⎠ 41 0 (44 − λ)     34 0 34 31 = 21 · det − (22 − λ) · det 0 (44 − λ) 41 (44 − λ)   31 0 = −31 · λ2 + [22 · 31 + 24 · det 41 0 + (31 · 44 − 41 · 34 )] · λ − 22 · (31 · 44 − 41 · 34 ) ϒ7 = −31 ; ϒ8 = 22 · 31 + (31 · 44 − 41 · 34 ) ϒ9 = −22 · (31 · 44 − 41 · 34 )

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

273

⎛

⎞ 24 21 (22 − λ) det ⎝ 31 0 34 ⎠ = ϒ7 · λ2 + ϒ8 · λ + ϒ9 41 0 (44 − λ) No.4 ⎞ 23 21 (22 − λ) det ⎝ 31 0 (33 − λ) ⎠ 0 43 41     0 (33 − λ) 31 (33 − λ) − (22 − λ) · det = 21 · det 43 0 43 41   31 0 + 23 · det 41 0 ⎞ ⎛ 23 21 (22 − λ) det ⎝ 31 0 (33 − λ) ⎠ = −(22 − λ) · [31 · 43 − 41 · (33 − λ)] 0 43 41 ⎞ ⎛ 23 21 (22 − λ) det ⎝ 31 0 (33 − λ) ⎠ 0 43 41 ⎛

= (−22 + λ) · [(31 · 43 − 41 · 33 ) + 41 · λ)] ⎛

⎞ 21 (22 − λ) 23 det ⎝ 31 0 (33 − λ) ⎠ 0 43 41 = 41 · λ2 + [(31 · 43 − 41 · 33 ) − 22 · 41 ] · λ − 22 · (31 · 43 − 41 · 33 ) ϒ10 = 41 ; ϒ11 = (31 · 43 − 41 · 33 ) − 22 · 41 ϒ12 = −22 · (31 · 43 − 41 · 33 ) ⎛

⎞ 21 (22 − λ) 23 det ⎝ 31 0 (33 − λ) ⎠ = λ2 · ϒ10 + λ · ϒ11 + ϒ12 0 43 41 We can summary our last results in the Table 3.3.3. det(A − λ · I ) = (11 − λ) · (−λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 ) − 12 · (ϒ4 · λ2 + ϒ5 · λ + ϒ6 )

274

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Table 3.3 System’s Jacobian matrix determinant elements No 1

2

3

Determinant ⎛ ⎞ 23 24 ( − λ) ⎜ 22 ⎟ det ⎜ 0 (33 − λ) 34 ⎟ ⎝ ⎠ 0 43 (44 − λ) ⎞ ⎛ 23 24  ⎟ ⎜ 21 det ⎜ 34 ⎟ ⎠ ⎝ 31 (33 − λ) 0 43 (44 − λ) ⎞ ⎛ 24 21 (22 − λ) ⎟ ⎜ det ⎜ 0 34 ⎟ ⎠ ⎝ 31 0

4

⎛

0 0

−λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3

ϒ4 · λ2 + ϒ5 · λ + ϒ6

ϒ7 · λ2 + ϒ8 · λ + ϒ9

(44 − λ)

0

21 (22 − λ)

⎜ det ⎜ ⎝ 31 0

Equivalent expression

23

⎞

⎟ (33 − λ) ⎟ ⎠ 43

λ2 · ϒ10 + λ · ϒ11 + ϒ12

+ 13 · (ϒ7 · λ2 + ϒ8 · λ + ϒ9 ) − 14 · (λ2 · ϒ10 + λ · ϒ11 + ϒ12 ) det(A − λ · I ) = −λ3 · 11 + ϒ1 · 11 · λ2 + ϒ2 · 11 · λ + ϒ3 · 11 + λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ − 12 · ϒ4 · λ2 − 12 · ϒ5 · λ − 12 · ϒ6 + 13 · ϒ7 · λ2 + 13 · ϒ8 · λ + 13 · ϒ9 − λ2 · 14 · ϒ10 − λ · 14 · ϒ11 − 14 · ϒ12 ) det(A − λ · I ) = λ4 − (11 + ϒ1 ) · λ3 + (ϒ1 · 11 − ϒ2 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ) · λ2 + (ϒ2 · 11 − ϒ3 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ) · λ + (ϒ3 · 11 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 ) We define new global parameters: 4 = 1; 3 = −(11 + ϒ1 ) 2 = ϒ1 · 11 − ϒ2 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 1 = ϒ2 · 11 − ϒ3 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11

3.1 Ion Channel Laser Perturbed Differential Equations of Motion Stability Analysis

275

0 = ϒ3 · 11 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 det( A − λ · I ) =

4 

k · λk ; det (A − λ · I ) = 0 ⇒

k=0

4 

k · λk = 0

k=0

Eigenvalues stability discussion: Our Ion channel laser perturbed system involving N  variables (N > 2, N = 4), the characteristic equation is of degree N = 4 ( 4k=0 k ·λk = 0) and must often be solve numerically. Except in some particular cases, such an equation has (N = 4) distinct roots that can be real or complex. These values are the eigenvalues of the 4 × 4 Jacobian matrix (A). The general rule is that the Steady State (SS) is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 4-variables (ψ qx q y qz ) Ion channel laser perturbed system has four eigenvalues. The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the four eigenvalues are real and negative (stable steady state), (2) the four eigenvalues are real, three of them are negative (unstable steady state), (3) and (4) two eigenvalues are complex conjugates with a negative real part and the other eigenvalues are real and negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) two eigenvalues are complex conjugates with a negative real part and at least one eigenvalue is positive (unstable steady state).

3.2 Ion Channel Laser Perturbed Differential Equations of Motion Stability Optimization Under Delayed Variables in Time We get already (3.1) the Ion channel laser perturbed system equations of motion (average over betatron period): (2 + qx2 + q y2 ) dψ = k z · c · [1 − ] − ω + ωβ dt 4 · qz2 1 1 · [1 − · A · cos(ψ)] + · k z 2 · qy 2 qy · c · 2 · A · cos(ψ) qz qy 1 dqz = − · kz · c · · A · sin(ψ) dt 2 qz

276

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz q y · qx dqx 1 · A · sin(ψ) = − · kz · c · dt 8 qz2 The mechanism is which a laser beam propagating through un-dense plasma produces a positively charged ion channels by expelling plasma electrons in the transverse direction. Basically the relativistic electron beam is injected and propagates in the axial (+z). The beam head is propagated through the plasma and expels plasma electrons from the beam volume. The result is focusing for the remainder of the beam. Self- focusing of intense laser pulses in plasmas can also be achieved by the expulsion of plasma electrons (cavitation) caused by the extreme ponderomotive force of a focused laser pulse. It is optical guiding of high intensity laser pulses in under dense plasmas and the creation of a plasma channel that guides subsequently injected laser pulses. The radial electrostatic force from charge separation expels the plasma ions from the electron cavitation regions. Due to their inertia, the ions continue to drift radially outward even after the passage of the laser pulse and a plasma channel is formed. The formation of a plasma channel can be most easily attributed to the effects of ponderomotive forces associated with the intense pump laser pulse as it propagates through the plasma. Initially, the pump laser pulses exert a ponderomotive force on the plasma electrons and expel them radially. There are four variables in our system ψ(phase variable in time) and qx ,q y ,qz are system variables in x, y, and z directions respectively. We assume that electrons with q y < A(A— Eikonal amplitude) contribute negligibly to the amplification, and that aβ < 1. Additionally, ω  ωβ , or ω − k z · Vz ∼ ωβ . The fast wave, with aβ  1, this corresponds to ω ∼ 2 · γ 2 · ωβ . The mechanism which a laser beam propagating through un-dense plasma is not ideal and there is phase variable (ψ) delay in time, ) is not effected [1]. ψ(t) → ψ(t − τ ) but the derivative of the phase in time ( dψ(t) dt We can define qx =qx (t); q y =q y (t); qz = qz (t) (2 + qx2 + q y2 ) dψ(t) = k z · c · [1 − ] − ω + ωβ dt 4 · qz2 1 · [1 − · A · cos(ψ(t − τ ))] 2 · qy qy 1 + · k z · c · 2 · A · cos(ψ(t − τ )) 2 qz qy 1 dqz (t) = − · kz · c · · A · sin(ψ(t − τ )) dt 2 qz q y2 dq y (t) 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ(t − τ )) dt 2 4 qz

3.2 Ion Channel Laser Perturbed Differential …

277

q y · qx 1 dqx (t) = − · kz · c · · A · sin(ψ(t − τ )) dt 8 qz2 At fixed points lim ψ(t − τ ) = ψ(t); t  τ ⇒ t − τ ≈ t ∀ τ ∈ R+ t→∞ then our system fixed points are the same as we find (3.1). The Ion channel laser perturbed system fixed values with arbitrarily small increments of exponential form [ψ, qx , q y , qz ] · eλ·t are; i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc., ψ(t) = ψ (i) + ψ · eλ·t ψ(t − τ ) = ψ (i) + ψ · eλ·(t−τ ) =ψ (i) + ψ · eλ·t · e−λ·τ qx (t) = qx(i) + qx · eλ·t q y (t) = q y(i) + q y · eλ·t ; qz (t) = qz(i) + qz · eλ·t dψ(t) dqx (t) = ψ · λ · eλ·t ; = qx · λ · eλ·t dt dt dq y (t) dqz (t) = q y · λ · eλ·t ; = qz · λ · eλ·t dt dt &&&& (2 + qx2 + q y2 ) dψ = k z · c · [1 − ] dt 4 · qz2 1 · A · cos(ψ(t − τ ))] − ω + ωβ · [1 − 2 · qy qy 1 + · k z · c · 2 · A · cos(ψ(t − τ )) 2 qz ψ · λ · eλ·t = k z · c · [1 − + ωβ · {1 − +

1 · kz · c 2

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 1

2 · [q y(i) + q y · eλ·t ] [q y(i) + q y · eλ·t ] · (i) · [qz + qz · eλ·t ]2

]−ω

· A · cos(ψ (i) + ψ · eλ·t · e−λ·τ )} A · cos(ψ (i) + ψ · eλ·t · e−λ·τ )

We take each part from the above expression and develop it mathematically differently.

278

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 (2 + (qx(i) )2 + 2 · qx(i) · qx · eλ·t + (qx )2 · e2·λ·t

(1) =

+(q y(i) )2 + 2 · q y(i) · q y · eλ·t + (q y )2 · e2·λ·t ) 4 · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t ]

We consider (qx )2 , (q y )2 , and (qz )2 are very small and tend to zero, then:(qx )2 → ε, (q y )2 → ε, and (qz )2 → ε. (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 )

=

(2 +

(i)

4 · [qz(i) + qz · eλ·t ]2 + (q y(i) )2 + 2 · qx(i) · qx · eλ·t + 2 · q y(i) · q y · eλ·t )

(qx(i) )2

4 · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t ]

(i)

(i)

(i)

(2 + (qx )2 + (q y )2 + 2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t )

(i) (i) 4 · [(qz )2 + 2 · qz · qz · eλ·t ] (i) (i) [(qz )2 − 2 · qz · qz · eλ·t ] · (i) (i) [(qz )2 − 2 · qz · qz · eλ·t ] (i) 2 (i) 2 (i) (i) (i) (i) [2 + (qx ) + (q y ) ] · (qz )2 − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i) (i) (i) +[2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t ] · (qz )2 (i) (i) (i) −[2 · qx · qx · eλ·t + 2 · q y · q y · eλ·t ] · 2 · qz · qz · eλ·t = (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ] (i) 2 (i) 2 (i) 2 (i) (i) (i) [2 + (qx ) + (q y ) ] · (qz ) − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i) (i) (i) (i) (i) (i) +2 · [qx · qx + q y · q y ] · (qz )2 · eλ·t − 2 · [qx · qx + q y · q y ] · 2 · qz · qz · e2·λ·t = (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ] (i) (i) (i) (i) (i) (i) [2 + (qx )2 + (q y )2 ] · (qz )2 − [2 + (qx )2 + (q y )2 ] · 2 · qz · qz · eλ·t (i)

=

(i)

(i)

(i)

(i)

(i)

+2 · [qx · qx + q y · q y ] · (qz )2 · eλ·t − 2 · [qx · qx · qz + q y · q y · qz ] · 2 · qz · e2·λ·t (i) (i) 4 · [(qz )4 − 4 · (qz )2 · (qz )2 · e2·λ·t ]

We consider qx · qz ≈ 0; q y · qz ≈ 0; (qz )2 ≈ 0 then (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 {[2 + (qx(i) )2 + (q y(i) )2 ] + 2 · [qx(i) · qx + q y(i) · q y ] · eλ·t } · (qz(i) )2 =

−[2 + (qx(i) )2 + (q y(i) )2 ] · 2 · qz(i) · qz · eλ·t 4 · (qz(i) )4 (2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2

3.2 Ion Channel Laser Perturbed Differential …

[2 + (qx(i) )2 + (q y(i) )2 ]

279

2 · qx(i)

· qx · eλ·t 4 · (qz(i) )2 4 · (qz(i) )2 2 · q y(i) [2 + (qx(i) )2 + (q y(i) )2 ] · 2 λ·t + · q · e − · qz · eλ·t y 4 · (qz(i) )2 4 · (qz(i) )3

=

1

=

2 · [q y(i) + q y · eλ·t ]

=

(2)

1

·

[q y(i) − q y · eλ·t ]

2 · [q y(i) + q y · eλ·t ] [q y(i) − q y · eλ·t ] q y(i) − q y · eλ·t 2 · [(q y(i) )2 − (q y )2 · e2·λ·t ]

(q y )2 ≈0

=

(3)

+

q y(i) − q y · eλ·t

1

=

−

2 · (q y(i) )2 2 · q y(i) (i) λ·t (i) [q y + q y · eλ·t ] [q y + q y · e ] = [qz(i) + qz · eλ·t ]2 (qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 [q y(i) + q y · eλ·t ] (qz )2 ≈0 = (qz(i) )2 + 2 · qz(i) · qz · eλ·t [q y(i) + q y · eλ·t ]

[q y(i) + q y · eλ·t ]

(qz )2 ≈0

=

[qz(i) + qz · eλ·t ]2

(qz(i) )2 + 2 · qz(i) · qz · eλ·t

·

1 2 · (q y(i) )2

· q y · eλ·t

· e2·λ·t

[(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

q y(i) · (qz(i) )2 − 2 · qz(i) · q y(i) · qz · eλ·t =

+(qz(i) )2 · q y · eλ·t − 2 · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

[q y(i) + q y · eλ·t ]

(qz )2 ≈0

[qz(i) + qz · eλ·t ]2 q y(i) · (qz(i) )2 − 2 ·

qz(i) · q y(i) · qz · eλ·t + (qz(i) )2 · q y · eλ·t − 2 · qz(i) · qz · q y · e2·λ·t

=

(qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t We assume that qz · q y ≈ 0; (qz )2 ≈ 0 then [q y(i) + q y · eλ·t ] [qz(i) + qz · eλ·t ]2

(qz )2 ≈0

=

=

q y(i) · (qz(i) )2 − 2 · qz(i) · q y(i) · qz · eλ·t + (qz(i) )2 · q y · eλ·t (qz(i) )4

q y(i) (qz(i) )2

−

2 · q y(i) (qz(i) )3

· qz · eλ·t +

1 (qz(i) )2

· q y · eλ·t

cos(ψ (i) + ψ · eλ·(t−τ ) ) = cos(ψ (i) ) · cos(ψ · eλ·(t−τ ) ) − sin(ψ (i) ) · sin(ψ · eλ·(t−τ ) )

280

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

cos(ψ (i) + ψ · eλ·t · e−λ·τ ) = cos(ψ (i) ) · cos(ψ · eλ·t · e−λ·τ ) − sin(ψ (i) ) · sin(ψ · eλ·t · e−λ·τ ) We use Taylor series to represent the functions cos(ψ ·eλ·(t−τ ) ) and sin(ψ ·eλ·(t−τ ) ) as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. The Taylor series are concentrated near to zero since (ψ → 0)·eλ·(t−τ ) which is called a Maclaurin series. The functions are approximated by using a finite number of terms of their Taylor series. It gives quantitative estimates on the error introduced by the use of such an approximations. cos(ψ · e

λ·(t−τ )

∞  (−1)n )= · ψ (2·n) · e2·n·λ·(t−τ ) (2 · n)! n=0

ψ 2 · e2·λ·(t−τ ) 2! 4 4·λ·(t−τ ) ψ ·e + − . . . ∀ ψ · eλ·(t−τ ) 4!

=1−

Since ψ 2 ≈ 0; ψ 4 ≈ 0 ; . . . . then cos(ψ · eλ·(t−τ ) ) ≈ 1 sin(ψ · eλ·(t−τ ) ) =

∞  n=0

(−1)n · ψ (2·n+1) · e(2·n+1)·λ·(t−τ ) (2 · n + 1)!

= ψ · eλ·(t−τ ) − +

ψ 3 · e3·λ·(t−τ ) 3!

ψ 5 · e5·λ·(t−τ ) − . . . . ∀ ψ · eλ·(t−τ ) 5!

Since ψ 3 ≈ 0; ψ 5 ≈ 0 ; . . . . then sin(ψ · eλ·(t−τ ) ) ≈ ψ · eλ·(t−τ ) cos(ψ (i) + ψ · eλ·t ) ≈ cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) sin(ψ (i) + ψ · eλ·(t−τ ) ) = sin(ψ (i) ) · cos(ψ · eλ·(t−τ ) ) + cos(ψ (i) ) · sin(ψ · eλ·(t−τ ) ) sin(ψ (i) + ψ · eλ·(t−τ ) ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) ψ · λ · eλ·t = k z · c · [1 − + ωβ · {1 − +

1 · kz · c 2

(2 + [qx(i) + qx · eλ·t ]2 + [q y(i) + q y · eλ·t ]2 ) 4 · [qz(i) + qz · eλ·t ]2 1

2 · [q y(i) + q y · eλ·t ] [q y(i) + q y · eλ·t ] · (i) · [qz + qz · eλ·t ]2

]−ω

· A · cos(ψ (i) + ψ · eλ·(t−τ ) )} A · cos(ψ (i) + ψ · eλ·(t−τ ) )

3.2 Ion Channel Laser Perturbed Differential …

ψ · λ · eλ·t = k z · c · [1 − {

281

[2 + (qx(i) )2 + (q y(i) )2 ]

4 · (qz(i) )2 2 · q y(i) 2 · qx(i) λ·t + · q · e + · q y · eλ·t x 4 · (qz(i) )2 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 − · qz · eλ·t }] − ω 4 · (qz(i) )3 1 1 + ωβ · {1 − [ − · q y · eλ·t ] (i) 2 · qy 2 · (q y(i) )2 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ]} q y(i) 2 · q y(i) 1 · k z · c · { (i) − (i) · qz · eλ·t 2 (qz )2 (qz )3 1 + (i) · q y · eλ·t } · A · [cos(ψ (i) ) (qz )2 − sin(ψ (i) ) · ψ · eλ·(t−τ ) ] +

ψ · λ · eλ·t = k z · c · [1 −

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz · eλ·t 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) λ·t − · q · e − · q y · eλ·t ] − ω x 4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · {1 − ( · A · [cos(ψ (i) ) − sin(ψ (i) ) 2 · q y(i) 1 · ψ · eλ·(t−τ ) ] − · q y · eλ·t · A · [cos(ψ (i) ) 2 · (q y(i) )2 + kz · c · [

− sin(ψ (i) ) · ψ · eλ·(t−τ ) ])} +

q y(i) 1 · k z · c · { (i) } · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ] 2 (qz )2

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ] +

ψ · λ · eλ·t = k z · c · [1 −

[2 + (qx(i) )2 + (q y(i) )2 ] 4 · (qz(i) )2

]

282

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz · eλ·t 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) λ·t − · q · e − · q y · eλ·t ] − ω x 4 · (qz(i) )2 4 · (qz(i) )2 1 1 + ωβ · {1 − ( · A · cos(ψ (i) ) − (i) 2 · qy 2 · q y(i) 1 · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) − · q y · eλ·t 2 · (q y(i) )2 + kz · c · [

· A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ])} +

q y(i) 1 1 · k z · c · { (i) } · A · cos(ψ (i) ) − · k z 2 2 (qz )2

·c·{

q y(i) (qz(i) )2

} · A · sin(ψ (i) ) · ψ · eλ·(t−τ )

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ]

+

ψ · λ · eλ·t = k z · c · [1 −

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz · eλ·t 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) λ·t − · q · e − · q y · eλ·t ] − ω x (i) 2 (i) 2 4 · (qz ) 4 · (qz ) 1 1 + ωβ · {1 − · A · cos(ψ (i) ) + (i) 2 · qy 2 · q y(i) 1 · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) + · q y · eλ·t 2 · (q y(i) )2 + kz · c · [

· A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ])} +

q y(i) 1 1 · k z · c · { (i) } · A · cos(ψ (i) ) − · k z 2 2 2 (qz )

·c·{

q y(i) (qz(i) )2

} · A · sin(ψ (i) ) · ψ · eλ·(t−τ )

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 (i) (i) λ·(t−τ ) · A · [cos(ψ ) − sin(ψ ) · ψ · e ]

+

3.2 Ion Channel Laser Perturbed Differential …

ψ · λ · eλ·t = k z · c · [1 −

283

[2 + (qx(i) )2 + (q y(i) )2 ]

] 4 · (qz(i) )2 1 − ω + ωβ · {1 − · A · cos(ψ (i) )} 2 · q y(i) +

q y(i) 1 · k z · c · { (i) } · A · cos(ψ (i) ) 2 (qz )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz · eλ·t 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) λ·t − · q · e − · q y · eλ·t ] x 4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · { · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) 2 · q y(i) 1 + · q y · eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ]} 2 · (q y(i) )2 + kz · c · [

−

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) 2 (qz )2

2 · q y(i) 1 1 · k z · c · { (i) · q y · eλ·t − (i) · qz · eλ·t } 2 (qz )2 (qz )3 (i) (i) λ·(t−τ ) ] · A · [cos(ψ ) − sin(ψ ) · ψ · e

+

At fixed points:

dψ dt



[2 + (qx(i) )2 + (q y(i) )2 ]

kz · c · 1 −  ·1 −

=0

4 · (qz(i) )2

1 2 · q y(i)

 − ω + ωβ



1 · A · cos(ψ ) + · k z · c · 2 (i)



q y(i)



(qz(i) )2

· A · cos(ψ (i) ) = 0  ψ ·λ·e

λ·t

= kz · c · −

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · qx(i) 4 · (qz(i) )2 

+ ωβ ·

2·

· qx · e 1 q y(i)

λ·t

−

2 · q y(i) 4 · (qz(i) )2

· qz · eλ·t  · qy · e

λ·t

· A · sin(ψ (i) ) · ψ · eλ·(t−τ ) +

1 2 · (q y(i) )2

· qy

284

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

· eλ·t · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ]}   q y(i) 1 − · kz · c · · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) (i) 2 2 (qz )   2 · q y(i) 1 1 λ·t λ·t + · kz · c · · q y · e − (i) · qz · e 2 (qz(i) )2 (qz )3 · A · [cos(ψ (i) ) − sin(ψ (i) ) · ψ · eλ·(t−τ ) ]  ψ ·λ·e

λ·t

= kz · c · ·e

λ·t

−

+ ωβ · + −

[2 + (qx(i) )2 + (q y(i) )2 ] · 2 4 · (qz(i) )3

2 · qx(i)

1

2 · (q y(i) )2

− +

1

· qy · e

 ·e

1 + · kz · c · 2 (qz(i) )2 2 · q y(i) (qz(i) )3 2 · q y(i) (qz(i) )3

4 · (qz(i) )2

λ·t

· q y · eλ·t · A · cos(ψ (i) )

2 · (q y(i) )2 1

−



2 · q y(i)

· A · sin(ψ (i) ) · ψ · eλ·(t−τ )

2 · q y(i)

1 − · kz · c · 2

−

· qx · e

4 · (qz(i) )2  1

λ·t

· qz

λ·t

 

(i)

· A · sin(ψ ) · q y · ψ · e

λ·(t−τ )



q y(i) (qz(i) )2 1 (qz(i) )2

· A · sin(ψ (i) ) · ψ · eλ·(t−τ ) · q y · eλ·t · A · cos(ψ (i) )

· eλ·t · A · sin(ψ (i) ) · ψ · q y · eλ·(t−τ ) · qz · eλ·t · A · cos(ψ (i) ) · eλ·t · A · sin(ψ (i) ) · qz · ψ · eλ·(t−τ ) }

Assumption: q y · ψ ≈ 0; qz · ψ ≈ 0 ψ · λ · eλ·t = k z · c · [ −

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

2 · qx(i) 4 · (qz(i) )2

4 · (qz(i) )3 · qx · eλ·t −

2 · q y(i) 4 · (qz(i) )2

· qz · eλ·t · q y · eλ·t ]

3.2 Ion Channel Laser Perturbed Differential …

+ ωβ · { +

1 2 · q y(i)

1 2·

(q y(i) )2

285

· A · sin(ψ (i) ) · ψ · eλ·(t−τ )

· q y · eλ·t · A · cos(ψ (i) )}

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·(t−τ ) 2 (qz )2 1 1 + · k z · c · { (i) · q y · eλ·t · A · cos(ψ (i) ) 2 (qz )2 −

− ψ · λ · eλ·t

2 · q y(i) (qz(i) )3

= kz · c · [

· qz · eλ·t · A · cos(ψ (i) )}

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

· qz 4 · (qz(i) )3 2 · q y(i) 2 · qx(i) − · q − · q y ] · eλ·t x 4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · { · A · sin(ψ (i) ) · ψ · e−λ·τ 2 · q y(i) 1 + · q y · A · cos(ψ (i) )} · eλ·t (i) 2 2 · (q y ) q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · eλ·t · e−λ·τ 2 (qz )2 1 1 + · k z · c · { (i) · q y · A · cos(ψ (i) ) 2 (qz )2 −

−

2 · q y(i) (qz(i) )3

· qz · A · cos(ψ (i) )} · eλ·t

Divide the two sides of the above equation by eλ·t term.  ψ · λ = kz · c · −

4 · (qz(i) )3

2 · qx(i) 4 · (qz(i) )2 

+ ωβ · +

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

2·

1 2 · (q y(i) )2

· qx − 1 q y(i)

· qz



2 · q y(i) 4 · (qz(i) )2

· qy

· A · sin(ψ (i) ) · ψ · e−λ·τ  (i)

· q y · A · cos(ψ )

286

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

1 − · kz · c · 2 1 + · kz · c · 2 −

2 · q y(i) (qz(i) )3

ψ · λ = kz · c ·

 



q y(i) (qz(i) )2 1 (qz(i) )2

· A · sin(ψ (i) ) · ψ · e−λ·τ · q y · A · cos(ψ (i) )  (i)

· qz · A · cos(ψ )

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

− kz · c ·

4 · (qz(i) )3 2 · qx(i)

· qx − k z · c ·

· qz 2 · q y(i)

4 · (qz(i) )2 4 · (qz(i) )2 1 + ωβ · · A · sin(ψ (i) ) · ψ · e−λ·τ 2 · q y(i) 1 + ωβ · · q y · A · cos(ψ (i) ) (i) 2 2 · (q y )

· qy

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) · ψ · e−λ·τ 2 (qz )2 1 1 + · k z · c · (i) · q y · A · cos(ψ (i) ) 2 (qz )2 2 · q y(i) 1 − · k z · c · (i) · qz · A · cos(ψ (i) ) 2 (qz )3 −

ψ · λ = [ωβ ·

1 2 · q y(i)

· A · sin(ψ (i) ) −

q y(i) 1 · k z · c · { (i) } 2 (qz )2

· A · sin(ψ (i) )] · e−λ·τ · ψ − k z · c ·

2 · qx(i)

· qx 4 · (qz(i) )2 2 · q y(i) 1 (i) + [ωβ · · A · cos(ψ ) − k · c · z 2 · (q y(i) )2 4 · (qz(i) )2 1 1 + · k z · c · (i) · A · cos(ψ (i) )] · q y 2 (qz )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 + [k z · c · 4 · (qz(i) )3 2 · q y(i) 1 − · k z · c · (i) · A · cos(ψ (i) )] · qz 2 (qz )3 [ωβ ·

1 2 · q y(i)

· A · sin(ψ (i) ) −

q y(i) 1 · k z · c · { (i) } · A 2 (qz )2

3.2 Ion Channel Laser Perturbed Differential …

287

· sin(ψ (i) )] · e−λ·τ · ψ − λ · ψ − k z · c ·

2 · qx(i)

· qx 4 · (qz(i) )2 2 · q y(i) 1 (i) + [ωβ · · A · cos(ψ ) − k · c · z 2 · (q y(i) )2 4 · (qz(i) )2 1 1 + · k z · c · (i) · A · cos(ψ (i) )] · q y 2 (qz )2 [2 + (qx(i) )2 + (q y(i) )2 ] · 2 1 + [k z · c · − · kz · c 2 4 · (qz(i) )3 (i) 2 · qy · (i) · A · cos(ψ (i) )] · qz = 0 (qz )3 We define the following global parameters for simplicity. 11 = ωβ · −

1 2 · q y(i)

· A · sin(ψ (i) )

q y(i) 1 · k z · c · { (i) } · A · sin(ψ (i) ) 2 (qz )2

12 = −k z · c · 13 = ωβ ·

2 · qx(i) 4 · [(qz(i) )2

1 2 · (q y(i) )2

· A · cos(ψ (i) )

1 1 · k z · c · (i) · A · cos(ψ (i) ) 2 (qz )2 2 · q y(i) − kz · c · 4 · [(qz(i) )2 +

14 = k z · c ·

[2 + (qx(i) )2 + (q y(i) )2 ] · 2

4 · (qz(i) )3 2 · q y(i) 1 − · k z · c · (i) · A · cos(ψ (i) ) 2 (qz )3

11 · e−λ·τ · ψ − ψ · λ + 12 · qx + 13 · q y + 14 · qz = 0 q y · qx dqx 1 = − · kz · c · · A · sin(ψ(t − τ )) dt 8 qz2

288

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

(q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) 1 qx · λ · eλ·t = − · k z · c · 8 (qz(i) + qz · eλ·t )2 · A · sin(ψ(t − τ )) sin(ψ(t − τ )) = sin(ψ (i) + ψ · eλ·(t−τ ) ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) 1 qx · λ · eλ·t = − · k z · c · 8 (qz(i) + qz · eλ·t )2 · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t )

=

(qz(i) + qz · eλ·t )2 q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t + q y · qx · e2·λ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t

Assumption q y · qx ≈ 0; (qz )2 ≈ 0 then (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) (qz(i) + qz · eλ·t )2

=

q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t

q y(i) · qx(i) + q y(i) · qx · eλ·t + qx(i) · q y · eλ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t

·

[(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

q y(i) · qx(i) · (qz(i) )2 − q y(i) · qx(i) · 2 · qz(i) · qz · eλ·t +(qz(i) )2 · q y(i) · qx · eλ·t − q y(i) · 2 · qz(i) · qz · qx · e2·λ·t =

+(qz(i) )2 · qx(i) · q y · eλ·t − qx(i) · 2 · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

Assumption qz · qx ≈ 0; qz · q y ≈ 0; (qz )2 then (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) (qz(i) + qz · eλ·t )2

=

q y(i) · qx(i) · (qz(i) )2 − q y(i) · qx(i) · 2 · qz(i) · qz · eλ·t +(qz(i) )2 · q y(i) · qx · eλ·t + (qz(i) )2 · qx(i) · q y · eλ·t (qz(i) )4 (q y(i) + q y · eλ·t ) · (qx(i) + qx · eλ·t ) (qz(i)

+ qz ·

eλ·t )2

=

q y(i) · qx(i) (qz(i) )2

−

q y(i) · qx(i) · 2 (qz(i) )3

· qz · eλ·t

3.2 Ion Channel Laser Perturbed Differential …

289

+

q y(i) (qz(i) )2

· qx · eλ·t +

qx(i) (qz(i) )2

· q y · eλ·t

q y(i) · qx(i) 1 qx · λ · eλ·t = − · k z · c · [ 8 (qz(i) )2 −

q y(i) · qx(i) · 2 (qz(i) )3

· qz · eλ·t +

q y(i) (qz(i) )2

· qx · eλ·t

qx(i)

· q y · eλ·t ] · A · [sin(ψ (i) ) (qz(i) )2 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ]

+

q y(i) · qx(i) 1 1 qx · λ · eλ·t = [− · k z · c · A · + · kz · c (i) 2 8 8 (qz ) (i) (i) q y(i) q y · qx · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · A· z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t ] · [sin(ψ (i) ) 8 (qz )2 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] q y(i) · qx(i) 1 1 qx · λ · eλ·t = [− · k z · c · A · + · kz · c · A (i) 8 8 (qz )2 (i) (i) q y(i) q y · qx · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) · k z · c · A · (i) · q y · eλ·t ] · sin(ψ (i) ) 2 8 (qz ) q y(i) · qx(i) 1 1 + · kz · c · A + [− · k z · c · A · (i) 2 8 8 (qz ) q y(i) q y(i) · qx(i) · 2 1 λ·t · k · q · e − · c · A · · qx · eλ·t · z z 8 (qz(i) )3 (qz(i) )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t ] · cos(ψ (i) ) · ψ · eλ·(t−τ ) 8 (qz )2 −



qx · λ · e

λ·t

q y(i) · qx(i) 1 1 = − · kz · c · A · + · kz · c · A (i) 2 8 8 (qz ) ·

q y(i) · qx(i) · 2 (qz(i) )3

· qz · eλ·t −

q y(i) 1 · k z · c · A · (i) · qx · eλ·t 8 (qz )2

290

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

 1 qx(i) λ·t − · k z · c · A · (i) · q y · e · sin(ψ (i) ) 8 (qz )2  q y(i) · qx(i) 1 1 + − · kz · c · A · · ψ + · kz · c · A (i) 2 8 8 (qz ) ·

q y(i) · qx(i) · 2 (qz(i) )3

· qz · ψ · eλ·t −

q y(i) 1 · k z · c · A · (i) · qx · ψ · eλ·t 8 (qz )2 

1 qx(i) − · k z · c · A · (i) · q y · ψ · eλ·t · cos(ψ (i) ) · eλ·(t−τ ) 8 (qz )2 Assumption: qx · ψ ≈ 0; qz · ψ ≈ 0; q y · ψ ≈ 0 then 

qx · λ · e

λ·t

q y(i) · qx(i) 1 1 = − · kz · c · A · + · kz (i) 8 8 (qz )2 ·c· A·

q y(i) · qx(i) · 2 (qz(i) )3

· qz · eλ·t

q y(i) 1 · k z · c · A · (i) · qx · eλ·t 8 (qz )2  1 qx(i) λ·t − · k z · c · A · (i) · q y · e · sin(ψ (i) ) 2 8 (qz )   q y(i) · qx(i) 1 + − · kz · c · A · · ψ · cos(ψ (i) ) · eλ·(t−τ ) 8 (qz(i) )2 −

q y(i) · qx(i) 1 qx · λ · eλ·t = − · k z · c · A · · sin(ψ (i) ) 8 (qz(i) )2 q y(i) · qx(i) · 2 1 · qz · eλ·t · sin(ψ (i) ) + · kz · c · A · 8 (qz(i) )3 q y(i) 1 − · k z · c · A · (i) · qx · eλ·t · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t · sin(ψ (i) ) 2 8 (qz ) q y(i) · qx(i) 1 · ψ] · cos(ψ (i) ) · eλ·(t−τ ) + [− · k z · c · A · 8 (qz(i) )2 At fixed point

dqx dt

= 0; −

1 8

· kz · c · A ·

q y(i) ·qx(i) (qz(i) )2

· sin(ψ (i) ) = 0 then

3.2 Ion Channel Laser Perturbed Differential …

q y(i) · qx(i) · 2 1 · kz · c · A · · qz · eλ·t · sin(ψ (i) ) 8 (qz(i) )3 q y(i) 1 − · k z · c · A · (i) · qx · eλ·t · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · eλ·t · sin(ψ (i) ) 8 (qz )2

qx · λ · eλ·t =

q y(i) · qx(i) 1 · ψ] · cos(ψ (i) ) · eλ·(t−τ ) + [− · k z · c · A · 8 (qz(i) )2 We divide the two sides of the above expression by eλ·t term. q y(i) · qx(i) · 2 1 · kz · c · A · · qz · sin(ψ (i) ) 8 (qz(i) )3 q y(i) 1 − · k z · c · A · (i) · qx · sin(ψ (i) ) 8 (qz )2 1 qx(i) − · k z · c · A · (i) · q y · sin(ψ (i) ) 2 8 (qz )

qx · λ =

−

q y(i) · qx(i) 1 · kz · c · A · · ψ · cos(ψ (i) ) · e−λ·τ 8 (qz(i) )2

q y(i) · qx(i) 1 · kz · c · A · · cos(ψ (i) ) · e−λ·τ · ψ 8 (qz(i) )2 q y(i) 1 − · k z · c · A · (i) · sin(ψ (i) ) · qx − qx · λ 8 (qz )2 1 qx(i) − · k z · c · A · (i) · sin(ψ (i) ) · q y 2 8 (qz )

−

+

q y(i) · qx(i) · 2 1 · kz · c · A · · sin(ψ (i) ) · qz = 0 (i) 3 8 (qz )

We define the following global parameters for simplicity q y(i) · qx(i) 1 21 = − · k z · c · A · · cos(ψ (i) ) 8 (qz(i) )2 q y(i) 1 22 = − · k z · c · A · (i) · sin(ψ (i) ) 8 (qz )2 1 qx(i) · sin(ψ (i) ) 23 = − · k z · c · A · (i) 8 (qz )2

291

292

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

24 =

q y(i) · qx(i) · 2 1 · kz · c · A · · sin(ψ (i) ) 8 (qz(i) )3

21 · ψ · e−λ·τ + 22 · qx − qx · λ + 23 · q y + 24 · qz = 0 q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ(t − τ )) dt 2 4 qz (q y(i) + q y · eλ·t )2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) ) · A · sin(ψ(t − τ )) 2 4 (qz + qz · eλ·t )2 sin(ψ(t − τ )) = sin(ψ (i) + ψ · eλ·(t−τ ) ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) (q y(i) + q y · eλ·t )2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) ) 2 4 (qz + qz · eλ·t )2 · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2

=

|(q y )2 ≈0;(qz )2 ≈0 ≈ (q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2

≈

(q y(i) )2 + 2 · q y(i) · q y · eλ·t + (q y )2 · e2·λ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t + (qz )2 · e2·λ·t (q y(i) )2 + 2 · q y(i) · q y · eλ·t (qz(i) )2 + 2 · qz(i) · qz · eλ·t

[(q y(i) )2 + 2 · q y(i) · q y · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ] · [(qz(i) )2 + 2 · qz(i) · qz · eλ·t ] [(qz(i) )2 − 2 · qz(i) · qz · eλ·t ]

(q y(i) )2 · (qz(i) )2 − (q y(i) )2 · 2 · qz(i) · qz · eλ·t =

+2 · q y(i) · q y · eλ·t · (qz(i) )2 − 4 · q y(i) · qz(i) · qz · q y · e2·λ·t (qz(i) )4 − 4 · (qz(i) )2 · (qz )2 · e2·λ·t

Assumption: qz · q y ≈ 0; (qz )2 ≈ 0 then (q y(i) + q y · eλ·t )2

≈ (qz(i) + qz · eλ·t )2 (q y(i) )2 · (qz(i) )2 − (q y(i) )2 · 2 · qz(i) · qz · eλ·t + 2 · q y(i) · q y · eλ·t · (qz(i) )2 (qz(i) )4

(q y(i) + q y · eλ·t )2 (qz(i) + qz · eλ·t )2

≈

(q y(i) )2 (qz(i) )2

−

(q y(i) )2 · 2 (qz(i) )3

· qz · eλ·t +

2 · q y(i) (qz(i) )2

· q y · eλ·t

(q y(i) )2 (q y(i) )2 · 2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · [ (i) − · qz · eλ·t 2 4 (qz )2 (qz(i) )3

3.2 Ion Channel Laser Perturbed Differential …

+

2 · q y(i) (qz(i) )2

293

· q y · eλ·t ]) · A · [sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) ]

(q y(i) )2 1 1 q y · λ · eλ·t = − · (ωβ + · k z · c · (i) 2 4 (qz )2 (q y(i) )2 · 2 1 − · kz · c · · qz · eλ·t 4 (qz(i) )3 2 · q y(i) 1 + · k z · c · (i) · q y · eλ·t ) · A · [sin(ψ (i) ) 4 (qz )2 (i) + cos(ψ ) · ψ · eλ·(t−τ ) ] (q y(i) )2 2 · q y(i) 1 1 1 q y · λ · eλ·t = − · ([ωβ + · k z · c · (i) ] + [ · k z · c · (i) · q y · eλ·t 2 4 4 (qz )2 (qz )2 (i) 2 (q y ) · 2 1 − · kz · c · · qz · eλ·t ]) · A · [sin(ψ (i) ) 4 (qz(i) )3 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · [sin(ψ (i) ) 2 4 (qz )2 2 · q y(i) 1 1 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] − · [ · k z · c · (i) · q y · eλ·t 2 4 (qz )2 (q y(i) )2 · 2 1 − · kz · c · · qz · eλ·t ] · A · [sin(ψ (i) ) 4 (qz(i) )3 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · sin(ψ (i) ) 2 4 (qz )2 (q y(i) )2 1 1 − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ 2 4 (qz )2 2 · q y(i) 1 1 · eλ·(t−τ ) − · [ · k z · c · (i) · q y · eλ·t 2 4 (qz )2 (i) 2 (q y ) · 2 1 − · kz · c · · qz · eλ·t ] · A · [sin(ψ (i) ) 4 (qz(i) )3 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ] At fixed points

dq y dt

= 0 ⇒ − 21 · [ωβ +

1 4

· kz · c ·

(q y(i) )2 ] (qz(i) )2

· A · sin(ψ (i) ) = 0 then

294

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

(q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A 2 4 (qz )2 · cos(ψ (i) ) · ψ · eλ·(t−τ ) −

2 · q y(i) 1 1 · [ · k z · c · (i) · q y · eλ·t 2 4 (qz )2

(q y(i) )2 · 2 1 · kz · c · · qz · eλ·t ] · A · [sin(ψ (i) ) 4 (qz(i) )3 + cos(ψ (i) ) · ψ · eλ·(t−τ ) ]

−

(q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ · eλ·(t−τ ) 2 4 (qz )2 2 · q y(i) 1 1 − · [A · · k z · c · (i) · q y · eλ·t 2 4 (qz )2 (i) 2 (q y ) · 2 1 − A · · kz · c · · qz · eλ·t ] · sin(ψ (i) ) 4 (qz(i) )3 2 · q y(i) 1 1 − · [A · · k z · c · (i) · q y · eλ·t 2 4 (qz )2 (q y(i) )2 · 2 1 − A · · kz · c · · qz · eλ·t ] · cos(ψ (i) ) · ψ · eλ·(t−τ ) 4 (qz(i) )3 (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ · eλ·(t−τ ) 2 4 (qz )2 2 · q y(i) 1 1 − · [A · · k z · c · (i) · q y 2 4 (qz )2 (i) 2 (q y ) · 2 1 − A · · kz · c · · qz ] · sin(ψ (i) ) · eλ·t 4 (qz(i) )3 2 · q y(i) 1 1 − · [A · · k z · c · (i) · q y · ψ · eλ·t 2 4 (qz )2 (i) 2 (q y ) · 2 1 − A · · kz · c · · qz · ψ · eλ·t ] · cos(ψ (i) ) · eλ·(t−τ ) 4 (qz(i) )3 Assumption q y · ψ ≈ 0; qz · ψ ≈ 0 then (q y(i) )2 1 1 q y · λ · eλ·t = − · [ωβ + · k z · c · (i) ] 2 4 (qz )2 · A · cos(ψ (i) ) · ψ · eλ·(t−τ )

3.2 Ion Channel Laser Perturbed Differential …

2 · q y(i) 1 1 · [A · · k z · c · (i) · q y 2 4 (qz )2 (q y(i) )2 · 2 1 − A · · kz · c · · qz ] · sin(ψ (i) ) · eλ·t 4 (qz(i) )3 −

We divide the two sides of the above equation by eλ·t term. (q y(i) )2 1 1 q y · λ = − · [ωβ + · k z · c · (i) ] 2 4 (qz )2 · A · cos(ψ (i) ) · ψ · e−λ·τ 2 · q y(i) 1 1 · [A · · k z · c · (i) · q y 2 4 (qz )2 (i) 2 (q y ) · 2 1 − A · · kz · c · · qz ] · sin(ψ (i) ) (i) 3 4 (qz ) −

(q y(i) )2 1 1 q y · λ = − · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ · e−λ·τ 2 4 (qz )2 (i) 2 · qy 1 − · A · k z · c · (i) · sin(ψ (i) ) · q y 8 (qz )2 (q y(i) )2 · 2 1 + · A · kz · c · · sin(ψ (i) ) · qz 8 (qz(i) )3 (q y(i) )2 1 1 · [ωβ + · k z · c · (i) ] · A · cos(ψ (i) ) · ψ · e−λ·τ 2 4 (qz )2 2 · q y(i) 1 − · A · k z · c · (i) · sin(ψ (i) ) · q y − q y · λ 8 (qz )2 (q y(i) )2 · 2 1 · sin(ψ (i) ) · qz = 0 + · A · kz · c · 8 (qz(i) )3

−

We define the following global parameters for simplicity 31

  (q y(i) )2 1 1 = − · ωβ + · k z · c · (i) · A · cos(ψ (i) ) 2 4 (qz )2

2 · q y(i) 1 32 = 0; 33 = − · A · k z · c · (i) · sin(ψ (i) ) 8 (qz )2 34 =

(q y(i) )2 · 2 1 · A · kz · c · · sin(ψ (i) ) (i) 3 8 (qz )

295

296

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

31 · ψ · e−λ·τ + 32 · qx + 33 · q y − q y · λ + 34 · qz = 0 qy dqz 1 · A · sin(ψ(t − τ )) = − · kz · c · dt 2 qz q y(i) + q y · eλ·t 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ(t − τ )) 2 qz + qz · eλ·t sin(ψ(t − τ )) = sin(ψ (i) + ψ · eλ·(t−τ ) ) ≈ sin(ψ (i) ) + cos(ψ (i) ) · ψ · eλ·(t−τ ) qz · λ · e

λ·t

  q y(i) + q y · eλ·t 1 = − · kz · c · · A · [sin(ψ (i) ) 2 qz(i) + qz · eλ·t + cos(ψ (i) ) · ψ · eλ·(t−τ ) ]

q y(i) + q y · eλ·t qz(i) + qz · eλ·t [q y(i) + q y · eλ·t ] [qz(i) − qz · eλ·t ] = (i) · [qz + qz · eλ·t ] [qz(i) − qz · eλ·t ] q y(i) · qz(i) − q y(i) · qz · eλ·t + qz(i) · q y · eλ·t − q y · qz · e2·λ·t = (qz(i) )2 − (qz )2 · e2·λ·t Assumption q y · qz ≈ 0; (qz )2 ≈ 0 then q y(i) + q y · eλ·t qz(i) + qz · eλ·t

≈ =

qz · λ · e

λ·t

q y(i) · qz(i) − q y(i) · qz · eλ·t + qz(i) · q y · eλ·t (qz(i) )2 q y(i) qz(i)

−

q y(i) (qz(i) )2

· qz · eλ·t +

1 qz(i)

· q y · eλ·t

 q y(i) q y(i) 1 = − · kz · c · − · qz · eλ·t 2 qz(i) (qz(i) )2  1 + (i) · q y · eλ·t · A · [sin(ψ (i) ) qz + cos(ψ (i) ) · ψ · eλ·(t−τ ) ]

qz · λ · e

λ·t

 q y(i) q y(i) 1 = − · kz · c · − · qz · eλ·t 2 qz(i) (qz(i) )2

3.2 Ion Channel Laser Perturbed Differential …

+

1 qz(i)

297

 · qy · e

λ·t

· A · sin(ψ (i) )

 q y(i) q y(i) 1 − · kz · c · − · qz · eλ·t (i) (i) 2 2 qz (qz )  1 + (i) · q y · eλ·t · A · cos(ψ (i) ) · ψ · eλ·(t−τ ) qz q y(i) 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ (i) ) 2 qz q y(i) 1 + · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz q y(i) q y(i) 1 · k z · c · [ (i) · ψ − (i) · qz · ψ · eλ·t 2 qz (qz )2 1 + (i) · q y · ψ · eλ·t ] · A · cos(ψ (i) ) · eλ·(t−τ ) qz −

Assumption qz · ψ ≈ 0; q y · ψ ≈ 0 then q y(i) 1 qz · λ · eλ·t = − · k z · c · (i) · A · sin(ψ (i) ) 2 qz q y(i) 1 + · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz −

At fixed point

dqz dt

q y(i) 1 · k z · c · [ (i) · ψ] · A · cos(ψ (i) ) · eλt · e−λ·τ 2 qz

= 0 ⇒ − 21 · k z · c ·

q y(i) qz(i)

· A · sin(ψ (i) ) = 0 then

q y(i) 1 · k z · c · (i) · qz · eλ·t · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · eλ·t · A · sin(ψ (i) ) 2 qz

qz · λ · eλ·t =

−

q y(i) 1 · k z · c · [ (i) · ψ] · A · cos(ψ (i) ) · eλt · e−λ·τ 2 qz

298

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

We divide the two sides of the above equation by eλ·t term. q y(i) 1 · k z · c · (i) · qz · A · sin(ψ (i) ) 2 (qz )2 1 1 − · k z · c · (i) · q y · A · sin(ψ (i) ) 2 qz

qz · λ =

−

q y(i) 1 · k z · c · [ (i) · ψ] · A · cos(ψ (i) ) · e−λ·τ 2 qz

q y(i) 1 · k z · c · (i) · A · cos(ψ (i) ) · e−λ·τ · ψ 2 qz 1 1 − · k z · c · (i) · A · sin(ψ (i) ) · q y 2 qz

−

+

q y(i) 1 · k z · c · (i) · A · sin(ψ (i) ) · qz − qz · λ = 0 2 (qz )2

We define the following global parameters for simplicity q y(i) 1 41 = − · k z · c · (i) · A · cos(ψ (i) ) 2 qz 1 42 = 0; 43 = − · k z · c · A · sin(ψ (i) ) (i) 2 · qz q y(i) 1 · k z · c · (i) · A · sin(ψ (i) ) 2 (qz )2 −λ·τ 41 · e · ψ + 42 · qx + 43 · q y + 44 · qz − λ · qz = 0

44 =

We can summary our Ion channel laser perturbed system arbitrarily small increments equations: 11 · e−λ·τ · ψ − ψ · λ + 12 · qx + 13 · q y + 14 · qz = 0 21 · ψ · e−λ·τ + 22 · qx − qx · λ + 23 · q y + 24 · qz = 0 31 · ψ · e−λ·τ + 32 · qx + 33 · q y − q y · λ + 34 · qz = 0 41 · e−λ·τ · ψ + 42 · qx + 43 · q y + 44 · qz − λ · qz = 0 The small increments Jacobian of Ion channel laser perturbed system is as follow:

3.2 Ion Channel Laser Perturbed Differential …

299

⎞ 12 13 14 (11 · e−λ·τ − λ) ⎜ 21 · e−λ·τ (22 − λ) 23 24 ⎟ ⎟ (A − λ · I ) = ⎜ ⎝ 31 · e−λ·τ 0 (33 − λ) 34 ⎠ 41 · e−λ·τ 0 43 (44 − λ) ⎛

det(A − λ · I ) = 0 ; D = D(λ, τ ) ⎛

⎞ 23 24 (22 − λ) det(A − λ · I ) = (11 · e−λ·τ − λ) · det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) ⎛ ⎞ 23 24 21 · e−λ·τ ⎝ − 12 · det 31 · e−λ·τ (33 − λ) 34 ⎠ −λ·τ 41 · e 43 (44 − λ) ⎛ ⎞ −λ·τ 21 · e (22 − λ) 24 + 13 · det ⎝ 31 · e−λ·τ 0 34 ⎠ ⎛

41 · e−λ·τ

0

(44 − λ)

⎞ (22 − λ) 23 21 · e − 14 · det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ 0 43 41 · e−λ·τ −λ·τ

No.1 ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ)   34 (33 − λ) = (22 − λ) · det 43 (44 − λ) = (22 − λ) · {(33 − λ) · (44 − λ) − 43 · 34 } ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = (22 − λ) · {(33 − λ) · (44 − λ) − 43 · 34 } = (22 − λ) · {λ2 − (33 + 44 ) · λ + (33 · 44 − 43 · 34 )} ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = (22 − λ) · {λ2 − (33 + 44 ) · λ + (33 · 44 − 43 · 34 )}

300

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

= λ2 · 22 − (33 + 44 ) · 22 · λ + (33 · 44 − 43 · 34 ) · 22 − λ3 + (33 + 44 ) · λ2 − (33 · 44 − 43 · 34 ) · λ ⎛

⎞ (22 − λ) 23 24 det ⎝ 0 (33 − λ) 34 ⎠ 0 43 (44 − λ) = −λ3 + [22 + (33 + 44 )] · λ2 − [(33 + 44 ) · 22 + (33 · 44 − 43 · 34 )] · λ + (33 · 44 − 43 · 34 ) · 22 We define for simplicity the following system global parameters: ϒ1 = 22 + (33 + 44 ) ϒ2 = −[(33 + 44 ) · 22 + (33 · 44 − 43 · 34 )] ϒ3 = (33 · 44 − 43 · 34 ) · 22 ⎛

⎞ 23 24 (22 − λ) det ⎝ 0 (33 − λ) 34 ⎠ = −λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 0 43 (44 − λ) No.2 ⎛

⎞ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ 41 · e−λ·τ 43 (44 − λ)   (33 − λ) 34 = 21 · e−λ·τ · det 43 (44 − λ)   −λ·τ 34 31 · e − 23 · det 41 · e−λ·τ (44 − λ)   31 · e−λ·τ (33 − λ) + 24 · det 43 41 · e−λ·τ ⎞ ⎛ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ 41 · e−λ·τ

= 21 · e

−λ·τ

43

(44 − λ)

· [(33 − λ) · (44 − λ) − 43 · 34 ]

− 23 · [31 · e−λ·τ · (44 − λ) − 41 · e−λ·τ · 34 ] + 24 · [31 · e−λ·τ · 43 − 41 · e−λ·τ · (33 − λ)]

3.2 Ion Channel Laser Perturbed Differential …

⎛

⎞ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ −λ·τ 41 · e 43 (44 − λ) = 21 · e−λ·τ · [33 · 44 − 43 · 34 − λ · (33 + 44 ) + λ2 ] − 23 · [(31 · 44 − 41 · 34 ) − 31 · λ] · e−λ·τ + 24 · [(31 · 43 − 41 · 33 ) + 41 · λ] · e−λ·τ ⎛

⎞ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ −λ·τ 41 · e 43 (44 − λ) = {21 · [33 · 44 − 43 · 34 − λ · (33 + 44 ) + λ2 ] − 23 · [(31 · 44 − 41 · 34 ) − 31 · λ] + 24 · [(31 · 43 − 41 · 33 ) + 41 · λ]} · e−λ·τ ⎛

⎞ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ −λ·τ 41 · e 43 (44 − λ) = {21 · (33 · 44 − 43 · 34 ) − λ · 21 · (33 + 44 ) + 21 · λ2 − 23 · (31 · 44 − 41 · 34 ) + 23 · 31 · λ + 24 · (31 · 43 − 41 · 33 ) + 24 · 41 · λ} · e−λ·τ ⎛

⎞ 23 24 21 · e−λ·τ det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ 41 · e−λ·τ 43 (44 − λ) = {21 · λ2 + [23 · 31 + 24 · 41 − 21 · (33 + 44 )] · λ + 21 · (33 · 44 − 43 · 34 ) − 23 · (31 · 44 − 41 · 34 ) + 24 · (31 · 43 − 41 · 33 )} · e−λ·τ ϒ4 = 21 ; ϒ5 = 23 · 31 + 24 · 41 − 21 · (33 + 44 ) ϒ6 = 21 · (33 · 44 − 43 · 34 ) − 23 · (31 · 44 − 41 · 34 ) + 24 · (31 · 43 − 41 · 33 ) ⎛

⎞ 21 · e−λ·τ 23 24 det ⎝ 31 · e−λ·τ (33 − λ) 34 ⎠ = {ϒ4 · λ2 + ϒ5 · λ + ϒ6 } · e−λ·τ 41 · e−λ·τ 43 (44 − λ) No.3

301

302

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

⎛

⎞ 24 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 34 ⎠ −λ·τ 41 · e 0 (44 − λ)   0 34 = 21 · e−λ·τ · det 0 (44 − λ)   31 · e−λ·τ 34 − (22 − λ) · det 41 · e−λ·τ (44 − λ)   31 · e−λ·τ 0 + 24 · det 41 · e−λ·τ 0 ⎞ ⎛ 24 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 34 ⎠ = −(22 − λ) · [31 · e−λ·τ 41 · e−λ·τ 0 (44 − λ) · (44 − λ) − 41 · e−λ·τ · 34 ] ⎛

⎞ 21 · e−λ·τ (22 − λ) 24 det ⎝ 31 · e−λ·τ 0 34 ⎠ = −(22 − λ) · [31 · 44 41 · e−λ·τ 0 (44 − λ) − 41 · 34 − 31 · λ] · e−λ·τ ⎞ 24 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 34 ⎠ −λ·τ 41 · e 0 (44 − λ) ⎛

= {−22 · (31 · 44 − 41 · 34 ) + 22 · 31 · λ + (31 · 44 − 41 · 34 ) · λ − 31 · λ2 } · e−λ·τ ⎛

⎞ 24 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 34 ⎠ −λ·τ 41 · e 0 (44 − λ) = {−31 · λ2 + [22 · 31 + (31 · 44 − 41 · 34 )] · λ − 22 · (31 · 44 − 41 · 34 )} · e−λ·τ ϒ7 = −31 ; ϒ8 = 22 · 31 + (31 · 44 − 41 · 34 ) ϒ9 = −22 · (31 · 44 − 41 · 34 ) ⎛

⎞ 21 · e−λ·τ (22 − λ) 24 det ⎝ 31 · e−λ·τ 0 34 ⎠ = {ϒ7 · λ2 + ϒ8 · λ + ϒ9 } · e−λ·τ 0 0 (44 − λ)

3.2 Ion Channel Laser Perturbed Differential …

303

No.4 ⎛

⎞ 23 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ −λ·τ 0 43 41 · e   0 (33 − λ) = 21 · e−λ·τ · det 0 43   31 · e−λ·τ (33 − λ) − (22 − λ) · det 43 41 · e−λ·τ   31 · e−λ·τ 0 + 23 · det 41 · e−λ·τ 0 ⎞ ⎛ 23 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ = −(22 − λ) · [31 · 43 0 43 41 · e−λ·τ − 41 · (33 − λ)] · e−λ·τ ⎛

⎞ 23 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ = −(22 − λ) · [(31 · 43 −λ·τ 0 43 41 · e − 41 · 33 ) + 41 · λ] · e−λ·τ ⎛

⎞ 23 21 · e−λ·τ (22 − λ) det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ = {41 · λ2 + [(31 · 43 0 43 41 · e−λ·τ − 41 · 33 ) − 22 · 41 ] · λ − 22 · (31 · 43 − 41 · 33 )} · e−λ·τ ϒ10 = 41 ; ϒ11 = (31 · 43 − 41 · 33 ) − 22 · 41 ϒ12 = −22 · (31 · 43 − 41 · 33 ) ⎛

⎞ 21 · e−λ·τ (22 − λ) 23 det ⎝ 31 · e−λ·τ 0 (33 − λ) ⎠ = {ϒ10 · λ2 + ϒ11 · λ + ϒ12 } · e−λ·τ 0 43 41 · e−λ·τ We can summary our last results in the following table (Table 3.4). det( A − λ · I ) = (11 · e−λ·τ − λ) · (−λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 ) − 12 · (ϒ4 · λ2 + ϒ5 · λ + ϒ6 ) · e−λ·τ

304

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Table 3.4 System’s Jacobian matrix determinant elements (with delay parameter τ ) No 1

2

3

4

Determinant ⎞ ⎛ 23 24 (22 − λ) ⎟ ⎜ det ⎜ 0 (33 − λ) 34 ⎟ ⎠ ⎝ 0 43 (44 − λ) ⎞ ⎛ 23 24 21 · e−λ·τ ⎟ ⎜ −λ·τ ( − λ) det ⎜ 34 ⎟ 33 ⎠ ⎝ 31 · e 41 · e−λ·τ 43 (44 − λ) ⎞ ⎛ 24  · e−λ·τ (22 − λ) ⎟ ⎜ 21 −λ·τ det ⎜ 0 34 ⎟ ⎠ ⎝ 31 · e 41 · e−λ·τ 0 (44 − λ) ⎞ ⎛ 23  · e−λ·τ (22 − λ) ⎟ ⎜ 21 −λ·τ det ⎜ 0 (33 − λ) ⎟ ⎠ ⎝ 31 · e 41 · e−λ·τ

0

Equivalent expression −λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3

(ϒ4 · λ2 + ϒ5 · λ + ϒ6 ) · e−λ·τ

(ϒ7 · λ2 + ϒ8 · λ + ϒ9 ) · e−λ·τ

(ϒ10 · λ2 + ϒ11 · λ + ϒ12 ) · e−λ·τ

43

+ 13 · (ϒ7 · λ2 + ϒ8 · λ + ϒ9 ) · e−λ·τ − 14 · (ϒ10 · λ2 + ϒ11 · λ + ϒ12 ) · e−λ·τ det(A − λ · I ) = 11 · (−λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 ) · e−λ·τ − λ · (−λ3 + ϒ1 · λ2 + ϒ2 · λ + ϒ3 ) − (12 · ϒ4 · λ2 + 12 · ϒ5 · λ + 12 · ϒ6 ) · e−λ·τ + (13 · ϒ7 · λ2 + 13 · ϒ8 · λ + 13 · ϒ9 ) · e−λ·τ − (14 · ϒ10 · λ2 + 14 · ϒ11 · λ + 14 · ϒ12 ) · e−λ·τ det( A − λ · I ) = (λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ) + (−11 · λ3 + 11 · ϒ1 · λ2 + 11 · ϒ2 · λ + 11 · ϒ3 ) · e−λ·τ − (12 · ϒ4 · λ2 + 12 · ϒ5 · λ + 12 · ϒ6 ) · e−λ·τ + (13 · ϒ7 · λ2 + 13 · ϒ8 · λ + 13 · ϒ9 ) · e−λ·τ − (14 · ϒ10 · λ2 + 14 · ϒ11 · λ + 14 · ϒ12 ) · e−λ·τ det(A − λ · I ) = (λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ) + (−11 · λ3 + 11 · ϒ1 · λ2 + 11 · ϒ2 · λ + 11 · ϒ3 − 12 · ϒ4 · λ2 − 12 · ϒ5 · λ − 12 · ϒ6 + 13 · ϒ7 · λ2 + 13 · ϒ8 · λ + 13 · ϒ9

3.2 Ion Channel Laser Perturbed Differential …

305

− 14 · ϒ10 · λ2 − 14 · ϒ11 · λ − 14 · ϒ12 ) · e−λ·τ det( A − λ · I ) = (λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ) + (−11 · λ3 + [11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ] · λ2 + [11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ] · λ + 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 ) · e−λ·τ The characteristics equation for our system is as follow: D(λ, τ ) = (λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ) + (−11 · λ3 + [11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ] · λ2 + [11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ] · λ + 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 ) · e−λ·τ We study the occurrence of any possible stability switching resulting from the increase the value of time delay τ for general characteristic equationD(λ, τ ). We choose τ parameter then D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ . The expression for Pn (λ, τ ) is Pn (λ, τ ) =

n 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + . . .

k=0

The expression for Q m (λ, τ ) is Q m (λ, τ ) =

m 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 + . . .

k=0

D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n = 4; m = 3; n > m The expression for Pn (λ, τ ): Pn=4 (λ, τ ) =

n=4 

pk (τ ) · λk

k=0

= p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3 + p4 (τ ) · λ4 p0 (τ ) = 0; p1 (τ ) = −ϒ3 ; p2 (τ ) = −ϒ2 ; p3 (τ ) = −ϒ1 ; p4 (τ ) = 1

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

The expression for Q m (λ, τ ): Q m=3 (λ, τ ) =

m=3 

qk (τ ) · λk

k=0

= q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 + q3 (τ ) · λ3 q0 (τ ) = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 q1 (τ ) = 11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 q2 (τ ) = 11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ; q3 (τ ) = −11 The homogeneous system for ψ qx q y qz leads to a characteristic equation for the  eigenvalue λ having the form P(λ) + Q(λ) · e−λ·τ ; P(λ) = 4j=0 a j · λ j ; Q(λ) = 3 j j=0 c j ·λ and the coefficients {a j (qi , qk ), c j (qi , qk )} ∈ R depend on qi , qk but not on τ ,qi , qk are any Ion channel laser perturbed system’s parameters, other parameters keep as a constant [4], 5, 6. a0 (τ ) = 0; a1 (τ ) = −ϒ3 ; a2 (τ ) = −ϒ2 ; a3 (τ ) = −ϒ1 ; a4 (τ ) = 1 c0 (τ ) = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 c1 (τ ) = 11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 c2 (τ ) = 11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ; c3 (τ ) = −11 Unless strictly necessary, the designation of the variation arguments (qi , qk ) will subsequently be omitted from P, Q, aj , and cj . The coefficients, aj , and cj are continuous and differential functions of their arguments, and direct substitution shows that a0 + c0 = 0 ∀ qi , qk ∈ R+ . λ = 0 is not a of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore,P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [BK] can also be verified: (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0. (b) | Q(λ) | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. P(λ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable respect to qi , qk . We assume that Pn (λ, τ ) = Pn (λ) and Q m (λ, τ ) = Q m (λ) cannot have common imaginary roots. That is for any real numberω: Pn (λ = i ·ω, τ )+ Q m (λ = i ·ω, τ ) =

3.2 Ion Channel Laser Perturbed Differential …

307

0. Pn (λ = i · ω, τ ) = −ϒ3 · i · ω + ϒ2 · ω2 + i · ϒ1 · ω3 + ω4 = i · (−ϒ3 · ω + ϒ1 · ω3 ) + ϒ2 · ω2 + ω4 Q m (λ = i · ω, τ ) = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 + (11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ) · i · ω − (11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ) · ω2 + 11 · i · ω3 Q m (λ = i · ω, τ ) = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 − (11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ) · ω2 + ((11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ) · ω + 11 · ω3 ) · i Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = i · (−ϒ3 · ω + ϒ1 · ω3 ) + ϒ2 · ω2 + ω4 + 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 − (11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ) · ω2 + ((11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ) · ω + 11 · ω3 ) · i = 0

Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = [(11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 − ϒ3 ) · ω + (11 + ϒ1 ) · ω3 ] · i + ω4 + [ϒ2 − (11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 )] · ω2 + 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 = 0 Pn (λ = i · ω, τ ) = i · (−ϒ3 · ω + ϒ1 · ω3 ) + ϒ2 · ω2 + ω4 Q m (λ = i · ω, τ ) = (11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 ) − (11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ) · ω2 + ((11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ) · ω + 11 · ω3 ) · i We define for simplicity new parameters which are related to Q m (λ = i · ω, τ ) = . . . . expression.

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

1 = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 2 = 11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 3 = 11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 Q m (λ = i · ω, τ ) = 1 − 2 · ω2 + (3 · ω + 11 · ω3 ) · i |Pn (λ = i · ω, τ )|2 = (ϒ2 · ω2 + ω4 )2 + (−ϒ3 · ω + ϒ1 · ω3 )2 |Pn (λ = i · ω, τ )|2 = ω8 + (2 · ϒ2 + ϒ12 ) · ω6 + (ϒ22 − 2 · ϒ3 · ϒ1 ) · ω4 + ϒ32 · ω2 |Q m (λ = i · ω, τ )|2 = (1 − 2 · ω2 )2 + (3 · ω + 11 · ω3 )2 |Q m (λ = i · ω, τ )|2 = 211 · ω6 + (22 + 2 · 3 · 11 ) · ω4 + (23 − 2 · 1 · 2 ) · ω2 + 21 F(ω) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω8 + (2 · ϒ2 + ϒ12 ) · ω6 + (ϒ22 − 2 · ϒ3 · ϒ1 ) · ω4 + ϒ32 · ω2 − [211 · ω6 + (22 + 2 · 3 · 11 ) · ω4 + (23 − 2 · 1 · 2 ) · ω2 + 21 ] F(ω) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω8 + [2 · ϒ2 + ϒ12 − 211 ] · ω6 + [ϒ22 − 2 · ϒ3 · ϒ1 − 22 − 2 · 3 · 11 ] · ω4 + [ϒ32 − 23 + 2 · 1 · 2 ] · ω2 − 21 8 = 1; 6 = 2 · ϒ2 + ϒ12 − 211 4 = ϒ22 − 2 · ϒ3 · ϒ1 − 22 − 2 · 3 · 11 2 = ϒ32 − 23 + 2 · 1 · 2 ; 0 = −21 F(ω) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 =

4  k=0

k · ω2·k

3.2 Ion Channel Laser Perturbed Differential …

309

 Hence F(ω) = 0 implies 4k=0 k · ω2·k = 0 and its roots are given by solving the above polynomial. Furthermore PR (i · ω, τ ) = ϒ2 · ω2 + ω4 ; PI (i · ω, τ ) = −ϒ3 · ω + ϒ1 · ω3 Q R (i · ω, τ ) = 1 − 2 · ω2 ; Q I (i · ω, τ ) = 3 · ω + 11 · ω3 sin θ (τ ) = Hence −PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) . |Q(i · ω, τ )|2 cos θ (τ ) = And PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) − |Q(i · ω, τ )|2 sin θ(τ ) =

−[ϒ2 · ω2 + ω4 ] · [3 · ω + 11 · ω3 ] + [−ϒ3 · ω + ϒ1 · ω3 ] · [1 − 2 · ω2 ] 211 · ω6 + (22 + 2 · 3 · 11 ) · ω4 + (23 − 2 · 1 · 2 ) · ω2 + 21

cos θ (τ ) = −

[ϒ2 · ω2 + ω4 ] · [1 − 2 · ω2 ] + [−ϒ3 · ω + ϒ1 · ω3 ] · [3 · ω + 11 · ω3 ] 211 · ω6 + (22 + 2 · 3 · 11 ) · ω4 + (23 − 2 · 1 · 2 ) · ω2 + 21

4 2·k Which jointly with F(ω) = 0 ⇒ = 0 that are contink=0 k · ω uous and differentiable in τ based on Lemma 1.1 (see Subchapter 2.1). Hence we use Theorem 1.2 (see Subchapter 2.1) and this proves the Theorem 1.3 (see Subchapter 2.1). We use different parameters terminology from our last characteristics parameters definition k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 4; m = 3; n > m additionally Pn (λ, τ ) → P(λ, τ ).   And Q m (λ, τ ) → Q(λ, τ ) then P(λ, τ ) = 4j=0 a j ·λ j ; Q(λ, τ ) = 3j=0 c j ·λ j . P(λ) = λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ Q(λ) = −11 · λ3 + [11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ] · λ2 + [11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ] · λ + 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in λ. The coefficients: {a j (k z , ω, ωβ , A, . . .) & c j (k z , ω, ωβ , A, v . . .)} ∈ R Depend on Ion channel laser perturbed system’s parameters k z , ω, ωβ , A, τ, . . . values. Unless strictly necessary, the designation of the variation arguments. System parameters k z , ω, ωβ , A, τ, . . . will subsequently be omitted from P, Q, aj , cj . The coefficients aj , cj are continuous, and differentiable functions of their arguments.

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

Direct substitution shows that a0 + c0 = 0. a0 + c0 = 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 = 0 i.e. λ = 0 is not a root of characteristic equation. Furthermore P(λ), Q(λ) are analytic function of λ for which the following requirements of the analysis (see kuang, 1993, Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire (k z , ω, ωβ , A, . . . system parameters) domain of interest. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (b) | Q(λ) P(λ) in the limit: − 11 · λ3 + [11 · ϒ1 − 12 · ϒ4 + 13 · ϒ7 − 14 · ϒ10 ] · λ2 + [11 · ϒ2 − 12 · ϒ5 + 13 · ϒ8 − 14 · ϒ11 ] · λ |

+ 11 · ϒ3 − 12 · ϒ6 + 13 · ϒ9 − 14 · ϒ12 Q(λ) |=| P(λ) λ4 − ϒ1 · λ3 − ϒ2 · λ2 − ϒ3 · λ

|

(c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 F(ω) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω8 + [2 · ϒ2 + ϒ12 − 211 ] · ω6 + [ϒ22 − 2 · ϒ3 · ϒ1 − 22 − 2 · 3 · 11 ] · ω4 + [ϒ32 − 23 + 2 · 1 · 2 ] · ω2 − 21 Has at most a finite number of zeros. Indeed, this is a polynomial in ω(degree in ω8 ). (d) Each positive rootω (k z , ω, ωβ , A, . . . system parameters) of F(ω) = 0 is continuous and differentiable with respect to k z , ω, ωβ , A, . . . system parameters. This condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i ·ω) thus λ = i ·ω; ω > 0 may be on eigenvalue of characteristic equation. The analysis consists in identifying the roots of characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the parameters k z , ω, ωβ , A, . . .. Re λ may, at the crossing, change its sign from (−) to (+), i.e. from a stable focus E (k) (ψ (k) , qx(k) , q y(k) , qz(k) ); k = 0, 1, 2, 3, 4 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with ) , ω, ωβ , A, τ, . . ., respect to k z , ω, ωβ , A, τ, . . . parameters. −1 (k z ) = ( ∂Reλ ∂k z λ=i·ω Ion channel laser perturbed system parameters are constant [4, 5]. −1 (ω) = (

∂Reλ )λ=i·ω ; k z , ωβ , A, τ, . . . = const ∂ω

3.2 Ion Channel Laser Perturbed Differential …

−1 (ωβ ) = ( −1 (A) = ( −1 (τ ) = (

311

∂Reλ )λ=i·ω ; k z , ω, A, τ, . . . = const ∂ωβ

∂Reλ )λ=i·ω ; k z , ω, ωβ , τ, . . . = const ∂A

∂Reλ )λ=i·ω ; k z , ω, ωβ , A, . . . = const ∂τ

Remark We have two ω parameters in our analysis, one is the system variable (ω— Frequency of radiation in the forward (+z) direction in the lab frame) and the second is related to the general geometric criterion for stability (see the article, Edoardo Beretta and Yang Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal, 2001), ω ∈ R+ . When writing P(λ) = PR (λ)+i · PI (λ); Q(λ) = Q R (λ)+i · Q I (λ), and inserting λ = i · ω into Ion channel laser perturbed system’s characteristic equation, ω must satisfy the following: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω)|2 = 0 in view of requirement (a) above; (g, h) ∈ R. Furthermore, it follows above equations sin(ω · τ ) and cos(ω · τ ) that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note F(ω) is dependent on τ . Now it is important to notice that if τ ∈ / I (assume that / I ; ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for τ ∈ not define. Then for all τ in I, ω(τ ) is satisfies that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For any τ ∈ I , where ω(τ ) is a positive solution of F(ω, τ ) = 0 we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of −PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2 PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) cos θ (τ ) = − |Q(i · ω)|2

sin θ (τ ) =

And the relation between the argument θ (τ ) and ω(τ ) · τ for τ ∈ I must be ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ N0 . Hence we can define the maps τn : I → R+0 )+2·n·π given by τn (τ ) = θ(τ ω(τ ; n ∈ N0 ; τ ∈ I . Let us introduce the functions ) I → R; Sn (τ ) = τ − τn (τ ) τ ∈ I ; n ∈ N0 that are continuous and differentiable in τ . In the following, the subscripts λ, ω, k z , ωβ , A, . . .Ion channel laser perturbed

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

system parameters indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(k z , ωβ , A, . . . ; ∈ R+ ); ω(k z , ωβ , A, . . . ; ∈ R+ ) and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Ion channel laser perturbed system characteristic equation P(λ) + Q(λ) · e−λ·τ with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: Remark x = k z , ωβ , A, . . . ; ∈ R+ 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where,Pλ = ∂∂λP , . . . etc. Substituting λ = i · ω, and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω) and i · Q λ (i · ω) = Q ω (i · ω) that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains. ∂λ −1 ) |λ=i·ω ∂x i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 = Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

(

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = QR + i · QI Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 When (x) can be any Ion channel laser perturbed system parameters k z , ωβ , A, . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i·ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] And ωx = − FFωx . U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ . PR (i · ω, τ ) = ϒ2 · ω2 + ω4 ; PI (i · ω, τ ) = −ϒ3 · ω + ϒ1 · ω3

3.2 Ion Channel Laser Perturbed Differential …

313

Q R (i · ω, τ ) = 1 − 2 · ω2 ; Q I (i · ω, τ ) = 3 · ω + 11 · ω3 PRω (i · ω, τ ) = 2 · ϒ2 · ω + 4 · ω3 ; PI ω (i · ω, τ ) = −ϒ3 + 3 · ϒ1 · ω2 Q Rω (i · ω, τ ) = −2 · 2 · ω; Q I ω (i · ω, τ ) = 3 + 3 · 11 · ω2 PRτ (i · ω, τ ) = 0; PI τ (i · ω, τ ) = 0 Q Rτ (i · ω, τ ) = 0; Q I τ (i · ω, τ ) = 0 PRω · PR = (2 · ϒ2 · ω + 4 · ω3 ) · (ϒ2 · ω2 + ω4 ) PI ω · PI = (−ϒ3 + 3 · ϒ1 · ω2 ) · (−ϒ3 · ω + ϒ1 · ω3 ) Fτ ; Q Rω · Q R = −2 · 2 · ω · (1 − 2 · ω2 ) Fω Q I ω · Q I = (3 + 3 · 11 · ω2 ) · (3 · ω + 11 · ω3 )

ωτ = −

Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0 PR · PI ω = (ϒ2 · ω2 + ω4 ) · (−ϒ3 + 3 · ϒ1 · ω2 ) Q I · Q Rω = (3 · ω + 11 · ω3 ) · (−2 · 2 · ω) V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 + Fτ = 0; τ ∈ I ⇒ Differentiating with respect to τ and we get Fω · ∂ω ∂τ

∂ω ∂τ

 ∂Reλ  (τ ) = ∂τ λ=i·ω   −2 · [U + τ · |P|2 ] + i · Fω −1 (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ] ∂ω Fτ = ωτ = − ∂τ Fω    ∂Reλ sign{−1 (τ )} = sign ∂τ λ=i·ω   U · ∂ω +V ∂ω ∂τ −1 +ω+ sign{ (τ )} = sign{Fω } · sign τ · ∂τ |P|2 −1



= − FFωτ

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

We shall presently examine the possibility of stability transitions (bifurcations) in a Ion channel laser perturbed system, about the equilibrium pointE (k) (ψ (k) , qx(k) , q y(k) , qz(k) ); k = 0, 1, 2, 3, 4 as a result of a variation of delay parameterτ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane where by increasing the delay parameterτ , Re λ may at the crossing, change its sign from “- “ to “ + ”, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ . ) , k , ωβ , A, . . . system parameters are constant where ω ∈ R+ . −1 (τ ) = ( ∂Reλ ∂τ λ=i·ω z We need to plot the stability switch diagram based on different delay values of our Ion channel laser perturbed system. Since it is a very complex function we recommend to solve it numerically rather than analytic.    ∂Reλ −2 · [U + τ · |P|2 ] + i · Fω = Re  (τ ) = ∂τ λ=i·ω Fτ + i · 2 · [V + ω · |P|2 ]   ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} −1 (τ ) = = ∂τ λ=i·ω Fτ2 + 4 · (V + ω · P 2 )2 −1



The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . ; cos θ (τ ) = . . . when ω = ω+ (τ ) if only ω+ is feasible. Additionally when all Ion channel laser perturbed system parameters are known and the stability switch due to various time delay values τ is describe in the following expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ ) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + |P(ω(τ ))|2 Remark we know F(ω, τ ) = 0 implies it roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which ωi are complex or imaginary numbers, then unable to analyze stability.

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time A semiconductor laser (LD) is a device that causes laser oscillation by the flowing an electric current to semiconductor. Practically, the mechanism of light emission is the same as a Lighting-emitting diode (LED). The optical gain is produced in a semiconductor material. By establishing the desired wavelength and other properties such as modulation speed we define the material. It is bulk semiconductor, quantum hetero

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

315

structure. The pumping mechanism is done electrically or optically (disk laser). The laser light is produced when electrons and photons interact in a p-n junction. The concentrated light beam is produced by “pumping” the light emitted from atoms repeatedly, between two mirrors. The semiconductor laser produces coherent radiation (wave are all at the same frequency and phase) in the visible or IR spectrum when current passes through it. Semiconductor lasers are widely using in optical fiber communication systems and optical disk system, optical subscriber networks. The mode competition phenomena are the phenomenon that different laser resonator modes experience laser amplification in the same gain medium, leading to crosssaturation effects. The resonator modes are characterized by the light circulating in an optical resonator. We characterize the light in each mode by parameters (electric field amplitude, optical phase, apart from optical frequency and polarization). The overall number of parameters is small in the lasing process. The different modes of a laser resonator experience optical amplification in the same gain medium. Different modes experience amplification in the same gain medium which leads to cross saturation effects. A stimulated emission by one mode causes gain saturation for itself (self- saturation) and for other modes. The power distribution over several modes is unstable. We need to investigate the influence of instantaneous mode-competition phenomena on the dynamics of semiconductor lasers. The system analysis and stability is based on the multimode rate equations which are superposed by Langevin noise sources that account for the intrinsic fluctuations associated with the spontaneous emission. The Langevin noise sources derive the laser rate equations keeping their cross-correlations satisfied. They are time-varying profiles of the fluctuating photon and carrier numbers and the instantaneous shift of the oscillating frequency. The gain saturation effects cause competition phenomena among lasing modes are based on a self-consistent model. The mode-competition phenomena are caused by nonlinear gain suppression effects and are characterized by the corresponding saturation coefficients. The gain suppression effect among different lasing modes is stronger than the suppression effect for an identical mode when the laser supports only the fundamental transverse mode. The latter effect is called self-saturation, and the former one is called cross-saturation. The semiconductor lasers show several types of extra noise, such as hopping noise, and optical feedback noise. These types of noise are results of dynamic behavior of the mode-competition phenomena. There are cases on mode-competition phenomena, the intensity fluctuations were assumed to be not sufficiently large to change the operation state. The fluctuations of mode intensities can change the state of operation as pointed out in. The model is the multimode rate equations in which both gain saturation effects and Langevin noise sources are presented. The nonlinear dynamics of the lasing nodes are described mathematically by multimode rate equations of the photon number Sp of the lasing modes and the injected electron number N [7]. d Sp a·ξ · N = (G p − G th ) · S p + + F p (t) 2·(λ p −λ0 ) 2 dt V · {[ δλ ] + 1}

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

 dN N I =− A p · Sp − + + FN (t) dt τs e p Cp ·

N a·ξ · N = ⇒ Cp 2·(λ p −λ0 ) 2 τs V · {[ δλ ] + 1} a · ξ · τs = 2·(λ p −λ0 ) 2 V · {[ δλ ] + 1}

Hint: The terms F p (t) and FN (t) are Langevin noise sources and are added to the rate equations to trigger fluctuations in the modal photon numbers and electron number. G p = A p − B · Sp −



(D p(q) + H p(q) ) · Sq

q = p

Ap = B=

a·ξ · {N − N g − b · V · (λ p − λ0 )2 } V

9 ξ · τin 2 π ·c ·( · ) · a · |Rcv |2 · (N − Ns ) 2 ε0 · n r2 ·  · λ0 V

B = B(N ); A p = A p (N ) D p(q) =

4 B · in 2 2 3 ( 2·π·c·τ ) · (λ p − λq ) + 1 λ2 p

D p(q)

H p(q) =

4 = · 3

9 2

·

π·c · ( ξ ·τVin )2 · a · |Rcv |2 · (N ε0 ·nr2 ··λ0 in 2 ( 2·π·c·τ ) · (λ p − λq )2 + 1 λ2p

⎧ ⎨

1 τs

+

3 a·ξ 2 ·( ) · (N − N g ) · ⎩( 1 + 4 V τs

·

3 2 3 2

·

a·ξ V a·ξ V

·S+α·

− Ns )

2·π·c λ2p

⎫ · (λ p − λq ) ⎬

· S)2 + ( 2·π·c )2 · (λ p − λq )2 ⎭ λ2 p

D p(q) = D p(q) ( p, q, N ); H p(q) = H p(q) ( p, q, N ) Index p = 0, ±1, ±2, . . . , ±M; M ∈ N0 is the mode number, G p is the optical gain of mode p. The mode p = 0 with wavelength λ0 is assumed to be at the center of the spectral profile of the gain. The lasing modes on the long-wavelength side of the central mode of the gain profile λ p > λ0 are indicated with positive numbers, 0 < p ≤ M, while the modes on the shorter side, λ p < λ0 , are indicated with negative numbers −M ≤ p < 0; λ p = λ0 + p · λ. λ2

λ2

0 0 Where, λ = nr ·L ; λ p = λ0 + p · nr ·L . n r —Refractive index of the active region, L—Length of the active region, λ—Mode wavelength spacing, A p · S p —Photon number after amplification (A p —linear gain coefficient) at mode p (Fig. 3.3). G th is the threshold gain level, and is determined by the loss coefficient κ of the 1 · ln( R f1·Rb )] where R f and Rb are power laser and mirror loss, G th = ncr · [κ + 2·L

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

317

Fig. 3.3 Typical spectral distribution of photon number after amplification for different wavelengths at mode (p)

reflectivity at the front and back facets, respectively. τs —Lifetime of the electron for spontaneous emission, e—electron charge, I —Injection current, B—self saturation coefficient, I th —Threshold current level, D p(q) , H p(q) —Coefficients for cross saturation effects of gain, a —Slope (or tangential) coefficient to give the local linear gain, b- Coefficient giving the wavelength dispersion of the linear gain, N g —Transparent electron number, τin —Intra-band relaxation time of the electron wave, Rcv —Dipole the saturation coefficient (B), c— moment, Ns —Electron number characterization  S —Total photon number summed for all Speed of light in free space, S = p p existing modes, δλ —Half width of the spontaneous emission profile, α—Linewidth enhancement factor, F p (t), FN (t)—Generating term of the fluctuation caused by the spontaneous emission (Langevin noise sources),V —Volume of the active region, ξ — Field confinement factor into the active region, D p(q) —Symmetric cross saturation coefficient, n—Injected density of electrons (n = VN ), C p —Spontaneous emission · S  2·π·c · |λq − λ p | and factor. Assumption for large model separation: τ1s + 23 · a·ξ V λ2 p

we get the reduce expression for H p(q) , H p(q) ≈

3·λ2p 8·π·c

· ( a·ξ )2 · V

α·(N −N g ) . (λq −λ p )

Function

coefficient D p(q) ∼ gives the similar value if λ p > λq or λ p < λq , then is called the symmetric cross saturation coefficient. On the other hand, the reduced 1 function H p(q) , H p(q) ∼ λq −λ is inversely proportional to λq − λ p .H p(q) is the p asymmetric cross saturation coefficient. Wavelengths  λ p , λq are at two modes p and > 0 λ p < λq q respectively ( p = q). Function H p(q) , H p(q) = . < 0 λ p > λq We can summary the asymmetric cross saturation coefficient H p(q) sign for λ p , λq conditions and the related meaning in the next table (Table 3.5). 1 (λ p −λq )2

Table 3.5 Asymmetric cross saturation coefficient H p(q) sign for λ p , λq conditions and the related meaning λ p , λq condition H p(q) value Meaning λ p < λq

H p(q) > 0

Suppress the lasing gain for the modes on the short wavelength side

λ p > λq

H p(q) < 0

Enhance the lasing gain for the modes on the longer wavelength side

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering



(D p(q) + H p(q) ) · Sq |λ p

q = p



(D p(q) + H p(q) ) · Sq |λ p >λq

q = p

G p |λ p λq

(Enhance lasing gain)

The periodic mode hopping (0 → 1 → 2 → 3 → 0 → . . .) is caused by the asymmetric cross saturation function H p(q) . The following flowchart shows our laser periodic mode hopping (Fig. 3.4). We need to analyze our system stability and stability switching for parameters variation. We take some assumptions before doing system analysis [2, 3, 7].

Fig. 3.4 Schematic for the rotating effect of the mode hopping among several longitudinal modes

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

319

Assumption A Terms F p (t) and FN (t) (Langevin noise sources) are very small compare to other elements on the rate equations.  a·ξ ·N  F p (t); − A p · S p − τNs + eI  FN (t) Then (G p − G th ) · S p + 2·(λ p −λ0 ) 2 V ·{[

δλ

] +1}

p

we can write our reduced rate equations: d Sp a·ξ · N = (G p − G th ) · S p + 2·(λ p −λ0 ) 2 dt V · {[ δλ ] + 1}  dN N I =− A p · Sp − + dt τ e s p D p(q) = D p(q) ( p, q, N ); H p(q) = H p(q) ( p, q, N ) G p = G p ( p, q, N ); B = B(N ); A p = A p (N ) G p = A p − B · Sp −



(D p(q) + H p(q) ) · Sq

q = p

G p = A p (N ) − B(N ) · S p −



{D p(q) ( p, q, N )

q = p

+ H p(q) ( p, q, N )} · Sq Assumption B Large model separation: H p(q) ≈

3 · λ2p 8·π ·c

·(

1 τs

+

3 2

·

a·ξ V

·S

2·π·c λ2p

· |λq − λ p | then.

a · ξ 2 α · (N − N g ) ) · ; H p(q) = H p(q) ( p, q, N ) V (λq − λ p )

Assumption C Laser operates  on a periodic mode hopping (0 → 1 → 2 → 3 →  0 → . . .) then p A p · S p = 3p=0 A p · S p ; 0 = 3p=0 A p · S p ; 0 ∈ R+ . The 3  total photon number summed for existing modes S = p Sp = p=0 S p ; 1 = 3 p=0 S p ; 1 ∈ R+ . Assumption D Laser modes are on the longer wavelength side and there is a periodically hopping (0 → 1 → 2 → 3 → 0 → . . .), increase the lasing gain G p (λ p > λq , H p(q) < 0). We define new variable λq− p = λq − λ p ; q = p; λq− p < 0; λq− p ∈ R and we define parameters

λq− p=0 = λq − λ0 ; λq− p=1 = λq − λ1 ; λq− p=2 = λq − λ2 ; λq− p=3 = λq − λ3 λq− p=0 = λq− p=0 (q); λq− p=1 = λq− p=1 (q); λq− p=2 =

λq− p=2 (q); λq− p=3 = λq− p=3 (q), λq− p=0 , λq− p=1 , λq− p=2 , λq− p=3 < 0(Index q is fixed). H p=0 ≈

3 · λ20 a · ξ 2 α · (N − N g ) ·( ) · 8·π ·c V (λq − λ0 )

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

H p=1 ≈

a · ξ 2 α · (N − N g ) 3 · λ21 ·( ) · 8·π ·c V (λq − λ1 )

H p=2 ≈

a · ξ 2 α · (N − N g ) 3 · λ22 ·( ) · 8·π ·c V (λq − λ2 )

H p=3 ≈

a · ξ 2 α · (N − N g ) 3 · λ23 ·( ) · 8·π ·c V (λq − λ3 )

H p=0 ≈

3 · λ20 α a·ξ 2 · (N − N g ) ·( ) · 8·π ·c V

λq− p=0

H p=1 ≈

3 · λ21 α a·ξ 2 · (N − N g ) ·( ) · 8·π ·c V

λq− p=1

H p=2 ≈

a·ξ 2 3 · λ22 α ·( ) · · (N − N g ) 8·π ·c V

λq− p=2

H p=3 ≈

a·ξ 2 3 · λ23 α ·( ) · · (N − N g ) 8·π ·c V

λq− p=3

We define new global parameters: 0 =

3·λ20 8·π·c

· ( a·ξ )2 · V

α

λq− p=0

1 =

a·ξ 2 3 · λ21 α ·( ) · 8·π ·c V

λq− p=1

2 =

a·ξ 2 3 · λ22 α ·( ) · 8·π ·c V

λq− p=2

3 =

a·ξ 2 3 · λ23 α ·( ) · 8·π ·c V

λq− p=3

H p=0 ≈ 0 · (N − N g ); H p=1 ≈ 1 · (N − N g ) H p=2 ≈ 2 · (N − N g ); H p=3 ≈ 3 · (N − N g ) D p(q) =

4 · 3

9 2

·

π·c · ( ξ ·τVin )2 · a · |Rcv |2 · (N ε0 ·nr2 ··λ0 in 2 ( 2·π·c·τ ) · (λ p − λq )2 + 1 λ2p

− Ns )

λ p−q = λ p − λq ; λ p−q > 0

λ p=0−q = λ0 − λq ; λ p=1−q = λ1 − λq

λ p=2−q = λ2 − λq ; λ p=3−q = λ3 − λq

D p(q)

⎡ ⎤ ξ ·τin 2 9 π·c 2 4 ⎣ 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ⎦ = · · (N − Ns ) in 2 3 ( 2·π·c·τ ) · ( λ p−q )2 + 1 λ2 p

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

⎡ ⎤ ξ ·τin 2 9 π·c 2 4 ⎣ 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ⎦ = · · (N − Ns ) in 2 3 ( 2·π·c·τ ) · ( λ p=0−q )2 + 1 λ2

D p=0

p

⎡

⎤ ξ ·τin 2 9 π·c 2 4 ⎣ 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ⎦ = · · (N − Ns ) in 2 3 ( 2·π·c·τ ) · ( λ p=1−q )2 + 1 λ2

D p=1

p

⎡

⎤ ξ ·τin 2 9 π·c 2 4 ⎣ 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ⎦ = · · (N − Ns ) in 2 3 ( 2·π·c·τ ) · ( λ p=2−q )2 + 1 λ2

D p=2

p

⎡

⎤ ξ ·τin 2 9 π·c 2 4 ⎣ 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ⎦ = · · (N − Ns ) in 2 3 ( 2·π·c·τ ) · ( λ p=3−q )2 + 1 λ2

D p=3

p

We define new global parameters: 0 =

4 3

·[

ξ ·τin 2 9 π·c 2 2 · ε ·n 2 ··λ ·( V ) ·a·|Rcv | 0 r 0 2·π·c·τin 2 ( ) ·( λ p=0−q )2 +1 λ2p

ξ ·τin 2 9 π·c 2 4 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | ] 1 = · [ 2·π·c·τin 3 ( λ2 )2 · ( λ p=1−q )2 + 1 p

4 2 = · [ 3 3 =

9 2

· ε0 ·nπ·c 2 r ··λ0 2·π·c·τin 2 ( λ2 ) p

· ( ξ ·τVin )2 · a · |Rcv |2 · ( λ p=2−q )2 + 1

]

ξ ·τin 2 9 π·c 2 4 2 · ε0 ·nr2 ··λ0 · ( V ) · a · |Rcv | · [ 2·π·c·τin ] 3 ( λ2 )2 · ( λ p=3−q )2 + 1 p

D p=0 = 0 · (N − Ns ); D p=1 = 1 · (N − Ns ) D p=2 = 2 · (N − Ns ); D p=3 = 3 · (N − Ns ) 

{D p(q) ( p, q, N ) + H p(q) ( p, q, N )} · Sq

q = p

=

3 

{D p(q) ( p, q, N ) + H p(q) ( p, q, N )} · Sq

p=0 p =q



{D p(q) ( p, q, N ) + H p(q) ( p, q, N )} · Sq

q = p

= {0 · (N − Ns ) + 1 · (N − Ns ) + 2 · (N − Ns ) + 3 · (N − Ns ) + 0 · (N − N g ) + 1 · (N − N g )

]

321

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

+ 2 · (N − N g ) + 3 · (N − N g )} · Sq 

{D p(q) ( p, q, N ) + H p(q) ( p, q, N )} · Sq

q = p

= {(N − Ns ) ·

3 

k + (N − N g ) ·

k=0

3 

k } · Sq

k=0

We can rewrite our system rate equations again based on assumptions A – D. d Sp a·ξ · N = (G p − G th ) · S p + 2·(λ p −λ0 ) 2 dt V · {[ δλ ] + 1} dN N I = −0 − + dt τs e  {D p(q) ( p, q, N ) + H p(q) ( p, q, N )} · Sq G p = A p (N ) − B(N ) · S p − q = p

a·ξ · {N − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · (N − Ns ) · S p − · ·( 2 ε0 · n r2 ·  · λ0 V

Gp =

− {(N − Ns ) ·

3 

k + (N − N g ) ·

k=0

3 

k } · Sq ; G p = G p (N , S p )

k=0

Our system rate equations variables are S p and N . At fixed points (equilibrium dS points) dtp = 0; ddtN = 0. (G p (N ∗ , S ∗p ) − G th ) · S ∗p + − 0 −

N∗ I + =0 τs e

−0 −

a · ξ · N∗ V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

= 0;

N∗ I I + = 0 ⇒ N ∗ = ( − 0 ) · τs τs e e

a·ξ · {N ∗ − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · (N ∗ − Ns ) · S ∗p − · ·( 2 2 ε0 · n r ·  · λ0 V

G p (N ∗ , S ∗p ) =

− {(N ∗ − Ns ) ·

3  k=0

k + (N ∗ − N g ) ·

3  k=0

k } · Sq

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

323

I G p (N ∗ = ( − 0 ) · τs , S ∗p ) e I a·ξ · {( − 0 ) · τs − N g − b · V · (λ p − λ0 )2 } = V e 9 π ·c ξ · τin 2 I − · ) · a · |Rcv |2 · {( − 0 ) · τs − Ns } · S ∗p ·( 2 2 ε0 · n r ·  · λ0 V e   I − {[( − 0 ) · τs − Ns ] · k + (N ∗ − N g ) · k } · Sq e k=0 k=0 3

3

We define for simplicity new global parameters: 1 = b · V · (λ p − λ0 )2 } 2 =

a·ξ V

· {( eI − 0 ) · τs − N g −

π ·c 9 ξ · τin 2 I · ) · a · |Rcv |2 · {( − 0 ) · τs − Ns } ·( 2 2 ε0 · n r ·  · λ0 V e

  I k + (N ∗ − N g ) · k } · Sq 3 = {[( − 0 ) · τs − Ns ] · e k=0 k=0 3

3

I G p (N ∗ = ( − 0 ) · τs , S ∗p ) = 1 − 2 · S ∗p − 3 e I ∗ G p (N = ( − 0 ) · τs , S ∗p ) = (1 − 3 ) − 2 · S ∗p e (G p (N ∗ , S ∗p ) − G th ) · S ∗p +

a · ξ · N∗ V · {[

((1 − 3 ) − 2 · S ∗p − G th ) · S ∗p + (1 − 3 − G th ) · S ∗p − 2 · (S ∗p )2 + 2 · (S ∗p )2 − (1 − 3 − G th ) · S ∗p −

S ∗p(1) =

S ∗p(2) =

(1 − 3 − G th ) +

2·(λ p −λ0 ) 2 ] δλ

+ 1}

=0

a · ξ · [( eI − 0 ) · τs ] V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

=0

a · ξ · [( eI − 0 ) · τs ] V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

a · ξ · [( eI − 0 ) · τs ] V · {[

2·(λ p −λ0 ) 2 ] δλ

(1 − 3 − G th )2 + 4 · 2 ·

+ 1}

=0 =0

a·ξ ·[( eI −0 )·τs ] V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

2 · 2 (1 − 3 − G th ) −

(1 − 3 − G th )2 + 4 · 2 · 2 · 2

a·ξ ·[( eI −0 )·τs ] V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

324

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

We have two fixed points in our system. (0) E (0) (S (0) p ,N )

(1 − 3 − G th ) + =(

(1 − 3 − G th )2 + 4 · 2 ·

a·ξ ·[( eI −0 )·τs ] V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

2 · 2

,

I ( − 0 ) · τs ) e (1) E (1) (S (1) p ,N )

(1 − 3 − G th ) − =(

(1 − 3 − G th )2 + 4 · 2 ·

a·ξ ·[( eI −0 )·τs ] V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

2 · 2

,

I ( − 0 ) · τs ) e Stability analysis: The standard local stability analysis about any one of the equilibrium points (fixed points) of long-wavelength semiconductor laser rate equations consists in adding to its coordinated [S p N ] arbitrarily small increments of exponential terms [s p n] · eλ·t , and retaining the first order terms in s p n. The system of two homogeneous equations leads to a polynomial characteristic equation in the eigenvalue λ. The polynomial characteristic equation accepts by set the longwavelength semiconductor laser rate equations. The long-wavelength semiconductor laser rate equations fixed values with arbitrarily small increments of exponential form [s p n] · eλ·t are i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc. [8−11]. Hint: please take care that we have one variable with two meanings, n—Injected density of electrons (n = VN ) and stability analysis arbitrarily small increment n. λ·t (i) + n · eλ·t S p (t) = S (i) p + s p · e ; N (t) = N d S p (t) d N (t) = s p · λ · eλ·t ; = n · λ · eλ·t dt dt

####### d Sp a·ξ · N = (G p − G th ) · S p + 2·(λ p −λ0 ) 2 dt V · {[ δλ ] + 1} a·ξ · {N − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · (N − Ns ) · S p − · ·( 2 ε0 · n r2 ·  · λ0 V

Gp =

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

− {(N − Ns ) ·

3 

k + (N − N g ) ·

k=0

3 

k } · Sq ; G p = G p (N , S p )

k=0

a·ξ · {N (i) + n · eλ·t − N g − b · V · (λ p − λ0 )2 } V 9 ξ · τin 2 π ·c − · ·( ) · a · |Rcv |2 2 2 ε0 · n r ·  · λ0 V

Gp =

λ·t · (N (i) + n · eλ·t − Ns ) · (S (i) p + sp · e ) 3 

− {(N (i) + n · eλ·t − Ns ) ·

k

k=0

+ (N (i) + n · eλ·t − N g ) ·

3 

k } · Sq

k=0

a·ξ a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } + · n · eλ·t V V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · ([N (i) − Ns ] − · ·( 2 2 ε0 · n r ·  · λ0 V

Gp =

(i) λ·t − Ns ] · s p · eλ·t + S (i) + n · s p · e2·λ·t ) · S (i) p + [N p ·n·e

− {[N (i) − Ns ] ·

3 

k + n · eλ·t ·

3 

k=0

·

3 

k + n · eλ·t ·

k=0

3 

k + [N (i) − N g ]

k=0

k } · Sq

k=0

a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · ([N (i) − Ns ] · S (i) − · ·( p ) 2 2 ε0 · n r ·  · λ0 V

Gp =

− {[N (i) − Ns ] ·

3 

k + [N (i) − N g ] ·

3 

k=0

k } · Sq +

k=0

a·ξ · n · eλ·t V

π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · ([N (i) − Ns ] · s p − · ·( 2 2 ε0 · n r ·  · λ0 V λ·t · eλ·t + S (i) + n · s p · e2·λ·t ) p ·n·e

−{

3  k=0

Gp =

k +

3 

k } · Sq · n · eλ·t ; n · s p ≈ 0

k=0

a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } V

325

326

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

−

π ·c 9 ξ · τin 2 · ) · a · |Rcv |2 · ([N (i) − Ns ] · S (i) ·( p ) 2 2 ε0 · n r ·  · λ0 V

− {[N (i) − Ns ] ·

3 

k + [N (i) − N g ] ·

k=0

3 

k } · Sq +

k=0

a·ξ · n · eλ·t V

9 ξ · τin 2 π ·c − · ·( ) · a · |Rcv |2 · ([N (i) − Ns ] 2 2 ε0 · n r ·  · λ0 V λ·t · s p · eλ·t + S (i) p ·n·e )−{

3 

k +

k=0

3 

k } · Sq · n · eλ·t

k=0

a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } V π ·c ξ · τin 2 9 ) · a · |Rcv |2 · ([N (i) − Ns ] · S (i) ·( − · p ) 2 ε0 · n r2 ·  · λ0 V

Gp =

− {[N (i) − Ns ] ·

3 

k + [N (i) − N g ] ·

k=0

+[ −{

3 

k } · Sq

k=0

9 π ·c a·ξ ξ · τin 2 − · ) · a · |Rcv |2 · S (i) ·( p V 2 ε0 · n r2 ·  · λ0 V 3 

k +

k=0

3 

k } · Sq ] · n · eλ·t

k=0

π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · [N (i) − Ns ] · s p · eλ·t − · ·( 2 2 ε0 · n r ·  · λ0 V We define two new global parameters for simplicity 1 , 2 . 1 = [

9 π ·c a·ξ ξ · τin 2 − · ) · a · |Rcv |2 · S (i) ·( p V 2 ε0 · n r2 ·  · λ0 V

−{

3 

k +

k=0

2 =

3 

k } · Sq ]

k=0

π ·c 9 ξ · τin 2 · ) · a · |Rcv |2 · [N (i) − Ns ] ·( 2 2 ε0 · n r ·  · λ0 V

a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 · ([N (i) − Ns ] · S (i) − · ·( p ) 2 ε0 · n r2 ·  · λ0 V

Gp =

− {[N

(i)

− Ns ] ·

3 

k + [N

k=0

+ 1 · n · eλ·t − 2 · s p · eλ·t

(i)

− Ng ] ·

3  k=0

k } · Sq

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

327

a·ξ · {N (i) − N g − b · V · (λ p − λ0 )2 } V π ·c 9 ξ · τin 2 ) · a · |Rcv |2 − · ·( 2 ε0 · n r2 ·  · λ0 V

G p (N ∗ , S ∗p ) =

· ([N (i) − Ns ] · S (i) p ) G p (S p , N ) = G p (N ∗ , S ∗p ) + 1 · n · eλ·t − 2 · s p · eλ·t d Sp a·ξ · N = (G p − G th ) · S p + 2·(λ p −λ0 ) 2 dt V · {[ δλ ] + 1} a · ξ · (N (i) + n · eλ·t )

λ·t s p · λ · eλ·t = (G p (S p , N ) − G th ) · (S (i) p + sp · e ) +

V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

s p · λ · eλ·t = (G p (N ∗ , S ∗p ) + 1 · n · eλ·t − 2 · s p · eλ·t − G th ) λ·t · (S (i) p + sp · e ) +

a · ξ · (N (i) + n · eλ·t ) V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

λ·t s p · λ · eλ·t = ([G p (N ∗ , S ∗p ) − G th ] + 1 · n · eλ·t − 2 · s p · eλ·t ) · (S (i) p + sp · e )

+

a · ξ · N (i) V·

2·(λ p −λ0 ) 2 {[ δλ ]

+ 1}

+

a · ξ · n · eλ·t V · {[

2·(λ p −λ0 ) 2 ] δλ

+ 1}

(i) λ·t λ·t s p · λ · eλ·t = [G p (N ∗ , S ∗p ) − G th ] · S (i) − 2 · S (i) p + 1 · S p · n · e p · sp · e ∗ ∗ λ·t 2·λ·t 2 2·λ·t +[G p (N , S p ) − G th ] · s p · e + 1 · n · s p · e − 2 · [s p ] · e a·ξ ·N (i) a·ξ ·n·eλ·t + 2·(λ p −λ0 ) 2 + 2·(λ p −λ0 ) 2 V ·{[

δλ

] +1}

V ·{[

δλ

] +1}

a · ξ · N (i)

s p · λ · eλ·t = [G p (N ∗ , S ∗p ) − G th ] · S (i) p + + 1 ·

S (i) p ∗

·n·e

+ [G p (N ,

S ∗p )

λ·t

− 2 ·

V · {[

S (i) p · sp λ·t

2·(λ p −λ0 ) 2 ] δλ λ·t

+ 1}

·e

− G th ] · s p · e

− 2 · [s p ]2 · e2·λ·t

+ 1 · n · s p · e2·λ·t a·ξ + · n · eλ·t 2·(λ p −λ0 ) 2 V · {[ δλ ] + 1}

At fixed points [G p (N ∗ , S ∗p ) − G th ] · S (i) p +

a·ξ ·N (i)

V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

=0

λ·t λ·t s p · λ · eλ·t = 1 · S (i) − 2 · S (i) p ·n·e p · sp · e

+ [G p (N ∗ , S ∗p ) − G th ] · s p · eλ·t + 1 · n · s p · e2·λ·t

328

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

− 2 · [s p ]2 · e2·λ·t +

a·ξ V·

2·(λ p −λ0 ) 2 {[ δλ ]

+ 1}

· n · eλ·t

Assumption: n · s p ≈ 0; [s p ]2 ≈ 0 λ·t λ·t − 2 · S (i) s p · λ · eλ·t = 1 · S (i) p ·n·e p · sp · e

+ [G p (N ∗ , S ∗p ) − G th ] · s p · eλ·t a·ξ + · n · eλ·t 2·(λ p −λ0 ) 2 V · {[ δλ ] + 1} By dividing the two sides of the above equation by eλ·t term we get. − s p · λ + [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] · sp a·ξ +[ + 1 · S (i) p ]·n =0 2·(λ p −λ0 ) 2 V · {[ δλ ] + 1} dN N I = −0 − + dt τs e N (i) + n · eλ·t I + n · λ · eλ·t = −0 − τs e n · eλ·t N (i) I + − n · λ · eλ·t = −0 − τs e τs (i)

At fixed points−0 − Nτs + eI = 0 then −n · λ − τns = 0. We summary our long-wavelength semiconductor laser rate small increments equations: − s p · λ + [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] · sp a·ξ +[ + 1 · S (i) p ]·n =0 2·(λ p −λ0 ) 2 V · {[ δλ ] + 1} n =0 −n·λ− τs !

−λ + [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] [ ⎛ (A − λ · I ) = ⎝

a·ξ

V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

−λ −

0 (i)

−λ + [G p (N ∗ , S ∗p ) − G th − 2 · S p ] [ 0

+ 1 · S (i) p ] 1 τs a·ξ

2·(λ p −λ0 ) 2 V ·{[ ] +1} δλ

−λ −

" ! ·

sp n (i)

"

+ 1 · S p ] 1 τs

det(A − λ · I ) = 0 ⇒ (−λ + [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ]) · (−λ −

=0 ⎞ ⎠

1 )=0 τs

3.3 Mode-Competition Phenomena in Long-Wavelength Lasers Instability in Time

329

The characteristic equation of our system: 1 − [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ]) τs 1 − [G p (N ∗ , S ∗p ) − G th − 2 · S (i) =0 p ]· τs

λ2 + λ · (

1 + [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ]) τs 1 ∗ ∗ + [G th + 2 · S (i) =0 p − G p (N , S p )] · τs

λ2 − λ · (−

! A=

[G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] [ 0

a·ξ

V ·{[

2·(λ p −λ0 ) 2 ] +1} δλ

+ 1 · S (i) p ]

"

− τ1s

We define 1 as Trace (A) and 2 as Det (A) elements for stability analysis. 1 + G p (N ∗ , S ∗p ) − G th − 2 · S (i) p τs 1 ∗ ∗

2 = [G th + 2 · S (i) p − G p (N , S p )] · τs  1 λ1,2 = ( 1 ± 21 − 4 · 2 ) 2

1 = −

1 + [G p (N ∗ , S ∗p ) + G th + 2 · S (i) p ] τs # $ $ (− 1 + G p (N ∗ , S ∗ ) − G th − 2 · S (i) )2 p p + % τs 1 +4 · [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] · τs

−

λ1 =

2 1 + [G p (N ∗ , S ∗p ) + G th + 2 · S (i) p ] τs # $ $ (− 1 + G p (N ∗ , S ∗ ) − G th − 2 · S (i) )2 p p − % τs 1 +4 · [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] · τs −

λ2 =

2

Are the solutions of the quadratic equation in λ, eigenvalues dependent only on the trace and determinant of the matrix A. We need to classify system fixed points. The following formulas exist:

330

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

1 + [G p (N ∗ , S ∗p ) + G th + 2 · S (i) p ] τs # $ $ (− 1 + G p (N ∗ , S ∗ ) − G th − 2 · S (i) )2 p p ± % τs 1 +4 · [G p (N ∗ , S ∗p ) − G th − 2 · S (i) p ] · τs −

λ1,2 =

2

 & 1 ∗ ∗

1 = 2k=1 λk ; 2 = 2k=1 λk . If 2 = [G th + 2 · S (i) p − G p (N , S p )] · τs < 0, the eigenvalues are real and have opposite signs; hence the fixed point is a saddle point. 1 ∗ ∗ If 2 = [G th + 2 · S (i) p − G p (N , S p )] · τs > 0, the eigenvalues are either real with the same sign (nodes), or complex conjugate (spiral and center). Nodes satisfy 2 ∗ ∗

21 −4· 2 > 0, (− τ1s +G p (N ∗ , S ∗p )−G th −2 · S (i) p ) +4·[G p (N , S p )−G th −2 · 1 1 2 (i) ∗ ∗ S p ] · τs > 0 And spirals satisfy 1 − 4 · 2 < 0; (− τs + G p (N , S p ) − G th − 2 · 1 2 2 ∗ ∗ (i) S (i) p ) +4·[G p (N , S p )− G th −2 · S p ]· τs < 0. The parabola 1 −4· 2 = 0 is the borderline between nodes and spirals; star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by 1 . When 1 < 0, both eigenvalues have negative real parts, the fixed point is stable. Unstable spirals and nodes have 1 > 0. Neutrally stable centers live on the borderline 1 = 0, where the eigenvalues are purely imaginary. If 2 = 0 then at least one of the eigenvalues is zero. Then the origin is not an isolated fixed point. There is either a whole line of fixed points, or a plane of fixed points, if A = 0. Saddle points, nodes, and spirals are the major types of fixed points; they occur in large open regions of the ( 1 , 2 ) plane. Centers, stars, degenerate nodes, and non-isolated fixed points are borderline cases that occur along curves in the ( 1 , 2 ) plane [10].

3.4 Questions 1. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. Due to additional external laser source which radiate inside our main Ion-channel path, there is additional element ( · tg(ψ);  ∈ R+ ) in phase = .....). All system parameters are as describe on subchapter change in time ( dψ dt (3.1).   ' ( (2 + qx2 + q y2 ) 1 dψ = kz · c · 1 − · 1 − · A · cos(ψ) − ω + ω β dt 4 · qz2 2 · qy +

qy 1 · k z · c · 2 · A · cos(ψ) +  · tg(ψ) 2 qz

q y2 qy dq y dqz 1 1 1 = − · kz · c · = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) · A · sin(ψ); dt 2 qz dt 2 4 qz q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2

3.4 Questions

331

1.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters. 1.2 Plot the appropriate flow charts that show how you find system fixed points. 1.3 Discuss stability and stability switching for different values of  ;  ∈ R+ parameter.  ) 1.4 Parameter  = |ψ| · ψ 3 , Find system fixed points and discuss stability. ) √ 1.5 We change the additional element to  · tg( ψ · ψ + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ . 2. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. Due to additional interference source which radiate inside our main Ion-channel path, there is additional element ( · sec(ψ);  ∈ R+ ) in qz variable z = .....). All system parameters are as describe on subchapter change in time ( dq dt (3.1). (2 + qx2 + q y2 ) 1 dψ = k z · c · [1 − ] − ω + ωβ · [1 − · A · cos(ψ)] dt 4 · qz2 2 · qy qy 1 + · k z · c · 2 · A · cos(ψ) 2 qz qy 1 dqz = − · kz · c · · A · sin(ψ) +  · sec(ψ) dt 2 qz q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz q y · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2 2.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters. 2.2 Plot the appropriate flow charts that show how you find system fixed points. 2.3 Discuss stability and stability switching for different values of  ;  ∈ R+ parameter.  ) 2.4 Parameter  = |ψ| · ψ 5 , Find system fixed points and discuss stability.  ) 2.5 We change the additional element to  · tg( ψ · ψ 2 + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ .

332

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

3. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. Due to additional interference source which √ radiate inside our main Ion-channel path, there is additional element ( · cosec( ψ);  ∈ R+ ) in qx varix able change in time ( dq = .....). All system parameters are as describe on subchapter dt (3.1). ' ( (2 + qx2 + q y2 ) 1 dψ ] − ω + ωβ · 1 − · A · cos(ψ) = k z · c · [1 − dt 4 · qz2 2 · qy qy 1 + · k z · c · 2 · A · cos(ψ) 2 qz qy 1 dqz = − · kz · c · · A · sin(ψ) dt 2 qz q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz ) q y · qx 1 dqx = − · kz · c · · A · sin(ψ) +  · cosec( ψ)

dt 8 qz2 3.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters. 3.2 Plot the appropriate flow charts that show how you find system fixed points. 3.3 Discuss stability and stability switching for different values of  ;  ∈ R+ parameter. √ 3.4 Parameter  = |ψ| · ψ, Find system fixed points and discuss stability.  ) 3.5 We change the additional element to  · tg( ψ · ψ 3 + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ . 4. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. The mechanism is which a laser beam propagating through un-dense plasma produces a positively charged ion channels by expelling plasma electrons in the transverse direction. Due to interference source which radiate inside our main √ Ion-channel path, there is additional element ( ·sec( ψ);  ∈ R+ ) in qx variable x = .....). Additionally, the mechanism is not ideal and there is change in time ( dq dt qx variable delay in time, qx (t) → qx (t − τ ) but the derivative of the qx (t) in time ( dqdtx (t) ) is not effected. We can define qx = qx (t); q y = q y (t); qz = qz (t).   (2 + qx2 (t − τ ) + q y2 ) dψ 1 · A · cos(ψ)] − ω + ωβ · [1 − = kz · c · 1 − dt 4 · qz2 2 · qy

3.4 Questions

+

333

qy 1 · k z · c · 2 · A · cos(ψ) 2 qz qy 1 dqz = − · kz · c · · A · sin(ψ) dt 2 qz q y2 dq y 1 1 = − · (ωβ + · k z · c · 2 ) · A · sin(ψ) dt 2 4 qz ) q y · qx (t − τ ) 1 dqx = − · kz · c · · A · sin(ψ) +  · sec( ψ) 2 dt 8 qz

4.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters. 4.2 Plot the appropriate flow charts that show how you find system fixed points. 4.3 Discuss stability and stability switching for different values of τ ; τ ∈ R+ parameter. √ 4.4 Parameter  = |ψ| · ψ, Find system fixed points and discuss stability for different values of τ ; τ ∈ R+ .  ) 4.5 We change the additional element to  · tg( ψ 2 · ψ 3 + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ (parameter τ ; τ ∈ R+ is fix). 5. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. The mechanism is which a laser beam propagating through un-dense plasma produces a positively charged ion channels by expelling plasma electrons in the transverse direction. Due to interference source ) which √ radiate inside our main Ion-channel path, there is additional element ( · sec( ψ);  ∈ R+ ) in qz variz = .....). Additionally, the mechanism is not ideal and there able change in time ( dq dt is qz variable delay in time, qz (t) → qz (t − τ ) but the derivative of the qz (t) in time ( dqdtz (t) ) is not effected. We can define qx =qx (t); q y =q y (t); qz = qz (t). (2 + qx2 + q y2 ) 1 dψ · A · cos(ψ)] = k z · c · [1 − ] − ω + ωβ · [1 − dt 4 · qz2 (t − τ ) 2 · qy qy 1 · A · cos(ψ) + · kz · c · 2 2 qz (t − τ )  ) qy 1 dqz = − · kz · c · · A · sin(ψ) +  · sec( ψ) dt 2 qz (t − τ ) ! " q y2 dq y 1 1 = − · ωβ + · k z · c · 2 · A · sin(ψ) dt 2 4 qz (t − τ )

334

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

q y · qx 1 dqx = − · kz · c · 2 · A · sin(ψ) dt 8 qz (t − τ ) 5.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters. 5.2 Plot the appropriate flow charts that show how you find system fixed points. 5.3 Discuss stability and stability switching for different values of τ ; τ ∈ R+ parameter. ) √ 5.4 Parameter  = |ψ| · 3 ψ, Find system fixed points and discuss stability for different values of τ ; τ ∈ R+ .  ) 5.5 We change the additional element to  · cos( ψ · ψ 3 + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ (parameter τ ; τ ∈ R+ is fix). 6. We have Ion-channel Laser system that is characterized by set of perturbed differential equations. The mechanism is which a laser beam propagating through un-dense plasma produces a positively charged ion channels by expelling plasma electrons in the transverse direction. Due to interference source)which radiate inside our main √ Ion-channel path, there is additional element ( · cosec( ψ);  ∈ R+ ) in q y dq variable change in time ( dty = .....). Additionally, the mechanism is not ideal and there is qz variable delay in time, q y (t) → q y (t − τ ) but the derivative of the q y (t) dq (t) in time ( dty ) is not effected. We can define qx = qx (t); q y =q y (t); qz = qz (t).   (2 + qx2 + q y2 (t − τ )) dψ = kz · c · 1 − dt 4 · qz2 1 · A · cos(ψ)] 2 · q y (t − τ ) q y (t − τ ) 1 + · kz · c · · A · cos(ψ) 2 qz2

− ω + ωβ · [1 −

q y (t − τ ) 1 dqz = − · kz · c · · A · sin(ψ) dt 2 qz q y (t − τ ) · qx 1 dqx = − · kz · c · · A · sin(ψ) dt 8 qz2 ! "  ) q y2 (t − τ ) dq y 1 1 = − · ωβ + · k z · c · · cosec( ψ) · A · sin(ψ) + 

2 dt 2 4 qz 6.1 Find system fixed points; plot the possible graphs that show how they change for different values of system parameters.

3.4 Questions

335

6.2 Plot the appropriate flow charts that show how you find system fixed points. 6.3 Discuss stability and stability switching for different values of τ ; τ ∈ R+ parameter. ) √ 6.4 Parameter  = |ψ| · 5 ψ, Find system fixed points and discuss stability for different values of τ ; τ ∈ R+ .  ) 6.5 We change the additional element to  · cos( ψ · ψ 2 + 1);  ∈ R+ , find System fixed points and discuss stability and stability switching for different values of  ;  ∈ R+ (parameter τ ; τ ∈ R+ is fix). 7. The simple case of two-mode competition describes the characterization of the mode-competition phenomena with the gain saturation effects. We describe the effect of the Langevin noise sources on the mode competition phenomena by considering the two modes p and q as follow. d Sp N = [A p − B · S p − {D p(q) + H p(q) } · Sq − G th ] · S p + C p · + F p (t) dt τs d Sq N = [Aq − B · Sq − {Dq( p) + Hq( p) } · S p − G th ] · Sq + Cq · + Fq (t) dt τs   dN N I =− A p · Sp − Aq · Sq − + + FN (t) dt τs e p q where λq > λ p and there is a competition phenomenon between the two modes. We consider F p (t) → ε; Fq (t) → ε(neglected functions), FN (t) → ε all above differential equation’s parameters are as describe in (3.3). The assumptions are the same as (3.3) for D p(q) , H p(q) and Dq( p) , Hq( p) respectively. 7.1 Find system fixed points and Basin of Attraction (BOA) for each fixed point. 7.2 Discuss the case when the injection current I is far above the threshold current level Ith (a bi-stable state), and when the injection current I is near to Ith (a near-threshold stable multimode state). 7.3 Discuss the effect of the asymmetric gain saturation in our system. 7.4 Discuss stability and stability switching for different values of system. parameters. 8. The simple case of two-mode competition describes the characterization of the mode-competition phenomena with the gain saturation effects. We describe the effect of the Langevin noise sources on the mode competition phenomena by considering the two modes p and q as follow. The two-mode competition mechanism is not ideal

336

3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

and there is a delay in time on S p , Sq variables, S p (t) → S p (t − τ1 ), Sq (t) → Sq (t − τ2 ) but the derivatives are not affected. d Sp N + F p (t) = [A p − B · S p (t − τ1 ) − {D p(q) + H p(q) } · Sq (t − τ2 ) − G th ] · S p (t − τ1 ) + C p · dt τs d Sq N = [Aq − B · Sq (t − τ2 ) − {Dq( p) + Hq( p) } · S p (t − τ1 ) − G th ] · Sq (t − τ2 ) + Cq · + Fq (t) dt τs

  dN N I A p · S p (t − τ1 ) − Aq · Sq (t − τ2 ) − + + FN (t) =− dt τs e p q where λq > λ p and there is a competition phenomenon between the two modes. We consider F p (t) → ε; Fq (t) → ε(neglected functions), FN (t) → ε all above differential equation’s parameters are as describe in (3.3). The assumptions are the same as (3.3) for D p(q) , H p(q) and Dq( p) , Hq( p) respectively. 8.1 Find system fixed points and Basin of Attraction (BOA) for each fixed point. 8.2 Discuss the stability and stability switching for different values of τ ; Case A τ1 = τ ; τ2 = 0. 8.3 Discuss the stability and stability switching for different values of τ ; Case B τ1 = 0; τ2 = τ . 8.4 Discuss the stability and stability switching for different values of τ ; Case C τ1 = τ ; τ2 = τ . 9. We have Ion-Channel Laser (ICL) system which relies on the injection of a relativistic electron beam in an Ion-Channel (IC) to create a coherent and highly amplified radiation source. The Ion-Channel (IC) can be produced in a plasma based wake field accelerator in the blowout or bubble regime. While propagating in plasma, a laser pulse or a particle beam pushes the electrons off-axis and lets an Ion-Channel in its wake. Analysis beam assumptions: No.1, Pierce parameter ρ is much smaller than one (ρ  1), FEL-like amplification, No.2, In an ICL the wiggler parameter K depends on the electron properties; it is different value for each electron. The beam energy spread and beam Wiggler parameter spread are limited and satisfy: 2

γ < 23 · ρ; K < ( 2+K ) · ρ; γ - Lorentz factor,K —Wiggler parameter,ρ—Pierce γ K 2·K 2 parameter, γη —Electron Lorentz factor after its interaction with the wave, z—Longitudinal position of the electron averaged over one betatron oscillation. The interaction between the electron beam and the radiation can lead to the amplification of the radiation and the interaction with the wave leads to the following DDEs (Delay Differential Equations) of motion for the electron in the (φ, η) phase space. τ1 and τ2 time delay parameters are due to non- ideal interaction with the wave process. dφ = dt



4 + K2 8 · γ02

 · η(t − τ1 );

A1 · K · [J J ] dη = · cos[φ(t − τ2 ) + ψ1 ] dt 2 · γ02

3.4 Questions

337

 [J J ] = J0 ·

K2 4 + 2 · K2



 − J1 ·

K2 4 + 2 · K2



J0 and J1 are the Bessel functions. Assumption: J0 and J1 Bessel functions are constant for our system. The beam of electrons is bunched by the Electro-Magnetic (EM) wave at the phase φ = −ψ1 + π2 + 2 · m · π ; m is integer which leads to a bunching at the position r = r0 · sin(k1 · z − ω1 · t + ψ1 ). φ—Electron phase, η— Relative electron in the EM wave, A1 and ψ1 are respectively the wave amplitude and phase. The electron oscillates in the (x, z) plane. K —Wiggler parameter. You γ −γ γη . Hint: In an Ion-Channel Laser (ICL), need to consider that η = ηγ0 0 ⇒ γ0 = η+1 the K parameter depends on r0 and γ0 which can be different for each electron and the radiation wavelength spread can be induced by both the beam energy spread and K spread. 9.1 Find ICL system fixed points (φ ∗ , η∗ ) and plot them as a function of K parameter. 9.2 Discuss stability and stability switching of our system for different values of τ parameter, Cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 9.3 There is a special case that parameter K (Wiggler parameter) is dependent on φ variable according the following function, K = ·sec(φ);  ∈ R+ . Find system fixed points and plot the graphs (φ ∗ , η∗ ) for different value of  parameter. Discuss Stability and Stability Switching (9.3), −1 (τ ); sign−1 (τ ) for the Following Cases: (1) τ1 = τ ; τ2 = 0(2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 10. In an ICL, K is a function of γ , K = K (γ ). K is a function of time. We define K 0 and the time-dependent longitudinal momentum pz and the maximum radius rm such  that K 0 = K (t = 0); pz (t = 0) = p0 ; rm (t = 0) = r0 ; γ0 = 1 + p02 . In presence of an EM wave, the energy and momentum change of an electron following betatron motion in the (x, z) plan is given by the following set of differential equations: K · (2 + K 2 ) dγ = βr · α − βr · · cos(θr ) dt 4 · γ2 dpr K · (2 + K 2 ) = (1 − βz ) · α − · cos(θr ) dt 4 · γ2 1 + K2 dK dpz =( = βr · α ) · α · sin(θr ); dt 2 · γ2 dt where α = A1 · sin(k1 · z − ω1 · t + ψ1 ).A1 and ψ1 are respectively the wave amplitude and phase. Hint: Betatron, a type of particle accelerator that uses the electric field induced by a varying magnetic field to accelerate electrons (beta particles) to high speeds in a

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3 Ion Channel and Long-Wavelength Lasers, Nonlinearity Applications in Engineering

circular orbit. … Modern compact betatron designs are used to produce high-energy X-ray beams for a variety of applications. βr and βz are the normalized transverse and longitudinal electron velocities (βr = vr ; βz = vcz ).vr and vz are the transverse and longitudinal electron velocities in c transverse and z direction respectively. r = rm · cos(θr ); pr = K · sin(θr ).   K = rm · k p · γ2 ; K = pr2 + γ2 · r 2 · k 2p . γ —Lorentz factor,K —Wiggler parameter,θr —Transverse angle of electron velocity vector.

10.1 Find system fixed points (equilibrium points) and plot fixed points (γ ∗ , pr∗ , K ∗ ) graphs for different values of βr , βz parameters (normalized transverse and longitudinal electron velocities). 10.2 Discuss stability of our system and plot the relevant graphs. 10.3 Due to some interference we get delay in time (τ ) of K parameter, K (t) → K (t − τ ). The derivative of K in time ( ddtK ) is not effected by τ parameter. Discuss stability and stability switching for different values of τ parameter. 10.4 Unexpected event happened in our system. It is represented by additional = .... differential equation ( dγ = .... +  · cosec(θr ),  ∈ R+ . element to dγ dt dt Find system fixed points and discuss stability and stability switching for parameters variation.

References 1. D.H. Whittum, A.M. Sessler, The Ion-Channel Laser, Dept. of Physics, Lawrence Berkeley Lab, Berkeley, California 2. M. Yamada, W. Ishimori, H. Sakaguchi, M. Ahmed, Time-dependent measurement of the Mode-competition phenomena among longitudinal modes in Long-wavelength lasers. IEEE J. Quantum Electronics 39(12) (2003) 3. M. Ahmed, M. Yamada, Influence of Instantaneous mode competition on the dynamics of semiconductor lasers. IEEE J. Quantum Electronics 38(6) (2002) 4. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993). 5. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay Differential systems with delay dependent parameters. SIAMJ. Math. Anal. 33, 1144–1165 (2002) 6. B. Balachandran, T. Kalmár-Nagy, D.E. Gilsinn, Delay Differential Equations: Recent Advances and New Directions (Hardcover). Springer, 1 edn (March 5, 2009) 7. M. Yamada, Theory of mode competition noise in semiconductor injection laser. IEEE J. Quantum Electronics QE-22(7) (1986) 8. Y.A. Kuznetsov, Elelments of applied bifurcation theory, in Applied Mathematical Sciences 9. J.K. Hale. Dynamics and bifurcations, in Texts in Applied Mathematics, vol. 3 10. S.H. Strogatz, Nonlinear Dynamics and Chaos. Westview Press 11. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, in Text in Applied Mathematics (Hardcover)

Chapter 4

Solid State Laser Nonlinearity Applications in Engineering

There are many different types of lasers. The laser medium can be solid, gas, liquid or semiconductor. A solid-state laser is a laser that uses a gain medium that is a solid. It is very different from liquid laser such as in dye lasers or a gas lasers. Lasers are commonly designed by the type of lasing material employed: solid-state lasers have lasing material distributed in a solid matrix such as the ruby or neodymium “YAG” lasers. Solid-state lasers use optical pumping and such pump sources are relatively cheap and can provide very high powers. It has low power efficiency, moderate lifetime, and strong thermal effects such as thermal lensing in the gain medium. Laser diodes are very often used for pumping solid-state lasers. Lasers use crystals like Nd:YAG crystal. The Neodymium doped Yttrium Aluminum Garnet-Nd:YAG laser crystal is the most popular lasing media for solid-state lasers. The gain medium in solid-state laser is a solid rather than a liquid laser. Semiconductor-based lasers are also in the solid-state, but are generally considered as a separate class from solidstate lasers. The solid-state lasers gain media can be crystals or glasses doped with rare earth or transition metal ions, or semiconductor lasers. Ion-doped solid-state lasers (Doped insulator lasers) can be made in the form of bulk lasers, fiber lasers or other types of waveguide lasers. Solid-state lasers may generate output powers between a few mill-watts and in high-power versions many kilowatts. Solid-state lasers are optically pumped with flash lamps or arc lamps. Such pump sources are good high power supply and lead to a low power efficiency, moderate lifetime, and strong thermal effects such as thermal lensing in the gain medium. Laser diodes are frequently used for pumping solid-state lasers. Diode-pumped solid-state lasers (DPSS lasers, all-solid-state lasers) have advantages as compact setup, long lifetime, and good beam quality. The laser transitions of rare-earth or transition-metal-doped crystal or glasses are normally weakly allowed transitions. The transitions are with very low oscillator strength, which leads to long radiative upper-state lifetime and consequently to good energy storage, with upper-state life-times of microseconds to milliseconds. The long upper-state lifetimes makes solid-state lasers very suitable for Q switching: the laser crystal can easily store an amount of energy which, when released in the form of a nanosecond pulse, leads to a peak power which is orders of © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_4

339

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4 Solid State Laser Nonlinearity Applications in Engineering

magnitude above the achievable average power. In mode-locked operation, solid-state lasers can generate ultrashort pulses with durations measured in picoseconds, with passive mode locking. They have a tendency for Q-switching instabilities, if there are not suppressed with suitable measures. Types of solid-state lasers: Small diodepumped Nd:YAG (YAG lasers) or Nd:YVO4 lasers (vanadate lasers) and Q-switched versions generate pulses with durations of a few nanoseconds, micro-joule pulse energies and peak powers of many kilowatts. Other types of solid-state laser are unidirectional ring lasers (single-frequency operation) or microchip lasers which allows for operation with very small linewidth in the lower kilohertz region. Larger lasers in side-pumped or end-pumped configurations, having the geometry of rod lasers, slab lasers, or thin-disk lasers, are suitable for output powers up to several kilo-watts. Particularly thin-disk lasers offer very high beam quality and a high power efficiency. An appropriately designed semiconductor saturate absorber device, the anti-resonant Fabry–Perot saturable absorber, can reliably start and sustain stable mode locking of solid state lasers such as Nd:YAG, Nd:YLF, Nd:Glass, Cr:LiSAF, and Ti:sapphire lasers. Solid-state lasers with long upper-state lifetimes, previous attempts to produce self-starting passive mode locking with saturate absorbers was accompanied by self Q-switching. There is a criteria that characterize the dynamic behavior of solidstate lasers in the important regimes of Q-switching, mode-locked Q-switching, and continuous-wave mode locking in the picosecond and femtosecond range for the pulse width. The rate equations of the laser are presented and the dynamics is analyze, semiconductor devices. By using laser-diode-pumped microchip solid-state laser we get nanometer vibration analysis of a target. It has been demonstrated by a self-aligned optical feedback vibrometry technique. The laser output waveform, which modulated through interference between a lasing field and an extremely week frequency-modulated (FM) feedback field is analyzed by the Hilbert transformation to yield the vibration waveform of the target. A simulation and analysis is done on the model differential equations, that is, laser with frequency-modulated optical feedback system. Light-injection-induced phenomena in laser, such as injection locking and return-light-induced instabilities, are profound interest because of the basic properties of laser dynamics as well as their practical importance. In Nd stoichiometric lasers the resonant excitation of periodic oscillations and chaotic oscillation in relaxation oscillation frequency regions, resulting from Dopper-shifted light injection from a moving light-scattering object. The phenomenon is the injection-induced modulation of lasers. The application of this phenomenon is to ultra-high sensitivity Dopper-shift measurements in wide velocity regions. A theoretical model that includes multimode effect is inspected and analyze for stability. The light-injection model differential equations for rotating-wave approximation fields are given by introducing injection field into spatially-hole-burned N-mode lasers. A novel scheme that combines gain switching with passive Q switching of a miniature diode-pumped solid-state laser is analyze and simulate for stability inspection. A composite pumping pulse, consisting of a long, low-intensity pulse and a short, high-intensity pulse, is used to reduce the timing Jitter. There is a simple model of a , Q-switched laser. The effective pumping rate of the upper laser level is Pe f f = h·vη·P L ·V where P, the electrical power input; η is the pumping efficiency, which is the product

4 Solid State Laser Nonlinearity Applications in Engineering

341

of laser-diode efficiency and optical coupling efficiency; v L is the laser frequency; V is the laser mode volume, and h is the Planck constant. The differential equations of the population density of the upper laser level, NG , and the photon density in the cavity are presented and analyze by using nonlinear dynamics for stability and stability switching under system parameter’s variation [1–4].

4.1 Solid State Laser Controlled by Semiconductor Devices Stability Analysis High power diode lasers array can be utilized to pump solid-state laser materials efficiently. Laser materials with high intrinsic quantum efficiencies and an improved scalable cavity design make high output powers. The powerful and compact laser sources have fundamental research application such as X-ray, plasma, and higher-harmonic generation, etc. The target is to get high laser peak power and ultrashort optical pulses and control the laser dynamics by various Q-switching and continuous-wave (CW) mode locking techniques. It is controlling the laser dynamics with compact scalable passive semiconductor devices. The semiconductor device is a saturate absorber. A saturate absorber can be used to passively Q-switch or mode-lock laser. Third regime is laser which is Q-switch-mode-locked. Q-switching is a technique for obtaining energetic short pulses from laser by modulating the intra-cavity losses and thus the Q-factor of the laser resonator. It generates nanosecond pulses of high energy and peak power with solid-state bulk lasers. Q-switch pulse generation: (1) the resonator losses are kept at a high level and the energy fed into the gain medium by the pumping mechanism accumulates there. The stored energy can be a multiple of the saturation energy (2) the losses are reduced to a small value, and the power of the laser radiation builds up in the laser resonator (3) the temporary integrated intra-cavity power reaches the order of the saturation energy of the gain medium and the gain starts to be saturated. When the gain equals the remaining (low) resonator losses then the peak of the pulse is reached. The large intra-cavity power is presented and there is a depletion of the stored energy during the time where the power decays. The pulse duration achieved with Q-switching is typically in the nanosecond range and above the resonator round-trip time. In the most cases, Q-switched lasers generate regular pulse trains via repetitive Q-switching. Basically Q-switching (giant pulse formation, Q-spooling) is a technique by which a laser can be made to produce a pulsed output beam. The production of light pulses is with high peak power (higher than if it be produced by the same laser operating in continuous wave mode). Compare to mode locking, Q-switching leads to much lower pulse repetition rates, higher pulse energies and longer pulse duration. The solid-state lasers (Nd:YAG or Nd:YLF) can be passively CW mode locked using saturate absorbers without Q-switching or at least Q-switched mode locking. The parameters of saturate absorber are recovery time of the absorption and the saturation energy, pure passive Q-switching or pure CW mode locking, Q-switched mode locking. There is a full control over the saturate absorber

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4 Solid State Laser Nonlinearity Applications in Engineering

device parameters and well-matched optimum values for a given solid-state laser material. Semiconductor absorbers have an intrinsic bit emporia pulse response, and the scales being given by intra band carrier-carrier scattering and thermal process. The absorber’s performances are ideal for the mode locking of solid-state lasers in the femtosecond and picosecond regime. The basic laser models describe the dynamic of solid-state laser. Based on the feature of the laser dynamics, there are criteria that characterize the laser dynamics with respect to Q-switching, mode-locked Q-switching, and self-starting of mode locking by a saturate absorber. A saturate absorber is an optical component with a certain optical loss, which is reduced at high optical intensities. The reduction can occur in a medium with absorbing dopant ions, when a strong optical intensity leads to depletion of the ground state of these ions. Artificial saturate absorbers are where there is no real absorption, but an optical loss which decreases for increasing optical power. The main applications of saturate absorbers are passive mode locking and Q-switching of lasers. Saturate absorbers are also useful for purposes of nonlinear filtering outside laser resonators like cleaning up pulse shapes, and in optical signal processing. The saturate absorbers with relaxation times (picosecond range) possibly generate stable picosecond pulses if the gain bandwidth is broad enough. A picosecond absorber can generate femtosecond pulses of soliton like pulse shaping is additionally employed. Additionally there is a soliton mode locking by a slow saturate absorber. This mode locking technique does not impose any requirements on the cavity design or operation of the cavity close to its stability limit. The soliton mode locking is a stable phenomenon, which allows for a scalable resonator design. The semiconductor elements with the required dynamical behavior can be fabricated due to the fact of existing Molecular Beam Epitaxy (MBE) technology. Molecular beam epitaxy (MBE) is an evaporation process performed in an ultra-high vacuum for the deposition of compounds of extreme regularity of layer thickness and composition from well-controlled deposition rates. We can explain molecular beam epitaxy (MBE) as an atomic layer by atomic layer crystal growth technique, based on reaction of molecular or atomic beams with a heated crystalline substrate, performed in an ultra-high vacuum (UHV) environment. The term “molecular beam” describes a unidirectional kinematic flow of atoms or molecules with no collisions among them, as opposed to a viscous, fluid-like flow. The term “epitaxy” is akin or upon, and taxi-meaning arrangement or order. Epitaxy refers to ordered growth of one crystalline layer on another crystalline layer, with the same or related crystal arrangement. The semiconductors tend to have optical loss, low a saturation intensity, and low a damage threshold for typical solid-state lasers then some semiconductor material parameters are not well matched to the characteristics required for solid-state lasers. Anti-resonant Fabry–Perot saturate absorber (A-FPSA) integrates the semiconductor saturate absorber inside a Fabry–Perot cavity operated at anti resonance with a free spectral range larger than the gain bandwidth of the solid-state laser. This can be possible solution to unmatched parameters of the semiconductor material. We can investigate and inspect the dynamics of the absorber within a simple two-level rate equation model. A large variety of dynamical effects occurring in a laser is sufficient to model the semiconductor absorber as an equivalent two-level

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

343

P+

P-

g , τ L , EL

q,τ A , E A

Aeff , L

Aeff , A

Tout

Fig. 4.1 Solid-state laser by semiconductor devices model corresponding to the rate Differential equation

system. We can then determine and modify the key parameters of the equivalent twolevel model and get a stable system. There are a system parameter variation ranges for CW Q-switching and CW mode locking and extension to the case of Q-switched mode locking [1]. The simplest laser model of a single-mode laser with homogeneously laser medium and saturate absorber is as shown in the next figure (Fig. 4.1). Tout —Small output coupling (laser power within one round trip does not change very much), P + , P − —Twice circulating power (related field density), neglecting standing-wave effects in the cavity [1]. Model assumptions: (1) the transverse relaxation times of the equivalent two-level models for the laser gain medium and for the saturate absorber are much faster than any other dynamics in the system. The outcome is that the model can be described by the rate equations to describe the laser dynamic (2) the changes in the laser intensity, gain, and saturate absorption on a time scale of the order of the round-trip time (TR ) in the cavity are small [1]. 2 dP = · (g − l − q) · P dt TR (g − g0 ) g · P dq (q − q0 ) q · P dg =− =− − ; − dt TL · T R E L dt T A · TR EA P = P(t); g = g(t); q = q(t) TR —Round-trip time in cavity, P—Laser power, g—Gain per round trip, l— Linear losses per round trip, q—Saturate losses per round trip, g0 —Small-signal gain per round trip, q0 —Un saturate but saturate losses per round trip, TL —Upper state lifetime of the gain medium (normalized to the round trip time of the cavity); TL = TτLR , T A —Absorber recovery time (normalized to the round trip time of the cavity); T A = Tτ AR , τ L —Upper state lifetime of the gain medium, τ A —Absorber recovery time, E L —Saturation energy of the gain; E L = energy of the absorber; E A =

h·ν·Ae f f,A . 2·σ A

h·ν·Ae f f,L , 2·σ L

E A —Saturation

Remark (1): the factor of 2 in the definition of the saturation energies is due to averaging over the standing wave effects in a linear cavity.

344

4 Solid State Laser Nonlinearity Applications in Engineering

Remark (2): For a ring cavity or if the laser is in pulsed operation and the media are much shorter than the equivalent pulse length, then the factor in the definition of the saturation energies is one. The laser model of a single-mode laser with homogeneously broadband laser medium and saturate absorber is not ideal and there are interferences. We present these interferences by time delay parameters τ1 , τ2 , τ3 for laser power, gain per round trip and saturate losses per round trip in time respectively. We can write our system Delay Differential Equations (DDEs): P(t) → P(t −τ1 ); g(t) → g(t −τ2 ); q(t) → q(t − τ3 ) dP 2 · [g(t − τ2 ) − l − q(t − τ3 )] · P(t − τ1 ) = dt TR dg (g(t − τ2 ) − g0 ) g(t − τ2 ) · P(t − τ1 ) − =− dt TL · T R EL (q(t − τ3 ) − q0 ) q(t − τ3 ) · P(t − τ1 ) dq =− − dt T A · TR EA We consider that time delay parameters τ1 , τ2 , τ3 do not affect the laser power,  , dq . gain per round trip and saturate losses per round trip derivatives in time ddtP , dg dt dt We have possible seven cases for system stability analysis: (1) τ1 = τ ; τ2 = 0; τ3 = 0 (2) τ1 = 0; τ2 = τ ; τ3 = 0 (3) τ1 = τ ; τ2 = τ ; τ3 = 0 (4) τ1 = 0; τ2 = 0; τ3 = τ (5) (6) τ1 = 0; τ2 = τ ; τ3 = τ (7) τ1 = τ ; τ2 = τ ; τ3 = τ . We analyze the stability of our system for the first case τ1 = τ ; τ2 = 0; τ3 = 0. We get the following system DDEs: 2 dP = · [g(t) − l − q(t)] · P(t − τ ); dt TR (g(t) − g0 ) g(t) · P(t − τ ) dg =− − dt TL · T R EL (q(t) − q0 ) q(t) · P(t − τ ) dq =− − dt T A · TR EA At fixed points (equilibrium points): τ ) =P(t)

dP dt

= 0; dg = 0; dq = 0; limt→∞ P(t − dt dt

2 (g ∗ − g0 ) g ∗ · P ∗ · [g ∗ − l − q ∗ ] · P ∗ = 0; − − =0 TR TL · T R EL (q ∗ − q0 ) q ∗ · P ∗ − − =0 T A · TR EA Case (1):

2 TR

Case (1.1):

· [g ∗ − l − q ∗ ] · P ∗ = 0.

2 TR

∗

∗

0) 0) · [g ∗ − l − q ∗ ] = 0; P ∗ = 0 ten we get − (gTL−g = 0; − (qTA−q =0 ·TR ·TR

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

345

TL · TR = 0; T A · TR = 0; g ∗ − g0 = 0 ⇒ g ∗ = g0 ; q ∗ − q0 = 0 ⇒ q ∗ = q0 The first fixed point: E (0) (P (0) , g (0) , q (0) ) = (0, g0 , q0 ). Case (1.2): T2R · [g ∗ − l − q ∗ ] = 0; P ∗ = 0; TR = 0 then g ∗ = l + q ∗ (l + q ∗ − g0 ) (l + q ∗ ) · P ∗ (q ∗ − q0 ) q ∗ · P ∗ − = 0 (∗∗) − − =0 TL · T R EL T A · TR EA  ∗  (q − q0 ) · E A + q ∗ · P ∗ · T A · TR (q ∗ − q0 ) q ∗ · P ∗ (∗∗) → − − =0⇒− T A · TR EA T A · TR · E A = 0; T A · TR · E A = 0

(∗) −

(q ∗ − q0 ) · E A + q ∗ · P ∗ · T A · TR = 0 ⇒ q ∗ · (E A + P ∗ · T A · TR ) q0 · E A − q0 · E A = 0; q ∗ = E A + P ∗ · T A · TR Submission to equation (∗) : − −

(l + q ∗ − g0 ) (l + q ∗ ) · P ∗ − =0 TL · T R EL

(l +

q0 ·E A E A +P ∗ ·T A ·TR

TL · T R

− g0 )

−

(l +

q0 ·E A E A +P ∗ ·T A ·TR

EL

) · P∗

=0

q0 · E A (l − g0 ) − TL · T R (E A + P ∗ · T A · TR ) · TL · TR   q0 · E A P∗ q0 · E A =0− − · l+ EL E A + P ∗ · T A · TR (E A + P ∗ · T A · TR ) · TL · TR   (l − g0 ) P∗ q0 · E A q0 · E A = − · l+ − EL E A + P ∗ · T A · TR TL · T R (E A + P ∗ · T A · TR ) · TL · TR   P∗ l · (E A + P ∗ · T A · TR ) + q0 · E A q0 · E A (l − g0 ) − · = EL E A + P ∗ · T A · TR TL · TR (E A + P ∗ · T A · TR ) · TL · TR   P∗ l · (E A + P ∗ · T A · TR ) + q0 · E A + · EL E A + P ∗ · T A · TR (g0 − l) q0 · E A · E L + P ∗ · [l · (E A + P ∗ · T A · TR ) + q0 · E A ] · TL · TR = TL · T R (E A + P ∗ · T A · TR ) · TL · TR · E L (g0 − l) = q0 · E A · E L + P ∗ · [l · (E A + P ∗ · T A · TR ) + q0 · E A ] · TL · TR TL · T R (g0 − l) = · (E A + P ∗ · T A · TR ) · TL · TR · E L TL · T R q0 · E A · E L + P ∗ · [l · E A + l · P ∗ · T A · TR + q0 · E A ] · TL · TR = (g0 − l) · (E A + P ∗ · T A · TR ) · E L q0 · E A · E L + P ∗ · [(l + q0 ) · E A + l · P ∗ · T A · TR ] · TL · TR = (g0 − l) · (E A · E L + P ∗ · T A · TR · E L ) q0 · E A · E L + P ∗ · [(l + q0 ) · E A · TL · TR + l · P ∗ · T A · TR · TL · TR ] = (g0 − l) · E A · E L + P ∗ · (g0 − l) · T A · TR · E L −

q0 · E A · E L + P ∗ · (l + q0 ) · E A · TL · TR + (P ∗ )2 · l · T A · TR · TL · TR

346

4 Solid State Laser Nonlinearity Applications in Engineering = (g0 − l) · E A · E L + P ∗ · (g0 − l) · T A · TR · E L (P ∗ )2 · l · T A · TL · TR2 + P ∗ · [(l + q0 ) · E A · TL · TR − (g0 − l) · T A · TR · E L ] + q0 · E A · E L − (g0 − l) · E A · E L = 0

We define for simplicity the following global parameters: 1 = l · T A · TL · TR2 ; 2 = (l + q0 ) · E A · TL · TR − (g0 − l) · T A · TR · E L 3 = q0 · E A · E L − (g0 − l) · E A · E L  −2 ± 22 − 4 · 1 · 3 (P ∗ )2 · 1 + P ∗ · 2 + 3 = 0 ⇒ P ∗ = 2 · 1 P ∗ Define two possible P variable fixed points (P (1) , P (2) ). P (1) =

−2 +



22 − 4 · 1 · 3 2 · 1

; P (2) =

−2 −



22 − 4 · 1 · 3 2 · 1

Then we get two possible q variable fixed points (q (1) , q (2) ). q0 · E A ⇒ E A + P ∗ · T A · TR q0 · E A q0 · E A = ; q (2) = (1) E A + P · T A · TR E A + P (2) · T A · TR q0 · E A = √ −2 + 22 −4·1 ·3 EA + [ ] · T A · TR 2·1

q∗ = q (1) q (1)

q (2) =

EA + [

q0 · E A √ 2

−2 −

2 −4·1 ·3 ] 2·1

· T A · TR

Then we get two possible g variable fixed points (g (1) , g (2) ). g ∗ = l + q ∗ ⇒ g (1) = l + q (1) ; g (2) = l + q (2) ; g (1) q0 · E A q0 · E A =l+ ; g (2) = l + E A + P (1) · T A · TR E A + P (2) · T A · TR q0 · E A   g (1) = l + √ 2 −2 + 2 −4·1 ·3 · T A · TR EA + 2·1 g (2) = l +

 EA +

q0 · E A √ 2

−2 −

2 −4·1 ·3 2·1

 · T A · TR

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

347

The second fixed point: ⎛ E (1) (P (1) , g (1) , q (1) ) = ⎝

−2 +

 22 − 4 · 1 · 3 2 · 1



l+ EA +

 EA +

q0 · E A √ 2

−2 +

2 −4·1 ·3 2·1

q0 · E A √ 2

−2 +

2 −4·1 ·3 2·1

,



, · T A · TR ⎞

 · T A · TR

⎟ ⎟ ⎠

The third fixed point: ⎛ E (2) (P (2) , g (2) , q (2) ) = ⎝

−2 −



22 − 4 · 1 · 3 2 · 1



l+ EA +

 EA +

q0 · E A √ 2

−2 −

2 −4·1 ·3 2·1

q0 · E A √ 2

−2 −

2 −4·1 ·3 2·1

,



, · T A · TR ⎞

 · T A · TR

⎟ ⎟ ⎠

Stability analysis: The standard local stability analysis about any one of the equilibrium points of the Solid-state laser by semiconductor devices model corresponding to the rate differential equation system consists in adding to coordinate [P, g, q] arbitrarily small increments of exponential form [ p, g, q]·eλ·t and retaining the first order terms in P, g, q. The system of three homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables (P, g, q) and variables derivatives with respect to time into system rate equations [5–7]. System fixed values with arbitrarily small increments of exponential form [ p, g, q] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. P(t) = P ( j) + p · eλ·t ; g(t) = g ( j) + g · eλ·t ; q(t) = q ( j) + q · eλ·t ; P(t − τ ) = P ( j) + p · eλ·(t−τ ) d P(t) dg(t) dq(t) = λ · p · eλ·t ; = λ · g · eλ·t ; = λ · q · eλ·t dt dt dt

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4 Solid State Laser Nonlinearity Applications in Engineering

We choose these expressions for ourselves P(t), g(t), q(t) as a small displacement [ p, g, q] from the system fixed points in time t = 0. P(t = 0) = P ( j) + p; g(t = 0) = g ( j) + g; q(t = 0) = q ( j) + q dP 2 · [g(t) − l − q(t)] · P(t − τ ) = dt TR 2 λ · p · eλ·t = · [g ( j) + g · eλ·t − l − (q ( j) + q · eλ·t )] · [P ( j) + p · eλ·(t−τ ) ] TR 2 λ · p · eλ·t = · [(g ( j) − l − q ( j) ) + g · eλ·t − q · eλ·t ] · [P ( j) + p · eλ·(t−τ ) ] TR 2 λ · p · eλ·t = · [(g ( j) − l − q ( j) ) · P ( j) TR + (g ( j) − l − q ( j) ) · p · eλ·(t−τ ) + P ( j) · g · eλ·t + g · p · eλ·(t−τ ) · eλ·t − P ( j) · q · eλ·t − q · p · eλ·(t−τ ) · eλ·t ] Assumption: g · p ≈ 0; q · p ≈ 0 2 · [(g ( j) − l − q ( j) ) · P ( j) + (g ( j) − l − q ( j) ) · p · eλ·(t−τ ) TR + P ( j) · g · eλ·t − P ( j) · q · eλ·t ]λ · p · eλ·t 2 = · [(g ( j) − l − q ( j) ) · P ( j) ] TR 2 + · [(g ( j) − l − q ( j) ) · p · e−λ·τ + P ( j) · g − P ( j) · q] · eλ·t TR

λ · p · eλ·t =

At fixed points: λ · p · eλ·t =

2 TR

· [(g ( j) − l − q ( j) ) · P ( j) ] = 0.

2 · [(g ( j) − l − q ( j) ) · p · e−λ·τ + P ( j) · g − P ( j) · q] · eλ·t TR

Divide the two sides of the above equation by eλ·t term. 2 λ· p = · [(g ( j) − l − q ( j) ) · p · e−λ·τ + P ( j) · g − P ( j) · q] TR   2 2 2 · (g ( j) − l − q ( j) ) · e−λ·τ − λ · p + · P ( j) · g − · P ( j) · q = 0 TR TR TR dg g(t) · P(t − τ ) (g(t) − g0 ) − =− dt TL · T R EL λ · g · eλ·t = −

(g ( j) + g · eλ·t − g0 ) (g ( j) + g · eλ·t ) · [P ( j) + p · eλ·(t−τ ) ] − TL · T R EL

λ · g · eλ·t = −

(g ( j) − g0 ) g · eλ·t − TL · T R TL · T R

4.1 Solid State Laser Controlled by Semiconductor Devices Stability … − λ · g · eλ·t = − −

[g ( j) · P ( j) + g ( j) · p · eλ·(t−τ ) + P ( j) · g · eλ·t + g · p · eλ·(t−τ ) · eλ·t ] EL (g ( j) − g0 ) g ( j) · P ( j) g · eλ·t − − TL · T R EL TL · T R [g ( j) · p · eλ·(t−τ ) + P ( j) · g · eλ·t + g · p · eλ·(t−τ ) · eλ·t ] EL

Assumption: g · p ≈ 0 (g ( j) − g0 ) g ( j) · P ( j) g − − · eλ·t TL · T R EL TL · T R [g ( j) · p · e−λ·τ + P ( j) · g] λ·t − ·e EL

λ · g · eλ·t = −

( j)

−g0 ) At fixed points: − (gTL ·T − R

λ · g · eλ·t = −

g ( j) ·P ( j) EL

= 0.

g [g ( j) · p · e−λ·τ + P ( j) · g] λ·t · eλ·t − ·e TL · T R EL

Divide the two sides of the above equation by eλ·t term. g g ( j) · e−λ·τ P ( j) g ( j) · e−λ·τ − · p− ·g⇒− TL · T R EL EL EL   ( j) 1 P dq · p+ − − −λ ·g =0 TL · T R EL dt

λ·g =−

=−

(q(t) − q0 ) q(t) · P(t − τ ) − λ · q · eλ·t T A · TR EA

=−

(q ( j) + q · eλ·t − q0 ) (q ( j) + q · eλ·t ) · [P ( j) + p · eλ·(t−τ ) ] − T A · TR EA

λ · q · eλ·t = − −

q ( j) − q0 q · eλ·t − T A · TR T A · TR [q ( j) · P ( j) + q ( j) · p · eλ·(t−τ ) + P ( j) · q · eλ·t + q · p · eλ·(t−τ ) · eλ·t ] EA

Assumption: q · p ≈ 0 q ( j) − q0 q · eλ·t − T A · TR T A · TR [q ( j) · P ( j) + q ( j) · p · eλ·(t−τ ) + P ( j) · q · eλ·t ] − EA q ( j) − q0 q · eλ·t q ( j) · P ( j) =− − − T A · TR T A · TR EA

λ · q · eλ·t = −

λ · q · eλ·t

349

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4 Solid State Laser Nonlinearity Applications in Engineering

[q ( j) · p · eλ·(t−τ ) + P ( j) · q · eλ·t ] q ( j) − q0 λ · q · eλ·t = − EA T A · TR ( j) ( j) ( j) −λ·τ ( j) + P · q] λ·t q ·P q [q · p · e − − · eλ·t − ·e EA T A · TR EA −

( j)

−q0 At fixed points: − qTA ·T − R

λ · q · eλ·t = −

q ( j) ·P ( j) EA

= 0.

q [q ( j) · p · e−λ·τ + P ( j) · q] λ·t · eλ·t − ·e T A · TR EA

Divide the two sides of the above equation by eλ·t term. q [q ( j) · p · e−λ·τ + P ( j) · q] q ( j) · e−λ·τ λ·q =− − ⇒− ·p T A · TR EA EA   P ( j) 1 − −λ ·q =0 + − T A · TR EA We can summarize our last results:   2 2 2 ( j) ( j) −λ·τ · (g − l − q ) · e −λ · p+ · P ( j) · g − · P ( j) · q = 0 TR TR TR   g ( j) · e−λ·τ 1 P ( j) − · p+ − − −λ ·g =0 EL TL · T R EL   ( j) −λ·τ q ·e 1 P ( j) − · p+ − − −λ ·q =0 EA T A · TR EA The small increments Jacobian of our system is as follow:

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

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4 Solid State Laser Nonlinearity Applications in Engineering



 2 · (g ( j) − l − q ( j) ) · e−λ·τ − λ TR     P ( j) 1 P ( j) 1 + +λ · + +λ · TL · T R EL T A · TR EA   g ( j) 1 P ( j) 2 · P ( j) · · + + λ · e−λ·τ − TR EL T A · TR EA  ( j)  2 1 P ( j) ( j) q + ·P · · + + λ · e−λ·τ TR EA TL · T R EL   2 det(A − λ · I ) = · (g ( j) − l − q ( j) ) · e−λ·τ − λ · TR       1 1 P ( j) P ( j) +λ · +λ + + TL · T R EL T A · TR EA ( j)  ( j)  g 1 P 2 · e−λ·τ · P ( j) · · + − TR EL T A · TR EA 2 g ( j) − · P ( j) · · λ · e−λ·τ TR EL  ( j)  2 1 P ( j) ( j) q · e−λ·τ + ·P · · + TR EA TL · T R EL 2 q ( j) + · P ( j) · · λ · e−λ·τ TR EA   2 det(A − λ · I ) = · (g ( j) − l − q ( j) ) · e−λ·τ − λ TR     1 P ( j) P ( j) 1 · + + · TL · T R EL T A · TR EA   ( j) ( j)  1 P P 1 2 ·λ+λ + + + + TL · T R T A · TR EA EL   g ( j) 1 P ( j) 2 · e−λ·τ · P ( j) · · + − TR EL T A · TR EA 2 g ( j) − · P ( j) · · λ · e−λ·τ TR EL   2 q ( j) 1 P ( j) · e−λ·τ + · P ( j) · · + TR EA TL · T R EL 2 q ( j) + · P ( j) · · λ · e−λ·τ TR EA det(A − λ · I ) =

#####

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …



 2 · (g ( j) − l − q ( j) ) · e−λ·τ − λ TR     1 P ( j) P ( j) 1 · + + · TL · T R EL T A · TR EA   ( j) ( j)  1 P P 1 2 ·λ+λ + + + + TL · T R T A · TR EA EL     1 2 P ( j) P ( j) 1 · · + + · (g ( j) − l − q ( j) ) · e−λ·τ = TL · T R EL T A · TR EA TR   2 1 1 P ( j) P ( j) ·λ + · (g ( j) − l − q ( j) ) · e−λ·τ · + + + TR TL · T R T A · TR EA EL 2 + · (g ( j) − l − q ( j) ) · e−λ·τ · λ2 TR     1 1 P ( j) P ( j) · −λ· + + TL · T R EL T A · TR EA   1 P ( j) P ( j) 1 · λ2 − λ3 + + + − TL · T R T A · TR EA EL   2 · (g ( j) − l − q ( j) ) · e−λ·τ − λ TR     1 P ( j) P ( j) 1 · + + · TL · T R EL T A · TR EA   ( j) ( j)  1 P P 1 2 )λ + λ + + + + TL · T R T A · TR EA EL     1 2 P ( j) P ( j) 1 · · + + · (g ( j) − l − q ( j) ) = TL · T R EL T A · TR EA TR   1 2 1 P ( j) P ( j) + · (g ( j) − l − q ( j) ) · + + + ·λ TR TL · T R T A · TR EA EL 2 + · (g ( j) − l − q ( j) ) · λ2 } · e−λ·τ TR     1 1 P ( j) P ( j) · −λ· + + TL · T R EL T A · TR EA   1 P ( j) P ( j) 1 · λ2 − λ3 + + + − TL · T R T A · TR EA EL

#####  det(A − λ · I ) =

1 P ( j) + TL · T R EL

  ·

1 P ( j) + T A · TR EA

 ·

2 · (g ( j) − l − q ( j) ) TR

353

354

4 Solid State Laser Nonlinearity Applications in Engineering 

1 1 P ( j) P ( j) + + + TL · T R T A · TR EA EL  2 + · (g ( j) − l − q ( j) ) · λ2 · e−λ·τ TR     1 P ( j) P ( j) 1 −λ· + + · TL · T R EL T A · TR EA   1 1 P ( j) P ( j) − + + + · λ2 − λ3 TL · T R T A · TR EA EL   ( j) 2 1 P ( j) ( j) g − ·P · · + · e−λ·τ TR EL T A · TR EA +

2 · (g ( j) − l − q ( j) ) · TR

 ·λ

2 g ( j) · P ( j) · · λ · e−λ·τ TR EL   ( j) 2 1 P ( j) ( j) q + ·P · · + · e−λ·τ TR EA TL · T R EL −

2 q ( j) + · P ( j) · · λ · e−λ·τ TR EA     1 2 1 P ( j) P ( j) · · det(A − λ · I ) = + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   1 2 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + ·λ TR TL · T R T A · TR EA EL  2 · (g ( j) − l − q ( j) ) · λ2 · e−λ·τ + TR     1 1 P ( j) P ( j) · −λ· + + TL · T R EL T A · TR EA   ( j) ( j) 1 1 P P − + + + · λ2 − λ3 TL · T R T A · TR EA EL    ( j) 2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   2 g ( j) 1 P ( j) − · P ( j) · · + · e−λ·τ TR EL T A · TR EA   ( j) ( j) 2 2 ( j) q ( j) g · λ · e−λ·τ + ·P · − ·P · TR EA TR EL

  1 1 P ( j) · + TL · T R EL T A · TR ( j)  2 q 1 + · P ( j) · · + TR EA TL · T R ( j)  1 2 ( j) g ·P · · + − TR EL T A · TR 

det(A − λ · I ) =

P ( j) EA ( j) 

+

P EL  P ( j) EA

 ·

2 · (g ( j) − l − q ( j) ) TR

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

355

  2 1 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + TR TL · T R T A · TR EA EL  ( j) ( j) q 2 g 2 ·λ · P ( j) · − · P ( j) · + TR EA TR EL 2 + · (g ( j) − l − q ( j) ) · λ2 ] · e−λ·τ TR     1 1 P ( j) P ( j) · −λ· + + TL · T R EL T A · TR EA  ( j) ( j)  1 P P 1 · λ2 − λ3 + + + − TL · T R T A · TR EA EL     1 2 1 P ( j) P ( j) · · D(λ, τ ) = + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   ( j) 1 P ( j) 2 ( j) g ·P · · + − TR EL T A · TR EA    1 1 P ( j) P ( j) 2 ( j) ( j) · (g − l − q ) · + + + + TR TL · T R T A · TR EA EL ( j) ( j)  q 2 g 2 ·λ · P ( j) · − · P ( j) · + TR EA TR EL  2 + · (g ( j) − l − q ( j) ) · λ2 · e−λ·τ TR     1 1 P ( j) P ( j) · −λ· + + TL · T R EL T A · TR EA  ( j) ( j)  1 P P 1 · λ2 − λ3 + + + − TL · T R T A · TR EA EL 

D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n > m    1 1 P ( j) P ( j) · + + Pn (λ, τ ) = −λ · TL · T R EL T A · TR EA  ( j) ( j)  1 P P 1 · λ2 − λ3 + + + − TL · T R T A · TR EA EL     1 2 1 P ( j) P ( j) · · Q m (λ, τ ) = + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL 

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4 Solid State Laser Nonlinearity Applications in Engineering

 ( j)  2 1 P ( j) ( j) g − ·P · · + TR EL T A · TR EA    1 1 P ( j) P ( j) 2 ( j) ( j) · (g − l − q ) · + + + + TR TL · T R T A · TR EA EL  ( j) ( j) q 2 g 2 ·λ · P ( j) · − · P ( j) · + TR EA TR EL 2 + · (g ( j) − l − q ( j) ) · λ2 TR D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n = 3, m = 2 ∈ N0 ; n > m Pn (λ, τ ) =

n=3 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3

k=0

    1 1 P ( j) P ( j) · ; p2 (τ ) p0 (τ ) = 0; p1 (τ ) = − + + TL · T R EL T A · TR EA   1 1 P ( j) P ( j) p3 (τ ) = −1 =− + + + TL · T R T A · TR EA EL Q m (λ, τ ) =

n=2 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2

k=0

   1 2 P ( j) · · + · (g ( j) − l − q ( j) ) T A · TR EA TR   1 P ( j) · + TL · T R EL   1 P ( j) · + T A · TR EA   2 1 1 P ( j) P ( j) ( j) ( j) q1 (τ ) = · (g − l − q ) · + + + TR TL · T R T A · TR EA EL ( j) ( j) q 2 g 2 · P ( j) · − · P ( j) · + TR EA TR EL 2 q2 (τ ) = · (g ( j) − l − q ( j) ) TR 

1 P ( j) q0 (τ ) = + TL · T R EL ( j) 2 q + · P ( j) · TR EA g ( j) 2 · P ( j) · − TR EL

The homogeneous system for P, g, q leads to a characteristic  equation for the λ; having the form P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; P(λ) = 3j=0 a j · λ j ; Q(λ) = 2 j j=0 c j · λ . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

357

and delayτ . qi , qk are any Solid-state laser by semiconductor devices model corresponding to the rate differential equation system parameters, other parameters kept as a constant [5–7]. Hint: We use “P” as the system Laser power variable and as a part of stability analysis criteria parameter (P(λ, τ )… characteristic equation). Reader must differentiate between the two meanings. Additionally, we use “q” as the system saturate losses per round trip and as a part of stability analysis criteria parameter (q0 , q1, … in characteristic equation).     1 1 P ( j) P ( j) · a0 = 0; a1 = − + + TL · T R EL T A · TR EA   1 1 P ( j) P ( j) + + + a2 = − TL · T R T A · TR EA EL a3 = −1    1 2 P ( j) · · + · (g ( j) − l − q ( j) ) T A · TR EA TR   1 P ( j) · + TL · T R EL   1 P ( j) · + T A · TR EA   1 2 1 P ( j) P ( j) ( j) ( j) c1 = · (g − l − q ) · + + + TR TL · T R T A · TR EA EL ( j) ( j) q 2 g 2 · P ( j) · − · P ( j) · + TR EA TR EL 2 c2 = · (g ( j) − l − q ( j) ) TR 

1 P ( j) + c0 = TL · T R EL ( j) 2 q + · P ( j) · TR EA g ( j) 2 · P ( j) · − TR EL

Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [5] can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) If | Q(λ) P(λ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk .

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4 Solid State Laser Nonlinearity Applications in Engineering

We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real numberω: Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0.  P ( j) 1 · Pn (λ = i · ω, τ ) = −i · ω · + T A · TR EA   1 1 P ( j) P ( j) + + + + · ω2 + i · ω3 TL · T R T A · TR EA EL      1 1 P ( j) P ( j) 3 · Pn (λ = i · ω, τ ) = i · ω − ω · + + TL · T R EL T A · TR EA   1 1 P ( j) P ( j) + + + + · ω2 TL · T R T A · TR EA EL     1 P ( j) P ( j) 1 2 Q m (λ = i · ω, τ ) = + + · (g ( j) − l − q ( j) ) · · TL · T R EL T A · TR EA TR   1 2 q ( j) P ( j) + · P ( j) · · + TR EA TL · T R EL   ( j) 1 P ( j) 2 2 ( j) g ·P · · + · (g ( j) − l − q ( j) ) · ω2 − − TR EL T A · TR EA TR    2 1 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + TR TL · T R T A · TR EA EL  2 q ( j) 2 g ( j) + · P ( j) · − · P ( j) · ·ω·i TR EA TR EL 

P ( j) 1 + TL · T R EL

 

Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )   1 P ( j) 1 P ( j) = i · ω3 − ω · ( + )·( + ) TL · T R EL T A · TR EA  ( j) ( j)  1 P P 1 · ω2 + + + + TL · T R T A · TR EA EL     1 2 1 P ( j) P ( j) · · + + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL  ( j)  2 g 1 P ( j) 2 − · P ( j) · · + · (g ( j) − l − q ( j) ) · ω2 − TR EL T A · TR EA TR    2 1 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + TR TL · T R T A · TR EA EL  ( j) ( j) q 2 g 2 ·i ·ω · P ( j) · − · P ( j) · + TR EA TR EL

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

359

Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )      1 1 P ( j) P ( j) · = i · ω3 − ω · + + TL · T R EL T A · TR EA    1 1 P ( j) P ( j) 2 ( j) ( j) · (g − l − q ) · + + + + TR TL · T R T A · TR EA EL  ( j) ( j) q 2 g 2 ·ω · P ( j) · − · P ( j) · + TR EA TR EL     1 1 2 P ( j) P ( j) + · · + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL  ( j)  g 1 P ( j) 2 · P ( j) · · + − TR EL T A · TR EA  ( j) 1 P P ( j) 1 + + + + TL · T R T A · TR EA EL  2 − · (g ( j) − l − q ( j) ) · ω2 TR   3 |P(i · ω, τ )| = ω − ω · 2



1 P ( j) + TL · T R EL

  ·

1 P ( j) + T A · TR EA  2 P ( j) + · ω4 EL

2

1 1 P ( j) + + TL · T R T A · TR EA  2 1 1 P ( j) P ( j) |P(i · ω, τ )|2 = ω6 + ω4 · + + + TL · T R T A · TR EA EL  ( j)   ( j)  1 1 P P · −2 · + + TL · T R EL T A · TR EA 2  ( j) 2  1 P 1 P ( j) 2 +ω · + · + TL · TR E  T A · TR E A L  +

1 2 P ( j) · + · (g ( j) − l − q ( j) ) T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL   2 1 2 g ( j) P ( j) 2 − · P ( j) · · + · (g ( j) − l − q ( j) ) · ω2 − TR EL T A · TR EA TR    2 1 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + TR TL · T R T A · TR EA EL

|Q(i · ω, τ )|2 =

1 P ( j) + TL · T R EL

·

360

4 Solid State Laser Nonlinearity Applications in Engineering 2 q ( j) 2 g ( j) + · P ( j) · − · P ( j) · TR EA TR EL



2 ·ω

 1 P ( j) + T A · TR EA  ( j)  2 1 P ( j) 2 ( j) ( j) ( j) q · (g − l − q ) + ·P · · + · TR TR EA TL · T R EL   2 2 g ( j) 1 P ( j) − · P ( j) · · + TR EL T A · TR EA   ( j)   1 1 P P ( j) · −2· + + TL · T R EL T A · TR EA 2 · (g ( j) − l − q ( j) ) · TR   q ( j) 1 P ( j) 2 · P ( j) · · + + TR EA TL · T R EL  ( j)  g 1 P ( j) 2 · P ( j) · · + − TR EL T A · TR EA 2 · (g ( j) − l − q ( j) ) · ω2 · TR 2 + ( )2 · (g ( j) − l − q ( j) )2 · ω4 TR    2 1 1 P ( j) P ( j) ( j) ( j) + · (g − l − q ) · + + + TR TL · T R T A · TR EA EL ( j) ( j) 2 2 q 2 g + · P ( j) · − · P ( j) · · ω2 T E T E A  R L  R 

|Q(i · ω, τ )|2 =

1 P ( j) + TL · T R EL

1 2 P ( j) · + · (g ( j) − l − q ( j) ) T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL  2 ( j) 1 2 P ( j) ( j) g − ·P · · + TR EL T A · TR EA    2 1 1 P ( j) P ( j) + · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL 2 2 q ( j) 2 g ( j) + · P ( j) · − · P ( j) · TR EA TR EL     ( j) 1 1 P P ( j) 2 −2· + + · (g ( j) − l − q ( j) ) · · TL · T R EL T A · TR EA TR

|Q(i · ω, τ )|2 =

1 P ( j) + TL · T R EL

  ·

·

4.1 Solid State Laser Controlled by Semiconductor Devices Stability … +

2 q ( j) · P ( j) · · TR EA



1 P ( j) + TL · T R EL

361



   ( j) 1 P ( j) 2 2 ( j) g ( j) ( j) ·P · · + · (g − l − q ) · ω2 − · TR EL T A · TR EA TR  2 2 + · (g ( j) − l − q ( j) )2 · ω4 TR

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω6  2 1 1 P ( j) P ( j) 4 +ω · + + + TL · T R T A · TR EA EL     1 1 P ( j) P ( j) · −2 · + + TL · T R EL T A · TR EA  2   2 1 P ( j) 1 P ( j) + · + + ω2 · TL · T R EL T A · TR EA   ( j)   ( j)  1 1 2 P P − · · + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL 2  2 g ( j) 1 P ( j) − · P ( j) · · + TR EL T A · TR EA    2 1 1 P ( j) P ( j) ( j) ( j) − · (g − l − q ) · + + + TR TL · T R T A · TR EA EL  2 q ( j) 2 g ( j) 2 · P ( j) · − · P ( j) · + TR EA TR EL   ( j)   1 2 1 P P ( j) · · −2· + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL    ( j) 2 1 P ( j) 2 ( j) g ( j) ( j) · ·P · · + · (g − l − q ) · ω2 − TR EL T A · TR EA TR  2 2 − · (g ( j) − l − q ( j) )2 · ω4 TR F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω6 + ω4  2 1 1 P ( j) P ( j) · + + + TL · T R T A · TR EA EL

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4 Solid State Laser Nonlinearity Applications in Engineering

   1 1 P ( j) P ( j) · −2· + + TL · T R EL T A · TR EA   2 2 − · (g ( j) − l − q ( j) )2 TR  2  2 1 P ( j) 1 P ( j) 2 + · + +ω · TL · T R EL T A · TR EA    2 1 1 P ( j) P ( j) ( j) ( j) − · (g − l − q )· + + + TR TL · T R T A · TR EA EL  2 2 q ( j) 2 g ( j) + · P ( j) · − · P ( j) · TR EA TR EL   ( j)   1 2 1 P P ( j) · · −2· + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL  ( j)  ( j)  2 g 1 P 2 · · P ( j) · · + · (g ( j) − l − q ( j) ) − TR EL T A · TR EA TR     1 2 P ( j) P ( j) 1 · · + + · (g ( j) − l − q ( j) ) − TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL 2 ( j)  g 1 P ( j) 2 · P ( j) · · + − TR EL T A · TR EA 

We define the following parameters for simplicity: 0 , 2 , 4 , 6 .    1 2 1 P ( j) P ( j) · · + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL 2 ( j)  1 P ( j) 2 ( j) g ·P · · + − TR EL T A · TR EA

 0 = −

2  2 1 P ( j) 1 P ( j) 2 = + · + TL · T R EL T A · TR EA    2 1 1 P ( j) P ( j) − · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL 

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363

2 2 q ( j) 2 g ( j) · P ( j) · − · P ( j) · TR EA TR EL     1 2 1 P ( j) P ( j) · · −2· + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL   ( j)  2 1 P ( j) 2 ( j) g ( j) ( j) · ·P · · + · (g − l − q ) − TR EL T A · TR EA TR

+

2   1 1 P ( j) P ( j) 1 P ( j) 4 = + + + −2· + TL · T R T A · TR EA EL TL · T R EL     2 2 1 P ( j) − · + · (g ( j) − l − q ( j) )2 6 = 1 T A · TR EA TR 

Hence F(ω, τ ) implies the above polynomial.

3 k=0

2·k · ω2·k = 0 and its roots are given by solving

 1 1 P ( j) P ( j) PR (i · ω, τ ) = · ω2 + + + TL · T R T A · TR EA EL     1 1 P ( j) P ( j) 3 · PI (i · ω, τ ) = ω − ω · + + E L  T A · TR EA  TL · TR 

P ( j) 1 2 + · (g ( j) − l − q ( j) ) · T A · TR EA TR   ( j) 1 2 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   ( j) 2 1 P ( j) 2 ( j) g − ·P · · + · (g ( j) − l − q ( j) ) · ω2 − TR EL T A · TR EA TR    2 1 1 P ( j) P ( j) Q I (i · ω, τ ) = · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL  ( j) ( j) 2 q 2 g + · P ( j) · − · P ( j) · ·ω TR EA TR EL

Q R (i · ω, τ ) =

1 P ( j) + TL · T R EL

·

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2 PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q i (i · ω, τ ) cos θ (τ ) = − |Q(i · ω, τ )|2 sin θ (τ ) =

We use different parameters terminology from our last characteristics parameters definition:

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4 Solid State Laser Nonlinearity Applications in Engineering

k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 3; m = 2; n > m; Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

3 

a j · λ j ; Q(λ) =

j=0

2 

c j · λ j ; P(λ) = a0 + a1 · λ + a2 · λ2

j=0

+ a3 · λ ; Q(λ) = c0 + c1 · λ + c2 · λ2 3

n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ . The coefficients a j (TL , TR , T A , E L , E A , l, τ, . . .) ∈ R. ∈ R depend on systems And c j (TL , TR , T A , E L , E A , l, τ, . . .) TL , TR , T A , E L , E A , l, τ, . . . values. Unless strictly, the designation of the varied arguments: (TL , TR , T A , E L , E A , l, τ, . . .) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0.    1 2 1 P ( j) P ( j) a0 = 0; c0 = · · + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  1 2 P ( j) ( j) q + ·P · · + TR EA TL · T R EL ( j)  ( j)  g 1 P 2 · P ( j) · · + − TR EL T A · TR EA   ( j)   1 1 2 P P ( j) · · + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   ( j) 1 P ( j) 2 ( j) g = 0 ·P · · + − TR EL T A · TR EA 

∀ TL , TR , T A , E L , E A , l, τ, . . . ∈ R+ i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang 1993, Sect. 3.4) can be also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire (TL , TR , T A , E L , E A , l, τ, . . .) domain of interest. P(λ) | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (b) | Q(λ) 1 ·λ+c2 ·λ in the limit: | Q(λ) | = | a0 +ac01+c |. P(λ) ·λ+a2 ·λ2 +a3 ·λ3 2

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

365

2 (c) The following expressions exist: F(ω) − |Q(i · ω)|2 F(ω, τ ) = 3 = |P(i · ω)| 2 2 2·k |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω = 0. Has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω6 ). (d) Each positive root ω(TL , TR , T A , E L , E A , l, τ, . . .) of F(ω) = 0 is continuous and differentiable with respect to TL , TR , T A , E L , E A , l, τ, . . . the condition can only be assessed numerically.

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω), and Q(−i · ω) = Q(i · ω) thus, ω > 0 maybe on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ—plane, whereby increasing the parameters: TL , TR , T A , E L , E A , l, τ, . . . Re λ may, at the crossing, change its sign from (−) to (+), i.e. from a stable focus E (∗) (P (∗) , g (∗) , q (∗) ) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to TL , TR , T A , E L , E A , l, τ, . . . and system parameters [5–7].  ∂Reλ ; TR , T A , E L , E A , l, τ, . . . = const  (TL ) = ∂ TL λ=i·ω   ∂Reλ −1 (T A ) = ; TL , TR , E L , E A , l, τ, . . . = const ∂ T A λ=i·ω   ∂Reλ −1 (E L ) = ; TL , TR , T A , E A , l, τ, . . . = const ∂ E L λ=i·ω   ∂Reλ −1 (E A ) = ; TL , TR , T A , E L , l, τ, . . . = const ∂ E A λ=i·ω   ∂Reλ −1 (l) = ; TL , TR , T A , E L , E A , τ, . . . = const ∂l λ=i·ω   ∂Reλ −1 (τ ) = ; TL , TR , T A , E L , E A , l, . . . = const ∂τ λ=i·ω −1



P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ) When writing and inserting λ = i · ω into system’s characteristic equation ω must satisfy the following equations. −PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2 PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) cos(ω · τ ) = h(ω) = − |Q(i · ω)|2 sin(ω · τ ) = g(ω) =

where |Q(i ·ω)|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is dependent on τ . Now it is important to notice that if τ ∈ / I (assume that

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4 Solid State Laser Nonlinearity Applications in Engineering

I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ / I ,ω(τ ) is not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ )solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin(θ ) = · · · ; cos(θ ) = · · · −PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2 PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) cos θ (τ ) = − |Q(i · ω)|2 sin θ (τ ) =

And the relation between the argument θ (τ ) and τ · ω(τ ) for τ ∈ I must be as describe τ · ω(τ ) = θ (τ ) + 2 · n · π ∀ N0 , hence we can define the maps τn : I → R+0 )+2·n·π given by τn (τ ) = θ(τ ω(τ ; n ∈ N0 ; τ ∈ I . Let us introduce the functions:I → ) R; Sn (τ ) = τ − τn (τ ). τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ . In the following, the subscribes λ, TL , TR , T A , E L , E A , l, . . . Indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(TL , TR , T A , E L , E A , l, . . .) and ω(λ, TL , TR , T A , E L , E A , l, . . .), and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating system characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = TL , TR , T A , E L , E A , l, . . . 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ (λ, x) = ∂∂λP ; Q λ (λ, x) = ∂∂λQ ; Px (λ, x) = ∂∂ Px ; Q x (λ, x) = ∂∂Qx , substituting λ = i ·ω and bearing P(−i · ω) = P(i ·ω); Q(−i · ω) = Q(i ·ω); i · Pλ (i ·ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtain: 

∂λ ∂x

=

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R +i · Q I and Pω = PRω +i · PI ω ; Q ω = Q Rω +i · Q I ω ; Q x = Q Rx +i · Q I x ; P 2 = PR2 + PI2 . When (x) can be any system parameter’s TL , TR , T A , E L , E A , l, . . . and time delayτ . Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )], Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V:

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameters as time delay x = τ .  1 1 P ( j) P ( j) · ω2 PR = + + + TL · T R T A · TR EA EL     1 1 P ( j) P ( j) 3 · PI = ω − ω · + + TL · T R EL T A · TR EA     1 1 P ( j) P ( j) 2 QR = + + · (g ( j) − l − q ( j) ) · · TL · T R EL T A · TR EA TR  ( j)  2 2 1 P ( j) g ( j) ( j) q − + ·P · · + · P ( j) · TR EA TL · T R EL TR EL  ( j)  2 1 P − · + · (g ( j) − l − q ( j) ) · ω2 T A · TR EA TR    2 1 1 P ( j) P ( j) QI = · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL ( j) ( j)  q 2 g 2 ·ω · P ( j) · − · P ( j) · + TR EA TR EL   1 1 P ( j) P ( j) · ω; PI ω = 2 · ω2 PRω = 2 · + + + TL · T R T A · TR EA EL     1 1 P ( j) P ( j) · − + + TL · T R EL T A · TR EA 

4 · (g ( j) − l − q ( j) ) · ω; PRτ = 0; PI τ = 0 TR Fτ Q Rτ = 0; Q I τ = 0; ωτ = − Fω   2 1 1 P ( j) P ( j) ( j) ( j) = · (g − l − q ) · + + + TR TL · T R T A · TR EA EL ( j) ( j) q 2 g 2 · P ( j) · − · P ( j) · + TR EA TR EL Q Rω = −

QIω

1 1 P ( j) P ( j) 2 3 + + + ) ·ω TL · T R T A · TR EA EL     4 1 P ( j) ( j) ( j) Q Rω · Q R = − · (g − l − q ) · ω · + TR TL · T R EL PRω · PR = 2 · (

367

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4 Solid State Laser Nonlinearity Applications in Engineering

 2 1 P ( j) · · + · (g ( j) − l − q ( j) ) T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   ( j) ( j)  2 g 1 P 2 ( j) ( j) ( j) 2 − ·P · · + · (g − l − q ) · ω − TR EL T A · TR EA TR     1 2 1 P ( j) P ( j) · · QR · QIω = + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR  ( j)  2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL  ( j) ( j)  g 1 P 2 ( j) ·P · · + − TR EL T A · TR EA  2 − · (g ( j) − l − q ( j) ) · ω2 · TR    2 1 1 P ( j) P ( j) · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL ( j) ( j)  q 2 g 2 · P ( j) · − · P ( j) · + TR EA TR EL  2 · (g ( j) − l − q ( j) ) Q I · Q Rω = − TR   2 1 1 P ( j) P ( j) q ( j) + · + + + · P ( j) · TL · T R T A · TR EA EL TR EA ( j)  4 2 g · − · P ( j) · · (g ( j) − l − q ( j) ) · ω2 TR EL TR 

V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 ⇒ F(ω, τ ) = 0 + Fτ = 0; τ ∈ I ⇒ Differentiating with respect to τ and we get Fω · ∂ω ∂τ −1





= − FFωτ

∂ω = ωτ λ=i·ω ∂τ   Fτ −2 · [U + τ · |P|2 ] + i · Fω = − ; −1 (τ ) = Re Fω Fτ + i · 2 · [V + ω · |P|2 ]    ∂Reλ sign{−1 (τ )} = sign ∂τ λ=i·ω   V + ∂ω ·U ∂ω ∂τ −1 ·τ sign{ (τ )} = sign{Fω } · sign +ω+ |P|2 ∂τ

 (τ ) =

∂Reλ ∂τ

∂ω ∂τ

;

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

369

We shall presently examine the possibility of stability transitions (bifurcations) of Solid-state laser by semiconductor devices model corresponding to the rate differential equation system, about the equilibrium point E (∗) (P (∗) , g (∗) , q (∗) ) as a result of a variation of delay parameterτ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complexλ— plane where by increasing the delay parameterτ , Reλ may at the crossing, changes its sign from “−” to “ + ”, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ . −1 (τ ) =



∂Reλ ∂τ

 λ=i·ω

; TL , TR , T A , E L , E A , l, . . . = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω )   1 1 P ( j) P ( j) PR · PI ω = + + + TL · T R T A · TR EA EL    ( j)   1 1 P P ( j) · · ω2 + + · 2 · ω2 − TL · T R EL T A · TR EA      1 1 P ( j) P ( j) · PI · PRω = 2 · ω2 · ω2 − + + TL · T R EL T A · TR EA  ( j) ( j)  1 P P 1 + + + · TL · T R T A · TR EA EL     1 2 1 P ( j) P ( j) · · QR · QIω = + + · (g ( j) − l − q ( j) ) TL · T R EL T A · TR EA TR   2 q ( j) 1 P ( j) + · P ( j) · · + TR EA TL · T R EL  ( j)  g 1 P ( j) 2 · P ( j) · · + − TR EL T A · TR EA  2 2 − · (g ( j) − l − q ( j) ) · ω2 } · · (g ( j) − l − q ( j) ) TR TR  1 1 P ( j) · + + TL · T R T A · TR EA ( j)  ( j) ( j)  2 P 2 ( j) q ( j) g + + ·P · − ·P · EL TR EA TR EL  2 · (g ( j) − l − q ( j) ) Q I · Q Rω = − TR   2 1 1 P ( j) P ( j) q ( j) + · + + + · P ( j) · TL · T R T A · TR EA EL TR EA

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4 Solid State Laser Nonlinearity Applications in Engineering ( j)  4 2 ( j) g · − ·P · · (g ( j) − l − q ( j) ) · ω2 TR EL TR



1 1 P ( j) P ( j) + + + · [2 · ω2 TL · T R T A · TR EA EL     1 1 P ( j) P ( j) · · ω2 − + + TL · T R EL T A · TR EA      1 P ( j) P ( j) 1 2 2 −2·ω · ω − + + · TL · T R EL T A · TR EA  1 1 P ( j) P ( j) + + + · TL · T R T A · TR EA EL     1 P ( j) P ( j) 1 − + + · TL · T R EL T A · TR EA   ( j) 1 2 2 P ( j) ( j) ( j) ( j) q · · (g − l − q ) + ·P · · + TR TR EA TL · T R EL   2 g ( j) 1 P ( j) − · P ( j) · · + TR EL T A · TR EA    1 2 2 1 P ( j) − · (g ( j) − l − q ( j) ) · ω2 · · (g ( j) − l − q ( j) ) · + + TR TR TL · T R T A · TR EA   ( j) ( j) ( j) 2 P q 2 g + + · P ( j) · − · P ( j) · EL TR EA TR EL   1 2 1 P ( j) P ( j) ( j) ( j) +[ · (g − l − q ) · + + + TR TL · T R T A · TR EA EL  2 4 q ( j) 2 g ( j) + · · P ( j) · · P ( j) · · (g ( j) − l − q ( j) ) · ω2 ) TR E A TR EL TR

U=

Then we get the expression for F(ω, τ ) system parameter values. We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore, complex, and imaginary values of ω for specific τ values, τ ∈ [0.001 . . . 10] second. We can plot the stability switch diagram based on different delay values of our system. −1 (τ ) = −1 (τ ) =

 

∂Reλ ∂τ ∂Reλ ∂τ

 

 λ=i·ω

λ=i·ω

= Re =

−2 · [U + τ · |P|2 + i · Fω Fτ + 2 · i · [V + ω · |P|2 ]



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = · · · ; cos θ (τ ) = · · · when ω = ω+ (τ ) if only ω+ is feasible. Additionally, when system parameters are known and the stability switch due to various time delay values τ is described in the following

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

371

expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) +ω(τ ) + |P(ω(τ ))|2



Remark We know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = · · · ; cos θ (τ ) = · · · I +PI ·Q R R +PI ·Q I ) ; cos(ω ·τ ) =− (PR ·Q|Q| ; |Q|2 = Q 2R + Q 2I . meaning sin(ω ·τ ) = −PR ·Q|Q| 2 2   . Finally, we plot the stability switch diagram g(τ ) = −1 (τ ) = ∂Reλ ∂τ λ=i·ω 



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2 λ=i·ω    ∂Reλ sign[g(τ )] = sign[−1 (τ )] = sign ∂τ λ=i·ω   2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2

g(τ ) = −1 (τ ) =

∂Reλ ∂τ

=

Fτ2 + 4 · (V + ω · P 2 )2 >; sign[−1 (τ )]

= sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )]    Fτ sign[−1 (τ )] = sign [Fω ] · (V + ω · P 2 ) − · (U + τ · P 2 ) ; ωτ Fω  −1 Fτ ∂ F/∂ω ∂ω = − ; ωτ = =− Fω ∂τ ∂ F/∂τ sign[−1 (τ )] = sign{[Fω ] · [(V + ω · P 2 ) + ωτ · (U + τ · P 2 )]} sign[−1 (τ )] = sign{[Fω ] · [V + ωτ · U + ω · P 2 + ωτ · τ · P 2 ]}    (V + ωτ · U ) 2 2 sign[−1 (τ )] = sign [Fω ] · · P + (ω + ω · τ ) · P τ P2    V + ωτ · U + ω + ω · τ sign[−1 (τ )] = sign [Fω ] · [P 2 ] · τ P2    V + ωτ · U + ω + ω · τ sign[P 2 ] > 0 ⇒ sign[−1 (τ )] = sign [Fω ] · τ P2 sign[P 2 ] > 0 ⇒ sign[−1 (τ )] = sign[Fω ]   V + ωτ · U · sign + ω + ω · τ τ P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]

We check the sign of −1 (τ ) according the following rule (Table 4.1).

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4 Solid State Laser Nonlinearity Applications in Engineering

Table. 4.1 Solid-state laser by semiconductor devices model corresponding to the rate differential equation system sign of sign[−1 (τ )] sign[Fω ]

τ ·U sign[ V +ω + ω + ωτ · τ ] P2

sign[−1 (τ )]

±

±

+

±

±

−

If sign[−1 (τ )] > 0; then the crossing proceeds from (−) to (+) respectively (stable to unstable). If sign[−1 (τ )] < 0; then the crossing proceeds from (+) to (−) respectively (unstable to stable) [7–9]. Appendix A A. Appendix A1 (Lemma 1.1) Assume that ω(τ ) is a positive and real root of F(ω, τ ) = 0 are defined for τ ∈ I , which is continuous and differentiable. Assume further that if λ = i · ω; ω ∈ R, then Pn (i · ω, τ ) + Q n (i · ω, τ ) = 0; τ ∈ R hold true. Then the functions Sn (τ ); n ∈ N0 , are continuous and differentiable on I . B. Appendix A2 (Theorem 1.2) Assume that ω(τ ) is a positive real root of F(ω, τ ) = 0 defined for τ ∈ I , I ⊆ R+0 and at some τ ∗ ∈ I , Sn (τ ∗ ) = 0. For some n ∈ N0 then a pair of simple conjugate pure imaginary roots λ+ (τ ∗ ) = i · ω(τ ∗ ) and λ− (τ ∗ ) = −i · ω(τ ∗ ) of D(λ, τ ) = 0 exists at τ = τ ∗ which crosses the imaginary axis from left to right if δ(τ ∗ ) > 0 and cross the imaginary axis from right to left if δ(τ ∗ ) < 0   n (τ ) ∗ | |τ =τ ∗ . = sign{Fω (ω(τ ∗ ), τ ∗ )} · sign d Sdτ where δ(τ ∗ ) = sign dReλ dτ λ=i·ω(τ )    dReλ  n (τ ) The theorem becomes sign dτ |λ=i·ω± = sign{±1/2 } · sign d Sdτ |τ =τ ∗ . C. Appendix A3 (Theorem 1.3) The characteristic equation has a pair of simple and conjugate pure imaginary rootsλ = ±ω(τ ∗ ), ω(τ ∗ ) real at τ ∗ ∈ I if Sn (τ ∗ ) = τ ∗ − τn (τ ∗ ) = 0 for some n ∈ N0 . If ω(τ ∗ ) = ω+ (τ ∗ ), this pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right if δ+ (τ ∗ ) > 0 andcrosses the imagi ∗ = | nary axis from right to left if δ+ (τ ∗ ) < 0 where δ+ (τ ∗ ) = sign dReλ dτ λ=i·ω+ (τ )   d Sn (τ ) ∗ ∗ sign dτ |τ =τ ∗ . If ω(τ ) = ω− (τ ), this pair of simple conjugate pure imaginary roots cross the imaginary axis from left to right if δ− (τ ∗ ) > 0 and   crosses the imagi∗ and | nary axis from right to left if δ− (τ ∗ ) < 0 where δ− (τ ∗ ) = sign dReλ dτ λ=i·ω− (τ )   d Sn (τ ) ∗ ∗ ∗ ∗ ∗ δ− (τ ) = −sign dτ |τ =τ ∗ . If ω+ (τ ) = ω− (τ ) = ω(τ ) then (τ ) = 0 and   ∗ | = 0 the same is true when Sn (τ ∗ ) = 0. The following result sign dReλ dτ λ=i·ω(τ ) can be useful in identifying values of τ where stability switches happened.

4.1 Solid State Laser Controlled by Semiconductor Devices Stability …

373

D. Appendix A4 (Theorem 1.4) Assume that for all τ ∈ I , ω(τ ) is defined as a function of F(ω, τ ) = 0 then δ± (τ ) = sign{±1/2 (τ )} · sign{D± (τ )}.

4.2 Nanometer-Vibration Measurement with Microchip Solid-State Laser Instability Under Parameters Variation In many applications we use laser-diode-pumped microchip solid-state laser (DPSSLs). One application is a self-aligned optical feedback vibrometry technique [2]. It can help to understand the nanometer vibration analysis. The laser-diodepumped microchip solid-state laser is useful for compact, durable, coherent light sources. If we use external optical feedback light we get an ultrahigh sensitivity response of DPSSL. They have short photon lifetimes compared with the population lifetimes that are inherent in microchip lasers. The results are in such significant applications as self-aligned laser Doppler velocimetry, vibrometry, and imaging. The laser acts as a high-efficiency mixer oscillator and a shot-noise-limited quantum detector. In laser-diode-pumped microchip solid-state lasers the effective long-term interference between a lasing field and a coherent component of a scattered field that is fed back to the micro cavity laser resonator occurs because of the high degree of coherence of the lasing field. The experimental system configuration of selfmixing laser Doppler vibrometry and optical micro-phony include the following elements: Laser diode (LD), Anamorphic prism pairs (APPs), Microscope objective lens (OLs), Glass-plate beam splitter (BS), Photodiode receiver (PD), RF spectrum analyzer (SA), Variable attenuator (VA), Digital oscilloscope (DO), Frequency demodulator circuit (DC), and Personal computer (PC). It is c plate Nd direct compound LiNdP4 O12 (LNP) laser with a thick plane-parallel Fabry–perot-cavity. An end surface is coated to be trans-missive at the laser-diode pump wavelength and highly reflective at specific lasing wavelength. The other surface is coated to be 1% transmissivity at the lasing wavelength. The collimated laser-diode pump light is passed through anamorphic prism pairs, which transformed the elliptical beam into a circular beam, and it is focused onto LNP crystal by a microscope objective lens of 20× magnification. An anamorphic prism pair is a pair of prisms which is usually used for reshaping the profile of a laser beam. In our case the elliptical beam from the laser diode is transformed into a beam with circular cross-section, by using a prism pair as a beam expander for only one direction. The principle of beam shaping with anamorphic prism pairs is not based on focusing effects (i.e. changes of wave front curvature), but rather an changed of the beam radius for reflection at plat prism interfaces. Such changes occur at the interfaces of any prism, because the angle of the beam against the surface-normal direction is different inside and outside according to Snell’s law. In the case of a symmetric beam path, where the

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4 Solid State Laser Nonlinearity Applications in Engineering

beam angles against input and output face of the prism are identical, the two changes in beam radius cancel each other. Therefore, one has to use an asymmetric configuration. A single prism is sufficient for changing the beam radius in one direction, but it also changes the beam direction. By using anamorphic prism pair, one can obtain an output beam with an unchanged direction, only a position offset. The two prisms are oriented such that they change the beam radius in the same direction. The overall magnification is then square of the refractive index, or the inverse of that. If the mentioned beam offset needs to be avoided, one may use a combination of four prisms. In an anamorphic prism the output beam is substantially narrower than the input beam. For example an anamorphic prism pair with refractive index 1.5, where Brewster’s angle is used on one side of each prism, and normal incidence on the other one. Two parallel beams passing through the prisms. The distance changes, and likewise their beam radii in the direction of the plane are changed. The prism pair thus works as a beam expander if the input beam comes from the left side. The beam radius in the direction perpendicular to the drawing plane is not changed. In our system a part (≈96%) of the output light is shifted by two Acoustic Optic Modulators (AOMs) and impinged upon a speaker that had an Al-coated surface with an average roughness of few hundred micro meters that is placed ≈1 m from the laser. An acoustic optic modulator (AOM) is a device which can be used for controlling the power, frequency or spatial direction of a laser beam with an electrical drive signal. It is based on the acousto-optic effect. The modification of the refractive index is by the oscillating mechanical pressure of a sound wave. The key element of an Acoustic optical modulator (AOM) is a transparent crystal through which the light propagates. A piezo electric traducer attached to the crystal is used to excite a sound wave with a frequency of the order of 100 MHz. The light can then experience Bragg diffraction at the traveling periodic refractive index grating generated by the sound wave; therefore, AOMs are Bragg cells. The optical frequency of the beam is increased or decreased by the frequency of the sound wave which is depending on the propagation direction of the acoustic wave relative to the beam and propagates in a slightly different direction. The frequency and direction of the scattered beam can be controlled via the frequency of the sound wave, whereas the acoustic power is the control for the optical powers. The acoustic wave may be absorbed at the other end of the crystal. It is possible to achieve a broad modulation bandwidth of many megahertz. In the experiment we use two AOMs to achieve appropriate frequency shifts. By changing the modulation frequencies of upshifted and downshifted AOMs, we shift the optical carrier frequency by specific frequency after a round trip for the feedback field. AOMs can also be used for cavity dumping of solidstate lasers, generating either nanosecond or ultrashort pulses. Active mode locking is often performed with an AOM for modulating the resonator losses at the roundtrip frequency or a multiple thereof (Fig. 4.2). In our system another part (≈4%) of the output light is detected by an InGaAs photo receiver. The additional units which used are frequency demodulation circuit, a digital oscilloscope connected to a Personal Computer (PC), and RF spectrum analyzer (Fig. 4.3). Microscope objective lens: The objective is the optical element that gathers light from the object being observed and focused the light rays to introduce a real image [2]. Objectives can be

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

375

To photo receiver diode 4% Anamorphic prism pairs (APPs)

Laser diode

Lens Speaker

Microscope objective lens Acousto Optic Modulator (AOM)

LNP crystal

96% Beam Splitter

Acousto Optic Modulator (AOM)

Variable Attenuator (VA)

Lens

Fig. 4.2 Experimental configuration of self-mixing laser-Doppler vibrometry and optical micro phony (flow chart for 96% of the output light)

Digital Oscilloscope (DO)

RF spectrum analyzer

Personal Computer (PC)

Frequency Demodulation Circuit (DC)

4% 96% Beam Splitter

Fig. 4.3 Experimental configuration of self-mixing laser-Doppler vibrometry and optical micro phony (flow chart for 4% of the output light)

a single lens or combinations of several optical elements. f m —Modulation  or mirror ωm , β—Frequency modulation index, Av,m —Maximum vibrafrequency f m = 2·π   λ·β tion amplitude Av,m = 2·π , Va —Applied voltage, Av (t)—Temporal evolutions of nanometer vibration (vibration amplitude), (t)—Phase difference between the reference (carrier) signed and the modulated signal ( Av (t) = λ · (t)),  A — Analytic phase, related to analytic signal V A , V A —Analytic signal time average, V A (t) = I (t) + i · I H (t), V A (t) − V A  = R A (t) · ei· A (t) , I (t)—Time series of the scalar intensity, I H —Hilbert transform. There are modulated output waveforms and the corresponding vibration waveforms for different feedback ratios. The variable optical attenuator (VA) is used to change the feedback ratio. There is an analytic and numerical simulation to measure

376

4 Solid State Laser Nonlinearity Applications in Engineering

the feedback ratio of a scattered field from the speaker into the laser experimentally. By analytic simulation we produce the experimental signal characteristics and estimate the feedback ratio accurately from the correspondence between experimental and numerical results. The model Delay Differential Equations (DDEs) is the simulation, that is, laser with frequency-modulated optical feedback [2]. E(t)—Normalized field amplitude, N (t)—Normalized excess population inver  P sion, ω—Relative pump power normalized by the threshold ω = Pth , K—Popula  tion to photon lifetime ration K = ττp , φ(t)—Phase of the lasing field, ψ(t)— Phase difference between the lasing and the feedback fields, —Normalized frequency shift, = (ωi − ω0 ) · τ p . (τ p —photon lifetime), m—Amplitude feedback coefficient, m —Normalized modulation frequency (m = ωm · τ p ), t—Time normalized by the photon lifetime, t D —Delay time normalized by the photon lifetime, ε—Spontaneous  emission rate,  ξ(t)—Gaussian white noise with zero mean. The value ξ(t) · ξ(t  ) = δ(t − t  ) is δ  √ correlated in time. 2 · ε · [N (t) + 1] · ξ(t) is the quantum spontaneous emission noise. The model DDEs (lasers with frequency-modulated optical feedback): d N (t) = K · {ω − 1 − N (t) − [1 + 2 · N (t)] · E 2 (t)} dt N = N (t); E = E(t); φ = φ(t) d E(t) = N (t) · E(t) + m · E(t − t D ) × cos ψ(t) dt   + 2 · ε · [N (t) + 1] · ξ(t)   dφ(t) E(t − t D ) = · sin ψ(t); ψ(t) dt E(t)

   · tD =  · t + β · sin(m · t) − φ(t) + φ(t − t D ) − 0 + 2

At fixed points d Ndt(t) = 0; d E(t) = 0; dφ(t) = 0; limt→∞ E(t − t D ) = dt dt E(t); limt→∞ φ(t − t D ) = φ(t). At fixed points: t  t D ⇒ t − t D ≈ t; ψ ∗ (t) = ψ(t, φ ∗ )    · tD ; ψ ∗ ψ =  · t + β · sin(m · t) − 0 + 2  · tD =  · t + β · sin(m · t) − 0 · t D − 2   1 ψ ∗ =  · t − · t D + β · sin(m · t) 2 1 1 − 0 · t D ; t  · t D ⇒ t − · t D ≈ t 2 2 ∗

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

377

dφ(t) dt = 0 ⇒ sin ψ ∗ = 0; ψ ∗ = n · π ∀ n = 0, 1, 2, . . .

ψ ∗ ≈  · t + β · sin(m · t) − 0 · t D ;

n ∈ N0 ; n · π =  · t + β · sin(m · t) − 0 · t D White Gaussian Noise (WGN): Gaussian white noise with zero mean (ξ(t)). White Gaussian Noise (WGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. It is additive because it is added to any noise that might be intrinsic to the information system. The noise is called “white” because it has the same intensity at every frequency. It called “white” when it contains all visible frequency. Gaussian White Noise (GWN) is a stationary and ergodic random process with zero mean that is defined by the fundamental property that two values of GWN are statistically independent no matter how close they are in time. The power spectrum of GWN is constant over all frequencies, hence the name “white noise”, in analogy to the white light that contains all visible wavelength with the same power. The additional adjective “Gaussian” in GWN indicates that the amplitude distribution of the white noise signal is Gaussian-like the independent steps in Brownian motion. The most common approximation for GWN is the band-limited GWN signal that has the “Sinc function” as autocorrelation function. Non-Gaussian white-noise signals have symmetric amplitude probability density functions and may exhibits, practical advantages in certain applications over bandlimited GWN. High-order autocorrelation functions of GWN signals have a specific structure that is suitable for nonlinear system identification. Specially, all the oddorder autocorrelation functions are uniformly zero and the even-order ones can be expressed in terms of sums of products of delta functions. Assumption: √ The quantum spontaneous√emission noise is very small and negligible ( 2 · ε · [N (t) + 1]) · ξ(t)  1; ( 2 · ε · [N (t) + 1]) · ξ(t) → 0. Then ddtN = 0 ⇒ K · {ω − 1 − N ∗ − [1 + 2 · N ∗ ] · (E ∗ )2 } = 0 dE = 0 ⇒ N ∗ · E ∗ + m · E ∗ × cos(ψ ∗ ) = 0; N ∗ · E ∗ + m · E ∗ × cos(ψ ∗ ) = 0 dt sin ψ ∗ = 0 ⇒ cos ψ ∗ = ±1; n = 0, 1, 2, . . . ; n ∈ N0 ; ψ ∗ = n · π ; cos ψ ∗ = (−1)n dE = 0 ⇒ N ∗ · E ∗ + m · E ∗ × cos(ψ ∗ ) = 0; N ∗ · E ∗ dt + m · E ∗ · (−1)n = 0 ∀ n = 0, 1, 2, . . . dN = 0 ⇒ K · {ω − 1 − N ∗ − [1 + 2 · N ∗ ] · (E ∗ )2 } = 0 dt K = 0 ⇒ ω − 1 − N ∗ − [1 + 2 · N ∗ ] · (E ∗ )2 = 0 We get two fixed points equations: N ∗ · E ∗ + m · E ∗ · (−1)n = 0 ∀ n = 0, 1, 2, . . . ω − 1 − N ∗ − [1 + 2 · N ∗ ] · (E ∗ )2 = 0

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4 Solid State Laser Nonlinearity Applications in Engineering

&&& ω − 1 − (E ∗ )2 = N ∗ · [1 + 2 · (E ∗ )2 ]; N ∗ =

ω − 1 − (E ∗ )2 1 + 2 · (E ∗ )2

N ∗ · E ∗ + m · E ∗ · (−1)n   ω − 1 − (E ∗ )2 · E ∗ + m · E ∗ · (−1)n = 0 =0⇒ 1 + 2 · (E ∗ )2   ω − 1 − (E ∗ )2 n E∗ · =0 + m · (−1) 1 + 2 · (E ∗ )2

Case 1: E ∗ = 0 then N ∗ = ω − 1. First fixed point: (N (0) , E (0) , φ (0) ) = (ω − 1, 0, φ ∗ ); φ ∗ ∈ R. ∗ 2 ) Case 2: ω−1−(E + m · (−1)n = 0 1+2·(E ∗ )2 ω − 1 − (E ∗ )2 ω − 1 − (E ∗ )2 n + m · (−1) = 0 ⇒ 1 + 2 · (E ∗ )2 1 + 2 · (E ∗ )2 n ∗ 2 = −m · (−1) ⇒ ω − 1 − (E ) = −m · (−1)n · [1 + 2 · (E ∗ )2 ] (E ∗ )2 · [m · (−1)n · 2 − 1] = 1 − ω − m · (−1)n ; (E ∗ )2 ! 1 − ω − m · (−1)n 1 − ω − m · (−1)n ∗ = = ± ; E m · (−1)n · 2 − 1 m · (−1)n · 2 − 1 Case 2.1: E ∗ =



1−ω−m·(−1)n m·(−1)n ·2−1

 2 1−ω−m·(−1)n ω − 1 − n m·(−1) ·2−1 ω − 1 − (E ) N∗ = ⇒ N∗ = 2  1 + 2 · (E ∗ )2 1−ω−m·(−1)n 1+2· n m·(−1) ·2−1 " # 1−ω−m·(−1)n ω − 1 − m·(−1)n ·2−1 # " N∗ = n 1 + 2 · 1−ω−m·(−1) n m·(−1) ·2−1 ∗ 2

(ω − 1) · (m · (−1)n · 2 − 1) − (1 − ω − m · (−1)n ) m · (−1)n · 2 − 1 + 2 · (1 − ω − m · (−1)n ) [ω · 2 − 1] · m · (−1)n ; N ∗ = −m · (−1)n N∗ = 1−2·ω  n Second fixed point: (N (1) , E (1) , φ (1) ) = (−m·(−1)n , 1−ω−m·(−1) , φ ∗ ); φ ∗ ∈ R. m·(−1)n ·2−1  n Case 2.2: E ∗ = − 1−ω−m·(−1) m·(−1)n ·2−1 N∗ =

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

379

  2 1−ω−m·(−1)n ω − 1 − − m·(−1)n ·2−1 ω − 1 − (E ) N∗ = ⇒ N∗ = 2   ∗ 2 n 1 + 2 · (E ) 1 + 2 · − 1−ω−m·(−1) n m·(−1) ·2−1 " # 1−ω−m·(−1)n ω − 1 − m·(−1)n ·2−1 # ; " N∗ = n 1 + 2 · 1−ω−m·(−1) n m·(−1) ·2−1 ∗ 2

[ω · 2 − 1] · m · (−1)n ; N ∗ = −m · (−1)n 1−2·ω    n ∗ ; φ∗ ∈ Third fixed point: (N (2) , E (2) , φ (2) ) = −m · (−1)n , − 1−ω−m·(−1) , φ m·(−1)n ·2−1 N∗ =

R.

We can summary our system fixed points: First fixed point: (N (0) , E (0) , φ (0) ) = (ω − 1, 0, φ ∗ ); φ ∗ ∈ R; ∀ n = 0, 1, 2, . . . ; n ∈ N0 .    n Second fixed point: (N (1) , E (1) , φ (1) ) = −m · (−1)n , 1−ω−m·(−1) , φ∗ ; φ∗ ∈ m·(−1)n ·2−1 R.    n ∗ ; φ∗ ∈ , φ Third fixed point: (N (2) , E (2) , φ (2) ) = −m · (−1)n , − 1−ω−m·(−1) m·(−1)n ·2−1 R. Stability analysis: The standard local stability analysis about any one of the equilibrium points of lasers with frequency-modulated optical feedback system consists in adding to coordinates [N E φ] arbitrarily small increments of exponential form [n E φ] · eλ·t , and retaining the first order terms in N E φ. The system of three homogenous equations leads to a polynomial characteristics equation in the eigenvalues λ. The polynomial characteristics equations accept by set the below variables and variables derivative respect to time into lasers with frequency-modulated optical feedback system equations. System fixed values with arbitrarily small increments of exponential form [n E φ] · eλ·t are i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc. [5–7]. N (t) = N (i) + n · eλ·t ; E(t) = E (i) + E · eλ·t ; φ(t) = φ (i) + φ · eλ·t ; E(t − t D ) = E (i) + E · eλ·(t−t D ) d N (t) φ(t − t D ) = φ (i) + φ · eλ·(t−t D ) ; = λ · n · eλ·t dt d E(t) dφ(t) = λ · E · eλ·t ; = λ · φ · eλ·t dt dt d N (t) = K · {ω − 1 − N (t) − [1 + 2 · N (t)] · E 2 (t)} dt λ · n · eλ·t = K · {ω − 1 − (N (i) + n · eλ·t ) − [1 + 2 · (N (i) + n · eλ·t )] · (E (i) + E · eλ·t )2 } λ · n · eλ·t = K · {ω − 1 − N (i) − n · eλ·t − [1 + (2 · N (i) + 2 · n · eλ·t )] · ([E (i) ]2 + 2 · E (i) · E · eλ·t + E 2 · e2·λ·t )}

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4 Solid State Laser Nonlinearity Applications in Engineering

Assumption: E 2 ≈ 0 λ · n · eλ·t = K · {ω − 1 − N (i) − n · eλ·t 3 − [(1 + 2 · N (i) ) + 2 · n · eλ·t ] · ([E (i) ]2 + 2 · E (i) · E · eλ·t )} λ · n · eλ·t = K · {ω − 1 − N (i) − n · eλ·t − [(1 + 2 · N (i) ) · [E (i) ]2 + (1 + 2 · N (i) ) · 2 · E (i) · E · eλ·t + [E (i) ]2 · 2 · n · eλ·t + 4 · E (i) · n · E · e2·λ·t ]} Assumption: n · E ≈ 0 λ · n · eλ·t = K · {ω − 1 − N (i) − (1 + 2 · N (i) ) · [E (i) ]2 } − K · n · eλ·t + K · (1 + 2 · N (i) ) · 2 · E (i) · E · eλ·t + K · [E (i) ]2 · 2 · n · eλ·t At fixed points: K · {ω − 1 − N (i) − (1 + 2 · N (i) ) · [E (i) ]2 } = 0. λ · n · eλ·t = −K · n · eλ·t + K · (1 + 2 · N (i) ) · 2 · E (i) · E · eλ·t + K · [E (i) ]2 · 2 · n · eλ·t − λ · n · eλ·t − K · n · eλ·t + K · [E (i) ]2 · 2 · n · eλ·t + K · (1 + 2 · N (i) ) · 2 · E (i) · E · eλ·t = 0

Divide the above equation by eλ·t term. (−λ − K + K · [E (i) ]2 · 2) · n + K · (1 + 2 · N (i) ) · 2 · E (i) · E = 0 (−λ + K · {2 · [E (i) ]2 − 1}) · n + 2 · K · (1 + 2 · N (i) ) · E (i) · E = 0   d E(t) = N (t) · E(t) + m · E(t − t D ) × cos ψ(t) + 2 · ε · [N (t) + 1] · ξ(t) dt Assumption: √ The quantum spontaneous√emission noise is very small and negligible ( 2 · ε · [N (t) + 1]) · ξ(t)  1; ( 2 · ε · [N (t) + 1]) · ξ(t) → 0. d E(t) = N (t) · E(t) + m · E(t − t D ) × cos ψ(t) dt λ · E · eλ·t = (N (i) + n · eλ·t ) · (E (i) + E · eλ·t ) + m · (E (i) + E · eλ·(t−t D ) ) × cos{ · t + β · sin(m · t)   ) · tD −φ(t) + φ(t − t D ) − (0 + 2 λ · E · eλ·t = (N (i) + n · eλ·t ) · (E (i) + E · eλ·t ) + m · (E (i) + E · eλ·(t−t D ) ) × cos{ · t + β · sin(m · t)

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

−(φ (i) + φ · eλ·t ) + (φ (i) + φ · eλ·(t−t D ) ) − (0 +

381

 ) · tD 2



First we do algebraic manipulation for the cos{. . .} expression.  cos  · t + β · sin(m · t) − (φ (i) + φ · eλ·t )     · tD +(φ (i) + φ · eλ·(t−t D ) ) − 0 + 2  (i) cos  · t + β · sin(m · t) − φ − φ · eλ·t   +φ (i) + φ · eλ·(t−t D ) − 0 · t D − · tD 2  λ·t cos  · t + β · sin(m · t) − φ · e   +φ · eλ·(t−t D ) − 0 · t D − · tD 2 cos{[ · t + β · sin(m · t) − 0 · t D ]   · tD +φ · eλ·(t−t D ) − φ · eλ·t − 2 cos{[ · t + β · sin(m · t) − 0 · t D ]    · tD + φ · eλ·t · (e−λ·t D − 1) − 2 We can write our second differential equation by adding to coordinates [N E φ] arbitrarily small increments of exponential form[n E φ] · eλ·t λ · E · eλ·t = (N (i) + n · eλ·t ) · (E (i) + E · eλ·t ) + m · (E (i) + E · eλ·(t−t D ) ) × cos{[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D · tD − 1) − + φ · e · (e 2 &&&    [ · t + β · sin(m · t) − 0 · t D ] + φ · eλ·t · (e−λ·t D − 1) − · tD 2    λ·t −λ·t D − 1) − = cos[ · t + β · sin(m · t) − 0 · t D ] · cos φ · e · (e · tD 2    λ·t −λ·t D − 1) − − sin[ · t + β · sin(m · t) − 0 · t D ] · sin φ · e · (e · tD 2 

cos

   λ·t −λ·t D · tD cos φ · e · (e − 1) − 2

382

4 Solid State Laser Nonlinearity Applications in Engineering

  · tD 2    · tD + sin[φ · eλ·t · (e−λ·t D − 1)] · sin 2    sin φ · eλ·t · (e−λ·t D − 1) − · tD 2    λ·t −λ·t D · tD = sin[φ · e · (e − 1)] · cos 2    λ·t −λ·t D · tD − cos[φ · e · (e − 1)] · sin 2  = cos[φ · eλ·t · (e−λ·t D − 1)] · cos



   cos [ · t + β · sin(m · t) − 0 · t D ] + φ · eλ·t · (e−λ·t D − 1) − · tD 2 = cos[ · t + β · sin(m · t) − 0 · t D ] · {cos[φ · eλ·t · (e−λ·t D − 1)]       · cos · t D + sin[φ · eλ·t · (e−λ·t D − 1)] · sin · tD 2 2     · tD − sin[ · t + β · sin(m · t) − 0 · t D ] · sin[φ · eλ·t · (e−λ·t D − 1)] · cos 2    λ·t −λ·t D − 1)] · sin · tD − cos[φ · e · (e 2

Assumption (A): ξ1 (φ, λ, t D ) = φ · eλ·t · (e−λ·t D − 1); sin[φ · eλ·t · (e−λ·t D − 1)] = sin(ξ1 (φ, λ, t D )) =

∞  n=0

sin(ξ1 (φ, λ, t D )) =

∞  n=0

(−1)n · [ξ1 (φ, λ, t D )]2·n+1 (2 · n + 1)!

(−1)n · [ξ1 (φ, λ, t D )]2·n+1 (2 · n + 1)!

[ξ1 (φ, λ, t D )]3 3! 5 [ξ1 (φ, λ, t D )]7 [ξ1 (φ, λ, t D )] − + ··· + 5! 7!

= ξ1 (φ, λ, t D ) −

sin(ξ1 (φ, λ, t D )) =

∞  n=0

(−1)n · [ξ1 (φ, λ, t D )]2·n+1 (2 · n + 1)! [φ · eλ·t · (e−λ·t D − 1)]3 3! [φ · eλ·t · (e−λ·t D − 1)]7 − 1)]5 − + ··· 7!

= φ · eλ·t · (e−λ·t D − 1) − +

[φ · eλ·t · (e−λ·t D 5!

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

sin(ξ1 (φ, λ, t D )) =

∞  n=0

383

(−1)n · [ξ1 (φ, λ, t D )]2·n+1 (2 · n + 1)! φ 3 · e3·λ·t · (e−λ·t D − 1)3 3! − 1)5 φ 7 · e7·λ·t · (e−λ·t D − 1)7 − + ··· 7!

= φ · eλ·t · (e−λ·t D − 1) − +

φ 5 · e5·λ·t · (e−λ·t D 5!

We consider φ m → ε ∀ m ≥ 2 then φ 3 · e3·λ·t · (e−λ·t D − 1)3 φ 5 · e5·λ·t · (e−λ·t D − 1)5 + 3! 5! φ 7 · e7·λ·t · (e−λ·t D − 1)7 − + ··· → ε 7!

−

sin(ξ1 (φ, λ, t D )) =

∞  n=0

(−1)n · [ξ1 (φ, λ, t D )]2·n+1 ≈ φ · eλ·t · (e−λ·t D − 1) (2 · n + 1)!

Assumption (B):     · t D ; sin · tD 2 2 ∞  = sin[ξ2 (, t D )] =

ξ2 (, t D ) =

n=0 ∞ 

sin(ξ2 (, t D )) =

n=0

(−1)n · [ξ2 (, t D )]2·n+1 (2 · n + 1)!

(−1)n · [ξ2 (, t D )]2·n+1 (2 · n + 1)!

[ξ2 (, t D )]3 3! [ξ2 (, t D )]7 [ξ2 (, t D )]5 − + ··· + 5! 7!

= ξ2 (, t D ) −

∞ 

(−1)n · [ξ2 (, t D )]2·n+1 (2 · n + 1)! n=0 $  $  $  %3 %5 %7 · tD · tD · tD  2 2 2 · tD − + − + ··· = 2 3! 5! 7! ∞  (−1)n sin(ξ2 (, t D )) = · [ξ2 (, t D )]2·n+1 (2 · n + 1)! n=0

sin(ξ2 (, t D )) =

 · tD − = 2

()3 23

3!

· t D3

+

()5 25

5!

· t D5

−

()7 27

7!

· t D7

+ ···

384

4 Solid State Laser Nonlinearity Applications in Engineering

We consider t Dm → ε ∀ m ≥ 2 then − sin(ξ2 (, t D )) =

∞  n=0

()3 23

·t D3

3!

+

()5 25

·t D5

5!

−

()7 27

·t D7

7!

+ ... → ε

(−1)n  · [ξ2 (, t D )]2·n+1 ≈ · tD (2 · n + 1)! 2

Assumption (C): ξ1 (φ, λ, t D ) = φ · eλ·t · (e−λ·t D − 1); cos[φ · eλ·t · (e−λ·t D − 1)] = cos(ξ1 (φ, λ, t D )) =

∞  (−1)n · [ξ1 (φ, λ, t D )]2·n (2 · n)! n=0

∞  (−1)n · [ξ1 (φ, λ, t D )]2·n cos(ξ1 (φ, λ, t D )) = (2 · n)! n=0

φ 4 · e4·λ·t · (e−λ·t D − 1)4 φ 2 · e2·λ·t · (e−λ·t D − 1)2 + − ··· 2! 4! ∀ φ · eλ·t · (e−λ·t D − 1)

=1−

φ 2 · e2·λ·t · (e−λ·t D − 1)2 2! We consider φ m → ε ∀ m ≥ 2 then φ 4 · e4·λ·t · (e−λ·t D − 1)4 − ··· → ε + 4! −

cos(ξ1 (φ, λ, t D )) =

∞  (−1)n · [ξ1 (φ, λ, t D )]2·n ≈ 1 (2 · n)! n=0

Assumption (D):     · t D ; cos · tD 2 2 ∞  (−1)n = cos(ξ2 (, t D )) = · [ξ2 (, t D )]2·n (2 · n)! n=0 2·n  ∞   (−1)n · · tD cos(ξ2 (, t D )) = (2 · n)! 2 n=0

ξ2 (, t D ) =

=1−

()2 22

2!

· t D2

+

We consider t Dm → ε ∀ m ≥ 2 then −

()4 24

· t D4

4!

()2 22

2!

·t D2

+

− ··· ∀

()4 24

4!

·t D4

 · tD 2

− ··· → ε

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

385

Table. 4.2 lasers with frequency-modulated optical feedback system assumptions A–D  (−1)n 2·n+1 ≈ φ · eλ·t · (e−λ·t D − 1) Assumption A sin(ξ1 (φ, λ, t D )) = ∞ n=0 (2·n+1)! · [ξ1 (φ, λ, t D )]  (−1)n 2·n+1 ≈  · t Assumption B sin(ξ2 (, t D )) = ∞ D n=0 (2·n+1)! · [ξ2 (, t D )] 2 ∞ (−1)n 2·n Assumption C cos(ξ1 (φ, λ, t D )) = n=0 (2·n)! · [ξ1 (φ, λ, t D )] ≈ 1 %2·n  (−1)n $  Assumption D cos(ξ2 (, t D )) = ∞ ≈1 n=0 (2·n)! · 2 · tD

2·n  ∞   (−1)n cos(ξ2 (, t D )) = ≈1 · · tD (2 · n)! 2 n=0 We can summary assumptions A–D in the next table (Table 4.2): We implement the above assumptions (A–D) in the below expression. cos{[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D · tD − 1) − + φ · e · (e 2 = cos[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D · tD − 1)] · cos · {cos[φ · e · (e 2    · tD + sin[φ · eλ·t · (e−λ·t D − 1)] · sin 2 − sin[ · t + β · sin(m · t) − 0 · t D ]    · tD · {sin[φ · eλ·t · (e−λ·t D − 1)] · cos 2    · tD − cos[φ · eλ·t · (e−λ·t D − 1)] · sin 2 cos{[ · t + β · sin(m · t) − 0 · t D ]    · tD + φ · eλ·t · (e−λ·t D − 1) − 2 = cos[ · t + β · sin(m · t) − 0 · t D ]    · tD · 1 + φ · eλ·t · (e−λ·t D − 1) · 2 − sin[ · t + β · sin(m · t) − 0 · t D ]    · tD · φ · eλ·t · (e−λ·t D − 1) − 2 We implement our last expression in the following formula

386

4 Solid State Laser Nonlinearity Applications in Engineering

λ · E · eλ·t = (N (i) + n · eλ·t ) · (E (i) + E · eλ·t ) + m · (E (i) + E · eλ·(t−t D ) ) × cos{[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D · tD − 1) − + φ · e · (e 2 λ · E · eλ·t = (N (i) + n · eλ·t ) · (E (i) + E · eλ·t ) + m · (E (i) + E · eλ·(t−t D ) ) × {cos[ · t + β · sin(m · t) − 0 · t D ]    · tD · 1 + φ · eλ·t · (e−λ·t D − 1) · 2 − sin[ · t + β · sin(m · t) − 0 · t D ]    · tD · φ · eλ·t · (e−λ·t D − 1) − 2 λ · E · eλ·t = N (i) · E (i) + N (i) · E · eλ·t + E (i) · n · eλ·t + n · E · e2·λ·t + [m · E (i) + m · E · eλ·(t−t D ) ] × {cos[ · t + β · sin(m · t) − 0 · t D ]    · tD · 1 + φ · eλ·t · (e−λ·t D − 1) · 2 − sin[ · t + β · sin(m · t) − 0 · t D ]    · tD · φ · eλ·t · (e−λ·t D − 1) − 2 Assumption: n · E ≈ 0 λ · E · eλ·t = N (i) · E (i) + N (i) · E · eλ·t + E (i) · n · eλ·t + [m · E (i) + m · E · eλ·(t−t D ) ] × {cos[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D · tD − 1) · · 1 + φ · e · (e 2 − sin[ · t + β · sin(m · t) − 0 · t D ]    λ·t −λ·t D − 1) − · φ · e · (e · tD 2 λ · E · eλ·t = N (i) · E (i) + N (i) · E · eλ·t + E (i) · n · eλ·t + [m · E (i) + m · E · eλ·(t−t D ) ] × {cos[ · t + β · sin(m · t) − 0 · t D ]  + φ · eλ·t · (e−λ·t D − 1) · · t D · cos[ · t + β · sin(m · t) − 0 · t D ] 2    − sin[ · t + β · sin(m · t) − 0 · t D ] · φ · eλ·t · (e−λ·t D − 1) − · tD 2

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

387

λ · E · eλ·t = N (i) · E (i) + N (i) · E · eλ·t + E (i) · n · eλ·t + m · E (i) × {cos[ · t + β · sin(m · t) − 0 · t D ]  + φ · eλ·t · (e−λ·t D − 1) · · t D · cos[ · t + β · sin(m · t) − 0 · t D ] 2 − sin[ · t + β · sin(m · t) − 0 · t D ] · [φ · eλ·t · (e−λ·t D − 1)   − · t D ] + m · E · eλ·(t−t D ) 2 × {cos[ · t + β · sin(m · t) − 0 · t D ]  + φ · eλ·t · (e−λ·t D − 1) · · t D · cos[ · t + β · sin(m · t) − 0 · t D ] 2    − sin[ · t + β · sin(m · t) − 0 · t D ] · φ · eλ·t · (e−λ·t D − 1) − · tD 2

λ · E · eλ·t = N (i) · E (i) + N (i) · E · eλ·t + E (i) · n · eλ·t + m · E (i) × cos[ · t + β · sin(m · t) − 0 · t D ]  · t D · cos[ · t + m · E (i) × φ · eλ·t · (e−λ·t D − 1) · 2 + β · sin(m · t) − 0 · t D ] − m · E (i) × sin[ · t + β · sin(m · t)    · t D + m · E · eλ·(t−t D ) − 0 · t D ] · φ · eλ·t · (e−λ·t D − 1) − 2 × {cos[ · t + β · sin(m · t) − 0 · t D ]  · t D · cos[ · t + φ · eλ·t · (e−λ·t D − 1) · 2 + β · sin(m · t) − 0 · t D ] − sin[ · t + β · sin(m · t)    λ·t −λ·t D · tD − 1) − −0 · t D ] · φ · e · (e 2 At fixed points: N (i) · E (i) + m · E (i) × cos[ · t + β · sin(m · t) − 0 · t D ] = 0. Remark ψ ∗ ≈  · t + β · sin(m · t) − 0 · t D ; dφ(t) = 0 ⇒ sin ψ ∗ = 0; ψ ∗ = dt n · π ∀ n = 0, 1, 2, . . . dE = 0 ⇒ N ∗ · E ∗ + m · E ∗ × cos(ψ ∗ ) dt = 0; n ∈ N0 ; n · π =  · t + β · sin(m · t) − 0 · t D λ · E · eλ·t = N (i) · E · eλ·t + E (i) · n · eλ·t + m · E (i) × φ · eλ·t · (e−λ·t D − 1)  · · t D · cos[ · t + β · sin(m · t) − 0 · t D ] 2 − m · E (i) × sin[ · t + β · sin(m · t) − 0 · t D ]

388

4 Solid State Laser Nonlinearity Applications in Engineering    · φ · eλ·t · (e−λ·t D − 1) − · t D + m · E · eλ·(t−t D ) 2  × cos[ · t + β · sin(m · t) − 0 · t D ] + φ · eλ·t · (e−λ·t D − 1)  · · t D · cos[ · t + β · sin(m · t) − 0 · t D ] 2    − sin[ · t + β · sin(m · t) − 0 · t D ] · φ · eλ·t · (e−λ·t D − 1) − · tD 2

n · π =  · t + β · sin(m · t) − 0 · t D ⇒ sin[ · t + β · sin(m · t) − 0 · t D ] = sin[n · π ] = 0 n · π =  · t + β · sin(m · t) − 0 · t D ⇒ cos[ · t + β · sin(m · t) − 0 · t D ] = cos[n · π ] = (−1)n = 0 ∀ n = 0, 1, 2, . . . Then we can write our second differential equation by adding to coordinates [N E φ] arbitrarily small increments of exponential form [n E φ] · eλ·t as follow. λ · E · eλ·t = N (i) · E · eλ·t + E (i) · n · eλ·t  · t D · (−1)n 2 + m · E · eλ·(t−t D ) × {(−1)n + φ · eλ·t · (e−λ·t D − 1)  · t D · (−1)n } ∀ n = 0, 1, 2, . . . · 2 + m · E (i) × φ · eλ·t · (e−λ·t D − 1) ·

λ · E · eλ·t = N (i) · E · eλ·t + E (i) · n · eλ·t  · t D · (−1)n 2 · (e−λ·t D − 1)

+ m · E (i) × φ · eλ·t · (e−λ·t D − 1) ·

+ m · E · eλ·(t−t D ) × {1 + φ · eλ·t  · t D } · (−1)n ∀ n = 0, 1, 2, . . . · 2 λ · E · eλ·t = N (i) · E · eλ·t + E (i) · n · eλ·t

 · t D · (−1)n + m · E (i) · φ · eλ·t · (e−λ·t D − 1) · 2  + m · E · eλ·(t−t D ) + m · E · φ · eλ·(t−t D ) · eλ·t · (e−λ·t D − 1)   · t D · (−1)n ∀ n = 0, 1, 2, . . . · 2 Assumption: E · φ ≈ 0 λ · E · eλ·t = N (i) · E · eλ·t + E (i) · n · eλ·t

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

389

 · t D · (−1)n 2 + m · E · eλ·(t−t D ) · (−1)n ∀ n = 0, 1, 2, . . . + m · E (i) · φ · eλ·t · (e−λ·t D − 1) ·

− λ · E · eλ·t + N (i) · E · eλ·t + E (i) · n · eλ·t  + m · E (i) · φ · eλ·t · (e−λ·t D − 1) · · t D · (−1)n 2 + m · E · eλ·t · e−λ·t D · (−1)n = 0 ∀ n = 0, 1, 2, . . . E (i) · n · eλ·t − λ · E · eλ·t + N (i) · E · eλ·t + m · E · eλ·t · e−λ·t D · (−1)n  · tD + m · E (i) · φ · eλ·t · (e−λ·t D − 1) · 2 n · (−1) = 0 ∀ n = 0, 1, 2, . . . Divide the two side of the above equation by eλ·t term, we get the expression: E (i) · n + [N (i) + m · e−λ·t D · (−1)n − λ] · E  · t D · (−1)n · φ= 0 ∀ n = 0, 1, 2, . . . + m · E (i) · (e−λ·t D − 1) · 2 We can write our Third differential equation by adding to coordinates [N E φ] arbitrarily small increments of exponential form [n E φ] · eλ·t dφ(t) = dt



 E(t − t D ) · sin ψ(t); ψ(t) =  · t E(t)

   + β · sin(m · t) − φ(t) + φ(t − t D ) − 0 + · tD 2   E (i) + E · eλ·(t−t D ) · sin[ · t + β · sin(m · t) λ · φ · eλ·t = E (i) + E · eλ·t     · tD −φ (i) − φ · eλ·t + φ (i) + φ · eλ·(t−t D ) − 0 + 2   (i) λ·(t−t ) D E + E ·e λ · φ · eλ·t = · sin[ · t E (i) + E · eλ·t     · tD +β · sin(m · t) − φ · eλ·t + φ · eλ·(t−t D ) − 0 + 2     (i) λ·(t−t ) (i) λ·t D E − E ·e E + E ·e λ · φ · eλ·t = · E (i) + E · eλ·t E (i) − E · eλ·t      · sin  · t + β · sin(m · t) − φ · eλ·t + φ · eλ·(t−t D ) − 0 + · tD 2   (i) λ·(t−t ) D E + E ·e λ · φ · eλ·t = · [E (i) − E · eλ·t ] (E (i) )2 − E 2 · e2·λ·t

390

4 Solid State Laser Nonlinearity Applications in Engineering      · tD · sin  · t + β · sin(m · t) − φ · eλ·t + φ · eλ·(t−t D ) − 0 + 2   (i) 2 (i) λ·t (i) λ·(t−t ) 2 λ·(t−t D D ) · e λ·t − E ·e (E ) − E · E · e + E · E · e λ · φ · eλ·t = (E (i) )2 − E 2 · e2·λ·t · sin[ · t + β · sin(m · t)     −φ · eλ·t + φ · eλ·(t−t D ) − 0 + · tD 2

Assumption: E 2 ≈ 0 

 (E (i) )2 − E (i) · E · eλ·t + E (i) · E · eλ·(t−t D ) λ·φ·e = · sin[ · t (E (i) )2     · tD +β · sin(m · t) − φ · eλ·t + φ · eλ·(t−t D ) − 0 + 2   (i) 2 (i) λ·t (i) λ·(t−t ) D (E ) − E · E · e + E · E · e λ · φ · eλ·t = (E (i) )2    · sin  · t + β · sin(m · t) + φ · eλ·t · (e−λ·t D − 1) − 0 · t D − · tD 2   λ·t λ·(t−t ) D E ·e E ·e + λ · φ · eλ·t = 1 − · sin[ · t + β · sin(m · t) E (i) E (i)   +φ · eλ·t · (e−λ·t D − 1) − 0 · t D − · tD 2   λ·t λ·(t−t D ) E · e E · e λ·t λ·φ·e = 1− · sin[( · t + E (i) E (i)   +β · sin(m · t) − 0 · t D ) + (φ · eλ·t · (e−λ·t D − 1) − · tD 2 λ·t

&&& sin[( · t + β · sin(m · t) − 0 · t D )    · tD + φ · eλ·t · (e−λ·t D − 1) − 2 = sin( · t + β · sin(m · t) − 0 · t D )    · tD · cos φ · eλ·t · (e−λ·t D − 1) − 2 + cos( · t + β · sin(m · t) − 0 · t D )    λ·t −λ·t D · tD − 1) − · sin φ · e · (e 2 &&&

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

391

Table. 4.3 lasers with frequency-modulated optical feedback system assumptions A–D Assumption A ξ1 (φ, λ, t D ) = φ · eλ·t · (e−λ·t D − 1)  (−1)n 2·n+1 ≈ φ · eλ·t · (e−λ·t D − 1) sin(ξ1 (φ, λ, t D )) = ∞ n=0 (2·n+1)! · [ξ1 (φ, λ, t D )] Assumption B ξ2 (, t D ) =

 2

(−1)n 2·n+1 n=0 (2·n+1)! · [ξ2 (, t D )] · eλ·t · (e−λ·t D − 1)

sin(ξ2 (, t D )) = Assumption C ξ1 (φ, λ, t D ) = φ

· tD ∞

cos(ξ1 (φ, λ, t D )) = Assumption D ξ2 (, t D ) =

∞

(−1)n n=0 (2·n)!

(−1)n (2·n)!

 2

· tD

· [ξ1 (φ, λ, t D )]2·n ≈ 1

 2

· tD  cos(ξ2 (, t D )) = ∞ n=0

≈

·

$  2

· tD

%2·n

≈1

  · tD cos φ · e · (e − 1) − 2    λ·t −λ·t D · tD = cos[φ · e · (e − 1)] · cos 2    λ·t −λ·t D · tD + sin[φ · e · (e − 1)] · sin 2    · tD sin φ · eλ·t · (e−λ·t D − 1) − 2    · tD = sin[φ · eλ·t · (e−λ·t D − 1)] · cos 2    · tD − cos[φ · eλ·t · (e−λ·t D − 1)] · sin 2 

λ·t

−λ·t D

We already got the following results by assumptions (Table 4.3). We implement the above assumptions (A–D) in the below expression.     λ·t −λ·t D · t D = 1 + φ · eλ·t · (e−λ·t D − 1) · · tD cos φ · e · (e − 1) − 2 2     · t D = φ · eλ·t · (e−λ·t D − 1) − · tD sin φ · eλ·t · (e−λ·t D − 1) − 2 2     λ·t −λ·t D · tD sin ( · t + β · sin(m · t) − 0 · t D ) + φ · e · (e − 1) − 2    λ·t −λ·t D · tD = sin( · t + β · sin(m · t) − 0 · t D ) · 1 + φ · e · (e − 1) · 2    λ·t −λ·t D · tD + cos( · t + β · sin(m · t) − 0 · t D ) · φ · e · (e − 1) − 2 Implement it in the third differential equation arbitrarily small increments, we get.

392

4 Solid State Laser Nonlinearity Applications in Engineering

λ·φ·e

λ·t

 λ·φ·e

λ·t

= 1−

  E · eλ·t E · eλ·(t−t D ) = 1− + E (i) E (i) · sin[( · t + β · sin(m · t) − 0 · t D )    λ·t −λ·t D − 1) − + φ · e · (e · t D λ · φ · eλ·t 2   E · eλ·t E · eλ·(t−t D ) = 1− + E (i) E (i) · {sin( · t + β · sin(m · t) − 0 · t D )    λ·t −λ·t D · tD − 1) · · 1 + φ · e · (e 2 + cos( · t + β · sin(m · t)    λ·t −λ·t D − 1) − −0 · t D ) · φ · e · (e · tD 2  E · eλ·t E · eλ·(t−t D ) + E (i) E (i)

   · 1 + φ · eλ·t · (e−λ·t D − 1) · · tD 2

· sin( · t + β · sin(m · t) − 0 · t D )     E · eλ·t E · eλ·(t−t D )  λ·t −λ·t D + 1− · φ · e + · (e − 1) − · t D 2 E (i) E (i) · cos( · t + β · sin(m · t) − 0 · t D )

    E · eλ·t  −λ·t D λ·t −λ·t D · t λ · φ · eλ·t = 1 + · [e − 1] · 1 + φ · e · (e − 1) · D E (i) 2 · sin( · t + β · sin(m · t) − 0 · t D )     E · eλ·t  −λ·t D λ·t −λ·t D + 1+ · tD · [e − 1] · φ · e · (e − 1) − E (i) 2 · cos( · t + β · sin(m · t) − 0 · t D ) λ·φ·e

λ·t

  E · eλ·t = 1 + φ · eλ·t · (e−λ·t D − 1) · · [e−λ·t D − 1] · tD + 2 E (i)  E · φ · eλ·t  −λ·t D λ·t −λ·t D + · tD · [e − 1] · e · (e − 1) · E (i) 2 · sin( · t + β · sin(m · t) − 0 · t D )   · tD + φ · eλ·t · (e−λ·t D − 1) − 2 E · φ · eλ·t + · [e−λ·t D − 1] · eλ·t · (e−λ·t D − 1) E (i)  E · eλ·t  −λ·t D − · t · [e − 1] · D E (i) 2

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

393

· cos( · t + β · sin(m · t) − 0 · t D ) Assumption: E · φ ≈ 0 λ · φ · eλ·t = {1 + φ · eλ·t · (e−λ·t D − 1) ·

 · tD 2

E · eλ·t · [e−λ·t D − 1]} · sin( · t + β · sin(m · t) − 0 · t D ) E (i)   E · eλ·t + φ · eλ·t · (e−λ·t D − 1) − · tD − · [e−λ·t D − 1] 2 E (i)   · t D · cos( · t + β · sin(m · t) − 0 · t D ) · 2 +

λ · φ · eλ·t = sin( · t + β · sin(m · t) − 0 · t D )    E · eλ·t −λ·t D · [e − 1] + φ · eλ·t · (e−λ·t D − 1) · · tD + 2 E (i) · sin( · t + β · sin(m · t) − 0 · t D )    E · eλ·t  −λ·t D · [e − 1] · + φ · eλ·t · (e−λ·t D − 1) − · tD − · t D 2 2 E (i) · cos( · t + β · sin(m · t) − 0 · t D )

At fixed point: t) − 0 · t D then

E∗ E∗

· sin(ψ ∗ ) = 0 ⇒ sin(ψ ∗ ) = 0; ψ ∗ ≈  · t + β · sin(m ·

λ · φ · eλ·t = {φ · eλ·t · (e−λ·t D − 1) ·

 · tD 2

E · eλ·t · [e−λ·t D − 1]} · sin( · t + β · sin(m · t) − 0 · t D ) E (i)  E · eλ·t · tD − · [e−λ·t D − 1] + {φ · eλ·t · (e−λ·t D − 1) − 2 E (i)  · t D } · cos( · t + β · sin(m · t) − 0 · t D ) · 2

+

dφ(t) = 0 ⇒ sin ψ ∗ = 0; ψ ∗ = n · π ∀ n = 0, 1, 2, . . . dt n ∈ N0 ; n · π =  · t + β · sin(m · t) − 0 · t D sin(n · π ) = sin( · t + β · sin(m · t) − 0 · t D ) = 0 sin ψ ∗ = 0 ⇒ cos ψ ∗ = ±1; n = 0, 1, 2, . . . ; n ∈ N0 ; ψ ∗ = n · π ; cos ψ ∗ = (−1)n cos(n · π ) = cos( · t + β · sin(m · t) − 0 · t D ) = (−1)n λ · φ · eλ·t    E · eλ·t  = φ · eλ·t · (e−λ·t D − 1) − · [e−λ·t D − 1] · · tD − · t D · (−1)n (i) 2 2 E

394

4 Solid State Laser Nonlinearity Applications in Engineering λ·t

Assumption: φ · eλ·t · (e−λ·t D − 1) − E·e · [e−λ·t D − 1] ·  · t D   · tD 2 2 E (i)   E   1  −λ·t D λ·t (e − 1) · e · φ − E (i) · 2 · t D  2 · t D ; φ  E · t D · E (i) · 2 then  E · eλ·t  · [e−λ·t D − 1] · · tD − · tD 2 E (i) 2 E · eλ·t  · (e−λ·t D − 1) − · [e−λ·t D − 1] · · tD (i) E 2   E  · t D · eλ·t · (−1)n = φ · (e−λ·t D − 1) − (i) · [e−λ·t D − 1] · E 2

φ · eλ·t · (e−λ·t D − 1) − ≈ φ · eλ·t λ · φ · eλ·t

Divide the above expression by eλ·t term. −

 1 · t D · (−1)n · E + [(e−λ·t D − 1) · (−1)n − λ] · φ = 0 · [e−λ·t D − 1] · (i) E 2

We can summary our arbitrary small increments equations. (−λ + K · {2 · [E (i) ]2 − 1}) · n + 2 · K · (1 + 2 · N (i) ) · E (i) · E = 0 E (i) · n + [N (i) + m · e−λ·t D · (−1)n − λ] · E + m · E (i) · (e−λ·t D − 1)  · · t D · (−1)n · φ= 0 ∀ n = 0, 1, 2, . . . 2 1  − (i) · [e−λ·t D − 1] · · t D · (−1)n · E + [(e−λ·t D − 1) E 2 · (−1)n − λ] · φ = 0 ∀ n = 0, 1, 2, . . . The small increments Jacobian of our lasers with frequency-modulated optical feedback system is as follow:

We inspect the occurrence of any possible stability switching resulting from the increase of value of time delay t D (delay time normalized by the photon lifetime) for lasers with frequency-modulated optical feedback system general characteristic equation D(λ, t D ) = det |A − λ · I |.

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1})  · N (i) + m · e−λ·t D · (−1)n − λ] · [(e−λ·t D − 1) · (−1)n − λ]    1 · t D · (−1)n − − (i) · [e−λ·t D − 1] · E 2    · t D · (−1)n · m · E (i) · (e−λ·t D − 1) · 2 − 2 · K · (1 + 2 · N (i) ) · E (i) · E (i) · [(e−λ·t D − 1) · (−1)n − λ] D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1}) · {[N (i) + m · e−λ·t D · (−1)n − λ] · [(e−λ·t D − 1) · (−1)n − λ]    2 −λ·t D 2 − 1] · · m · t D2 · (−1)n · (−1)n } + [e 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · [(e−λ·t D − 1) · (−1)n − λ] Hint: (−1)n · (−1)n = 1 ∀ n ∈ N0 D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1})  · [N (i) + m · e−λ·t D · (−1)n − λ] · [(e−λ·t D − 1) · (−1)n − λ]  2   +[e−λ·t D − 1]2 · · m · t D2 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · [(e−λ·t D − 1) · (−1)n − λ] D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1})

395

396

4 Solid State Laser Nonlinearity Applications in Engineering

 · [N (i) + m · e−λ·t D · (−1)n − λ] · [e−λ·t D · (−1)n − (−1)n − λ]     2 −2·λ·t D −λ·t D 2 −2·e + 1] · · m · tD +[e 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · [(e−λ·t D − 1) · (−1)n − λ] D(λ, t D ) = det |A − λ · I |

 = (−λ + K · {2 · [E (i) ]2 − 1}) · N (i) · e−λ·t D · (−1)n − N (i) · (−1)n − N (i) · λ + e−λ·t D · m · e−λ·t D · (−1)n · (−1)n − m · e−λ·t D · (−1)n · (−1)n − λ · m · e−λ·t D · (−1)n − λ · e−λ·t D · (−1)n +λ · (−1) + λ + [e n

2

−2·λ·t D

−2·e

−λ·t D



+ 1] ·

 2



2 ·m·

t D2

− 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · [(e−λ·t D − 1) · (−1)n − λ] Hint: (−1)n · (−1)n = 1 ∀ n ∈ N0 D(λ, t D ) = det |A − λ · I |

 = (−λ + K · {2 · [E (i) ]2 − 1}) · N (i) · e−λ·t D · (−1)n − N (i) · (−1)n − N (i) · λ + m · e−λ·t D · e−λ·t D − m · e−λ·t D − λ · m · e−λ·t D · (−1)n − λ · e−λ·t D · (−1)n + λ · (−1)n + λ2     2 −2·λ·t D −λ·t D 2 +[e −2·e + 1] · · m · tD 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · [e−λ·t D · (−1)n − (−1)n − λ]

D(λ, t D ) = det |A − λ · I |

 = (−λ + K · {2 · [E (i) ]2 − 1}) · N (i) · e−λ·t D · (−1)n − m · e−λ·t D − λ · m · e−λ·t D · (−1)n − λ · e−λ·t D · (−1)n − N (i) · (−1)n + λ · (−1)n − N (i) · λ + λ2 + m · e−2·λ·t D     2 −2·λ·t D −λ·t D 2 +[e −2·e + 1] · · m · tD 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

397

D(λ, t D ) = det |A − λ · I |

 = (−λ + K · {2 · [E (i) ]2 − 1}) · N (i) · e−λ·t D · (−1)n − m · e−λ·t D − λ · m · e−λ·t D · (−1)n − λ · e−λ·t D · (−1)n    2 −λ·t D −2·e · · m · t D2 2    2 + · m · t D2 − N (i) · (−1)n + λ · (−1)n 2     2 (i) 2 −2·λ·t D −2·λ·t D 2 −N · λ + λ + m · e +e · · m · tD 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ

D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1}) ·

$

N (i) · (−1)n − m    2 − λ · m · (−1)n − λ · (−1)n − 2 · 2 2  %  ·m · t D2 · e−λ·t D + · m · t D2 − N (i) · (−1)n 2      2 2 n (i) 2 −2·λ·t D +[(−1) − N ] · λ + λ + m · e · 1+ · tD 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n 4 + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ

Assumption: K =

τ τp

is the population-to-photon lifetime ratio (τ p ≈ 600 ps) then

typical value for K is K = 2 × 105 .  = (ωi − ω0 ) · τ p is the normalized frequency −5 amplitude feedback coefficient, m ≈ 10 shift (τ p is the photon lifetime). m is the # "   2 2   2 2 #   2  then 2 · t D  1, 1 + 2 · t D ≈ 1; and m · e−2·λ·t D · 1 + 2 · t D2 is very small compare to other elements in the above expression and we can ignore it and assume as a very small expression value. Then we get. D(λ, t D ) = det |A − λ · I | = (−λ + K · {2 · [E (i) ]2 − 1}) ·

$

N (i) · (−1)n − m

398

4 Solid State Laser Nonlinearity Applications in Engineering

 −λ · m · (−1) − λ · (−1) − 2 · n

 +

 2

2

n

 2



2 ·m·

t D2

· e−λ·t D

· m · t D2 − N (i) · (−1)n

+ [(−1)n − N (i) ] · λ + λ2 } − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ $

N (i) · (−1)n − m     2 n n 2 · m · tD −λ · m · (−1) − λ · (−1) − 2 · 2    2 · m · t D2 − N (i) · (−1)n · e−λ·t D + 2  +[(−1)n − N (i) ] · λ + λ2 + K · (2 · [E (i) ]2 − 1) · {[N (i) · (−1)n − m    2 n n − λ · m · (−1) − λ · (−1) − 2 · 2 2  %  ·m · t D2 · e−λ·t D + · m · t D2 − N (i) · (−1)n 2

D(λ, t D ) = det |A − λ · I | = −λ ·

+ [(−1)n − N (i) ] · λ + λ2 } − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ  D(λ, t D ) = det |A − λ · I | = − λ · [N (i) · (−1)n − m − λ · m · (−1)n − λ · (−1)n       2  2 2 −λ·t D −2 · · m · tD · e +λ· · m · t D2 2 2  −λ · N (i) · (−1)n + [(−1)n − N (i) ] · λ2 + λ3 + {K · (2 · [E (i) ]2 − 1) $ · (N (i) · (−1)n − m) − λ · (−1)n     2 2 ·(m + 1) − 2 · · m · t D · e−λ·t D 2

4.2 Nanometer-Vibration Measurement with Microchip Solid-State … (i) 2

+ K · (2 · [E ] − 1) ·



 2

399

2 · m · t D2

− K · (2 · [E (i) ]2 − 1) · N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ] · λ + K · (2 · [E (i) ]2 − 1) · λ2 } − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ $  D(λ, t D ) = det |A − λ · I | = −λ · N (i)  2   · (−1)n − m − 2 · · m · t D2 2 + λ2 · m · (−1)n + λ2 · (−1)n ] · e−λ·t D    2 −λ· · m · t D2 + λ · N (i) · (−1)n − [(−1)n − N (i) ] · λ2 − λ3 2 + [K · (2 · [E (i) ]2 − 1) · (N (i) · (−1)n − m) − K · (2 · [E (i) ]2 − 1) · λ · (−1)n · (m + 1)     2 (i) 2 2 · m · t D · e−λ·t D −K · (2 · [E ] − 1) · 2 · 2    2 + K · (2 · [E (i) ]2 − 1) · · m · t D2 2 − K · (2 · [E (i) ]2 − 1) · N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ] · λ + K · (2 · [E (i) ]2 − 1) · λ2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · e−λ·t D · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ $ D(λ, t D ) = det |A − λ · I | = −λ · (N (i) · (−1)n    2 −m−2· · m · t D2 ) + λ2 · m · (−1)n + λ2 · (−1)n ] · e−λ·t D 2 + [K · (2 · [E (i) ]2 − 1) · (N (i) · (−1)n − m) − K · (2 · [E (i) ]2 − 1) · λ · (−1)n · (m + 1)

400

4 Solid State Laser Nonlinearity Applications in Engineering (i) 2

− K · (2 · [E ] − 1) · 2 ·



 2

2

· m · t D2 ] · e−λ·t D

− 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n · e−λ·t D    2 + K · (2 · [E (i) ]2 − 1) · · m · t D2 − K · (2 · [E (i) ]2 − 1) 2 · N (i) · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + λ · N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ]    2 · m · t D2 · λ + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · λ − λ · 2 + K · (2 · [E (i) ]2 − 1) · λ2 − [(−1)n − N (i) ] · λ2 − λ3 D(λ, t D ) = det |A − λ · I | = [−λ · (N (i) · (−1)n − m − 2     2 2 · m · t D + λ2 · m · (−1)n + λ2 · (−1)n ] · e−λ·t D · 2 + [K · (2 · [E (i) ]2 − 1) · (N (i) · (−1)n − m) − K · (2 · [E (i) ]2 − 1) · λ · (−1)n · (m + 1)     2 (i) 2 2 · m · t D · e−λ·t D −K · (2 · [E ] − 1) · 2 · 2 − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n · e−λ·t D  2   +K · (2 · [E (i) ]2 − 1) · · m · t D2 − K · (2 · [E (i) ]2 − 1 2 · N (i) · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + {N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ]     2 (i) (i) 2 2 · m · tD · λ +2 · K · (1 + 2 · N ) · (E ) − 2 + {K · (2 · [E (i) ]2 − 1) − [(−1)n − N (i) ]} · λ2 − λ3 D(λ, t D ) = det |A − λ · I | = {K · (2 · [E (i) ]2 − 1) · (N (i) · (−1)n − m) − K · (2 · [E (i) ]2 − 1) · 2 · − 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n    2 (i) n − [(N · (−1) − m − 2 · · m · t D2 ) 2 + K · (2 · [E (i) ]2 − 1) · (−1)n · (m + 1)]



 2

2 · m · t D2

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

· λ + [m · (−1)n + (−1)n ] · λ2 } · e−λ·t D    2 + K · (2 · [E (i) ]2 − 1) · · m · t D2 − K · (2 · [E (i) ]2 − 1) 2 · N (i) · (−1)n + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n + {N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ]    2 (i) (i) 2 · m · t D2 } · λ + 2 · K · (1 + 2 · N ) · (E ) − 2 + {K · (2 · [E (i) ]2 − 1) − [(−1)n − N (i) ]} · λ2 − λ3 We define for simplicity new global parameters: ϒ1 (n) = K · (2 · [E (i) ]2 − 1) · (N (i) · (−1)n − m); ϒ2    2 = K · (2 · [E (i) ]2 − 1) · 2 · ·m 2 ϒ3 (n) = 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n ; ϒ4 (n) = N (i) · (−1)n − m; ϒ5 (n) = K · (2 · [E (i) ]2 − 1) · (−1)n · (m + 1) ϒ6 (n) = m · (−1)n + (−1)n ; ϒ7 = K · (2 · [E (i) ]2 − 1)    2 · m; ϒ8 (n) = K · (2 · [E (i) ]2 − 1) · N (i) · (−1)n · 2 ϒ9 (n) = 2 · K · (1 + 2 · N (i) ) · (E (i) )2 · (−1)n ; ϒ11 (n) = K · (2 · [E (i) ]2 − 1) − [(−1)n − N (i) ] ϒ10 (n) = N (i) · (−1)n + K · (2 · [E (i) ]2 − 1) · [(−1)n − N (i) ] + 2 · K · (1 + 2 · N (i) ) · (E (i) )2 ϒ1 = ϒ1 (n); ϒ3 = ϒ3 (n); ϒ4 = ϒ4 (n); ϒ5 = ϒ5 (n); ϒ6 = ϒ6 (n); ϒ8 = ϒ8 (n); ϒ9 = ϒ9 (n) ϒ11 = ϒ11 (n); ϒ10 = ϒ10 (n)  D(λ, t D ) = det |A − λ · I | = ϒ1 − ϒ2 · t D2 − ϒ3       2 2 2 − ϒ4 − 2 · · m · t D + ϒ5 · λ + ϒ6 · λ · e−λ·t D 2   2   + ϒ7 · t D2 − ϒ8 + ϒ9 + ϒ10 − · m · t D2 · λ 2 + ϒ11 · λ2 − λ3

401

402

4 Solid State Laser Nonlinearity Applications in Engineering

Table. 4.4 ϒk (n); k = 1, 2, . . . , 11 expression for n parameter values (n = 0, 1, 2, . . .) ϒk (n); k = n = 0, 2, 4, 6, 8, . . .(n is even number) 1, 2, . . . , 11

n = 1, 3, 5, 7, 9, . . .(n is odd number)

ϒ2 (n)

K · (2 · [E (i) ]2 − 1) · (N (i) − m)  2 K · (2 · [E (i) ]2 − 1) · 2 ·  ·m 2

−K · (2 · [E (i) ]2 − 1) · (N (i) + m)  2 K · (2 · [E (i) ]2 − 1) · 2 ·  ·m 2

ϒ3 (n)

2 · K · (1 + 2 · N (i) ) · (E (i) )2

−2 · K · (1 + 2 · N (i) ) · (E (i) )2

ϒ4 (n)

N (i)

−(N (i) + m)

ϒ1 (n)

−m · [E (i) ]2

ϒ5 (n)

K · (2

ϒ6 (n)

m+1

−K · (2 · [E (i) ]2 − 1) · (m + 1)

− 1) · (m + 1)   2

−(m + 1)

ϒ8 (n)

K · (2

· [E (i) ]2

ϒ9 (n)

2 · K · (1 + 2 · N (i) ) · (E (i) )2

−2 · K · (1 + 2 · N (i) ) · (E (i) )2

ϒ10 (n)

N (i) + K · (2 · [E (i) ]2 − 1) · [1 − N (i) ]

−N (i) + K · (2 · [E (i) ]2 − 1) · [−1 − N (i) ]

ϒ7 (n)

− 1) · − 1) ·

+2 · K · (1 + 2 · N ϒ11 (n)

(i)

2 N (i)

K · (2 · [E (i) ]2 − 1) ·

  2

K · (2

· [E (i) ]2

·m

−K

(i) 2

· (2 · [E (i) ]2

− 1) ·

2

·m

N (i)

+2 · K · (1 + 2 · N (i) ) · (E (i) )2

) · (E )

K · (2 · [E (i) ]2 − 1) − [1 − N (i) ]

K · (2 · [E (i) ]2 − 1) + [1 + N (i) ]

We can summary our ϒk (n); k = 1, 2, . . . , 11 expression for n parameter values (n = 0, 1, 2, . . .) in the next table (Table 4.4): D(λ, t D ) = det |A − λ · I | = [ϒ7 · t D2 − ϒ8 + ϒ9 ]    2 · m · t D2 ] · λ + ϒ11 · λ2 − λ3 + [ϒ10 − 2 % $ + ϒ1 − ϒ2 · t D2 − ϒ3       2 2 2 · m · t D + ϒ5 · λ + ϒ6 · λ · e−λ·t D − ϒ4 − 2 · 2 D(λ, t D ) = Pn (λ, t D ) + Q m (λ, t D ) · e−λ·t D ; n, m ∈ N0 ; n > m; n = 3; m = 2  Pn=3 (λ, t D ) = ϒ7 ·

t D2

− ϒ8 + ϒ9 + ϒ10 −



 2



2 ·m·

t D2

· λ + ϒ11 · λ2 − λ3 Q m=2 (λ, t D ) = ϒ1 − ϒ2 · t D2 − ϒ3      2 2 − ϒ4 − 2 · · m · t D + ϒ5 · λ + ϒ6 · λ2 2 Pn=3 (λ, t D ) =

3  k=0

pk (t D ) · λk = p0 (t D ) + p1 (t D ) · λ + p2 (t D ) · λ2 + p3 (t D ) · λ3

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

 p0 (t D ) = ϒ7 ·

t D2

− ϒ8 + ϒ9 ; p1 (t D ) = ϒ10 −

 2

403

2

· m · t D2 ; p2 (t D ) = ϒ11 ; p3 (t D ) = −1 Q m=2 (λ, t D ) =

2 

qk (t D ) · λk = q0 (t D ) + q1 (t D ) · λ + q2 (t D ) · λ2

k=0

q0 (t D ) = ϒ1 − ϒ2 · t D2 − ϒ3 ; q1 (t D )      2 2 = − ϒ4 − 2 · · m · t D + ϒ5 ; q2 (t D ) = ϒ6 2 The homogeneous system for N E φ leads to a characteristic equation  for the eigenvalue λ having the form P(λ, t D ) + Q(λ, t D ) · e−λ·t D ; P(λ, t D ) = 3j=0 a j · λ j ;  Q(λ, t D ) = 2j=0 c j ·λ j . The coefficients {a j (qi , qk , t D ), c j (qi , qk , t D )} ∈ R depend on qi , qk and delayt D .qi , qk are any lasers with frequency-modulated optical feedback system’s parameters, other parameters kept as a constant [5–7]. a0 (t D ) = ϒ7 · t D2 − ϒ8 + ϒ9 ; a1 (t D )    2 · m · t D2 ; a2 (t D ) = ϒ11 ; a3 (t D ) = −1 = ϒ10 − 2 c0 (t D ) = ϒ1 − ϒ2 · t D2 − ϒ3 ; c1 (t D )      2 2 · m · t D + ϒ5 ; c2 (t D ) = ϒ6 = − ϒ4 − 2 · 2 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous, and differentiable functions of their arguments and direct substitution show that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ, t D ) + Q(λ, t D ) · e−λ·t D = 0. Furthermore,P(λ, t D ), Q(λ, t D ). Are analytic functions of λ, for which the following requirements of the analysis [5] can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. D) (b) If | Q(λ,t | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. P(λ,t D ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, t D ), Q m (λ, t D ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i · ω, t D ) + Q m (λ = i · ω, t D ) = 0. 2 Pn (λ = i · ω, t D ) = ϒ7 · t D − ϒ8 + ϒ9

404

4 Solid State Laser Nonlinearity Applications in Engineering 



+ ϒ10 −

 2



2

2 · i · ω − ϒ11 · ω2 − i · ω3 · m · tD

  2 Pn (λ = i · ω, t D ) = ϒ7 · t D − ϒ8 + ϒ9  − ϒ11 · ω + i · 2

ϒ10 −



 2



2 ·m

2 · tD

 ·ω−ω

3

# " 2 Q m (λ = i · ω, t D ) = ϒ1 − ϒ2 · t D − ϒ3 − ϒ6 · ω2      2 2 − ϒ4 − 2 · · m · t D + ϒ5 · i · ω 2

|P(i

|P(i

|Q(i

|Q(i

Pn (λ = i · ω, t D ) + Q m (λ = i · ω, t D )   2 2 = ϒ7 · t D − ϒ8 + ϒ9 + ϒ1 − ϒ2 · t D − ϒ3 − (ϒ11 + ϒ6 ) · ω2     2 2 +i · ϒ10 − · m · tD 2      2 2 3 −ϒ4 + 2 · · m · t D − ϒ5 · ω − ω 2  #2 " 2 · ω, t D )|2 = ϒ7 · t D − ϒ8 + ϒ9 − ϒ11 · ω2   2    2 2 3 + ϒ10 − · m · tD · ω − ω 2 2  2 · ω, t D )|2 = ϒ7 · t D − ϒ8 + ϒ9 ⎡ 2    2 2 ⎣ + · m · tD ϒ10 − 2  #  2 −2 · ϒ7 · t D − ϒ8 + ϒ9 · ϒ11 · ω2       2 2 2 + ϒ11 − 2 · ϒ10 − · m · tD · ω4 + ω6 2 " # 2 2 · ω, t D )|2 = ϒ1 − ϒ2 · t D − ϒ3 − ϒ6 · ω2  2    2 2 + ϒ4 − 2 · · m · t D + ϒ5 · ω2 2 #2 " 2 · ω, t D )|2 = ϒ1 − ϒ2 · t D − ϒ3 2     2 2 + ϒ4 − 2 · · m · t D + ϒ5 2 #  " 2 −2 · ϒ1 − ϒ2 · t D − ϒ3 · ϒ6 · ω2 + ϒ62 · ω4

F(ω, t D ) = |P(i · ω, t D )|2 − |Q(i · ω, t D )|2

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

405

2 $ %2  = ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ1 − ϒ2 · t D2 − ϒ3 ⎡ ⎤ 2 2     + ⎣ ϒ10 − · m · t D2 − 2 · ϒ7 · t D2 − ϒ8 + ϒ9 · ϒ11 ⎦ · ω2 2 ⎛ 2  2  2 − ⎝ ϒ4 − 2 · · m · t D + ϒ5 2  − −2 · [ϒ1 − ϒ2 · t D2 − ϒ3 ] · ϒ6 · ω2    2   2 + ϒ11 − 2 · ϒ10 − · m · t D2 · ω4 − ϒ62 · ω4 + ω6 2 F(ω, t D ) = |P(i · ω, t D )|2 − |Q(i · ω, t D )|2 = (ϒ7 · t D2 − ϒ8 + ϒ9 )2 − [ϒ1 − ϒ2 · t D2 − ϒ3 ]2    2 + [(ϒ10 − · m · t D2 )2 − 2 · (ϒ7 · t D2 − ϒ8 + ϒ9 ) 2  2    2 · ϒ11 − ϒ4 − 2 · · m · t D2 + ϒ5 2 $ % % −2 · ϒ1 − ϒ2 · t D2 − ϒ3 · ϒ6 · ω2        2 2 2 2 + ϒ11 − 2 · ϒ10 − · m · tD − ϒ6 · ω4 + ω6 2 We define the following parameters for simplicity: 0 , 2 , 4 , 6 . 2 $ %2  0 = ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ1 − ϒ2 · t D2 − ϒ3  2    2 2 2 = ϒ10 − · m · tD 2 ⎛ 2 2     − 2 · ϒ7 · t D2 − ϒ8 + ϒ9 · ϒ11 − ⎝ [ϒ4 − 2 · · m · t D2 + ϒ5 2 %  $ −2 · ϒ1 − ϒ2 · t D2 − ϒ3 · ϒ6      2 2 2 · m · t D − ϒ62 ; 6 = 1 4 = ϒ11 − 2 · ϒ10 − 2 Hence F(ω, t D ) = 0 implies solving the above polynomial.

3 k=0

2·k · ω2·k = 0 and its roots are given by

406

4 Solid State Laser Nonlinearity Applications in Engineering  1 2 P ( j) · + · (g ( j) − l − q ( j) ) T A · TR EA TR   ( j) 2 1 P ( j) ( j) q + ·P · · + TR EA TL · T R EL   ( j) 1 2 P ( j) 2 ( j) g − ·P · · + · (g ( j) − l − q ( j) ) · ω2 − TR EL T A · TR EA TR    2 1 1 P ( j) P ( j) Q I (i · ω, τ ) = · (g ( j) − l − q ( j) ) · + + + TR TL · T R T A · TR EA EL  2 q ( j) 2 g ( j) + · P ( j) · − · P ( j) · ·ω TR EA TR EL 

Q R (i · ω, τ ) =

1 P ( j) + TL · T R EL

  ·

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2 PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q i (i · ω, τ ) cos θ(τ ) = − |Q(i · ω, τ )|2 sin θ(τ ) =

We use different parameters terminology from our last characteristics parameters definition: k → j; pk (t D ) → a j ; qk (t D ) → c j ; n = 3; m = 2; n > m; a j = a j (t D ); c j = c j (t D ) Pn (λ, t D ) → P(λ, t D ); Q m (λ, t D ) → Q(λ, t D ); P(λ, t D ) =

3 

a j · λ j ; Q(λ, t D ) =

j=0

P(λ, t D ) =

2 

cj · λj

j=0 3 

a j · λ j = a 0 + a 1 · λ + a 2 · λ2

j=0

+ a3 · λ3 ; Q(λ, t D ) =

2 

c j · λ j = c0 + c1 · λ + c2 · λ2

j=0

n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable function of t D such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ, t D ), Q(λ, t D ) are analytic functions in λ and differentiable in t D . The coefficients a j (K , ω, ε, , β, t D , . . .) ∈ R and c j (K , ω, ε, , β, t D , . . .) ∈ R depend on lasers with frequency-modulated optical feedback system’s K , ω, ε, , β, m , 0 , , t D , . . . values. Remark we use ω as system relative pump power normalized by the threshold   ω = PPth and also for Geometric stability switch criteria ω parameter (λ = i ·ω; ω ∈ R). Reader must differentiate between the two utilizations. Additionally ε is used as a system spontaneous emission rate and as a small positive infinitesimal quantity in math.

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

407

Unless strictly necessary, the designation of the varied arguments: (K , ω, ε, , β, t D , . . .) will subsequently be omitted from P, Q, a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0. a0 (t D ) = ϒ7 · t D2 − ϒ8 + ϒ9 ; c0 (t D ) = ϒ1 − ϒ2 · t D2 − ϒ3 a0 + c0 = −ϒ8 + ϒ9 + ϒ1 − ϒ3 + (ϒ7 − ϒ2 ) · t D2 = 0; t D2 ! ϒ8 − ϒ9 − ϒ1 + ϒ3 ϒ8 − ϒ9 − ϒ1 + ϒ3 = ; t D = ϒ7 − ϒ2 ϒ7 − ϒ2 ∀ K , ω, ε, , β, m , 0 , , t D , . . . ∈ R+ i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ, t D ), Q(λ, t D ) are analytic functions of λ for which the following requirements of the analysis (see Kuang 1993, Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0 i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire (K , ω, ε, , t D , . . .) domain of interest. P(λ,t D ) | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) | Q(λ,t D) Indeed, in the limit: * * * P(λ, t D ) * * * * Q(λ, t ) * D * * * ϒ · t 2 − ϒ + ϒ + [ϒ −   2 · m · t 2 ] · λ + ϒ · λ2 − λ3 * 8 9 10 11 * 7 D * D 2 =* * * ϒ1 − ϒ2 · t 2 − ϒ3 − [ϒ4 − 2 ·   2 · m · t 2 + ϒ5 ] · λ + ϒ6 · λ2 * D D 2 (c) The following expressions exist: F(ω, t D ) = |P(i · ω, t D )|2 − |Q(i · ω, t D )|2 F(ω, t D ) = |P(i · ω, t D )|2 − |Q(i · ω, t D )|2 =

3 

2·k · ω2·k

k=0

Has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω6 ). (d) Each positive root ω(K , ω, ε, , t D , . . .) of F(ω, t D ) = 0 is continuous and differentiable with respect to K , ω, ε, , t D , . . . The condition can only be assessed numerically. In addition, since the coefficients P and Q are real, we have P(−i · ω) = P(i · ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 maybe on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ—plane, whereby increasing the parameters: K , ω, ε, , β, m , 0 , , t D , . . . Re λ may, at the crossing, change

408

4 Solid State Laser Nonlinearity Applications in Engineering

its sign from (−) to (+), i.e. from stable focus(N (k) , E (k) , φ (k) ); k = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to K , ω, ε, , β, m , t D , . . . and system parameters [5–7]. ∂Reλ )λ=i·ω ; K , ω, ε, , β, m , 0 , , . . . = const ∂t D ∂Reλ −1 (K ) = ( )λ=i·ω ; t D , ω, ε, , β, m , 0 , , . . . = const ∂K ∂Reλ )λ=i·ω ; K , t D , ε, , β, m , 0 , , . . . = const −1 (ω) = ( ∂ω ∂Reλ )λ=i·ω ; K , ω, t D , , β, m , 0 , , . . . = const −1 (ε) = ( ∂ε ∂Reλ )λ=i·ω ; K , ω, ε, t D , β, m , 0 , , . . . = const −1 () = ( ∂ ∂Reλ )λ=i·ω ; K , ω, ε, , t D , m , 0 , , . . . = const −1 (β) = ( ∂β ∂Reλ −1 (m ) = ( )λ=i·ω ; K , ω, ε, , β, t D , 0 , , . . . = const ∂m ∂Reλ )λ=i·ω ; K , ω, ε, , β, m , t D , , . . . = const −1 (0 ) = ( ∂0 ∂Reλ )λ=i·ω ; K , ω, ε, , β, m , 0 , t D , . . . = const −1 () = ( ∂ P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ) −1 (t D ) = (

When writing and inserting λ = i ·ω into lasers with frequency-modulated optical feedback system’s characteristic equation ω must satisfy the following equations. −PR (i · ω, t D ) · Q I (i · ω, t D ) + PI (i · ω, t D ) · Q R (i · ω, t D ) |Q(i · ω, t D )|2 PR (i · ω, t D ) · Q R (i · ω, t D ) + PI (i · ω, t D ) · Q I (i · ω, t D ) cos(ω · t D ) = h(ω) = − |Q(i · ω, t D )|2 sin(ω · t D ) = g(ω) =

where |Q(i · ω, t D )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω·t D ) and cos(ω·t D ) equations that, by squaring and adding the sides, ω must be a positive root of F(ω, t D ) = |P(i ·ω, t D )|2 −|Q(i ·ω, t D )|2 = 0. / I (assume Note: F(ω, t D ) is dependent on t D . Now it is important to notice that if t D ∈ / I, that I ⊆ R+0 is the set where ω(t D ) is a positive root of F(ω, t D ) and for, t D ∈ ω(t D ) is not defined. Then for all t D in I , ω(t D ) is satisfied that F(ω, t D ) = 0. Then there are no positive ω(t D ) solutions for F(ω, t D ) = 0, and we cannot have stability switches. For t D ∈ I where ω(t D ) is a positive solution of F(ω, t D ) = 0, we can define the angle θ (t D ) ∈ [0, 2·π ] as the solution of sin θ (t D ) = . . . ; cos θ (t D ) = . . ..

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

409

−PR (i · ω, t D ) · Q I (i · ω, t D ) + PI (i · ω, t D ) · Q R (i · ω, t D ) |Q(i · ω, t D )|2 PR (i · ω, t D ) · Q R (i · ω, t D ) + PI (i · ω, t D ) · Q I (i · ω, t D ) cos θ (t D ) = − |Q(i · ω, t D )|2 sin θ (t D ) =

And the relation between the argument θ (t D ) and t D · ω(t D ) for t D ∈ I must be as describe below. ω(t D ) · t D = θ (t D ) + 2 · n · π ∀ n ∈ N0 Hence we can define the maps t D(n) : I → R+0 given by t D(n) (t D ) = ∈ N0 ; t D ∈ I . Let us introduce the functions: I → R; Sn (t D ) = t D − ∈ I ; n ∈ N0 , that is continuous and differentiable in t D . In the following, the subscripts λ, K , ω, ε, , β, m , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(K , ω, ε, , β, m , . . .) and ω(K , ω, ε, , β, m , . . .), and keeping all parameters except one (x) and t D . The derivation closely follows that in reference [BK], Differentiating system characteristic equation P(λ, t D ) + Q(λ, t D ) · e−λ·t D = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculate: x = K , ω, ε, , β, m , . . . θ(t D )+2·n·π ;n ω(t D ) t D(n) (t D ); t D



∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − t D · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) and i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(i · ω, t D )|2 = |Q(i · ω, t D )|2 , one obtains: 

∂λ ∂x =

−1 

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − t D · |P(i · ω, x)|2



Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R +i · Q I and Pω = PRω +i · PI ω ; Q ω = Q Rω +i · Q I ω ; Px = PRx +i · PI x ; Q x = Q Rx +i · Q I x ; P 2 = PR2 + PI2 . When (x) can be and lasers with frequency-modulated optical feedback system’s parameters K , ω, ε, , β, m , 0 , , . . . and the time delayt D etc. Where for convenience, we dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] and Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ); V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ). We choose our specific parameter as time delay x = t D .

410

4 Solid State Laser Nonlinearity Applications in Engineering

  PR (i · ω, t D ) = ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ11 · ω2      2 2 PI (i · ω, t D ) = ϒ10 − · m · t D · ω − ω3 2 % $ Q R (i · ω, t D ) = ϒ1 − ϒ2 · t D2 − ϒ3 − ϒ6 · ω2      2 2 Q I (i · ω, t D ) = − ϒ4 − 2 · · m · t D + ϒ5 ] · ω 2 PR = PR (i · ω, t D ); PI =PI (i · ω, t D ); Q R = Q R (i · ω, t D ); Q I =Q I (i · ω, t D )   PRω = −2 · ϒ11 · ω; PI ω = ϒ10 −

 2 



2 ·m·

− 3 · ω2 ; Q Rω = −2 · ϒ6 · ω; Q I ω = − ϒ4 − 2 ·

t D2 

 2



2 · m · t D2 + ϒ5

  2 · m · t D · ω; Q Rt D PRt D = 2 · ϒ7 · t D ; PI t D = −2 · 2    2 = −2 · ϒ2 · t D ; Q I t D = 4 · · m · tD · ω 2 $  % PRω · PR = −2 · ϒ11 · ω · ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ11 · ω2 % $ Q Rω · Q R = −2 · ϒ6 · ω · ϒ1 − ϒ2 · t D2 − ϒ3 ] − ϒ6 · ω2 

Ft D = 2 · [(PRt D · PR + PI t D · PI ) − (Q Rt D · Q R + Q I t D · Q I )] $  % $ = 2 · 2 · ϒ7 · t D · ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ11 · ω2         2  2 · m · tD · ω · ϒ10 − · m · t D2 · ω − ω3 −2 · 2 2 − (−2 · ϒ2 · t D · {[ϒ1 − ϒ2 · t D2 − ϒ3 ] − ϒ6 · ω2 }         2  2 2 · m · t D · ω · ϒ4 − 2 · · m · t D + ϒ5 · ω −4 · 2 2 $ % Q R · Q I ω = − ϒ1 − ϒ2 · t D2 − ϒ3       2 2 2 · m · t D + ϒ5 −ϒ6 · ω · ϒ4 − 2 · 2      2 2 · m · t D + ϒ5 · 2 · ϒ6 · ω2 Q I · Q Rω = ϒ4 − 2 · 2 V = (PR · PI t D − PI · PRt D ) − (Q R · Q I t D − Q I · Q Rt D )

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …





% $ = − (ϒ7 · t D2 − ϒ8 + ϒ9 ) − ϒ11 · ω2 · 2 ·  −



ϒ10 −

 2



2 ·m·

t D2

 ·ω−ω

3

411

 2

2 · m · tD · ω 

· 2 · ϒ7 · t D

   %  $  2 2 2 · m · tD · ω − ϒ1 − ϒ2 · t D − ϒ3 − ϒ6 · ω · 4 · 2       2 2 − ϒ4 − 2 · · m · t D + ϒ5 · ω · 2 · ϒ2 · t D 2 F(ω, t D ) = 0. Differentiating with respect to t D and we get Fω ·

∂ω ∂t D

+ Ft D = 0.

∂ω Ft tD ∈ I ⇒ = − D ; −1 (t D ) ∂t D Fω   ∂ω Ft ∂Reλ ; = ωt D = − D = ∂t D λ=i·ω ∂t D Fω   2 −2 · (U + t D · |P| ) + i · Fω −1  (t D ) = Re Ft D + i · 2 · (V + ω · |P|2 )    ∂Reλ −1 sign{ (t D )} = sign ∂t D λ=i·ω   ∂ω V + · U ∂ω ∂t D sign{−1 (t D )} = sign{Fω } · sign +ω+ · tD |P|2 ∂t D We shall presently examine the possibility of stability transitions (bifurcations) lasers with frequency-modulated optical feedback system, about the equilibrium point (N (k) , E (k) , φ (k) ); k = 0, 1, 2 as a result of a variation of delay parametert D . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane where by increasing the delay parametert D , Reλ may at the crossing, changes its sign from “−” to “+”, i.e. from a stable focus (N (k) , E (k) , φ (k) ); k = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to t D . Hint: t D is delay time normalized by the photon lifetime (delay parameter). −1 (t D ) =



∂Reλ ∂t D

 λ=i·ω

; −1 (t D ) =



∂Reλ ∂t D

 λ=i·ω

;

K , ω, ε, , β, m , 0 , , . . . = const U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω )

412

4 Solid State Laser Nonlinearity Applications in Engineering

 % $ ϒ7 · t D2 − ϒ8 + ϒ9 − ϒ11 · ω2       2 2 2 · m · tD − 3 · ω · ϒ10 − 2        2 2 3 · m · t D · ω − ω · 2 · ϒ11 · ω) − − ϒ10 − 2  $ %  − − ϒ1 − ϒ2 · t D2 − ϒ3 − ϒ6 · ω2      2 2 · m · t D + ϒ5 · ϒ4 − 2 · 2       2 2 · m · t D + ϒ5 · ω · 2 · ϒ6 · ω − ϒ4 − 2 · 2

U=

Then we get the expression for F(ω, t D ) lasers with frequency-modulated optical feedback system parameters values. We find those ω, t D values which fulfil F(ω, t D ) = 0. We ignore negative, complex, and imaginary values of ω for specific t D values. We can be express by 3D function F(ω, t D ) = 0. We plot the stability switch diagram based on different delay values of our system. ∂Reλ −2 · (U + t D · |P|2 ) + i · Fω } )λ=i·ω = Re{ ∂t D Ft D + i · 2 · (V + ω · |P|2 ) ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Ft D · (U + t D · P 2 )} −1 (t D ) = ( )λ=i·ω = ∂t D Ft2D + 4 · (V + ω · P 2 )2 −1 (t D ) = (

The stability switch occurs only on those delay values (t D ) which fit the equation: t D = ωθ++(t(tDD)) and θ+ (t D ) is the solution of sin θ (t D ) = · · · ; cos θ (t D ) = · · · when ω = ω+ (t D ) if only ω+ is feasible. Additionally, when all lasers with frequencymodulated optical feedback system parameters are known and the stability switch due to various time delay values t D is described in the following expression: sign{−1 (t D )} = sign{Fω (ω(t D ), t D )} U (ω(t D )) · ωt D (ω(t D )) + V (ω(t D )) · sign{t D · ωt D (ω(t D )) + ω(t D ) + } |P(ω(t D ))|2 Remark we know F(ω, t D ) = 0 implies its roots ωi (t D ) and finding those delay values t D which ωi is feasible. There are t D values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, t D ) function is dependent on t D the parameter F(ω, t D ) = 0. The results: We find those ω, t D values which fulfill F(ω, t D ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, t D values which fulfill sin θ (t D ) = · · · ; cos θ (t D ) = · · ·

4.2 Nanometer-Vibration Measurement with Microchip Solid-State …

413

−PR · Q I + PI · Q R ; cos(ω · t D ) |Q|2 (PR · Q R + PI · Q I ) =− ; |Q|2 = Q 2R + Q 2I |Q|2

sin(ω · t D ) =

Finally, we plot the stability switch diagramg(t D ) = −1 (t D ) = −1

g(t D ) =  (t D ) =



∂Reλ ∂t D





∂Reλ . ∂t D λ=i·ω

 λ=i·ω

2 · {Fω · (V + ω · P 2 ) − Ft D · (U + t D · P 2 )} = Ft2D + 4 · (V + ω · P 2 )2    ∂Reλ sign{g(t D )} = sign{−1 (t D )} = sign ∂t D λ=i·ω   2 2 · {Fω · (V + ω · P ) − Ft D · (U + t D · P 2 )} = sign Ft2D + 4 · (V + ω · P 2 )2 Ft2D + 4 · (V + ω · P 2 )2 > 0 sign{−1 (t D )} = sign{Fω · (V + ω · P 2 ) − Ft D · (U + t D · P 2 )}    Ft D −1 2 2 · (U + t D · P ) sign{ (t D )} = sign [Fω ] · (V + ω · P ) − Fω   ∂ω −1 Ft ∂ F/∂ω ωt D = − D ; ωt D = =− Fω ∂t D ∂ F/∂t D sign{−1 (t D )} = sign{[Fω ] · [V + ωt D · U + ω · P 2 + ωt D · t D · P 2 ]}    V + ωt D · U + ω + ω · t sign{−1 (t D )} = sign [Fω ] · [P 2 ] · tD tD D P2 sign[P 2 ] > 0 ⇒ sign{−1 (t D )}    V + ωt D · U + ω + ω · t = sign [Fω ] · tD D P2   V + ωt D · U sign{−1 (t D )} = sign[Fω ] · sign + ω + ω · t tD D P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] Fω = 2 · [(−2 · ϒ11 · ω · {(ϒ7 · t D2 − ϒ8 + ϒ9 ) − ϒ11 · ω2 }       2 2 2 · m · tD − 3 · ω + ϒ10 − 2       2 · m · t D2 · ω − ω3 · ϒ10 − 2 $ %  − (−2 · ϒ6 · ω · ϒ1 − ϒ2 · t D2 − ϒ3 − ϒ6 · ω2

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4 Solid State Laser Nonlinearity Applications in Engineering

Table. 4.5 lasers with frequency-modulated optical feedback system sign of −1 (t D ). V +ωt D ·U P2

sign{−1 (t D )}

sign[Fω ]

sign[

±

±

+

±

±

−

 +

ϒ4 − 2 ·



+ ω + ωt D · t D ]

 2

2

2 · m · t D2 + ϒ5

⎞⎤ · ω ⎠⎦

We check the sign of −1 (t D ) according the following rule (Table 4.5): If sign{−1 (t D )} > 0 then the crossing proceeds from (−) to (+) respectively (stable to unstable). If sign{−1 (t D )} < 0 then the crossing proceeds from (+) to (−) respectively (unstable to stable).

4.3 Doppler-Shift with a Microchip Solid State Laser Stability Analysis Under Parameters Variation In laser we have the phenomena of light-injection-induced. It is injection locking and return-light-induced instabilities. Important is the basic properties of laser dynamics as well as their practical importance. One group of lasers is Nd stoichiometric lasers. Stoichiometric laser materials are pure chemical compounds capable of emitting coherent light in the undiluted state as opposed to conventional laser materials where the active ions or molecules are dispersed in a host. One chemical compound is Neodymium compounds. The stoichiometric laser materials can store several kilojoules of energy per cm3 for times ranging from 100 ns to 10 ms. Continuous room temperature laser operation can be achieved with just a few hundred microwatts of pump power. Optical gains up to 10 dB per optical wavelength can be expected. Stoichiometric systems are related to spectroscopic properties. Threshold, pump power, and dynamical behavior (relaxation oscillations) can be calculated including the effects of anisotropy, saturation, inversion profile, reabsorption, pump profile, and mode profile effects that are important for materials with a high concentration of active ions. Structural aspects such as phase transitions, Ferro elasticity, ionic spacing, probability density of rare earth electrons, and energy transfer can be discussed using NdP5 O14 . Hybrid exchange can be a possible mechanism for energy transfer between rare earth ions. Their wave functions overlap appreciably in stoichiometric materials. There is a resonant excitation of periodic oscillations and chaotic oscillations in relaxation oscillation, resulting from Doppler-shifted light injection from a moving light-scattering object. It is injection-induced modulation of lasers phenomenon. One application of this phenomenon is ultra-high sensitivity Doppler-shift measurements in wide velocity regions. Light waves from a moving source experience the Doppler Effect to result in either a red shift or blue shift in the

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

415

light’s frequency. Light waves do not require a medium for travel. This is one effect on light which shows that its source is moving with respect to the observer; its color changes. The Doppler Effect can be observed to occur with all types of waves and light waves. The relativistic Doppler Effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer, when taking into account effects described by the special theory of relativity. The Doppler Effect is caused when the source of a waveform light-send out waves at a regular rate or frequency, but there is a constant relative motion between the source and observer, causing the observed frequency to change. Doppler Effect Redshift and Blue shift describe how light shifts toward shorter or longer wavelengths as objects move closer or farther away from us. When an object moves away from us, the light is shifted to the red end of the spectrum, as its wavelengths get longer. The model includes multimode effects which are in correlation with the experimental results. Our experimental system includes the following elements: (1) An argon laser which served as a pump laser to NLP laser, (2) LiNdP4 O12 (LNP) crystal which is coated with dielectric mirrors, (3) Infrared transmission filter (IR), (4) Beam splitter element (BS), (5) InGaAs photodiode (~120 MHz bandwidth), (6) Variable Attenuator (VA), (7) Lens to focus the main beam on a rotation circular paper sheet stuck to a metal plate which is attached to the rotating arm. Argon ion laser is powerful gas laser, which typically generate multiple watts of optical power in a green or blue output beam with high beam quality. The core component of an argon ion laser is an argonfiled tube, made of beryllium oxide ceramics, in which an intense electrical discharge between two hollow electrodes generates plasma with a high density of argon (Ar+ ) ions. A solenoid around the tube can be used for generating a magnetic field, which increases the output power by better confining the plasma. Multi-watt argon ion lasers can be used for pumping titanium-sapphire lasers and dye lasers, or for laser light activities. They are rivaled by frequency-doubled diode-pumped solid-state lasers. The latter are far more power efficient and have longer lifetimes. Argon tubes have a limited lifetime of the order of a few thousand hours. An argon laser may thus be preferable if it is used only during a limited number of hours, whereas a diodepumped solid-state laser is the better solution for reliable and efficient long-term operation. The argon laser pumps the LNP laser which emits light through infrared transmission filter. Then the light goes through Beam Splitter (BS) which splits the light to the InGaAs photo diode and to Variable Attenuator (VA), lens and finally hit paper sheet placed on rotating arm. The LNP laser radiation is linearly polarized along the pseudo-orthorhombic c axis. Pseudo-orthorhombic shape is having unequal axes at right angles (Fig. 4.4). Orthorhombic crystal system is orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base [3, 10. Two-longitudinal-mode oscillation is obtained above w = PPth = 1.09 (Pth is threshold pump power) and three-longitudinal-mode oscillation is obtained above w = PPth = 1.7. Part of the output light is used for monitoring output waveforms and power spectra with an InGaAs photo diode (120 MHz bandwidth). The impinged laser power on the paper sheet is changed by a Variable Attenuator (VA). Our system

416

4 Solid State Laser Nonlinearity Applications in Engineering

Beam Splier

Argon Laser (pump)

LNP laser

Variable Aenuator (VA)

Infrared transmission filter

InGaAs Photo diode Spectrum Analyzer

Lens Paper sheet

Oscilloscope

Rotating Arm

Fig. 4.4 Experimental structure of Doppler-shift with Microchip solid-state laser

observes the injection induce modulation, even the fact that the impinged light is attenuated [11]. The system phenomenon can be inspected in terms of injection-induced intensity +i and Doppler modulation resulting from interference between oscillating field E , shifted field E i,s injected into the laser cavity, where “i” denotes the model index. Laser cavity is a means of optical confinement intended to increase the gain length of radiation prior to emission from the device. The means of optical confinement used to increase gain path length vary depending upon the properties of the beam desired within the lasing medium. High light intensities occur within a laser cavity and dielectric mirrors coated for the lasing wavelength are used. The position and curvature of the optical cavity elements is alerted in order to optimize the laser performance as needed for a particular application. A similar phenomenon is expected outside the injection-locking region in single-mode lasers. The injection-locking properties of a semiconductor laser can be analyze and the injected carrier density dependent refractive index in the action region. We can characterize the light-injection model for rotating-wave approximation fields by the following set of differential equations. It is done by introducing injection fields into spatially-hole-burned N-mode lasers.   N N0  d Ni Nk d N0 =P− − ; Bk · Sk · N0 − dt τ 2 dt k=1 N  Ni + Bi · N0 · Si − Ni · Bk · Sk τ k=1       1 κ Ei Ni = Bi · (N0 − )− + · · E i,s · cos[ψi (t) + φi,s (t)] 2 τp 2 τl     E i,s κ · · sin[ψi (t) + φi,s (t)]; i = 1, 2, . . . , N = ωi − ωi,s − τl Ei

=−

d Ei dt dψi dt

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

417

N0 - Space average of population inversion density, Ni —First spatial Fourier component of population inversion for the i − th mode, i—Modal index, P—Pump power, τ —Population lifetime, τ p —Photon lifetime, κ—Effective amplitude transmission coefficient of the output mirror, τl —Round trip time in the cavity, ψi — Phase different between the two fields, φi,s —Express phase fluctuations of scattered +i , E i —Lasing field, , E i,s —Scattered field, Bi —Gain, Si —Photon density, light, E 2 Si = E i2 ; E i,s (t) ∝ E i (t − td ); td —Delay time, EEi,si < 10−4 (extremely small). In the     ω lock-in region: 2·π = τκl · EEi,si < 100 kHz. Where “i” denotes the model index (laser mode number). N —Number of relaxation oscillations which each mode may exhibit. Hint: The phase interaction between longitudinal modes are omitted since the longitudinal mode spacing is extremely large compared with the Doppler-shift frequencies and the lasing time scales. P ; Pth,1 is the first lasing mode pump threshold, n 0 , n i —are the space w = Pth,1 average and the first Fourier components of population inversion normalized by the   mode threshold value N0 − N21 th . si —Photon density normalized by the steady   state S1 value at w = 2. γi —Gain ratio to the first lasing mode ( γi = BBi ), m = 2·E i,s ; (ωi − ωi,s ) · τ ≡  D (time is scaled by τ ), K = ττp . ω D —Doppler-shift κ·E i RO frequency. We define the relaxation oscillation frequency f R O as f R O = ω2·π =  1   (w−1) · | = 540 kHz. k · v—Angular frequency of Doppler 2·π τ ·τ p τ =120 µs,τ p =310 ps shift and increases with increasing angular moving velocity v and with decreasing the scattering angle θs , where θs is the angle between the laser axis and velocity vector. λ—Oscillation wavelength. Assumption: The delay time td is extremely small compared with modulation time scales. The simplest case of our system is when the number of relaxation oscillations which each mode exhibits is N = 2 and N = 3. The system same collective behavior is featuring antiphase dynamics for larger number of modes. We analyze the system for the simplest case N = 2 and get differential equations [3, 10].   N =2  d N0 Nk N0 Bk · Sk · N0 − =P− − ; dt τ 2 k=1

N =2

 d Ni Ni Bk · Sk =− + Bi · N0 · Si − Ni · dt τ k=1       Ni d Ei 1 Ei = Bi · N0 − − · dt 2 τp 2   κ · E i,s · cos[ψi (t) + φi,s (t)] + τl     κ E i,s dψi · = ωi − ωi,s − dt τl Ei · sin[ψi (t) + φi,s (t)]; i = 1, 2, . . . , N

418

4 Solid State Laser Nonlinearity Applications in Engineering

We write the above differential equations for N = 2.     N0 N1 N2 d N0 =P− − B1 · S1 · N0 − − B2 · S2 · N0 − dt τ 2 2 d Ni Ni =− + Bi · N0 · Si − Ni · (B1 · S1 + B2 · S2 ) dt     τ   Ei 1 d Ei Ni · = Bi · N0 − − dt 2 τp 2   κ · E i,s · cos[ψi (t) + φi,s (t)] + τl     dψi E i,s κ · = ωi − ωi,s − dt τl Ei · sin[ψi (t) + φi,s (t)]; i = 1, 2, . . . , N At fixed points:

d N0 dt

i = 0; ddtNi = 0; ddtEi = 0; dψ =0 dt

  N∗ d N0 N1 = 0 ⇒ P − 0 − B1 · S1 · N0∗ − dt τ 2   N2 − B2 · S2 · N0∗ − =0 2   1 − N0∗ · + B1 · S1 + B2 · S2 + P τ  2  P + 2k=1 Bk · Sk · N2k Nk  + Bk · Sk · = 0; N0∗ =  2 1 2 k=1 k=1 Bk · Sk τ + N∗ d Ni = 0 ⇒ − i + Bi · N0∗ · Si dt τ   2 2  1  − Ni∗ · Bk · Sk = 0; Ni∗ · Bk · Sk = Bi · N0∗ · Si + τ k=1

Ni∗

k=1

Bi · N0∗ · Si ; =  2 1 k=1 Bk · Sk τ +  Bi · Si ·

Ni∗



  + 2k=1 Bk · Sk   Bi · Si · P + 2k=1 Bk · Sk · =  2 2 1 k=1 Bk · Sk τ +

=



 N P+ 2k=1 Bk ·Sk · 2k   1 2 τ + k=1 Bk ·Sk

1 τ

Nk 2



  ∗    Ei Ni∗ 1 d Ei ∗ · − = 0 ⇒ Bi · N0 − dt 2 τp 2

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

419

  $ % κ · E i,s · cos ψi∗ + φi,s (t) = 0 τl   $ % 2 · τκl · E i,s · cos ψi∗ + φi,s (t) "  #  E i∗ = − ⇒ E i∗ N∗ Bi · N0∗ − 2i − τ1p   $ % 2 · τκl · E i,s · cos ψi∗ + φi,s (t) " #  = Ni∗ 1 ∗ − B · N − i 0 τp 2 +

E i∗ 

= 1 τp

− Bi ·

E i∗

2·

1 τp

κ τl

$ % · E i,s · cos ψi∗ + φi,s (t)   

 N P+ 2k=1 Bk ·Sk · 2k    2 1 τ + k=1 Bk ·Sk

−

1 2

·

Bi ·Si · P+ 2k=1 Bk ·Sk ·  2 1 2 τ + k=1 Bk ·Sk

Nk 2

τ

 

 

$ % · E i,s · cos ψi∗ + φi,s (t)      N P+ 2k=1 Bk ·Sk · 2k Bi ·Si 1   · 1 − − Bi ·  1  · 2 1 2 2 2·

=

 

κ τl

+

k=1

Bk ·Sk

τ

+

k=1



Bk ·Sk



    $ % E i,s dψi κ · · sin ψi∗ + φi,s (t) = 0; i = 1, 2, . . . , N = 0 ⇒ ωi − ωi,s − ∗ dt τl Ei     E i,s κ E i,s · · sin[ψi∗ + φi,s (t)] = ωi − ωi,s ⇒ ∗ ∗ τl Ei Ei   κ E i,s · τl · sin[ψi∗ + φi,s (t)] ωi − ωi,s ; E i∗ = =  κ ωi − ωi,s · sin[ψ ∗ + φ (t)] τl

i,s

i

Then we get the following equation for fixed point’s coordinate ψi∗ :

1 τp

  κ τl

· E i,s · cos[ψi∗ + φi,s (t)]      N P+ 2k=1 Bk ·Sk · 2k Bi ·Si 1   · 1 − − Bi ·  1  · 2 1 2 2 2· τ

=

E i,s ·

+

  κ τl

k=1

Bk ·Sk

τ

+

k=1

Bk ·Sk

 

· sin[ψi∗ + φi,s (t)]

ωi − ωi,s

The above equation needs to solve numerically for finding ψi∗ then submit the value in one of E i∗ = . . . equations.

420

4 Solid State Laser Nonlinearity Applications in Engineering

Our system fixed points: 

 N0∗ , Ni∗ , E i∗ , ψi∗ ⎛  P + 2k=1 Bk · Sk · N2k  , =⎝  2 1 + B · S k k k=1 τ  2 Bi · Si · P + k=1 Bk · Sk ·  2  1 + 2k=1 Bk · Sk τ

Nk 2

 ,

2·

  κ τl

"

1 τp

⎞ · E i,s · cos[ψi∗ + φi,s (t)] ⎟ #  , ψi∗ ⎠ Ni∗ ∗ − Bi · N0 − 2

Stability analysis: The standard local stability analysis about any one of the equilibrium points of the light-injection model for rotating-wave approximation fields differential equation system consists in adding to coordinate [N0 , Ni , E i , ψi ] arbitrarily small increments of exponential form [n 0 , n i , E i , ψi ] · eλ·t and retaining the first order terms in N0 , Ni , E i , ψi . The system of four homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables (N0 , Ni , E i , ψi ) and variables derivatives with respect to time into system equations [12–14]. System fixed values with arbitrarily small increments of exponential form [n 0 , n i , E i , ψi ]·eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. N0 = N0 (t); Ni = Ni (t); E i = E i (t); ψi = ψi (t) ( j)

( j)

+ n i · eλ·t ; E i (t)

( j)

+ ψi · eλ·t

N0 (t) = N0 + n 0 · eλ·t ; Ni (t) = Ni ( j)

= Ei

+ E i · eλ·t ; ψi (t) = ψi

d N0 (t) d Ni (t) d E i (t) = n 0 · λ · eλ·t ; = n i · λ · eλ·t ; dt dt dt (t) dψ i = E i · λ · eλ·t ; = ψi · λ · eλ·t dt We choose these expressions for ourselves N0 (t), Ni (t), E i (t), ψi (t) as a small displacement [n 0 , n i , E i , ψi ] from the system fixed points in time t = 0. ( j)

( j)

( j) Ei

( j) ψi

N0 (t = 0) = N0 + n 0 ; Ni (t = 0) = Ni =

+ E i ; ψi (t = 0) =

+ n i ; E i (t = 0) + ψi

N0 d N0 N1 N2 d N0 =P− − B1 · S1 · (N0 − ) − B2 · S2 · (N0 − ) dt τ 2 2 dt  2  2   N0 Nk d N0 − N0 · ; =P− Bk · Sk + Bk · Sk · τ 2 dt k=1 k=1

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

421



  2 2   1 Nk = P − N0 · Bk · Sk Bk · Sk · + + τ 2 k=1 k=1

 2  2   1 Nk + =P− + n0 · e ] · Bk · Sk Bk · Sk · + τ 2 k=1 k=1  2    1 ( j) + = P − N0 · Bk · Sk τ k=1  2   2   1 Nk − n0 · + Bk · Sk Bk · Sk · · eλ·t + τ 2 k=1 k=1 

n0 · λ · e

λ·t

n 0 · λ · eλ·t

( j) [N0

λ·t

At fixed points: 

  2 2   1 Nk P− · Bk · Sk Bk · Sk · + + τ 2 k=1 k=1  2    1 + = 0 − n0 · λ − n0 · Bk · Sk τ k=1 ( j) N0

Ni d Ni =− + Bi · N0 · Si − Ni · (B1 · S1 + B2 · S2 ) dt τ   ( j)   Ni + n i · eλ·t ( j) + Bi · N0 + n 0 · eλ·t =−   τ ( j) · Si − Ni + n i · eλ·t · (B1 · S1 + B2 · S2 )n i · λ · eλ·t

=0

n i · λ · eλ·t

( j)

Ni n i · eλ·t ( j) ( j) + Bi · N0 · Si − Ni · (B1 · S1 + B2 · S2 ) − τ τ λ·t λ·t − n i · e · (B1 · S1 + B2 · S2 ) + Bi · n 0 · e · Si

=−

At fixed points: −

( j)

Ni τ

( j)

( j)

+ Bi · N0 · Si − Ni

· (B1 · S1 + B2 · S2 ) = 0

ni − n i · (B1 · S1 + B2 · S2 ) = 0 Bi · Si · n 0 − n i · λ −         τ Ei 1 κ d Ei Ni · = Bi · N0 − − + dt 2 τp 2 τl · E i,s · cos[ψi (t) + φi,s (t)]   Ei · λ · e

λ·t

= Bi ·

( j) N0

+ n0 · e

λ·t

−

( j)

[Ni

+ n i · eλ·t ] 2



1 − τp



422

4 Solid State Laser Nonlinearity Applications in Engineering

 ·

( j)

+ E i · eλ·t 2

Ei

( j)

· cos[ψi

 +

  κ · E i,s τl

+ ψi · eλ·t + φi,s (t)]

First we do algebraic manipulation for the cos{. . .} expression ( j)

+ ψi · eλ·t + φi,s (t)] = cos[ψi

( j)

+ φi,s (t) + ψi · eλ·t ] = cos[ψi

cos[ψi cos[ψi

· cos[ψi · eλ·t ] −

( j) sin[ψi

( j)

+ φi,s (t) + ψi · eλ·t ]

( j)

+ φi,s (t)]

+ φi,s (t)] · sin[ψi · eλ·t ]

Assumption (A): ξ1 (ψi , λ) = ψi · eλ·t ; cos[ξ1 (ψi , λ)] = cos[ψi · eλ·t ]. cos(ξ1 (ψi , λ)) =

∞  (−1)n · [ξ1 (ψi , λ)]2·n ; cos(ξ1 (ψi , λ)) (2 · n)! n=0

=1−

ψi2 · e2·λ·t ψ 4 · e4·λ·t + i − · · · ∀ ψi · eλ·t 2! 4!

We consider ψim → ε ∀ m ≥ 2 then − cos(ξ1 (ψi , λ)) =

ψi2 ·e2·λ·t 2!

+

ψi4 ·e4·λ·t 4!

− ... → ε

∞  (−1)n · [ξ1 (ψi , λ)]2·n ≈ 1 (2 · n)! n=0

Assumption (B): ξ1 (ψi , λ) = ψi · eλ·t ; sin[ξ1 (ψi , λ)] = sin[ψi · eλ·t ]. sin(ξ1 (ψi , λ)) =

∞  n=0

(−1)n · [ξ1 (ψi , λ)]2·n+1 (2 · n + 1)!

= ξ1 (ψi , λ) − sin(ξ1 (ψi , λ)) =

∞  n=0

[ξ1 (ψi , λ)]5 [ξ1 (ψi , λ)]7 [ξ1 (ψi , λ)]3 + − + ··· 3! 5! 7!

(−1)n · [ξ1 (ψi , λ)]2·n+1 (2 · n + 1)!

[ψi · eλ·t ]3 [ψi · eλ·t ]5 [ψi · eλ·t ]7 + − + ··· 3! 5! 7! ∞  (−1)n · [ξ1 (ψi , λ)]2·n+1 sin(ξ1 (ψi , λ)) = (2 · n + 1)! n=0 = ψi · eλ·t −

= ψi · eλ·t −

ψi3 · e3·λ·t ψ 5 · e5·λ·t ψ 7 · e7·λ·t + i − i + ··· 3! 5! 7!

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

We consider ψim → ε ∀ m ≥ 2 then − sin(ξ1 (ψi , λ)) =

∞  n=0

ψi3 ·e3·λ·t 3!

+

ψi5 ·e5·λ·t 5!

423

−

ψi7 ·e7·λ·t 7!

+ ··· → ε

(−1)n · [ξ1 (ψi , λ)]2·n+1 ≈ ψi · eλ·t (2 · n + 1)!

Implementing our last results in cos{. . .} expression # " # " ( j) ( j) cos ψi + φi,s (t) + ψi · eλ·t = cos ψi + φi,s (t) " # ( j) − sin ψi + φi,s (t) · ψi · eλ·t Then we get  Ei · λ · e

λ·t

 ( j) N0

= Bi · 

+ n0 · e

λ·t

( j)

−

[Ni

+ n i · eλ·t ] 2

 + E i · eλ·t · 2   κ ( j) + · E i,s · (cos[ψi + φi,s (t)] τl

1 − τp



( j)

Ei

( j)

− sin[ψi 



+ φi,s (t)] · ψi · eλ·t )



  ( j) Ni n i · eλ·t 1 λ·t E i · λ · e = Bi · + n0 · e − − − 2 2 τp  ( j)    Ei E i · eλ·t κ ( j) · E i,s · cos[ψi + φi,s (t)] + · + 2 2 τl   κ ( j) · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t − τl      ( j)  Ni n i  λ·t 1 ( j) λ·t ·e Ei · λ · e = Bi · N0 − + Bi · n 0 − − 2 τp 2  ( j)    Ei E i · eλ·t κ ( j) · E i,s · cos[ψi + φi,s (t)] + · + 2 2 τl   κ ( j) · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t − τl      ( j) ( j)  Ni E n i  λ·t 1 ( j) λ·t ·e Ei · λ · e = Bi · N0 − + Bi · n 0 − · i − 2 τp 2 2 λ·t

( j) N0

424

4 Solid State Laser Nonlinearity Applications in Engineering

 +

Bi ·

  κ · E i,s + τl ( j)

· sin[ψi 

 ni E i · eλ·t λ·t + Bi · (n 0 − ) · e · 2 2   κ ( j) · E i,s · cos[ψi + φi,s (t)] − τl

( j) (N0

( j)

N 1 − i )− 2 τp



+ φi,s (t)] · ψi · eλ·t



 ( j) ( j)  E n i  λ·t E i 1 ·e · E i · λ · eλ·t = Bi · · i + Bi · n 0 − − τp 2 2 2     ( j) N E i · eλ·t 1 ( j) · + Bi · N0 − i − 2 τp 2     λ·t κ ni Ei · e ( j) · E i,s · cos[ψi + φi,s (t)] · eλ·t · + + Bi · n 0 − 2 2 τl   κ ( j) − · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t τl     ( j) ( j) ( j)  Ni E ni  Ei 1 ( j) λ·t · · eλ·t E i · λ · e = Bi · N0 − · i + Bi · n 0 − − 2 τp 2 2 2     ( j) Ni eλ·t 1 ( j) + Bi · N0 − · Ei · − 2 τp 2  2·λ·t  e Ei · ni · + Bi · E i · n 0 − 2 2   κ ( j) · E i,s · cos[ψi + φi,s (t)] + τl   κ ( j) · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t − τl ( j) N0

( j)

N − i 2



Assumption: E i · n 0 ≈ 0; E i · n i ≈ 0  Ei · λ · e

λ·t

= Bi ·

 ( j) N0

( j)

N − i 2



1 − τp





( j)

·

Ei 2 

+

κ τl



" # ( j) · E i,s · cos ψi + φi,s (t)

   ( j) ( j)  N 1 ni  Ei eλ·t ( j) · − · Ei · + Bi · n 0 − · eλ·t + Bi · N0 − i 2 2 2 τp 2   κ ( j) − · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t τl

"  ( j) At fixed points: Bi · N0 − 0.

( j)

Ni 2



−

1 τp

#

·

( j)

Ei 2

+

  κ τl

( j)

· E i,s ·cos[ψi +φi,s (t)] =

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

425

    ( j) ( j)  N 1 ni  Ei eλ·t ( j) · − · Ei · E i · λ · eλ·t = Bi · n 0 − · eλ·t + Bi · N0 − i 2 2 2 τp 2   κ ( j) · E i,s · sin[ψi + φi,s (t)] · ψi · eλ·t − τl     ( j) ( j)  Ni 1 ni  Ei ( j) · − E i · λ = Bi · n 0 − + Bi · N0 − 2 2 2 τp   " # 1 κ ( j) · Ei · − · E i,s · sin ψi + φi,s (t) · ψi 2 τl    ( j) ( j) ( j) E N ni Ei ( j) − E i · λ + Bi · n 0 · i − Bi · · + Bi · N0 − i 2 2 2 2    " # 1 1 κ ( j) · Ei · − · E i,s · sin ψi + φi,s (t) · ψi = 0 Bi − τp 2 τl     ( j) ( j) ( j) Ei Ei Ni 1 1 ( j) · · n 0 − Bi · · n i − E i · λ + · Bi · N0 − − 2 4 2 2 τp   " # κ ( j) · E i,s · sin ψi + φi,s (t) · ψi = 0 · Ei − τl     dψi E i,s κ · · sin[ψi (t) + φi,s (t)]ψi · λ · eλ·t = ωi − ωi,s − dt τl Ei     E i,s κ · = ωi − ωi,s − ( j) τl E i + E i · eλ·t   " # κ ( j) · sin ψi + ψi · eλ·t + φi,s (t) ψi · λ · eλ·t = ωi − ωi,s − τl ⎛  ( j) ⎞ # " E i − E i · eλ·t E i,s ⎠ · sin ψ ( j) + ψi · eλ·t + φi,s (t) #· · ⎝" i ( j) ( j) E i − E i · eλ·t E i + E i · eλ·t # "   E i,s · E ( j) − E i · eλ·t i κ  ψi · λ · eλ·t = ωi − ωi,s − ·   τl ( j) 2 Ei − E i2 · e2·λ·t # " ( j) · sin ψi + ψi · eλ·t + φi,s (t)

Assumption: E i2 ≈ 0 ψi · λ · eλ·t = ωi − ωi,s − ( j)

− E i · eλ·t ]

( j)

· sin[ψi + ψi · eλ·t + φi,s (t)] ( j) (E i )2     E i,s κ E i,s λ·t · = ωi − ωi,s − − ( j) · E i · e ( j) τl Ei (E i )2 ·

ψi · λ · eλ·t

E i,s · [E i

  κ τl

( j)

· sin[ψi

+ ψi · eλ·t + φi,s (t)]

426

4 Solid State Laser Nonlinearity Applications in Engineering

First we do algebraic manipulation for the sin{. . .} expression. ( j)

+ ψi · eλ·t + φi,s (t)] = sin[ψi

( j)

+ φi,s (t) + ψi · eλ·t ] = sin(ψi

sin[ψi sin[ψi

( j)

+ cos(ψi

( j)

+ φi,s (t) + ψi · eλ·t ]

( j)

+ φi,s (t)) · cos(ψi · eλ·t )

+ φi,s (t)) · sin(ψi · eλ·t )

Based on assumptions A and B:ξ1 (ψi , λ) = ψi · eλ·t cos(ξ1 (ψi , λ)) = cos(ψi · eλ·t ) =

∞  (−1)n · [ξ1 (ψi , λ)]2·n ≈ 1 (2 · n)! n=0

sin(ξ1 (ψi , λ)) = sin(ψi · eλ·t ) =

∞  n=0

(−1)n · [ξ1 (ψi , λ)]2·n+1 ≈ ψi (2 · n + 1)!

( j)

( j)

· eλ·t sin[ψi

( j)

+ φi,s (t) + ψi · eλ·t ] = sin(ψi + φi,s (t)) + cos(ψi + φi,s (t)) · ψi     E i,s κ E i,s λ·t λ·t λ·t · · e ψi · λ · e = ωi − ωi,s − − · Ei · e ( j) ( j) τl Ei (E i )2 ( j)

· [sin(ψi

( j)

+ φi,s (t)) + cos(ψi   κ = ωi − ωi,s − τl

+ φi,s (t)) · ψi · eλ·t ]

ψi · λ · eλ·t  E i,s E i,s ( j) ( j) · sin(ψi + φi,s (t)) + ( j) · cos(ψi + φi,s (t)) · ψi · eλ·t · ( j) Ei Ei E i,s ( j) − · E i · eλ·t · sin(ψi + φi,s (t)) ( j) (E i )2  E i,s ( j) − ( j) · E i · ψi · e2·λ·t · cos(ψi + φi,s (t)) (E i )2

Assumption: E i · ψi ≈ 0 ψi · λ · e

λ·t

= ωi − ωi,s + −

ψi · λ · e

λ·t

E i,s ( j) Ei

( j)

· cos(ψi

E i,s ( j)

   E i,s κ ( j) · − · sin(ψi + φi,s (t)) ( j) τl Ei

(E i )2

+ φi,s (t)) · ψi · eλ·t 

· Ei · e

λ·t

·

( j) sin(ψi

+ φi,s (t))

  E i,s κ ( j) · ( j) · sin(ψi + φi,s (t)) = ωi − ωi,s − τl Ei   E i,s κ ( j) · ( j) · cos(ψi + φi,s (t)) · ψi · eλ·t − τl Ei

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

+

  E i,s κ ( j) · ( j) · E i · eλ·t · sin(ψi + φi,s (t)) 2 τl (E i )

At fixed points: ωi − ωi,s − ( τκl ) ·

E i,s ( j) Ei

( j)

· sin(ψi

+ φi,s (t)) = 0.

  E i,s κ ( j) · ( j) · sin(ψi + φi,s (t)) · E i − ψi · λ 2 τl (E )  i κ E i,s ( j) − · ( j) · cos(ψi + φi,s (t)) · ψi = 0 τl Ei We can summarize our last arbitrarily small increments system equations:  2   1 + − n0 · λ − n0 · Bk · Sk =0 τ k=1 ni Bi · Si · n 0 − n i · λ − − n i · (B1 · S1 + B2 · S2 ) = 0 τ ( j) ( j) E E 1 Bi · i · n 0 − Bi · i · n i − E i · λ + 2 4 2     ( j) Ni 1 κ ( j) )− · Bi · (N0 − · Ei − 2 τp τl 

( j)

· E i,s · sin[ψi + φi,s (t)] · ψi = 0   κ E i,s ( j) · ( j) · sin(ψi + φi,s (t)) · E i − ψi · λ τl (E i )2 κ E i,s ( j) − ( ) · ( j) · cos(ψi + φi,s (t)) · ψi = 0 τl Ei

The small increments Jacobian of our system is as follow:Jacobian

427

428

4 Solid State Laser Nonlinearity Applications in Engineering

⎛

 2  2   2 ⎞   1 1 ⎠ + + det(A − λ · I ) = ⎝λ2 + 2 · λ · Bk · Sk Bk · Sk + τ τ k=1 k=1    ( j) Ni 1 1 ( j) )− · −λ + · Bi · (N0 − 2 2 τp     E i,s κ ( j) · ( j) · cos(ψi + φi,s (t)) · −λ − τl Ei    κ E i,s ( j) · ( j) · sin(ψi + φi,s (t)) − τl (E i )2    " # κ ( j) · − · E i,s · sin ψi + φi,s (t) τl 

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

429

Part I:         ( j)   Ni κ 1 E i,s 1 ( j) ( j) −λ + · Bi · N0 − · −λ − · ( j) · cos ψi + φi,s (t) − 2 2 τp τl Ei         ( j) N E i,s κ 1 1 ( j) · ( j) · −λ − − −λ + · Bi · N0 − i 2 2 τp τl Ei       E κ i,s ( j) ( j) · ( j) · cos ψi + φi,s (t) · cos ψi + φi,s (t) = λ2 + λ · τl Ei         ( j) ( j) Ni Ni 1 1 1 1 ( j) ( j) − · Bi · N0 − − − − λ · · Bi · N0 − 2 2 τp 2 2 τp     κ E i,s ( j) · · ( j) · cos ψi + φi,s (t) τl Ei          ( j)   Ni E i,s 1 1 κ ( j) ( j) · ( j) · cos ψi + φi,s (t) · −λ − − −λ + · Bi · N0 − 2 2 τp τl Ei     E i,s κ ( j) · ( j) · cos ψi + φi,s (t) = λ2 + λ · τl Ei     ( j) Ni 1 1 ( j) − − · Bi · N0 − 2 2 τp       ( j) Ni 1 κ 1 ( j) − · Bi · N0 − · − 2 2 τp τl   E i,s ( j) · ( j) · cos ψi + φi,s (t) Ei



Part II: ⎧ ⎪ ⎨ κ 

⎫ ⎪ ⎬  κ   " # E i,s ( j) ( j) · · E i,s · sin ψi + φi,s (t) 2 · sin ψi + φi,s (t) ⎪ · − ⎪ τl ( j) ⎩ τl ⎭ Ei ⎫ ⎧ ⎪ ⎪ ⎬  κ  ⎨ κ    " # E i,s ( j) ( j) · · E i,s · sin ψi + φi,s (t) 2 · sin ψi + φi,s (t) ⎪ · − ⎪ τl ( j) ⎭ ⎩ τl Ei 2  2  " # E i,s κ ( j) 2 ψ · · sin + φ (t) =− i,s i ( j) τl Ei 

Implementing Part I, II in system eigenvalue equation:

430

4 Solid State Laser Nonlinearity Applications in Engineering ⎛

  2 ⎞  2  2   1 1 ⎠ det(A − λ · I ) = ⎝λ + 2 · λ · Bk · Sk Bk · Sk + + + τ τ k=1 k=1      κ E i,s ( j) 2 · λ +λ· · ( j) · cos ψi + φi,s (t) τl Ei         ( j) ( j) Ni Ni 1 1 1 1 ( j) ( j) − · Bi · N0 − − − − · Bi · N0 − 2 2 τp 2 2 τp     E i,s κ ( j) · ( j) · cos ψi + φi,s (t) · τl Ei ⎫ 2  2  " #⎬ E i,s κ ( j) · · sin2 ψi + φi,s (t) + ( j) ⎭ τl E 

2

i

We define for simplicity the following global parameters:  2  2   2   1 1 1 = 2 · + + Bk · Sk ; 2 = Bk · Sk τ τ k=1 k=1       ( j)   1 Ni E i,s κ 1 ( j) ( j) · ( j) · cos ψi + φi,s (t) − · Bi · N0 − 3 = − τl 2 2 τp Ei       ( j) N E i,s 1 κ 1 ( j) · ( j) 4 = · Bi · N0 − i · − 2 2 τp τl Ei  2   "   # E i,s κ 2 ( j) ( j) · cos ψi + φi,s (t) + · · sin2 ψi + φi,s (t) ( j) τl Ei 

det(A − λ · I ) = (λ2 + λ · 1 + 2 ) · (λ2 + λ · 3 − 4 ) = λ4 + λ3 · 3 − λ2 · 4 + λ3 · 1 + λ2 · 1 · 3 − λ · 1 · 4 + λ2 · 2 + λ · 3 · 2 − 4 · 2 det(A − λ · I ) = (λ2 + λ · 1 + 2 ) · (λ2 + λ · 3 − 4 ) = λ4 + λ3 · (3 + 1 ) + λ2 (1 · 3 + 2 − 4 ) + λ · (3 · 2 − 1 · 4 ) − 4 · 2 det(A − λ · I ) =

4 

λk · ϒk ; ϒ4 = 1; ϒ3

k=0

= 3 + 1 ; ϒ2 = 1 · 3 + 2 − 4

4.3 Doppler-Shift with a Microchip Solid State Laser Stability …

ϒ1 = 3 · 2 − 1 · 4 ; ϒ0 = −4 · 2     E i,s κ ( j) · ( j) · cos ψi + φi,s (t) ϒ3 = 3 + 1 = τl Ei   2      ( j)  Ni 1 1 1 ( j) + − · Bi · N0 − Bk · Sk +2· − 2 2 τp τ k=1

431

 2   1 ϒ2 = 1 · 3 + 2 − 4 = 2 · + Bk · Sk τ k=1     κ E i,s ( j) · ( j) · cos ψi + φi,s (t) · τl Ei   2   2 ( j)  Ni 1 1 1 ( j) )− + Bk · Sk + − · Bi · (N0 − 2 2 τp τ k=1        ( j)   Ni E i,s 1 κ 1 ( j) ( j) · ( j) · cos ψi + φi,s (t) − · Bi · N0 − · − 2 2 τp τl Ei ⎫   2  2 " #⎬ E i,s κ ( j) 2 ψ + · · sin + φ (t) i,s i ( j) ⎭ τl E 

i

    κ E i,s ( j) · ( j) · cos ψi + φi,s (t) ϒ1 = 3 · 2 − 1 · 4 = τl Ei   2    2  ( j)  Ni 1 1 1 ( j) + Bk · Sk · − · Bi · N0 − − 2 2 τp τ k=1  2        ( j)  Ni 1 1 1 ( j) + · Bi · N0 − −2· Bk · Sk · − τ 2 2 τp k=1     E i,s κ ( j) · ( j) · cos ψi + φi,s (t) · τl Ei ⎫ 2  2  ⎬ " # E i,s κ ( j) · · sin2 ψi + φi,s (t) + ( j) ⎭ τl Ei      ( j) Ni 1 1 ( j) · Bi · N0 − − ϒ0 = −4 · 2 = − 2 2 τp     κ E i,s ( j) · · ( j) · cos ψi + φi,s (t) τl Ei

432

4 Solid State Laser Nonlinearity Applications in Engineering

⎫  2 2  2  2  " #⎬ 1  E i,s κ ( j) 2 + · · sin ψi + φi,s (t) · Bk · Sk + ( j) ⎭ τ τl Ei k=1 Eigenvalues stability discussion: Our light-injection model for rotating-wave approximation fields system involving N variables (N > 2, N = 4, arbitrarily small increments), the characteristic equation is of degree N = 4 and must often be solved numerically. Except in some particular cases, such an equation has (N = 4) distinct roots that can be real or complex. These values are the eigenvalues of the (4 × 4) Jacobian matrix (A). The general rule is that the light-injection model for rotatingwave approximation fields system is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our 4 variables (n 0 , n i , E i , ψi ) system has four eigenvalues (four system’s arbitrarily small increments). The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-degenerate cases can be distinguished: (1) the four eigenvalues are real and negative (stable steady state), (2) the four eigenvalues are real, at least one of them is positive (unstable steady state), (3) and (4) two eigenvalues are complex conjugates with a negative real part and other eigenvalues real are negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) two eigenvalues are complex conjugates with a negative real part and at least one of the other eigenvalues real is positive (unstable steady state) [12, 13]. det(A − λ · I ) =

4  k=0

λ · ϒk ; det(A − λ · I ) = 0 ⇒ k

4 

λk · ϒk = 0

k=0

Remark We use N as the Number of relaxation oscillations which each mode may exhibit, and also as system number of variables (eigenvalue stability discussion). Reader must consider it.

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation Our system implements and combines gain switching with passive Q switching of a miniature diode-pumped solid-state laser. A composite pumping pulse, consisting of a long, low-intensity pulse and a following short, high-intensity pulse, and it is used to reduce the timing jitter. A greater-than-tenfold reduction in timing jitter is exists. Compact all-solid-state lasers play a crucial role in applications as laser radars and range finders for smart munitions, for which precision, light weight, low power consumption, and ruggedness are achieved simultaneously. Many applications require powerful nanosecond pulses whose firing times can be controlled as well.

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

433

These pulses can be obtained by Q switching. Passive Q switching with saturate absorbers has the advantage of ruggedness, simplicity, compactness, and good control of transverse mode structure. Q switching is a method for obtaining energetic pulses from lasers by modulating the intra cavity losses. Q switching is a technique for obtaining energetic short pulses from a laser by modulating the intra cavity losses and thus the Q factor of the laser resonator. The technique is mainly applied for the generation of nanosecond pulses of high energy and peak power with solid-state bulk lasers. A generation of a Q switched pulse follows the following steps: (1) initially, the resonator losses are kept at a high level. As lasing cannot occur at that time, the energy fed into the gain medium by the pumping mechanism accumulates there. The amount of stored energy is limited only by spontaneous emission, in other cases with strong enough gain by the onset of lasing or strong ASE, or by the pump energy available. The stored energy can be a multiple of the saturation energy, (2) the losses are suddenly reduced to a small value, and the power of the laser radiation builds up very quickly in the laser resonator. The process starts with noise from spontaneous emission, which is amplified to macroscopic power levels within hundreds or thousands of resonator roundtrips, (3) once, the temporally integrated intra cavity power has reached the order of the saturation energy of the gain medium, the gain starts to be saturated. The peak of the pulse is reached when the gain equals the remaining resonator losses. The large intra cavity power present at that time leads to further depletion of the stored energy during the time where the power decays. The pulse duration achieved with Q switching is typically in nanosecond range, and usually well above the resonator round-trip time. The energy of the generated pulse is typically higher than the saturation energy of the gain medium and can be in the mill joule range even for small lasers. The peak power can be orders of magnitude higher than the power which is achievable in continuous-wave operation. Regularly, Q-switched lasers generate regular pulse trains via repetitive Q switching. The pulse repetition rate is typically in the range from 1 to 100 kHz, sometimes higher. Passively Q-switched microchip lasers have reached pulse durations from below 1nsec and repetition rates up to several MHz, whereas large laser systems can deliver pulses with many kilojoules of energy and durations in the nanosecond range. Q switched lasers are lasers which applied Q-switching technique [4]. Active Q switching: The losses are modulated with an active control element (active Q switch). The pulse is formed shortly after an electrical trigger signal arrives. There are mechanical Q-switches such as spinning mirrors, used as end mirrors of laser resonators. The achieved pulse energy and pulse duration depends on the energy stored in the gain medium (pump power and the pulse repetition rate). The pulse repetition rate of an actively Q-switched laser can be controlled via the modulator. Higher repetition rates typically lead to lower pulse energies, if the pump power is kept constant. At the same time, the pulses then become longer, as the initial laser gain become lower. Once the pulse period exceeds the upper-state lifetime, increasing losses via spontaneous emission limit the possible pulse energy. The duration of the generated pulses is at least of the order of the resonator round-trip time, and often substantially longer than that, if the laser gain and/or the resonator losses are low. For high pulse repetition rates, it can be difficult to obtain very short pulses. Cavity

434

4 Solid State Laser Nonlinearity Applications in Engineering

dumping method can solve the problem. Once most of the stored energy has been transferred into the circulating pulse, the energy is suddenly released with the cavity dumper, which is a fast optical switch. The optical energy in the resonator can be extracted within one resonator round-trip time, independent of the time required for pulse build-up. Passive Q switching: For passive Q switching (self Q switching), the losses are automatically modulated with a saturate absorber. The pulse is formed as soon as the energy stored in the gain medium has reached a sufficiently high level. In many cases, the pulse energy and duration are then fixed, and changes of the pump power only influence the pulse repetition rate. However, the performance of a passively Q switched laser can be substantially degraded by timing jitter. The jitter is caused by fluctuations (temperature, pump power and wavelength, and intra cavity loss, within the system). Timing jitter in a Q-switched laser is inevitable because the first photon of the oscillation mode comes from spontaneous emission of the gain medium. This initial timing jitter imposes a fundamental lower limit on the timing jitter of all Qswitched lasers. A second limitation derives from the inevitable fluctuations of the pump. Temperature fluctuation, which affects pump power and wavelength and the absorption coefficient of the gain medium, usually contributes to the long-term drift of the repetition rate. An actively Q-switched solid-state laser has substantially less jitter but requires a number of additional components, which increases the size, weight, complexity, and ultimately the cost of the laser. We discuss a novel scheme to reduce timing jitter that combines gain switching with passive Q switching of a miniature diodepumped solid-state laser by use of a composite pumping pulse that consists of a long, low-intensity pulse and a following short, high intensity pulse. The simple model of a passively Q-switched laser is presented. The effective pumping rate of , where P is the electrical power input; η is the upper laser level is pe f f = h·vη·P L ·V the pumping efficiency, which is the product of laser-diode efficiency and optical coupling efficiency; v L is the laser frequency; V is the laser mode volume, and h is the Planck constant. Then the delay differential equations (DDEs) of our system is for population density of the upper laser level (NG ) and the photon density in the cavity (n p ) are as follow. Due to jitter interferences there is a delay in time for NG and n p system variables: NG (t) → NG (t − τ1 ), n p (t) → n p (t − τ2 ). There is no effects of time   delay parameters (τ1 , τ2 ) on NG and n p variables derivatives in time d NG dn p , dt dt

. Remark: NG = NG (t); n p = n p (t)

d NG 1 = pe f f − · NG (t − τ1 ) − c · σG · n p (t − τ2 ) · NG (t − τ1 ) dt τg dn p = c · σG · n p (t − τ2 ) · NG (t − τ1 ) − c · σ A · n p (t − τ2 ) · N A dt 1 − · n p (t − τ2 ) + rsp τp

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

435

n p —Density of photons in the laser mode, NG —Population density of the upper laser level, τg —Lifetime of the upper laser level, σG —Emission cross section of the gain medium, c—Speed of light in vacuum, N A —Density of the saturate-absorber atoms (in the absence of laser light it is N A0 ), σ A —Effective absorption cross section (it takes into account the ratio between the volumes of the active mode in the absorber and in the gain medium), τ p —Photon cavity lifetime, rsp —Rate of spontaneous 0 —Threshold density of the upper laser level (with emission into the laser mode, NG,th dn the unsaturated saturate absorber, by setting dtp = 0 and assuming that rsp = 0. Assuming that rsp = 0 because few photons generated by spontaneous emission 0 = c·σG1 ·τ p + contribute to the laser mode that is defined by the cavity mirrors; NG,th −

t

0 , as N A0 · σσGA . By replacing NG in equation NG = pe f f · τg · (1 − e τg ) with NG,th p ·τ ef f g α τd = τg · ln( α−1 ), where α = N 0 . τd —Time when the laser reaches its threshold, G,th α—Ratio of pump power to the threshold pump power.

Remark 1 The larger the pump power is the shorter the delay time. dn

0 Remark 2 The threshold density of the upper laser level is NG,th ( dtp = 0; rsp = 0) and we get the time when the laser reaches it threshold (τd ) by the equation NG = − t 0 pe f f · τg · (1 − e τg ) and replacing t → τd ; NG → NG,th .

NG = pe f f · τg · (1 − e

− τtg

0 ) ⇒ NG,th = pe f f · τg · (1 − e

τ

− τdg

);

pe f f · τ g 0 NG,th

τ τ pe f f · τ g − d − d α · (1 − e τg ) = 1 ⇒ 1 − e τg 0 NG,th     τ τ α−1 τd 1 − τdg 1 α−1 α−1 − τdg ; ln[e ] = ln ; − = ln = ;e =1− = α α α α τg α     τd α−1 α−1 α ] ⇒ τd = −τg · ln ; τd = τg · ln − = ln[ τg α α α−1

· (1 − e

τ

− τdg

) = 1; α =

Remark 3 Timing jitter differential; dτd .      dα · (α − 1) − α · dα α α−1 ⇒ dτd = τg · · ; dτd τd = τg · ln α−1 α (α − 1)2 1 · (α · dα − dα − α · dα) = τg · α · (α − 1) 1 1 dτd = −τg · · dα ⇒ dτd = τg · α · (α − 1) α · (1 − α) 1 α P · dα; d   ⇒ τd = τg · · α  ⇒ τd α · (1 − α) α P α P 1 1 · ; τd = τg · · = τg · (1 − α) α (1 − α) P 

436

4 Solid State Laser Nonlinearity Applications in Engineering

Remark 4 A small change in pump power causes very large charge in the delay unless α is very large. It is difficult to maintain a pump power that is so large for a prolonged period of time. Remark 5 The solution of the differential equation for the population density of the NG upper laser level ddt = . . . in the absence of lasing (n p = 0) can be calculated as follow: we neglect the time delay parameters τ1 and τ2 then we get the differential equation: 1 d NG 1 d NG = pe f f − = pe f f − · N G − c · σG · n p · N G ; n p = 0 ⇒ · NG dt τg dt τg Assumption: The population density of the upper laser level (NG ) at t = 0 is equal to zero (NG (t = 0) = 0). d NG 1 d NG d NG = pe f f − = pe f f · τ g − N G ; · N G ⇒ τg · dt τg dt pe f f · τ g − N G  4  1 1 · d NG = · dt τg pe f f · τ g − N G 4 1 = · dt ⇒ − ln( pe f f · τg − NG ) τg 1 = · t + const1 τg 1 · t − const1; const2 = −const1; ln( pe f f · τg − N G ) τg 1 − 1 ·t+const2 − 1 ·t+const2 = − · t + const2e τg = pe f f · τ g − N G ; e τ g τg

ln( pe f f · τg − N G ) = −

=e

− τ1g ·t

· econst2 ; e

N G = pe f f · τ g − e

− τ1g

− τ1g ·t

·t

· econst2 = pe f f · τg − N G

· econst2

According to the assumption:NG (t = 0) = 0. NG (t = 0) = pe f f · τg − econst2 = 0 ⇒ pe f f · τg = econst2 ; ln( pe f f · τg ) = ln(econst2 ) ⇒ const2 = ln( pe f f · τg ) econst2 = eln( pe f f ·τg ) = pe f f · τg −

1

·t

−

1

·t

Then we get NG = pe f f · τg − e τg · pe f f · τg ; NG (t) = pe f f · τg · (1 − e τg ). Q-switched laser scheme system: The Q-switched diode-pumped solid-state laser with low timing jitter scheme is uses composite pump pulses. Each composite pump pulse consists of two rectangular pulses: a long, low-power pulse followed by a short, high power pulse. The first pump pulse brings the system close to lasing, and the second one raises the upper-level population in a much faster manner and thus trigger

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

437

NG the Q-switch pulse. Timing jitter is in reverse proportion to the slope, ddt , within the window ( pe f f ). Incorporating two pulses into the pump pulse one make the slop much higher within the lasing window, leading to a substantial reduction in timing jitter. The experimental system is constructed from a composite pulse from two separate lasers rather than one laser because the two laser arrangement makes the electronics much simpler. Two laser diodes (LDs) emitting at 808 nm with orthogonal polarization were used to generate long and short pulses, respectively. Two pulses were combined by a polarization beam splitter (PBS) unit. A variable delay generator is used to set the correct timing of the second pulse relative to the first pulse. A comparison of Q-switched pulses obtained with simple pumping when the short pulse is timed to be delayed relative to the Q-switched pulse. The composite pulse, the Q-switched pulse is timed to be triggered by the short pulse. The effect of the power of pumping on the timing jitter of a passively Q-switched laser can be also demonstrated by use of only one pumping laser diode but with constant DC injection currents of different magnitude (Fig. 4.5). The performance of the composite pumping scheme can be further improved by use of only one pump laser to provide both long and short pulses and by improvement of the electronics to yield a sharper short pulse. It is a simple way to reduce the timing jitter of a passively Q-switched laser with a composite pump pulse that is composed of a long, low-power pulse and a short, high power pulse (Fig. 4.6). The target is to obtain a reduction in timing jitter for Nd:YAG/Cr4+ :YAG laser. Polarizing Beam Splitter (PBS): Polarizing beam splitter is used to split unpolarized light into polarized parts. Polarizing beam splitter is beam splitter designed to split light by polarization state rather than by wavelength or intensity. Polarizing beam splitter is used in semiconductor or photonics instrumentation to transmit ppolarized light while reflecting s-polarized light. Polarizing beam splitter is typically designed for 00 to 450 angle of incident with 900 separations of the beams, depending

PBS LED driver and LED (1)

Nd:YAG Len (1)

Beam Shaper

Len (3) Beam Shaper

Trigger source

Len (2)

Delay Unit

LED driver and LED (2)

Fig. 4.5 Experimental Q-switched laser scheme system

Cr4+:YAG

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4 Solid State Laser Nonlinearity Applications in Engineering

Fig. 4.6 Experimental Q-switched laser system signals in time

LED (1) Intensity

0 LED (2) Intensity Time Delay 0 Len (3) Output

0

t [sec]

t [sec]

t [sec]

on the configuration. In other way we define beam splitter as an optical component used to split incident light at a designated ration into two separate beams. Additionally, beam splitter can be used in reverse to combine two different beams into a single one (our case). Beam splitter can be cube or plate. Cube beam splitter: It is constructed using two typically right angle prisms. The hypotenuse surface of the prism is coated, and the two prisms are centered together so that they cubic shape. The light is transmitted into the coated prism, which often features a reference mark on the ground surface. Plate beam splitter: It is consist of a thin, flat glass plate that has been coated on the first surface of the substrate. Most plate beam splitters feature an anti-reflection coating on the second surface to remove unwanted Fresnel reflections. Beam splitter construction: It is made from two triangular glass prisms which are glued together at their base using polyester, epoxy, or urethane-based adhesives. The thickness of the resin layer is adjusted such that half of the light incident through one “port” is reflected and the other half is transmitted due to frustrated total internal reflection. There are other design configurations for beam splitter. A beam splitter that consists of a glass plate with a reflective dielectric coating on one side gives a phase shift of 0 to π , depending on the side from which it is incident. Transmitted waves have no phase shift. Reflected waves entering from the reflective side are phaseshifted by π , whereas reflected waves entering from the glass side have no phase shift. The explanation is Fresnel equations, according to which reflection causes a phase shift only when light passing through a material of low refractive index is reflected at a material of high refractive index.

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

439

Optical beam shaper: A beam shaper (beam converter) is an optical device which reshapes a light beam by modifies its spatial properties. It can be refractive and microoptic beam shaper which creates a flat-top beam from a Gaussian input beam. Another option is to use in conjunction with a high-power laser diode. It makes both its beam radius and beam quality more symmetric with respect to two orthogonal directions. It can be a device, which based on two highly reflective mirrors and largely preserves the brightness or a device which based on micro-optical structures, containing arrays of small prisms, to perform a similar function with a smaller device. System fixed points (equilibrium points): lim t→∞ NG (t −τ1 ) = NG (t); t  τ1 ⇒ t − τ1 ≈ t lim n p (t − τ2 ) = n p (t); t  τ2 ⇒ t − τ2 ≈ t;

t→∞

dn p d NG = 0; =0 dt dt

1 d NG · N G∗ − c · σG · n ∗p · N G∗ = 0 = 0 ⇒ pe f f − dt τg dn p 1 (∗∗) · n ∗p + rsp = 0 = 0 ⇒ c · σG · n ∗p · N G∗ − c · σ A · n ∗p · N A − dt τp 1 · N G∗ − c · σG · n ∗p · N G∗ = 0 ⇒ n ∗p (∗) pe f f − τg 1 1 = · pe f f − c · σG · N G∗ c · σG · τg 1 (∗) → (∗∗) : c · σG · n ∗p · N G∗ − c · σ A · n ∗p · N A − · n ∗p + rsp = 0 τp   1 + rsp = 0 n ∗p · c · σG · N G∗ − c · σ A · N A − τp (∗)



   1 1 ∗ · c · σ + rsp = 0 · N − c · σ · N − G A A G c · σG · NG∗ c · σG · τg τp 1 1 ∗ · pe f f · c · σ A · N A ∗ · pe f f · c · σ G · N G − c · σG · N G c · σG · NG∗ 1 1 1 − − · c · σG · NG∗ ∗ · pe f f · c · σG · N G τp c · σG · τg 1 1 1 + · c · σA · NA + · + rsp = 0 c · σG · τg c · σG · τg τ p 1

· pe f f −

pe f f 1 1 · NG∗ ∗ · pe f f · σ A · N A − ∗ − σG · N G c · σG · τ p · N G τg σA · NA 1 + + + rsp = 0 σG · τg c · σG · τg · τ p σA · NA 1 1 + pe f f + + rsp = · pe f f · σ A · N A σG · τg c · σG · τg · τ p σG · NG∗ pe f f 1 + · NG∗ ∗ + c · σG · τ p · N G τg

pe f f −

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4 Solid State Laser Nonlinearity Applications in Engineering

We define for simplicity a global parameter: σA · NA 1 + pe f f + σG · τg c · σG · τg · τ p pe f f 1 1 + rsp · pe f f · σ A · N A + + · NG∗ σG · NG∗ c · σG · τ p · NG∗ τg

1 =

= 1

c · τ p · τg · pe f f · σ A · N A + τg · pe f f + c · σG · τ p · (NG∗ )2 c · σG · τ p · NG∗ · τg

= 1 c · τ p · τg · pe f f · σ A · N A + τg · pe f f + c · σG · τ p · (NG∗ )2 = 1 · c · σG · τ p · NG∗ · τg c · σG · τ p · (NG∗ )2 − 1 · c · σG · τ p · τg · NG∗ + c · τ p · τg · pe f f · σ A · N A + τg · pe f f = 0 We define for simplicity global parameters: 2 = c · σG · τ p ; 3 = −1 · c · σG · τ p · τg   σA · NA 1 3 = −1 · c · σG · τ p · τg = − + pe f f + + rsp σG · τg c · σG · τg · τ p · c · σG · τ p · τg 3 = −1 · c · σG · τ p · τg = −(σ A · N A · c · τ p + pe f f · c · σG · τ p · τg + 1 + rsp · c · σG · τ p · τg ) 4 = c · τ p · τg · pe f f · σ A · N A + τg · pe f f ; 2 · (NG∗ )2 + 3 · NG∗ + 4 = 0 2 · (NG∗ )2 + 3 · NG∗ + 4  −3 ± 23 − 4 · 2 · 4 = 0 ⇒ NG∗ = 2 · 2 23 − 4 · 2 · 4 = (σ A · N A · c · τ p + pe f f · c · σG · τ p · τg + 1 + rsp · c · σG · τ p · τg )2 − 4 · c · σ G · τ p · c · τ p · τ g · pe f f · σ A · N A + τ g · pe f f 23 − 4 · 2 · 4 = (σ A · N A · c · τ p + pe f f · c · σG · τ p · τg + 1 + rsp · c · σG · τ p · τg )2 − 4 · c2 · σG · τ p2 · τg · pe f f · σ A · N A + τg · pe f f NG∗ is the population density of the upper laser level—fixed points and must be real and positive value then NG∗ ≥ 0(except specific  cases when is equal to zero). NG∗ ≥ 0 ⇒ 23 − 4 · 2 · 4 > 0 and −3 ±

23 − 4 · 2 · 4 > 0

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

n ∗p =

1 c · σG ·

NG∗ 

= c · σG ·

· pe f f − 1 √

−3 ±

441

1 c · σG · τg

23 −4·2 ·4

 · pe f f −

1 c · σG · τg

2·2

1 c · σG · c · σG · τg 2 · 2 · pe f f 1 −  =  c · σG · τg c · σG · −3 ± 23 − 4 · 2 · 4

n ∗p =

1

NG∗

· pe f f −

We can define our system two fixed points:   (0) E (0) N G , n (0) p ⎛  ⎜ −3 + =⎜ ⎝

23 − 4 · 2 · 4 2 · 2

⎞ ,

⎟ 2 · 2 · pe f f 1 ⎟ −   c · σG · τg ⎠ c · σG · −3 + 23 − 4 · 2 · 4

  (1) E (1) N G , n (1) p ⎛  ⎜ −3 − =⎜ ⎝

23 − 4 · 2 · 4 2 · 2

⎞ ,

⎟ 2 · 2 · pe f f 1 ⎟  −  ⎠ c · σ · τ G g c · σG · −3 − 23 − 4 · 2 · 4

     (1) NG(1) , n (1) must be nonRemark System fixed points E (0) NG(0) , n (0) p ,E p negative and real numbers, if for specific system parameters values fixed point is complex or negative you need to ignore it (mathematical fixed point and not system physical fixed point). Stability analysis: The standard local stability analysis about any one of the equilibrium points ofthe Q-switched laser system corresponding to the rate differential  dn p d NG equation system dt = · · · ; dt = · · · consists in adding to coordinate [NG , n p ] arbitrarily small increments of exponential form [n G , n p ] · eλ·t and retaining the first order terms in NG , n p . The system of two homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables (NG , n p ) and variables derivatives with respect to time into system rate equations [5–7]. System fixed values with arbitrarily small increments of exponential form [n G , n p ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. ( j)

NG (t) = NG + n G · eλ·t ; n p (t) = n (pj) + n p · eλ·t ; NG (t − τ1 )

442

4 Solid State Laser Nonlinearity Applications in Engineering ( j)

= NG + n G · eλ·(t−τ1 ) n p (t − τ2 ) = n (pj) + n p · eλ·(t−τ2 ) ; = n G · λ · eλ·t ;

dn p (t) = n p · λ · eλ·t dt

d NG (t) dt

We choose these expressions for ourselves NG (t), n p (t) as a small displacement [n G , n p ] from the system fixed points in time t = 0. ( j)

NG (t = 0) = NG + n G ; n p (t = 0) = n (pj) + n p d NG 1 · NG (t − τ1 ) − c · σG · n p (t − τ2 ) · NG (t − τ1 ) = pe f f − dt τg  1  ( j) n G · λ · eλ·t = pe f f − · NG + n G · eλ·(t−τ1 ) τg    ( j)  − c · σG · n (pj) + n p · eλ·(t−τ2 ) · NG + n G · eλ·(t−τ1 ) 1 1 ( j) · NG − · n G · eλ·(t−τ1 ) τg τg  ( j) − c · σG · n (pj) · NG + n (pj) · n G · eλ·(t−τ1 )

n G · λ · eλ·t = pe f f −

( j)

+NG · n p · eλ·(t−τ2 ) + n p · n G · eλ·(t−τ2 ) · eλ·(t−τ1 )



Assumption: n p · n G ≈ 0 1 1 ( j) · NG − · n G · eλ·(t−τ1 ) τg τg   ( j) ( j) − c · σG · n (pj) · NG + n (pj) · n G · eλ·(t−τ1 ) + NG · n p · eλ·(t−τ2 )

n G · λ · eλ·t = pe f f −

n G · λ · eλ·t = pe f f −

1 1 ( j) · NG − · n G · eλ·(t−τ1 ) τg τg ( j)

− c · σG · n (pj) · NG − c · σG · n (pj) · n G · eλ·(t−τ1 ) ( j)

n G · λ · eλ·t

− c · σG · NG · n p · eλ·(t−τ2 ) 1 ( j) ( j) = pe f f − · NG − c · σG · n (pj) · NG τg 1 − · n G · eλ·(t−τ1 ) − c · σG · n (pj) · n G · eλ·(t−τ1 ) τg ( j)

− c · σG · NG · n p · eλ·(t−τ2 ) At fixed points: pe f f − n G · λ · eλ·t = −

1 τg

( j)

( j)

( j)

· NG − c · σG · n p · NG = 0.

1 · n G · eλ·(t−τ1 ) − c · σG · n (pj) · n G · eλ·(t−τ1 ) τg

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation ( j)

− c · σG · NG · n p · eλ·(t−τ2 ) n G · λ · eλ·t = −

443

1 · n G · eλ·t · e−λ·τ1 τg

( j)

− c · σG · n (pj) · n G · eλ·t · e−λ·τ1 − c · σG · NG · n p · eλ·t · e−λ·τ2 Divide the two sides by eλ·t term. nG · λ = −

1 ( j) · n G · e−λ·τ1 − c · σG · n (pj) · n G · e−λ·τ1 − c · σG · NG · n p · e−λ·τ2 τg

We have there possible cases: (1) τ1 = τ ; τ2 = 0, (2) τ1 = 0; τ2 = τ , (3) τ1 = τ ; τ2 = τ . We analyze our system for the first case τ1 = τ ; τ2 = 0. 1 ( j) ( j) · n G · e−λ·τ − c · σG · n p · n G · e−λ·τ − c · σG · N G · n p τg   1 ( j) ( j) − nG · λ − + c · σG · n p · n G · e−λ·τ − c · σG · N G · n p = 0 τg

nG · λ = −

dn p = c · σG · n p (t − τ2 ) · N G (t − τ1 ) − c · σ A · n p (t − τ2 ) dt 1 · NA − · n p (t − τ2 ) + rsp τp     ( j) ( j) n p · λ · eλ·t = c · σG · n p + n p · eλ·(t−τ2 ) · N G + n G · eλ·(t−τ1 )   ( j) − c · σ A · n p + n p · eλ·(t−τ2 ) · N A   1  ( j) ( j) ( j) − · n p + n p · eλ·(t−τ2 ) + rsp n p · λ · eλ·t = c · σG · n p · N G τp ( j)

( j)

+n p · n G · eλ·(t−τ1 ) + N G · n p · eλ·(t−τ2 ) + n p · n G · eλ·(t−τ2 ) · eλ·(t−τ1 )



( j)

− c · σ A · n p · N A − c · σ A · n p · N A · eλ·(t−τ2 ) 1 1 ( j) − · np − · n p · eλ·(t−τ2 ) + rsp τp τp ( j)

n p · λ · eλ·t = c · σG · n (pj) · NG + c · σG · n (pj) · n G · eλ·(t−τ1 ) ( j)

+ c · σG · NG · n p · eλ·(t−τ2 ) + c · σG · n p · n G · eλ·(t−τ2 ) · eλ·(t−τ1 ) − c · σ A · n (pj) · N A 1 1 − c · σ A · n p · N A · eλ·(t−τ2 ) − · n (pj) − · n p · eλ·(t−τ2 ) + rsp τp τp Assumption: n p · n G ≈ 0 ( j)

n p · λ · eλ·t = c · σG · n (pj) · NG + c · σG · n (pj) · n G · eλ·t · e−λ·τ1 ( j)

+ c · σG · NG · n p · eλ·t · e−λ·τ2

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4 Solid State Laser Nonlinearity Applications in Engineering

− c · σ A · n (pj) · N A − c · σ A · n p · N A · eλ·t · e−λ·τ2 1 1 − · n (pj) − · n p · eλ·t · e−λ·τ2 + rsp τp τp We analyze our system for the first case τ1 = τ ; τ2 = 0. ( j)

n p · λ · eλ·t = c · σG · n (pj) · NG + c · σG · n (pj) · n G · eλ·t · e−λ·τ1 ( j)

+ c · σG · NG · n p · eλ·t · e−λ·τ2 − c · σ A · n (pj) · N A − c · σ A · n p · N A · eλ·t · e−λ·τ2 1 1 − · n (pj) − · n p · eλ·t · e−λ·τ2 + rsp τp τp ( j)

n p · λ · eλ·t = c · σG · n (pj) · NG − c · σ A · n (pj) · N A −

1 · n (pj) + rsp τp ( j)

+ c · σG · n (pj) · n G · eλ·t · e−λ·τ + c · σG · NG · n p · eλ·t 1 − c · σ A · n p · N A · eλ·t − · n p · eλ·t τp ( j)

( j)

( j)

At fixed points: c · σG · n p · NG − c · σ A · n p · N A −

1 τp

( j)

· n p + rsp = 0 ( j)

n p · λ · eλ·t = c · σG · n (pj) · n G · eλ·t · e−λ·τ + c · σG · NG · n p · eλ·t 1 − c · σ A · n p · N A · eλ·t − · n p · eλ·t τp Divide the two sides by eλ·t term. ( j)

n p · λ = c · σG · n (pj) · n G · e−λ·τ + c · σG · NG · n p − c · σ A · n p · N A − ( j)

− n p · λ + c · σG · n (pj) · n G · e−λ·τ + c · σG · NG · n p 1 − c · σA · n p · NA − · n p = 0c · σG · n (pj) · n G · e−λ·τ τp   1 ( j) · np = 0 − n p · λ + c · σG · N G − c · σ A · N A − τp We can summarize our last results:   1 ( j) + c · σG · n (pj) · n G · e−λ·τ − c · σG · NG − nG · λ − τg · n p = 0 c · σG · n (pj) · n G · e−λ·τ − n p · λ

1 · np τp

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

445

  1 ( j) · np = 0 + c · σG · N G − c · σ A · N A − τp The small increments Jacobian of our system is as follow: ⎛   ( j) ( j) −c · σG · N G − τ1g + c · σG · n p · e−λ·τ − λ  ⎝ ( j) ( j) c · σG · N G − c · σ A · N A − c · σG · n p · e−λ·τ   nG · =0 np

⎞ 1 τp



−λ

(A − λ · I )  ⎛  ( j) ( j) −c · σG · N G − τ1g + c · σG · n p · e−λ·τ − λ  =⎝ ( j) ( j) c · σG · n p · e−λ·τ c · σG · N G − c · σ A · N A −

⎠

⎞ 1 τp



−λ

⎠

det(A − λ · I ) = 0 ⎛  ⎞  ( j) ( j) − τ1g + c · σG · n p · e−λ·τ − λ −c · σG · N G   ⎠=0 det ⎝ ( j) ( j) c · σG · N G − c · σ A · N A − τ1p − λ c · σG · n p · e−λ·τ     1 1 ( j) ( j) det(A − λ · I ) = − + c · σG · n p · c · σG · N G − c · σ A · N A − · e−λ·τ τg τp   1 ( j) + + c · σG · n p · e−λ·τ · λ τg   1 ( j) − c · σG · N G − c · σ A · N A − · λ + λ2 τp ( j)

( j)

( j)

( j)

+ c2 · σG2 · n p · N G · e−λ·τ     1 1 ( j) ( j) · e−λ·τ det(A − λ · I ) = − + c · σG · n p · c · σG · N G − c · σ A · N A − τg τp 1 ( j) + ( + c · σG · n p ) · e−λ·τ · λ τg   1 ( j) ( j) ( j) − c · σG · N G − c · σ A · N A − · λ + λ2 + c2 · σG2 · n p · N G · e−λ·τ τp     1 1 ( j) ( j) det(A − λ · I ) = − + c · σG · n p · c · σG · N G − c · σ A · N A − · e−λ·τ τg τp   1 ( j) + + c · σG · n p · e−λ·τ · λ τg   1 ( j) · λ + λ2 − c · σG · N G − c · σ A · N A − τp + c2 · σG2 · n p · N G · e−λ·τ

  " 1 ( j) ( j) · λ + λ2 + c2 · σG2 · n (pj) · NG D(λ, τ ) = − c · σG · NG − c · σ A · N A − τp     1 1 ( j) − + c · σG · n (pj) · c · σG · NG − c · σ A · N A − τg τp

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4 Solid State Laser Nonlinearity Applications in Engineering

   1 + + c · σG · n (pj) · λ · e−λ·τ τg D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n > m   1 ( j) · λ + λ2 Pn (λ, τ ) = − c · σG · NG − c · σ A · N A − τp   1 ( j) Q m (λ, τ ) = c2 · σG2 · n (pj) · NG − + c · σG · n (pj) τg   1 ( j) · c · σG · N G − c · σ A · N A − τp   1 + + c · σG · n (pj) · λ τg n  Pn (λ, τ ) = pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ k=0

+ p2 (τ ) · λ2 ; Q m (λ, τ ) =

m 

qk · λk = q0 (τ ) + q1 (τ ) · λ

k=0

D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n = 2; m = 1; n > m   1 ( j) ; p2 (τ ) = 1 p0 (τ ) = 0; p1 (τ ) = − c · σG · NG − c · σ A · N A − τp   1 ( j) + c · σG · n (pj) q0 (τ ) = c2 · σG2 · n (pj) · NG − τg     1 1 ( j) ( j) ; q1 (τ ) = · c · σG · N G − c · σ A · N A − + c · σG · n p τp τg The homogeneous system for NG , n p leads to a characteristic equation for the  eigenvalue λ having the form P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; P(λ) = 2k=0 a j ·  λk ; Q(λ) = 1k=0 c j · λk . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and τ . qi , qk are any Q-switched laser system parameters, other parameters kept as a constant.   1 ( j) ; a2 = 1 a0 = 0; a1 = − c · σG · NG − c · σ A · N A − τp   1 ( j) 2 2 ( j) ( j) + c · σG · n p c0 = c · σG · n p · NG − τg     1 1 ( j) ( j) ; c1 = · c · σG · N G − c · σ A · N A − + c · σG · n p τp τg Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

447

and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0. Furthermore P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [5] can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) If | Q(λ) P(λ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0. ( j)

Pn (λ = i · ω, τ ) = −(c · σG · NG − c · σ A · N A −

1 ) · i · ω − ω2 τp

( j)

Q m (λ, τ ) = c2 · σG2 · n (pj) · NG     1 1 ( j) ( j) − + c · σG · n p · c · σG · N G − c · σ A · N A − τg τp   1 + + c · σG · n (pj) · i · ω τg Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )    1 ( j) ( j) 2 2 ( j) ( j) + c · σG · n p · c · σG · N G = c · σG · n p · N G − τg  1 − ω2 −c · σ A · N A − τp     1 1 ( j) · i · ω = 0 + + c · σG · n (pj) − c · σG · NG − c · σ A · N A − τg τp ( j)

|P(i · ω, τ )|2 = (c · σG · NG − c · σ A · N A −

1 2 2 ) · ω + ω4 τp

" ( j) |Q(i · ω, τ )|2 = c2 · σG2 · n (pj) · NG   2  1 1 ( j) − + c · σG · n (pj) · c · σG · NG − c · σ A · N A − τg τp 2  1 + + c · σG · n (pj) · ω2 τg F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2

448

4 Solid State Laser Nonlinearity Applications in Engineering

 1 2 2 = c · σG · − c · σA · NA − · ω + ω4 τp    1 ( j) 2 2 ( j) ( j) − c · σG · n p · N G − + c · σG · n p τg 2  1 ( j) · c · σG · N G − c · σ A · N A − τp 2  1 − + c · σG · n (pj) · ω2 τg 

( j) NG

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2     1 ( j) ( j) = − c2 · σG2 · n (pj) · NG − + c · σG · n (pj) · c · σG · NG τg 2   1 1 2 ( j) c · σG · N G − c · σ A · N A − − c · σA · NA − + τp τp 2   1 − + c · σG · n (pj) · ω2 + ω4 τg We define the following parameters for simplicity: 0 , 2 , 4 .    1 ( j) 2 2 ( j) ( j) + c · σG · n p 0 = − c · σG · n p · NG − τg  2 1 ( j) · c · σG · N G − c · σ A · N A − τp   1 2 ( j) 2 = c · σ G · N G − c · σ A · N A − τp 2  1 − + c · σG · n (pj) ; 4 = 1 τg Hence F(ω, τ ) = 0 implies solving the above polynomial.

2 k=0

2·k · ω2·k = 0 and its roots are given by

PR (i · ω, τ ) = −ω2 ; PI (i · ω, τ )   1 ( j) · ω; Q I (i · ω, τ ) = − c · σG · N G − c · σ A · N A − τp   1 + c · σG · n (pj) · ω = τg ( j)

Q R (i · ω, τ ) = c2 · σG2 · n (pj) · NG

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

 − sin θ (τ ) =

449

   1 1 ( j) + c · σG · n (pj) · c · σG · NG − c · σ A · N A − τg τp

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2

cos θ (τ ) = −

PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) |Q(i · ω, τ )|2

We use different parameters terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 2; m = 1; n > m; Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

2 

a j · λ j ; Q(λ) =

j=0

1 

c j · λ j ; P(λ)

j=0

= a0 + a1 · λ + a2 · λ ; Q(λ) = c0 + c1 · λ 2

n, m ∈ N0 ; n > m and a j , c j :R+0 → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ . The coefficients a j ( pe f f , τg , σG , σ A , N A , τ, . . .) ∈ R and c j ( pe f f , τg , σG , σ A , N A , τ, . . .) ∈ R depend on Q-switched laser system’s pe f f , τg , σG , σ A , N A , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: ( pe f f , τg , σG , σ A , N A , τ, . . .) will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0. ( j)

a0 = 0; c0 = c2 · σG2 · n (pj) · NG     1 1 ( j) − + c · σG · n (pj) · c · σG · NG − c · σ A · N A − τg τp   1 ( j) + c · σG · n (pj) c2 · σG2 · n (pj) · NG − τg   1 ( j) = 0 · c · σG · N G − c · σ A · N A − τp ∀ pe f f , τg , σG , σ A , N A , τ, . . . ∈ R+ , i.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ); Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang, 1993, Sect. 3.4) can be varied in the present case.

450

4 Solid State Laser Nonlinearity Applications in Engineering

(a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire ( pe f f , τg , σG , σ A , N A , τ, . . .) domain of interest. P(λ) (b) | Q(λ) | is bounded for |λ| → ∞; Reλ ≥ 0. No roots P(λ) bifurcation from ∞. Indeed, in the limit: | Q(λ) | = |

( j)

−(c·σG ·NG −c·σ A ·N A − τ1p )·λ+λ2

( j)

c2 · σG2 · n (pj) · NG − (

1 1 | ( j) + c · σG · n (pj) ) · (c · σG · NG − c · σ A · N A − ) τg τp

1 + c · σG · n (pj) ) · λ τg (c)  The following expressions exist: F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 ; F(ω) = 2 2·k k=0 2·k · ω Has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω4 ). (d) Each positive root ω( pe f f , τg , σG , σ A , N A , τ, . . .) of F(ω) = 0 is continuous and differentiable with respect to pe f f , τg , σG , σ A , N A , τ, . . .. The condition can only be assessed numerically. +(

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 maybe on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ—plane, whereby increasing the parameters: the crossing, change its sign from (−) to pe f f , τg , σG , σ A , N A , τ, . . . Reλ  may, at  to an unstable one, or vice versa. This (+), i.e. from a stable focus E (∗) NG(∗) , n (∗) p feature may be further assessed by examining the sign of the partial derivatives with respect to pe f f , τg , σG , σ A , N A , τ, . . . and system parameters [5, 6].  ∂Reλ ; pe f f , τg , σG , σ A , N A , . . . = const ∂τ λ=i·ω   ∂Reλ −1 ( pe f f ) = ; τ, τg , σG , σ A , N A , . . . = const ∂ pe f f λ=i·ω   ∂Reλ −1 (τg ) = ; pe f f , τ, σG , σ A , N A , . . . = const ∂τg λ=i·ω   ∂Reλ −1  (σG ) = ; pe f f , τg , τ, σ A , N A , . . . = const ∂σG λ=i·ω   ∂Reλ −1 (σ A ) = ; pe f f , τg , σG , τ, N A , . . . = const ∂σ A λ=i·ω   ∂Reλ −1 (N A ) = ; pe f f , τg , σG , σ A , τ, . . . = const ∂ N A λ=i·ω −1 (τ ) =



P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ)

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

451

When writing and inserting λ = i · ω into system’s characteristic equation ω must satisfy the following equations. −PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2 PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) cos(ω · τ ) = h(ω) = − |Q(i · ω, τ )|2

sin(ω · τ ) = g(ω) =

where |Q(i · ω, τ )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω·τ ) and cos(ω·τ ) equations, that by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ . It is important to notice that if τ ∈ / I (assume that I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for:τ ∈ / I ;ω(τ ) is not defined. Then for all τ in I , ω(τ ) is satisfied that F(ω) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = · · · ; cos θ (τ ) = · · · −PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2 PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) cos θ (τ ) = − |Q(i · ω)|2

sin θ (τ ) =

And the relation between the argument θ (τ ) and τ · ω(τ ) for τ ∈ I must be as describe below. ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ N0 )+2·n·π Hence we can define the maps τn : I → R+0 given by τn (τ ) = θ(τ ω(τ ;n ∈ ) N0 ; τ ∈ I . Let us introduce the functions: I → R; Sn (τ ) = τ −τn (τ ); τ ∈ I ; n ∈ N0 . That is continuous and differentiable in τ . In the following, the subscripts λ, ω, pe f f , τg , σG , σ A , N A , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ( pe f f , τg , σG , σ A , N A , . . .) and ω( pe f f , τg , σG , σ A , N A , . . .), and keeping all parameters except (x) and τ . The derivation closely follows that in [BK]. Differentiating system characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates:



x = pe f f , τ g , σ G , σ A , N A , . . . −1 ∂λ −Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) = ∂x Px (λ, x) · Q(λ, x) − Q X (λ, x) · P(λ, x)

452

4 Solid State Laser Nonlinearity Applications in Engineering

where Pλ = ∂∂λP ; Q λ = ∂∂λQ ; Px = ∂∂ Px ; Q x = ∂∂Qx , substituting λ = i · ω and bearing P(−i · ω) = P(i · ω) and Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains: 

∂λ ∂x 

=

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(i · ω, x) − τ · |P(i · ω, x)|2



Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I ; Pω = PRω + i · PI ω Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , when (x) can be any Q-switched laser system parameter pe f f , τg , σG , σ A , N A , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x) and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ); V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ .   1 ( j) · ω; PR = −ω2 ; PI = − c · σG · NG − c · σ A · N A − τp   1 ( j) QI = + c · σG · n (pj) · ω Q R = c2 · σG2 · n (pj) · NG τg     1 1 ( j) − + c · σG · n (pj) · c · σG · NG − c · σ A · N A − τg τp   1 PRω = −2 · ω; Q I ω = + c · σG · n (pj) ; τg   1 ( j) ; Q Rω = 0 PI ω = − c · σG · NG − c · σ A · N A − τp PRτ = 0; PI τ = 0; Q I τ = 0; Q Rτ = 0; PRω · PR = 2 · ω3 ; Q Rω · Q R = 0 Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0;

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

453

  1 ( j) · 2 · ω2 PI · PRω = c · σG · NG − c · σ A · N A − τp    1 ( j) Q R · Q I ω = c2 · σG2 · n (pj) · NG − + c · σG · n (pj) τg     1 1 ( j) ( j) · · c · σG · N G − c · σ A · N A − + c · σG · n p τp τg Q I · Q Rω = 0; V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0; F(ω, τ ) = 0 Differentiating with respect to τ and we get ∂ω ∂ω + Fτ = 0; τ ∈ I ⇒ ∂τ ∂τ   Fτ −2 · [U + τ · |P|2 ] + i · Fω −1 =− ;  (τ ) = Re Fω Fτ + 2 · i · [V + ω · |P|2 ]    ∂Reλ −1 ; sign{−1 (τ )} sign{ (τ )} = sign ∂τ λ=i·ω   V + ∂ω ·U ∂ω ∂τ +ω+ = sign{Fω } · sign ·τ |P|2 ∂τ Fω ·

We shall presently examine the possibility of stability transitions (bifurcation) Q-switched laser system, about the equilibrium point E (∗) (NG(∗) , n (∗) p ) as a result of a variation of delay parameterτ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ— plane, where by increasing the delay parameterτ , Reλ may at the crossing, changes its sign from − to + , i.e. from a stable focus E ∗ to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ . −1

 (τ ) =



∂Reλ ∂τ

 λ=i·ω

;

pe f f , τg , σG , σ A , . . . = const; ω ∈ R+  ( j) PR · PI ω = ω2 · c · σG · NG  1 ; Q I · Q Rω = 0 −c · σ A · N A − τp    1 ( j) + c · σG · n (pj) Q R · Q I ω = c2 · σG2 · n (pj) · NG − τg     1 1 ( j) · · c · σG · N G − c · σ A · N A − + c · σG · n (pj) τp τg

454

4 Solid State Laser Nonlinearity Applications in Engineering

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω )    1 ( j) = ω 2 · c · σG · N G − c · σ A · N A − τp    1 ( j) · 2 · ω2 − c · σG · N G − c · σ A · N A − τp    1 ( j) − c2 · σG2 · n (pj) · NG − + c · σG · n (pj) τg     1 1 ( j) · · c · σG · N G − c · σ A · N A − + c · σG · n (pj) τp τg Then we get the expression for F(ω, τ ) circuit parameter values. We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ ∈ [0.001 . . . 10] second. We plot the stability switch diagram based on different delay values of our Q-switched laser system [5, 6]. sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) · sign{τ · ωτ (ω(τ )) + ω(τ ) + |P(ω(τ ))|2 Remark We know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability.F—Function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = · · · ; cos θ (τ ) = · · · . −PR · Q I + PI · Q R ; cos(ω · τ ) |Q|2 PR · Q R + PI · Q I ; |Q|2 = Q 2R + Q 2I =− |Q|2

sin(ω · τ ) =

Finally, we plot the stability switch diagram. −1

g(τ ) =  (τ ) =



∂Reλ ∂τ

 λ=i·ω

2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = Fτ2 + 4 · (V + ω · P 2 )2    ∂Reλ −1 sign[g(τ )] = sign[ (τ )] = sign ∂τ λ=i·ω   2 2 · {Fω · (V + ω · P ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2

4.4 Q-Switched Diode-Pumped Solid-State Laser Stability Under Parameters Variation

455

Fτ2 + 4 · (V + ω · P 2 )2 > 0; sign[−1 (τ )] = sign{Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )}  $ sign[−1 (τ )] = sign {[Fω ] · (V + ω · P 2 )  Fτ Fτ 2 · (U + τ · P ) ; ωτ = − − Fω Fω  −1 ∂ω ∂ F/∂ω ωτ = =− ∂τ ∂ F/∂τ sign[−1 (τ )] = sign{[Fω ] · [V + ωτ · U + ω · P 2 + ωτ · τ · P 2 ]}sign[−1 (τ )]    V + ωτ · U + ω + ω · τ = sign [Fω ] · [P 2 ] · τ P2 sign[P 2 ] > 0 ⇒ sign[−1 (τ )]    V + ωτ · U + ω + ω · τ = sign [Fω ] · τ P2   V + ωτ · U + ω + ω · τ sign[−1 (τ )] = sign[Fω ] · sign τ P2 2  1 ( j) PI ω · PI = c · σG · NG − c · σ A · N A − · ω; τp 2  1 + c · σG · n (pj) · ω; PRω · PR = 2 · ω3 QIω · QI = τg Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]     1 2 ( j) 3 ·ω =2· 2 · ω + c · σG · N G − c · σ A · N A − τp 2   1 − + c · σG · n (pj) · ω τg We check the sign −1 (τ ) according the following rule (Table 4.6): If sign[−1 (τ )] > 0 then the crossing proceeds from (−) to (+), respectively (stable to unstable). If sign[−1 (τ )] < 0 then the crossing proceeds from (+) to (−) respectively (unstable to stable). Table. 4.6 Q-switched laser system sign of sign[−1 (τ )]

sign[Fω ]

τ ·U sign[ V +ω + ω + ωτ · τ ] P2

sign[−1 (τ )]

±

±

+

±

±

−

456

4 Solid State Laser Nonlinearity Applications in Engineering

4.5 Questions 1. We have a model of solid-state laser by semiconductor devices and the corresponding to the rate equations. We have two model assumptions: (1) the transverse relaxation times of the equivalent two-level models for the laser gain medium and for the saturate absorber are much faster than any other dynamics in the system (2) the changes in the laser intensity, gain, and saturate absorption on a time scale of the order of the round-trip time (TR ) in the cavity are small. The laser model of a single-mode laser with homogeneously broadband laser medium and saturate absorber is not ideal and there are interferences. We represent these interferences by time delay parameterτ . We can write our system Delay Differential Equations (DDEs): 2 (g(t) − g0 ) g(t) · P(t) dg dP = =− · (g(t) − l − q(t − τ )) · P(t); − dt TR dt TL · T R EL (q(t − τ ) − q0 ) q(t − τ ) · P(t) dq =− − dt T A · TR EA TR —Round-trip time in cavity, P—Laser power, g—Gain per round trip, l— Linear losses per round trip, q—Saturate losses per round trip, g0 —Small-signal gain per round trip, q0 —Un saturate but saturate losses per round trip, TL —Upper state lifetime of the gain medium (normalized to the round trip time of the cavity) ; TL = TτLR , T A —Absorber recovery time (normalized to the round trip time of the cavity) ; T A = Tτ AR , τ L —Upper state lifetime of the gain medium, τ A —Absorber recovery time, E L —Saturation energy of the gain ; E L = energy of the absorber ; E A =

h·ν·Ae f f,A . 2·σ A

h·ν·Ae f f,L , 2·σ L

E A —Saturation

1.1 Find system fixed points, plot the related graphs of fixed point’s coordinates as a function of system parameters (l, T A , TL , TR , E A , E L ). 1.2 Discuss stability and stability switching, find small increments Jacobian. 1.3 Find homogeneous system characteristic equation for the λ. 1.4 How our results change if there is no linear losses per round trip in our system (l = 0)? Find fixed points and discuss stability and stability switching. 1.5 How our results change if the small-signal gain per round trip is very small (g0 ≈ 0) and the un-saturate but saturate losses per round trip is very small (q0 ≈ 0)? Find fixed points and discuss stability and stability switching. 2. We have a model of solid-state laser by semiconductor devices and the corresponding to the rate equations. We control the laser dynamics by various Qswitching and continuous-wave (CW) mode-locking techniques which increase the laser peak power and obtaining ultra-short optical pulses. Solid-state lasers such as Nd:YAG or Nd:YLF, with their long upper-state lifetimes, can hardly be passively CW mode locked using saturable absorbers without Q-switching or at least Q-switched mode locking. To understand the regime of Q-switched mode locking we consider the basic system rate equations. The approximate laser

4.5 Questions

457

6 T /2  power is P(T, t) = E p (T ) · n f (t − n · TR )with −TR R /2 f (t − n · TR ) · dt = 1where E p (T = n · TR ); E p (T )is the laser phase energy; TR is the round trip time in the cavity; n—number of pulse. The pulse energy of the n-th pulse is changes appreciably over many cavity round trips. Function f (t)is the shape of the mode-locked pulses. We assume that the mode-locked pulses are much shorter than the recovery time of the absorber. We can consider that the relaxation term for the absorber in q = 1+q0P ; PA = Eτ AA can be neglected for the PA

duration of the mode-locked pulses, q =

q0 P·τ 1+ E A

; q0 —Un saturate but saturate

A

losses per round trip, q—Saturate losses per round trip, PA is the saturation power of the absorber, τ A —Absorber recovery time. We take the assumption that since the absorber recovery time is assumed to be much shorter than the cavity round-trip time, the absorber is unsaturated before the arrival of the pulse. The of the absorber # during one pulse is q(T, t); q(T, t) = " saturation 6t )   therefore the loss in the pulse energy per · f (t ) · dt q0 · exp − E PE(T −TR /2 A   E p (T )   − E 6 TR /2 A round trip is q p (T ); q p (T ) = −TR /2 f (t) · q(T, t) · dt = q0 · 1−e . E p (T )/E A The saturable absorber essentially saturates with the pulse energy. We average the basic laser power (P) and the gain per round trip (g) rate equations of the laser over one round-trip and obtain the following two DDEs for the dynamics of the pulse energy and the gain on a coarse-grained time scale T. dEp = 2 · [g(T − τ1 ) − l − q p (E p (T − τ2 ))] · E p (T − τ2 ) dT g(T − τ1 ) · E p (T − τ2 ) g(T − τ1 ) − g0 dg =− TR · − dT TL EL

TR ·

τ1 , τ2 are delay parameters for the gain per round trip (g) and the laser pulse energy (E p ) respectively. g(T ) → g(T − τ1 ) and E p (T ) → E p (T − τ2 ). The delay parameters are due to non-ideal averaging process for the basic rate equations. Non ideal averaging: The saturation of the gain medium within one pulse is not completely negligible due to the small interaction cross section of the solid state laser material. 2.1 Find system fixed points, plot the related graphs of fixed points coordinates as a function of system parameters (l, TL , E L , . . .). 2.2 Discuss stability and stability switching and find small increment Jacobian for the cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ . 2.3 Find the homogeneous system characteristic equations for the λ, cases: (1) τ1 = τ ; τ2 = 0(2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ .nn 2.4 Return 2.1–2.3 if there is no linear losses per round trip in our system (l = 0). Explain how the results change and the behavior of the system. 2.5 Return 2.1–2.3 for the case that the small signal gain per round trip is neglected (g0 ≈ 0). Explain how the results change and the behavior of the system.

458

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4 Solid State Laser Nonlinearity Applications in Engineering

We have a model of solid-state laser by semiconductor devices and the corresponding to the rate equations. We control the laser dynamics by various Qswitching and continuous-wave (CW) mode-locking techniques which increase the laser peak power and obtaining ultrashort optical pulses. For ultrashort pulse generation; a fast saturable absorber or the interplay between a slow saturable absorber and gain saturation is usually used. Both mechanisms open a window of net gain (only the pulse itself experiences gain per round trip). This allows the system to discriminate against noise that may grow in the wings of the pulse, and therefore the pulse is kept stable against perturbations or noise. Soliton pulse shaping—the combined action of negative group velocity dispersion (GVD) and self-phase-modulation (SPM) may lead to additional pulse shortening and stabilization in actively and passively mode-locked lasers. Soliton like pulse formation in actively mode-locked lasers allows considerable pulse shortening beyond that due to the usual active mode locking. In the presence of soliton like pulse shaping even a slow saturable absorber with a recovery time much longer than the pulse width can stabilize the pulse. For the soliton the nonlinear effects due to SPM and the linear effects due to the negative GVD are in balance. In contrast, the noise or instabilities that might grow within the longer time window are not intense enough to experience the Kerr nonlinearity and are therefore spread in time. When they spread in time they experience the higher absorption due to the slowly recovering absorber after passage of the much shorter soliton like pulse. The instabilities see less gain per round trip than the soliton and will decay with time. For femtosecond pulse generation there is an extending of our laser model to include SPM and GVD. The laser by Hau’s master equation of mode locking is as follow:   ∂2 ∂ A(T, t) 2 = −i · D · 2 + i · δ · |A(T, t)| A(T, t) TR · ∂T ∂t   ∂2 + g − l + Dg · 2 − q(T, t) A(T, t) ∂t

A(T, t) is the slowly varying field envelope of the pulse, D is the intra cavity GVD, Dg = g2 is the gain dispersion, δ is the nonlinear coefficient due to SPM, and g l is the amplitude loss and g is the gain. g is the HWHM gain bandwidth. The Hau’s master equation of mode locking describes the laser dynamics on two time scales: T, which is on the order of the round trip time in the laser cavity and t, which is on the order of the pulse width and which resolves the pulse shape. In term of pulse shaping, the action of the saturable absorption q(t)on short time 2 0 = − q−q − |A(T,t)| · q. Where we have assumed that the scale is as follow: ∂q(T,t) ∂t τA EA recovery time τ A of the saturable absorber is much shorter than the round-trip time of the pulse, and the between two consecutive pulses, # " absorber6recovers completely t )   . The loss in pulse energy per round · f (t ) · dt q(T, t) = q0 · exp − E PE(T −TR /2 A trip can be written as follow.

4.5 Questions

459



T 4R /2

q p (T ) =

f (t) · q(T, t) · dt = q0 · −TR /2

1 − exp[−

E p (T )  ] EA

E P (T ) EA

3.1 In term of laser by Hau’s master equation of mode locking, find system fixed points (A = A(T, t), A∗ ). Plot the related fixed point’s graphs as a function of d ; ∂t∂ ↔ dtd . See Hint 1. system parameters (D, δ, g, l, Dg ). Consider ∂∂T ↔ dT 3.2 Due to non-linearity of our system there is a time delay τ in the slow varying field envelope of the pulse A(T, t) in time A(T, t) → A(T, t − τ ). The time delay does not affect the derivatives of A(T, t). We can rewrite our laser by Hau’s master equation of mode locking and get the partial differential equation: ∂ 2 A(T, t) ∂ A(T, t) = −i · D · + i · δ · |A(T, t − τ )|2 · A(T, t − τ ) ∂T ∂t 2 ∂ 2 A(T, t) +(g − l − q(T, t)) · A(T, t − τ ) + Dg · ∂t 2 TR ·

Discuss stability and stability switching for different values of time delay parameter τ . 3.3 Return (3.1) and (3.2) if the gain dispersion Dg is negligible (Dg ≈ 0). 3.4 Return (3.1) and (3.2) if the system amplitude loss is negligible (l ≈ 0). 3.5 Return (3.1) and (3.2) if the nonlinear coefficient due to SPM is negligible (δ ≈ 0). Hint 1:   ∂2 ∂ A(T, t) 2 = −i · D · 2 + i · δ · |A(T, t)| A(T, t) TR · ∂T ∂t   ∂2 + g − l + Dg · 2 − q(T, t) A(T, t) ∂t   ∂ 2 A(T, t) 2 + δ · |A(T, t)| · A(T, t) i · −D · ∂t 2 + (g − l) · A(T, t) − q(T, t) · A(T, t) + Dg ·

∂ 2 A(T, t) ∂ A(T, t) =0 − TR · ∂t 2 ∂T

We define two functions: f 1 ( ∂ A(T,t) , A(T, t)) = −D · ∂ A(T,t) + δ · |A(T, t)|2 · ∂t 2 ∂t 2 A(T, t). 2 2 , A(T, t)) = 0 & f 2 ( ∂ A(T,t) , ∂ A(T,t) , A(T, t)) = 0 Then f 1 ( ∂ A(T,t) ∂t 2 ∂T ∂t 2 2

 f1

2

 ∂ 2 A(T, t) ∂ 2 A(T, t) , A(T, t) = 0 ⇒ −D · + δ · |A(T, t)|2 · A(T, t) = 0 ∂t 2 ∂t 2

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4 Solid State Laser Nonlinearity Applications in Engineering

D·

∂ 2 A(T, t) ∂ 2 A(T, t) 2 = δ · |A(T, t)| · A(T, t) ⇒ ∂t 2 ∂t 2 1 · δ · |A(T, t)|2 · A(T, t) = D

Inserting into f2 equation: f 2 ( ∂ A(T,t) ,∂ ∂T

2

A(T,t) , ∂t 2

A(T, t)) = 0.

∂ 2 A(T, t) ∂ A(T, t) =0 − TR · 2 ∂t ∂T 1 · δ · |A(T, t)|2 · A(T, t) (g − l) · A(T, t) − q(T, t) · A(T, t) + Dg · D ∂ A(T, t) =0 − TR · ∂T (g − l) · A(T, t) − q(T, t) · A(T, t) + Dg ·

TR ·

∂ A(T, t) = (g − l) · A(T, t) − q(T, t) · A(T, t) ∂T 1 ∂ A(T, t) · δ · |A(T, t)|2 · A(T, t) + Dg · D ∂T 1 1 = · (g − l) · A(T, t) − · q(T, t) · A(T, t) TR TR 1 + Dg · · δ · |A(T, t)|2 · A(T, t) D · TR

4. We have a system of nanometer-vibration measurement with microchip solidstate laser. The experiment configuration is a self-mixing Doppler vibrometry and optical micro phony. The model Delay Differential Equations (DDEs) is the simulation, that is, laser with frequency-modulated optical feedback. We define three variables, N (t)—normalized excess population inversion, E(t)— normalized field amplitude and φ(t)—phase of the lasing field. We define delay time parameter τ as a delay in time of the normalized excess population inversion due to non-ideal system model. We define the system model DDEs (laser with frequency-modulated optical feedback): d N (t) = K · {ω − 1 − N (t − τ ) − [1 + 2 · N (t − τ )] · E 2 (t)} dt N = N (t) ; E = E(t) ; φ = φ(t) d E(t) = N (t − τ ) · E(t) + m · E(t − t D ) × cos ψ(t) dt   + 2 · ε · [N (t − τ ) + 1] · ξ(t)   dφ(t) E(t − t D ) · sin ψ(t) ; ψ(t) =  · t + β · sin(m · t) = dt E(t)    · tD − φ(t) + φ(t − t D ) − 0 + 2

4.5 Questions

461

ω—Relative pump power normalized by the threshold (ω = PPth ), K—Population to photon lifetime ration (K = ττp ), φ(t)—Phase of the lasing field, ψ(t)—Phase difference between the lasing and the feedback fields, —Normalized frequency shift,  = (ωi − ω0 ) · τ p (τ p —photon lifetime), m—Amplitude feedback coefficient, m —Normalized modulation frequency (m = ωm · τ p ), t—Time normalized by the photon lifetime, t D —Delay time normalized by the photon lifetime, ε— Spontaneous noise with zero mean. The value  emission rate, ξ(t)—Gaussian white √  ξ(t) · ξ(t  ) = δ(t − t  ) is δcorrelated in time. ( 2 · ε · [N (t) + 1]) · ξ(t) is the quantum spontaneous emission noise. Assumption emission noise is very small and √ √ 1: The quantum spontaneous negligible ( 2 · ε · [N (t) + 1]) · ξ(t)1; ( 2 · ε · [N (t) + 1]) · ξ(t) → 0. Assumption 2: The delay time normalized by the photon lifetime t D is very small (t D ≈ 0). 4.1 Find system fixed points, plot the related graphs of fixed point’s coordinates as a function of system parameters (K , ω, , β, . . .). 4.2 Discuss stability and stability switching for different values of delay parameter τ . Find the small increments Jacobian. 4.3 Find the homogeneous system characteristic equation for the λ. 4.4 How our results (4.1–4.3) change if the normalized frequency shift is very small ( ≈ 0). Discuss stability and stability switching for different values of delay parameter τ . 4.5 Return (4.1–4.3) if the frequency modulation index is 100% (β = 1). 5. We have a system of nanometer-vibration measurement with microchip solidstate laser. The experiment configuration is a self-mixing Doppler vibrometry and optical micro phony. The model Delay Differential Equations (DDEs) is the simulation, that is, laser with frequency-modulated is the simulation, that is, laser with frequency-modulated optical feedback. We define three variables, N (t)—normalized excess population inversion, E(t)—normalized field amplitude and φ(t)—phase of the lasing field. Due to non-ideal system model there is an additional shift in the field amplitude  E ;  E ∈ R. The additional shift in the field amplitude  E can be a positive or negative number. We define the system model DDEs: d N (t) = K · {ω − 1 − N (t) − [1 + 2 · N (t)] · [E(t) +  E ]2 } dt N = N (t) ; E = E(t) ; φ = φ(t) d E(t) = N (t) · [E(t) +  E ] + m · [E(t − t D ) +  E ] dt   × cos ψ(t) + 2 · ε · [N (t) + 1] · ξ(t)   dφ(t) E(t − t D ) +  E · sin ψ(t) = dt E(t) +  E



 ψ(t) =  · t + β · sin(m · t) − φ(t) + φ(t − t D ) − 0 + 2

 · tD

462

4 Solid State Laser Nonlinearity Applications in Engineering

ω—Relative pump power normalized by the threshold (ω = PPth ), K—Population to photon lifetime ration (K = ττp ), φ(t)—Phase of the lasing field, ψ(t)—Phase difference between the lasing and the feedback fields, —Normalized frequency shift,  = (ωi − ω0 ) · τ p (τ p —photon lifetime), m—Amplitude feedback coefficient, m —Normalized modulation frequency (m = ωm · τ p ), t—Time normalized by the photon lifetime, t D —Delay time normalized by the photon lifetime, ε— Spontaneous noise with zero mean. The value  emission rate, ξ(t)—Gaussian white √  ξ(t) · ξ(t  ) = δ(t − t  ) is δcorrelated in time. ( 2 · ε · [N (t) + 1]) · ξ(t) is the quantum spontaneous emission noise. Assumption √ emission noise is very small and √ 1: The quantum spontaneous negligible ( 2 · ε · [N (t) + 1]) · ξ(t) → 1; ( 2 · ε · [N (t) + 1]) · ξ(t) → 0. 5.1 Find system fixed points, plot the related graphs of fixed point’s coordinates as a function of additional shift in the field amplitude parameter  E ;  E ∈ R and (N ∗ , E ∗ , φ ∗ ) = f ( E ). 5.2 Discuss stability and stability switching for different values of t D parameter (delay time normalized by the photon lifetime). Find the small increments Jacobian. 5.3 Discuss stability and stability switching for different values of  E ;  E ∈ R parameter (additional shift in the field amplitude). Find the small increments Jacobian. 5.4 How our results (5.1–5.3) change if we consider that the delay time normalized by the photon lifetime is very small (t D ≈ 0)? Discuss stability and stability switching for different values of  E ;  E ∈ Rparameter. 5.5 How our results (5.1–5.3) change if the frequency modulation index is 100% (β = 1)? Discuss stability and stability switching for different values of  E ;  E ∈ Rparameter. 6. We have two systems of Doppler shift with a microchip solid state laser. In the first system, the number of relaxation oscillations which each mode exhibits is 4 (N = 4) and in the second system is 5 (N = 5). The differential equations which characterize our systems are as follow.   N N0  d Ni d N0 Nk =P− − ; Bk · Sk · N0 − dt τ 2 dt k=1 N  Ni + Bi · N0 · Si − Ni · Bk · Sk τ k=1       Ei 1 Ni · = Bi · N0 − − 2 τp 2   κ · E i,s · cos[ψi (t) + φi,s (t)] + τl     E i,s κ · · sin[ψi (t) + φi,s (t)]; i = 1, 2, . . . , N = ωi − ωi,s − τl Ei

=−

d Ei dt

dψi dt

4.5 Questions

463

N0 —Space average of population inversion density, Ni —First spatial Fourier component of population inversion for the i − thmode, i—Modal index, P—Pump power, τ —Population lifetime, τ p —Photon lifetime, κ—Effective amplitude transmission coefficient of the output mirror, τl —Round trip time in the cavity, ψi — Phase different between the two fields, φi,s —Express phase fluctuations of scattered +i , E i —Lasing field, , E i,s —Scattered field, Bi —Gain, Si —Photon density, light, E 2 Si = E i2 ; E i,s (t) ∝ E i (t − td ) ; td —Delay time, EEi,si < 10−4 (extremely small).     ω < 100 kHz. Where “i” denotes the In the lock-in region: 2·π = τκl · EEi,si model index (laser mode number). N —Number of relaxation oscillations which P each mode may exhibit. w = Pth,1 ; Pth,1 is the first lasing mode pump threshold, n 0 , n i —are the space average and the first Fourier components of population inversion normalized by the mode threshold value (N0 − N21 )th . si —Photon density normalized by the steady state S1 value at w = 2. γi —Gain ratio to the first lasing mode i,s ; (ωi − ωi,s ) · τ ≡  D (time is scaled by τ ), K = ττp . (γi = BBi ), m = 2·E κ·E i We define the relaxation oscillation frequency f R O as ω D — Doppler-shift frequency.  1   (w−1) ωR O f R O = 2·π = 2·π · τ ·τ p |τ =120 µs,τ p =310 ps = 540 kHz. k · v—Angular frequency of Doppler shift and increases with increasing angular moving velocity vand with decreasing the scattering angle θs , where θs is the angle between the laser axis and velocity vector. λ—Oscillation wavelength. Assumptions: (1) The delay time td is extremely small compared with modulation time scales, (2) Two systems parameters are the same except the number of relaxation oscillations which each mode exhibits (N). 6.1 Find the fixed points of the two systems and draw the related graphs of the different function between each similar fixed point coordinate. System (1) fixed point: (N0∗ , Ni∗ , E i∗ , ψi∗ ), System (2) fixed point: (N0∗∗ , Ni∗∗ , E i∗∗ , ψi∗∗ ), Systems different functions: (|N0∗ − N0∗∗ |, |Ni∗ − Ni∗∗ |, |E i∗ − E i∗∗ |, |ψi∗ −ψi∗∗ |). 6.2 Discuss stability and stability switching of the two systems for different parameters values. How the stability behavior changes if we move from the first system to the second one? Why? 6.3 How our results (6.1) and (6.2) are changed if the round trip time in the cavity (τl ) is equal to the population life time (τ ), (τl = τ )? Discuss stability and stability switching for different values of τ for the two systems. 6.4 How our results (6.1) and (6.2) are changed if the round trip time in the cavity (τl ) is equal to the photon lifetime (τ p ), (τl = τ p )? Discuss stability and stability switching for different values of τl for the two systems. 6.5 In the first system the express phase fluctuations of the scattered light (φi,s ) is very small and we can ignore it, φi,s → ε; ψi (t)  φi,s (t) ⇒ ψi (t)+φi,s (t) ≈ ψi (t). In the second system there is no change. Repeat (6.1) and (6.2) for that case. 7. We have two systems of Doppler shift with a microchip solid state laser. The express phase fluctuations of scattered light in first system (φi,s−1 (t)) is mtimes the express phase fluctuations of scattered light in second system (φi,s−2 (t)), φi,s−1 (t) = m · φi,s−1 (t); m ≥ 2; m ∈ N. All other parameters are the same for

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4 Solid State Laser Nonlinearity Applications in Engineering

the first and second systems. The differential equations which characterize our systems are as follow.   N N0  d N0 Nk =P− − Bk · Sk · N0 − dt τ 2 k=1  d Ni Ni Bk · Sk =− + Bi · N0 · Si − Ni · dt τ k=1         Ei κ d Ei Ni 1 · = Bi · N0 − − + dt 2 τp 2 τl N

· E i,s · cos[ψi (t) + φi,s−k (t)]     dψi E i,s κ = ωi − ωi,s − · · sin[ψi (t) + φi,s−k (t)] dt τl Ei i = 1, 2, . . . , N ; k = 1, 2 N0 —Space average of population inversion density, Ni —First spatial Fourier component of population inversion for the i − thmode, i—Modal index, P—Pump power, τ —Population lifetime, τ p —Photon lifetime, κ—Effective amplitude transmission coefficient of the output mirror, τl —Round trip time in the cavity, ψi — Phase different between the two fields, φi,s−k —Express phase fluctuations of scattered light (φi,s−1 (t) in the first system (k = 1) and φi,s−2 (t)in the second system +i , E i —Lasing field, , E i,s —Scattered field, Bi —Gain, Si —Photon density, (k = 2) ), E 2 2 Si = E i ; E i,s (t) ∝ E i (t − td ) ; td —Delay time, EEi,si < 10−4 (extremely small).     ω < 100 kHz. Where “i” denotes the In the lock-in region: 2·π = τκl · EEi,si model index (laser mode number). N —Number of relaxation oscillations which P each mode may exhibit. w = Pth,1 ; Pth,1 is the first lasing mode pump threshold, n 0 , n i —are the space average and the first Fourier components of population inversion normalized by the mode threshold value (N0 − N21 )th . si —Photon density normalized by the steady state S1 value at w = 2. γi —Gain ratio to the first lasing mode i,s (γi = BBi ), m = 2·E ; (ωi − ωi,s ) · τ ≡  D (time is scaled by τ ), K = ττp . κ·E i We define the relaxation oscillation frequency f R O as ω D — Doppler-shift frequency.  1   (w−1) ωR O f R O = 2·π = 2·π · τ ·τ p |τ =120 µs,τ p =310 ps = 540 kHz. k · v—Angular frequency of Doppler shift and increases with increasing angular moving velocityvand with decreasing the scattering angle θs , where θs is the angle between the laser axis and velocity vector. λ—Oscillation wavelength. Assumptions: The delay time td is extremely small compared with modulation time scales. 7.1 Plot the systems fixed points coordinates as a function of m parameter (m ≥ 2; m ∈ N). Keep all other parameters constant in your analysis. 7.2 Discuss stability and stability switching of the two systems for different m parameter values (m ≥ 2; m ∈ N). How the stability behavior changes if we move from the first system to the second one? Why?

4.5 Questions

465

7.3 How our results (7.1) and (7.2) are changed if the round trip time in the cavity (τl ) is equal to the population life time (τ ), (τl = τ )? Discuss stability and stability switching for different values ofτ for the two systems. 7.4 How our results (7.1) and (7.2) are changed if the round trip time in the cavity (τl ) is equal to the photon lifetime (τ p ), (τl = τ p )? Discuss stability and stability switching for different values ofτl for the two systems. 7.5 How our stability analysis results are changed ifφi,s−1 (t) = m1 · φi,s−1 (t)and m ≥ 2; m ∈ N? Plot the related stability switching diagrams for different values of m parameter. 8. Laser feedback interferometry has become a fast-developing precision measurement modality with many kinds of lasers. By employing the frequency shifted optical feedback; microchip laser feedback interferometry has been widely researched due to its advantages of high sensitivity, simple structure, and easy alignment. Laser confocal feedback tomography combines the high sensitivity of laser frequency-shift feedback effect and the axial positioning ability of confocal microscopy. Laser feedback is a laser self-mixing interference which it is a physical phenomenon where part of the output laser is reflected or scattered by the external object returning back into the laser resonator to modulate the laser output power, phase, polarization states, and so on. Laser feedback interference occurs inside the laser resonator. The external optical feedback of LiNdP4 O12 lasers produced using a rotating glass plate. It belongs to the class-B laser, of which the population decays slowly compared with the field. In this case, the feedback signal can be amplified by a gain factor of the ratio γγ1c , where γc the damping rate of the laser cavity and γ1 is the damping rate of the population inversion. The ratio γγ1c of a microchip laser can be high as 106 . The ratio is only 103 for a laser diode (LD) laser, and much lower in a gas laser. The microchip laser has a high sensitivity compared with other kinds of lasers. In the weak feedback level, the laser power fluctuation under the effect of frequency-shifted feedback can be simulated using the modified Lang-Kobayashi equation. Due to system interferences there is a shift in time for the population inversion N(t), N (t) → N (t − τ ); but there is not effect on the derivative of the population in time ( d Ndt(t) ). Our system Delay Differential Equations (DDEs) are as follow: d N (t) = γ · (N0 − N (t − τ )) − B · N (t − τ ) · |E(t)|2 dt 1 d E(t) = · (B · N (t − τ ) − γc ) · E(t) dt 2 + γc · κ · cos(2 · π ·  · t − ω · τ ) · E(t) N (t) is the population inversion, E(t)is the amplitude of the laser electric field, N0 is the inversion particle number under a small signal, γ is the decay rate of the population inversion, Bis the Einstein coefficient, γc is the laser cavity decay rate, κis the effective laser feedback level, ω is the optical running laser frequency, and τ is the photon round-trip time between the laser and the target.

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4 Solid State Laser Nonlinearity Applications in Engineering

8.1 Find system fixed points, plot the fixed points coordinate 3D graphs as a function of laser cavity decay rate (γc ) and the decay rate of the population inversion (γ ). 8.2 Discuss stability and stability switching for different values ofτ parameter. If the effective laser feedback level (κ) is very low (κ → ε), How the stability and stability switching of the system is influenced? Discuss it deeply. 8.3 Discuss (8.1) and (8.2) for the case that d Ndt(t) = . . . differential equation has cubed of E(t) element, d Ndt(t) = . . . − B · N (t − τ ) · |E(t)|3 . 8.4 Discuss (8.1) and (8.2) for the case that d Ndt(t) = . . . differential equation has √ square root of E(t) element, d Ndt(t) = . . . − B · N (t − τ ) · |E(t)|. 9. We have a system of frequency shifted optical feedback measurement using a solid-state microchip laser. In the weak feedback level, the laser power fluctuation under the effect of frequency-shifted feedback is simulated using the modified Lang-Kobayashi equation (see question 8). The stationary solution of the modified Lang-Kobayashi equation can be obtained by setting the electric field E and population inversion N to be constant. Without feedback (κ = 0), the steady laser solutions are: Ns = γBc ; Is = |E s |2 = γB ·(η−1). Where Is , is the stationary is the normalized pumping rate. The intensity of the laser field, and η = B·N γc power spectrum of the continuous pumped Nd : YVO4 laser can be considered as slight fluctuations of the stable solutions. The electric field E(t) and the population inversion N(t) are: N (t) = Ns + n(t); E(t) = E s + e(t). Combining the steady laser solutions equation and the electric field E(t) and the population inversion N(t) by Lang-Kobayashi equation gives the next differential equations (by neglecting the second-order terms): dn ∗ (t) = −γ · η · n ∗ (t) − 2 · γ · (η − 1) · e(t) dt 1 de∗ (t) = · γc · n ∗ (t) + γc · κ · cos(2 · π ·  · t − ω · τ ) dt 2 + γc · κ · cos(2 · π ·  · t − ω · τ ) · e∗ (t) ; e∗ (t) = e(t) are the normalized variations of the population where n ∗ (t) = n(t) Ns Es inversion and the electric field. The variation of laser intensity versus time can be = 2E(t)·e(t) ≈ 2·e(t) = 2 · e∗ (t).γ is the decay rate of the derived accordingly, I I E 2 (t) Es population inversion, γc is the laser cavity decay rate, κis the effective laser feedback level, ωis the optical running laser frequency, and τ is the photon round-trip time between the laser and the target. 9.1 Find system fixed points and plot the relative coordinate 3D graphs as a function of γc , κ parameters. 9.2 Discuss stability and stability switching for different values of γc , κ parameters. 9.3 The normalized pumping rate η is double his value, how the stability of the system is changed?

4.5 Questions

467

9.4 The normalized pumping rate η is very small (η → ε), how the stability of the system is changed? Find fixed points. 9.5 The normalized pumping rate η is very close to one (η → 1), how the stability of the system is changed? Find fixed points. 10. We have a system of Q-switched diode-pumped solid-state laser. It implements and combines gain switching with passive Q switching of a miniature diode pumped solid-state laser. The simple model of a passively Q-switched laser is , presented. The effective pumping rate of the upper laser level is pe f f = h·vη·P L ·V where P is the electrical power input; ηis the pumping efficiency, which is the product of laser-diode efficiency and optical coupling efficiency; v L is the laser frequency; V is the laser mode volume, and his the Planck constant. Then the delay differential equations (DDEs) of our system is for population density of the upper laser level (NG ) and the photon density in the cavity (n p ) are as follow. Due to jitter interferences there is a delay in time for NG and n p system variables: NG (t) → NG (t −τ1 ), n p (t) → n p (t −τ2 ). There is no effects of time NG dn p , dt ). delay parameters (τ1 , τ2 ) on NG and n p variables derivatives in time ( ddt Remark: NG = NG (t); n p = n p (t) d NG 1 = pe f f − · NG (t − τ1 ) − c · σG · n p (t − τ2 ) · NG (t − τ1 ) dt τg 1 − · ξ · k NG (t − τ1 ) τg dn p = c · σG · n p (t − τ2 ) · NG (t − τ1 ) dt 1 − c · σ A · n p (t − τ2 ) · N A − · n p (t − τ2 ) + rsp τp nn p —Density of photons in the laser mode, NG —Population density of the upper laser level, τg —Lifetime of the upper laser level, σG —Emission cross section of the gain medium, c—Speed of light in vacuum, N A —Density of the saturate-absorber atoms (in the absence of laser light it is N A0 ), σ A —Effective absorption cross section (it takes into account the ratio between the volumes of the active mode in the absorber NG = · · · delay differand in the gain medium), ξ, k—are setting coefficients for ddt ential equation (ξ ∈ R; k > 2; k ∈ N), τ p —Photon cavity lifetime, rsp —Rate of 0 spontaneous emission into the laser mode, NG,th —Threshold density of the upper dn laser level (with the unsaturated saturate absorber, by setting dtp = 0 and assuming that rsp = 0. Assuming that rsp = 0 because few photons generated by spontaneous emission contribute to the laser mode that is defined by the cavity mirrors; − t 0 = c·σG1 ·τ p + N A0 · σσGA . By replacing NG in equation NG = pe f f · τg · (1 − e τg ) NG,th ·τ

α 0 with NG,th , as τd = τg · ln( α−1 ), where α = Ne f0f g . τd —Time when the laser reaches G,th its threshold, α—Ratio of pump power to the threshold pump power. p

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4 Solid State Laser Nonlinearity Applications in Engineering

10.1 Find system fixed points for k = 3, 4, 5, draw the related graph of fixed points coordinates as a function of k parameter. All other system parameters are constant. 10.2 Discuss stability and stability switching for different values ofτ parameter, the case is τ1 = 0; τ2 = τ . 10.3 How the results (10.1) and (10.2) are changed if ξ = 0? Discuss stability and stability switching for different values ofτ parameter, the case is τ1 = 0; τ2 = τ. 10.4 Parameter k → ∞, find system fixed points and discuss stability and stability switching (τ1 = 0; τ2 = τ ). 10.5 Plot the fixed points coordinate graphs (ξ = k, setting coefficients are equal) as a function of k parameter. Discuss stability and stability switching (τ1 = 0; τ2 = τ ).

References 1. F.X. Kartner, L.R. Brovelli, D. Kopf, M. Kamp, I.G. Calasso, U. Keller, Control of solid state laser dynamics by semiconductor devices. Opt. Eng. 34(7) (1995) 2. K. Otsuka, K. Abe, J.-Y. Ko, Real-time nanometer vibration measurement with a self-mixing microchip solid-state laser. Opt. Lett. 27(15) (2002) 3. K. Otsuka, Highly sensitive measurement of Doppler-shift with a microchip solid-state laser. Jpn. J. Appl. Phys. 31, L1546–L1548, part 2, No. 11A (1992) 4. J.B. Khurgin, F. Jin, G. Solyar, C.-C. Wang, S. Trivedi, Cost-effective low timing jitter passively Q-switched diode-pumped solid-state laser with composite pumping pulses. Appl. Opt. 41(6) (2002) 5. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993). 6. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002) 7. J. Kuang, Y. Cong, Stability of Numerical Methods for Delay Differential Equations Elsevier science; 1 edition (December 12, 2007). 8. E. Beretta, Y. Kuang, Geometric stability switch criteria In delay differential systems with delay dependent parameters. SIAM J.MATH. ANAL. 33(5), 1144–1165 (Published electronically February 14, 2002). 9. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, Vol. 191 (1993 by Academic Press, Inc). 10. J. Gu, S.-C. Tam, …, Novel use of GaAs as a passive Q- switch as well as an output coupler for diode-pumped infrared solid-state lasers, in Proceedings of SPIE 3929, Solid State Lasers IX, 2000. 11. U. Levy, Y. Silberberg, Second and third harmonic waves excited by focused Gaussian beams. Opt. Express 27795, 23(21) (2015) 12. Y.A. Kuznetsov, Elelments of Applied Bifurcation Theory (Applied Mathematical Sciences) 13. J.K. Hale, Dynamics and Bifurcations. Texts in Applied Mathematics, Vol. 3 14. S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press)

Chapter 5

Nd:YAG, Mid-Infrared and Q-Switched Microchip Lasers Stability Analysis

A diode-pumped Nd:YAG laser is Q-switched by a GaAs saturable absorber and intracavity frequency-doubled by a KTP crystal. At pump power the device produces high quality pulses at specific pulse repetition rate. We can characterize the dynamics of the pulse formation by rate equations and an energy-level model which accounts for the various energy transfer process in GaAs. High power laser diode technology brings the ability to build efficient, reliable, and compact laser sources with good beam quality. Short pulses can be generated from diode-pumped, solid-state lasers by Q-switching or model locking. Passively methods are based on nonlinear optical properties of solid-state materials. We can use efficient nonlinear wavelength conversion techniques to extend the wavelength coverage of these devices to the visible region. Frequency doubling of a Q-switched, diode-pumped laser is a method to generating compact, pulsed laser sources in the blue-green region. Second-harmonic generation is realized either inside or outside of the cavity. Advantages of externalcavity doubling are that is simplifies the cavity design and results in shorter pulses. Intra cavity frequency doubling utilizes the high peak power that is present inside the cavity and results in higher conversion efficiency. In Q-switched operation, the generated pulses are longer than those produced by a comparable Q-switched laser with no intra cavity doubling. Fully utilization of the high power inside the cavity is reachable by finding an optimal compromise between SHG and the saturable absorption that leads to Q-switching. We adjusting the spot size, the polarization, or the phase mismatch at the doubling crystal. Higher or lower second-harmonic coupling results in frequency-doubled pulses with lower peak power. The compact green-pulse source is based entirely on solid-state components. The device is a diode-pumped Nd:YAG laser that is passively Q-switched by a GaAs saturable absorber and intracavity frequency-doubled by a KTP crystal. One type is passive Q-switching of a diode-pump Nd:YAG laser with GaAs. Nonlinear losses in GaAs limit the obtainable pulse energies, especially at high peak powers. The frequency-doubled system operates at lower peak powers and has a performance that is similar to a comparable actively Q-switched device. KTP (KTiOPO4) is a nonlinear optical crystal, which possess excellent nonlinear, electro optical and acoustic-optical properties. © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_5

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

It is potassium titanyl phosphate KTP crystals. Potassium titanyl phosphate (KTP) is an inorganic compound with the formula KTiOPO4 and it is a white solid. KTP is a nonlinear optical material that is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. Crystals of KTP are highly transparent for wavelengths between 350 and 2700 nm with a reduced transmission out to 4500 nm where the crystal is effectively opaque. It is second harmonic generation (SHG) coefficient which is about three times higher than KDP. KTP can be used as an optical parametric oscillator for near IR generation up to 4 μm. It is suited to high power operation as an optical parametric oscillator due to its high damage threshold and large crystal aperture. Estimation for the performance of Q-switched laser is obtained from a four-level rate equation model. Pulse formation with GaAs saturable absorber is analyzing by includes both two-photon absorption (TPA) and free-carrier absorption (FCA) into the model. Dynamic and stability is inspected. TPA generates free electrons and holes whereas FCA promotes electrons higher into the conduction band and holes deeper into the valence band. Combination of all these effects with the accompanying recombination process derives the set of intra-cavity rate equations. Semiconductor saturable absorber mirror (SESAM) devices are a key component of ultrafast passive mode-locked-laser sources. The key SESAM parameters such as saturation fluency, modulation depth, and non saturable losses are measured and inspected. They are important and controllable to obtain stable pulse generation for a given laser. Nonlinear reflectivity of saturable absorbers is measured for getting a high precision, wide dynamic range setup. An accurate calibration is achievable by measures a low modulation depth. The model function for the nonlinear reflectivity is based on a simple two-level traveling wave system. Spatial beam profiles, non saturable losses and high-order absorption (two-photon absorption) are included in the model. We measure nonlinear reflectivity to fit a model function to obtain the key parameters of the saturable absorber. The model function is described the nonlinear behavior of real SESAMs. A band-structure is approximated by two level systems. The approximation model neglects intra-band relaxations as well as trapping or recombination. Effects of standing wave patterns and carrier diffusion and temperature effects in the device are not included. The traveling wave model is based on rate equations for a two-level system without relaxations. It is a good approximation for a slow saturable absorber, where the recovery time is longer then the pulse bleaching the absorber. The two-level system is characterized by set of differential equations and dynamics is inspected. Ion-doped crystalline lasers operating in the mid-IR spectral range (2–5 μm) is inspected. It is rare earth and transition metal based ionic crystals as well as color-center lasers. They are compact all-solid-state room-temperature tunable sources, belonging to the broad class of vibrionic lasers. The efficient high-power room-temperature operation and super-broad tenability, and the possibility of generating ultrashort pulses from the novel class of chromium doped chalcogenide laser led to an analysis in vibrionic laser systems involving 3dn transition-metal ions. The mid-IR lasers and spectroscopic and laser characteristics are allowing assessment of the suitability of laser materials to serve as active media in diode pumped laser systems. The process influencing formation of the inversion

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

471

population in mid-IR crystalline solid-state laser is inspected and the effect of these processes on laser operation (laser threshold and efficiency). It is continuous-wave operation by assuming the four-level scheme the balance equation for the population of the upper laser level n and the rate equation for photon flux at the laser wavelength. Q-switched lasers are ideal for applications requiring the highest available peak power and/or single pulse energy. These lasers are commonly used in medical, military defense, ranging and LiDAR, scientific research, material processing and industrial applications. Passively Q-switched Nd:YAG microchip lasers can produce up to 250 μJ/pulse at 1.064 μm, with a pulse duration of 380 ps. The infrared output has been converted to 532, 355, and 266 nm with high efficiency [1, 2].

5.1 Nd:YAG Laser Passively Q-Switched with GaAs Stability Optimization We discuss all-solid-state green-pulse laser. A diode-dumped Nd:YAG laser is Qswitched by a GaAs saturable absorber and intra-cavity frequency-doubled by a KTP crystal. The dynamics of a pulse formation can be described by rate equations and an energy-level model which accounts for the various energy-transfer processes in GaAs. High-power diode-laser technology helps as to build efficient, reliable and compact laser sources with beam quality. We generate short pulses from diode-pumped, solid-state lasers by Q-switching or model locking. The fully passive methods are based on nonlinear optical properties of solid-state materials. The efficient nonlinear wavelength conversion techniques can be utilized to extend the wavelength coverage of these devices to the visible region. Frequency doubling of a Q-switched, diode pumped laser is a convenient method of generating compact, pulsed laser sources in the blue-green region. The devices are applicable in communication, displays, chemical processing, and as pump sources for laser materials. We can realize the second-harmonic generation either inside or outside of the cavity. The external-cavity doubling has the advantages which simplifies the cavity design and results in shorter pulses. This method is applicable to laser sources that operate with high output powers, such that acceptable conversion efficiencies which can be realize. These devices typically require tens of watts of pump power. Intra-cavity frequency doubling utilizes the high peak power that is present inside the cavity and results in higher conversion efficiency. For Q-switched operation, the generated pulses are longer than those produced by a comparable Q-switched laser with no intra cavity doubling. Intensity dependent loss prevents fast lasing build-up. We fully utilize the high power inside the cavity by finding an optimal compromise between SHG and the saturable absorption that leads to Q-switching. We can get it by adjusting the spot size, the polarization, or the phase mismatch at the doubling crystal. There are high or low second-harmonic coupling results in frequency-doubled pulses with lower peak power. Actually it is a compact green-pulse source that is based on solid-state components [1]. KTP (potassium titanyl phosphate) crystal is commonly used to

472

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip … M3 end Mirror

GaAs Saturable Absorber

KTP Crystal

M1 (Nd:YAG) Diode pumped Nd:YAG laser M2 fold Mirror L1 AR Coated Len

L2 AR Coated Len

λ

, (AR-coated 4 1064nm quarter Wave plate)

Fig. 5.1 Cavity configuration of the intra-cavity frequency doubled passively Q-switched Nd:YAG laser

frequency-double diode-pumped solid state laser. If the optical power becomes too high, KTP shows photo-induced “gray-tracks”. The diode-pumped Nd:YAG laser is passively Q-switched by a GaAs saturable absorber and intra-cavity frequency doubled by a KTP crystal. Another possible is a passive Q-switching of a diodepumped Nd:YAG laser with GaAs. Practically nonlinear losses in GaAs limit the obtaining pulse energies, mainly at high peak powers. The frequency-doubled system operates at lower peak powers and has a performance that is similar to a comparable actively Q-switched device. The cavity configuration of the intra-cavity frequencydoubled, passively Q-switched Nd:YAG laser is presented in the next figure (Fig. 5.1). First the system has partially collimated outputs from two 2 W laser diodes. They are combined by a polarizing beam splitter and focused onto the end of a 1 cm, 1.1% Nd-doped YAG crystal. The laser diodes offer high cw optical power and high brightness with unsurpassed reliability. The small emitting aperture combined with low beam divergence makes it among the highest brightness cw laser diodes available for our system. The diodes comprised of partially coherent broad area emitters with relatively uniform emission over the emitting aperture. Operation is multi-longitudinal mode with a spectral envelope width of approximately 2 nm FWHM. The far field beam divergence in the plane perpendicular to the diode junction is nearly Gaussian while the lateral beam profile exhibits the complex pattern typical of broad area emitters. Emitting apertures can be in the range of 50–500 μm giving cw power output capability of up to 4 W with superlative reliability. There is power for 100 and 200 μmaperture devices or fiber-coupled lasers. The high efficiency of the quantum well structure combined with low thermal resistance epi-down chip mounting provides minimum junction temperature at high optical power. Low junction temperature and low thermal resistance packages extend lifetime and increase reliability. The pumping end of the Nd-YAG crystal has an anti-reflectance (AR) coating at 808 nm and a high-reflectance (HR) coating at 1064 nm (M1). The L-cavity design, with proper dichroic coatings on the end mirror (M3) and on the fold mirror (M2), allows

5.1 Nd:YAG Laser Passively Q-Switched with GaAs Stability …

473

one to extract ~90% of the green light that is generated during the double pass through the crystal. The end mirror (M3) has a radius of curvature of 10 cm and reflects 99.5% of both 1064 and 532 nm light. The fold mirror (M2) reflects approximately 99% of the fundamental light and transmits more than 90% of the second harmonic light at a 45° angle of incidence. Two anti-reflectance (AR) coated (1064 nm) 5 cm lenses are each used to form intra-cavity foci at the GaAs saturable absorber and at the 5 mm, dual AR-coated KTP crystal. The polished single-crystal GaAs wafer has a thickness of 450 μm and is oriented at Brewster’s angle. The wafer 450 μm thickness produces the best compromise between pulse energy and stability. An AR-coated (1064 nm) quarter-wave plate is placed in the cavity to compensate the birefringence caused by the KTP crystal. Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent. This cavity design is stable and allows considerable flexibility in positioning the elements for optimum performance. Green pulses are detected using a 1.5 GHz photodiode and a digital oscilloscope. At 3.7 W of incident pumping power 250 mW of cw green light is generated without the GaAs saturable absorber in the cavity. At the maximum pump power the pulse width is shortened, and the repetition rate decreased. This behavior is similar to what we observed with the GaAs saturable absorber without frequency doubling. We can plot the corresponding pulse energies and peak powers, as determined from the average power, repetition rate, and pulse width. The theoretical estimate for the performance of this Q-switched laser can be obtained from a four-level rate equation model. We discuss the pulse formation with the GaAs saturable absorber, and present both twophoton absorption (TPA) and free-carrier absorption (FCA) into the model. The rate equations for photon density φ and population inversion density N x are as follow. Due to uncomplete of photon generation and interferences, there is a delay  in time . (τ ) for photon density φ(t) → φ(t − τ ); it is not density (φ) in time dφ dt dφ = [(2 · N x (t) · σ · L 0 − γ − 2 · αq · L q ) · φ(t − τ ) dt 1 − (K · L q2 + B · L c ) · φ 2 (t − τ )] · T  d Nx N x (t) = −2 · N x (t) · σ · c · φ(t − τ ) − + P · (Ntot − N x (t) dt τ where, σ is the stimulated emission cross section, T is cavity round-trip time, P is the pump rate, Ntot is the density of active atoms, and L 0 is the length of the laser medium, L c is the length of the doubling crystal, L q is the length of the GaAs wafer. The linear losses of the cavity are included in γ , and the saturable absorption of the GaAs wafer is described by αq . φ—Photon density in time. The terms quadratic in φ(t − τ )represent SHG and TPA in GaAs, and the corresponding coupling coefficients are as describe below.

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …



   w0 2 2 k · L c K = 32 · π · h · (η · v) · c · (ε0 · d · L c ) · · sinc wc 2  2 w0 1 · B = 6 · β · h · v · c · Lq · wq nq 2

3

2

Respectively, where v is the frequency of the fundamental field, d is the effective second order nonlinear coefficient, β is the TPA coefficient, and k is the wavevector mismatch. The parameters w0 , wc and wq are the spot sizes of the beam at the laser rod, doubling crystal and GaAs wafer, respectively. The refractive index of GaAs appears in B = · · · because the wafer is at Brewster’s angle. c is the speed of light in vacuum. Remark Brewster’s angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When un-polarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. If we have a system with a boundary between two media with different refractive indices, some of it is usually reflected. The fraction that is reflected is described by the Fresnel equations, and is dependent upon the incoming light’s polarization and angle of incidence. The Fresnel equations predict that light with the p polarization (electric field polarized in the same plane as the incident ray and the surface normal at the point of incidence) will not be reflected if the angle of incidence is θ B = arctan nn 21 . n 1 is the refractive index of the initial medium through which the light propagates, and n 2 is the index of the other medium. We define θ1 as the angle of reflection (or incidence) and θ2 is the angle of refraction then using Snell’s law n 1 · sin θ1 = n 2 · sin θ2 . We consider the incident angle, θ1 = θ B at which nolight is reflected, then n 1 · sin θ B = n 2 · sin(90◦ − θ B ) = n 2 · cos θ B ; θ B = arctan nn 21 . Gas lasers typically use a window tilted at Brewster’s angle to allow the beam to leave the laser tube. Since the window reflects some s-polarized light but no p-polarized light, the round trip loss for the s polarization is higher than that of the p polarization. This causes the laser’s output to be p polarized due to competition between the two modes. When the reflecting surface is absorbing, reflectivity at parallel polarization (p) goes through a non-zero minimum at the so-called pseudo-Brewster’s angle [2]. The beam inside the GaAs wafer is elliptical and its area is larger, by a factor of n q , than what the spot size suggests. The initial inversion N x−0 at the operating pulse repetition rate f is given by      1 P · Ntot 1 · 1 − exp − +P · N x−0 ( f ) = 1 τ f +P τ At fixed points:

dφ dt

= 0; ddtNx = 0; limt→∞ φ(t − τ ) = φ(t); T > 0; T ∈ R

5.1 Nd:YAG Laser Passively Q-Switched with GaAs Stability …

475

 dφ = 0 ⇒ (2 · N x∗ · σ · L 0 − γ − 2 · αq · L q ) · φ ∗ dt 1 −(K · L q2 + B · L c ) · (φ ∗ )2 · = 0 T

∗  1

= 0 ⇒ (2 · N x∗ · σ · L 0 − γ − 2 · αq · L q ) · φ ∗ T − (K · L q2 + B · L c ) · (φ ∗ )2 = 0

d Nx N∗ = 0 ⇒ −2 · N x∗ · σ · c · φ ∗ − x + P · (Ntot − N x∗ ) = 0 dt τ ∗ N − 2 · N x∗ · σ · c · φ ∗ − x + P · Ntot − P · N x∗ = 0; P · N x∗ τ N x∗ ∗ + + 2 · N x · σ · c · φ ∗ = P · Ntot τ  

∗∗  ∗ 1 P · Ntot N x · P + + 2 · σ · c · φ ∗ = P · Ntot ⇒ N x∗ = τ P + τ1 + 2 · σ · c · φ ∗  

∗∗ 

∗  P · Ntot → : 2· · σ · L 0 − γ − 2 · αq · L q · φ ∗ P + τ1 + 2 · σ · c · φ ∗ − (K · L q2 + B · L c ) · (φ ∗ )2 = 0

2·

P+

1 τ

P · Ntot + 2 · σ · c · φ∗

 · σ · L 0 · φ ∗ − γ · φ ∗ − 2 · αq · L q · φ ∗

− (K · L q2 + B · L c ) · (φ ∗ )2 = 0 

 1 ∗ 2 · P · Ntot · σ · L 0 · φ − γ · φ · P + + 2 · σ · c · φ τ   1 ∗ ∗ − 2 · αq · L q · φ · P + + 2 · σ · c · φ τ   1 2 ∗ 2 ∗ − (K · L q + B · L c ) · (φ ) · P + + 2 · σ · c · φ = 0 τ   1 ∗ ∗ − γ · 2 · σ · c · [φ ∗ ]2 2 · P · Ntot · σ · L 0 · φ − γ · φ · P + τ   1 − 2 · αq · L q · P + · φ ∗ − 4 · αq · L q · σ · c · [φ ∗ ]2 τ   1 2 ∗ 2 − (K · L q + B · L c ) · (φ ) · P + τ ∗

∗

− (K · L q2 + B · L c ) · 2 · σ · c · [φ ∗ ]3 = 0

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …



   1 1 − 2 · αq · L q · P + · φ∗ 2 · P · Ntot · σ · L 0 − γ · P + τ τ  γ · 2 · σ · c + 4 · αq · L q · σ · c   1 2 − ·(φ ∗ )2 +(K · L q + B · L c ) · P + τ



− (K · L q2 + B · L c ) · 2 · σ · c · [φ ∗ ]3 = 0     1 1 − 2 · αq · L q · P + 2 · P · Ntot · σ · L 0 − γ · P + τ τ    1 − γ · 2 · σ · c + 4 · αq · L q · σ · c + (K · L q2 + B · L c ) · P + · φ∗ τ



− (K · L q2 + B · L c ) · 2 · σ · c · [φ ∗ ]2 ) · φ ∗ = 0 For simplicity we define ξ(φ ∗ ) function:      1 1 − 2 · αq · L q · P + ξ(φ ∗ ) = 2 · P · Ntot · σ · L 0 − γ · P + τ τ    1 − γ · 2 · σ · c + 4 · αq · L q · σ · c + (K · L q2 + B · L c ) · P + · φ∗ τ − (K · L q2 + B · L c ) · 2 · σ · c · [φ ∗ ]2 Then we get the equation ξ(φ ∗ ) · φ ∗ = 0, first fixed pointis φ ∗ = 0 then N x∗ = P·Ntot tot . We can define the first fixed point E (0) (φ (0) , N x(0) ) = 0, P·N . Additional P+ τ1 P+ τ1 ∗ fixed points we find numerically by ξ(φ ) = 0. First we need to calculate parameters K , B.  B = 6 · β · h · v · c · Lq ·

w0 wq

2 ·

1 nq



K = 32 · π · h · (η · v) · c · (ε0 · d · L c ) · 2

3

2

w0 wc

2

 · sinc

2

k · L c 2



αq = σe · (N − N + ) + σh · N + + σ f c · n N—is the total EL2 density, N0+ —is the initial ionized EL2 (EL2+ ) density, N + — is the ionized EL2 (EL2+ ) density, σe —is the EL2 absorption cross section, σh is the EL2+ absorption cross section, σ f c —is the free-carrier absorption cross section, n—is p—is the produce free holes, c—Speed of light   in Vacuum

the produce free electron, h is c = c = 3 × 108 ms , h—is Planck’s constant h = 6.626 × 10−34 J s ,  = 2·π

 −34 J s the reduced Planck constant  = 1.054 × 10 , ε is the Vacuum permittivity 0 rad

5.1 Nd:YAG Laser Passively Q-Switched with GaAs Stability …



477

 ε0 = 8.85 × 10−12 mF , v—is the frequency of the fundamental field, k—is the wave vector mismatch [3]. Saturable Absorber (SA): The saturable absorption of the GaAs wafer is described by parameter αq . The SA consists of a saturable absorber layer on semi-insulating (transparent) GaAs wafer. Both sides of the SA are anti-reflect (AR) coated for the laser wavelength. The saturable absorber is a nonlinear absorber and the transmittance increases with increasing pulse energy. The typical saturation fluency of a SA is higher than that of a SAM. After a relaxation time m The expression for Pn (λ, τ ):Pn=2 (λ, τ ) = λ2 + g4 · λ; p0 (τ ) = 0; p1 (τ ) = g4 ; p2 (τ ) = 1 Pn=2 (λ, τ ) =

n=2 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2

k=0

The expression for Q m (λ, τ ): Q m=1 (λ, τ ) = −g1 · λ + [g2 · g3 − g1 · g4 ] Q m=1 (λ, τ ) = −g1 · λ + [g2 · g3 − g1 · g4 ]; q0 (τ ) = g2 · g3 − g1 · g4 ; q1 (τ ) = −g1 Q m=1 (λ, τ ) =

m=1 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ

k=0

The homogenous system for φ, N x lead to characteristic equation for the eigenvalue form P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; 2 λ having the 1 j j P(λ, τ ) = j=0 a j (τ ) · λ ; Q(λ, τ ) = j=0 c j (τ ) · λ and the coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delay τ , qi , qk are any Nd: YAG laser passively Q-switched with GaAs system parameters, other parameters

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

keep constant. a0 = 0; a1 = g4 = 2 · σ · c · φ (i) +

1 + P; a2 = 1 τ

 c1 = − g1 = − [2 · N x(i) · σ · L 0 − γ − 2 · αq · L q ]  1 − (K · L q2 + B · L c ) · 2 · φ (i) · T 1 · σ 2 · L 0 · φ (i) · c · N x(i) T − {[2 · N x(i) · σ · L 0 − γ − 2 · αq · L q ]   1 1 2 (i) (i) − (K · L q + B · L c ) · 2 · φ } · · 2 · σ · c · φ + + P T τ

c0 = g2 · g3 − g1 · g4 = 4 ·

Unless strictly necessary, the designation of the variation arguments (qi , qk ) will subsequently by omitted from P, Q, a j and c j [8, 9]. The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0. 1 · σ 2 · L 0 · φ (i) · c · N x(i) − {[2 · N x(i) · σ · L 0 − γ − 2 · αq · L q ] T   1 1 2 (i) (i) − (K · L q + B · L c ) · 2 · φ } · · 2 · σ · c · φ + + P T τ  = 0 ∀ qi , qk ∈ R+

4·

λ = 0, is not a P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0. We assume that Pn (λ, τ ), Q m (λ, τ ) can’t have common imaginary roots. That is for any real number ω: Pn (λ = i · ω, τ ), Q m (λ = i · ω, τ ) = 0. Pn (λ = i · ω, τ ) = −ω2 + i · g4 · ω Q m (λ = i · ω, τ ) = −i · g1 · ω + [g2 · g3 − g1 · g4 ] Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = −ω2 + (g2 · g3 − g1 · g4 ) + (g4 − g1 ) · i · ω |Pn (λ = i · ω, τ )|2 = ω4 + g42 · ω2 ; |Q m (λ = i · ω, τ )|2 = g12 · ω2 + (g2 · g3 − g1 · g4 )2

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493

We find the expression:F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 = ω4 + g42 · ω2 − g12 · ω2 − (g2 · g3 − g1 · g4 )2 F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 = ω4 + (g42 − g12 ) · ω2 − (g2 · g3 − g1 · g4 )2 We define the following parameters for simplicity: 0 = −(g2 · g3 − g1 · g4 )2 2 = g42 − g12 ; 4 = 1; F(ω, τ ) = |P(i · ω)|2 − |Q(i · ω)|2 =

2 

2·k · ω2·k

k=0

 F(ω, τ ) = 2k=0 2·k · ω2·k = 0 + 2 · ω2 + 4 · ω4 . Hence F(ω, τ ) = 0  implies 2k=0 2·k ·ω2·k = 0 and its roots are given by solving the above polynomial. Furthermore, PR (ω, τ ) = −ω2 ; PI (ω, τ ) = g4 · ω and Q R (ω, τ ) = g2 · g3 − g1 · g4 ; Q I (ω, τ ) = −g1 · ω. Hence sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

sin θ (τ ) =

−ω3 · g1 + g4 · ω · (g2 · g3 − g1 · g4 ) g12 · ω2 + (g2 · g3 − g1 · g4 )2

cos θ (τ ) = −

−ω2 · (g2 · g3 − g1 · g4 ) − g4 · g1 · ω2 g12 · ω2 + (g2 · g3 − g1 · g4 )2

Above expressions are continuous and differentiable in τ based on Lemma 1.1. Hence we use Theorem 1.2 and this proves Theorem 1.3. We use different parameters terminology from our last characteristics parameter definition k → j; pk (τ ) → a j and qk (τ ) → c j ; n = 2; m = 1; n > m, additionally Pn (λ, τ ) → 2 j P(λ, τ ); Q m (λ, τ ) → Q(λ, τ ) then P(λ, τ ) = j=0 a j · λ ; Q(λ, τ ) = 1 j 2 j=0 c j · λ ; P(λ) = λ + g4 · λ; Q(λ) = −g1 · λ + [g2 · g3 − g1 · g4 ]. n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable functions of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ) and Q(λ) are analytic functions in λ and differentiable in τ . The coefficients:

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{a j (σ, L 0 , γ , αq , L q , K , B, L c , T, . . .) & c j (σ, L 0 , γ , αq , L q , K , B, L c , T, . . .)} ∈ R Depend on Nd: YAG laser passively Q-switched with GaAs system’s parameters σ, L 0 , γ , αq , L q , τ , . . . values. Unless strictly necessary, the designation of the variation arguments, σ, L 0 , γ , αq , L q , τ , . . . system parameters will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0  = 0. 1 · {4 · σ 2 · L 0 · φ (i) · c · N x(i) − {[2 · N x(i) · σ · L 0 − γ − 2 · αq · L q ] T 1 − (K · L q2 + B · L c ) · 2 · φ (i) } · (2 · σ · c · φ (i) + + P)}  = 0 τ 1  = 0; 4 · σ 2 · L 0 · φ (i) · c · N x(i) − {[2 · N x(i) · σ · L 0 − γ − 2 · αq · L q ] T   1 2 (i) (i) − (K · L q + B · L c ) · 2 · φ } · 2 · σ · c · φ + + P  = 0 τ ∀ σ, L 0 , γ , αq , L q , . . . ∈ R+ i.e. λ = 0 is not a root of characteristic equation. Furthermore, P(λ), Q(λ) are analytic function of λ for which the following requirements of the analysis (see Kuang 1993, Sect. 3.4) can also be verified in the present case [9, 10]. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω)  = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire (σ, L 0 , γ , αq , L q , . . . system parameters) domain of interest.   (b)  Q(λ) is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, P(λ)        −g1 ·λ+[g2 ·g3 −g1 ·g4 ]   = in the limit:  Q(λ)   . 2 P(λ) λ +g4 ·λ (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = ω4 + (g42 − g12 ) · ω2 − (g2 · g3 − g1 · g4 )2 has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω4 ). (d) Each positive root ω(σ, L 0 , γ , αq , L q , τ , . . .) of F(ω) = 0 is continuous and differentiable with respect to σ, L 0 , γ , αq , L q , τ , . . . parameters. This condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus λ = i · ω ∀ ω > 0 may be on eigenvalue of characteristic equation. The analysis consists in identifying the roots of characteristic equation situated on the imaginary axis of the complex λ—plane, where by increasing the parameters σ, L 0 , γ , αq , L q , τ , . . . (system parameters), Reλ may, at the crossing, change its sign from (−) to (+), i.e. from a stable focus E (k) (φ (k) , N x(k) ) ∀ k = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to σ, L 0 , γ , αq , L q , τ , . . . parameters.

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495



 ∂Reλ ; L 0 , γ , αq , L q , τ , . . . = const ∂σ λ=i·ω   ∂Reλ −1 ; σ, γ , αq , L q , τ , . . . = const  (L 0 ) = ∂ L 0 λ=i·ω   ∂Reλ ; σ, γ , L 0 , L q , τ , . . . = const −1 (αq ) = ∂αq λ=i·ω   ∂Reλ ; σ, γ , L 0 , αq , τ , . . . = const −1 (L q ) = ∂ L q λ=i·ω   ∂Reλ −1 ; σ, γ , L 0 , αq , L q , . . . = const  (τ ) = ∂τ λ=i·ω

−1 (σ ) =

where ω ∈ R+ . Where writing P(λ) = PR (λ) + i · PI (λ) and Q(λ) = Q R (λ) + i · Q I (λ), and inserting λ = i · ω into Nd: YAG laser passively Q-switched with GaAs system’s characteristic equation, ω must satisfy the following: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i ·ω)|2  = 0 in the view of requirement (a) above, (g, h) ∈ R. Furthermore, it follows the above equations sin(ω ·τ ) and cos(ω ·τ ) that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i ·ω)|2 −|Q(i ·ω)|2 = 0. Note F(ω) / I (assume that I ⊆ R+0 is independent of τ . Now it is important to notice that if τ ∈ is the set where ω(τ ) is a positive root of F(ω) and for τ ∈ / I ; ω(τ ) is not define. Then for all τ ∈ I is satisfies that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switch. For any τ ∈ I , where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ I (i·ω)·Q R (i·ω) ; cos θ (τ ) = [0, 2 · π ] as the solution of sin θ (τ ) = −PR (i·ω)·Q I (i·ω)+P |Q(i·ω)|2

I (i·ω)·Q I (i·ω) − PR (i·ω)·Q R (i·ω)+P and the relation between the argument θ (τ ) and ω(τ )· |Q(i·ω)|2 τ for τ ∈ I must be ω(τ )·τ = θ (τ )+2·n·π ∀ n ∈ N0 . Hence we can define the )+n·2·π ; n ∈ N0 , τ ∈ I . Let introduce maps τn : I → R+0 given by τn (τ ) = θ(τω(τ ) the functions I → R; Sn (τ ) = τ −τn (τ ), n ∈ N0 , τ ∈ I that are continuous and differentiable in τ . In the following, the subscripts λ, ω, L 0 , γ , αq , L q , τ , . . . system parameters indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(L 0 , γ , αq , L q , τ , . . . ∈ R+ ); ω(L 0 , γ , αq , L q , τ , . . . ∈ R+ ) and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Nd: YAG laser passively Q-switched with GaAs system characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to

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specific parameter (x), and inverting the derivative, for convenience, one calculates, x = σ, γ , L 0 , αq , L q , . . . ∈ R+ , 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., Substituting λ = i · ω, and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω) and i · Q λ (i · ω) = Q ω (i · ω) that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains, 

∂λ ∂x =

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 . When (x) can be an Nd: YAG laser passively Q-switched with GaAs system parameters L 0 , γ , αq , L q , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] and Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ . Pn (λ = i · ω, τ ) = −ω2 + i · g4 · ω; Q m (λ = i · ω, τ ) = −i · g1 · ω + [g2 · g3 − g1 · g4 ] PR (i · ω, τ ) = −ω2 ; PI (i · ω, τ ) = g4 · ω Q R (i · ω, τ ) = g2 · g3 − g1 · g4 ; Q I (i · ω, τ ) = −g1 · ω PRω (i · ω, τ ) = −2 · ω; PI ω (i · ω, τ ) = g4 Q Rω (i · ω, τ ) = 0; Q I ω (i · ω, τ ) = −g1 PRω = PRω (i · ω, τ ); PI ω = PI ω (i · ω, τ ); Q Rω = Q Rω (i · ω, τ ) Q I ω = Q I ω (i · ω, τ )

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497

PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; PRω · PR = 2 · ω3 PI ω · PI = g42 · ω; Q Rω · Q R = 0; Q I ω · Q I = g12 · ω Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0 Fτ ωτ = −  = 0 Fω PR · PI ω = −ω2 · g4 ; PI · PRω = −g4 · 2 · ω2 ; Q R · Q I ω = −(g2 · g3 − g1 · g4 ) · g1 ; Q I · Q Rω = 0 PR · PI x |x=τ = 0; PI · PRx |x=τ = 0; Q R · Q I x |x=τ = 0; Q I · Q Rx |x=τ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 ∂ω ∂ω Fτ Fω · + Fτ = 0; τ ∈ I ; =−  ∂τ ∂τ Fω   ∂Reλ −1 (τ ) = ; −1 (τ ) ∂τ λ=i·ω   −2 · [U + τ · |P|2 ] + i · Fω ∂ω = Re ; Fτ + i · 2 · [V + ω · |P|2 ] ∂τ Fτ = ωτ = −  Fω    ∂Reλ −1 sign{ (τ )} = sign ; sign{−1 (τ )} ∂τ λ=i·ω ! ∂ω U · ∂τ +V ∂ω  +ω+ = sign{Fω } · sign τ · ∂τ |P|2 We shall presently examine the possibility of stability transitions (bifurcations) in an Nd: YAG laser passively Q-switched with GaAs system, about equilibrium points E (k) (φ (k) , N x(k) ) ∀ k = 0, 1, 2 as a result of a variation of delay parameter τ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane where by increasing the delay parameter τ , Reλ may at the crossing, change its sign from “−” to “+”, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ , −1 (τ ) =   ∂Reλ and L 0 , γ , αq , L q , . . . system parameters are constant where ω ∈ R+ . ∂τ λ=i·ω We need to plot the stability switch diagram based on different delay values of our Nd: YAG laser passively Q-switched with GaAs system. Since it is very complex

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function we recommend to solve it numerically rather than analytic [9, 10]. 





−2 · [U + τ · |P|2 ] + i · Fω  (τ ) = ;  (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ] λ=i·ω   ∂Reλ −1 (τ ) = ; −1 (τ ) ∂τ λ=i·ω −1

∂Reλ ∂τ

=

−1



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = · · · ; cos θ (τ ) = · · · , when ω = ω+ (τ ) if only ω+ is feasible. Additionally when all Nd: YAG laser passively Q-switched with GaAs system parameters are known and the stability switch due to various time delay values τ is describe in the following expression: sign{∧−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + ω(τ ) + } |P(ω(τ ))|2 Remark we know F(ω, τ ) = 0 implies it roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which ωi are complex or imaginary numbers, then unable to analyse stability. Lemma 1.1 Assume that ω(τ ) is a positive and real root of F(ω, τ ) = 0 defined for τ ∈ I , which is continuous and differentiable. Assume further that if λ = i ·ω, ω ∈ R, then Pn (i · ω, τ ) + Q n (i · ω, τ ) = 0, τ ∈ R hold true. The functions Sn (τ ), n ∈ N0 , are continuous and differentiable on I. Theorem 1.2 Assume that ω(τ ) is a positive real root of F(ω, τ ) = 0 defined for τ ∈ I, I ⊆ R+0 , and at some τ ∗ ∈ I , Sn (τ ∗ ) = 0 for some n ∈ N0 then a pair of simple conjugate pure imaginary roots λ+ (τ ∗ ) = i · ω(τ ∗ ), λ− (τ ∗ ) = −i · ω(τ ∗ ) of D(λ, τ ) = 0 exist at τ = τ ∗ which crosses the imaginary axis from left to right if δ(τ ∗ ) > 0 and cross the imaginary axis from right to left if δ(τ ∗ ) < 0 where δ(τ ∗ ) = sign



   dReλ d Sn (τ ) |λ=iω(τ ∗ ) = sign{Fω (ω(τ ∗ ), τ ∗ )} · sign |τ =τ ∗ dτ dτ

Theorem 1.3 The characteristic equation has a pair of simple and conjugate pure imaginary roots λ = ±ω(τ ∗ ), ω(τ ∗ ) real at τ ∗ ∈ I if Sn (τ ∗ ) = τ ∗ − τn (τ ∗ ) = 0 for some n ∈ N0 . If ω(τ ∗ ) = ω+ (τ ∗ ), this pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right if δ+ (τ ∗ ) > 0and crosses the  ∗ ; | imaginary axis from right to left if δ+ (τ ∗ ) < 0 where δ+ (τ ∗ ) = sign dReλ dτ λ=iω+ (τ ) " #   dReλ d Sn (τ ) ∗ ∗ ∗ δ+ (τ ) = sign dτ |λ=iω+ (τ ) = sign dτ |τ =τ .

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If ω(τ ∗ ) = ω− (τ ∗ ), this pair of simple conjugate pure imaginary roots cross the imaginary the imaginary axis from left to right, if δ− (τ ∗ ) > 0 and crosses   ∗ = | axis from right to left. If δ− (τ ∗ ) < 0 where δ− (τ ∗ ) = sign dReλ dτ λ=iω− (τ ) " # d Sn (τ ) −sign dτ |τ =τ ∗ .   ∗ If ω+ (τ ∗ ) = ω− (τ ∗ ) = ω(τ ∗ ) then (τ ∗ ) = 0 and sign dReλ = 0, the | dτ λ=iω(τ ) same is true when Sn (τ ∗ ) = 0 the following result can be useful in identifying values of τ where stability switches happened. Remark: Lemma 1.1 and Theorems 1.2, 1.3: In the first and second cases we discuss delay parameter τ and in the third case we discuss delay parameter τ [8].

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers Rate Equations Stability Analysis Semiconductor saturable absorber mirror (SESAM) is a key component of ultrafast passive mode-locked laser sources. The key SESAM parameters are saturation fluence, modulation depth, and non saturable losses are measurable with a high accuracy. We can control these parameters to obtain stable pulse generation for a given laser. The model function for the nonlinear reflectivity is based on a simple two-level travelling wave system. SESAM is a mirror structure with an incorporated saturable absorber, all modes in semiconductor technology. We use SESAM for the generation of ultrafast pulses by passive mode locking of various types of lasers. A SESAM contains a semiconductor Bragg mirror and near the surface a single quantum well absorber layer (Fig. 5.4). The materials of the Bragg mirror have larger bandgap energy and no absorption occurs in that region (such SESAMs also called saturable Bragg reflectors (SBRs). We can obtain a large modulation depth for passive Q switching by using a thicker absorber layer. The suitable passivation layer on the top surface increases the device lifetime [11].

Pulses

GaAs Substrate

GaAs/AlAs Bragg mirror

InGaAs Quantum well absorbet Fig. 5.4 Structure of a typical SESAM (semiconductor saturable absorber mirror)

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

The structure of a typical SESAM is designed to operate around 1064 nm. On a GaAs substrate a GaAs/AlGaAs bragg mirror is grown, and within the top layers, there is an InGaAs quantum well absorber layer, which may be 10 nm thick. We calculate the optical field into a SESAM with the same matrix method as applied to other types of dielectric mirrors. The optical intensity in the region where the saturable material is placed influences the modulation depth and the saturation fluence. The design of the structure also influences the bandwidth and the chromatic dispersion. The absorber layer is placed in an anti-node of the electric field and then we get a maximum saturable absorption and the smallest possible saturation fluence. If multiple absorber layers are required for a high modulation depth, they may be placed in separate anti-nodes, or possible several of them near one anti-node. Other types of semiconductor saturable absorbers can be based on quantum dots embedded in glass or on carbon nanotubes or graphene. We can tailor the macroscopic parameters of a SESAM for operation in very different regime. The macroscopic parameters of a SESAM can be the operation wavelength, of the modulation depth, the saturation fluence, and the recovery time. The most important characteristics of a SESAM for passive mode locking or Q-switching are the following: modulation depth, saturation fluence, recovery time and non saturable losses. The saturation fluence is the fluence of an incident short pulse which is required for causing significant absorption saturation. It depends on the absorber material, the wavelength, and the field penetration into the absorber structure. A “roll-over” of saturation curve (reduction in reflectivity for high fluence values) can be caused by two-photon absorption for sub-picosecond pulses or by other effects. The recovery time is the exponential time constant of absorption recovery after a saturating pulse. Typical values are between a few picoseconds and hundreds of picoseconds. The recovery time is strongly influenced by the defect density in the absorber, and possibly in nearby structures. SESAMs non saturable losses lead to device heating while not contributing to the pulse shaping. Non saturable losses tend to be higher for SESAMs with a larger modulation depth and faster recovery. There are SESAMs for high-power operation which the average output powers of well above 100w (passively mode-locked high-power laser) and the intra cavity average power of above 1 kW. There can be a substantial thermal lensing, which affects the mode properties of the laser resonator. It is recommended to optimize SESAMs for use in high-power lasers. A SESAM which generated heat must be conducted through some hundreds of micrometer of semiconductor materials which are determined by the thickness of the used wafer. A substantial local temperature increases can occur even in low-power lasers (around 1w of average output power) if the pulse repetition rate is very high (≈GHz). We need to use a strong focusing of the radiation in order to achieve sufficiently strong saturation of the absorption despite the small pulse energy. High power lasers with much lower pulse repetition rate (tens of MHz) able to use of much larger beam areas and then the handle substantial absorber powers is easier. The dispersion of any sign can be engineered into a SESAM via the multilayer structure. Dispersive SESAMs serve the purpose of dispersion compensation in a laser resonator, which is in addition to the function of a passive mode locker. SESAMs are widely used for passive mode locking of lasers and for solid-state bulk and fiber lasers. They work with a wide range of

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

501

laser parameters and allow for reliable self-starting mode locking for the right device and operation parameters. Another application of SESAMs is passive Q-switching (microchip lasers and fiber lasers). The key decisions using SESAMs are the selection of a suitable SESAM design and the adjustment of a numbers of laser parameters, especially the resonator mode size on the absorber. We can face problems in the form of various instabilities or SESAM damage by using inappropriate device and operation parameters. The engineering areas of nonlinear filtering and signal processing can be used SESAMs (optical fiber communications). The performance of SESAMs in solid-state lasers worked out design guidelines for application to practical laser systems, and the performance improvements in shortest pulse widths, highest average and peak power from a passively mode-locked laser, and extending the pulse repetition rate to 160 GHz. We can adjust and get stable Q-switching of compact microchip lasers with pulse durations as short as 37 ps, or pulse energies as high as 1.1 μJ at a wavelength of around 1um. We integrate the semiconductor saturable absorber into a mirror structure, which results in a device whose reflectivity increases as the incident optical intensity increases. SESAMs both the linear and nonlinear optical properties can be engineered over a wide range, allowing for more freedom in the specific laser cavity design. Semiconductor saturable absorbers are excellent solution for passive mode locking solid-state lasers because the large absorber’s cross section, and therefore small saturation fluence, and is ideally suited for suppressing Q-switching instabilities. The key parameters for a saturable absorber are its wavelength range (where it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity and pulse energy density it saturated). The SESAM needs to be characterized to obtain the required parameters such as saturation fluence, modulation depth, non saturable absorption, and recovery time. There are different SESAM designs. Concern the design, a high modulation depth is suited for passive Q-switching and a smaller modulation depth for passive continuous wave (cw) mode locking. The SESAM nonlinear reflectivity R versus the incident pulse energy fluence Fp is described by three parameters: (1) the linear reflectivity Rlin for pulses with “zero” pulse energy fluence, (2) the reflectivity Rns for “infinitely” high pulse energy fluences when all saturable absorption is bleached, and (3) the saturation fluence Fsat . The modulation depth R and the non saturable losses Rns in reflectivity are defined as: R = Rns − Rlin ; Rns = 1 − Rns . It is implying that Rlin and Rns are not experimentally accessible but extrapolated values from the measured data using a proper model function. The significant and stability of these parameters depend on the quality of the model function. The pulse energy E fluence F p = Ap is the incident pulse energy per unit surface area. The saturation fluence Fsat is the fluence required to begin absorption saturation. In the case of infinitely thin absorber, the reflectivity for a pulse with fluence FP = Fsat is increase by 1e (≈37%) of R with respect to Rlin . In SESAM mode-locked solid-state lasers (picosecond regime) there is a simple design guideline which prevents Q-switching instabilities. The inequality E 2p > E sat,L · E sat,A · R describes the condition to prevent Q-switching instabilities, where E p is the intra cavity pulse energy, E sat,L is the saturation energy of the laser medium, and E sat,A the saturation energy of the

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

saturable absorber. Assumption is taken for a fully saturated, slow saturable absorber that fully recovers between consecutive pulses. The modified criteria to prevent Qswitching instabilities is E 2p >  Esat,A1  · R, where a2 (scaling constant) is related a2 + E

sat,L

to nonlinear coefficient that increases linearity with the pulse energy and with a scaling constant. The nonlinear loss coefficient is given by q2 (E p ) = a2 · E p which includes a certain roll-off at higher fluence. The roll-off (femtosecond regime) is β ·z dominated by two-photon absorption (TPA) for which a2 = TτPpA·A Ae f f , where βT P A is the two-photon absorption coefficient, z e f f is the effective TPA-absorber thickness, τ p and A A is the laser mode area on the saturable absorber. The roll-off even in the picosecond regime is observed, where TPA is negligible. Our target is to measure the most reliable nonlinear reflectivity. The experimental system is built from an incident laser beam which is focused onto a device under test (DUT), can be SESAM. The attenuators control the incident fluence. A beam-splitter feeds a small part of the incident and reflector beam onto two photodiodes. After proper calibration the applied fluence is measured by the photocurrent of the incident photodiode. The reflectivity is calculated from the ratio of the two photodiodes photocurrents. We divide system setup into five functional groups: (1) an isolated laser source, (2) power adjustment, (3) focusing system and DUT holder, (4) beam-splitter and photodiode optics, and (5) imaging system. The system setup allows us to measure the reflectivity with accuracy close to 10–4 over a dynamic input fluence range of more than 4 orders of magnitude. The experimental system to measure R(F) is as follow: The experimental system (Fig. 5.5) elements are as follow: First the isolated laser source which should match the parameters (wavelength, pulse duration, repetition rate, etc.), of the laser in which the SESAM will be used. The laser needs to be stable over time and good beam quality. We need to compensate for group delay dispersion (GDD) for measurements with pulse durations below 100 fs. GDD comp element: group delay dispersion pre-compensation. An external cavity resonator between the device under test (DUT) and the laser’s output coupler. It will disturb the laser source especially at the lowest attenuation of highest fluences. Therefor we use an isolator. The optical isolator prevents back reflections. A λ2 computer L2

OUT (Photo diode reflected beam)

CCD Laser diode

Laser GDD comp

λ/2 Isolator

L3

Polarizing Beam Splitter

1 AOM

Beam L1 Splitter DUT

0 IN (Photo diode incident beam)

Fig. 5.5 SESAM’s experimental system to measure R(F)

A

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

503

controlled, half  motorized

λ  wave plate; PBS: Polarizing beam splitter; Attenuation

λ + PBS : + PBS . We need a computer-controlled attenuator which operates 2 2 over 4 orders of magnitude. The possible attenuator systems are: (1) motorized graded neutral density filters, (2) rotating half-wave plates in combination with a polarizing beam-splitter, (3) an Acoustic-optical modulator (AOM), or other optics. AOM: acoustic-optic modulator: chopper and attenuation: OD 0.1–3.0; 0, 1: zero and first deflection order. One option is to use a graded neutral density (ND) filters, must avoid thermal lenses inside the ND filters. An acoustic-optical modulator (AOM) is not change the shape and direction of the diffracted beam. Possible thermal effects inside the AOM caused by different RF powers applied to change the diffraction efficiency. We need to avoid strange attenuations. The half wave plate operating and at stronger attenuations reduces repeatability, and increase the sensitivity to mechanical vibrations, changes the spectrum of short pulses with broad spectra, and reduces the quality of polarization, which directly affects the calibration. By using AOM, a stronger attenuation increases amplitude noise in the diffracted beam due to electrical noise at the input side. A combination of attenuators is needed to increase the dynamic range of system setup. The half-wavelength λ2 is used to slower coarse adjustment and the AOM for fast modulation. Then beam splitter (BS): low percentage beam splitter; L1: focusing lens; Device under test (DUT); LD: Laser diode; A: aperture; IN: photodiode incident beam; OUT: photodiode reflected beam; L2: lens which is part of imaging system; CCD: imaging laser spot on sample; L3: flip-in “illumination” lens, used for calibration of imaging system. We need to lock-in detection to detect the signals and reject the photodiodes dark currents and environmental background light. The best place to put the chopper is at a focal point of the beam. If we place the chopper in other places than the focal point of the beam the rising and falling edges of the laser power will result in measuring an average reflectivity over a fluence range instead of measuring the reflectivity at a specific fluence. We use AOM as a chopper with negligible edge effects. If we need to get larger spot sizes then a smaller duty cycles reduce the average thermal load on the sample. The higher chopping frequencies do not reduce laser noise, because the modulation induced sidebands of the lasers noise spectrum is always shifted to the detection frequency. The device under test (DUT) is mounted on a xyz-stage and a mirror mount. Reproducible surface orientation from device to device is maintained by a tilt control consisting of an alignment laser diode and an aperture. The focusing lens is implemented as a zoom lens system which allow for variable spot size. The beam splitter (BS) is an uncoated small-angle wedged glass plate. The wedge is placed close to normal incident in s-polarized beam to get the minimum sensitivity to polarization effects. Transfer optics deliver these few percent of reflected beams to large-area photo-detectors for the incident beam “IN” and reflected beam “OUT”. For larger beam areas the wings of the beam profile are cut off at the edge of the diode making calibration sensitive to beam steering and lensing effects. Small focused spots on the diodes might be the origin of unnoticed local saturation or nonlinear absorption (TPA) at higher Fp and short pulse durations or deviation from linearity due to in homogeneities of the detector area. To adjust the power level at the diodes, neutral density filter wheels are placed in the transfer optics. The signal of each photo-diode

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

is fed into a lock-in amplifier. This discriminates background light and electric DC offsets. Obviously it does not discriminate against scattered light from the setup. The scattered light is coded with the same chopping frequency and will add to the signal in the lock-in amplifiers. The diodes need to be shielded against scattered light. The imaging system (two lenses L1 and L2 with focal lengths f1 and f2 ) is used to position the sample in the focal plane and to image bad spots on the surface. The magnification is the ratio of f2 /f1 and a typical SESAM has a homogenous flat surface, like any other mirror in the setup. The image on the CCD is not an image of the beam’s spot on the SESAM surface. If we move the device under test (DUT) in the direction of the beam propagation changes the beam size on the CCD camera. It is recommended that the position of lens L1 is fixed and the device under test (DUT) is moved along the beam axis to obtain a minimum spot size on the device under test (DUT). The CCD can be placed such that the beam diameter on the CCD is minimal when the sample is placed in the focus. We put additional lens (L3) which is flipped into the beam. It increases the illuminated area on the sample and makes surface irregularities such as scratches, pitches, or dust particles visible. The focal length f3 of L3 should be 30–50 times f1 and placed at a distance of about f3 in the front of L1. Len L3 has to be placed in front of the beam-splitter (BS). By calibration the image system can be used to measure the actual spot size, if the beam is collimated between the beam-splitter (BS) and the focusing lens (L1). We need to fit the model function to obtain the key parameters of the saturable absorber. It is done by measuring the nonlinear reflectivity. We need to characterize the model function R(Fp ) and the model function is well suited to describe to nonlinear behavior of real SESAMs. The two levels system helps us to approximate a band-structure. Assumption: neglect intra band relaxations, neglect trapping or recombination, not include effects of standing wave pattern in the device, and not include carrier diffusion and temperature effects [11–15]. The traveling wave model is based on rate equations for two levels system without relaxations, and it is a good approximation for a slow saturable absorber, where the recovery time is longer than the pulse bleaching the absorber. We can characterize the two levels system by the following equations: n 1 + n 2 = n;

dn 1 σ dn 2 σ =− = −(n 1 − n 2 ) · · I = −n · ·I dt dt ·ω ·ω

dn dn 1 2·σ dα 2·σ =2· = −n · · I ; α = σ · n; = −α · ·I dt dt ·ω dt ·ω n 1 and n 2 are the occupation numbers of level 1 and 2, respectively, σ is the absorption and emission cross-section, ω is the angular frequency of the optical field, α is the absorption coefficient. The microscopic definition of Fsat is given by dα ·ω α·I α·I ; Fsat = · dt ⇒ =− ; dα = − dt F 2·σ Fsat $ sat I I · dt ⇒ ln α = − ·t =− Fsat Fsat

$

1 · dα α

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

505

The saturation of absorption α for times after the pulse has passed: α(F p ) = Fp

αlin · e− Fsat . The target is to get the transmitted pulse energy through a saturable absorber; we get it by solving the following system of differential equations: α(z, t) · I (z, t) d I (z, t) dα(z, t) ; =− = −α(z, t) · I (z, t) dt Fsat dt Assumption: No standing wave patters or Fabry-Perot effects in the SESAM. The reflectivity is calculated as twice the transmission through an absorber of length L2 . The deviations scale linearly with the field enhancement factor for the modulation depth and inverse linearly for the saturation fluence. We get from the model function SESAM parameters and microscopic values for material comparison or extracted SESAM design. Finally we get the internal transmission for a pulse  twicethrough F L an absorber of length 2 . Parameter S is the saturation parameter S = Fsatp , Tlin is define as the linear transmission. We replace Tlin by Rlin [11].  Fp  %∞ ln(1 + Tlin · e Fsat − 1) I (L , t) · dt Fout R(F p ) = = %−∞ = ∞ Fp FI N −∞ I (0, t) · dt Fsat

ln(1 + Tlin · (e S − 1)) = S There are saturable losses (reflectivity less than 100%) causes by residual transmission losses through the Bragg mirror, scattering losses from rough interfaces, non saturable defect absorption, free carrier absorption, Auger recombination, etc. These losses are homogeneously distributed over the absorber layer or transmission losses. We evaluate these losses by a scaling factor Rns . R(F p ) = Rns ·

 ln 1 +

Rlin Rns



· (e S − 1) S

At higher fluences the reflectivity deceases with increasing fluence according to a second order process like two-photon absorption (TPA), leading to a roll-off in the reflectivity curve. This additional absorption coefficient increases more or less linearity with fluence and depends on the pulse duration as well. In the femtosecond regime the most significant part is due to two-photon absorption in the SESAM structure, not limited to the absorber layer only. There are other terms like thermal effects, free-carrier absorption, and other sources of induced absorption. The roll-off −

Fp

is taken into account by multiplying the model function R(F p ) = · · · with e F2 term. F2 is the fluence where SESAM reflectivity is 37% (1/e) due to induced absorption. A smaller F2 value is corresponding to a stronger roll-off. We drive the relation between F2 and the material parameter a2 nn(TPA), q2 (E p ) = a2 · E p ; q2 (E p ) =β · F E I · z ≈ βT P A · τ pp · z = βT P A · τ p ·Ap A · z, βT P A is TPA coefficient, τ p is the pulse

506

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

duration, and A A is the mode area. Absorber with thickness z the reduced transmission T2 is T2 (E p ) = e−q2 (E p ) = e−a2 ·E p . The characteristics of standing wave pattern, penetration depth of the optical field into the Bragg mirror, and different two-photon absorption cross-sections for different materials are consider for SESAM structure. We approximate these effects by an effective absorber thickness z e f f . Parameter z e f f need to be calculated numerically for specific SESAM design. a2 =

βT P A · z e f f τp 1 ; F2 = ≈ τp · AA a2 · A A βT P A · z e f f

F p is the constant fluence for  flattop spot,  ω is the radius. We calculate F p from

the measured pulse energy E p F p =

Ep π ·ω2

. In most application we use a Gaussian

2

− 2r ω2

where F0 is the peak fluence and ω is e12 beam profile: F pGauss (r ) = F0 · e % radius. The pulse energy E p is E p = F pGauss (r ) · 2 · π · r · dr = 21 · F0 (π · E

ω2 ); F p = 21 · F0 = π ·ωp 2 . The peak fluence in the Gaussian beam is 2 · F p , and the saturation occurs at lower fluences. Integration over the spatial energy distribution   ln 1+

Rlin Rns

x

·(e Fsat −1)

−

x

(x) = Rns · × e F2 . gives: R x Fsat The system model function describes the nonlinear behavior of real SESAMs under some approximations which describe before. Due to interferences and experimental system restrictions there are delays in time (τ1 , τ2 , τ3 , τ4 ) for systems varidelays in ables n 1 , n 2 , n, α respectively.  time do not affect the derivatives

1 The dn 2 dn dα , n 1 = n 1 (t); n 2 = n 2 (t); n = , , , in time of these variables dn dt dt dt dt n(t); α = α(t). The two levels system is characterized by set of delay differential equations (DDEs). Flat T op

σ dn 2 dn 1 = −[n 1 (t − τ1 ) − n 2 (t − τ2 )] · · I; dt ·ω dt σ ·I = [n 1 (t − τ1 ) − n 2 (t − τ2 )] · ·ω dn 2·σ 1 ·I = −n(t − τ3 ) · · I = −n(t − τ3 ) · dt ·ω Fsat dα 2·σ 1 ·ω dα · I ; Fsat = = −α(t − τ4 ) · ·I = = −α(t − τ4 ) · dt ·ω dt Fsat 2·σ n 1 —Occupation number of level 1 in time, n 2 —Occupation number of level 1 in time, n—Occupation difference between level 1 and level 2 in time, α—Absorption coefficient in time, σ —Absorption and emission crosssection, ω—Angular frequency of the optical field,  —Reduced Planck constant

h Js s . There are 16 sub cases: (1) τi = = 1.054 × 10−34 rad = 6.58 × 10−16 eV  = 2·π rad 0; i = 1, 2, 3, 4, (2) τ1 = τ ; τi = 0; i = 2, 3, 4, (3) τ2 = τ ; τi = 0; i = 1, 3, 4, (4) τ3 = τ ; τi = 0; i = 1, 2, 4, (4) τ4 = τ ; τi = 0; i = 1, 2, 3, (5) τ1 = τ2 = τ ; τi =

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

507

0; i = 3, 4, (6) …, (10) τ1 = τ4 = τ ; τi = 0; i = 2, 3. We choose to analyze the (2) sub case, the two levels system delay differential equations (DDEs) are σ σ dn 2 dn 1 = −[n 1 (t − τ ) − n 2 (t)] · · I; = [n 1 (t − τ ) − n 2 (t)] · ·I dt ·ω dt ·ω dn 2·σ 1 ·I = −n(t) · · I = −n(t) · dt ·ω Fsat dα ·ω 2·σ dα 1 · I ; Fsat = = −α(t) · ·I = = −α(t) · dt ·ω dt Fsat 2·σ 1 2 At fixed points: dn = 0; dn = 0; dn = 0; dα = 0; limt→∞ n 1 (t − τ ) = n 1 (t) dt dt dt dt

t → ∞ ⇒ t τ; t − τ ≈ t dn 1 σ σ (∗) (∗) = 0 ⇒ −[n (∗) · I = 0; · I  = 0 ⇒ n (∗) 1 − n2 ] · 1 = n2 dt ·ω ·ω dn 2 σ σ (∗) (∗) = 0 ⇒ [n (∗) · I = 0; · I  = 0 ⇒ n (∗) 1 − n2 ] · 1 = n2 dt ·ω ·ω dn 2·σ 1 ·I =0 = 0 ⇒ −n (∗) · · I = −n (∗) · dt ·ω Fsat 2·σ 1 · I = 0; n (∗) = 0 · I = 0; ·ω Fsat dα = 0 ⇒ −α (∗) · dt dα = 0 ⇒ −α (∗) · dt

2·σ 2·σ · I = 0; · I  = 0 ⇒ α (∗) = 0 ·ω ·ω 1 1 · I = 0; · I  = 0; α (∗) = 0 Fsat Fsat

(∗) (∗) (∗) Our system fixed point: = (n (∗) 1 , n 1 , 0, 0) ∀ k = 0, 1, 2= (n 1 , n 1 , 0, 0) ∀ k = 0, 1, 2. Stability analysis: The standard local stability analysis about any one of the equilibrium points of Q-switched microchip lasers semiconductors saturable absorbers system consists in adding to its coordinates [n 1 n 2 n α] arbitrarily small increments of exponential terms [n 1 n 2 n α] · eλ·t , and retaining the first order terms in n 1 n 2 n α. The system of four homogenous equations leads to a polynomial characteristic equation in the eigenvalue λ [6, 7]. The polynomial characteristic equation accepts by set the Q-switched microchip lasers semiconductors saturable absorbers system equations. The system fixed values with arbitrarily small increments of exponential form [n 1 n 2 n α] · eλ·t are; i = 0 (first fixed point), i = 1 (second fixed point), i = 2 (third fixed point), etc.

n 1 (t) = n 1(i) + n 1 · eλ·t ; n 2 (t) = n 2(i) + n 2 · eλ·t

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

n(t) = n (i) + n · eλ·t ; α(t) = α (i) + α · eλ·t n 1 (t − τ ) = n 1(i) + n 1 · eλ·(t−τ ) ;

dn 1 (t) dn 2 (t) = n 1 · λ · eλ·t ; = n 2 · λ · eλ·t dt dt

dn(t) dα(t) = n · λ · eλ·t ; = α · λ · eλ·t dt dt dn 1 σ = −[n 1 (t − τ ) − n 2 (t)] · · I ; n 1 · λ · eλ·t dt ·ω σ = −[n 1(i) + n 1 · eλ·(t−τ ) − n 2(i) − n 2 · eλ·t ] · ·I ·ω n 1 · λ · eλ·t = −[n 1(i) − n 2(i) ] ·

σ σ · I − [n 1 · eλ·(t−τ ) − n 2 · eλ·t ] · ·I ·ω ·ω

σ σ At fixed points: −[n 1(i) −n 2(i) ]· ·ω ·I = 0; n 1 ·λ·eλ·t = −[n 1 ·eλ·(t−τ ) −n 2 ·eλ·t ]· ·ω ·I

σ ·I ·ω σ − n1 · λ + n2 · ·I =0 ·ω

n 1 · λ · eλ·t = −[n 1 · e−λ·τ − n 2 · eλ·t ] · eλ·t · ⇒ −n 1 ·

σ · I · e−λ·τ ·ω

dn 2 σ = [n 1 (t − τ ) − n 2 (t)] · · I ; n 2 · λ · eλ·t dt ·ω σ = [n 1(i) + n 1 · eλ·(t−τ ) − n 2(i) − n 2 · eλ·t ] · ·I ·ω n 2 · λ · eλ·t = (n 1(i) − n 2(i) ) ·

σ σ · I + (n 1 · eλ·(t−τ ) − n 2 · eλ·t ) · ·I ·ω ·ω

σ σ At fixed points: (n 1(i) −n 2(i) )· ·ω · I = 0; n 2 ·λ·eλ·t = (n 1 ·eλ·(t−τ ) −n 2 ·eλ·t )· ·ω ·I

n 2 · λ · eλ·t = (n 1 · eλ·(t−τ ) − n 2 · eλ·t ) · ⇒ n1 ·

σ ·I ·ω

σ σ · I · e−λ·τ − n 2 · · I − n2 · λ = 0 ·ω ·ω

dn 2·σ 2·σ = −n(t) · · I ; n · λ · eλ·t = −(n (i) + n · eλ·t ) · ·I dt ·ω ·ω 2·σ · I − n · n · λ · eλ·t = −n (i) · ·ω (i) 2·σ At fixed points: −n · ·ω · I = 0

n · λ · eλ·t = −n ·

2·σ ·ω

· I · eλ·t ;

2·σ 2·σ · I · eλ·t ⇒ −n · · I − n · λ = 0 ·ω ·ω

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

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dα 2·σ 2·σ = −α(t) · · I ; α · λ · eλ·t = −(α (i) + α · eλ·t ) · ·I dt ·ω ·ω 2·σ · I − α · eλ·t · α · λ · eλ·t = −α (i) · ·ω 2·σ ·I =0 At fixed points: −α (i) · ·ω

α · λ · eλ·t = −α · eλ·t ·

2·σ ·ω

·I

2·σ 2·σ · I ⇒ −α · · I −α·λ=0 ·ω ·ω

We can summary our Q-switched microchip lasers semiconductors saturable absorbers system arbitrarily small increments equations: σ σ · I · e−λ·τ − n 1 · λ + n 2 · ·I =0 ·ω ·ω σ σ n1 · · I · e−λ·τ − n 2 · · I − n2 · λ = 0 ·ω ·ω − n1 ·

−n ·

2·σ 2·σ · I − n · λ = 0; −α · · I −α·λ=0 ·ω ·ω

Our eigenvalue matrix is ⎛

σ σ · I · e−λ·τ − λ ·I − ·ω ·ω σ σ −λ·τ ⎜ · I · e − · I −λ ·ω ·ω ⎜ 2·σ ⎝ 0 0 − ·ω 0 0

⎛

0 0 · I −λ 2·σ 0 − ·ω

σ σ · I · e−λ·τ − λ ·I − ·ω ·ω σ σ −λ·τ ⎜ · I · e − · I −λ ·ω ·ω A−λ· I =⎜ 2·σ ⎝ 0 0 − ·ω 0 0

⎞ ⎛ n1 ⎞ 0 ⎟ ⎜ n2 ⎟ 0 ⎟ ⎟·⎜ ⎟=0 ⎜ ⎠ 0 ⎝ n ⎠ · I −λ α 0 0 · I −λ 2·σ 0 − ·ω

⎞ 0 ⎟ 0 ⎟ ⎠ 0 · I −λ

To get the system characteristic equation: det(A − λ · I ) = 0 det(A − λ · I )

⎞ ⎛ σ − ·ω · I − λ 0 0  σ −λ·τ 2·σ ⎠ · I ·e − λ · det ⎝ =0⇒ − 0 0 − ·ω · I − λ ·ω 2·σ 0 0 − ·ω · I −λ ⎞ ⎛ σ −λ·τ 0 0 ·ω · I · e σ 2·σ ⎠=0 − · I · det ⎝ 0 0 − ·ω · I − λ ·ω 2·σ 0 0 − ·ω · I −λ 

−

  σ     2·σ σ · I · e−λ·τ + λ · − · I −λ · − · I −λ ·ω ·ω ·ω

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

  2·σ σ σ · − · I −λ − ·I· · I · e−λ·τ ·ω ·ω ·ω     2·σ 2·σ · − · I −λ · − · I −λ =0 ·ω ·ω      σ  2 · σ  σ 2·σ −λ·τ +λ · · I ·e · I +λ · · I +λ · · I +λ ·ω ·ω ·ω ·ω       2·σ 2·σ σ · I 2 −λ·τ ·e · · I +λ · · I +λ =0 − ·ω ·ω ·ω 2   σ  2 · σ  σ −λ·τ +λ · · I ·e · I +λ · · I +λ ·ω ·ω ·ω    2 σ · I 2 −λ·τ 2·σ − ·e · · I +λ =0 ·ω ·ω   2 σ σ σ −λ·τ −λ·τ 2 + ·λ+ · I ·e · I ·e · I ·λ+λ ·ω ·ω ·ω   2 2·σ 4·σ · ·I + · I · λ + λ2 ·ω ·ω     2 σ · I 2 −λ·τ 2·σ 4·σ 2 − ·e · ·I + · I ·λ+λ =0 ·ω ·ω ·ω  2  2 2  σ 2·σ σ 4·σ · I · e−λ·τ · ·I + · I · e−λ·τ · · I ·λ ·ω ·ω ·ω ·ω  2 2  σ 2·σ −λ·τ 2 σ −λ·τ ·λ ·λ· · I ·e · I ·e ·I + ·ω ·ω ·ω σ 4·σ σ + · I · e−λ·τ · λ · · I ·λ+ · I · e−λ·τ · λ · λ2 ·ω ·ω ·ω  2 2·σ 4·σ + λ2 · · I + λ2 · · I · λ + λ4 ) ·ω ·ω     2 σ · I 2 −λ·τ 2·σ 4·σ 2 − ·e · ·I + · I ·λ+λ =0 ·ω ·ω ·ω  2  2 2  σ 2·σ σ 4·σ · I · e−λ·τ · ·I + · I · e−λ·τ · · I ·λ ·ω ·ω ·ω ·ω  2 2  σ 2·σ −λ·τ 2 σ −λ·τ ·λ ·λ· · I ·e · I ·e ·I + ·ω ·ω ·ω σ 4·σ σ + · I · e−λ·τ · λ · · I ·λ+ · I · e−λ·τ · λ · λ2 ·ω ·ω ·ω

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

511



  2  2 2·σ σ · I 2 −λ·τ 2·σ 2 4·σ 4 +λ · ·e · · I +λ · · I ·λ+λ )− ·I ·ω ·ω ·ω ·ω     σ · I 2 −λ·τ 4 · σ σ · I 2 −λ·τ 2 − ·e · ·e ·λ =0 · I ·λ− ·ω ·ω ·ω       σ · I 4 −λ·τ σ · I 3 −λ·τ σ · I 2 −λ·τ 2 4· ·e +4· ·e ·λ+ ·e ·λ ·ω ·ω ·ω       σ · I 3 −λ·τ σ · I 2 −λ·τ 2 σ·I +4· ·e ·λ+4· ·e ·λ + · e−λ·τ · λ3 ·ω ·ω ·ω       2·σ · I 2 4·σ · I σ · I 4 −λ·τ 2 3 4 +λ · +λ · ·e +λ −4· ·ω ·ω ·ω     σ · I 3 −λ·τ σ · I 2 −λ·τ 2 −4· ·e ·λ− ·e ·λ =0 ·ω ·ω     2·σ · I 2 4·σ · I λ2 · + λ3 · + λ4 ·ω ·ω !       σ·I 3 σ·I 2 2 σ·I 3 + 4· ·λ+4· ·λ + · λ · e−λ·τ = 0 ·ω ·ω ·ω 2



   2·σ · I 2 4·σ · I + λ3 · ·ω ·ω !    3   σ·I σ·I 2 2 σ·I 4 3 +λ + 4· ·λ+4· ·λ + · λ · e−λ·τ ·ω ·ω ·ω

D(λ, τ ) = λ2 ·

D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n = 4; m = 3; n > m 

   2·σ · I 2 4·σ · I 3 Pn=4 (λ, τ ) = λ · +λ · + λ4 ; Q m=3 (λ, τ ) ·ω ·ω       σ·I 2 2 σ·I σ·I 3 ·λ+4· ·λ + · λ3 =4· ·ω ·ω ·ω 2

The expression for Pn (λ, τ ): Pn=4 (λ, τ ) =

n=4 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2

k=0

+ p3 (τ ) · λ3 + p4 (τ ) · λ4  p0 (τ ) = 0; p1 (τ ) = 0; p2 (τ ) =

2·σ · I ·ω

2 ; p3 (τ ) =

4·σ · I ; p4 (τ ) = 1 ·ω

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

Q m=3 (λ, τ ) =

n=3 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 + q3 (τ ) · λ3

k=0

 q0 (τ ) = 0; q1 (τ ) = 4 ·

σ·I ·ω



3 ; q2 (τ ) = 4 ·

σ·I ·ω

2 ; q3 (τ ) =

σ·I ·ω

The homogeneous system for n 1 n 2 n α leads to a characteristic equation for  the eigenvalue λ having the form P(λ)+ Q(λ)·e−λ·τ ; P(λ) = 4j=0 a j ·λ j ; Q(λ) = 3 j j=0 c j ·λ and the coefficients {a j (qi , qk ), c j (qi , qk )} ∈ R depend on qi , qk but not on τ , qi , qk are any Q-switched microchip lasers semiconductors saturable absorbers system parameters, other parameters keep as a constant. 

 2·σ · I 2 4·σ · I ; a3 = ; a4 = 1 a0 = 0; a1 = 0; a2 = ·ω ·ω     σ·I 3 σ·I 2 σ·I ; c2 = 4 · ; c3 = c0 = 0; c1 = 4 · ·ω ·ω ·ω Unless strictly necessary, the designation of the variation argument (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j and c j are continuous and differential functions of their arguments, and direct substitution shows that a0 + c0 = 0 ∀ qi , qk ∈ R+ . λ = 0 is not a of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [BK] can also be verified in the present case: (a) (b)

If  λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0.  Q(λ)   P(λ)  is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞.

(c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω(degree in ω4 ). (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk parameters. Hint: We define ω parameter as the system angular frequency of the optical field and as a ω parameter in our stability analysis. We assume that Pn (λ, τ ) = Pn (λ); Q m (λ, τ ) = Q m (λ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i ·ω, τ )+ Q m (λ = i ·ω, τ )  = 0.  Pn=4 (λ = i · ω, τ ) = ω4 − ω2 ·

2·σ · I ·ω



2 − i · ω3 ·

4·σ · I ·ω



5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

513

  σ·I 2 2 Q m=3 (λ = i · ω, τ ) = −4 · ·ω ·ω

     σ·I 3 σ·I 3 +i · 4· ·ω− ·ω ·ω ·ω  Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = ω − ω · 4



2

   4·σ · I σ·I 2 2 ·ω −4· ·ω ·ω     σ·I 3 σ·I +4· ·i ·ω−i · · ω3  = 0 ·ω ·ω

2·σ · I ·ω

2

− i · ω3 ·

Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )

  2 2  2 · σ · I σ · I +4· · ω2 = ω4 − ·ω ·ω       σ·I 3 4·σ · I σ·I 3 +4· ·i ·ω−i ·ω · + = 0 ·ω ·ω ·ω Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )   σ·I 2 2 ·ω = ω4 − 8 · ·ω   σ·I 3 5·σ · I +4· · i · ω − i · ω3 · = 0 ·ω ·ω

   2  2·σ · I 2 4·σ · I 2 2 4 2 + ω6 · |Pn (λ = i · ω, τ )| = ω − ω · ·ω ·ω 

 σ·I 4 4 ·ω ·ω

 2    σ·I 3 σ·I + 4· ·ω− · ω3 ·ω ·ω

|Q m (λ = i · ω, τ )|2 = 16 ·

F(ω) = |P(i · ω)|2 − |Q(i · ω)|2

   2  2·σ · I 2 4·σ · I 2 4 2 = ω −ω · + ω6 · ·ω ·ω

2       σ·I 4 4 σ·I 3 σ·I 3 − 16 · ·ω − 4· ·ω− ·ω ·ω ·ω ·ω

514

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

 F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = ω8 − 2 · ω6 · 

2·σ · I ·ω

2

   2·σ · I 4 4·σ · I 2 + ω6 · ·ω ·ω  4 σ·I − 16 · · ω4 ·ω

       σ·I 6 2 σ·I 4 4 σ·I 2 6 − 16 · ·ω −8· ·ω + ·ω ·ω ·ω ·ω + ω4 ·

 F(ω) = |P(i · ω)| − |Q(i · ω)| = ω − 2 · ω · 2

2

8

6



   2·σ · I 4 4·σ · I 2 + ω6 · ·ω ·ω   4  σ·I σ·I 6 2 4 − 16 · · ω − 16 · ·ω ·ω ·ω     σ·I 4 4 σ·I 2 6 +8· ·ω − ·ω ·ω ·ω

2·σ · I ·ω

2

+ ω4 ·

F(ω) = |P(i · ω)|2 − |Q(i · ω)|2

       4·σ · I 2 2·σ · I 2 σ·I 2 = ω8 + ω6 · −2· − ·ω ·ω ·ω

 4 4       2·σ · I σ·I σ·I 4 σ·I 6 2 4 − 16 · +8· ·ω +ω · − 16 · ·ω ·ω ·ω ·ω

F(ω) = |P(i · ω)|2 − |Q(i · ω)|2       σ·I 2 σ·I 4 σ·I 6 = ω8 + ω6 · 7 · + ω4 · 8 · − ω2 · 16 · ·ω ·ω ·ω We define the following parameters for simplicity: 0 = 0; 2 = −16 ·  4 = 8 ·

σ·I ·ω



4 ; 6 = 7 ·

σ·I ·ω

σ ·I 6 ·ω

2 ; 8 = 1

F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 =  8 · ω +  6 · ω + 4 · ω +  2 · ω = 8

6

4

2

4 

2·k · ω2·k

k=0

Hence F(ω) = 0 implies the above polynomial.

4 k=0

2·k · ω2·k = 0 and its roots are given by solving

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

515



   2·σ · I 2 4·σ · I 3 Pn=4 (i · ω) = ω − ω · −i ·ω · ·ω ·ω  2 σ·I · ω2 Q m=3 (i · ω) = −4 · ·ω

     σ·I 3 σ·I 3 +i · 4· ·ω− ·ω ·ω ·ω 4

2

Furthermore PR (i · ω) = ω4 − ω2 ·

2·σ ·I 2 ·ω

; PI (i · ω) = −ω3 ·

4·σ ·I  ·ω



 σ·I 2 2 Q R (i · ω) = −4 · · ω ; Q I (i · ω) ·ω     σ·I σ·I 3 ·ω− · ω3 =4· ·ω ·ω Hence −PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2 PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) cos θ (τ ) = − |Q(i · ω)|2 sin θ (τ ) =

And sin θ (τ ) 

·I 2   σ ·I 3

σ ·I  3 

σ ·I 3 − ω4 − ω2 · 2·σ · ω + 16 · ω5 · ·ω · 4 · ·ω · ω − ·ω ·ω =   2

σ ·I 4

σ ·I  σ ·I 3 · ω3 16 · ·ω · ω4 + 4 · ·ω · ω − ·ω cos θ(τ )



σ ·I 2 2

·I 2 

·I   σ ·I 3

σ ·I  3  − ω4 − ω2 · 2·σ ·ω · 4 · ·ω · ω − ω3 · 4·σ · ω − ·ω ·ω ·ω · 4 · ·ω =− 2  3 

σ ·I 4

σ ·I σ ·I · ω3 16 · ·ω · ω4 + 4 · ·ω · ω − ·ω

 Which jointly with F(ω) = 0 ⇒ 4k=0 2·k · ω2·k = 0, that are continuous and differentiable in τ . It based on Lemma 1.1, hence we use Theorem 1.2 and this proves the Theorem 1.3. We use different parameters terminology from our last characteristics parameters definition k → j; pk (τ ) → a j and qk (τ ) → c j ; n = 4; m = 3; n > m,  additionally Pn (λ, τ ) → P(λ, τ ); Q m (λ, τ ) → Q(λ, τ ) then P(λ, τ ) = 2j=0 a j ·  λ j ; Q(λ, τ ) = 1j=0 c j · λ j

516

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …





 4·σ · I P(λ, τ ) = λ · +λ · + λ4 ; Q(λ, τ ) ·ω       σ·I 2 2 σ·I σ·I 3 ·λ+4· ·λ + · λ3 =4· ·ω ·ω ·ω 2

2·σ · I ·ω

2

3

n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable functions of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ) and Q(λ) are analytic functions in λ and differentiable in τ . The coefficients {a j (σ, ω, I )& c j (σ, ω, I )} ∈ R depend on Q-switched microchip lasers semiconductors saturable absorbers system parameters σ, ω, I, τ, . . . values. Unless strictly necessary, the designation of the variation arguments, σ, ω, I, τ, . . . system parameters will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0. Furthermore, P(λ), Q(λ) are analytic function of λ for which the following requirements of the analysis (see Kuang 1993, Sect. 3.4) can also be verified in the present case [9, 10]. (e) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire (σ, ω, I, τ  parameters) domain of interest. system   is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (f)  Q(λ) P(λ)     σ ·I 3  σ ·I 2 2 σ ·I ·λ+4·( ·ω ·λ +( ·ω ·λ3   Q(λ)   4·( ·ω ) ) ) in the limit:  P(λ)  =  ·I 2 . 3 4·σ ·I 4 λ2 ·( 2·σ ·ω ) +λ ·( ·ω )+λ 

σ ·I 4 σ ·I 2 +ω4 · 8 · ·ω − ω2 · (g) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = ω8 + ω6 · 7 · ·ω

σ ·I 6 16 · ·ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω8 ). (h) Each positive root ω(σ, ω, I, τ ) of F(ω) = 0 is continuous and differentiable with respect to σ, ω, I, τ parameters. This condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus λ = i · ω ∀ ω > 0 may be on eigenvalue of characteristic equation. The analysis consists in identifying the roots of characteristic equation situated on the imaginary axis of the complex λ—plane, where by increasing the parameters σ, ω, I, τ (system parameters), Reλ may, at the crossing, (k) (k) (k) change its sign from (−) to (+), i.e. from a stable focus E (k) (n (k) 1 , n 2 , n , α ) = (∗) (∗) (n 1 , n 1 , 0, 0) ∀ k = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to σ, ω, I, τ parameters (ω ∈ R+ ). −1

 (σ ) =



∂Reλ ∂σ

 λ=i·ω

; ω, I, τ = const

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

517



 ∂Reλ ; σ, I, τ = const ∂ω λ=i·ω   ∂Reλ −1 ; σ, ω, τ = const  (I ) = ∂ I λ=i·ω   ∂Reλ ; σ, ω, I = const −1 (τ ) = ∂τ λ=i·ω −1 (ω) =

where writing P(λ) = PR (λ)+i · PI (λ) and Q(λ) = Q R (λ)+i · Q I (λ), and inserting λ = i · ω into Q-switched microchip lasers semiconductors saturable absorbers system’s characteristic equation ω must satisfy the following: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i ·ω)|2  = 0 in the view of requirement (a) above, (g, h) ∈ R. Furthermore, it follows the above equations sin(ω · τ ) and cos(ω · τ ) that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note F(ω) is independent of τ . Now it is important to notice that if τ ∈ / I (assume that / I ; ω(τ ) I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for τ ∈ is not define. Then for all τ ∈ I is satisfies that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switch. For any τ ∈ I , where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the I (i·ω)·Q R (i·ω) ; angle θ (τ ) ∈ [0, 2 · π] as the solution of sin θ (τ ) = −PR (i·ω)·Q I (i·ω)+P |Q(i·ω)|2

I (i·ω)·Q I (i·ω) cos θ (τ ) = − PR (i·ω)·Q R (i·ω)+P and the relation between the argument θ (τ ) |Q(i·ω)|2 and ω(τ )·τ for τ ∈ I must be ω(τ )·τ = θ (τ )+2·n·π ∀ n ∈ N0 . Hence we can define )+n·2·π ; n ∈ N0 , τ ∈ I . Let introduce the maps τn : I → R+0 given by τn (τ ) = θ(τ ω(τ ) the functions I → R; Sn (τ ) = τ − τn (τ ), n ∈ N0 , τ ∈ I , that are continuous and differentiable in τ . In the following, the subscripts λ, ω, σ, ω, I, τ, . . . system parameters (first, ω is related to parameter in our stability analysis and second ω is related to system angular frequency of the optical field) indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(σ, ω, I, τ, . . . ∈ R+ ); ω(σ, ω, I, τ, . . . ∈ R+ ) and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Q-switched microchip lasers semiconductors saturable absorbers system characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates, x = σ, ω, I, τ, . . . ∈ R+ ,



∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

518

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

where Pλ = ∂∂λP , . . . etc., Substituting λ = i · ω, and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω) and i · Q λ (i · ω) = Q ω (i · ω) that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains, 

∂λ ∂x =

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R +i · Q I and Pω = PRω +i · PI ω ; Q ω = Q Rω +i · Q I ω ; Px = PRx +i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 . When (x) can be an Q-switched microchip lasers semiconductors saturable absorbers system parameters σ, ω, I, . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] and Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ . Hint: We define the system angular frequency of the optical field as ωo f . 

   2·σ · I 2 4·σ · I − i · ω3 ·  · ωo f  · ωo f   σ·I 2 2 ·ω Q m=3 (i · ω) = −4 ·  · ωo f

     σ·I 3 σ·I 3 +i · 4· ·ω− ·ω  · ωo f  · ωo f Pn=4 (i · ω) = ω4 − ω2 ·



   2·σ · I 2 4·σ · I ; PI (i · ω) = −ω3 ·  · ωo f  · ωo f   σ·I 2 2 · ω ; Q I (i · ω) Q R (i · ω) = −4 ·  · ωo f     σ·I 3 σ·I =4· ·ω− · ω3  · ωo f  · ωo f   2·σ · I 2 3 PRω (i · ω, τ ) = 4 · ω − 2 · ω · ; PI ω (i · ω, τ )  · ωo f

PR (i · ω) = ω4 − ω2 ·

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

519



 4·σ · I  · ωo f   σ·I 2 · ω; Q I ω (i · ω, τ ) Q Rω (i · ω, τ ) = −8 ·  · ωo f     σ·I 3 σ·I =4· −3· · ω2  · ωo f  · ωo f = −3 · ω2 ·

PRω = PRω (i · ω, τ ); PI ω = PI ω (i · ω, τ ); Q Rω = Q Rω (i · ω, τ ); Q I ω = Q I ω (i · ω, τ ) PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0

  2   2 · σ · I 2·σ · I 2 3 4 2 ·ω −ω · PRω · PR = 4 · ω − 2 · ω ·  · ωo f  · ωo f  PI ω · PI = 3 · ω · 5

QIω · QI

 = 4·

σ·I  · ωo f

4·σ · I  · ωo f



3 −3·

σ·I  · ωo f



2 ; Q Rω · Q R = 32 · 





· ω2 · 4 ·

σ·I  · ωo f

σ·I  · ωo f 

3 ·ω−

4 · ω3

σ·I  · ωo f



 · ω3

Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] Fτ = 0; ωτ = − =0 Fω

     2·σ · I 2 4·σ · I 2 4 2 · PR · PI ω = −3 · ω · ω − ω ·  · ωo f  · ωo f    2  4 · σ · I 2 · σ · I s PI · PRω = −ω3 · · 4 · ω3 − 2 · ω ·  · ωo f  · ωo f QR · QIω



= −4 · Q I · Q Rω 4 = −8 ·

σ·I  · ωo f 

2

σ·I  · ωo f



·ω · 4· 2

2

·ω 4·



σ·I  · ωo f

σ·I  · ωo f



3 −3·



3 ·ω−

σ·I  · ωo f

σ·I  · ωo f



 ·ω

2



 ·ω

3

520

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

PR · PI x |x=τ = 0; PI · PRx |x=τ = 0; Q R · Q I x |x=τ = 0; Q I · Q Rx |x=τ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 ∂ω ∂ω Fτ Fω · + Fτ = 0; τ ∈ I ; =− ∂τ ∂τ Fω   ∂Reλ −1 (τ ) = ; −1 (τ ) ∂τ λ=i·ω   −2 · [U + τ · |P|2 ] + i · Fω = Re Fτ + i · 2 · [V + ω · |P|2 ] Fτ ∂ω = ωτ = − ∂τ Fω    ∂Reλ −1 sign{ (τ )} = sign ∂τ λ=i·ω +V U · ∂ω ∂ω ∂τ sign{ (τ )} = sign{Fω } · sign τ · +ω+ 2 ∂τ |P|

!

−1

We shall presently examine the possibility of stability transitions (bifurcations) in a Q-switched microchip lasers semiconductors saturable absorbers system, about (k) (∗) (∗) (k) (k) equilibrium points E (k) (n (k) 1 , n 2 , n , α ) = (n 1 , n 1 , 0, 0) ∀ k = 0, 1, 2 as a result of a variation of delay parameter τ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane where by increasing the delay parameter τ , Reλ may at the crossing, change its sign from “−” to “+”, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed by  examining the sign of the

and σ, ω, I, . . . system partial derivatives with respect to τ , −1 (τ ) = ∂Reλ ∂τ λ=i·ω parameters are constant where ω ∈ R+ . We need to plot the stability switch diagram based on different delay values of our Q-switched microchip lasers semiconductors saturable absorbers system. Since it is very complex function we recommend to solve it numerically rather than analytic [9, 10]. 

 ∂Reλ ; −1 (τ ) ∂τ λ=i·ω   −2 · [U + τ · |P|2 ] + i · Fω = Re Fτ + i · 2 · [V + ω · |P|2 ]   ∂Reλ −1  (τ ) = ; −1 (τ ) ∂τ λ=i·ω −1 (τ ) =

5.2 Q-Switched Microchip Lasers Semiconductors Saturable Absorbers …

=

521

2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . ; cos θ (τ ) = . . ., when ω = ω+ (τ ) if only ω+ is feasible. Additionally when all Q-switched microchip lasers semiconductors saturable absorbers system parameters are known and the stability switch due to various time delay values τ is describe in the following expression: sign{∧−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign   U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) τ · ωτ (ω(τ )) + ω(τ ) + |P(ω(τ ))|2 Remark Lemma 1.1, Theorem 1.2 and Theorem 1.3 are described in Sect. 5.1.

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations Instability Under Delayed Variables in Time Ion-doped crystalline laser mainly operate in the mid-IR spectral range between 2 and 5 μm. Types are rare-earth and transition-metal based ionic crystals, and colorcenter lasers. They are compact all-solid-state room-temperature tunable sources, belonging to class of vibrionic lasers. These lasers have the efficient high-power room-temperature operation and super-broad tenability, the possibility of generating ultrashort pulses from the novel class of chromium doped chalcogenide lasers led to an analysis in vibrionic laser systems, involving 3dn transition-metal ions. They are mid-IR lasers which have spectroscopic and laser characteristics. Mainly they are continuous-wave (CW), diode-pumped and tunable lasers which are based on both rare-earth ion doped crystals (operate at fixed wavelengths and allow high-power operation) and transition-metal ™ ion doped-and color-center crystals, allowing broad tuning and ultrashort pulse generation. When working in operation threshold the important parameter is bandwidth of the gain medium and tuning is done. The bandwidth definition is as the full width at half maximum relative to the  v ≈ . The definition is the same in the wavelength and gain maximum at λ0 λ λ0 v0 frequency domains. Then we can compare gain media with different central wavelengths. Crystalline active media have broadest gain in a certain wavelength range at room temperature, from near-infrared to mid-IR range. The number of optical cycles  −1 . The mid-IR (MIR) per pulse for ultrashort pulse generation is equal to λ λ0 wavelength region (“molecular fingerprint” region) and the range between 2 and 5 μm is characterized by a strong fundamental vibrational absorption lines of atmospheric constituents, vapors and other gases. Mid-IR radiation helps us to identify many materials by their characteristic spectra. Fingerprint region in IR spectroscopy,

522

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

are the regions where all the bending vibrations are seen in spectroscopy. We can identify two different organic compounds by comparing the troughs in the right hand side of spectroscopy graph. Molecules can be in different excited states: stretch along their bands, vibrate around their center of mass and rotate around one of their axes. These excitations store energy and, energy can be expressed in wavelengths and the energy of many of these rotational and vibrational (ro-vibrational) excitations correspond to mid-IR wavelengths. We use these spectral fingerprints to identify materials using spectroscopy (by the characteristic of the energy stored in each of these excitations). Typical lasers in mid-IR region are semiconductor lasers, heterojunction lasers, and lead-salt, antimonide and quantum cascade laser sources. They require cryogenic cooling and have limited output power levels. Crystalline solid-state lasers can provide very high power levels retaining good beam quality and narrow spectral line width. Near infrared diode lasers is function as a pump sources to these lasers. We get stability, efficiency and compactness, broad spectral coverage and tuning ranges. Remote sensing and trace gas detection are possible by using room temperature diode-pumped tunable solid-state lasers. The ultra-broad gain bandwidth of laser crystals allows generation of ultrashort pulses of only a few optical cycles. These lasers fit application as mid-IR free space communications, optical frequency standards, and optical coherence tomography (OCT). Additional medical applications of mid-IR solid-state lasers are tissue cutting and welding, ophthalmology, neurosurgery, etc. we emphasis on the basics of lasers operation of tunable crystalline lasers. Laser techniques used in the visible and near-infrared regions and extended to mid-IR spectral regions. We handle request for modification the techniques and analyze the Kerr-lens mode-locking. Kerr lens mode locking is a technique of passive mode locking a laser. It use an artificial saturable absorber based on Kerr lensing in the gain medium. The effect causes a reduction in the beam size for high optical intensities by two mechanisms which act as a fast saturable absorber. The first mechanism is related to the case of hard aperture KLM, the Kerr lens reduces the optical losses at an aperture which the beam must pass in each resonator round trip. The second mechanism is in the case of soft aperture KLM, the Kerr lens leads to a better overlap of laser and pump beam, and thus to a higher gain for the peak of the pulse. The increase of gain is equivalent to decrease of losses (similar effects). Both effects increase the net round trip gain. Kerr lens mode locking has enabled the generation of the short pulses with durations down to 5fsec in Ti:sapphire lasers. It is a very fast response and that no special saturable absorber medium is required. The laser operates close to stability limit of its resonator which is disadvantage, because otherwise the Kerr lensing effect is too weak. We get difficulties for long-term stable operation and resonator design. The operation procedure for such laser starts in noisy operation mode, and after being turned on and switch to mode-locked operation only after an external trigger. KLM is called also self-mode locking since there is no request for a visible saturable absorber device. Crystalline lasers are key elements in mid-IR lasers. Laser devices key element is a chromium ion in a sapphire. A ruby laser is a three-level system and operated in the pulsed regime. Another type of mid-IR system is dysprosium doped calcium fluoride which operated both pulsed and continuouswave (CW). It is pumped by a Xenon flash-lamp. We discuss the process in fluencing

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

523

formation of the inversion population in mid-IR crystalline solid-state lasers. These processes effect on the laser operation (laser threshold and efficiency). We assume four level scheme which characterized by balance equation for the population of the upper laser level n [16]. n dn = F pump · σG S A · (Nt − n) − F las · σem · n − − α · n 2 dt τ The term for pumping is F pump · σG S A · (Nt − n), the term for stimulated emission is F las · σem · n, and the term for spontaneous decay is τn . The term for up conversion with the up conversion macro parameter (α) is α ·n 2 . Nt —is the total concentration of the active ions, n—is the excited ion concentration, σG S A —is the GSA cross section at the pumping wavelength (pump absorption cross-section), σem —is the emission cross section at the laser wavelength, F pump —is the pumping photon flux, F las —is the intra cavity emission photon flux, and τ —is the temperature dependent lifetime of the upper laser level. Remark ESA terms are not under the assumption that the quantum yield from the upper states to the upper laser level is equal to unity. ESA does not change the excited ion concentration. The rate equation for the photon flux at the laser wavelength is F las c c d F las − · F las · σ ElasS A · n · la = · F las · σem · n · la − dt l τc l las

The stimulate emission term is cl · F las · σem · n · la , the loss term is Fτc , and the ESA term is cl · F las · σ ElasS A · n · la . The photon lifetime in the resonator is l , where l is the effective cavity round-trip length, and T + L are the τc = (T +L)·c logarithmic round-trip losses, T —output coupling, L—other losses. The threshold , where, la is the active medium length. We get the population n th is n th = l ·(σ T +L las a em −σ E S A ) expression for the threshold absorbed power in longitudinally pumped configuration: h · vlas λlas · A pump · σem · τ λ pump   −1  pump σ las σ E S A · n th × (1 + α · n th · τ ) · 1 − E S A · 1+ σem σG S A · (Nt − n th )

Pthabs = (T + L) ·

The above expression fulfills the following: Pthabs ∼ (T + L); Pthabs ∼ las h·v ; Pthabs ∼ λλpump , and Pthabs ∼ A pump . The last three terms represent the up converσem ·τ sion, ESA at the laser wavelength, and ESA at the pump wavelength, respectively. The slope efficiency with respect to the absorbed pump power is las

ηslope

λ pump = las · λ



T T +L

−1     pump σ ElasS A σ E S A · n th · 1− · 1 + pump σem σ E S A · (Nt − n th )

524

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

where the first two terms are the stokes shift and output coupling efficiency, and the last two stand for the efficiency loss due to the ESA at the laser  and pump  wavelengths, σ ElasS A λ pump respectively. The intrinsic slope efficiency is η0 = λlas · 1 − σem . Remark The up conversion and temperature reduction of the active ion lifetime increase the laser threshold but do not affect the slope efficiency. Our target is to keep the up conversion losses at minimum (α · n th · τ  1), keep overall losses T + L low to reduce the threshold population. The losses and ESA affect the threshold and slope efficiency. In four-level lasers, where n th < Nt we can ignore the ESA at the pump wavelength [16]. Laser transitions in the mid-IR tend to be broadband. The widths of the individual states (energy units) are determined by the particular crystal-lattice interaction and inhomogeneous broadening. They are characteristic for the given ion host. The transition bandwidth h · v (energy units) is the sum of the widths of the upper and lower states. The transition energy h · v is inversely proportional to the wavelength. The ≈ λ increases towards the infrared. The relative bandwidth of the transition v v λ transition bandwidth is determining the ability of the laser material to lase and to be tuned. Additional characteristics are transition lifetime and cross-section. Both are connected with the bandwidth of the laser transition. The spontaneous emission cross section σem (λ) is obtained from the fluorescence intensity signal I (λ) by using (λ)·λ5 the relation: σem (λ) = 8·π A·c·n2 · % II (λ)·λ·dλ ; A—is the full spontaneous emission probability (relative decay rate) from the upper laser level, c—is the speed of light and n is the index of refraction. The formula for σem (λ) does not contain assumptions; we neglect the polarization dependence of fluorescence. We get radiative decay rate by taking inverse life time at low temperature. Then we get the effective emission crosssection σem (λ) by radiative decay rate. Ion-host interaction outcome is non radiative decay. The luminescence life time τ consists of purely radiative and non-radiative 1 + τ1nr . components, τ1 = τrad 1 τrad

$ = A = 8 · c · n2 ·

σem (λ) · dλ 1 λ ≈ 8 · c · n 2 · σem (λ0 ) · 3 · 4 λ λ0 λ0

We get the fact that lifetime inversely scales with emission bandwidth. The minimum absorbed pump power density at threshold is proportional to the loss coefficient and the gain saturation intensity. The incident pumping intensity (Ith ) h·v · σ (v)·τ , is inversely proportional to the pump absorption coefficient: Ith ∝ ααloss abs where αloss include the absorption from the ground state in the case of three and quasi three level schemes. The pump intensity at threshold is proportional   1 1 λ n2 τ is the to the following: Ith ∝ F O M · η Q E · λ0 · λ4 , where η Q E η Q E = τrad 0 quantum yield of the transition [16]. We can summary our system differential equations:n = n(t); F las = F las (t) dn n = F pump · σG S A · (Nt − n) − F las · σem · n − − α · n 2 dt τ

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

525

d F las F las c c − · F las · σ ElasS A · n · la = · F las · σem · n · la − dt l τc l At fixed points:

∗ 

dn dt

las

= 0; d Fdt = 0

F pump · σG S A · (Nt − n ∗ ) − F las∗ · σem · n ∗ −

∗∗  c l

F las∗ c − · F las∗ · σ ElasS A · n ∗ · la = 0 τc l  1 c · n ∗ · la − − · σ ElasS A · n ∗ · la = 0 τc l

· F las∗ · σem · n ∗ · la −

∗∗ 

 : F las∗ ·

c · σem l

Case I: F las∗ = 0 ⇒ F pump · σG S A · (Nt − n ∗ ) − ∗∗

n∗ − α · (n ∗ )2 = 0 τ

∗

∗ 2



( ) → ( ) : α · (n ) + F ∗

Case Ia: n =

pump

1 · σG S A + τ



n∗ τ

− α · (n ∗ )2 = 0.

· n ∗ − F pump · σG S A · Nt = 0.

 2 −( F pump ·σG S A + τ1 )+ ( F pump ·σG S A + τ1 ) +4·α·F pump ·σG S A ·Nt . 2·α  1 1 2 pump pump pump −( F ·σG S A + τ )− ( F ·σG S A + τ ) +4·α·F ·σG S A ·Nt . 2·α

Case Ib: n ∗ = We can summary our first and second fixed points:

E 0 (n (0) , F las(0) ) ⎛

⎞  

2 F pump · σG S A + τ1 + 4 · α · F pump · σG S A · Nt − F pump · σG S A + τ1 + =⎝ , 0⎠ 2·α E 1 (n (1) , F las(1) ) ⎛

⎞  

2 F pump · σG S A + τ1 + 4 · α · F pump · σG S A · Nt − F pump · σG S A + τ1 − =⎝ , 0⎠ 2·α

Case II:

c l

· σem · n ∗ · la −

n ∗ · la ·

1 τc

−

c l

· σ ElasS A · n ∗ · la = 0

c 1 l ⇒ n∗ = · (σem − σ ElasS A ) = l τc τc · la · c · (σem − σ ElasS A )

∗ 

∗∗  → : F pump · σG S A · (Nt − n ∗ ) n∗ − α · (n ∗ )2 = 0 − F las∗ · σem · n ∗ − τ

F las∗ = F pump · σG S A · (Nt − n ∗ ) ·

1 1 α · n∗ − − ∗ σem · n τ · σem σem

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F

las∗

=F

pump

l · σG S A · Nt − τc · la · c · (σem − σ ElasS A )



τc · la · c · (σem − σ ElasS A ) σem · l 1 α l − − · τ · σem σem τc · la · c · (σem − σ ElasS A )   F pump · σG S A · l pump = F · σG S A · Nt − τc · la · c · (σem − σ ElasS A ) ·

F las∗

τc · la · c · (σem − σ ElasS A ) σem · l 1 α l − − · τ · σem σem τc · la · c · (σem − σ ElasS A )

·

τc · la · c · (σem − σ ElasS A ) σem · l 1 α l − − · τ · σem σem τc · la · c · (σem − σ ElasS A )

F las∗ = F pump · σG S A · Nt · −

F pump · σG S A σem

The third fixed point:

(2)



E (n , F 2

las(2)

)=

l , τc · la · c · (σem − σ ElasS A ) τc · la · c · (σem − σ ElasS A ) F pump · σG S A − σem · l σem  α l − · σem τc · la · c · (σem − σ ElasS A )

F pump · σG S A · Nt · −

1 τ · σem

We can summary our system three fixed points: E 0 (n (0) , F las(0) ) ⎛

⎞  

2 F pump · σG S A + τ1 + 4 · α · F pump · σG S A · Nt − F pump · σG S A + τ1 + =⎝ , 0⎠ 2·α E 1 (n (1) , F las(1) ) 

⎛ ⎞ 2 F pump · σG S A + τ1 + 4 · α · F pump · σG S A · Nt −(F pump · σG S A + τ1 ) − =⎝ , 0⎠ 2·α

E 2 (n (2) , F las(2) ) =



l , τc · la · c · (σem − σ ElasS A )

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

527

τc · la · c · (σem − σ ElasS A ) F pump · σG S A − σem · l σem  α l − · σem τc · la · c · (σem − σ ElasS A )

F pump · σG S A · Nt · −

1 τ · σem

We need to do linearization to our system. The general form of our Crystalline Mid-infrared laser system (no delay variables in time) is dn d F las = f 1 (n, F las ); = f 2 (n, F las ) dt dt f 1 (n, F las ) = F pump · σG S A · (Nt − n) − F las · σem · n − f 2 (n, F las ) =

n − α · n2 τ

c F las c − · F las · σ ElasS A · n · la · F las · σem · n · la − l τc l

System any fixed point (we have three fixed points) is n ∗ , F las∗ and exist f 1 (n ∗ , F las∗ ) = 0; f 2 (n ∗ , F las∗ ) = 0. Let u 1 = n − n ∗ ; u 2 = F las − F las∗ denote the components of a small disturbance from the fixed point. To see whether the disturbance grows or decays, we need to derive differential equations for u 1 1 and u 2 . The u 1 equation is du = dn since n ∗ is a constant and by substitudt dt du 1 dn ∗ las∗ + u 2 ). By using Taylor series expansion, tion dt = dt = f 1 (n + u 1 , F du 1 dn ∗ las∗ 1 = dt = f 1 (n + u 1 , F + u 2 ) = f 1 (n ∗ , F las∗ ) + u 1 · ∂∂nf1 + u 2 · ∂∂Fflas + dt ∂ f1 du 1 dn 2 2 ∗ las∗ O(u 1 , u 2 , u 1 · u 2 ) and since f 1 (n , F ) = 0 then dt = dt = u 1 · ∂n + 1 + O(u 21 , u 22 , u 1 · u 2 ). To simplify the notation, we have written ∂∂nf1 and u 2 · ∂∂Fflas ∂ f1 ; these partial derivatives are to be evaluated at the fixed point (n ∗ , F las∗ ); thus ∂ F las they are umbers, not functions. The shorthand notation O(u 21 , u 22 , u 1 · u 2 ) denotes quadratic term in u 1 and u 2 . Since u 1 and u 2 are small, these quadratic terms are 2 2 = u 1 · ∂∂nf2 + u 2 · ∂∂Fflas + O(u 21 , u 22 , u 1 · u 2 ) extremely small. Similarly we find du dt   las du 2 2 = d Fdt = u 1 · ∂∂nf2 + u 2 · ∂∂Fflas + O(u 21 , u 22 , u 1 · u 2 ) . Hence the disturbance dt ⎛ du ⎞ 1  d f1 d f1   u1 ⎜ dt ⎟ las · + quadratic terms. (u 1 , u 2 ) evolved according to ⎝ ⎠ = ddnf2 ddFf2 du 2 u2 dn d F las  dt  The matrix A =

d f1 d f1 dn d F las d f2 d f2 dn d F las

is called the Jacobian matrix at the fixed (n ∗ ,F las∗ )

⎛ du ⎞

1  d f1 d f1   u1 ⎜ dt ⎟ las point (n , F ). We obtain the linearized system ⎝ · . ⎠ = ddnf2 ddFf2 du 2 u2 dn d F las dt The linearized system gives a qualitatively correct picture of the phase portrait near (n ∗ , F las∗ ), as long as the fixed point for the linearized system is not one of the ∗

las∗

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

borderline cases (centers, degenerate nodes, stars, or non-isolated fixed points) [6, 7]. 1 d f1 d f1 = −σem · n = −F pump · σG S A − F las · σem − − 2 · α · n; dn τ d F las d f2 d f2 c 1 c = · (σem − σ ElasS A ) · n · la − = · F las · (σem − σ ElasS A ) · la ; dn l d F las l τc The eigenvalues of a matrix A are given by the characteristic equation det(A − λ · I ) = 0, where I is the identity matrix.

The characteristic equation becomes

We define for simplicity:ϒ1 = −F pump · σG S A − F las∗ · σem −

1 τ

− 2 · α · n∗

c c 1 · (σem − σ ElasS A ) · la ; ϒ3 = · (σem − σ ElasS A ) · n ∗ · la − l l τc   ϒ1 − λ −σem · n ∗ det = 0; (ϒ1 − λ) · (ϒ3 − λ) + ϒ2 · σem · n ∗ = 0 ϒ2 ϒ3 − λ

ϒ2 = F las∗ ·

λ2 − λ · (ϒ1 + ϒ3 ) + ϒ1 · ϒ3 + ϒ2 · σem · n ∗ = 0 trace(A) = ϒ1 + ϒ3 ; det(A) = ϒ1 · ϒ3 + σem · n ∗ · ϒ2 Then λ1 =

√

ϒ1 +ϒ3 +

λ2 =

(ϒ1 +ϒ3 )2 −4·(ϒ1 ·ϒ3 +σem ·n ∗ ·ϒ2 ) 2

ϒ1 + ϒ3 −

& (ϒ1 + ϒ3 )2 − 4 · (ϒ1 · ϒ3 + σem · n ∗ · ϒ2 ) 2

The solutions of the quadratic equation λ2 − trace(A) · λ + det(A) = 0 are λ1 , λ2 . The eigenvalues depend only on the trace and determinant of the matrix A. Classification of fixed points: If det(A) < 0, the eigenvalues are real and have opposite sign; hence the fixed point is a saddle point. If det(A) > 0, the eigenvalues

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

529

are either real with the same sign (nodes), or complex conjugate (spirals and centers). Nodes satisfy (trace(A))2 − 4 · det(A) > 0 and spirals satisfy (trace(A))2 − 4 · det(A) < 0. The parabola (trace(A))2 −4·det(A) = 0 is a borderline between nodes and spirals; star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by trace(A). When trace(A) < 0, both eigenvalues have negative real parts, so the fixed point is stable. Unstable spirals and nodes have trace(A) > 0. Neutrally stable centers line on the borderline trace(A) = 0, where the eigenvalues are purely imaginary. If det(A) = 0, at least one of the eigenvalues is zero. Then the origin is not an isolated fixed point. There is either a whole line of fixed points, or a plane of fixed points, if A = 0. Saddle points, nodes, and spirals are the major types of fixed points and they occur in large open regions of the (det(A), trace(A)) plane. Centers, stars, degenerate nodes, and non-isolated fixed points are borderline cases that occur along curves in the (det(A), trace(A)) plane. Stability analysis with delay parameters in time (τ1 , τ2 ): The four level schemes gives the balance equation for the population of the upper laser level (n) and since the processes influences formation is not ideal there is a delay in time τ1 for variable n, n(t) → n(t − τ1 ) and a delay in time τ2 for the photon flux at the laser wavelength F las , F las (t) → F las (t − τ2 ). The derivatives of the population of the upper laser level (n) and photon flux (F las ) do not effected by time delay parameters τ1 , τ2 . dn = F pump · σG S A · [Nt − n(t − τ1 )] dt − F las (t − τ2 ) · σem · n(t − τ1 ) n(t − τ1 ) − − α · n 2 (t − τ1 ) τ c d F las = · F las (t − τ2 ) · σem · n(t − τ1 ) · la dt l F las (t − τ2 ) c − − · F las (t − τ2 ) · σ ElasS A · n(t − τ1 ) · la τc l System fixed points (equilibrium points) are the same for the case of no delays (τ1 = 0, τ2 = 0) since at fixed points limt→∞ n(t − τ1 ) = n(t); t τ1 ⇒ t − τ1 ≈ t and limt→∞ F las (t − τ2 ) = F las (t); t τ2 ⇒ t − τ2 ≈ t. Stability analysis: The standard local stability analysis about any one of the equilibrium points of Crystalline Mid-infrared laser system consists in adding to its coordinated [n F las ] arbitrarily small increments of exponential term [n f las ] · eλ·t , and retaining the first order terms in n f las . The system of two homogeneous equations leads to a polynomial characteristic equation in the eigenvalue λ [8, 9]. The polynomial characteristic equation accepts by set of Crystalline Mid-infrared laser system equations. The system fixed values with arbitrarily small increments of exponential form [n f las ] · eλ·t are; j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc.

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We have three cases: (1) τ1 = τ ; τ2 = 0, (2) τ1 = 0; τ2 = τ , (3) τ1 = τ ; τ2 = τ . We analyze the stability for the first case τ1 = τ ; τ2 = 0. We get the following delay differential equations: dn = F pump · σG S A · [Nt − n(t − τ )] dt n(t − τ ) − F las (t) · σem · n(t − τ ) − − α · n 2 (t − τ ) τ c d F las = · F las (t) · σem · n(t − τ ) · la dt l F las (t) c − − · F las (t) · σ ElasS A · n(t − τ ) · la τc l n(t) = n ( j) + n · eλ·t ; F las (t) = F las( j) + f las · eλ·t ; n(t − τ ) = n ( j) + n · eλ·(t−τ ) dn(t) d F las (t) = n · λ · eλ·t ; = f las · λ · eλ·t dt dt Implement it in the first DDE (delay differential equation): n · λ · eλ·t = F pump · σG S A · [Nt − (n ( j) + n · eλ·(t−τ ) )] − (F las( j) + f las · eλ·t ) · σem · (n ( j) + n · eλ·(t−τ ) )   ( j) n + n · eλ·(t−τ ) − α · (n ( j) + n · eλ·(t−τ ) )2 − τ n · λ · eλ·t = F pump · σG S A · (Nt − n ( j) ) − F pump · σG S A · n · eλ·(t−τ ) − (F las( j) + f las · eλ·t ) · σem · n ( j) n ( j) n · eλ·(t−τ ) − τ τ − α · ([n ( j) ]2 + 2 · n ( j) · n · eλ·(t−τ ) + n 2 · eλ·(t−τ )·2 )

− (F las( j) + f las · eλ·t ) · σem · n · eλ·(t−τ ) −

n · λ · eλ·t = F pump · σG S A · (Nt − n ( j) ) − F pump · σG S A · n · eλ·(t−τ ) − F las( j) · σem · n ( j) − f las · σem · n ( j) · eλ·t − F las( j) · σem · n · eλ·(t−τ ) n ( j) n · eλ·(t−τ ) − − α · [n ( j) ]2 τ τ − 2 · α · n ( j) · n · eλ·(t−τ ) − α · n 2 · eλ·(t−τ )·2 − f las · n · eλ·t · σem · eλ·(t−τ ) −

We assume f las · n ≈ 0; n 2 ≈ 0 then

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

531

n · λ · eλ·t = F pump · σG S A · (Nt − n ( j) ) − F pump · σG S A · n · eλ·(t−τ ) − F las( j) · σem · n ( j) − f las · σem · n ( j) · eλ·t − F las( j) · σem · n · eλ·(t−τ ) n ( j) n · eλ·(t−τ ) − − α · [n ( j) ]2 τ τ − 2 · α · n ( j) · n · eλ·(t−τ ) −

n ( j) − α · [n ( j) ]2 τ · n ( j) · eλ·t

n · λ · eλ·t = F pump · σG S A · (Nt − n ( j) ) − F las( j) · σem · n ( j) − − F pump · σG S A · n · eλ·t · e−λ·τ − f las · σem − F las( j) · σem · n · eλ·t · e−λ·τ −

n · eλ·t · e−λ·τ − 2 · α · n ( j) · n · eλ·t · e−λ·τ τ ( j)

At fixed point: F pump ·σG S A ·(Nt −n ( j) )− F las( j) ·σem ·n ( j) − nτ −α ·[n ( j) ]2 = 0. n · λ = −F pump · σG S A · n · e−λ·τ − f las · σem · n ( j) − F las( j) · σem · n · e−λ·τ −  −F pump · σG S A − F las( j) · σem

n · e−λ·τ − 2 · α · n ( j) · n · e−λ·τ τ   1 − − 2 · α · n ( j) · e−λ·τ − λ τ

· n − f las · σem · n ( j) = 0 Implement it in the second DDE (delay differential equation): c · (F las( j) + f las · eλ·t ) · σem · (n ( j) + n · eλ·(t−τ ) ) · la l F las( j) + f las · eλ·t c − − · (F las( j) + f las · eλ·t ) τc l las ( j) λ·(t−τ ) ) · la · σ E S A · (n + n · e

f las · λ · eλ·t =

c · σem · la · (F las( j) + f las · eλ·t ) · (n ( j) + n · eλ·(t−τ ) ) l F las( j) f las · eλ·t − − τc τc c las − · σ E S A · la · (F las( j) + f las · eλ·t ) · (n ( j) + n · eλ·(t−τ ) ) l

f las · λ · eλ·t =

Hint: we develop the expression (F las( j) + f las · eλ·t ) · (n ( j) + n · eλ·(t−τ ) ) (F las( j) + f las · eλ·t ) · (n ( j) + n · eλ·(t−τ ) )

532

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

= F las( j) · n ( j) + F las( j) · n · eλ·(t−τ ) + n ( j) · f las · eλ·t + f las · n · eλ·t · eλ·(t−τ ) Assumption f las · n ≈ 0 (F las( j) + f las · eλ·t ) · (n ( j) + n · eλ·(t−τ ) ) = F las( j) · n ( j) + F las( j) · n · eλ·t · e−λ·τ + n ( j) · f las · eλ·t ### c · σem · la · (F las( j) · n ( j) + F las( j) · n · eλ·t · e−λ·τ + n ( j) · f las · eλ·t ) l F las( j) f las · eλ·t c − − − · σ ElasS A · la τc τc l

f las · λ · eλ·t =

· (F las( j) · n ( j) + F las( j) · n · eλ·t · e−λ·τ + n ( j) · f las · eλ·t )

c c · σem · la · F las( j) · n ( j) + · σem · la · F las( j) · n · eλ·t · e−λ·τ l l c F las( j) f las · eλ·t + · σem · la · n ( j) · f las · eλ·t − − l τc τc c c las las( j) ( j) las − · σ E S A · la · F · n − · σ E S A · la · F las( j) · n · eλ·t · e−λ·τ l l c − · σ ElasS A · la · n ( j) · f las · eλ·t l

f las · λ · eλ·t =

c F las( j) c − · σ ElasS A · la · F las( j) · n ( j) · σem · la · F las( j) · n ( j) − l τc l c las( j) λ·t −λ·τ + · σem · la · F ·n·e ·e l c f las · eλ·t + · σem · la · n ( j) · f las · eλ·t − l τc c − · σ ElasS A · la · F las( j) · n · eλ·t · e−λ·τ l c − · σ ElasS A · la · n ( j) · f las · eλ·t l

f las · λ · eλ·t =

At fixed point: cl · σem · la · F las( j) · n ( j) −

F las( j) τc

− cl · σ ElasS A · la · F las( j) · n ( j) = 0.

c · σem · la · F las( j) · n · eλ·t · e−λ·τ l c f las · eλ·t + · σem · la · n ( j) · f las · eλ·t − l τc c las las( j) λ·t −λ·τ − · σ E S A · la · F ·n·e ·e l

f las · λ · eλ·t =

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

533

c · σ ElasS A · la · n ( j) · f las · eλ·t l   c c 1 las las( j) −λ·τ las ( j) − σ E S A ) · · la · F ·e · n + (σem − σ E S A ) · · la · n − l l τc −

(σem

· f las − f las · λ = 0 We can summary our Crystalline Mid-infrared laser system arbitrarily small increments equations.  −F

pump

· σG S A − F

las( j)

· σem

  1 ( j) −λ·τ − −2·α·n −λ ·e τ

· n − f las · σem · n ( j) = 0 (σem −

σ ElasS A )

  c c 1 las( j) −λ·τ las ( j) · · la · F ·e · n + (σem − σ E S A ) · · la · n − l l τc

· f las − f las · λ = 0 The small increments Jacobian of our Crystalline Mid-infrared laser system is as below:

We define for simplicity: 1 = −F pump · σG S A − F las( j) · σem −

1 τ

− 2 · α · n ( j)

c c 1 · la · F las( j) ; 3 = (σem − σ ElasS A ) · · la · n ( j) − l l τc   1 · e−λ·τ − λ −σem · n ( j) (A − λ · I ) = ; det(A − λ · I ) = 0 2 · e−λ·τ 3 − λ

2 = (σem − σ ElasS A ) ·

det(A − λ · I ) = (1 · e−λ·τ − λ) · (3 − λ) + 2 · σem · n ( j) · e−λ·τ

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

det(A − λ · I ) = λ2 − 3 · λ + (1 · 3 + 2 · σem · n ( j) − 1 · λ) · e−λ·τ D(λ, τ ) = λ2 − 3 · λ + (1 · 3 + 2 · σem · n ( j) − 1 · λ) · e−λ·τ n = 2; m = 1; n > m D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ n = 2; m = 1; n > m The expression for Pn (λ, τ ): Pn=2 (λ, τ ) = Pn=2 (λ, τ ) =

2 

2 k=0

pk (τ ) · λk = λ2 − 3 · λ

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2

k=0

p0 (τ ) = 0; p1 (τ ) = −3 ; p2 (τ ) = 1 The expression for Q m (λ, τ ): Q m (λ, τ ) = −1 · λ + (1 · 3 + 2 · σem · n ( j) ) Q m=1 (λ, τ ) =

1 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ

k=0

q0 (τ ) = 1 · 3 + 2 · σem · n ( j) ; q1 (τ ) = −1 The homogenous system for n F las lead to a characteristic equation for τ ) + Q m=1 (λ, τ ) · e−λ·τ = 0; the eigenvalue 2λ having thej form Pn=2 (λ, 1 j P(λ, τ ) = j=0 a j (τ ) · λ ; Q(λ, τ ) = j=0 c j (τ ) · λ and the coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delay τ , qi , qk are any Crystalline Mid-infrared laser system’s parameters, other parameters keep as a constant. a0 = 0; a1 = −3 ; a2 = 1; c0 = 1 · 3 + 2 · σem · n ( j) ; c1 = −1 Unless strictly necessary, the designation of the variation arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0  = 0, 1 · 3 + 2 · σem · n ( j) = 0; ∀ qi , qk ∈ R+ , λ = 0 is not a P(λ, τ )+Q(λ, τ ) · e−λ·τ = 0. We assume that Pn=2 (λ, τ ),Q m=1 (λ, τ ) can’t have common imaginary roots. That is for any real number ω; ω ∈ R: Pn=2 (λ = i · ω, τ )  = 0; Q m=1 (λ, τ ) = 0. Pn=2 (λ = i · ω, τ ) = −ω2 − i · ω · 3 ; Q m=1 (λ, τ ) = −i · ω · 1 + (1 · 3 + 2 · σem · n ( j) )

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

535

Pn=2 (λ = i · ω, τ ) + Q m=1 (λ = i · ω, τ ) = −ω2 + (1 · 3 + 2 · σem · n ( j) ) − i · ω · (3 + 1 ) |P(i · ω, τ )|2 = ω4 + ω2 · 23 ; |Q(i · ω, τ )|2 = ω2 · 21 + (1 · 3 + 2 · σem · n ( j) )2

We need to find the expression: F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω4 + ω2 · 23 − ω2 · 21 − (1 · 3 + 2 · σem · n ( j) )2 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω4 + ω2 · (23 − 21 ) − (1 · 3 + 2 · σem · n ( j) )2 We define the following parameters: ϒ0 = −(1 · 3 + 2 · σem · n ( j) )2 ; ϒ2 = 23 − 21 ; ϒ4 = 1 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 =

2 

ϒ2·k · ω2·k = ϒ0 + ϒ2 · ω2 + ϒ4 · ω4

k=0

 Hence F(ω, τ ) = 0 implies 2k=0 ϒ2·k · ω2·k = 0 and its roots are given by solving the above polynomial. Furthermore, PR = −ω2 ; PI = −ω · 3 ; Q R = 1 · 3 + 2 · σem · n ( j) ; Q I = −ω · 1 . Hence sin θ (τ ) =

−PR (i · ω, τ ) · Q I (i · ω, τ ) + PI (i · ω, τ ) · Q R (i · ω, τ ) |Q(i · ω, τ )|2

cos θ (τ ) = −

PR (i · ω, τ ) · Q R (i · ω, τ ) + PI (i · ω, τ ) · Q I (i · ω, τ ) |Q(i · ω, τ )|2

sin θ (τ ) =

−ω3 · 1 − ω · 3 · (1 · 3 + 2 · σem · n ( j) ) ω2 · 21 + (1 · 3 + 2 · σem · n ( j) )2

cos θ (τ ) = −

−ω2 · (1 · 3 + 2 · σem · n ( j) ) + ω2 · 3 · 1 ω2 · 21 + (1 · 3 + 2 · σem · n ( j) )2

Above expressions are continuous and differentiable in τ based on Lemma 1.1. Hence we use Theorem 1.2 and this proves Theorem 1.3. We use different parameters terminology from our last characteristics parameters definition k → j; pk (τ ) → a j ; qk (τ ) → c j ; n = 2; m = 1; n > m additionally  Pn (λ, τ ) → P(λ, τ ) and Q m (λ, τ ) → Q(λ, τ ) then P(λ, τ ) = 2j=0 a j ·

536

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

λ j ; Q(λ, τ ) =

1 j=0

cj · λj.

P(λ) = λ2 − 3 · λ; Q(λ) = 1 · 3 + 2 · σem · n ( j) − 1 · λ n, m ∈ N0 ; n > m and a j , c j : R+0 → R are continuous and differentiable functions of τ such that a0 + c0 = 0. In the following “−” denotes complex and conjugate. P(λ) and Q(λ) are analytic functions in λ and differentiable in τ . & The coefficients {a j (F pump , σG S A , σem , τ, α, σ ElasS A , . . .) las pump , σG S A , σem , τ, α, σ E S A , . . .)} ∈ R depend on Crystalline Midc j (F infrared laser system parameters F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . values. Unless strictly necessary, the designation of the variation arguments, F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . system parameters will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0  = 0. 

1 · σG S A − F · σem − − 2 · α · n ( j) a0 + c0 = −F τ   c 1 · (σem − σ ElasS A ) · · la · n ( j) − l τc c las las( j) · σem · n ( j)  = 0 + (σem − σ E S A ) · · la · F l pump



las( j)

Furthermore, P(λ), Q(λ) are analytic function of λ for which the following requirements of the analysis (see Kuang 1993, Sect. 3.4) can also be verified in the present case [9, 10]. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω)  = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire las pump (F  , σG S A , σem , τ, α, σ E S A , τ , . . . system parameters) domain of interest.   is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. Indeed, (b)  Q(λ) P(λ)        1 ·3 +2 ·σem ·n ( j) −1 ·λ   = in the limit:  Q(λ)   . 2 P(λ) λ −3 ·λ

(c) F(ω) = |P(i ·ω)|2 −|Q(i ·ω)|2 =ω4 +ω2 ·(23 −21 )−(1 ·3 +2 ·σem ·n ( j) )2 has at most a finite number of zeros. Indeed, this is a polynomial in ω(degree in ω4 ). of (d) Each positive root ω(F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . .) F(ω) = 0 is continuous and differentiable with respect to F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . parameters. This condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus λ = i · ω ∀ ω > 0 may be on eigenvalue of characteristic equation. The analysis consists in identifying the roots of characteristic equation situated on the imaginary axis of the complex λ—plane, where

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

537

by increasing the parameters F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . .(system parameters), Reλ may, at the crossing, change its sign from (−) to (+), i.e. from a stable focus E k (n (k) , F las(k) ) ∀ k = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . parameters (ω ∈ R+ ). 

 ∂Reλ  (F )= ; σG S A , σem , τ, α, σ ElasS A , τ , · · · = const ∂ F pump λ=i·ω   ∂Reλ ; F pump , σem , τ, α, σ ElasS A , τ , · · · = const −1 (σG S A ) = ∂σG S A λ=i·ω   ∂Reλ ; F pump , σG S A , σem , τ, α, τ , · · · = const −1 (σ ElasS A ) = ∂σ ElasS A λ=i·ω   ∂Reλ −1 ; F pump , σG S A , σem , τ, α, σ ElasS A , · · · = const  (τ ) = ∂τ λ=i·ω −1

pump

where writing P(λ) = PR (λ)+i · PI (λ) and Q(λ) = Q R (λ)+i · Q I (λ), and inserting λ = i · ω into Crystalline Mid-infrared laser system’s characteristic equation ω must satisfy the following: sin(ω · τ ) = g(ω) =

−PR (i · ω) · Q I (i · ω) + PI (i · ω) · Q R (i · ω) |Q(i · ω)|2

cos(ω · τ ) = h(ω) = −

PR (i · ω) · Q R (i · ω) + PI (i · ω) · Q I (i · ω) |Q(i · ω)|2

where |Q(i · ω)|2 = 0 in the view of requirement (a) above, (g, h) ∈ R. Furthermore, it follows the above equations sin(ω · τ ) and cos(ω · τ ) that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. / I Note F(ω) is independent of τ . Now it is important to notice that if τ ∈ (assume that I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for / I ; ω(τ ) is not define. Then for all τ ∈ I is satisfies that F(ω, τ ) = 0. τ ∈ Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switch. For any τ ∈ I , where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π] as the solution of I (i·ω)·Q R (i·ω) I (i·ω)·Q I (i·ω) ; cos θ (τ ) = − PR (i·ω)·Q R (i·ω)+P sin θ (τ ) = −PR (i·ω)·Q I (i·ω)+P |Q(i·ω)|2 |Q(i·ω)|2 and the relation between the argument θ (τ ) and ω(τ ) · τ for τ ∈ I must be ω(τ ) · τ = θ (τ ) + 2 · n · π ∀ n ∈ N0 . Hence we can define the )+n·2·π ; n ∈ N0 , τ ∈ I . maps τ ,n : I → R+0 given by τ ,n (τ ) = θ(τω(τ ) Let introduce the functions I → R; Sn (τ ) = τ − τ ,n (τ ), n ∈ N0 , τ ∈ I , that are continuous and differentiable in τ . In the following, the subscripts λ, ω, F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . system parameters ( ω is related to

538

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

parameter in our stability analysis) indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . ∈ R+ ), ω(F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . ∈ R+ ) and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Crystalline Mid-infrared laser system characteristic equation P(λ)+ Q(λ)· e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates, x = F pump , σG S A , σem , τ, α, σ ElasS A , τ , . . . ∈ R+ , 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., Substituting λ = i · ω, and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω) and i · Q λ (i · ω) = Q ω (i · ω) that on the surface |P(i · ω)|2 = |Q(i · ω)|2 , one obtains, 

∂λ ∂x =

−1

|λ=i·ω

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2 Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 . When (x) can be a Crystalline Mid-infrared laser system parameters F pump , σG S A , σem , τ, α, σ ElasS A , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] and Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )]; ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ) We choose our specific parameter as time delay x = τ . Pn=2 (i · ω) = −ω2 − i · ω · 3 ; Q m=1 (i · ω) = −i · ω · 1 + (1 · 3 + 2 · σem · n ( j) ) PR (i · ω) = −ω2 ; PI (i · ω) = −ω · 3 ; Q R (i · ω) = 1 · 3 + 2 · σem · n ( j) ; Q I (i · ω) = −ω · 1 PRω (i · ω, τ ) = −2 · ω; PI ω (i · ω, τ ) = −3 ; Q Rω (i · ω, τ ) = 0; Q I ω (i · ω, τ ) = −1

5.3 Crystalline Mid-Infrared Laser Balance and Rate Equations …

539

PRω = PRω (i · ω, τ ); PI ω = PI ω (i · ω, τ ) Q Rω = Q Rω (i · ω, τ ); Q I ω = Q I ω (i · ω, τ ) PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; PRω · PR = 2 · ω3 PI ω · PI = ω · 23 ; Q Rω · Q R = 0; Q I ω · Q I = ω · 21 Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] Fτ = 0; ωτ = −  = 0 Fω PR · PI ω = ω2 · 3 ; PI · PRω = 3 · 2 · ω2 Q R · Q I ω = −(1 · 3 + 2 · σem · n ( j) ) · 1 Q I · Q Rω = 0; PR · PI x |x=τ = 0; PI · PRx |x=τ = 0 Q R · Q I x |x=τ = 0; Q I · Q Rx |x=τ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) ∂ω ∂ω Fτ = 0; Fω · + Fτ = 0; τ ∈ I ; =−  ∂τ ∂τ Fω   ∂Reλ −1 (τ ) = ; −1 (τ ) ∂τ λ=i·ω   −2 · [U + τ · |P|2 ] + i · Fω Fτ ∂ω = Re = ωτ = −  ; 2 Fτ + i · 2 · [V + ω · |P| ] ∂τ Fω    ∂Reλ sign{−1 (τ )} = sign ; sign{−1 (τ )} ∂τ λ=i·ω ! ∂ω U · ∂τ +V ∂ω  +ω+ = sign{Fω } · sign τ · ∂τ |P|2 We shall presently examine the possibility of stability transitions (bifurcations) in a Crystalline Mid-infrared laser system, about equilibrium points E k (n (k) , F las(k) ) ∀ k = 0, 1, 2 as a result of a variation of delay parameter τ . The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complexλ-plane where by increasing the delay parameter τ , Re λ may at the crossing, change its sign from “−” to “+”, i.e. from a stable focus E (∗) to an unstable one, or vice versa. This feature may be further assessed  byexamining the sign of the partial derivatives with respect to τ , −1  (τ ) = ∂Reλ and F pump , σG S A , σem , τ, α, σ ElasS A , . . . system parameters ∂τ λ=i·ω

540

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

are constant where ω ∈ R+ . We need to plot the stability switch diagram based on different delay values of our Crystalline Mid-infrared laser system. Since it is very complex function we recommend to solve it numerically rather than analytic [9, 10]. 





−2 · [U + τ · |P|2 ] + i · Fω ;  (τ ) = Re  (τ ) = Fτ + i · 2 · [V + ω · |P|2 ] λ=i·ω   ∂Reλ ; −1 (τ ) −1 (τ ) = ∂τ λ=i·ω −1

∂Reλ ∂τ

=

−1



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . ; cos θ (τ ) = . . ., when ω = ω+ (τ ) if only ω+ is feasible. Additionally when all Crystalline Mid-infrared laser system parameters are known and the stability switch due to various time delay values τ is describe in the following expression: sign{∧−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ ) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + } |P(ω(τ ))|2 Remark Lemma 1.1, Theorem 1.2 and Theorem 1.3 are described in Sect. 5.1.

5.4 Questions 1. We have a system of Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs. The saturable absorption in the GaAs wafer needs to apply the energy-level model of Valley and Smirl for energy transfer process in GaAs. In this model, schematically, the level responsible for absorption around 1um is believed to be EL2 defect that forms a deep level 0.82 eV below the band edge. This level has a total density N, part of which (N0+ ) is positively charged. Transitions from EL2 to the conduction band absorb optical energy and produce free electrons (n), while valence to EL2+ transistors produce free holes (p) and neutral EL2 donors. Two-photon absorption (TPA) generates free electrons and holes, whereas free-carrier absorption (FCA) promotes electrons higher into the conduction band and holes deeper into the valence band. Combining all these effects with the accompanying recombination process, we get the set of intracavity rate equations, n = n(t); p = p(t); N + = N + (t).

5.4 Questions

541

dn φ 2 (t − τ ) = 2 · φq · (N − N + ) · σe + B · − γet · n · N + − γeh · n · p dt 2 dp φ 2 (t − τ ) = 2 · φq · N + · σh + B · − γhd · p · (N − N + ) − γeh · n · p dt 2 dN+ = 2 · φq · [(N − N + ) · σe + N + · σh ] − γet · n · N + + γhd · p · (N − N + ) dt αq = σe · (N − N + ) + σh · N + + σ f c · n Which need to be solved simultaneously with the rate equations for photon density φ and population inversion density N x . Due to uncomplete of photon generation and interferences, there is a delay in time (τ ) for photon density φ(t) → φ(t − τ ). σ —is stimulated emission cross-section, Ntot —is the total density of active atoms, τ —is spontaneous fluorescence lifetime, γ —is the cavity round trip loss coefficient, L 0 —is the length of the laser rod, L c —is the length of the doubling crystal, L q —is the GaAs wafer thickness, T —is the cavity round trip, P—is the , n c —is the refractive index of the pump rate, η—is the plane wave impedance 377 nc doubling crystal, n q —is the refractive index of GaAs, d—the effective nonlinear coefficient, w0 —is the spot size at the laser rod, wc —is the spot size at the doubling crystal, wq —is the spot size at GaAs wafer, N—is the total EL2 density, N + —is the ionized EL2 (EL2+ ) density, N0+ —is the initial ionized EL2 (EL2+ ) density, γet — is the electron-EL2 recombination coefficient, γhd —is the hole-EL2 recombination coefficient, γeh —is the direct electron-EL2 recombination coefficient, σe —is the EL2 absorption cross section, σh is the EL2+ absorption cross section, σ f c —is the free-carrier absorption cross section, β—is the two-photon absorption coefficient.  2 Here φq = φ(t − τ ) · wwq0 · n1q is the photon density at the GaAs wafer and the coefficients γet , γhd and γeh describe electron—EL2+ , EL2-hole, and direct electron– hole recombination process, respectively. In αq = · · · equation n(≈ p) represents the number of generated electron–hole pairs. The model assumes that all the elements are homogeneously distributed throughout the cavity, which is valid under the condition that the intensity does not change substantially during a single round trip. Transverse beam shape effects including the refraction caused by free carriers in GaAs. Hint: Some system parameters are not explain in the question, see Sect. 5.1. 1.1 Find system fixed points. Draw fixed points coordinates as a function of σe (EL2 absorption cross section) and σh (EL2+ absorption cross section)—3D graph. 1.2 Discuss stability and stability switching for different values of τ parameter. 1.3 Parameter L q (length of the GaAs wafer) is very small (L q → ε), return (1.1) and (1.2). Explain how our results change. 1.4 Parameter L c (length of the doubling crystal) is very small (L c → ε), return (1.1) and (1.2). Explain how our results change. 1.5 Discuss stability and stability switching for different values of γhd parameter values (hole—EL2 recombination coefficient).

542

2.

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

The model presented the passively Q-switched lasers. By using the secondthreshold criterion and numerical fitting procedure, we get the analytical function of output pulse energy. An analytical expression for the optimal output reflectivity maximized the output pulse energy of a passively Q-switched laser with a given initial transmission of the saturable absorber. The analysis of the passively Q-switched laser used the variable z, z = L1 · 2 · σ · n i · l, where n i is the initial population density in the gain medium, σ is the stimulated emission cross section of the gain medium, l is the length of the gain medium, L is the non saturable intra cavity round-trip dissipative optical loss. In the cavity Qswitched laser, n i is proportional to the pump rate. We characterize the output pulse energy of the cavity Q-switch laser with the variable z. To is the initial transmission of the saturable absorber, and R is the reflectivity of the output mirror. The second threshold of passively Q-switches laser is accepted when the photon density turns upward and produces a giant pulse. The gain can saturate first and the photon density never turns upward to develop a giant pulse. The model takes the assumptions: uniform pumping of the gain medium, the intra cavity optical intensity as axially uniform, and the complete recovery of the saturable absorber. The coupled rate equations describe the model of a passively Q-switched laser. The coupled equations for three or four level gain media are as follow:

φ(t − τ )  dφ · 2 · σ · n(t) · l − 2 · σgs · n gs (t) · ls − 2 · σes · n es · ls = dt tr     1 − ln +L R dn gs dn = −γ · c · σ · φ(t − τ ) · n(t); dt dt A · c · σgs · φ(t − τ ) · n gs (t); n gs + n es = n so =− As φ(t) = φ; n(t) = n; n gs = n gs (t). The parameters table is described below. System parameter Description φ

Intra cavity photon density with respect to the effective cross-section area of the laser beam in the gain medium

n

Population density of the gain medium

ls

Length of the saturable absorber

A As

Ratio of the effective area in the gain medium and in the saturable absorber

n gs

Absorber ground

n es

Excited state

n so

Total population densities (continued)

5.4 Questions

543

(continued) System parameter Description σgs , σes

GSA and ESA cross sections in the saturable absorber, respectively

R

Reflectivity of the output mirror

γ

Inversion reduction factor (γ = 1 and γ = 2 correspond to, respectively, four-level and three-level systems

tr =

2·l  c

Round trip transit time of light in the cavity optical length l  , where c is the speed of light

The passively Q-switched laser model is not ideal, and there is a shift in time in the interactivity photon density variable, φ(t) → φ(t − τ ). It is not affect the derivative . τ is the delay parameter which can affect the dynamic of the system. of φ(t), dφ dt 2.1 Find system fixed points, How the results are changed if τ = 0? 2.2 Discuss stability and stability switching for different values of τ parameter. 2.3 The reflectivity of the output mirror system parameter (R) is double in his value (R → 2 · R), How it effect the behavior of the system? 2.4 There is a situation where, R → 1, How the system dynamic change? Find fixed points and discuss stability, stability switching. What happened when R = 1 ? 2.5 Discuss the system stability switching for different values of the ratio of the effective area, AAs . 3. We get for the passively Q-switched   lasers model (question 2), for four-level α

σ

saturable absorber. n gs = n so · nni ; α = γ1 · σgs · AAs and n i is determined from the condition that the round-trip gain is exactly equal to theround-trip

 losses before the Q-switch opens. Thus 2 · σ · n i ·l − 2 · σgs · n so ·ls − ln R1 + L = 0. The parallel between the two analyses is done by T0 = e−σgs ·n so ·ls , where T0 is the initial   transmission of the saturable absorber. We define n i = f (T0 ), ln 12 +ln( R1 )+L T0 and the differential equation: ni = 2·σ ·l

   α−1 dφ (1 − β) 1 l n · 1 − · ln =− · dn γ · l 2 · σ · l · ni ni T02   ⎤

 β · ln T12 + ln R1 + L 0 ⎦; β = σes − 2·σ ·l ·n σgs

The first derivative of φ with respective to n at n = n i is equal to zero, the criterion for Q-switching behavior is whether the second derivative of φ with respective to n@ n = n i has a positive or a negative sign. If positive, the growth curve for the photon intensity will turn increasingly upward. We take account of the influence of excited-state absorption and intra cavity focusing.

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

3.1 Plot the function between dφ and T0 parameter, dφ = ξ(T0 ), find extreme points dn dn and explore the behavior of the function. 2 3.2 Find the expression for ddnφ2 and find by substitution n = n i , the criterion for a giant pulse to occur and the criterion for the second threshold. 3.3 Discuss the stability of our system for different values of n i parameter. 2 3.4 In the case β = 0; AAs = ;  ∈ R+ , plot the function between ddnφ2 and   2  parameter ddnφ2 = ζ () . Discuss stability and stability switching for different values of  parameter. 3.5 The parameters α and β are determined from the cavity configuration and the physical properties of the gain medium and the saturable absorber. The initial transmission of the saturable absorber has an upper bound T0−upper for

 producing a giant pulse for a given ln RI + L, a given α, and a given value of β, " # ln 1 +L T = exp − ( R ) . On the other hand, the reflectivity of the output 0−upper

2·[α·(1−β)−1]

coupler has a lower bound Rupper to build up " a giant pulse for a given  L,a given #

T0 , a given α, and a given β, Rlower = exp −[α · (1 − β) − 1] · ln T12 + L . 0 The parameters T0−upper and Rlower are used to express the output pulse energy and optimum output reflectivity as an analytical function, respectively. Express 2 T0 by Rlower and R by T0−upper in the ddnφ2 = ζ (), second order differential equation. Discuss stability and stability switching for different values of T0−upper and Rlower parameters. 4. We have a device of semiconductor saturable absorber mirror (SESAM). The measured nonlinear reflectivity is used to fit a model function to obtain the key parameters of the saturable absorber. The model function is R(F p ) and this model function describes the nonlinear behavior of real SESAMs. The two level system differential equation is discussed, when the reflectivity is given by where, c R is established by using a dielectric high-reflector R(F p ) = c R · VVout IN (R = 100%) as reference. Parameter c R is a constant, and using a constant c R and plotting R(F p ) of the high-reflector does not show 100% flat curve. A real absorber device shows non-saturable losses (the device not reach 100% reflectivity) even for arbitrarily high fluences. The losses are homogeneously distributed over the absorber layer or transmission losses. These losses are accounted by a scaling factor Rns . " # R   ln 1+ Rlin ·(e S −1) Fp ns R(F p ) = Rns · , where S is the saturation parameter S = , S Fsat Tlin is the linear transmission, Fsat is the saturation fluence. We can replace Tlin by Rlin in R(F p ) equation. At higher fluences the reflectivity decreases with increasing fluence according to a second order process which leads to a roll-off in the reflectivity curve. The roll-off is taken into account by multiplying the model function with " # 

−

Fp



ln 1+

Rlin R

·(e S −1)



−

Fp



ns · e F2 . A smaller F2 value is e F2 which gives R(F p ) = Rns · S is the fluence where the SESAM reflectivity corresponded to a stronger roll-off. F 2

 is dropped to 37% 1e due to induced absorption. We define a2 as the material

5.4 Questions

545 

− 2·r

2



parameter. For a Gaussian laser beam F pGauss (r ) = F0 · e ω2 , where F0 is the peak

 fluence and ω is the e12 beam radius. A Gaussian laser beam is focused to a small spot spreads out rapidly as it propagates away from the spot. The laser beam is kept collimated by larger diameter. The Gaussian laser is propagated in z direction   beam & (z – axis), r is the radial distance from the center r = x 2 + y 2 . The peak fluence in a Gaussian beam is 2· F p which gives the outcome that saturation already occurs at lower fluences. We define R Flat T op (χ ), where it is a transformation for any function R Gauss (F p ) and R Flat T op (F p ) as long as lateral diffusion is neglected.

R Flat T op (χ ) = Rns ·  χ is F p variable S =

χ Fsat

" ln 1 +

Rlin Rns

 χ # · e Fsat − 1 χ Fsat



·e

 − Fχ , 2

 .

4.1 Get the expression for d R dχ (χ ) , and plot the graph d R dχ (χ ) versus χ. 4.2 χ = 0 + 1 · R Flat T op (χ ) + 2 · [R Flat T op (χ )]2 ; 0 , 1 , 2 ∈ R+ . Find Flat T op fixed points of the system d R dχ (χ ) = ξ(R Flat T op (χ )) and plot fixed points coordinates as a function of Fsat parameter. 4.3 Discuss the stability and stability switching for different values of Fsat parameter. 4.4 Discuss stability and stability switching for different values of 0 , 1 , 2 parameters. variable 4.5 There is a shifting in χ for R Flat T op

Flat T op  d R Flat T op (χ ) Flat T op R (χ ) → R (χ − χ ) , it does not affect . Discuss dχ stability and stability switching for different values of χ parameter. 5. We have a passively Q-switched laser system. The system is characterized by rate equations, the intra cavity photon density and the initial population inversion density in the diode-pumped passively Q-switched laser. We assume that the rate equations are characterized by Gaussian spatial distributions. We assume that the intra cavity photon density is a Gaussian spatial distribution during the entire formatting process of the Laser-Diode (LD) pumped passively Qswitched laser pulse. The population-inversion density at t = 0 is a Gaussian spatial distribution, and the ground-state population density of the saturable absorber at t = 0 is uniform. The intra cavity photon density φ(r, t) for TEM00  Flat T op

Flat T op

2

− 2·r2

mode is φ(r, t) = φ(0, t) · e ωL , where r is the radial coordinate, ω L is the laser-mode radius, which determined by the geometry of the resonator. r = 0 is the laser axis, and φ(0, t) is the photon density in the laser axis. The differential is integrated over the beam cross section to guarantee equation describing dφ(r,t) dt the beam Gaussian spatial distribution during the entire formatting process of the Q-switched pulse.

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5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

dn sg (r, t) dn(r, t) = −γ · σ · c · φ(r, t) · n(r, t); dt dt Sg = − · σgsa · c · φ(r, t) · n sg (r, t) Ss where, n(r, t) is the population-inversion density, n sg (r, t) and n s0 are, respectively, the ground-state and total population densities of the saturable absorber. σ and l are, respectively, the simulated-emission cross section and length of the gain medium. σgsa and σasa are, respectively, the ground-state and excited-state absorption cross  sections of the saturable absorber. ls is the length of the saturable absorber, tr = 2·lc is the round-trip transit time of light in the resonator of optical length l  , c is the light speed in vacuum. R is the reflectivity of the output mirror, L is the remaining round-trip dissipative optical loss; Sg and Ss are, respectively, the beam cross-section area in the gain medium and in the saturable absorber; γ is the inversion-reduction factor which corresponds to the net reduction in the population inversion resulting from the stimulated emission of a single photon. We neglect the pumping and spontaneous emission during the pulse formation. The solid-state saturable absorbers are of slow recovery then the ground-state recovery of the saturable absorber during the formation of the Q-switched pulse is neglected. The pump-light intensity in the gain medium is attenuated along the longitudinal axis z, and n(r, t) is a function of z. We K (r, t); K > 0; K ∈ choose the radial coordinate, r as a function of n sg (r, t), r = n sg 

−

K (r,t) 2·n sg ω2L



N then φ(r, t) = φ(0, t) · e ; t = t (n sg , K ); φ(0, t) ∈ R+ is a constant in our analysis and for specific r value (r = r0 ), we can write our rate equations: dn sg (r0 , t) dn(r0 , t) = −γ · σ · c · φ(r0 , t) · n(r0 , t); dt dt Sg = − · σgsa · c · φ(r0 , t) · n sg (r0 , t) Ss 

φ(r0 , n sg ) = φ(0, t) · e

−

K (r ,t) 2·n sg 0 ω2L



; t = t (n sg , K ); φ(0, t) ∈ R+

5.1 Find system fixed points and discuss stability. 5.2 There is a delay in time only in n sg (r0 , t) variable, n sg (r0 , t)→ n sg (r0 , t − τ ), dn (r ,t)

sg 0 but it is not affect the derivative of n sg (r0 , t) in time . Discuss stability dt and stability switching for different values of τ parameter. 5.3 How the system dynamic is changed if we change K parameter value from K = 1 to K > 1? refer to (5.2). Discuss stability for two cases, (1) K = 1, (2) K > 1.

5.4 Questions

547

5.4 There is a delay in time only in n(r0 , t) variable,  n(r0 , t) → n(r0 , t − ), but it is not affect the derivative of n(r0 , t) in time dn(rdt0 ,t) . Discuss stability and stability switching for different values of  parameter. 5.5 How the system dynamic is changed if we change K parameter value from K > 4 to K = 2 ? refer to (5.4). Discuss stability for two cases, (1) K > 4, (2) K = 2. 6. We have a system of passively Q-switched microchip laser at specific wavelength (λ). It produces a laser beam in the eye-safe wavelength regime that produces peak power at specific pulse durations and repetition rate. An Er:Yb:glass microchip laser is passively Q-switched with a semiconductor saturable absorber mirror (SESAM). The reflectivity of a SESAM as a function of the incident pulse fluence is very important. The modulation depth R is defined as the maximum reflectivity change between low and high intensity. Non saturable

losses are the remaining losses at high intensity. The fluence which switch 1e of the saturable losses is the saturation fluence Fsat . τp When the recovery time τ A of the absorber is larger than the pulse duration  and smaller than the pulse-to-pulse duration fr1ep , fr1ep > τ A > τ p , then the dynamic response of the absorber is not dominant in the Q switching. Shorter recovery time than the pulse duration reduces the efficiency of the laser. One system SESAM design for Q switching with pulse energies is the high-finesse anti resonant Fabry–Perot saturable absorber. The semiconductor absorber is embedded in an anti-resonant cavity between a semiconductor bottom Bragg mirror of high reflectivity and a sputtered dielectric top reflector with a transmittance Tt . The modulation depth of the AR-coated sample R A R and the non saturable losses lns,A R , the reflectivity in the saturated state Rns,A R = 1 −lns,A R . The transmittance of the top reflector Tt , tunes adjust the intensity that enters to the absorber. Decreasing the transmittance Tt reduces the intensity on the absorber. It increases the damage fluence and reduces the non saturable losses. Other effect is decreasing the modulation depth and increasing the saturation fluence. The fluence on the absorber is Fabs , (Fabs = ξ · FA R ), where FA R is the fluence in the AR-coated reference sample. The low fluences in anti-resonance, t #2 . In saturated condition we get ξ fraction is define as ξlow = "  T1−T 1+

t Rns,A R −R A R

expression for ξhigh ; which is ξhigh =

√

{1+

Tt . For small non saturable (1−Tt )·Rns,A R }2

losses lns,A R  1 (Rns,A R ≈ 1) the modulation depth is scaling with the same factor as the fluence for low intensities, R ≈ ξlow · R A R where R is the modulation depth of the sample with the top reflector. 6.1 Plot the graphs of two functions, ξlow = g1 (Tt ); ξhigh = g2 (Tt ). Investigate the behavior. Are there any intersection points between the two graphs? If yes find them (consider that Rns,A R , R A R are constants and Rns,A R  = R A R ). 6.2 How the graphs (6.1) change for small non saturable losses lns,A R  1, (Rns,A R ≈ 1)? Find if exist intersection points between two graphs.

548

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

 6.3 Experimental results show that ξhigh = ξlow −1;  > 1;  ∈ N, find the possible values Tt for specific Rns,A R , R A R parameters values (Rns,A R  = R A R ). How your results change for different values of  parameter? Plot the graph of Tt versus . t as a function of Tt , Rns,A R , R A R , and . Find fixed 6.4 Find the expression for dT dt points and explore the dynamic of the system. 6.5 For (6.4) discuss stability and stability switching for different values of  parameter. 7. We have two SESAMs that have different transmittance Tt of the top reflector and the same modulation depth of the final structure. The SESAM with higher top-mirror reflectivity has a thicker absorber. The resulting laser performance is the same for SESAMs are fully saturated. The lower transmittance Tt reduces the pulse fluence on the absorber and increases the damage threshold. Fd is ξ E · F p , where F p = A Lp then the damage fluence of the absorber. Fabs = Thigh OC ξ E E √ Tt Fabs = Thigh · A p , Fabs = · A p . F p is the pulse fluence 2 OC

TOC ·{1+

L

(1−Tt )·Rns,A R }

L

outside the cavity. The transmittance of the output coupler, TOC is related when the SESAM is exposed to the intra cavity intensity.  The pulse fluence, the TOC L pulse energy E p , E p = 2 · A L · FL · R · TOC +l par and FL = 2·σ h·v abs L +2·σ L is the saturation fluence of the gain material, v L is the laser frequency, σ L and σ Labs are the cross section for stimulated emission and re absorption at the is for the output coupling laser wavelength, respectively. The fraction (TOCTOC +l par

, where n is the index of efficiency. The pulse duration is given by τ p = 7·n·L c·R refraction of the gain material, L is the cavity length, and c is the speed of light . The fluence on the absorber layer in vacuum. The cavity round trip time is 2·n·L c (Rns,A R − R A R = 0) is Fabs = 2 · FL · R A R · ·

{1 +

&

1 (TOC + l par )

Tt2 & . (1 − Tt ) · Rns,A R }2 · {1 + (1 − Tt ) · (Rns,A R − R A R )}2

The condition Fabs < Fd avoids damage, keeping the fluence on the absorber below the damage fluence. l par is the parasitic losses, it is small compared with the transmittance of the output coupler and then can be neglected. The condition for the 

·TOC . transmittance of the top reflector to prevent the SESAM damage is Tt < 2 · FFLd·R AR The way to achieve high modulation depth and avoiding damage for a Q switched microchip laser is by growing the semiconductor part of the SESAM with higher modulation depth which is limited by the critical thickness for strained layers and for lattice match growth. The non saturable losses reduce the modulation depth with increasing of the absorber thickness.

7.1 Plot the graph of Fabs as a function of Tt (transmittance), Fabs = ψ1 (Tt ) where all other parameters are constants. Explore the behavior of the function.

5.4 Questions

549

7.2 The transmittance of the output coupler TOC is dependent on Tt (transmittance), & TOC = Tt2 + ;  ∈ R+ . Plot the graph Fabs |T =√T 2 +;∈R = ψ2 (Tt ) and OC + t discuss the behavior of the graph for different values of  parameter (all other parameters are constants). K (t); K ∈ N+ . Discuss fixed 7.3 Find the expression for d Fdtabs , where Tt (t) = Fabs points, How fixed points behave for different values of K parameter? 7.4 Discuss stability and stability switching (7.3) for different values of K parameter. 7.4 There is a delay in time, 1 for variable Fabs (t), Fabs (t) → Fabs (t − 1 ) which do not affect the derivative of d Fdtabs . Discuss stability and stability switching for different values of 1 parameter. 8. We have ion-doped crystalline laser which operates in the mid-IR spectral range. The bandwidth of the gain medium is the key parameter for establishing the operation threshold.  The full  width at half maximum is relative to the gain λ v maximum at λ0 ; λ0 ≈ v0 . We define parameter n ocp as the number of optical cycles per pulse, and it is especially for ultra-short pulse generation, n ocp =  −1 λ . We consider a system continuous-wave operation and assume fourλ0 level scheme, balance equation for the population of the upper laser level n: dn = F pump · σG S A · [Nt − n(t)] − F las (t) · σem · n(t) − n(t) − α · n 2 (t). dt τ The rate equation for the photon flux at the laser wavelength: las d F las = cl · F las (t)·σem ·n(t)·la − F τc(t) − cl · F las (t)·σ ElasS A ·n(t)·la . We define the dt   las pump σ las intrinsic slop efficiency as η0 , where η0 = λλlas · 1 − σEemS A . λλpump is the stokes shift and we consider it as a constant,  =

λlas λ pump

 → ε, then we can define η0 = discuss in this chapter.

1 

8.1 8.2 8.3

8.4 8.5 9.

with very small variation,  →  ± ,   σ las · 1 − σEemS A . Hint: All parameters are as

Express σ ElasS A as a function of η0 ,  (3D graph). Find fixed points and discuss the behavior of fixed points for different values of  parameter. What happened for  →  ± ;  ∈ R+ ; 0 <  < 1? Investigate the behavior of the system for different values of  parameter. There is a shift in time for F las (photon flux at the laser wavelength), F las (t) → las F las (t − ) but it is not affect the derivative in time of F las (t); d Fdt . Discuss the stability and stability switching for different values of  parameter. What happened (8.3), if  parameter double his value,  → 2 · ? How the behavior of the system is changed? √ What happened (8.3), if  → 3 ? How the stability map is changed? We have crystalline and fiber Raman laser system which based on the Simulated Raman Scattering (SRS) effect in crystals and silica-based fibers. The laser system operates in the CW, nano second, and picosecond regimes with low and high repetition rates. The spontaneous Raman scattering (RS) process is a two-photon resonance, when the difference of the absorption (v L ) and emission (v S ) optical frequencies is equal to the frequency of an atomic or molecular vibration in Raman media v R = v L − v S . Due to the two-photon

550

9.1

9.2

9.3 9.4

9.5

5 Nd:YAG, Mid-Infrared and Q-Switched Microchip …

nature, RS cross-sections are very small, σ ≈ 10−30 cm2 . The intensity of stimulated Raman scattering is proportional to the occupation densities of the states of three interacting fields: laser pump photons (N L ), photons of the generated stokes wave (N S ), and crystal phonons (molecular vibrations) in the ground (N g ) or excited (Ne ) state (N = N g − Ne ). The rate equation is d NS ∼ N L · N S ·(N g − Ne ). In the steady-state regime, when the pump duration dz t p is much longer than the Raman mode dephasing time TR , (t p TR ), the intensity Is of the stimulated Raman scattering (SRS) stokes beam passing through the Raman medium with length l is given by Is (l) = Is (0) · e[g·I L ·l] , where I L is the intensity of the incident pump laser beam and g is the gain

dσ  λ L ·λ2s ·N coefficient g = π ·c··n 2 ·v · d . λ L and λs are the wavelengths of the pump R s laser beam and the stokes Raman beam, respectively;  c is the speed of light in

h , n s is the refractive index vacuum,  is the reduced Planck constant  = 2·π is the full width at half maximum of the Raman at the stokes wavelength; v R 1 units; and  is the solid scattering angle. The gain is very spectral line in sec high for materials with a high density of scatters N, large Raman scattering dσ , and small Raman linewidth v R . integral cross-section d The crystalline and fiber Raman laser system is not ideal, then there is a shift in z coordinate for photons of the generated stokes

N S  wave (N S ), N S (z) → . Discover the dynamic N S (z − z) but not affect the derivative of N S ; ddz of the system (fixed points, stability, and stability switching for different values of z parameter). Hint: consider that N L , N g and Ne are constants with some approximation. of (N S ), photons of the generated The excited (Ne ) state is a function & √ stokes wave, Ne = ξ1 (N S ) = 1 · N S + 2 · N S ; 1 , 2 ∈ R+ and there is a shift in z parameter (z → z + z) as discuss in (9.1). The NS is d NdzS (z) ∼ N L · N S (z − z)·(N g − dynamic differential equation for ddz & √ 1 · N S (z − z) + 2 · N S (z − z)). Find system fixed points and plot 3D graph of fixed points coordinates as a function of 1 , 2 parameters. Hint: consider N g and N L as a constants. Discuss stability and stability switching (9.2) for different values of z parameter. √ The laser pump photons (N L ) is a function of N S , N L = ξ2 (N S ) = 3 N S + 1, NS ∼ ξ2 (N S (z − z)) · N S (z − z) · (N g − Ne ) and there is a shift in then ddz z parameter (z → z + z) as discuss in (9.1). Find system fixed points, and discuss stability and stability switching for different values of z parameter. Hint: consider N g and Ne as a constants.

dσ  λ L ·λ2s ·N  g is the system gain coefficient g = π ·c··n 2 ·v · d , g = σ ( −  );  ∈ R s   −1 λ L ·λ2s ·N dσ = π ·c··n · σ  ( −  );  > 0. Find fixed N;  > 1 then d 2 ·v R s points and plot fixed points coordinates as a function of  parameter. Discuss stability and stability switching for different values of  parameter. How the stability map is changed if we increase the value of  parameter?

5.4 Questions

10.

551

A crystalline Raman laser is based on self-Raman laser materials. It is a passive Q-switched intra-cavity Raman laser, which emits a yellow orange light beam. The laser frequency conversion is based on stimulated Raman scattering (SRS) in crystalline materials. Crystalline Raman laser generates both high average power (5 W) and high pulse energies ( 0). 10.5 What happened (10.4), if q parameter doubles his values q → 2 · q ? How the stability map is changed?

References 1. T.T. Kajava, A.L. Gaeta, Intra-cavity frequency-doubling of a Nd:YAG laser passively Qswitched with GaAs. Opt. Commun. 137, 93–97 (1997) 2. R. Zhou, W. Wen, Z. Cai, X. Ding, P. Wang, J. Yao, Efficient stable simultaneous CW dualwavelength diode-end-pumped Nd:YAG laser operating at 1.319 and 1.338 μm. Chin. Opt. Lett. 3(10), (2005) 3. R.C. Eckardt, R.L. Byer, Measurement of nonlinear optical coefficients by phase-matched harmonic generation, in SPIE Vol. 1561 Inorganic Crystals for Optics, Electro-Optics, and Frequency Conversion (1991). 4. J. Gu, S.-C. Tam et al., Novel use of GaAs as a passive Q-switch as well as an output coupler for diode-pumped infrared solid-state lasers, Proc. SPIE 3929, Solid State Lasers IX, 2000. 5. W. Su-Mei, Z. Qiu-Lin et al., Diode-pumped passive Q-switched 946nm Nd:YAG laser with a GaAs saturable absorber. Chin. Phys Lett. 23(3), 619 (2006) 6. J.K. Hale, Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3 7. S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press) 8. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993). 9. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAMJ. Math. Anal. 33, 1144–1165 (2002) 10. J. Kuang, Y. Cong, Stability of Numerical Methods for Delay Differential Equations (Elsevier Science, 2007) 11. M. Haiml, R. Grange, U. Keller, Optical characterization of semiconductor saturable absorbers. Appl. Phys. B 79, 331–339 (2004) 12. G.J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, U. Keller, Experimentally confirmed design guidelines for passive Q-switched microchip lasers using semiconductor saturable absorbers. J. Opt. Soc. Am. B/ 16(3), (1999) 13. R. Haring, R. Paschotta, R. Fluck, E. Gini, H. Melchior, U. Keller, Passively Q-switched microchip laser at 1.5 um. J. Opt. Soc. Am. B 18(12) (2001) 14. X. Zhang, S. Zhao, Q. Wang, Modeling of passively Q-switched lasers. J. Opt. Soc. Am. B/ 17(7) (2000). 15. M.C. Stumpf, S.C. Zeller, A. Schlatter, T. Okuno, T. Sudmeyer, U. Keller, Compact Er:Yb:glasslaser-based supercontinuum source for high-resolution optical coherence tomography. Opt. Express 16(14), 10572–11057 (2008) 16. I.T. Sorokina, Crystalline Mid-infrared laser. Topics Appl. Phys. 89, 255–351 (2003)

Chapter 6

Gas Laser Systems Stability Analysis Under Parameters Variation

Lasing from molecular nitrogen is used in many scientific and industrial applications. The discharge pumped nitrogen laser, operating in a broad range of gas pressures, from several mill bars to the atmospheric pressure, and repetition rates from several hertz to several kilo hertz. It is robust source of high-power near-UV radiation. Achieving nitrogen lasing via remote excitation would pave the way to many potential applications. It is remotely initiated lasing from molecular gases by femtosecond filaments. A mid infrared femtosecond laser filament can introduce backward-directed lasing of molecular nitrogen via a resonant excitation transfer mechanism. The characterization of the filament ignited nitrogen laser, including the spatial and temporal properties of the generated UV emission and generation thresholds. A filamentassisted nitrogen laser is efficient as its conventional discharge-pumped counterpart. Mid-IR ultrashort laser pulses radically enhance fila mentation-assisted lasing of N2 relative to ultrashort pulses in the near-IR. We need a full analysis of multiple fila mentation. Another area is a remotely pumped N2 laser in the atmosphere and optimization of the pumping via electron-N2 collisions and evolving long-pulse light or microwave sources in combination with the femtosecond filament. Fila mentation is modeled by numerical solving the cylindrically symmetric nonlinear Schrodinger equation, which accounts for the impact of plasma dispersion and refraction, beam diffraction, Kerr, Raman, and plasma nonlinearities. We describe a change of concentration of different neutral and ionic atomic and molecular species in the filament s] = [G s ]−[L s ]; Ns is the density of plasma by the following set of rate equations: d[N dt species of type-s; G s and L s are the relevant generation and loss rates. The rate equations are solved jointly with the equations for the electron temperature and vibrational temperature of ground-state nitrogen molecules and stability is inspected. To include the lase effects, the set of plasma kinetic equations is inspected by the rate equations for the population of the lase levels and number of generated photons, stability analysis is done. The Quasi-two level analytic model described the end pumped Alkali metal vapor laser. We consider the steady state rate equations for the number densities of the, 2 s1/2 , 2 p3/2 and 2 p1/2 , energy states for the three level laser system. We consider that the relaxation between the two upper levels, 2 p3/2 and 2 p1/2 , caused by © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_6

553

554

6 Gas Laser Systems Stability Analysis …

collisions with additive ethane is much faster (infinity fast), by comparison with any other process in the system including stimulated emission. We get the ratio of the number densities for the upper two levels, 2 p3/2 and 2 p1/2 , and its statistical equilibrium value and the mathematical description. A quasi-two level system is from which an analytic solution is extracted. The analytic model description gives expressions for the threshold pump power and the slope efficiency including intra-cavity losses. The model compares with the steady state three level models. The system of coupled differential rate equations describes the kinetics and radiation process in the system at steady state and stability analysis is done under parameters variation. A high repetition rate copper vapor lasers can be describe by a self-consistent model. We simulate the equations for the discharge pulse, laser pulse, and inter pulse afterglow period. It simultaneously integrated over many discharge cycles until a consistent solution is obtained. By modeling both an ideal and real laser we get to conclusion that the peak electron temperature and the initial density of metastable copper are factors which dominate the performance of the laser. We can describe the laser by the processes which affect these two quantities. High repetition rate (HRR) electric discharge metal vapor lasers are a source of high peak and high average power in the visible. The most sophisticated HRR copper lasers are resonantly charged electric discharge devices. These lasers are self-heated; the energy required to heat the laser tube and vaporize to copper comes from the discharge itself. Tube temperature and the copper vapor pressure are established by the insulation surrounding the discharge tube. All for   a given repetition rate, buffer gas pressure and discharge pulse energy 21 · C · V 2 . The copper laser is a cyclic or self-terminating laser. The upper laser level is resonant to the ground state while the lower laser level is metastable. The discharge circuit for the real laser simulates the actual circuit used for the laser. The discharge circuitry is inspected for stability under parameters variation. The gas temperature relaxes to the wall temperature with a time constant proportional to the radius of the discharge tube. We derive the time rate of change of Pg and Pcu inspect the stability under parameters variation. Parameters Tg , Pcu and Pg are thermodynamic quantities during the operation of the lasers, the temperature of the discharge tube, which determines the copper vapor pressure, can be measured by an optical pyrometer. If the gas density stays in equilibrium with the local temperature and pressure, we derive the related equations for the change in gas density due to changes in temperature and pressure and inspect the dynamic.

6.1 Nitrogen Gas Laser Filament Plasma Kinetic Equations Stability Analysis Under Parameters Variation Nitrogen is a chemical element with symbol N and atomic number 7. Nitrogen is the lightest member of group 15 of the periodic table and is called pnictogens. Many industrially compound, such as ammonia, nitric acid, organic nitrates, and cyanides, contain nitrogen. Nitrogen is a constituent of organic compounds as diverse as Kevlar

6.1 Nitrogen Gas Laser Filament Plasma …

555

used in high-strength fabric and cyanoacrylate used in superglue. The molecular nitrogen (N2 ) has some properties: inter molecular potential and the equation of state. We use Quantum mechanics (QM) calculations to determine N2 -N2 inter molecular interaction potential. We get the anisotropic potential surface for all orientations of the pair of the nitrogen molecules in the rotation invariant form. The molecular dynamics (MD) of the N2 molecules has been used to determine nitrogen state. The classical motion of N2 molecules is integrated and it accounted only translational and rotational degrees of freedom. We use the molecular dynamics (MD) modeling of the nitrogen pressure to get the fifth order predictor-corrector, or Runge-kutta. The nitrogen molecule is two nitrogen atoms stuck together by a “covalent bond”. They are bound together tightly and it is very difficult to separate them, except in certain chemical reactions. In our laser system a laser generation in molecular nitrogen in an argon-nitrogen gas mixture is remotely excited at a specific distance above in a femtosecond laser filament. Mid-IR laser pulses enable radical enhancement of fila mentation assisted lasing by N2 molecules. High energies are achieved from laser pulses which generated through the second-positive-band transitions of N2 . It is corresponding to a 0.5% total conversion efficiency from mid infrared laser energy to the energy of UV lasing. Lasing from molecular nitrogen is used in many applications. The discharge-pumped nitrogen laser, operating in a broad range of gas pressures, from several millibars to the atmospheric pressure, and repetition rates from several hertz to several kilohertz. It is robust source of high-power near-UV radiation. Achieving nitrogen via remote excitation would pave the way to many potential applications. A narrow band source of stimulated emission would provide coherent, highly directional radiation for highly selective and sensitive remote spectroscopy of the atmosphere. The possibility for remote ignition of a free localized nitrogen laser is demonstrated previously in a microwave discharge. By using focused laser beams we can excite free-space nitrogen lasing. The process of femtosecond fila mentation in gases is related to remotely excited free-space nitrogen laser. Fila mentation of high-power femtosecond laser radiation in gases is connected the formation of a self-guided, high-intensity field structure, accompanied by a significant spectral broadening, super continuum generation, and creation of a plasma channel in the wake of the pulse. The femtosecond filaments can be generated at standoff distances of up to tens of kilometers and from ionized channels of up to hundreds of meters. Formation of plasma in the filament, similar to a gas discharge, initiates a chain of plasma chemical reactions in the atmosphere which lead to the appearance of a large variety of neutral and ionic species in rotationally, vibrational, and electronically excited states. We get a situation that favorable conditions for population inversion and lasing between different electronic levels in nitrogen and in oxygen is created. We can use the standoff lasing with a femtosecond filament as an advantage in comparison to scheme due to substantially less scattering and absorption losses. There is a possibility of stimulated emission from nitrogen in femtosecond filaments. The proof of lasing is based on the exponential fit of the dependence of fluorescence on the filament length. The exponential dependence is valid under the condition of a stationary population inversion only. In filament driven laser the pump is the femtosecond laser pulse and amplified spontaneous emission (ASE) is about

556

6 Gas Laser Systems Stability Analysis …

nanosecond [1]. The laser generation is build up under the condition of a decaying population inversion which has a decay rate on the same nanoscale time scale. The filament length is calculated from the laser power assuming the direct proportionality between these values. Multiple filament formation is expected at high levels of laser power, and the assumption is that the filament length is proportional to the laser power. The generation of micro joule pulses from a cavity-free nitrogen laser which uses the plasma channel traced by a femtosecond laser filament as its active medium. In the lasing process, the electronic transitions of nitrogen are the same as in the conventional discharge-pumped nitrogen laser. In the discharge-driven scheme, the upper lasing triplet C 3 u in the three-level nitrogen laser is populated by electron impact excitation from the singlet ground state X 1 g . This gives rise to an efficient fluorescence in the near-UV spectral range due to radiative transitions between the vibrational manifolds of C 3 u and B 3 g states. The control over the density and temperature of electrons is of key importance for obtaining population inversion and lasing between these states. The impulsive RF or capacitor discharge caused to rapid joule heating of electrons due to collisions with neutrals leads to the formation of nonequilibrium energy distribution of electrons. The non-equilibrium energy distribution and the plasma concentration are maintained throughout the discharge, resulting in a sufficient amount of hot electrons—which is needed for efficient pumping. In a laser filament, the electronic energy distribution is formed on the time scale of the femtosecond laser pulse by the optical field ionization process. After the femtosecond pulse, the resultant energy distribution function evolves freely as the plasma can concentration decays. Therefore, the required C 3 u excitation by electron impact can be achieved within the average lifetime of the plasma electrons. By using a single femtosecond pulse, we cannot control the electron temperature during the buildup of population inversion, nor to control the duration of the time window within which the nitrogen laser is pumped. The electric discharge pumping, which maintains hot plasma, in the femtosecond filament the optimal electronic energy distribution is governed by optical field ionization and therefore is determined by the intensity and the optical cycle duration. The effect of intensity clamping precluded the possibility of direct scaling of the temperature of electrons in a filament via an increasing of the energy pulse. The plasma density is limited by the self-consistent balance between self-focusing and plasma refraction when focusing conditions are fixed. The coherence length of ASE is determined by the radiative lifetime. We need to inspect and analyze the efficient lasing of molecular nitrogen in a femtosecond filament. In this process the excitation energy of argon atoms transfers to molecular nitrogen. Then, excited argon atoms provide a collisional pump for laser transitions of N2 in a laser-induced filament, as same as hot electrons in a discharge-pumped nitrogen laser [1]. The analytical model is used to analyze the lasing in a laserinduced filament which includes the fila mentation dynamics and of plasma kinetics in the wake of the filament. We get the electron density and the intensity of the laser pulses. The target is to solve the system of plasma kinetics equations jointly with the equations for the electron temperature and the vibrational temperature of ground-state nitrogen molecules. The set of plasma kinetic equations is associated with the relevant rate equations for the population of the lasing levels and number of

6.1 Nitrogen Gas Laser Filament Plasma …

557

emitted photons. The change of concentration of different neutral and ionic atomic and molecular species in the filament plasma is described by set of rate equations d[Ns ] = [G s ]−[L s ]. Where the subscript s stands for e, Ar + , Ar2+ , N2+ , N4+ , N3+ and dt + N , N , N2 (A3 u ), N2 (B 3 g ), N2 (C 3 u ), Ar ∗ (43 P2 ); Ns is the density of species of type-s; and G s , L s are the relevant generation and loss rates. The rate equations are solved jointly with the equations for the electron temperature and the vibrational temperature of ground-state nitrogen molecules. System No.1: The first rate equation: d 3 3 · k · (Ne · Te ) = − · Ne · k · (Te − Tvibr ) · v∗ 2 dt 2 3 − · Ne · k · (Te − T ) · [δ N2 2 1 · (v N2 + v N2+ + · v N4+ ) 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 + ke · Ne · N Ar ∗ · Iexc − 23 · Ne ·k ·(Te −Tvibr )·v∗ —is the electron energy transfer to vibrational excitation of nitrogen molecules. − 23 · Ne ·k ·(Te −T )·[δ N2 ·(v N2 +v N2+ + 21 ·v N4+ )+δ Ar ·(v Ar +v Ar+ + 21 ·v Ar + )]—is the 2 electron energy transfer in elastic and Coulombic collisions to translational energy of nitrogen and argon molecules, atoms, and ions. k is the Boltzmann constant, T, Te , Tvibr are gas, electronic, and vibrational temperatures, respectively. vs are the frequencies of elastic collisions of electrons and corresponding species, v∗ is the frequency of inelastic collisions of electrons with N2 in the ground state which are responsible for vibrational heating; 2·m ; δ Ar = M ; ke is the de excitation rate of excited argon atoms by δ N2 = M2·m N2 Ar electrons. Assumption G e and L e are the relevant generation and loss rates, and they are very close in their values G e − L e → ε then ddtNe → ε. 3 d d Ne 3 3 · k · (Ne · Te ) = · k · · Te + · k 2 dt 2 dt 2 dTe ddtNe →ε 3 dTe = · Ne · · k · Ne · dt 2 dt The reduced first rate equation: 3 dTe 3 3 · k · Ne · = − · Ne · k · (Te − Tvibr ) · v∗ − · Ne · k 2 dt 2 2

558

6 Gas Laser Systems Stability Analysis …

· (Te − T ) · [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 · v Ar + )] + ke · Ne · N Ar ∗ · Iexc 2 2

The second rate equation: dTvibr = dt



Ne N N2 



· (Te − Tvibr ) · v∗ − (Tvibr − T )    1 N Ar /τV T,Ar × + τV T,N2 N2

τV T,N2 and τV T,Ar are the vibrational relaxation times due to N2 -N2 and N2 -Ar collisions, respectively. They are calculated by using the Landau-Teller approximation; Iexc ≈ 11.8eV stands for the argon excitation energy into Ar ∗ (43 P2 ) [1]. Rigorous bounds on the rate of energy exchange between vibrational and translational degrees of freedom are described by simple classical models of diatomic molecules. The outcomes are known as the elementary approximation introduced by Landau and Teller. The initial energy of electrons in the wake of the mid-IR laser pulse is taken equal to 1 eV (Te = 0.66 eV). The initial translational temperature of atoms and molecules in the gas mixture and the vibrational temperature of nitrogen molecules are set equal to T = TV = 290 K. We have two variables in our systems: Te , electronic temperature and Tvibr , vibrational temperature which change in time. 2 . We multiply the reduced first rate equation’s two sides by 3·k·N e The reduced first rate equation: dTe = −(Te − Tvibr ) · v∗ − (Te − T ) dt 1 1 · [δ N2 · (v N2 + v N2+ + · v N4+ ) + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 2 + · ke · N Ar ∗ · Iexc 3·k The second rate equation: dTvibr = dt



Ne NN  2



· (Te − Tvibr ) · v∗ − (Tvibr − T )    1 N Ar /τV T,Ar × + τV T,N2 N2

At fixed points (equilibrium points): (*)

dTe dt

= 0; dTdtvibr = 0

∗ − (Te∗ − Tvibr ) · v∗ − (Te∗ − T ) · [δ N2 · (v N2 + v N2+ +

+ δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 2 · v Ar + )] + · ke · N Ar ∗ · Iexc = 0 2 2 3·k

6.1 Nitrogen Gas Laser Filament Plasma …

559

1 ∗ ∗ (**) ( NNNe ) · (Te∗ − Tvibr ) · v∗ − (Tvibr − T ) × [ τV T,N + ( NNA2r )/τV T,Ar ] = 0 2 2 1 ∗ (Te∗ · v∗ − Tvibr · v∗ ) + (Te∗ − T ) · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 (*) → 1 2 · ke · N Ar ∗ · Iexc + δ Ar · (v Ar + v Ar+ + · v Ar + )] = 2 2 3·k ∗ Tvibr · v∗ = Te∗ · v∗ + Te∗ · [δ N2 · (v N2 + v N2+ +

1 · v +) 2 N4

1 · v Ar + )] 2 2 1 − T · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 − · ke · N Ar ∗ · Iexc 3·k

+ δ Ar · (v Ar + v Ar+ +

∗ Tvibr · v∗ = Te∗ · {v∗ + δ N2 · (v N2 + v N2+ +

1 · v +) 2 N4

1 · v Ar + )} 2 2 1 − T · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 − · ke · N Ar ∗ · Iexc 3·k + δ Ar · (v Ar + v Ar+ +

1 1 · {v∗ + δ N2 · (v N2 + v N2+ + · v N4+ ) v∗ 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )} 2 2 1 T · [δ N2 · (v N2 + v N2+ + · v N4+ ) − v∗ 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 − · ke · N Ar ∗ · Iexc 3 · k · v∗

∗ Tvibr = Te∗ ·

∗ We define for simplicity two global parameters: 1 , 2 ; Tvibr = Te∗ · 1 + 2

1 = 1 (v∗ , δ N2 , v N2 , v N2+ , v N4+ , δ Ar , v Ar , v Ar+ , v Ar + ) 2

2 = 2 (T, v∗ , δ N2 , v N2 , v N2+ , v N4+ , δ Ar , v Ar , v Ar+ , v Ar + , k, ke , N Ar ∗ , Iexc ) 2

560

6 Gas Laser Systems Stability Analysis …

1 1 · {v∗ + δ N2 · (v N2 + v N2+ + · v N4+ ) v∗ 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )} 2 2 T 1 2 = − · [δ N2 · (v N2 + v N2+ + · v N4+ ) v∗ 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 − · ke · N Ar ∗ · Iexc 3 · k · v∗ 1 =

1 ∗ ∗ (*) → (**) ( NNNe ) · (Te∗ − Tvibr ) · v∗ − (Tvibr − T ) × [ τV T,N + ( NNA2r )/τV T,Ar ] = 0 2

2

Ne Ne ∗ ∗ · v∗ − Tvibr · · v∗ − (Tvibr − T) N N2 N N2 1 NA ×[ + ( r )/τV T,Ar ] = 0 τV T,N2 N2

Te∗ ·

Ne Ne ∗ ∗ · v∗ = Tvibr · · v∗ + (Tvibr − T) N N2 N N2     1 N Ar /τV T,Ar × + τV T,N2 N2     N N2 1 N Ar ∗ ∗ /τV T,Ar + · (Tvibr − T) × + Te∗ = Tvibr Ne · v∗ τV T,N2 N2     N N2 1 N Ar ∗ ∗ /τV T,Ar + · Tvibr × + Te∗ = Tvibr Ne · v∗ τV T,N2 N2     1 N Ar N N2 /τV T,Ar ·T × + − Ne · v∗ τV T,N2 N2      N N2 1 N Ar ∗ ∗ /τV T,Ar × + Te = Tvibr · 1 + Ne · v∗ τV T,N2 N2     1 N Ar N N2 /τV T,Ar ·T × + − Ne · v∗ τV T,N2 N2 Te∗ ·

∗ We define for simplicity two global parameters: 3 , 4 ; Te∗ = Tvibr · 3 + 4

3 = 3 (N N2 , Ne , v∗ , τV T,N2 , N Ar , N2 , τV T,Ar ) 4 = 4 (N N2 , Ne , v∗ , T, τV T,N2 , N Ar , N2 , τV T,Ar ) 3 = 1 +

    N N2 1 N Ar /τV T,Ar × + Ne · v∗ τV T,N2 N2

6.1 Nitrogen Gas Laser Filament Plasma …

4 = −

561

    N N2 1 N Ar /τV T,Ar ·T × + Ne · v∗ τV T,N2 N2

∗ ∗ Result: Tvibr = Te∗ · 1 + 2 ; Te∗ = Tvibr · 3 + 4 ∗ ∗ ∗ Tvibr = (Tvibr · 3 + 4 ) · 1 + 2 ⇒ Tvibr

∗ ∗ = Tvibr · 3 · 1 + 4 · 1 + 2 ⇒ Tvibr 4 · 1 + 2 = 1 − 3 · 1   4 · 1 + 2 ∗ · 3 + 4 Te∗ = Tvibr · 3 + 4 = 1 − 3 · 1 ∗ Our system fixed point is E (∗) (Te∗ , Tvibr )= The reduced two rate equations are:



4 ·1 +2 1−3 ·1



· 3 + 4 ,



4 ·1 +2 1−3 ·1

dTe 1 = −(Te − Tvibr ) · v∗ − (Te − T ) · [δ N2 · (v N2 + v N2+ + · v N4+ ) dt 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 + · ke · N Ar ∗ · Iexc 3·k Ne dTvibr =( ) · (Te − Tvibr ) · v∗ − (Tvibr − T ) dt N N2 1 NA ×[ + ( r )/τV T,Ar ] τV T,N2 N2 We define two functions: f 1 (Te , Tvibr ); f 2 (Te , Tvibr ) f 1 (Te , Tvibr ) = −(Te − Tvibr ) · v∗ − (Te − T ) 1 · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 + · ke · N Ar ∗ · Iexc 3·k   Ne · (Te − Tvibr ) · v∗ − (Tvibr − T ) f 2 (Te , Tvibr ) = NN    2  1 N Ar /τV T,Ar × + τV T,N2 N2



562

6 Gas Laser Systems Stability Analysis …

Our system set of differential equations: f 2 (Te , Tvibr )

dTe dt

=

f 1 (Te , Tvibr ); dTdtvibr

=

∗ ∗ Linearization: suppose that (Te∗ , Tvibr ) is a fixed point, f 1 (Te∗ , Tvibr ) = ∗ ∗ ∗ ∗ 0; f 2 (Te , Tvibr ) = 0 and let u 1 = Te − Te ; u 2 = Tvibr − Tvibr denote the components of a small disturbance from the fixed point. To see whether the disturbance grows or decays, we need to derive differential equations for u 1 and u 2 . e e 1 1 = dT since Te∗ is constant and du = dT = Let’s do the u 1 -equation first: du dt dt dt dt ∗ ∗ f 1 (Te + u 1 , Tvibr + u 2 ) by substitution, then by using Taylor series expansion f1 du 1 ∗ e = dT = f 1 (Te∗ , Tvibr ) + u 1 · ∂∂ Tf1e + u 2 · ∂ T∂ vibr + O(u 21 , u 22 , u 1 · u 2 ) and since dt dt ∂ f f1 ∗ e 1 ) = 0 then du = dT = u 1 · ∂ T1e + u 2 · ∂ T∂ vibr + O(u 21 , u 22 , u 1 · u 2 ). To f 1 (Te∗ , Tvibr dt dt ∂ f1 ∂ f1 simplify the notation, we have written ∂ Te and ∂ Tvibr , these partial derivatives are to ∗ ); thus they are numbers, not functions. The be evaluated at the fixed point (Te∗ , Tvibr 2 2 shorthand notation, O(u 1 , u 2 , u 1 · u 2 ) denotes quadratic terms in u 1 and u 2 are small, these quadratic terms are extremely small [2, 3]. f2 du 2 = dTdtvibr = u 1 · ∂∂ Tf2e + u 2 · ∂ T∂ vibr + O(u 21 , u 22 , u 1 · u 2 ). Hence the disturbance dt ⎛ du ⎞ 1     ∂ f1 ∂ f1 u1 ⎜ dt ⎟ ∂ Te ∂ Tvibr (u 1 , u 2 ) evolves according to ⎝ · + quadratic terms. ⎠ = ∂ f2 ∂ f2 du 2 u2 ∂ Te ∂ Tvibr  dt 

The matrix A =

∂ f1 ∂ f1 ∂ Te ∂ Tvibr ∂ f2 ∂ f2 ∂ Te ∂ Tvibr

is called the Jacobian matrix at the fixed point ∗ (Te∗ ,Tvibr )

∗ (Te∗ , Tvibr ). It is multivariable analog to f 1 (Te∗ ). Since the quadratic terms are tiny, we neglect them altogether. We obtain the linearized system:

⎛ du ⎞ 1

⎜ dt ⎟ ⎠= ⎝ du 2 dt



∂ f1 ∂ f1 ∂ Te ∂ Tvibr ∂ f2 ∂ f2 ∂ Te ∂ Tvibr

  ·

 u1 . u2

f 1 = f 1 (Te , Tvibr ) = −(Te − Tvibr ) · v∗ − (Te − T ) 1 · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 1 2 + δ Ar · (v Ar + v Ar+ + · v Ar + )] + · ke · N Ar ∗ · Iexc 2 2 3·k ∂ f1 1 = −v∗ − [δ N2 · (v N2 + v N2+ + · v N4+ ) ∂ Te 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 ∂ f1 = v∗ ∂ Tvibr

6.1 Nitrogen Gas Laser Filament Plasma …

563

Ne ) · (Te − Tvibr ) · v∗ − (Tvibr − T ) N N2     1 N Ar /τV T,Ar × + τV T,N2 N2     1 N Ar ∂ f2 Ne ∂ f2 Ne = · v∗ ; =− · v∗ − + /τV T,Ar ∂ Te N N2 ∂ Tvibr N N2 τV T,N2 N2 f 2 (Te , Tvibr ) = (

⎛

⎜ ⎜ A=⎜ ⎝

⎞ 1 − v∗ − [δ N2 · (v N2 + v N + + · v N + ) 2 4 ⎟ 2 v∗ ⎟ 1 ⎟ + δ Ar · (v Ar + v Ar+ + · v Ar + )] ⎠ 2 2 N Ne Ne Ar 1 · v − · v − [ + ( )/τ ] ∗ ∗ V T,Ar NN NN τV T,N N2 2

2

∗ ) (Te∗ ,Tvibr

2

The eigenvalues of the matrix A are given by the characteristic equationdet(A − λ · I ) = 0, where I is the identity matrix. For our 2 × 2 matrix the characteristic equation becomes: ⎛ ⎜ ⎜ det ⎜ ⎝

⎞ 1 − v∗ − [δ N2 · (v N2 + v N + + · v N + ) 2 4 ⎟ 2 v∗ ⎟ 1 ⎟=0 + δ Ar · (v Ar + v Ar+ + · v Ar + )] − λ ⎠ 2 2 N Ne 1 − NNNe · v∗ − [ τV T,N + ( NA2r )/τV T,Ar ] − λ N N · v∗ 2

2

2

For simplicity we define two global parameters: ϒ1 , ϒ2 ϒ1 = −v∗ − [δ N2 · (v N2 + v N2+ +

1 1 · v N4+ ) + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2

1 ϒ2 = − NNNe · v∗ − [ τV T,N + ( NNA2r )/τV T,Ar ], then we get the determinant: 2

2



ϒ1 − λ v∗ det Ne · v∗ ϒ2 − λ NN

 = 0 ⇒ (ϒ1 − λ) · (ϒ2 − λ)

2

−

Ne · v∗ · v∗ = 0 N N2

Expanding the determinant yields: λ2 − λ · (ϒ1 + ϒ2 ) + (ϒ1 · ϒ2 − NNNe · v∗2 ) = 0 2 Another way to get the system characteristic equation and explore stability is by adding to coordinates [Te Tvibr ] arbitrarily small increments of exponential form [te tvibr ] · eλ·t , and retaining the first order terms in Te Tvibr . The system of two homogeneous equations leads to a polynomial characteristics equation in the eigenvalues λ. Nitrogen gas laser filament plasma kinetic system Te , electronic temperature and Tvibr , vibrational temperature fixed values with arbitrarily small increments of expo∗ + tvibr · eλ·t . We nential form [te tvibr ] · eλ·t are Te = Te∗ + te · eλ·t ; Tvibr = Tvibr choose these expressions for our Te (t), Tvibr (t) as small [te tvibr ] from the system ∗ . fixed points at time t = 0; Te (t = 0) = Te∗ + te ; Tvibr (t = 0) = Tvibr

564

6 Gas Laser Systems Stability Analysis …

dTe dTvibr = te · λ · eλ·t ; = tvibr · λ · eλ·t dt dt dTe 1 = −(Te − Tvibr ) · v∗ − (Te − T ) · [δ N2 · (v N2 + v N2+ + · v N4+ ) dt 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 2 + · ke · N Ar ∗ · Iexc 3·k ∗ te · λ · eλ·t = −(Te∗ + te · eλ·t − Tvibr − tvibr · eλ·t ) · v∗

− (Te∗ + te · eλ·t − T ) · [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 2 · v Ar + )] + · ke · N Ar ∗ · Iexc 2 2 3·k

∗ te · λ · eλ·t = −(Te∗ − Tvibr ) · v∗ − (te − tvibr ) · v∗ · eλ·t 1 − (Te∗ − T ) · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 1 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] − te · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 2 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] · eλ·t 2 2 2 · ke · N Ar ∗ · Iexc + 3·k ∗ te · λ · eλ·t = −(Te∗ − Tvibr ) · v∗ − (Te∗ − T ) · [δ N2 · (v N2 + v N2+ +

+ δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 · v Ar + )] 2 2

2 · ke · N Ar ∗ · Iexc − (te − tvibr ) · v∗ · eλ·t − te 3·k 1 1 · [δ N2 · (v N2 + v N2+ + · v N4+ ) + δ Ar · (v Ar + v Ar+ + · v Ar + )] · eλ·t 2 2 2 +

At fixed point:

∗ − (Te∗ − Tvibr ) · v∗ − (Te∗ − T ) · [δ N2 · (v N2 + v N2+ +

+ δ Ar · (v Ar + v Ar+ +

1 2 · v Ar + )] + · ke · N Ar ∗ · Iexc = 0 2 2 3·k

te · λ = −(te − tvibr ) · v∗ − te · [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 · v Ar + )] 2 2

1 · v +) 2 N4

6.1 Nitrogen Gas Laser Filament Plasma …

565

te · {−v∗ − [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 · v Ar + )] − λ} + tvibr · v∗ = 0 2 2

Ne dTvibr =( ) · (Te − Tvibr ) · v∗ − (Tvibr − T ) dt N N2 1 NA ×[ + ( r )/τV T,Ar ] τV T,N2 N2 tvibr · λ · eλ·t = (

Ne ∗ ) · (Te∗ + te · eλ·t − Tvibr N N2

∗ − tvibr · eλ·t ) · v∗ − (Tvibr + tvibr · eλ·t − T ) NA 1 + ( r )/τV T,Ar ] ×[ τV T,N2 N2

Ne Ne ∗ ) · (Te∗ − Tvibr ) · v∗ + ( ) · (te − tvibr ) · v∗ · eλ·t N N2 N N2 1 NA ∗ − (Tvibr − T) × [ + ( r )/τV T,Ar ] − tvibr τV T,N2 N2 1 NA ×[ + ( r )/τV T,Ar ] · eλ·t τV T,N2 N2

tvibr · λ · eλ·t = (

Ne ∗ ∗ ) · (Te∗ − Tvibr ) · v∗ − (Tvibr − T) N N2 1 NA ×[ + ( r )/τV T,Ar ] τV T,N2 N2 Ne +( ) · (te − tvibr ) · v∗ · eλ·t − tvibr N N2 1 NA ×[ + ( r )/τV T,Ar ] · eλ·t τV T,N2 N2

tvibr · λ · eλ·t = (

1 ∗ ∗ At fixed point: ( NNNe ) · (Te∗ − Tvibr ) · v∗ − (Tvibr − T ) × [ τV T,N + ( NNA2r )/τV T,Ar ] = 0 2

     Ne 1 N Ar · (te − tvibr ) · v∗ − tvibr × /τV T,Ar + N N2 τV T,N2 N2     Ne Ne · te · v∗ − · tvibr · v∗ + tvibr N N2 N N2     1 N Ar /τV T,Ar − tvibr · λ = 0 × + τV T,N2 N2

 tvibr · λ =

2

566

6 Gas Laser Systems Stability Analysis …



       Ne Ne 1 N Ar · v∗ · te − · v∗ + /τV T,Ar + N N2 N N2 τV T,N2 N2 · tvibr − tvibr · λ = 0

The small increment Jacobian of our Nitrogen gas laser filament plasma kinetic system is as follow: te · {−v∗ − [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ + (

1 · v +) 2 N4

1 · v Ar + )] − λ} + tvibr · v∗ = 0 2 2

Ne Ne 1 NA ) · v∗ · te − {( ) · v∗ + [ + ( r )/τV T,Ar ]} N N2 N N2 τV T,N2 N2 · tvibr − tvibr · λ = 0

⎛

⎞ 1 −v∗ − [δ N2 · (v N2 + v N + + · v N + ) 2 4 ⎜ ⎟ 2 v∗ ⎜ ⎟ 1 ⎜ +δ Ar · (v Ar + v A+ + · v A + )] − λ ⎟ r ⎝ ⎠ r2 2 N 1 ( NNNe ) · v∗ −{( NNNe ) · v∗ + [ τV T,N + ( NA2r )/τV T,Ar ]} − λ 2 2 2     0 te = · tvibr 0 A−λ· I ⎛ ⎞ 1 −v∗ − [δ N2 · (v N2 + v N + + · v N + ) 2 4 ⎜ ⎟ 2 v∗ ⎜ ⎟ = ⎜ +δ Ar · (v Ar + v + + 1 · v A + )] − λ ⎟ Ar ⎝ ⎠ r2 2 N Ar Ne Ne 1 · v − · v − [ + ( )/τ ] − λ ∗ ∗ V T,Ar NN NN τV T,N N2

⎛ ⎜ ⎜ A=⎜ ⎝

2

2

2

⎞ 1 −v∗ − [δ N2 · (v N2 + v N2+ + · v N4+ ) ⎟ 2 v∗ ⎟ 1 ⎟ +δ Ar · (v Ar + v Ar+ + · v Ar + )] ⎠ 2 2 N Ar Ne Ne 1 · v − · v − [ + ( )/τ ] ∗ ∗ V T,Ar NN NN τV T,N N2 2

2

2

For simplicity we define two global parameters: ϒ1 , ϒ2 ϒ1 = −v∗ − [δ N2 · (v N2 + v N2+ + + δ Ar · (v Ar + v Ar+ +

1 · v +) 2 N4

1 · v Ar + )] 2 2

1 ϒ2 = − NNNe · v∗ − [ τV T,N + ( NNA2r )/τV T,Ar ], then we get the determinant: 2

2

6.1 Nitrogen Gas Laser Filament Plasma …



ϒ1 − λ v∗ det Ne · v∗ ϒ2 − λ NN

567

 = 0 ⇒ (ϒ1 − λ) · (ϒ2 − λ)

2

−

Ne · v∗ · v∗ = 0 N N2

Expanding the determinant yields: λ2 − λ · (ϒ1 + ϒ2 ) + (ϒ1 · ϒ2 − We define ℘1 = trace(A); ℘2 = det(A)

Ne N N2

· v∗2 ) = 0

   1 ℘1 = trace(A) = −v∗ − δ N2 · v N2 + v N2+ + · v N4+ 2   1 +δ Ar · v Ar + v Ar+ + · v Ar + 2 2     Ne 1 N Ar /τV T,Ar − · v∗ − + N N2 τV T,N2 N2 Ne 1 ] − [δ N2 · (v N2 + v N2+ + · v N4+ ) N N2 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] 2 2 N Ar 1 +( )/τV T,Ar ] −[ τV T,N2 N2

℘1 = trace(A) = −v∗ · [1 +

℘2 = det(A) = {−v∗ − [δ N2 · (v N2 + v N2+ +

1 · v +) 2 N4

1 Ne · v Ar + )]} · {− · v∗ 2 2 N N2 NA Ne + ( r )/τV T,Ar ]} − · v∗ · v∗ N2 N N2

+ δ Ar · (v Ar + v Ar+ + −[

1 τV T,N2

℘2 = det(A) = {v∗ + [δ N2 · (v N2 + v N2+ +

1 · v +) 2 N4

1 Ne · v Ar + )]} · { · v∗ 2 2 N N2 NA Ne + ( r )/τV T,Ar ]} − · v2 N2 N N2 ∗

+ δ Ar · (v Ar + v Ar+ + +[

1 τV T,N2

√ √ ℘ + ℘12 −4·℘2 ℘1 − ℘12 −4·℘2 Then λ1 = 1 ; λ = are the solutions of the quadratic 2 2 2 equationλ2 − λ · (ϒ1 + ϒ2 ) + (ϒ1 · ϒ2 − NNNe · v∗2 ) = 0; ℘1 = ϒ1 + ϒ2 ; ℘2 = 2

ϒ1 · ϒ2 − NNNe · v∗2 . The eigenvalues depend only on the trace and determinant of 2 the matrix A. The typical situation is for the eigenvalues to be distinct: λ1 = λ2 . In this case, the theorem of linear algebra states that the corresponding eigenvectors are linearly independent and hence span the entire plane [2, 3].

568

6 Gas Laser Systems Stability Analysis …

Classification of fixed points: We need to know the type and stability of all the system different fixed points on a single diagram. The axes are the trace ℘1 and the determinant ℘2 of the matrix A. All of √ the information in the diagram is implied by   ℘1 ± ℘12 −4·℘2 the following formulas: λ1,2 = ; ℘2 = 2k=1 λk ; ℘1 = 2k=1 λk and 2 (λ − λ1 ) · (λ − λ2 ) = 0(λ − λ1 ) · (λ − λ2 ) = λ2 − λ · ℘1 + ℘2 = 0. If ℘2 < 0, the eigenvalues are real and have opposite sign; hence the fixed point is a saddle point. If ℘2 > 0, the eigenvalues are either real with the same sign (nodes), or complex conjugate (spirals and centers). Nodes satisfy ℘12 − 4 · ℘2 > 0 and spirals satisfy ℘12 − 4 · ℘2 < 0. The parabola ℘12 − 4 · ℘2 = 0 is the borderline between nodes and spirals; star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by ℘1 . When ℘1 < 0, both eigenvalues have negative real parts, the fixed point is stable. Unstable spirals and nodes have ℘1 > 0. Neutrally stable center live on the borderline ℘1 = 0, where the eigenvalues are purely imaginary. If ℘2 = 0 then at least one of the eigenvalues is zero. Then the origin is not an isolated fixed point. There is either a whole line of fixed points, or a plane of fixed points, if A = 0. Saddle points, nodes, and spirals are the major types of fixed points; they occur in large open regions of the (℘2 , ℘1 ) plane. Centers, stars, degenerate nodes, and non-isolated fixed points are borderline cases that occur along the curves in the (℘2 , ℘1 ) plane. System No.2: We include lase effects, and the set of plasma kinetic equations include the populations of the lasing levels and number of generated photons: dn c = Sc − n c · (AC B1 + AC B2 + σC B1 · c · n ph1 + σC B2 · c · n ph2 ) dt + n B1 · σ B1C · c · n ph1 + n B2 · σ B2C · c · n ph2 dn B1 = S B1 + n c · (AC B1 + σC B1 · c · n ph1 ) − n B1 · σ B1C · c · n ph1 dt dn B2 = S B2 + n c · (AC B2 + σC B2 · c · n ph2 ) − n B2 · σ B2C · c · n ph2 dt σC B1 · (n c − n B1 ) · n ph1 · c n ph1 dn ph1 = − dt τ ph 1 + II L s1

σC B2 · (n c − n B2 ) · n ph2 · c n ph2 dn ph2 = − dt τ ph 1 + II L s2

n c is the population of the C 3 u state, n B1 , n B2 are the populations of B 3 g in the ground and first excited vibrational states, Sc , S B1 , S B2 are the rates of generation and loss through all the non-radiative channels; σC B1 , σC B2 , σ B1C , σ B2C are the stimulated h·v ; k = 1, 2 are the emission cross sections; c is the speed of light; Isk = σ (v)·τ sk

6.1 Nitrogen Gas Laser Filament Plasma …

569

saturation intensities where the radiative lifetimes τs ∝ A−1 and are determined from the corresponding Einstein coefficients; I L = n ph ·h·v·c are the lasing intensities; and n ph1 , n ph2 are the photon number densities at specific wavelength (λ = 337nm; λ = 357 nm); τ ph is a photon lifetime related with the filament length L as τ ph = Lc . The broadening of each of the laser modes is assumed equal to λ = 0.1nm. The , where g0 (v) is the pressure-induced line broadening is taken as g(v) ≈ gp0Ar(v)·760 + p N2 spectral line shape at the gas pressure p = 760 Torr and T = 300 K, and p Ar , p N2 are the partial pressures of argon and molecular nitrogen [1]. The time dependent model is the standard model for description of chemical and plasma-chemical kinetics. The variables in the model are functions of time only, and we neglect the spatial effects (diffusion and spatial gain inhomogeneity). The plasma kinetic model is not ideal and there is a time delay for n c (t) → n c (t − τ D ); τ D > 0, then the set of plasma kinetic equations: parameters (Sc , S B1 , S B2 , σC B1 , σC B2 , σ B1C , σ B2C , τ D , . . .) dn c = Sc − n c (t − τ D ) · (AC B1 + AC B2 + σC B1 · c · n ph1 dt + σC B2 · c · n ph2 ) + n B1 · σ B1C · c · n ph1 + n B2 · σ B2C · c · n ph2 dn B1 = S B1 + n c (t − τ D ) · (AC B1 + σC B1 · c · n ph1 ) dt − n B1 · σ B1C · c · n ph1 dn B2 = S B2 + n c (t − τ D ) · (AC B2 + σC B2 · c · n ph2 ) dt − n B2 · σ B2C · c · n ph2 σC B1 · (n c (t − τ D ) − n B1 ) · n ph1 · c n ph1 dn ph1 = − dt τ ph 1 + IIs1L σC B2 · (n c (t − τ D ) − n B2 ) · n ph2 · c n ph2 dn ph2 = − dt τ ph 1 + IIs2L n c = n c (t); n B1 = n B1 (t) n B2 = n B2 (t); n ph1 = n ph1 (t); n ph2 = n ph2 (t) Hint: time delay τ D does not affect the derivative of n c = n c (t) in time, dn dn c = 0; dndtB1 = 0; dndtB2 = 0; dtph1 = 0; dtph2 = 0 At fixed points: dn dt lim n c (t − τ D ) = n c (t); t τ D ; t − τ D ≈ t ∀ t → ∞

t→0

dn c . dt

570

6 Gas Laser Systems Stability Analysis …

(*)

Sc − n ∗c · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 )

+ n ∗B1 · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 (**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 (***) S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 (****)

σC B1 ·(n ∗c −n ∗B1 )·n ∗ph1 ·c I

1+ I L

−

s1

n ∗ph1 τ ph

= 0; (*****)

σC B2 ·(n ∗c −n ∗B2 )·n ∗ph2 ·c I

1+ I L

s2

Some mathematical manipulations: σ ·(n ∗ −n ∗ )·c (****) n ∗ph1 ·[ C B1 c I L B1 − τ1ph ] = 0; (1.1) n ∗ph1 = 0; (1.2) 1+ I

0

n ∗c = n ∗B1 + (1 +

n ∗ph2

IL ) Is1

·

1 ; τ ph ·σC B1 ·c

σC B2 ·(n ∗c −n ∗B2 )·c I

1+ I L

s2

− τ1ph = 0;

(*****); n ∗ph2 · [

σC B2 ·(n ∗c −n ∗B2 )·c I

1+ I L

n ∗ph2 τ ph

I

1+ I L

s1

−

s2

σC B2 ·(n ∗c −n ∗B2 )·c I

1+ I L

=

s2

1 ; n ∗c τ ph

=0

σC B1 ·(n ∗c −n ∗B1 )·c

s1

=0

(2.2)

−

1 ] τ ph

− τ1ph =

= 0; (2.1)

= n ∗B2 + (1 + IIs2L ) · σC B21·τ ph ·c

We can summary the above partial fixed points discussion in the next table (Table 6.1). Case A, (1.1) & (2.1) n ∗ph1 = 0; n ∗ph2 = 0 Sc − n ∗c · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 ) (*) + n ∗B1 · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 n ∗ph1 = 0; n ∗ph2 = 0 ⇒ Sc − n ∗c · (AC B1 + AC B2 ) Sc = 0 ⇒ n ∗c = AC B1 + AC B2 Table 6.1 Plasma kinetic equations (populations of the lasing levels and number of generated photons), partial fixed points discussion Case

Description

A: (1.1) & (2.1)

n ∗ph1 = 0; n ∗ph2 = 0

B: (1.1) & (2.2)

n ∗ph1 = 0; n ∗c = n ∗B2 + (1 +

IL 1 Is2 ) · σC B2 ·τ ph ·c

= n ∗B2 + 2

We define for simplicity 2 parameter: 2 = (1 + C: (1.2) & (2.1)

IL 1 Is2 ) · σC B2 ·τ ph ·c ; 2

n ∗c = n ∗B1 + (1 +

= 2 (I L , Is2 , σC B2 , τ ph )

IL 1 Is1 ) · τ ph ·σC B1 ·c

= n ∗B1 + 1 ; n ∗ph2 = 0

We define for simplicity 1 parameter: 1 = (1 + D (1.2) & (2.2)

n ∗c

=

n ∗c

=

Then

IL 1 Is1 ) · τ ph ·σC B1 ·c ; 1

= 1 (I L , Is1 , τ ph , σC B1 )

n ∗B1

+ (1 + IIs1L ) · τ ph ·σ1C B1 ·c ; n ∗c = n ∗B2 + (1 + IIs2L ) · σC B21·τ ph ·c n ∗B1 + 1 ; n ∗c = n ∗B2 + 2 n ∗B1 + (1 + IIs1L ) · τ ph ·σ1C B1 ·c = n ∗B2 + (1 + IIs2L ) · σC B21·τ ph ·c

n ∗B1 = n ∗B2 + (1 +

IL 1 Is2 ) · σC B2 ·τ ph ·c

n ∗B1 = n ∗B2 + 2 − 1

− (1 +

IL 1 Is1 ) · τ ph ·σC B1 ·c

6.1 Nitrogen Gas Laser Filament Plasma …

(**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 c n ∗ph1 = 0; n ∗ph2 = 0 ⇒ n ∗c = − ASCB1B1 (Conclusion: AC B1 S+A = − ASCB1B1 ). C B2 ∗ ∗ ∗ (***) S B2 + n c · (AC B2 + σC B2 · c · n ph2 ) − n B2 · σ B2C · c · n ∗ph2 = 0 n ∗ph1 = 0; n ∗ph2 = 0 ⇒ S B2 + n ∗c · AC B2 S B2 = 0 ⇒ n ∗c = − AC B2 Conclusion:

Sc AC B1 +AC B2

= − ASCB1B1 = − ASCB2B2

n ∗B1 = n ∗c − 1 = n ∗c − 1 (I L , Is1 , τ ph , σC B1 ) Sc = AC B1 + AC B2 − 1 (I L , Is1 , τ ph , σC B1 ) n ∗B2 = n ∗c − 2 = n ∗c − 2 (I L , Is2 , σC B2 , τ ph ) Sc = AC B1 + AC B2 − 2 (I L , Is2 , σC B2 , τ ph ) First fixed point:

(0) (0) (0) (0) , n , n , n , n E (0) = n (0) c B1 B2 ph1 ph2  Sc Sc , − 1 (I L , Is1 , τ ph , σC B1 ) , = AC B1 + AC B2 AC B1 + AC B2  Sc − 2 (I L , Is2 , σC B2 , τ ph ), 0, 0 AC B1 + AC B2 Case B, (1.1) & (2.2) n ∗ph1 = 0; n ∗c = n ∗B2 + 2 Sc − n ∗c · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 ) (*) + n ∗B1 · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − (n ∗B2 + 2 ) · (AC B1 + AC B2 + σC B2 · c · n ∗ph2 ) + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − (n ∗B2 + 2 ) · (AC B1 + AC B2 ) − (n ∗B2 + 2 ) · (σC B2 · c · n ∗ph2 ) + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − n ∗B2 · (AC B1 + AC B2 ) − 2 · (AC B1 + AC B2 )

571

572

6 Gas Laser Systems Stability Analysis …

− n ∗B2 · n ∗ph2 · σC B2 · c − 2 · σC B2 · c · n ∗ph2 + n ∗B2 · n ∗ph2 · σ B2C · c = 0 (B.1)

Sc − 2 · (AC B1 + AC B2 ) + n ∗B2 · n ∗ph2 · (σ B2C − σC B2 ) · c

− 2 · σC B2 · c · n ∗ph2 − n ∗B2 · (AC B1 + AC B2 ) = 0 (**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 S B1 + (n ∗B2 + 2 ) · AC B1 = 0 ⇒ S B1 + n ∗B2 · AC B1 + 2 · AC B1 (S B1 + 2 · AC B1 ) = 0 ⇒ n ∗B2 = − AC B1 2 ·AC B1 ) 2 ·AC B1 ) n ∗c = n ∗B2 + 2 = − (SB1 + + 2 ; n ∗B2 = − (SB1 + → (B.1): AC B1 AC B1

(S B1 + 2 · AC B1 ) AC B1 · n ∗ph2 · (σ B2C − σC B2 ) · c − 2 · σC B2 · c · n ∗ph2 (S B1 + 2 · AC B1 ) + · (AC B1 + AC B2 ) = 0 AC B1

Sc − 2 · (AC B1 + AC B2 ) −

Sc − 2 · (AC B1 + AC B2 ) + =

(S B1 + 2 · AC B1 ) · (AC B1 + AC B2 ) AC B1

(S B1 + 2 · AC B1 ) · (σ B2C − σC B2 ) · c · n ∗ph2 + 2 · σC B2 · c · n ∗ph2 AC B1

(S B1 + 2 · AC B1 ) · (σ B2C − σC B2 ) + 2 · σC B2 ] · c · n ∗ph2 AC B1 (S B1 + 2 · AC B1 ) = Sc − 2 · (AC B1 + AC B2 ) + · (AC B1 + AC B2 ) AC B1

[

n ∗ph2

=

Sc − 2 · (AC B1 + AC B2 ) +

(S B1 +2 ·AC B1 ) AC B1

· (AC B1 + AC B2 )

2 ·AC B1 ) [ (SB1 + · (σ B2C − σC B2 ) + 2 · σC B2 ] · c AC B1

(***) S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 (S B1 + 2 · AC B1 ) + 2 ) · (AC B2 + σC B2 · c AC B1 2 ·AC B1 ) Sc − 2 · (AC B1 + AC B2 ) + (SB1 + · (AC B1 + AC B2 ) AC B1

S B2 + (− ·{ +

2 ·AC B1 ) [ (SB1 + · (σ B2C − σC B2 ) + 2 · σC B2 ] · c AC B1

(S B1 + 2 · AC B1 ) · σ B2C · c AC B1

})

6.1 Nitrogen Gas Laser Filament Plasma …

·{

Sc − 2 · (AC B1 + AC B2 ) + 2 ·AC B1 ) [ (SB1 + AC B1

573 (S B1 +2 ·AC B1 ) AC B1

· (AC B1 + AC B2 )

· (σ B2C − σC B2 ) + 2 · σC B2 ] · c

}=0

Remark The above mathematical relation must fulfill for the second fixed point. (**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 n ∗B1 =

S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) σ B1C · c · n ∗ph1

|n ∗ph1 =0 → ∞

Second fixed point:

(1) (1) (1) (1) , n , n , n , n E (1) = n (1) c B1 B2 ph1 ph2  (S B1 + 2 · AC B1 ) (S B1 + 2 · AC B1 ) + 2 , n (1) , 0, = − B1 → ∞, − AC B1 AC B1 ⎞ 2 ·AC B1 ) Sc − 2 · (AC B1 + AC B2 ) + (SB1 + · (AC B1 + AC B2 ) AC B1 ⎠  (S B1 +2 ·AC B1 ) · c · (σ − σ ) +  · σ B2C C B2 2 C B2 AC B1 3 Since n (1) B1 → ∞ (population of B g in the ground state), the second fixed point is only “mathematical” and Not “physical”. Case C, (1.2) & (2.1) n ∗c = n ∗B1 + 1 ; n ∗ph2 = 0 Sc − n ∗c · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 ) (*) + n ∗B1 · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0

Sc − (n ∗B1 + 1 ) · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 ) + n ∗B1 · σ B1C · c · n ∗ph1 = 0 Sc − n ∗B1 · (AC B1 + AC B2 ) − 1 · (AC B1 + AC B2 ) − (n ∗B1 + 1 ) · σC B1 · c · n ∗ph1 + n ∗B1 · σ B1C · c · n ∗ph1 = 0 Sc − n ∗B1 · (AC B1 + AC B2 ) − 1 · (AC B1 + AC B2 ) − n ∗B1 · σC B1 · c · n ∗ph1 − 1 · σC B1 · c · n ∗ph1 + n ∗B1 · σ B1C · c · n ∗ph1 = 0 (C.1)

Sc − 1 · (AC B1 + AC B2 ) − n ∗B1 · (AC B1 + AC B2 ) − 1 · σC B1 · c · n ∗ph1

+ n ∗B1 · n ∗ph1 · (σ B1C − σC B1 ) · c = 0 (**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0

574

6 Gas Laser Systems Stability Analysis …

S B1 + (n ∗B1 + 1 ) · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 S B1 + (n ∗B1 + 1 ) · AC B1 + (n ∗B1 + 1 ) · σC B1 · c · n ∗ph1 − σ B1C · c · n ∗B1 · n ∗ph1 = 0 S B1 + 1 · AC B1 + n ∗B1 · AC B1 + 1 · σC B1 · c · n ∗ph1 + c · (σC B1 − σ B1C ) · n ∗B1 · n ∗ph1 = 0 We get two equations, two fixed coordinates: n ∗B1 , n ∗ph1 Sc − 1 · (AC B1 + AC B2 ) − n ∗B1 · (AC B1 + AC B2 ) − 1 · σC B1 · c · n ∗ph1 (C.2) + n ∗B1 · n ∗ph1 · (σ B1C − σC B1 ) · c = 0 S B1 + 1 · AC B1 + n ∗B1 · AC B1 + 1 · σC B1 · c · n ∗ph1 (C.3) + c · (σC B1 − σ B1C ) · n ∗B1 · n ∗ph1 = 0 ### (***) S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 S B2 + n ∗c · AC B2 = 0 ⇒ n ∗c = −

S B2 AC B2

  S B2 n ∗c = n ∗B1 + 1 ⇒ n ∗B1 = n ∗c − 1 = − + 1 AC B2 n ∗B1 = −( ASCB2B2 + 1 ) ⇒ (C.2) : S B2 + 1 ) AC B2 · (AC B1 + AC B2 ) − 1 · σC B1 · c · n ∗ph1 S B2 −( + 1 ) · n ∗ph1 · (σ B1C − σC B1 ) · c = 0 AC B2

Sc − 1 · (AC B1 + AC B2 ) + (

n ∗ph1 =

Sc − 1 · (AC B1 + AC B2 ) + ( ASCB2B2 + 1 ) · (AC B1 + AC B2 ) [1 · σC B1 + ( ASCB2B2 + 1 ) · (σ B1C − σC B1 )] · c

n ∗B1 = −( ASCB2B2 + 1 ); n ∗ph1 = . . . ⇒ (C.3): S B2 + 1 )) · AC B1 + 1 · σC B1 · c AC B2 Sc − 1 · (AC B1 + AC B2 ) + ( ASCB2B2 + 1 ) · (AC B1 + AC B2 )

S B1 + 1 · AC B1 + (−( ·{

[1 · σC B1 + ( ASCB2B2 + 1 ) · (σ B1C − σC B1 )] · c

}

6.1 Nitrogen Gas Laser Filament Plasma …

575

S B2 + 1 ) AC B2 Sc − 1 · (AC B1 + AC B2 ) + ( ASCB2B2 + 1 ) · (AC B1 + AC B2 )

− c · (σC B1 − σ B1C ) · ( ·{

[1 · σC B1 + ( ASCB2B2 + 1 ) · (σ B1C − σC B1 )] · c

}=0

Remark The above mathematical relation must fulfill for the third fixed point. (***) S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 n ∗B2 =

S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) σ B2C · c · n ∗ph2

|n ∗ph2 =0 → ∞

Second fixed point:

(2) (2) (2) (2) E (2) = n (2) c , n B1 , n B2 , n ph1 , n ph2   S B2 S B2 , −( + 1 , n (1) = − B2 → ∞, AC B2 AC B2

⎞ Sc − 1 · (AC B1 + AC B2 ) + ASCB2B2 + 1 · (AC B1 + AC B2 ) 

, 0⎠ 1 · σC B1 + ASCB2B2 + 1 · (σ B1C − σC B1 ) · c 3 Since n (1) B2 → ∞ (population of B g in the first excited vibrational state), the third fixed point is only “mathematical” and Not “physical”. Second and third fixed points discussion: For both Second and Third fixed points, (1) (2) (2) n (1) c < 0; n c < 0, respectively and n B1 < 0 or n B1 < 0, respectively. Population of specific state cannot be negative; it is evidence that these fixed points are theoretical. Case D, (1.2) & (2.2) n ∗c = n ∗B1 + 1 ; n ∗c = n ∗B2 + 2 ; n ∗B1 = n ∗B2 + 2 − 1 Sc − n ∗c · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 ) (*) + n ∗B1 · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0

Sc − (n ∗B2 + 2 ) · (AC B1 + AC B2 + σC B1 · c · n ∗ph1 + σC B2 · c · n ∗ph2 ) + (n ∗B2 + 2 − 1 ) · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − (n ∗B2 + 2 ) · [AC B1 + AC B2 ] − (n ∗B2 + 2 ) · σC B1 · c · n ∗ph1 − (n ∗B2 + 2 ) · σC B2 · c · n ∗ph2 + n ∗B2 · σ B1C · c · n ∗ph1 + [2 − 1 ] · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − (n ∗B2 · [AC B1 + AC B2 ] + 2 · [AC B1 + AC B2 ])

576

6 Gas Laser Systems Stability Analysis …

− (n ∗B2 · σC B1 · c · n ∗ph1 + 2 · σC B1 · c · n ∗ph1 ) − (n ∗B2 · σC B2 · c · n ∗ph2 + 2 · σC B2 · c · n ∗ph2 ) + n ∗B2 · σ B1C · c · n ∗ph1 + [2 − 1 ] · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − n ∗B2 · [AC B1 + AC B2 ] − 2 · [AC B1 + AC B2 ] − n ∗B2 · σC B1 · c · n ∗ph1 − 2 · σC B1 · c · n ∗ph1 − n ∗B2 · σC B2 · c · n ∗ph2 − 2 · σC B2 · c · n ∗ph2 + n ∗B2 · σ B1C · c · n ∗ph1 + [2 − 1 ] · σ B1C · c · n ∗ph1 + n ∗B2 · σ B2C · c · n ∗ph2 = 0 Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + n ∗B2 · n ∗ph1 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c · n ∗ph1 + n ∗B2 · n ∗ph2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c · n ∗ph2 = 0 (**) S B1 + n ∗c · (AC B1 + σC B1 · c · n ∗ph1 ) − n ∗B1 · σ B1C · c · n ∗ph1 = 0 S B1 + (n ∗B2 + 2 ) · (AC B1 + σC B1 · c · n ∗ph1 ) − (n ∗B2 + 2 − 1 ) · σ B1C · c · n ∗ph1 = 0 S B1 + 2 · AC B1 + n ∗B2 · AC B1 + n ∗B2 · n ∗ph1 · [σC B1 − σ B1C ] · c + (2 · σC B1 − [2 − 1 ] · σ B1C ) · c · n ∗ph1 = 0 (***) S B2 + n ∗c · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 S B2 + (n ∗B2 + 2 ) · (AC B2 + σC B2 · c · n ∗ph2 ) − n ∗B2 · σ B2C · c · n ∗ph2 = 0 S B2 + 2 · AC B2 + n ∗B2 · AC B2 + n ∗B2 · n ∗ph2 · (σC B2 − σ B2C ) · c + 2 · σC B2 · c · n ∗ph2 = 0 We can summary the three fixed points equations: (n ∗B2 , n ∗ph1 , n ∗ph2 ) Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] (D.1)

(D.2)

+ n ∗B2 · n ∗ph1 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c · n ∗ph1 + n ∗B2 · n ∗ph2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c · n ∗ph2 = 0 S B1 + 2 · AC B1 + n ∗B2 · AC B1 + n ∗B2 · n ∗ph1 · [σC B1 − σ B1C ] · c + (2 · σC B1 − [2 − 1 ] · σ B1C ) · c · n ∗ph1 = 0

6.1 Nitrogen Gas Laser Filament Plasma …

577

(D.3) S B2 +2 · AC B2 +n ∗B2 · AC B2 +n ∗B2 ·n ∗ph2 ·(σC B2 −σ B2C )·c+2 ·σC B2 ·c·n ∗ph2 = 0

(D.3) → n ∗ph2 =

(D.3) → (D.1):

−S B2 −2 ·AC B2 −n ∗B2 ·AC B2 n ∗B2 ·(σC B2 −σ B2C )·c+2 ·σC B2 ·c

Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + n ∗B2 · n ∗ph1 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c · n ∗ph1 + n ∗ph2 · {n ∗B2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c} = 0

Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + n ∗B2 · n ∗ph1 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c · n ∗ph1 −S B2 − 2 · AC B2 − n ∗B2 · AC B2 } ∗ n B2 · (σC B2 − σ B2C ) · c + 2 · σC B2 · c {n ∗B2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c} =

+{ ·

0

We get two equations with variables (n ∗ph1 , n ∗B2 ): Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + n ∗B2 · n ∗ph1 · (σ B1C − σC B1 ) · c (D.3) → (D.1):

(D.2)

+ ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c · n ∗ph1  −S B2 − 2 · AC B2 − n ∗B2 · AC B2 + n ∗B2 · (σC B2 − σ B2C ) · c + 2 · σC B2 · c

· {n ∗B2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c} = 0 S B1 + 2 · AC B1 + n ∗B2 · AC B1 + n ∗B2 · n ∗ph1 · [σC B1 − σ B1C ] · c + (2 · σC B1 − [2 − 1 ] · σ B1C ) · c · n ∗ph1 = 0 S B1 + 2 · AC B1 + n ∗B2 · AC B1 + {n ∗B2 · [σC B1 − σ B1C ] + (2 · σC B1 − [2 − 1 ] · σ B1C )} · c · n ∗ph1 = 0

n ∗ph1 =

−n ∗B2 · AC B1 − 2 · AC B1 − S B1 {n ∗B2 · [σC B1 − σ B1C ] + (2 · σC B1 − [2 − 1 ] · σ B1C )} · c

Inserting it to (D.3) → (D.1) equation gives Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + {n ∗B2 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c} · n ∗ph1 +{

−S B2 − 2 · AC B2 − n ∗B2 · AC B2 } · (σC B2 − σ B2C ) · c + 2 · σC B2 · c

n ∗B2

578

6 Gas Laser Systems Stability Analysis …

· {n ∗B2 · [σ B2C − σC B2 ] · c − 2 · σC B2 · c} = 0 Sc − 2 · [AC B1 + AC B2 ] − n ∗B2 · [AC B1 + AC B2 ] + {n ∗B2 · (σ B1C − σC B1 ) · c + ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c} −n ∗ · AC B1 − 2 · AC B1 − S B1 } · { ∗ B2 {n B2 · [σC B1 − σ B1C ] · c + (2 · σC B1 − [2 − 1 ] · σ B1C ) · c} −S B2 − 2 · AC B2 − n ∗B2 · AC B2 } · {n ∗B2 · [σ B2C − σC B2 ] +{ ∗ n B2 · (σC B2 − σ B2C ) · c + 2 · σC B2 · c · c − 2 · σC B2 · c} = 0 We define some parameters for simplicity: A1 = Sc − 2 · [AC B1 + AC B2 ]; A2 = AC B1 + AC B2 A3 = (σ B1C − σC B1 ) · c A4 = ([2 − 1 ] · σ B1C − 2 · σC B1 ) · c A5 = −2 · AC B1 − S B1 A6 = −S B2 − 2 · AC B2 ; [σC B1 − σ B1C ] · c = −A3 (2 · σC B1 − [2 − 1 ] · σ B1C ) · c = −A4 A7 = (σC B2 − σ B2C ) · c; A8 = 2 · σC B2 · c A9 = [σ B2C − σC B2 ] · c; A10 = 2 · σC B2 · c A1 − n ∗B2 · A2 + (n ∗B2 · A3 + A4 ) · ( +(

−n ∗B2 · AC B1 + A5 ) −n ∗B2 · A3 − A4

A6 − n ∗B2 · AC B2 ) · (n ∗B2 · A9 − A10 ) = 0 n ∗B2 · A7 + A8

(n ∗B2 · A3 + A4 ) · (−n ∗B2 · AC B1 + A5 ) −n ∗B2 · A3 − A4 (A6 − n ∗B2 · AC B2 ) · (n ∗B2 · A9 − A10 ) + =0 n ∗B2 · A7 + A8

A1 − n ∗B2 · A2 +

(n ∗B2 · A3 + A4 ) · (−n ∗B2 · AC B1 + A5 ) = −(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 (A6 − n ∗B2 · AC B2 ) · (n ∗B2 · A9 − A10 ) = n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9

6.1 Nitrogen Gas Laser Filament Plasma …

579

Then we get A1 − n ∗B2 · A2 +

−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 −n ∗B2 · A3 − A4

+

n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 =0 n ∗B2 · A7 + A8

(A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 )

+ {−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) +{n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) =0

−n ∗B2 · A3 − A4 = 0 ⇒ n ∗B2 = −

A4 ∗ A8 ; n · A7 + A8 = 0 ⇒ n ∗B2 = − A3 B2 A7

(A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) + {−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) + {n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = 0 Result 1: (A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) = (A1 − n ∗B2 · A2 ) · (−[n ∗B2 ]2 · A3 · A7 − n ∗B2 · [A3 · A8 + A4 · A7 ] − A4 · A8 ) = −[n ∗B2 ]2 · A3 · A7 · A1 − n ∗B2 · [A3 · A8 + A4 · A7 ] · A1 − A4 · A8 · A1 + [n ∗B2 ]3 · A3 · A7 · A2 + [n ∗B2 ]2 · [A3 · A8 + A4 · A7 ] · A2 + n ∗B2 · A4 · A8 · A2 (A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) = (A1 − n ∗B2 · A2 ) · (−[n ∗B2 ]2 · A3 · A7 − n ∗B2 · [A3 · A8 + A4 · A7 ] − A4 · A8 ) = [n ∗B2 ]3 · A3 · A7 · A2 + [n ∗B2 ]2 · {[A3 · A8 + A4 · A7 ] · A2 − A3 · A7 · A1 } + n ∗B2 · {A4 · A8 · A2 − [A3 · A8 + A4 · A7 ] · A1 } − A4 · A8 · A1

580

6 Gas Laser Systems Stability Analysis …

We define some parameters for simplicity: 1 = A3 · A7 · A2 ; 2 = [A3 · A8 + A4 · A7 ] · A2 − A3 · A7 · A1 3 = A4 · A8 · A2 − [A3 · A8 + A4 · A7 ] · A1 ; 4 = −A4 · A8 · A1 (A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) = [n ∗B2 ]3 · 1 + [n ∗B2 ]2 · 2 + n ∗B2 · 3 + 4 Result 2: {−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) = −(n ∗B2 )3 · A3 · AC B1 · A7 + (n ∗B2 )2 · [(A3 · A5 − AC B1 · A4 ) · A7 − A3 · AC B1 · A8 ] + n ∗B2 · (A3 · A5 − AC B1 · A4 ) · A8 + A4 · A5 · n ∗B2 · A7 + A4 · A5 · A8 = −(n ∗B2 )3 · A3 · AC B1 · A7 + (n ∗B2 )2 · [(A3 · A5 − AC B1 · A4 ) · A7 − A3 · AC B1 · A8 ] + n ∗B2 · {(A3 · A5 − AC B1 · A4 ) · A8 + A4 · A5 · A7 } + A4 · A5 · A8 We define some parameters for simplicity: 5 = −A3 · AC B1 · A7 ; 6 = (A3 · A5 − AC B1 · A4 ) · A7 − A3 · AC B1 · A8 7 = (A3 · A5 − AC B1 · A4 ) · A8 + A4 · A5 · A7 ; 8 = A4 · A5 · A8 {−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) = −(n ∗B2 )3 · 5 + (n ∗B2 )2 · 6 + n ∗B2 · 7 + 8 Result 3: {n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = (n ∗B2 )3 · AC B2 · A9 · A3 + (n ∗B2 )2 · AC B2 · A9 · A4 − (n ∗B2 )2 · (A9 · A6 + AC B2 · A10 ) · A3 + n ∗B2 · A6 · A10 · A3 − n ∗B2 · (A9 · A6 + AC B2 · A10 ) · A4 + A6 · A10 · A4 {n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = (n ∗B2 )3 · AC B2 · A9 · A3 + (n ∗B2 )2 · {AC B2 · A9 · A4 − (A9 · A6 + AC B2 · A10 ) · A3 } + n ∗B2 · {A6 · A10 · A3 − (A9 · A6 + AC B2 · A10 ) · A4 } + A6 · A10 · A4

6.1 Nitrogen Gas Laser Filament Plasma …

581

Table 6.2 System fixed points, results 1–3 and expressions Result No.

Expression

1

(A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) = [n ∗B2 ]3 · 1 + [n ∗B2 ]2 · 2 + n ∗B2 · 3 + 4

2

{−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) = −(n ∗B2 )3 · 5 + (n ∗B2 )2 · 6 + n ∗B2 · 7 + 8

3

{n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = (n ∗B2 )3 · 9 + (n ∗B2 )2 · 10 + n ∗B2 · 11 + 12

We define some parameters for simplicity: 9 = AC B2 · A9 · A3 ; 10 = AC B2 · A9 · A4 − (A9 · A6 + AC B2 · A10 ) · A3 11 = A6 · A10 · A3 − (A9 · A6 + AC B2 · A10 ) · A4 ; 12 = A6 · A10 · A4 {n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = (n ∗B2 )3 · 9 + (n ∗B2 )2 · 10 + n ∗B2 · 11 + 12 We can summary our three results in the next table (Table 6.2). (A1 − n ∗B2 · A2 ) · (−n ∗B2 · A3 − A4 ) · (n ∗B2 · A7 + A8 ) + {−(n ∗B2 )2 · A3 · AC B1 + n ∗B2 · (A3 · A5 − AC B1 · A4 ) + A4 · A5 } · (n ∗B2 · A7 + A8 ) + {n ∗B2 · (A9 · A6 + AC B2 · A10 ) − A6 · A10 − (n ∗B2 )2 · AC B2 · A9 } · (−n ∗B2 · A3 − A4 ) = 0 [n ∗B2 ]3 · 1 + [n ∗B2 ]2 · 2 + n ∗B2 · 3 + 4 − (n ∗B2 )3 · 5 + (n ∗B2 )2 · 6 + n ∗B2 · 7 + 8 + (n ∗B2 )3 · 9 + (n ∗B2 )2 · 10 + n ∗B2 · 11 + 12 = 0 [n ∗B2 ]3 · [1 − 5 + 9 ] + [n ∗B2 ]2 · [2 + 6 + 10 ] + n ∗B2 · [3 + 4 + 7 + 8 + 11 ] + 12 = 0 For specific parameters values, we can solve the above equation numerically and find

582

6 Gas Laser Systems Stability Analysis …

The possible values for fixed point coordinate n ∗B2 (the population of B 3 g in the first excited vibrational states for t → ∞). Then we get the fixed point coordinates: n ∗ph1 , n ∗ph2 . Stability analysis: The standard local stability analysis about any one of the equilibrium point of the plasma kinetic system for the populations of the lasing levels and number of generated photons consists in adding to coordinate [n c , n B1 , n B2 , n ph1 , n ph2 ] arbitrarily small increments of exponential form [n c , n B1 , n B2 , n ph1 , n ph2 ] · eλ·t and retaining the first order terms in n c , n B1 , n B2 , n ph1 , n ph2 . The system of five homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into plasma kinetic system equations. The plasma kinetic system fixed values with arbitrarily small increments of exponential form [n c , n B1 , n B2 , n ph1 , n ph2 ] · eλ·t are: j = 0(first fixed point), j = 1(second fixed point), j = 2(third fixed point), etc. [4, 5]. ( j)

n c (t) = n (c j) + n c · eλ·t ; n B1 (t) = n B1 + n B1 · eλ·t ( j)

( j)

n B2 (t) = n B2 + n B2 · eλ·t ; n ph1 (t) = n ph1 + n ph1 · eλ·t dn c (t) = n c · λ · eλ·t dt dn B1 (t) dn B2 (t) = n B1 · λ · eλ·t ; = n B2 · λ · eλ·t dt dt ( j)

n ph2 (t) = n ph2 + n ph2 · eλ·t ;

dn ph1 (t) = n ph1 · λ · eλ·t dt dn ph2 (t) = n ph2 · λ · eλ·t dt n c (t − τ D ) = n (c j) + n c · eλ·(t−τ D ) We choose these expressions for ourselves n c (t), n B1 (t), n B2 (t), n ph1 (t), n ph2 (t) as a small displacement [n c , n B1 , n B2 , n ph1 , n ph2 ] from the plasma kinetic system fixed points in time t = 0. ( j)

n c (t = 0) = n (c j) + n c ; n B1 (t = 0) = n B1 + n B1 ( j)

( j)

n B2 (t = 0) = n B2 + n B2 ; n ph1 (t = 0) = n ph1 + n ph1 ( j)

n ph2 (t = 0) = n ph2 + n ph2 dn c = Sc − n c (t − τ D ) dt · (AC B1 + AC B2 + σC B1 · c · n ph1 + σC B2 · c · n ph2 ) + n B1 · σ B1C · c · n ph1 + n B2 · σ B2C · c · n ph2

6.1 Nitrogen Gas Laser Filament Plasma …

583

n c · λ · eλ·t = Sc − [n (c j) + n c · eλ·(t−τ D ) ] ( j)

· (AC B1 + AC B2 + σC B1 · c · [n ph1 + n ph1 · eλ·t ] ( j)

( j)

+ σC B2 · c · [n ph2 + n ph2 · eλ·t ]) + [n B1 + n B1 · eλ·t ] ( j)

( j)

· σ B1C · c · [n ph1 + n ph1 · eλ·t ] + [n B2 + n B2 · eλ·t ] ( j)

· σ B2C · c · [n ph2 + n ph2 · eλ·t ] n c · λ · eλ·t = Sc − [n (c j) + n c · eλ·(t−τ D ) ] ( j)

( j)

· (AC B1 + AC B2 + σC B1 · c · n ph1 + σC B2 · c · n ph2 + σC B1 · c · n ph1 · eλ·t + σC B2 · c · n ph2 · eλ·t ) ( j)

( j)

( j)

( j)

+ [n B1 + n B1 · eλ·t ] · [n ph1 + n ph1 · eλ·t ] · σ B1C · c + [n B2 + n B2 · eλ·t ] · [n ph2 + n ph2 · eλ·t ] · σ B2C · c ( j)

n c · λ · eλ·t = Sc − (n (c j) · [AC B1 + AC B2 ] + n (c j) · n ph1 ( j)

· σC B1 · c + n (c j) · n ph2 · σC B2 · c + n (c j) · σC B1 · c · eλ·t · n ph1 + n (c j) · σC B2 · c · eλ·t · n ph2 ) ( j)

− ([AC B1 + AC B2 ] · eλ·(t−τ D ) · n c + σC B1 · c · n ph1 ( j)

· eλ·(t−τ D ) · n c + σC B2 · c · n ph2 · eλ·(t−τ D ) · n c + σC B1 · c · n ph1 · n c · eλ·t · eλ·(t−τ D ) + σC B2 · c · n ph2 · n c · eλ·t · eλ·(t−τ D ) ) ( j)

( j)

( j)

( j)

+ [n B1 · n ph1 + n B1 · n ph1 · eλ·t + n ph1 · n B1 · eλ·t ( j)

( j)

+ n B1 · n ph1 · eλ·t · eλ·t ] · σ B1C · c + [n B2 · n ph2 ( j)

( j)

+ n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t + n B2 · n ph2 · eλ·t · eλ·t ] · σ B2C · c We consider n ph1 · n c ≈ 0; n ph2 · n c ≈ 0; n B1 · n ph1 ≈ 0; n B2 · n ph2 ≈ 0 then ( j)

n c · λ · eλ·t = Sc − (n (c j) · [AC B1 + AC B2 ] + n (c j) · n ph1 · σC B1 · c ( j)

+ n (c j) · n ph2 · σC B2 · c + n (c j) · σC B1 · c · eλ·t · n ph1 + n (c j) · σC B2 · c · eλ·t · n ph2 ) − ([AC B1 + AC B2 ] · eλ·(t−τ D ) · n c ( j)

( j)

+ σC B1 · c · n ph1 · eλ·(t−τ D ) · n c + σC B2 · c · n ph2 · eλ·(t−τ D ) · n c ) ( j)

( j)

( j)

( j)

( j)

( j)

( j)

( j)

+ [n B1 · n ph1 + n B1 · n ph1 · eλ·t + n ph1 · n B1 · eλ·t ] · σ B1C · c + [n B2 · n ph2 + n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c

584

6 Gas Laser Systems Stability Analysis … ( j)

n c · λ · eλ·t = Sc − n (c j) · [AC B1 + AC B2 ] − n (c j) · n ph1 · σC B1 · c ( j)

− n (c j) · n ph2 · σC B2 · c − n (c j) · σC B1 · c · eλ·t · n ph1 − n (c j) · σC B2 · c · eλ·t · n ph2 − [AC B1 + AC B2 ] · eλ·(t−τ D ) · n c ( j)

( j)

− σC B1 · c · n ph1 · eλ·(t−τ D ) · n c − σC B2 · c · n ph2 · eλ·(t−τ D ) · n c ( j)

( j)

( j)

( j)

+ [n B1 · n ph1 ] · σ B1C · c + [n B1 · n ph1 · eλ·t + n ph1 · n B1 · eλ·t ] ( j)

( j)

· σ B1C · c + [n B2 · n ph2 ] · σ B2C · c ( j)

( j)

+ [n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c ( j)

n c · λ · eλ·t = Sc − n (c j) · [AC B1 + AC B2 ] − n (c j) · n ph1 · σC B1 · c ( j)

( j)

( j)

( j)

( j)

− n (c j) · n ph2 · σC B2 · c + [n B1 · n ph1 ] · σ B1C · c + [n B2 · n ph2 ] · σ B2C · c − n (c j) · σC B1 · c · eλ·t · n ph1 − n (c j) · σC B2 · c · eλ·t · n ph2 ( j)

− [AC B1 + AC B2 ] · eλ·t · e−λ·τ D · n c − σC B1 · c · n ph1 · eλ·t · e−λ·τ D ( j)

( j)

· n c − σC B2 · c · n ph2 · eλ·t · e−λ·τ D · n c + [n B1 · n ph1 · eλ·t ( j)

( j)

+ n ph1 · n B1 · eλ·t ] · σ B1C · c + [n B2 · n ph2 · eλ·t ( j)

+ n ph2 · n B2 · eλ·t ] · σ B2C · c ( j)

n c · λ · eλ·t = Sc − n (c j) · [AC B1 + AC B2 + n ph1 · σC B1 · c ( j)

( j)

( j)

( j)

( j)

+ n ph2 · σC B2 · c] + n B1 · n ph1 · σ B1C · c + n B2 · n ph2 · σ B2C · c − n (c j) · σC B1 · c · eλ·t · n ph1 − n (c j) · σC B2 · c · eλ·t ( j)

· n ph2 − [AC B1 + AC B2 ] · eλ·t · e−λ·τ D · n c − σC B1 · c · n ph1 ( j)

· eλ·t · e−λ·τ D · n c − σC B2 · c · n ph2 · eλ·t · e−λ·τ D · n c ( j)

( j)

( j)

( j)

+ [n B1 · n ph1 · eλ·t + n ph1 · n B1 · eλ·t ] · σ B1C · c + [n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c At fixed point:

dn c dt

= 0; lim n c (t − τ D ) = n c (t) ∀ t τ D t→0

( j)

( j)

Sc − n (c j) · [AC B1 + AC B2 + n ph1 · σC B1 · c + n ph2 · σC B2 · c] ( j)

( j)

( j)

( j)

+ n B1 · n ph1 · σ B1C · c + n B2 · n ph2 · σ B2C · c = 0 n c · λ · eλ·t = −n (c j) · σC B1 · c · eλ·t · n ph1 − n (c j) · σC B2 · c · eλ·t · n ph2 ( j)

− [AC B1 + AC B2 ] · eλ·t · e−λ·τ D · n c − σC B1 · c · n ph1

6.1 Nitrogen Gas Laser Filament Plasma …

585 ( j)

· eλ·t · e−λ·τ D · n c − σC B2 · c · n ph2 · eλ·t · e−λ·τ D · n c ( j)

( j)

( j)

( j)

+ [n B1 · n ph1 · eλ·t + n ph1 · n B1 · eλ·t ] · σ B1C · c + [n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c Divide the two sides of the above equation by eλ·t gives n c · λ = −n (c j) · σC B1 · c · n ph1 − n (c j) · σC B2 · c · n ph2 ( j)

− [AC B1 + AC B2 ] · e−λ·τ D · n c − σC B1 · c · n ph1 · e−λ·τ D · n c ( j)

( j)

( j)

− σC B2 · c · n ph2 · e−λ·τ D · n c + [n B1 · n ph1 + n ph1 · n B1 ] ( j)

( j)

· σ B1C · c + [n B2 · n ph2 + n ph2 · n B2 ] · σ B2C · c ( j)

− [AC B1 + AC B2 ] · e−λ·τ D · n c − σC B1 · c · n ph1 · e−λ·τ D · n c ( j)

( j)

− σC B2 · c · n ph2 · e−λ·τ D · n c − n c · λ + n ph1 · σ B1C · c · n B1 ( j)

( j)

+ n ph2 · σ B2C · c · n B2 − n (c j) · σC B1 · c · n ph1 + n B1 · σ B1C · c · n ph1 ( j)

− n (c j) · σC B2 · c · n ph2 + n B2 · σ B2C · c · n ph2 = 0 ( j)

( j)

{(−[AC B1 + AC B2 ] − σC B1 · c · n ph1 − σC B2 · c · n ph2 ) ( j)

( j)

· e−λ·τ D − λ} · n c + n ph1 · σ B1C · c · n B1 + n ph2 · σ B2C ( j)

· c · n B2 + [n B1 · σ B1C − n (c j) · σC B1 ] · c · n ph1 ( j)

+ [n B2 · σ B2C − n (c j) · σC B2 ] · c · n ph2 = 0 We define for simplicity some parameters: 1 = −[AC B1 + AC B2 ] − σC B1 · c · ( j) ( j) n ph1 − σC B2 · c · n ph2 ( j)

( j)

2 = n ph1 · σ B1C · c; 3 = n ph2 · σ B2C · c ( j)

( j)

4 = [n B1 · σ B1C − n (c j) · σC B1 ] · c; 5 = [n B2 · σ B2C − n (c j) · σC B2 ] · c The first arbitrarily small increments equation is (1 · e−λ·τ D − λ) · n c + 2 · n B1 + 3 · n B2 + 4 · n ph1 + 5 · n ph2 = 0 dn B1 = S B1 + n c (t − τ D ) · (AC B1 + σC B1 · c · n ph1 ) dt − n B1 · σ B1C · c · n ph1 n B1 · λ · eλ·t = S B1 + [n (c j) + n c · eλ·(t−τ D ) ] ( j)

· (AC B1 + σC B1 · c · [n ph1 + n ph1 · eλ·t ])

586

6 Gas Laser Systems Stability Analysis … ( j)

( j)

− (n B1 + n B1 · eλ·t ) · σ B1C · c · [n ph1 + n ph1 · eλ·t ] ( j)

n B1 · λ · eλ·t = S B1 + [n (c j) + n c · eλ·(t−τ D ) ] · ([AC B1 + σC B1 · c · n ph1 ] ( j)

( j)

+ σC B1 · c · n ph1 · eλ·t ) − [(n B1 + n B1 · eλ·t ) · σ B1C · c · n ph1 ( j)

+ (n B1 + n B1 · eλ·t ) · σ B1C · c · n ph1 · eλ·t ] n B1 · λ · eλ·t = S B1 + [n (c j) + n c · eλ·(t−τ D ) ] ( j)

· [AC B1 + σC B1 · c · n ph1 ] + [n (c j) + n c · eλ·(t−τ D ) ] ( j)

· σC B1 · c · n ph1 · eλ·t − (n B1 + n B1 · eλ·t ) · σ B1C ( j)

( j)

· c · n ph1 − (n B1 + n B1 · eλ·t ) · σ B1C · c · n ph1 · eλ·t ( j)

n B1 · λ · eλ·t = S B1 + n (c j) · AC B1 + n (c j) · σC B1 · c · n ph1 ( j)

+ AC B1 · n c · eλ·(t−τ D ) + σC B1 · c · n ph1 · n c · eλ·(t−τ D ) + n (c j) · σC B1 · c · n ph1 · eλ·t + n c · n ph1 · eλ·(t−τ D ) · σC B1 · c ( j)

( j)

( j)

· eλ·t − n B1 · σ B1C · c · n ph1 − n B1 · σ B1C · c · n ph1 · eλ·t ( j)

− n B1 · σ B1C · c · n ph1 · eλ·t − n B1 · n ph1 · eλ·t · σ B1C · c · eλ·t We consider n c · n ph1 ≈ 0; n B1 · n ph1 ≈ 0 then ( j)

n B1 · λ · eλ·t = S B1 + n (c j) · AC B1 + n (c j) · σC B1 · c · n ph1 ( j)

+ AC B1 · n c · eλ·(t−τ D ) + σC B1 · c · n ph1 · n c · eλ·(t−τ D ) ( j)

( j)

+ n (c j) · σC B1 · c · n ph1 · eλ·t − n B1 · σ B1C · c · n ph1 ( j)

( j)

− n B1 · σ B1C · c · n ph1 · eλ·t − n B1 · σ B1C · c · n ph1 · eλ·t ( j)

( j)

n B1 · λ · eλ·t = S B1 + n (c j) · [AC B1 + σC B1 · c · n ph1 ] − n B1 ( j)

( j)

· σ B1C · c · n ph1 + AC B1 · n c · eλ·(t−τ D ) + σC B1 · c · n ph1 · n c · eλ·(t−τ D ) + n (c j) · σC B1 · c · n ph1 · eλ·t ( j)

( j)

− n B1 · σ B1C · c · n ph1 · eλ·t − n B1 · σ B1C · c · n ph1 · eλ·t At fixed point:

dn B1 dt

= 0; lim n c (t − τ D ) = n c (t) ∀ t τ D t→0

( j)

( j)

( j)

S B1 + n (c j) · [AC B1 + σC B1 · c · n ph1 ] − n B1 · σ B1C · c · n ph1 = 0 ( j)

n B1 · λ · eλ·t = AC B1 · n c · eλ·(t−τ D ) + σC B1 · c · n ph1 · n c · eλ·(t−τ D )

6.1 Nitrogen Gas Laser Filament Plasma …

587 ( j)

+ n (c j) · σC B1 · c · n ph1 · eλ·t − n B1 · σ B1C · c · n ph1 · eλ·t ( j)

− n B1 · σ B1C · c · n ph1 · eλ·t ( j)

AC B1 · n c · eλ·t · e−λ·τ D + σC B1 · c · n ph1 · n c · eλ·t · e−λ·τ D ( j)

+ n (c j) · σC B1 · c · n ph1 · eλ·t − n B1 · σ B1C · c · n ph1 · eλ·t ( j)

− n B1 · σ B1C · c · n ph1 · eλ·t − n B1 · λ · eλ·t = 0 Divide the two sides of the above equation by eλ·t gives ( j)

AC B1 · n c · e−λ·τ D + σC B1 · c · n ph1 · n c · e−λ·τ D ( j)

+ n (c j) · σC B1 · c · n ph1 − n B1 · σ B1C · c · n ph1 ( j)

− n B1 · σ B1C · c · n ph1 − n B1 · λ = 0 ( j)

( j)

[AC B1 + σC B1 · c · n ph1 ] · n c · e−λ·τ D − σ B1C · c · n ph1 · n B1 ( j)

− n B1 · λ + [n (c j) · σC B1 − n B1 · σ B1C ] · c · n ph1 = 0 We define for simplicity some parameters: ( j)

( j)

6 = AC B1 + σC B1 · c · n ph1 ; 2 = n ph1 · σ B1C · c ( j)

− 4 = [n (c j) · σC B1 − n B1 · σ B1C ] · c The second arbitrarily small increments equation is 6 · n c · e−λ·τ D − 2 · n B1 − n B1 · λ − 4 · n ph1 = 0 dn B2 = S B2 + n c (t − τ D ) · (AC B2 + σC B2 · c · n ph2 ) dt − n B2 · σ B2C · c · n ph2 n B2 · λ · eλ·t = S B2 + [n (c j) + n c · eλ·(t−τ D ) ] · (AC B2 + σC B2 ( j)

( j)

· c · [n ph2 + n ph2 · eλ·t ]) − [n B2 + n B2 · eλ·t ] ( j)

· σ B2C · c · [n ph2 + n ph2 · eλ·t ] n B2 · λ · eλ·t = S B2 + [n (c j) + n c · eλ·(t−τ D ) ] · ([AC B2 + σC B2 ( j)

( j)

· c · n ph2 ] + σC B2 · c · n ph2 · eλ·t ) − [n B2 + n B2 · eλ·t ] ( j)

· [n ph2 + n ph2 · eλ·t ] · σ B2C · c

588

6 Gas Laser Systems Stability Analysis … ( j)

n B2 · λ · eλ·t = S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] ( j)

+ n (c j) · σC B2 · c · n ph2 · eλ·t + [AC B2 + σC B2 · c · n ph2 ] · n c · eλ·(t−τ D ) + eλ·(t−τ D ) · σC B2 · c · n c · n ph2 · eλ·t ( j)

( j)

( j)

( j)

− [n B2 · n ph2 + n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t + eλ·t · n B2 · n ph2 · eλ·t ] · σ B2C · c We consider n c · n ph2 ≈ 0; n B2 · n ph2 ≈ 0 then ( j)

n B2 · λ · eλ·t = S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] + n (c j) ( j)

· σC B2 · c · n ph2 · eλ·t + [AC B2 + σC B2 · c · n ph2 ] ( j)

( j)

( j)

( j)

· n c · eλ·(t−τ D ) − [n B2 · n ph2 + n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c ( j)

S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] + n (c j) · σC B2 · c · n ph2 · eλ·t ( j)

( j)

( j)

+ [AC B2 + σC B2 · c · n ph2 ] · n c · eλ·(t−τ D ) − [n B2 · n ph2 ( j)

( j)

+ n B2 · n ph2 · eλ·t + n ph2 · n B2 · eλ·t ] · σ B2C · c − n B2 · λ · eλ·t = 0 ( j)

S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] + n (c j) · σC B2 · c · n ph2 ( j)

( j)

( j)

· eλ·t + [AC B2 + σC B2 · c · n ph2 ] · n c · eλ·(t−τ D ) − n B2 · n ph2 ( j)

( j)

· σ B2C · c − n B2 · σ B2C · c · n ph2 · eλ·t − n ph2 · σ B2C · c · n B2 · eλ·t − n B2 · λ · eλ·t = 0 ( j)

( j)

( j)

S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] − n B2 · n ph2 · σ B2C · c ( j)

+ n (c j) · σC B2 · c · n ph2 · eλ·t + [AC B2 + σC B2 · c · n ph2 ] · n c ( j)

( j)

· eλ·(t−τ D ) − n B2 · σ B2C · c · n ph2 · eλ·t − n ph2 · σ B2C · c · n B2 · eλ·t − n B2 · λ · eλ·t = 0 At fixed point:

dn B2 dt

= 0; lim n c (t − τ D ) = n c (t) ∀ t τ D t→0

( j)

( j)

( j)

S B2 + n (c j) · [AC B2 + σC B2 · c · n ph2 ] − n B2 · n ph2 · σ B2C · c = 0 ( j)

n (c j) · σC B2 · c · n ph2 · eλ·t + [AC B2 + σC B2 · c · n ph2 ] ( j)

( j)

· n c · eλ·(t−τ D ) − n B2 · σ B2C · c · n ph2 · eλ·t − n ph2 · σ B2C · c · n B2 · eλ·t − n B2 · λ · eλ·t = 0

6.1 Nitrogen Gas Laser Filament Plasma …

589 ( j)

n (c j) · σC B2 · c · n ph2 · eλ·t + [AC B2 + σC B2 · c · n ph2 ] · n c ( j)

( j)

· eλ·t · e−λ·τ D − n B2 · σ B2C · c · n ph2 · eλ·t − n ph2 · σ B2C · c · n B2 · eλ·t − n B2 · λ · eλ·t = 0 Divide the two sides of the above equation by eλ·t gives ( j)

n (c j) · σC B2 · c · n ph2 + [AC B2 + σC B2 · c · n ph2 ] · n c · e−λ·τ D ( j)

( j)

− n B2 · σ B2C · c · n ph2 − n ph2 · σ B2C · c · n B2 − n B2 · λ = 0 ( j)

( j)

[AC B2 + σC B2 · c · n ph2 ] · n c · e−λ·τ D − n ph2 · σ B2C · c · n B2 ( j)

− n B2 · λ + [n (c j) · σC B2 − n B2 · σ B2C ] · c · n ph2 = 0 We define for simplicity some parameters: ( j)

( j)

7 = AC B2 + σC B2 · c · n ph2 ; 3 =n ph2 · σ B2C · c ( j)

− 5 = [n (c j) · σC B2 − n B2 · σ B2C ] · c The third arbitrarily small increments equation is 7 · n c · e−λ·τ D − 3 · n B2 − n B2 · λ − 5 · n ph2 = 0 dn ph1 σC B1 · (n c (t − τ D ) − n B1 ) · n ph1 · c n ph1 = − dt τ ph 1 + IIs1L n ph1 · λ · eλ·t ( j)

=

( j)

( j)

σC B1 · ([n c + n c · eλ·(t−τ D ) ] − [n B1 + n B1 · eλ·t ]) · [n ph1 + n ph1 · eλ·t ] · c 1+

IL Is1

( j)

−

[n ph1 + n ph1 · eλ·t ] τ ph n ph1 · λ · eλ·t =

σC B1 · c (1 +

IL ) Is1

( j)

· ([n (c j) − n B1 ] + [n c · e−λ·τ D ( j)

− n B1 ] · eλ·t ) · (n ph1 + n ph1 · eλ·t ) ( j)

− n ph1 · λ · eλ·t =

σC B1 · c (1 +

IL ) Is1

n ph1 τ ph

−

n ph1 · eλ·t τ ph ( j)

( j)

( j)

· ([n (c j) − n B1 ] · n ph1 + [n (c j) − n B1 ] ( j)

· n ph1 · eλ·t + n ph1 · [n c · e−λ·τ D − n B1 ] · eλ·t

590

6 Gas Laser Systems Stability Analysis … ( j)

+ [n c · e−λ·τ D − n B1 ] · eλ·t · n ph1 · eλ·t ) − n ph1 · λ · eλ·t =

σC B1 · c (1 +

( j)

n ph1 τ ph

−

n ph1 · eλ·t τ ph

( j)

· ([n (c j) − n B1 ] · n ph1

IL ) Is1

( j)

( j)

+ [n (c j) − n B1 ] · n ph1 · eλ·t + n ph1 · [n c · e−λ·τ D − n B1 ] · eλ·t + [n c · n ph1 · e−λ·τ D − n B1 · n ph1 ] · eλ·t · eλ·t ) ( j)

n ph1

−

τ ph

n ph1 · eλ·t τ ph

−

We consider n c · n ph1 ≈ 0; n B1 · n ph1 ≈ 0 then σC B1 · c

n ph1 · λ · eλ·t =

(1 +

IL ) Is1

( j)

( j)

( j)

· ([n (c j) − n B1 ] · n ph1 + [n (c j) − n B1 ] ( j)

· n ph1 · eλ·t + n ph1 · [n c · e−λ·τ D − n B1 ] · eλ·t ) ( j)

− n ph1 · λ · eλ·t =

n ph1

−

τ ph

σC B1 · c (1 +

IL ) Is1

n ph1 · eλ·t τ ph ( j)

( j)

· [n (c j) − n B1 ] · n ph1 + ( j)

· [n (c j) − n B1 ] · n ph1 · eλ·t

σC B1 · c

(1 + IIs1L ) σC B1 · c ( j) + · n ph1 (1 + IIs1L ) ( j)

· [n c · e n ph1 · λ · eλ·t =

−λ·τ D

σC B1 · c

− n B1 ] · e

λ·t

( j)

n ph1

−

τ ph

−

n ph1 · eλ·t τ ph

( j)

( j)

· [n (c j) − n B1 ] · n ph1 −

n ph1

τ ph (1 + IIs1L ) σC B1 · c σC B1 · c ( j) + · [n (c j) − n B1 ] · n ph1 · eλ·t + IL (1 + Is1 ) (1 + IIs1L ) ( j)

· n ph1 · [n c · e−λ·τ D − n B1 ] · eλ·t − At fixed point:

dn ph1 dt

= 0; lim n c (t − τ D ) = n c (t) ∀ t τ D t→0

σC B1 · c (1 + then

n ph1 · eλ·t τ ph

IL ) Is1

( j)

( j)

· [n (c j) − n B1 ] · n ph1 −

( j)

n ph1 τ ph

= 0,

6.1 Nitrogen Gas Laser Filament Plasma …

n ph1 · λ · eλ·t =

σC B1 · c IL ) Is1

591 ( j)

· [n (c j) − n B1 ] · n ph1 · eλ·t

(1 + σC B1 · c ( j) + · n ph1 · [n c · e−λ·τ D − n B1 ] · eλ·t IL (1 + Is1 ) −

n ph1 · eλ·t τ ph

Divide the two sides of the above equation by eλ·t gives σC B1 · c

( j)

· [n (c j) − n B1 ] · n ph1 (1 + IIs1L ) n ph1 σC B1 · c ( j) + · n ph1 · [n c · e−λ·τ D − n B1 ] − IL τ ph (1 + Is1 )

n ph1 · λ =

σC B1 · c (1 +

IL ) Is1

( j)

· [n (c j) − n B1 ] · n ph1 +

· [n c · e−λ·τ D − n B1 ] − σC B1 · c (1 +

IL ) Is1

( j)

· n ph1 · n c · e−λ·τ D −

· n B1 + (

σC B1 · c (1 +

IL ) Is1

σC B1 · c (1 +

IL ) Is1

( j)

· n ph1

n ph1 − n ph1 · λ = 0 τ ph σC B1 · c (1 + ( j)

IL ) Is1

· [n (c j) − n B1 ] −

( j)

· n ph1

1 ) · n ph1 − n ph1 · λ τ ph

=0 We define for simplicity some parameters: 8 = ( j)

σC B1 ·c I (1+ I L )

( j)

· n ph1 ; 9 =

s1

( j)

σC B1 ·c I (1+ I L )

·

s1

[n c − n B1 ] − τ1ph The fourth arbitrarily small increments equation is 8 · n c · e−λ·τ D − 8 · n B1 + 9 · n ph1 − n ph1 · λ = 0 dn ph2 σC B2 · (n c (t − τ D ) − n B2 ) · n ph2 · c n ph2 = − dt τ ph 1 + IIs2L n ph2 · λ · eλ·t ( j)

=

( j)

( j)

σC B2 · ([n c + n c · eλ·(t−τ D ) ] − [n B2 + n B2 · eλ·t ]) · [n ph2 + n ph2 · eλ·t ] · c 1+

IL Is2

( j)

−

[n ph2 + n ph2 · eλ·t ] τ ph n ph2 · λ · eλ·t =

σC B2 · c 1+

IL Is2

( j)

· ([n (c j) − n B2 ] + [n c · e−λ·τ D − n B2 ] · eλ·t )

592

6 Gas Laser Systems Stability Analysis … ( j)

n ph2

( j)

· [n ph2 + n ph2 · eλ·t ] − n ph2 · λ · eλ·t =

σC B2 · c 1+

τ ph

( j)

n ph2 · eλ·t τ ph

−

( j)

( j)

· ([n (c j) − n B2 ] · n ph2 + [n (c j) − n B2 ] · n ph2

IL Is2

( j)

· eλ·t + n ph2 · [n c · e−λ·τ D − n B2 ] · eλ·t + [n c · e−λ·τ D − n B2 ] · eλ·t · n ph2 · eλ·t ) ( j)

−

n ph2 τ ph

n ph2 · λ · eλ·t =

n ph2 · eλ·t τ ph

−

σC B2 · c 1+

IL Is2

( j)

( j)

( j)

· ([n (c j) − n B2 ] · n ph2 + [n (c j) − n B2 ] ( j)

· n ph2 · eλ·t + n ph2 · [n c · e−λ·τ D − n B2 ] · eλ·t + [n c · n ph2 · e−λ·τ D − n B2 · n ph2 ] · eλ·t · eλ·t ) ( j)

−

n ph2 τ ph

−

n ph2 · eλ·t τ ph

We consider n c · n ph2 ≈ 0; n B2 · n ph2 ≈ 0 then n ph2 · λ · eλ·t =

σC B2 · c 1+

IL Is2

( j)

( j)

( j)

· ([n (c j) − n B2 ] · n ph2 + [n (c j) − n B2 ] ( j)

· n ph2 · eλ·t + n ph2 · [n c · e−λ·τ D − n B2 ] · eλ·t ) ( j)

− n ph2 · λ · eλ·t =

n ph2 τ ph

−

σC B2 · c 1+

IL Is2

n ph2 · eλ·t τ ph ( j)

( j)

· [n (c j) − n B2 ] · n ph2 + ( j)

· [n (c j) − n B2 ] · n ph2 · eλ·t

σC B2 · c

1 + IIs2L σC B2 · c ( j) + · n ph2 1 + IIs2L ( j)

· [n c · e n ph2 · λ · eλ·t =

−λ·τ D

σC B2 · c 1+

IL Is2

− n B2 ] · e ( j)

λ·t

−

n ph2 τ ph

( j)

· [n (c j) − n B2 ] · n ph2 − ( j)

· [n (c j) − n B2 ] · n ph2 · eλ·t + · [n c · e−λ·τ D − n B2 ] · eλ·t −

− ( j)

n ph2

τ ph σC B2 · c 1+

n ph2 · eλ·t τ ph

IL Is2

+

σC B2 · c 1+

( j)

· n ph2

n ph2 · eλ·t τ ph

IL Is2

6.1 Nitrogen Gas Laser Filament Plasma …

At fixed point:

dn ph2 dt

593

= 0; lim n c (t − τ D ) = n c (t) ∀ t τ D t→0

σC B2 · c 1+

IL Is2

n ph2 · λ · eλ·t =

( j)

( j)

( j)

· [n (c j) − n B2 ] · n ph2 −

σC B2 · c

n ph2

=0

τ ph

( j)

· [n (c j) − n B2 ] · n ph2 · eλ·t 1 + IIs2L σC B2 · c ( j) + · n ph2 · [n c · e−λ·τ D − n B2 ] · eλ·t 1 + IIs2L −

n ph2 · eλ·t τ ph

Divide the two sides of the above equation by eλ·t gives σC B2 · c

( j)

· [n (c j) − n B2 ] · n ph2 1 + IIs2L n ph2 σC B2 · c ( j) + · n ph2 · [n c · e−λ·τ D − n B2 ] − τ ph 1 + IIs2L

n ph2 · λ =

σC B2 · c 1+

IL Is2

+(

( j)

· n ph2 · e−λ·τ D · n c −

σC B2 · c 1+

IL Is2

( j)

σC B2 · c 1+

· [n (c j) − n B2 ] −

IL Is2

1 ) · n ph2 − n ph2 · λ = 0 τ ph

We define for simplicity some parameters:10 = ( j) [n c

( j) n B2 ]

( j)

· n ph2 · n B2

σC B2 ·c I 1+ I L

( j)

· n ph2 ; 11 =

s2

σC B2 ·c I 1+ I L

·

s2

− − τ1ph The fives arbitrarily small increments equation is 10 · e−λ·τ D · n c − 10 · n B2 + 11 · n ph2 − n ph2 · λ = 0 We can summary our plasma kinetic system five arbitrarily small increments equations: (1 · e−λ·τ D − λ) · n c + 2 · n B1 + 3 · n B2 + 4 · n ph1 + 5 · n ph2 = 0 6 · e−λ·τ D · n c − 2 · n B1 − λ · n B1 − 4 · n ph1 = 0 7 · e−λ·τ D · n c − 3 · n B2 − λ · n B2 − 5 · n ph2 = 0 8 · e−λ·τ D · n c − 8 · n B1 + 9 · n ph1 − λ · n ph1 = 0 10 · e−λ·τ D · n c − 10 · n B2 + 11 · n ph2 − λ · n ph2 = 0 The small increments Jacobian of our plasma kinetic system is as follow:

594

6 Gas Laser Systems Stability Analysis …

⎛ ⎛

ϒ11 . . . ⎜ .. . . ⎝ . . ϒ51 · · ·

nc

⎞

⎞ ⎜ ⎟ ϒ15 ⎜ n B1 ⎟ ⎜ .. ⎟ · ⎜ n ⎟ −λ·τ D ⎟ − λ; ϒ12 = 2 . ⎠ ⎜ B2 ⎟ = 0; ϒ11 = 1 · e ⎟ ⎜ ⎝ n ph1 ⎠ ϒ55 n ph2

ϒ13 = 3 ; ϒ14 = 4 ; ϒ15 = 5 ϒ21 = 6 · e−λ·τ D ; ϒ22 = −2 − λ; ϒ23 = 0 ϒ24 = −4 ; ϒ25 = 0; ϒ31 = 7 · e−λ·τ D ; ϒ32 = 0 ϒ33 = −3 − λ; ϒ34 = 0; ϒ35 = −5 ϒ41 = 8 · e−λ·τ D ; ϒ42 = −8 ; ϒ43 = 0; ϒ44 = 9 − λ ϒ45 = 0; ϒ51 = 10 · e−λ·τ D ; ϒ52 = 0 ϒ53 = −10 ; ϒ54 = 0; ϒ55 = 11 − λ ⎛

ϒ11 . . . ⎜ .. . . A−λ· I =⎝ . .

⎞ ϒ15 .. ⎟ . ⎠

ϒ51 · · · ϒ55 ⎛ ϒ11 . . . ⎜ .. . . det(A − λ · I ) = det ⎝ . . ϒ51 · · ·

⎞ ϒ15 .. ⎟ . ⎠

ϒ55

det(A − λ · I ) = 0 ⎞ ϒ22 ϒ23 ϒ24 ϒ25 ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎟ det(A − λ · I ) = ϒ11 · det⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ23 ϒ24 ϒ25 ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ − ϒ12 · det ⎜ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟ + ϒ13 · det ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ⎛

ϒ51 ϒ52 ϒ54 ϒ55

6.1 Nitrogen Gas Laser Filament Plasma …

595

⎛

ϒ21 ⎜ ϒ31 − ϒ14 · det ⎜ ⎝ ϒ41 ϒ51 ⎛ ϒ21 ⎜ ϒ31 + ϒ15 · det ⎜ ⎝ ϒ41 ϒ51 ⎛

ϒ22 ⎜ ϒ32 Step 1: Find the expression for ϒ11 · det ⎜ ⎝ ϒ42 ϒ52

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53 ϒ54

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎞ ⎛ 0 −4 0 (−2 − λ) ⎜ 0 −5 ⎟ 0 (−3 − λ) ⎟ = (1 · e−λ·τ D − λ) · det ⎜ ⎠ ⎝ −8 0 (9 − λ) 0 0 (11 − λ) 0 −10 ⎛ ⎞ 0 −5 (−3 − λ) ⎠ = (1 · e−λ·τ D − λ) · {(−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 ⎛ ⎞⎫ 0 (−3 − λ) −5 ⎬ ⎠ −4 · det ⎝ −8 0 0 ⎭ 0 −10 (11 − λ) ⎛

ϒ22 ⎜ ϒ32 ϒ11 · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎛

⎞ 0 −5 (−3 − λ) ⎠ Step 1.1: (−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 ⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = (−2 − λ) · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} = (−2 − λ) · {}

596

6 Gas Laser Systems Stability Analysis …

⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = (−2 − λ) · {(−3 · 9 + (3 − 9 ·)λ + λ2 ) · (11 − λ) − 5 · 10 · 9 + 5 · 10 · λ} = (−2 − λ) · {−3 · 9 · 11 + 3 · 9 · λ + 11 · (3 − 9 ·) · λ − (3 − 9 ·) · λ2 + 11 · λ2 − λ3 − 5 · 10 · 9 + 5 · 10 · λ} ⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = (−2 − λ) · {(−3 · 9 + (3 − 9 ·)λ + λ2 ) · (11 − λ) − 5 · 10 · 9 + 5 · 10 · λ} = (−2 − λ) · {−3 · 9 · 11 − 5 · 10 · 9 + (3 · 9 + 11 · (3 − 9 ) + 5 · 10 ) · λ + (11 − 3 + 9 ) · λ2 − λ3 } ⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = (−2 − λ) · {(−3 · 9 + (3 − 9 ·)λ + λ2 ) · (11 − λ) − 5 · 10 · 9 + 5 · 10 · λ} = (−2 − λ) · {−(3 · 9 · 11 + 5 · 10 · 9 ) + (3 · 9 + 11 · (3 − 9 ) + 5 · 10 ) · λ + (11 − 3 + 9 ) · λ2 − λ3 } = (3 · 9 · 11 + 5 · 10 · 9 ) · 2 − 2 · (3 · 9 + 11 · [3 − 9 ] + 5 · 10 ) · λ − 2 · (11 − 3 + 9 ) · λ2 + 2 · λ3 + (3 · 9 · 11 + 5 · 10 · 9 ) · λ − (3 · 9 + 11 · (3 − 9 ) + 5 · 10 ) · λ2 − (11 − 3 + 9 ) · λ3 + λ4 ⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = (3 · 9 · 11 + 5 · 10 · 9 ) · 2 + {3 · 9 · 11 + 5 · 10 · 9 − 2 · (3 · 9 + 11 · [3 − 9 ] + 5 · 10 )} · λ − {2 · (11 − 3 + 9 ) + 3 · 9 + 11 · (3 − 9 ) + 5 · 10 } · λ2 + {2 − 11 + 3 − 9 }

6.1 Nitrogen Gas Laser Filament Plasma …

597

· λ3 + λ4 We define some global parameters: ι1 = (3 · 9 · 11 + 5 · 10 · 9 ) · 2 ι2 = 3 · 9 · 11 + 5 · 10 · 9 − 2 · (3 · 9 + 11 · [3 − 9 ] + 5 · 10 ) ι3 = −{2 · (11 − 3 + 9 ) + 3 · 9 + 11 · (3 − 9 ) + 5 · 10 }; ι4 =2 − 11 + 3 − 9 ⎛

⎞ 0 −5 (−3 − λ) ⎠ (−2 − λ) · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 = ι1 + ι2 · λ + ι3 · λ2 + ι4 · λ3 + λ4 ⎛

⎞ 0 (−3 − λ) −5 ⎠ Step 1.2: −4 · det ⎝ −8 0 0 0 −10 (11 − λ) ⎛

⎞ 0 (−3 − λ) −5 ⎠ − 4 · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 = −4 · {(3 + λ) · det 0 (11 − λ)   −8 0 } − 5 · det 0 −10 = −4 · {−(3 + λ) · 8 · (11 − λ) − 5 · 8 · 10 } = −4 · {−5 · 8 · 10 − 8 · 3 · 11 − 8 · [11 − 3 ] · λ + 8 · λ2 } = 4 · 8 · (5 · 10 + 3 · 11 ) + 4 · 8 · [11 − 3 ] · λ − 4 · 8 · λ2 We define some global parameters: ι5 = 4 · 8 · (5 · 10 + 3 · 11 ); ι6 = 4 · 8 · [11 − 3 ] ⎛

⎞ 0 (−3 − λ) −5 ⎠ ι7 = −4 · 8 ; −4 · det ⎝ −8 0 0 0 −10 (11 − λ) = ι5 + ι6 · λ + ι7 · λ2 We can summary:

598

6 Gas Laser Systems Stability Analysis …

⎛

ϒ22 ⎜ ϒ32 ϒ11 · det ⎜ ⎝ ϒ42 ϒ52 = (1 · e−λ·τ D

= (1 · e−λ·τ D

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎞ ⎛ 0 −4 0 (−2 − λ) ⎜ 0 −5 ⎟ 0 (−3 − λ) ⎟ − λ) · det ⎜ ⎠ ⎝ −8 0 (9 − λ) 0 0 (11 − λ) 0 −10 ⎛ ⎞ 0 −5 (−3 − λ) ⎠ − λ) · {(−2 − λ) · det ⎝ 0 (9 − λ) 0

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

−10 ⎞

⎛

0

(11 − λ)

0 (−3 − λ) −5 ⎠} − 4 · det ⎝ −8 0 0 0 −10 (11 − λ)

= (1 · e−λ·τ D − λ) · {ι1 + ι2 · λ + ι3 · λ2 + ι4 · λ3 + λ4 + ι5 + ι6 · λ + ι7 · λ2 } = (1 · e−λ·τ D − λ) · {ι1 + ι5 + (ι2 + ι6 ) · λ + (ι3 + ι7 ) · λ2 + ι4 · λ3 + λ4 } ⎛

ϒ22 ⎜ ϒ32 ϒ11 · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= (1 · e−λ·τ D − λ) · {(ι1 + ι5 ) + (ι2 + ι6 ) · λ + (ι3 + ι7 ) · λ2 + ι4 · λ3 + λ4 } = (ι1 + ι5 ) · 1 · e−λ·τ D + (ι2 + ι6 ) · λ · 1 · e−λ·τ D + (ι3 + ι7 ) · λ2 · 1 · e−λ·τ D + ι4 · λ3 · 1 · e−λ·τ D + λ4 · 1 · e−λ·τ D − (ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 ⎛

ϒ22 ⎜ ϒ32 ϒ11 · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= −(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 + {(ι1 + ι5 ) · 1 + (ι2 + ι6 ) · λ · 1 + (ι3 + ι7 ) · λ2 · 1 + ι4 · λ3 · 1 + λ4 · 1 } · e−λ·τ D ⎛

ϒ21 ⎜ ϒ31 Step 2: Find the expression for ϒ12 · det ⎜ ⎝ ϒ41 ϒ51

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

6.1 Nitrogen Gas Laser Filament Plasma …

599

⎞ ϒ21 ϒ23 ϒ24 ϒ25 ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ϒ12 · det ⎜ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎛ 0 −4 0 6 · e−λ·τ D ⎜ 7 · e−λ·τ D (−3 − λ) 0 −5 ⎟ ⎟ = 2 · det ⎜ ⎠ ⎝ 8 · e−λ·τ D 0 (9 − λ) 0 −λ·τ D −10 0 (11 − λ) 10 · e ⎛ ⎞ 0 −5 (−3 − λ) ⎠ = 2 · {6 · e−λ·τ D · det ⎝ 0 (9 − λ) 0 ⎛

−10

⎛

0

(11 − λ) ⎞

7 · e−λ·τ D (−3 − λ) −5 ⎝ ⎠} − 4 · det 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e ⎛

Step 2.1: 6 · e−λ·τ D

⎞ 0 −5 (−3 − λ) ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

⎛

⎞ 0 −5 (−3 − λ) ⎠ 6 · e−λ·τ D · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10   0 (9 − λ) −λ·τ D = 6 · e · {(−3 − λ) · det 0 (11 − λ)   0 (9 − λ) } − 5 · det 0 −10 = 6 · e−λ·τ D · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} ⎛

6 · e−λ·τ D

⎞ (−3 − λ) 0 −5 ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

= 6 · e−λ·τ D · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} = 6 · e−λ·τ D · {(−3 · 9 + [3 − 9 ] · λ + λ2 ) · (11 − λ) − 5 · 10 · 9 + 5 · 10 · λ}

600

6 Gas Laser Systems Stability Analysis …

⎛

6 · e−λ·τ D

⎞ 0 −5 (−3 − λ) ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

= 6 · e−λ·τ D · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} = 6 · e−λ·τ D · {−3 · 9 · 11 + 3 · 9 · λ + 11 · [3 − 9 ] · λ − [3 − 9 ] · λ2 + 11 · λ2 − λ3 − 5 · 10 · 9 + 5 · 10 · λ} ⎛

6 · e−λ·τ D

⎞ 0 −5 (−3 − λ) ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

= 6 · e−λ·τ D · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} = 6 · e−λ·τ D · {(−3 · 11 − 5 · 10 ) · 9 + (3 · 9 + 11 · [3 − 9 ] + 5 · 10 ) · λ + (11 − 3 + 9 ) · λ2 − λ3 } ⎛

6 · e−λ·τ D

⎞ 0 −5 (−3 − λ) ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

= 6 · e−λ·τ D · {(−3 − λ) · (9 − λ) · (11 − λ) − 5 · 10 · (9 − λ)} = {6 · (−3 · 11 − 5 · 10 ) · 9 + 6 · (3 · 9 + 11 · [3 − 9 ] + 5 · 10 ) · λ + 6 · (11 − 3 + 9 ) · λ2 − 6 · λ3 } · e−λ·τ D We define some global parameters: ι8 = 6 · (−3 · 11 − 5 · 10 ) · 9 ι9 = 6 · (3 · 9 + 11 · [3 − 9 ] + 5 · 10 ) ι10 = 6 · (11 − 3 + 9 ); ι11 = −6 ⎛

6 · e

−λ·τ D

⎞ 0 −5 (−3 − λ) ⎠ · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10

= {ι8 + ι9 · λ + ι10 · λ2 + ι11 · λ3 } · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

601

⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ Step 2.2: 4 · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 −10 (11 − λ) 10 · e−λ·τ D   0 0 = 4 · {7 · e−λ·τ D · det −10 (11 − λ)   0 8 · e−λ·τ D − (−3 − λ) · det 10 · e−λ·τ D (11 − λ)   8 · e−λ·τ D 0 } − 5 · det 10 · e−λ·τ D −10 ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 10 · e−λ·τ D

−10

(11 − λ)

 0 8 · e 10 · e−λ·τ D (11 − λ)   8 · e−λ·τ D 0 } − 5 · det 10 · e−λ·τ D −10 ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 −10 (11 − λ) 10 · e−λ·τ D 

= 4 · {(3 + λ) · det

−λ·τ D

= 4 · {(3 + λ) · 8 · e−λ·τ D · (11 − λ) + 5 · 8 · e−λ·τ D · 10 } ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {4 · (3 + λ) · 8 · (11 − λ) + 4 · 5 · 8 · 10 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {4 · 8 · (3 · 11 + [11 − 3 ] · λ − λ2 )

602

6 Gas Laser Systems Stability Analysis …

+ 4 · 5 · 8 · 10 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ 4 · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {(3 · 11 + 5 · 10 ) · 8 · 4 + 4 · 8 · [11 − 3 ] · λ − 4 · 8 · λ2 } · e−λ·τ D We define some global parameters: ι12 = (3 · 11 + 5 · 10 ) · 8 · 4 ; ι13 = 4 · 8 · [11 − 3 ] ⎛

ι14

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ = −4 · 8 ; 4 · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {ι12 + ι13 · λ + ι14 · λ2 } · e−λ·τ D

We can summary: ⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎞ ⎛ 0 −5 (−3 − λ) ⎠ = 2 · {6 · e−λ·τ D · det ⎝ 0 (9 − λ) 0 0 (11 − λ) −10 ⎛ ⎞⎫ −λ·τ D (−3 − λ) −5 7 · e ⎬ ⎠ −4 · det ⎝ 8 · e−λ·τ D 0 0 ⎭ −10 (11 − λ) 10 · e−λ·τ D ⎛

ϒ21 ⎜ ϒ31 ϒ12 · det ⎜ ⎝ ϒ41 ϒ51

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

= 2 · {(ι8 + ι9 · λ + ι10 · λ2 + ι11 · λ3 ) · e−λ·τ D − (ι12 + ι13 · λ + ι14 · λ2 ) · e−λ·τ D } ⎛

ϒ21 ⎜ ϒ31 ϒ12 · det ⎜ ⎝ ϒ41 ϒ51

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= 2 · {(ι8 − ι12 ) + (ι9 − ι13 ) · λ + (ι10 − ι14 ) · λ2 + ι11 · λ3 } · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

⎛

ϒ21 ⎜ ϒ31 ϒ12 · det ⎜ ⎝ ϒ41 ϒ51

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

603

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= {2 · (ι8 − ι12 ) + 2 · (ι9 − ι13 ) · λ + 2 · (ι10 − ι14 ) · λ2 + 2 · ι11 · λ3 } · e−λ·τ D ⎛

ϒ21 ⎜ ϒ31 Step 3: Find the expression for ϒ13 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

⎞ 0 6 · e−λ·τ D (−2 − λ) −4 ⎜ 7 · e−λ·τ D 0 0 −5 ⎟ ⎟ 3 · det ⎜ ⎠ ⎝ 8 · e−λ·τ D −8 (9 − λ) 0 −λ·τ D 0 0 (11 − λ) 10 · e ⎛ ⎞ 0 0 −5 ⎠ = 3 · {6 · e−λ·τ D · det ⎝ −8 (9 − λ) 0 0 0 (11 − λ) ⎛ ⎞ −λ·τ D 0 −5 7 · e ⎠ − (−2 − λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 ⎛

⎛

10 · e−λ·τ D

−λ·τ D

0

⎞

(11 − λ)

0 −5 7 · e ⎠} − 4 · det⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e ⎞ ⎛ 0 0 −5 ⎠ det ⎝ −8 (9 − λ) 0 0 0 (11 − λ)   −8 (9 − λ) = −5 · det =0 0 0 ⎞ ⎛ 0 6 · e−λ·τ D (−2 − λ) −4 ⎜ 7 · e−λ·τ D 0 0 −5 ⎟ ⎟ 3 · det ⎜ −λ·τ ⎠ ⎝ 8 · e D −8 (9 − λ) 0 −λ·τ D 0 0 (11 − λ) 10 · e ⎛ ⎞ −λ·τ D 0 −5 7 · e ⎠ = 3 · {(2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e

604

6 Gas Laser Systems Stability Analysis …

⎛

⎞ −5 7 · e−λ·τ D 0 ⎠} − 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ Step 3.1: (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 0 (11 − λ) 10 · e−λ·τ D ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 0 (11 − λ) 10 · e−λ·τ D   0 (9 − λ) = (2 + λ) · {7 · e−λ·τ D · det 0 (11 − λ)   −λ·τ D (9 − λ) 8 · e } − 5 · det 0 10 · e−λ·τ D = (2 + λ) · {7 · e−λ·τ D · (9 − λ) · (11 − λ) + 5 · 10 · e−λ·τ D · (9 − λ)} ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e = (2 + λ) · {7 · e−λ·τ D · (9 · 11 − [9 + 11 ] · λ + λ2 ) + 9 · 5 · 10 · e−λ·τ D − λ · 5 · 10 · e−λ·τ D } ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e = (2 + λ) · {7 · 9 · 11 · e−λ·τ D − 7 · [9 + 11 ] · λ · e−λ·τ D + 7 · λ2 · e−λ·τ D + 9 · 5 · 10 · e−λ·τ D − λ · 5 · 10 · e−λ·τ D } ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e = (2 + λ) · {(7 · 9 · 11 + 9 · 5 · 10 ) − (7 · [9 + 11 ] + 5 · 10 ) · λ + 7 · λ2 } · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

605

⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e = {2 · (7 · 9 · 11 + 9 · 5 · 10 ) + [(7 · 9 · 11 + 9 · 5 · 10 ) − 2 · (7 · [9 + 11 ] + 5 · 10 )] · λ + [2 · 7 − (7 · [9 + 11 ] + 5 · 10 )] · λ2 + 7 · λ3 } · e−λ·τ D We define some global parameters: ι15 = 2 · (7 · 9 · 11 + 9 · 5 · 10 ) ι16 = (7 · 9 · 11 + 9 · 5 · 10 ) − 2 · (7 · [9 + 11 ] + 5 · 10 ) ι17 = 2 · 7 − (7 · [9 + 11 ] + 5 · 10 ); ι18 = 7 ⎛

⎞ 0 −5 7 · e−λ·τ D ⎠ (2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e = {ι15 + ι16 · λ + ι17 · λ2 + ι18 · λ3 } · e−λ·τ D ⎛

⎞ −5 7 · e−λ·τ D 0 ⎠ Step 3.2: 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e ⎛

⎞ −5 7 · e−λ·τ D 0 ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e   0 −8 = 4 · {7 · e−λ·τ D · det 0 (11 − λ)   −λ·τ D −8 8 · e } − 5 · det 10 · e−λ·τ D 0 = 4 · {7 · e−λ·τ D · (−8 ) · (11 − λ) − 5 · 8 · 10 · e−λ·τ D } ⎛

⎞ −5 7 · e−λ·τ D 0 ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e   0 −8 −λ·τ D = 4 · {7 · e · det 0 (11 − λ)   8 · e−λ·τ D −8 } − 5 · det 10 · e−λ·τ D 0

606

6 Gas Laser Systems Stability Analysis …

= {−4 · 8 · (7 · 11 + 5 · 10 ) + 4 · 7 · 8 · λ} · e−λ·τ D We define some global parameters: ι19 = −4 · 8 · (7 · 11 + 5 · 10 ); ι20 = 4 · 7 · 8 ⎛

⎞ −5 7 · e−λ·τ D 0 ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e   0 −8 = 4 · {7 · e−λ·τ D · det 0 (11 − λ)   −λ·τ D −8 8 · e } = {ι19 + ι20 · λ} · e−λ·τ D − 5 · det 10 · e−λ·τ D 0 We can summary: ⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎛ ⎞ 0 −5 7 · e−λ·τ D ⎠ == 3 · {(2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e ⎛ ⎞ −λ·τ D 0 −5 7 · e ⎠} − 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 (11 − λ) 10 · e ⎛

ϒ21 ⎜ ϒ31 ϒ13 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

= 3 · {(ι15 + ι16 · λ + ι17 · λ2 + ι18 · λ3 ) · e−λ·τ D − (ι19 + ι20 · λ) · e−λ·τ D } ⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎛ ⎞ 0 −5 7 · e−λ·τ D ⎠ == 3 · {(2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e ⎛ ⎞ −λ·τ D 0 −5 7 · e −λ·τ D ⎝ ⎠} − 4 · det 8 · e −8 0 −λ·τ D 0 (11 − λ) 10 · e ⎛

ϒ21 ⎜ ϒ31 ϒ13 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

= 3 · {(ι15 − ι19 ) + (ι16 − ι20 ) · λ + ι17 · λ2 + ι18 · λ3 } · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

607

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎛ ⎞ 0 −5 7 · e−λ·τ D ⎠ == 3 · {(2 + λ) · det ⎝ 8 · e−λ·τ D (9 − λ) 0 −λ·τ D 0 (11 − λ) 10 · e ⎛ ⎞ −λ·τ D 0 −5 7 · e ⎠} − 4 · det ⎝ 8 · e−λ·τ D −8 0 ⎛

ϒ21 ⎜ ϒ31 ϒ13 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

10 · e−λ·τ D

0

(11 − λ)

= {3 · (ι15 − ι19 ) + 3 · (ι16 − ι20 ) · λ + 3 · ι17 · λ2 + 3 · ι18 · λ3 } · e−λ·τ D ⎛

ϒ21 ⎜ ϒ31 Step 4: Find the expression for ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

⎞ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55 ⎞ ⎛ 0 0 6 · e−λ·τ D (−2 − λ) ⎜ 7 · e−λ·τ D 0 (−3 − λ) −5 ⎟ ⎟ = 4 · det ⎜ ⎠ ⎝ 8 · e−λ·τ D −8 0 0 −λ·τ D 0 −10 (11 − λ) 10 · e ⎞ ⎛ 0 0 6 · e−λ·τ D (−2 − λ) ⎜ 7 · e−λ·τ D 0 (−3 − λ) −5 ⎟ ⎟ 4 · det ⎜ ⎠ ⎝ 8 · e−λ·τ D −8 0 0 ⎛

10 · e−λ·τ D

⎛

0

−10

(11 − λ) ⎞ −5 ⎠ 0

0 (−3 − λ) = 4 · {6 · e−λ·τ D · det ⎝ −8 0 0 −10 (11 − λ) ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠} − (−2 − λ) · det⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e

608

6 Gas Laser Systems Stability Analysis …

⎛

Step 4.1: 6 · e−λ·τ D

⎞ 0 (−3 − λ) −5 ⎠ · det ⎝ −8 0 0 0 −10 (11 − λ) ⎛

⎞ 0 (−3 − λ) −5 ⎠ 6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 = 6 · e−λ·τ D · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } − 5 · det 0 −10 = 6 · e−λ·τ D · {(3 + λ) · (−8 ) · (11 − λ) − 5 · 8 · 10 } ⎛

⎞ 0 (−3 − λ) −5 ⎠ 6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 −λ·τ D = 6 · e · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } − 5 · det 0 −10 = 6 · e−λ·τ D · {(−8 ) · (3 · 11 + (11 − 3 ) · λ − λ2 ) − 5 · 8 · 10 } ⎛

⎞ 0 (−3 − λ) −5 ⎠ 6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 = 6 · e−λ·τ D · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } − 5 · det 0 −10 = 6 · e−λ·τ D · {−8 · 3 · 11 + 8 · (−11 + 3 ) · λ + 8 · λ2 − 5 · 8 · 10 } ⎛

6 · e−λ·τ D

⎞ 0 (−3 − λ) −5 ⎠ · det ⎝ −8 0 0 0 −10 (11 − λ)

6.1 Nitrogen Gas Laser Filament Plasma …

609

 0 −8 = 6 · e · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } = 6 · {−8 · (3 · 11 − 5 · det 0 −10 

−λ·τ D

+ 5 · 10 ) + 8 · (−11 + 3 ) · λ + 8 · λ2 } · e−λ·τ D ⎛

⎞ 0 (−3 − λ) −5 ⎠ 6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 = 6 · e−λ·τ D · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } = {−8 · 6 · (3 · 11 + 5 · 10 ) − 5 · det 0 −10 + 8 · 6 · (−11 + 3 ) · λ + 6 · 8 · λ2 } · e−λ·τ D We define some global parameters: ι21 = −8 · 6 · (3 · 11 + 5 · 10 ); ι22 = 8 · 6 · (−11 + 3 ) ι23 = 6 · 8 ⎛

⎞ 0 (−3 − λ) −5 ⎠ 6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ)   0 −8 −λ·τ D = 6 · e · {−(−3 − λ) · det 0 (11 − λ)   −8 0 } − 5 · det 0 −10 = (ι21 + ι22 · λ + ι23 · λ2 ) · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ Step 4.2: (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e   0 0 −λ·τ D = (−2 − λ) · {7 · e · det −10 (11 − λ)

610

6 Gas Laser Systems Stability Analysis …

 0 8 · e−λ·τ D − (−3 − λ) · det 10 · e−λ·τ D (11 − λ)   8 · e−λ·τ D 0 } − 5 · det 10 · e−λ·τ D −10   0 0 −λ·τ D 7 · e =0 · det −10 (11 − λ) ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e   0 8 · e−λ·τ D = (−2 − λ) · {(3 + λ) · det 10 · e−λ·τ D (11 − λ)   8 · e−λ·τ D 0 } − 5 · det 10 · e−λ·τ D −10 ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e 

= (−2 − λ) · {(3 + λ) · 8 · e−λ·τ D · (11 − λ) − 5 · 8 · e−λ·τ D · (−10 )} ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = (−2 − λ) · {(3 + λ) · 8 · (11 − λ) + 5 · 8 · 10 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = (−2 − λ) · {8 · (3 · 11 + 5 · 10 ) + 8 · [11 − 3 ] · λ − 8 · λ2 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = −{2 · 8 · (3 · 11 + 5 · 10 ) + [2 · 8 · [11 − 3 ] + 8 · (3 · 11 + 5 · 10 )] · λ + (8 · [11 − 3 ] − 2 · 8 )

6.1 Nitrogen Gas Laser Filament Plasma …

· λ2 − 8 · λ3 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {−2 · 8 · (3 · 11 + 5 · 10 ) − (2 · 8 · [11 − 3 ] + 8 · [3 · 11 + 5 · 10 ]) · λ − (8 · [11 − 3 ] − 2 · 8 ) · λ2 + 8 · λ3 } · e−λ·τ D We define some global parameters: ι24 = −2 · 8 · (3 · 11 + 5 · 10 ) ι25 = −(2 · 8 · [11 − 3 ] + 8 · [3 · 11 + 5 · 10 ]) ι26 = −(8 · [11 − 3 ] − 2 · 8 ); ι27 = 8 ⎛

⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠ (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = (ι24 + ι25 · λ + ι26 · λ2 + ι27 · λ3 ) · e−λ·τ D We can summary: ⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎛

⎞ 0 (−3 − λ) −5 ⎠ = 4 · {6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ) ⎛ ⎞ 7 · e−λ·τ D (−3 − λ) −5 ⎠} − (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = 4 · {(ι21 + ι22 · λ + ι23 · λ2 ) · e−λ·τ D − (ι24 + ι25 · λ + ι26 · λ2 + ι27 · λ3 ) · e−λ·τ D } ⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

611

612

6 Gas Laser Systems Stability Analysis …

⎛

⎞ 0 (−3 − λ) −5 ⎠ = 4 · {6 · e−λ·τ D · det ⎝ −8 0 0 0 −10 (11 − λ) ⎛ ⎞ −λ·τ D (−3 − λ) −5 7 · e ⎠} − (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 0 −λ·τ D −10 (11 − λ) 10 · e = {4 · (ι21 − ι24 ) + 4 · (ι22 − ι25 ) · λ + 4 · (ι23 − ι26 ) · λ2 − 4 · ι27 · λ3 } · e−λ·τ D ⎛

ϒ21 ⎜ ϒ31 Step 5: Find the expression for ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

⎞ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 6 · e−λ·τ D (−2 − λ) 0 −4 ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ⎛

ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 0 −4 6 · e−λ·τ D (−2 − λ) ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎛ ⎞ 0 0 (−3 − λ) = 5 · {6 · e−λ·τ D · det ⎝ −8 0 (9 − λ) ⎠ 0 0 −10 ⎛ ⎞ 0 7 · e−λ·τ D (−3 − λ) − (−2 − λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e

6.1 Nitrogen Gas Laser Filament Plasma …

613

⎛

⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠} − (−4 ) · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e ⎛ ⎞ 0 0 (−3 − λ) 6 · e−λ·τ D · det ⎝ −8 0 (9 − λ) ⎠ 0 0 −10   −8 (9 − λ) −λ·τ D = 6 · e · {−(−3 − λ) · det }=0 0 0 ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 0 −4 6 · e−λ·τ D (−2 − λ) ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎞ ⎛ −λ·τ D (−3 − λ) 0 7 · e = 5 · {−(−2 − λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −10 0 10 · e−λ·τ D ⎛ ⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎝ ⎠} − (−4 ) · det 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 0 −4 6 · e−λ·τ D (−2 − λ) ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎞ ⎛ −λ·τ D (−3 − λ) 0 7 · e = 5 · {(2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ ⎛

10 · e−λ·τ D

−λ·τ D

7 · e + 4 · det ⎝ 8 · e−λ·τ D 10 · e−λ·τ D

−10 ⎞ 0 (−3 − λ) ⎠} −8 0 0 −10

0

614

6 Gas Laser Systems Stability Analysis …

⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) Step 5.1: (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −10 0 10 · e−λ·τ D   0 (9 − λ) = (2 + λ) · {7 · e−λ·τ D · det 0 −10   −λ·τ D (9 − λ) 8 · e } − (−3 − λ) · det 0 10 · e−λ·τ D ⎞ ⎛ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = (2 + λ) · {7 · e−λ·τ D · 10 · (9 − λ) − (−3 − λ) · (−10 · e−λ·τ D ) · (9 − λ)} ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = (2 + λ) · {7 · 10 · (9 − λ) · e−λ·τ D − (3 + λ) · 10 · (9 − λ) · e−λ·τ D } ⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e ⎛

= (2 + λ) · {7 · 10 · (9 − λ) − (3 + λ) · 10 · (9 − λ)} · e−λ·τ D ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = (2 + λ) · {7 · 10 · 9 − 7 · 10 · λ − 10 · (3 · 9 + [9 − 3 ] · λ − λ2 )} · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

615

⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = (2 + λ) · {7 · 10 · 9 − 7 · 10 · λ − 10 · 3 · 9 − 10 · [9 − 3 ] · λ + 10 · λ2 } · e−λ·τ D ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = (2 + λ) · {(7 − 3 ) · 10 · 9 − (7 · 10 + 10 · [9 − 3 ]) · λ + 10 · λ2 } · e−λ·τ D ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = {2 · (7 − 3 ) · 10 · 9 − 2 · (7 · 10 + 10 · [9 − 3 ]) · λ + 2 · 10 · λ2 + (7 − 3 ) · 10 · 9 · λ − (7 · 10 + 10 · [9 − 3 ]) · λ2 + 10 · λ3 } · e−λ·τ D ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = {2 · (7 − 3 ) · 10 · 9 + [(7 − 3 ) · 10 · 9 − 2 · (7 · 10 + 10 · [9 − 3 ])] · λ + [2 · 10 − (7 · 10 + 10 · [9 − 3 ])] · λ2 + 10 · λ3 } · e−λ·τ D We define some global parameters: ι28 = 2 · (7 − 3 ) · 10 · 9 ; ι31 = 10 ι29 = (7 − 3 ) · 10 · 9 − 2 · (7 · 10 + 10 · [9 − 3 ]) ι30 = 2 · 10 − (7 · 10 + 10 · [9 − 3 ]) ⎛

⎞ 0 7 · e−λ·τ D (−3 − λ) (2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ −λ·τ D −10 0 10 · e = {ι28 + ι29 · λ + ι30 · λ2 + ι31 · λ3 } · e−λ·τ D

616

6 Gas Laser Systems Stability Analysis …

⎛

⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠ Step 5.2: 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e ⎛

⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −10 10 · e−λ·τ D 0   −8 0 = 4 · {7 · e−λ·τ D · det 0 −10   8 · e−λ·τ D −8 } + (−3 − λ) · det 10 · e−λ·τ D 0 ⎛ ⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e = 4 · {7 · 8 · 10 + (−3 − λ) · 8 · 10 } · e−λ·τ D ⎛

⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e = {4 · 8 · 10 · [7 − 3 ] − 4 · 8 · 10 · λ} · e−λ·τ D We define some global parameters: ι32 = 4 · 8 · 10 · [7 − 3 ]; ι33 = −4 · 8 · 10 ⎛

⎞ 7 · e−λ·τ D 0 (−3 − λ) ⎠ 4 · det ⎝ 8 · e−λ·τ D −8 0 −λ·τ D 0 −10 10 · e = {ι32 + ι33 · λ} · e−λ·τ D We can summary: ⎞ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 0 −4 6 · e−λ·τ D (−2 − λ) ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎛

6.1 Nitrogen Gas Laser Filament Plasma …

617

= 5 · {{ι28 + ι29 · λ + ι30 · λ2 + ι31 · λ3 } · e−λ·τ D + (ι32 + ι33 · λ) · e−λ·τ D } ⎞ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 ⎞ ⎛ 0 −4 6 · e−λ·τ D (−2 − λ) ⎟ ⎜ 7 · e−λ·τ D 0 (−3 − λ) 0 ⎟ = 5 · det ⎜ ⎝ 8 · e−λ·τ D −8 0 (9 − λ) ⎠ 0 −10 0 10 · e−λ·τ D ⎛ ⎞ −λ·τ D (−3 − λ) 0 7 · e = 5 · {(2 + λ) · det ⎝ 8 · e−λ·τ D 0 (9 − λ) ⎠ ⎛

⎛

10 · e−λ·τ D

−λ·τ D

7 · e + 4 · det ⎝ 8 · e−λ·τ D 10 · e−λ·τ D

−10 ⎞ 0 (−3 − λ) ⎠} −8 0 0 −10

0

= {5 · [ι28 + ι32 ] + 5 · [ι29 + ι33 ] · λ + 5 · ι30 · λ2 + 5 · ι31 · λ3 } · e−λ·τ D We can summary steps 1–5 results (Table 6.3): ⎛

ϒ22 ϒ23 ⎜ ϒ32 ϒ33 −λ·τ D det(A − λ · I ) = (1 · e − λ) · det ⎜ ⎝ ϒ42 ϒ43 ϒ52 ϒ53 ⎞ ⎛ ϒ21 ϒ23 ϒ24 ϒ25 ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ − ϒ12 · det ⎜ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟ + ϒ13 · det ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ − ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

4

3

2

1

Step No.

Determinant ⎞ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ϒ11 · det ⎜ ⎟ ⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ23 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ϒ12 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟ ϒ13 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55 ⎛ ⎞ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ⎟ ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ϒ14 · det ⎜ ⎜ ⎟ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55 + 4 · (ι23 − ι26 ) · λ2 − 4 · ι27 · λ3 } · e−λ·τ D

{4 · (ι21 − ι24 ) + 4 · (ι22 − ι25 ) · λ

+ 3 · ι18 · λ3 } · e−λ·τ D

{3 · (ι15 − ι19 ) + 3 · (ι16 − ι20 ) · λ + 3 · ι17 · λ2

+ 2 · ι11 · λ3 } · e−λ·τ D

{2 · (ι8 − ι12 ) + 2 · (ι9 − ι13 ) · λ + 2 · (ι10 − ι13 ) · λ2

+ ι4 · λ3 · 1 + λ4 · 1 } · e−λ·τ D

+ {(ι1 + ι5 ) · 1 + (ι2 + ι6 ) · λ · 1 + (ι3 + ι7 ) · λ2 · 1

− (ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5

Expression

Table 6.3 Small increments Jacobian of plasma kinetic system determinants, results 1–5 and expressions

(continued)

618 6 Gas Laser Systems Stability Analysis …

5

Step No.

Table 6.3 (continued)

Determinant ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎟ ⎜ ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54 + 5 · ι30 · λ2 + 5 · ι31 · λ3 } · e−λ·τ D

{5 · [ι28 + ι32 ] + 5 · [ι29 + ι33 ] · λ

Expression

6.1 Nitrogen Gas Laser Filament Plasma … 619

620

6 Gas Laser Systems Stability Analysis …

⎛

ϒ21 ⎜ ϒ31 + ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

det(A − λ · I ) = [−(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 + {(ι1 + ι5 ) · 1 + (ι2 + ι6 ) · λ · 1 + (ι3 + ι7 ) · λ2 · 1 + ι4 · λ3 · 1 + λ4 · 1 } · e−λ·τ D ] − [{2 · (ι8 − ι12 ) + 2 · (ι9 − ι13 ) · λ + 2 · (ι10 − ι13 ) · λ2 + 2 · ι11 · λ3 } · e−λ·τ D ] + [{3 · (ι15 − ι19 ) + 3 · (ι16 − ι20 ) · λ + 3 · ι17 · λ2 + 3 · ι18 · λ3 } · e−λ·τ D ] − [{4 · (ι21 − ι24 ) + 4 · (ι22 − ι25 ) · λ + 4 · (ι23 − ι26 ) · λ2 − 4 · ι27 · λ3 } · e−λ·τ D ] + [{5 · [ι28 + ι32 ] + 5 · [ι29 + ι33 ] · λ + 5 · ι30 · λ2 + 5 · ι31 · λ3 } · e−λ·τ D ] det(A − λ · I ) = −(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 + {(ι1 + ι5 ) · 1 + (ι2 + ι6 ) · λ · 1 + (ι3 + ι7 ) · λ2 · 1 + ι4 · λ3 · 1 + λ4 · 1 − 2 · (ι8 − ι12 ) − 2 · (ι9 − ι13 ) · λ − 2 · (ι10 − ι13 ) · λ2 − 2 · ι11 · λ3 + 3 · (ι15 − ι19 ) + 3 · (ι16 − ι20 ) · λ + 3 · ι17 · λ2 + 3 · ι18 · λ3 − 4 · (ι21 − ι24 ) − 4 · (ι22 − ι25 ) · λ − 4 · (ι23 − ι26 ) · λ2 + 4 · ι27 · λ3 + 5 · [ι28 + ι32 ] + 5 · [ι29 + ι33 ] · λ + 5 · ι30 · λ2 + 5 · ι31 · λ3 } · e−λ·τ D det(A − λ · I ) = −(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 + {(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) + [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · λ + [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · λ2 + [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · λ3 + λ4 · 1 } · e−λ·τ D

6.1 Nitrogen Gas Laser Filament Plasma …

D(λ, τ D ) = −(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 + {(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) + [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · λ + [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · λ2 + [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · λ3 + λ4 · 1 } · e−λ·τ D D(λ, τ D ) = Pn (λ, τ D ) + Q m (λ, τ D ) · e−λ·τ D n, m ∈ N0 ; n = 5; m = 4; n > m Pn (λ, τ D ) = −(ι1 + ι5 ) · λ − (ι2 + ι6 ) · λ2 − (ι3 + ι7 ) · λ3 − ι4 · λ4 − λ5 ; n = 5 Q m (λ, τ D ) = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) + [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · λ + [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · λ2 + [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · λ3 + 1 · λ4 ; m = 4 Pn (λ, τ D ) =

n=5 

pk (τ D ) · λk

k=0

= p0 (τ D ) + p1 (τ D ) · λ + p2 (τ D ) · λ2 + p3 (τ D ) · λ3 + p4 (τ D ) · λ4 + p5 (τ D ) · λ5 p0 = p0 (τ D ); p1 = p1 (τ D ); p2 = p2 (τ D ) p3 = p3 (τ D ); p4 = p4 (τ D ); p5 = p5 (τ D ) p0 = 0; p1 = − (ι1 + ι5 ); p2 = − (ι2 + ι6 ) p3 = − (ι3 + ι7 ); p4 = − ι4 ; p5 = − 1

621

622

6 Gas Laser Systems Stability Analysis …

Q m (λ, τ D ) =

m=4 

qk (τ D ) · λk

k=0

= q0 (τ D ) + q1 (τ D ) · λ + q2 (τ D ) · λ2 + q3 (τ D ) · λ3 + q4 (τ D ) · λ4 q0 = q0 (τ D ); q1 =q1 (τ D ); q2 = q2 (τ D ) q3 = q3 (τ D ); q4 = q4 (τ D ) q0 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) q1 =(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 ) q2 = (ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 q3 = ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ; q4 = 1 The homogeneous system for n c , n B1 , n B2 , n ph1 , n ph2 leads to a characteristic equation for the eigenvalue  λ having the form D(λ, τ D ) = P(λ, τ D ) + Q(λ, τ D ) · 5 4 j j a · λ ; Q(λ) = e−λ·τ D = 0; and P(λ) = j j=0 j=0 c j · λ . The coefficients {a j (qi , qk , τ D ), c j (qi , qk , τ D )} ∈ R depend on qi , qk and delay τ D . qi , qk are any plasma kinetic system’s parameters, other parameters kept as a constant [4, 5]. a0 = 0; a1 = − (ι1 + ι5 ); a2 = − (ι2 + ι6 ) a3 = − (ι3 + ι7 ); a4 = − ι4 ; a5 = − 1 c0 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) c1 =(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 ) c2 = (ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30

6.1 Nitrogen Gas Laser Filament Plasma …

623

c3 = ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ; c4 = 1 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ D = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang J and Cong Y 2005; Kuang Y 1993) can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) If | Q(λ) P(λ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ D ) and Q m (λ, τ D ) cannot have common imaginary roots. That is for any real numberω:Pn (λ = i · ω, τ D ) + Q m (λ = i · ω, τ D ) = 0 and Pn (λ = i · ω, τ D ) = p0 (τ D ) + p1 (τ D ) · i · ω − p2 (τ D ) · ω2 − p3 (τ D ) · i · ω3 + p4 (τ D ) · ω4 + p5 (τ D ) · i · ω5 Pn (λ = i · ω, τ D ) = p0 (τ D ) − p2 (τ D ) · ω2 + p4 (τ D ) · ω4 + [ p1 (τ D ) · ω − p3 (τ D ) · ω3 + p5 (τ D ) · ω5 ] · i Pn (λ = i · ω, τ D ) = (ι2 + ι6 ) · ω2 − ι4 · ω4 + [−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 ] · i Q m (λ = i · ω, τ D ) = q0 (τ D ) + q1 (τ D ) · i · ω − q2 (τ D ) · ω2 − q3 (τ D ) · i · ω3 + q4 (τ D ) · ω4 Q m (λ = i · ω, τ D ) = q0 (τ D ) − q2 (τ D ) · ω2 + q4 (τ D ) · ω4 + [q1 (τ D ) · ω − q3 (τ D ) · ω3 ] · i Q m (λ = i · ω, τ D ) = [(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 )

624

6 Gas Laser Systems Stability Analysis …

· 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 + {[(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 } · i Pn (λ = i · ω, τ D ) + Q m (λ = i · ω, τ D ) = (ι2 + ι6 ) · ω2 − ι4 · ω4 + {−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 } · i + [(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 + {[(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 } · i = 0 Pn (λ = i · ω, τ D ) + Q m (λ = i · ω, τ D ) = (ι2 + ι6 ) · ω2 − ι4 · ω4 + [(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 + {−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 + [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 } · i = 0 |P(i · ω, τ D )|2 = [(ι2 + ι6 ) · ω2 − ι4 · ω4 ]2 + [−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 ]2 |Q(i · ω, τ D )|2 = {[(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 }2 + {[(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 }2

6.1 Nitrogen Gas Laser Filament Plasma …

625

Table 6.4 Plasma kinetic system’s stability analysis F(ω, τ D ) elements Expression with ω

Equivalent expression

[(ι2 + ι6 ) · ω2 − ι4 · ω4 ]2

(ι2 + ι6 )2 · ω4 − 2 · (ι2 + ι6 ) · ι4 · ω6 + ι24 · ω8

[−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 ]2

(ι1 + ι5 )2 · ω2 − 2 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω4

[q0 (τ D ) − q2 (τ D ) · ω2 + q4 (τ D ) · ω4 ]2

q02 (τ D ) − 2 · q0 (τ D ) · q2 (τ D ) · ω2

+ [2 · (ι1 + ι5 ) + (ι3 + ι7 )2 ] · ω6 − 2 · (ι3 + ι7 ) · ω8 + ω10

+ [q0 (τ D ) · q4 (τ D ) + q22 (τ D ) + q0 (τ D ) · q4 (τ D )] · ω4 − 2 · q2 (τ D ) · q4 (τ D ) · ω6 + q42 (τ D ) · ω8

[q1 (τ D ) · ω − q3 (τ D ) · ω3 ]2

q12 (τ D ) · ω2 − 2 · q1 (τ D ) · q3 (τ D ) · ω4 + q32 (τ D ) · ω6

F(ω, τ D ) = |P(i · ω, τ D )|2 − |Q(i · ω, τ D )|2 = [(ι2 + ι6 ) · ω2 − ι4 · ω4 ]2 + [−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 ]2 − {[q0 (τ D ) − q2 (τ D ) · ω2 + q4 (τ D ) · ω4 ]2 + [q1 (τ D ) · ω − q3 (τ D ) · ω3 ]2 } The plasma kinetic system’s stability analysis F(ω, τ D ) elements are (Table 6.4): F(ω, τ D ) = |P(i · ω, τ D )|2 − |Q(i · ω, τ D )|2 = (ι2 + ι6 )2 · ω4 − 2 · (ι2 + ι6 ) · ι4 · ω6 + ι24 · ω8 + (ι1 + ι5 )2 · ω2 − 2 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω4 + [2 · (ι1 + ι5 ) + (ι3 + ι7 )2 ] · ω6 − 2 · (ι3 + ι7 ) · ω8 + ω10 − q02 (τ D ) + 2 · q0 (τ D ) · q2 (τ D ) · ω2 − [q0 (τ D ) · q4 (τ D ) + q22 (τ D ) + q0 (τ D ) · q4 (τ D )] · ω4 + 2 · q2 (τ D ) · q4 (τ D ) · ω6 − q42 (τ D ) · ω8 − q12 (τ D ) · ω2 + 2 · q1 (τ D ) · q3 (τ D ) · ω4 − q32 (τ D ) · ω6 F(ω, τ D ) = |P(i · ω, τ D )|2 − |Q(i · ω, τ D )|2 = −q02 (τ D ) + [(ι1 + ι5 )2 + 2 · q0 (τ D ) · q2 (τ D ) − q12 (τ D )] · ω2 + {(ι2 + ι6 )2 − 2 · (ι1 + ι5 ) · (ι3 + ι7 ) − [q0 (τ D ) · q4 (τ D ) + q22 (τ D ) + q0 (τ D ) · q4 (τ D )] + 2 · q1 (τ D ) · q3 (τ D )}

626

6 Gas Laser Systems Stability Analysis …

· ω4 + {[2 · (ι1 + ι5 ) + (ι3 + ι7 )2 ] − q32 (τ D ) + 2 · q2 (τ D ) · q4 (τ D )} · ω6 − {2 · (ι2 + ι6 ) · ι4 − ι24 + 2 · (ι3 + ι7 ) + q42 (τ D )} · ω8 + ω10 We define the following parameters for simplicity: 0 , 2 , 4 , 6 .8 , 10 0 = −q02 (τ D ); 2 = (ι1 + ι5 )2 + 2 · q0 (τ D ) · q2 (τ D ) − q12 (τ D ) 4 = (ι2 + ι6 )2 − 2 · (ι1 + ι5 ) · (ι3 + ι7 ) − [q0 (τ D ) · q4 (τ D ) + q22 (τ D ) + q0 (τ D ) · q4 (τ D )] + 2 · q1 (τ D ) · q3 (τ D ) 6 = [2 · (ι1 + ι5 ) + (ι3 + ι7 )2 ] − q32 (τ D ) + 2 · q2 (τ D ) · q4 (τ D ) 8 = −{2 · (ι2 + ι6 ) · ι4 − ι24 + 2 · (ι3 + ι7 ) + q42 (τ D )}10 = 1 Hence F(ω, τ D ) = 0 implies solving the above polynomial.

5 k=0

2·k · ω2·k = 0 and its roots are given by

PR (iω, τ D ) = (ι2 + ι6 ) · ω2 − ι4 · ω4 ; PI (iω, τ D ) = −(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 Q R (iω, τ D ) = q0 (τ D ) − q2 (τ D ) · ω2 + q4 (τ D ) · ω4 Q I (iω, τ D ) = q1 (τ D ) · ω − q3 (τ D ) · ω3 Q R (iω, τ D ) = [(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 Q I (iω, τ D ) = [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3

6.1 Nitrogen Gas Laser Filament Plasma …

627

sin θ (τ D ) −PR (iω, τ D ) · Q I (iω, τ D ) + PI (iω, τ D ) · Q R (iω, τ D ) = |Q(iω, τ D )|2 cos θ (τ D ) PR (iω, τ D ) · Q R (iω, τ D ) + PI (iω, τ D ) · Q I (iω, τ D ) =− |Q(iω, τ D )|2 We use different terminology from our last characteristics parameters definition: k → j; pk (τ D ) → a j ; qk (τ D ) → c j n, m ∈ N0 ; n = 5; m = 4; n > m Pn (λ, τ D ) → P(λ); Q m (λ, τ D ) → Q(λ) P(λ) =

5  j=0

a j · λ j ; Q(λ) =

4 

cj · λj

j=0

P(λ) = a0 + a1 · λ + a2 · λ2 + a3 · λ3 + a4 · λ4 + a5 · λ5 Q(λ) = c0 + c1 · λ + c2 · λ2 + c3 · λ3 + c4 · λ4 n, m ∈ N0 ; n > m and a j , c j : R0+ → R are continuous and differentiable function of τ D such that a0 + c0 = 0. In the following “−” denoted complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ D . The coefficients a j (Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . .) ∈ R and c j (Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . .) ∈ R depend on plasma kinetic system’s parameters, Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . values. Unless strictly necessary, the designation of the varied arguments: Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0[4, 5]. a0 = 0 c0 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) a0 + c0 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) = 0 ∀ Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following

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6 Gas Laser Systems Stability Analysis …

requirements of the analysis (see Kuang, 1993, Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . domain of interest. P(λ) | is bounded for |λ| → ∞ ; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (b) | Q(λ) 1 ·λ+c2 ·λ +c3 ·λ +c4 ·λ in the limit: | Q(λ) | = | a0 +ac01+c |. P(λ) ·λ+a2 ·λ2 +a3 ·λ3 +a4 ·λ4 +a5 ·λ5 2 2 (c) The following expressions exist: F(ω) = |P(i · ω)| 5 − |Q(i · ω)| 2 2 2·k F(ω, τ D ) = |P(i · ω, τ D )| − |Q(i · ω, τ D )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω10 ). (d) Each positive root ω( Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . .) of F(ω) = 0 is continuous and differentiable with respect to Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . and the condition can only be assessed numerically. 2

3

4

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . ., Reλ may, at the crossing, change its sign from (-) ( j) ( j) ( j) ( j) ( j) to (+). i.e. from a stable focus E ( j) = (n c , n B1 , n B2 , n ph1 , n ph2 ); j = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . and any system parameters.  ∂Reλ ; S B1 , S B2 , σC B1 , σC B2 , τ D , . . . = const ∂ Sc λ=i·ω   ∂Reλ −1 (S B1 ) = ; Sc , S B2 , σC B1 , σC B2 , τ D , . . . = const ∂ S B1 λ=i·ω   ∂Reλ −1  (S B2 ) = ; Sc , S B1 , σC B1 , σC B2 , τ D , . . . = const ∂ S B2 λ=i·ω   ∂Reλ −1 (σC B1 ) = ; S B1 , S B2 , Sc , σC B2 , τ D , . . . = const ∂σC B1 λ=i·ω   ∂Reλ −1 (σC B2 ) = ; S B1 , S B2 , Sc , σC B1 , τ D , . . . = const ∂σC B2 λ=i·ω   ∂Reλ −1  (τ D ) = ; S B1 , S B2 , Sc , σC B1 , σC B2 , . . . = const ∂τ D λ=i·ω −1 (Sc ) =



P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ), When writing and inserting λ = i · ω into plasma kinetic system’s characteristic equation ω must satisfy the following equations:

6.1 Nitrogen Gas Laser Filament Plasma …

629

sin(ω · τ D ) = g(ω) −PR (iω, τ D ) · Q I (iω, τ D ) + PI (iω, τ D ) · Q R (iω, τ D ) = |Q(iω, τ D )|2 cos(ω · τ D ) = h(ω) PR (iω, τ D ) · Q R (iω, τ D ) + PI (iω, τ D ) · Q I (iω, τ D ) =− |Q(iω, τ D )|2 where |Q(iω, τ D )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω·τ D ) and cos(ω·τ D ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: / I (assume that F(ω) is independent on τ D . Now it is important to notice that if τ D ∈ / I , ω(τ D ) is I ⊆ R+0 is the set where ω(τ D ) is a positive root of F(ω) and for, τ D ∈ not defined. Then for all τ D in I, ω(τ D ) is satisfied that F(ω, τ D ) = 0. Then there are no positive ω(τ D ) solutions for F(ω, τ D ) = 0, and we cannot have stability switches. For τ D ∈ I where ω(τ D ) is a positive solution of F(ω, τ D ) = 0, we can define the angle θ (τ D ) ∈ [0, 2 · ] as the solution of sin θ (τ D ) = . . . and cos θ (τ D ) = . . .; the relation between the arguments θ (τ D ) and τ D · ω(τ D ) for τ D ∈ I must be describing below. τ D · ω(τ D ) = θ (τ D ) + 2 · n ·  ∀ n ∈ N0 Hence we can define the maps: )+2·n· ; n ∈ N0 ; τ D ∈ I . τ D(n) : I → R+0 , is given by τ D(n) (τ D ) = θ(τ Dω(τ D) Let us introduce the function I → R ; Sn (τ D ) = τ D − τ D(n) (τ D ); τ D ∈ I ; n ∈ N0 that is continuous and differentiable in τ D . In the following, the subscripts λ, ω, Sc , S B1 , S B2 , σC B1 , σC B2 , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(Sc , S B1 , S B2 , σC B1 , σC B2 , . . .) and ω(Sc , S B1 , S B2 , σC B1 , σC B2 , . . .), and keeping all parameters except one (x) and τ D . The derivation closely follows that in reference [BK]. Differentiating plasma kinetic system’s characteristic equation P(λ) + Q(λ) · e−λ·τ D = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = Sc , S B1 , S B2 , σC B1 , σC B2 , τ D , . . . 

 ∂λ −1 ∂x −Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ D · P(λ, x) · Q(λ, x) = Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω = λ (i·ω,x)·Q(λ,x)−τ D ·|P(i·ω,x)| ( i·Pω (i·ω,x)·P(i·ω,x)+i·Q ) P (i·ω,x)·P(i·ω,x)−Q (i·ω,x)·Q(i·ω,x) 2

x

x

630

6 Gas Laser Systems Stability Analysis …

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any plasma kinetic system’s parameters Sc , S B1 , S B2 , σC B1 , σC B2 , . . . and time delay τ D etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )];Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delayx = τ D : PR = (ι2 + ι6 ) · ω2 − ι4 · ω4 ; PI = −(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 Q R = [(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 We define for simplicity the following global parameters: ϑ2 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) ϑ1 = (ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 Q R = ϑ2 − ϑ1 · ω2 + 1 · ω4 ⇒ Q Rω = −2 · ϑ1 · ω + 4 · 1 · ω3 Q I = [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 We define for simplicity the following global parameters: ϑ3 = (ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 ) ϑ4 = ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31

6.1 Nitrogen Gas Laser Filament Plasma …

Q I = ϑ3 · ω − ϑ4 · ω3 ⇒ Q I ω = ϑ3 − 3 · ϑ4 · ω2 PRω = 2 · (ι2 + ι6 ) · ω − 4 · ι4 · ω3 PI ω = −(ι1 + ι5 ) + 3 · (ι3 + ι7 ) · ω2 − 5 · ω4 Q Rω = −2 · [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω + 4 · 1 · ω3 Q I ω = [(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] − 3 · [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω2 PRτ D = 0; PI τ D = 0; Q Rτ D = 0; Q I τ D = 0 Fτ D = 0; V = (PR · PI τ D − PI · PRτ D ) − (Q R · Q I τ D − Q I · Q Rτ D ) = 0 Fω = . . . . Elements: PRω · PR = [2 · (ι2 + ι6 ) · ω − 4 · ι4 · ω3 ] · [(ι2 + ι6 ) · ω2 − ι4 · ω4 ] = 2 · (ι2 + ι6 )2 · ω3 − 6 · ι4 · (ι2 + ι6 ) · ω5 + 4 · ι4 · ι4 · ω7 PI ω · PI = {−(ι1 + ι5 ) + 3 · (ι3 + ι7 ) · ω2 − 5 · ω4 } · {−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 } = (ι1 + ι5 )2 · ω − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω3 + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 · ω5 − 8 · (ι3 + ι7 ) · ω7 + 5 · ω9 Q Rω · Q R = {−2 · [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω + 4 · 1 · ω3 } · {[(ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 ) − 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 )] − [(ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ] · ω2 + 1 · ω4 } ϑ1 = (ι3 + ι7 ) · 1 − 2 · (ι10 − ι13 ) + 3 · ι17 − 4 · (ι23 − ι26 ) + 5 · ι30 ϑ2 = (ι1 + ι5 ) · 1 − 2 · (ι8 − ι12 ) + 3 · (ι15 − ι19 )

631

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6 Gas Laser Systems Stability Analysis …

− 4 · (ι21 − ι24 ) + 5 · (ι28 + ι32 ) Q Rω · Q R = {−2 · ϑ1 · ω + 4 · 1 · ω3 } · {ϑ2 − ϑ1 · ω2 + 1 · ω4 } = −2 · ϑ1 · ϑ2 · ω + (2 · ϑ12 + 4 · 1 · ϑ2 ) · ω3 − 6 · ϑ1 · 1 · ω5 + 4 · 1 · 1 · ω7 Q I ω · Q I = {[(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] − 3 · [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω2 } · {[(ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 )] · ω − [ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 ] · ω3 } ϑ3 = (ι2 + ι6 ) · 1 − 2 · (ι9 − ι13 ) + 3 · (ι16 − ι20 ) − 4 · (ι22 − ι25 ) + 5 · (ι29 + ι33 ) ϑ4 = ι4 · 1 − 2 · ι11 + 3 · ι18 + 4 · ι27 + 5 · ι31 Q I ω · Q I = (ϑ3 − 3 · ϑ4 · ω2 ) · (ϑ3 · ω − ϑ4 · ω3 ) = ϑ32 · ω − 4 · ϑ4 · ϑ3 · ω3 + 3 · ϑ4 · ϑ4 · ω5 U = . . . .Elements: PR · PI ω = {(ι2 + ι6 ) · ω2 − ι4 · ω4 } · {−(ι1 + ι5 ) + 3 · (ι3 + ι7 ) · ω2 − 5 · ω4 } = −(ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + {3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )} · ω4 − {(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )} · ω6 + ι4 · 5 · ω8 PI · PRω = {−(ι1 + ι5 ) · ω + (ι3 + ι7 ) · ω3 − ω5 } · {2 · (ι2 + ι6 ) · ω − 4 · ι4 · ω3 } = −2 · (ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + 2 · {2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )} · ω4 − 2 · {2 · (ι3 + ι7 ) · ι4 + (ι2 + ι6 )} · ω6 + 4 · ι4 · ω8 Q R · Q I ω = (ϑ2 − ϑ1 · ω2 + 1 · ω4 ) · (ϑ3 − 3 · ϑ4 · ω2 ) = ϑ2 · ϑ3 − (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) · ω2 + (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) · ω4 − 3 · 1 · ϑ4 · ω6

6.1 Nitrogen Gas Laser Filament Plasma …

633

Table 6.5 Plasma kinetic system’s stability analysis Fω , U elements Fω , U elements

Expression

PRω · PR

2 · (ι2 + ι6 )2 · ω3 − 6 · ι4 · (ι2 + ι6 ) · ω5 + 4 · ι4 · ι4 · ω7

PI ω · PI

(ι1 + ι5 )2 · ω − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω3 + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 · ω5 − 8 · (ι3 + ι7 ) · ω7 + 5 · ω9

Q Rω · Q R

− 2 · ϑ1 · ϑ2 · ω + (2 · ϑ12 + 4 · 1 · ϑ2 ) · ω3 − 6 · ϑ1 · 1 · ω5 + 4 ·  1 · 1 · ω 7

QIω · QI PR · PI ω

ϑ32 · ω − 4 · ϑ4 · ϑ3 · ω3 + 3 · ϑ4 · ϑ4 · ω5 − (ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + {3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )} · ω4 − {(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )} · ω6 + ι4 · 5 · ω8

PI · PRω

− 2 · (ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + 2 · {2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )} · ω4 − 2 · {2 · (ι3 + ι7 ) · ι4 + (ι2 + ι6 )} · ω6 + 4 · ι4 · ω8

Q R · Q I ω ϑ2 · ϑ3 − (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) · ω2 + (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) · ω4 − 3 · 1 · ϑ4 · ω6 Q I · Q Rω −2 · ϑ1 · ϑ3 · ω2 + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 ) · ω4 − 4 · ϑ4 · 1 · ω6

Q I · Q Rω = (ϑ3 · ω − ϑ4 · ω3 ) · (−2 · ϑ1 · ω + 4 · 1 · ω3 ) = −2 · ϑ1 · ϑ3 · ω2 + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 ) · ω4 − 4 · ϑ4 · 1 · ω6 We can summary our last results in the next table (Table 6.5). F(ω, τ D ) = 0, differentiating with respect to τ D and we get Fω · 0; τ D ∈ I ⇒

∂ω ∂τ D

=

F − FτωD

∂ω ∂τ D

+ Fτ D =

∂Reλ )λ=iω ∂τ D  ∂ω Fτ −2 · [U + τ D · |P|2 + i · Fω = ωτ D = − D ; −1 (τ D ) = Re ∂τ D Fω Fτ D + i · 2 · [V + ω · |P|2 ]   ∂Reλ sign{−1 (τ D )} = sign ∂τ D λ=iω   ∂ω V + ∂τ ·U ∂ω −1 D sign{ (τ D )} = sign{Fω } · sign +ω+ · τD |P|2 ∂τ D

−1 (τ D ) = (

We shall presently examine the possibility of stability transitions (bifurcations) plasma kinetic system, about the equilibrium points E ( j) = ( j) ( j) ( j) ( j) ( j) (n c , n B1 , n B2 , n ph1 , n ph2 ); j = 0, 1, 2, . . .. The analysis consists in identifying

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6 Gas Laser Systems Stability Analysis …

the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameterτ D . Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ D [4, 5]. −1 (τ D ) = (

∂Reλ )λ=iω ; S B1 , S B2 , Sc , σC B1 , σC B2 , . . . = const ∂τ D

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = −(ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + {3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )} · ω4 − {(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )} · ω6 + ι4 · 5 · ω8 − [−2 · (ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + 2 · {2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )} · ω4 − 2 · {2 · (ι3 + ι7 ) · ι4 + (ι2 + ι6 )} · ω6 + 4 · ι4 · ω8 ] − [ϑ2 · ϑ3 − (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) · ω2 + (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) · ω4 − 3 · 1 · ϑ4 · ω6 − {−2 · ϑ1 · ϑ3 · ω2 + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 ) · ω4 − 4 · ϑ4 · 1 · ω6 }] U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = −(ι1 + ι5 ) · (ι2 + ι6 ) · ω2 + [3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )] · ω4 − [(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )] · ω6 + ι4 · 5 · ω8 + 2 · (ι1 + ι5 ) · (ι2 + ι6 ) · ω2 − 2 · [2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )] · ω4 + 2 · [2 · (ι3 + ι7 ) · ι4 − (ι2 + ι6 )] · ω6 − 4 · ι4 · ω8 − ϑ2 · ϑ3 + (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) · ω2 − (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) · ω4 + 3 · 1 · ϑ4 · ω6 − 2 · ϑ1 · ϑ3 · ω2 + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 ) · ω4 − 4 · ϑ4 · 1 · ω6 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = −ϑ2 · ϑ3 + {2 · (ι1 + ι5 ) · (ι2 + ι6 ) + (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) − 2 · ϑ1 · ϑ3 − (ι1 + ι5 ) · (ι2 + ι6 )} · ω2 + {[3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )] − 2 · [2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )] − (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 )} · ω4 + {2 · [2 · (ι3 + ι7 ) · ι4 − (ι2 + ι6 )] − [(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )] + 3 · 1 · ϑ4 − 4 · ϑ4 · 1 } · ω6 + {ι4 · 5 − 4 · ι4 } · ω8

6.1 Nitrogen Gas Laser Filament Plasma …

We define for simplicity the following global parameters: A0 = −ϑ2 · ϑ3 A2 = 2 · (ι1 + ι5 ) · (ι2 + ι6 ) + (3 · ϑ2 · ϑ4 + ϑ1 · ϑ3 ) − 2 · ϑ1 · ϑ3 − (ι1 + ι5 ) · (ι2 + ι6 ) A4 = [3 · (ι2 + ι6 ) · (ι3 + ι7 ) + ι4 · (ι1 + ι5 )] − 2 · [2 · (ι1 + ι5 ) · ι4 + (ι3 + ι7 ) · (ι2 + ι6 )] − (3 · ϑ1 · ϑ4 + 1 · ϑ3 ) + 2 · (2 · ϑ3 · 1 + ϑ4 · ϑ1 ) A6 = 2 · [2 · (ι3 + ι7 ) · ι4 − (ι2 + ι6 )] − [(ι2 + ι6 ) · 5 + 3 · ι4 · (ι3 + ι7 )] + 3 · 1 · ϑ4 − 4 · ϑ4 · 1 A8 = ι4 · 5 − 4 · ι4 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) =

4 

A2k · ω2·k

k=0

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [(2 · (ι2 + ι6 )2 · ω3 − 6 · ι4 · (ι2 + ι6 ) · ω5 + 4 · ι4 · ι4 · ω7 + (ι1 + ι5 )2 · ω − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω3 + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 · ω5 − 8 · (ι3 + ι7 ) · ω7 + 5 · ω9 ) − (−2 · ϑ1 · ϑ2 · ω + (2 · ϑ12 + 4 · 1 · ϑ2 ) · ω3 − 6 · ϑ1 · 1 · ω5 + 4 · 1 · 1 · ω7 + ϑ32 · ω − 4 · ϑ4 · ϑ3 · ω3 + 3 · ϑ4 · ϑ4 · ω5 )] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [2 · (ι2 + ι6 )2 · ω3 − 6 · ι4 · (ι2 + ι6 ) · ω5 + 4 · ι4 · ι4 · ω7 + (ι1 + ι5 )2 · ω − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) · ω3 + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 · ω5 − 8 · (ι3 + ι7 ) · ω7 + 5 · ω9 + 2 · ϑ1 · ϑ2 · ω − (2 · ϑ12 + 4 · 1 · ϑ2 ) · ω3 + 6 · ϑ1 · 1 · ω5 − 4 · 1 · 1 · ω7 − ϑ32 · ω + 4 · ϑ4 · ϑ3 · ω3 − 3 · ϑ4 · ϑ4 · ω5 ] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]

635

636

6 Gas Laser Systems Stability Analysis …

= 2 · [(ι1 + ι5 )2 + 2 · ϑ1 · ϑ2 − ϑ32 ] · ω + 2 · [2 · (ι2 + ι6 )2 − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) − (2 · ϑ12 + 4 · 1 · ϑ2 ) + 4 · ϑ4 · ϑ3 ] · ω3 + 2 · [6 · ϑ1 · 1 − 6 · ι4 · (ι2 + ι6 ) + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 − 3 · ϑ4 · ϑ4 ] · ω5 + 2 · [4 · ι4 · ι4 − 8 · (ι3 + ι7 ) − 4 · 1 · 1 ] · ω7 + 10 · ω9 We define for simplicity the following global parameters: B1 = 2 · [(ι1 + ι5 )2 + 2 · ϑ1 · ϑ2 − ϑ32 ] B7 = 2 · [4 · ι4 · ι4 − 8 · (ι3 + ι7 ) − 4 · 1 · 1 ]; B9 = 10 B3 = 2 · [2 · (ι2 + ι6 )2 − 4 · (ι1 + ι5 ) · (ι3 + ι7 ) − (2 · ϑ12 + 4 · 1 · ϑ2 ) + 4 · ϑ4 · ϑ3 ] B5 = 2 · [6 · ϑ1 · 1 − 6 · ι4 · (ι2 + ι6 ) + {(ι3 + ι7 )2 + 2 · (ι1 + ι5 )} · 3 − 3 · ϑ4 · ϑ4 ] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

4 

B2·k+1 · ω2·k+1

k=0

 Then we get the expression for Fω = 4k=0 B2·k+1 · ω2·k+1 plasma kinetic system parameter values. We find those ω, τ D values which fulfill Fω (ω, τ D ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ D values. τ D ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ D ) = 0. We plot the stability switch diagram based on different delay values of our plasma kinetic system. ∂Reλ )λ=iω ∂τ D −2 · [U + τ D · |P|2 ] + i · Fω } = Re{ Fτ D + 2i · [V + ω · |P|2 ]

−1 (τ D ) = (

∂Reλ )λ=iω ∂τ D 2 · {Fω · (V + ω · P 2 ) − Fτ D · (U + τ D · P 2 )} = Fτ2D + 4 · (V + ω · P 2 )2

−1 (τ D ) = (

The stability switch occurs only on those delay values (τ D ) which fit the equation: τ D = ωθ++(τ(τDD)) and θ+ (τ D ) is the solution of sin θ (τ D ) = . . . . and cos θ (τ D ) = . . . .

6.1 Nitrogen Gas Laser Filament Plasma …

637

when ω = ω+ (τ D ) If only ω+ is feasible. Additionally, when all plasma kinetic system’s parameters are known and the stability switch due to various time delay values τ D is described in the following expression: sign{−1 (τ D )} = sign{Fω (ω(τ D ), τ D )} · sign{τ D · ωτ D (ω(τ D )) + ω(τ D ) U (ω(τ D )) · ωτ D (ω(τ D )) + V (ω(τ D )) + } |P(ω(τ D ))|2 Remark we know F(ω, τ D ) = 0 implies its roots ωi (τ D ) and finding those delays values τ D which ωi is feasible. There are τ D values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ D ), function is independent on τ D the parameter F(ω, τ D ) = 0. The results: We find those ω, τ D values which fulfill F(ω, τ D ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ D values which fulfill sin θ (τ D ) = . . . . and cos θ (τ D ) = . . . .; I +PI ·Q R R +PI ·Q I ) ; cos(ω · τ D ) = − (PR ·Q|Q| |Q|2 = Q 2R + Q 2I . Finally sin(ω · τ D ) = −PR ·Q|Q| 2 2 ) we plot the stability switch diagram g(τ D ) = −1 (τ D ) = ( ∂Reλ ∂τ D λ=iω g(τ D ) = −1 (τ D ) ∂Reλ )λ=iω =( ∂τ D 2 · {Fω · (V + ω · P 2 ) − Fτ D · (U + τ D · P 2 )} = Fτ2D + 4 · (V + ω · P 2 )2 sign[g(τ D )] = sign[−1 (τ D )]    ∂Reλ = sign ∂τ D λ=iω   2 · {Fω · (V + ω · P 2 ) − Fτ D · (U + τ D · P 2 )} = sign Fτ2D + 4 · (V + ω · P 2 )2 Fτ2D + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ D )] = sign[Fω · (V + ω · P 2 ) − Fτ D · (U + τ D · P 2 )] sign[−1 (τ D )] = sign{[Fω ] · [(V + ω · P 2 ) Fτ − D · (U + τ D · P 2 )]} Fω Fτ D ∂ω −1 ∂ F/∂ω ; ωτ D = ( ) =− ωτ D = − Fω ∂τ D ∂ F/∂τ D V + ωτ D · U P2 · τ D ]} ; sign[P 2 ] > 0

sign[−1 (τ D )] = sign{[Fω ] · [P 2 ] · [ + ω + ωτ D

638

6 Gas Laser Systems Stability Analysis …

Table 6.6 Plasma kinetic system sign of sign[−1 (τ D )] V +ωτ D ·U P2

sign[−1 (τ D )]

sign[Fω ]

sign[

+ ω + ωτ D · τ D ]

±

±

+

±

∓

–

V + ωτ D · U P2 · τ D ]}

sign[−1 (τ D )] = sign{[Fω ] · [ + ω + ωτ D

sign[−1 (τ D )] = sign[Fω ] · sign[

V + ωτ D · U P2

+ ω + ωτ D · τ D ] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

4 

B2·k+1 · ω2·k+1

k=0

We check the sign of −1 (τ D ) according to the following rule (Table 6.6): If, sign[−1 (τ D )] > 0, then the crossing proceeds from (−) to (+) respectively (stable to unstable). If, sign[−1 (τ D )] < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

6.2 A Quasi-Two Level Analytic Model for Metal Vapor Laser System Stability Analysis A quasi-two level analytic model for end pumped Alkali metal vapor laser describes precisely the mechanism between two upper levels, 2 P3/2 and 2 P1/2 , caused by collision with additive ethane which is much faster (infinity) by comparison with any other process in the system including stimulated emission. The Alkali metals (lithium, sodium, potassium, rubidium and cesium) are “visible hydrogen”, readily ionized, and strongly interacting with laser light. It is systems which describe the microscopic inter conversion mechanisms between photon (laser), chemical, electrical and thermal energy. There are many practical applications such as sodium lamps, thermionic converters, magneto hydrodynamic devices and lasers (“lithium water fall”). Alkali-vapor lasers (rubidium) have a very high efficiency and the flowinggas versions have a very high energy. The optical pumping of Alkali-metal vapors

6.2 A Quasi-Two Level Analytic Model …

639

Fig. 6.1 Diode pumping along the laser axis (flowing gas through an Alkali-metal laser)

Flow direction

Laser resonator axis Pump array

has only a single valence electron in their outer shells. Alkali metals have pair of low-lying energy levels of their single valence electron. Flowing gas through an Alkali-metal laser is enable the generation of much higher powers. Diode pumping can be either along the laser axis or transverse to the laser axis. Operating at higher power levels requires that Alkali vapor be flowed transversely through the pump volume, and waste heat is removed outside of the laser cavity. There are two diode pumping structures, one is along the laser axis and the second is transverse to the laser axis, when flowing gas through an Alkali-metal laser. The diode pumping along the laser axis structure is as follow (Fig. 6.1). The quasi-two level analytic model takes the assumption of steady state rate equations for the number of densities of the, 2 S1/2 , 2 P3/2 and 2 P1/2 , energy states for the three level laser system. The ratio of the number densities for the upper two levels, 2 P3/2 and 2 P1/2 , is given by its statistical equilibrium value. We extract analytical solution from the quasi-two system. We get the threshold pump power and the slope efficiency including intra-cavity losses [6]. The diode pumping transverse to the laser axis structure is as follow (Fig. 6.2). The diode pumped Alkali metal vapor laser (DPAL) is a single narrow bond diodes partially narrow bonded low power diode arrays, and surrogate pump sources such as Ti-Sapphire as pump sources. The three-level laser devices are pumped from the Alkali metal 2 S1/2 ground state to the lowest excited 2 P3/2 state, the D2 transition. Rapid collision induced spin-orbit relaxation with additive ethane, (200 Torr) populated the lower 2 P1/2 state. Lasing then occurs between 2 P1/2 and the 2 S1/2 ground state, the D1 transition (Fig. 6.3). The difference in energy between the 2 P3/2 and 2 P1/2 fine structure spin-orbit components is 554 cm−1 (Cs ), 237 cm−1 (Rb ), 57.7 cm−1 (K). The small energy defects lead to intrinsic quantum efficiencies (>95%) and to Fig. 6.2 Diode pumping transverse to the laser axis (flowing gas through on Alkali-metal laser)

Pump array

Flow direction

Laser resonator axis

640

6 Gas Laser Systems Stability Analysis …

P3/2 , 2 P1/2 Spin-orbit Relaxation 2

n3

2

P3/2

∆E32

n2

2

D2 transition

2

P3/ 2 Spontaneous

Emission/ Quenching

Pump stimulated Absorption/ Emission

D1 transition 2

Laser Stimulated Absorption/ Emission

Ground state

n1

P1/2

P1/ 2 Spontaneous

Emission/ Quenching 2

S1/2

Fig. 6.3 Schematic energy level diagram of Alkali metal vapor laser

the possibility of very efficient laser systems. A quasi-two level analytic model can be applied to longitudinal laser pumped Alkali metal vapor lasers. The model includes with modeling three level solid state laser systems that are pumped longitudinally with diodes or other lasers. An analytic DPAL model extracts an analytical solution for the DPAL from the rate equations. The energy balance methodology is used longitudinally averaged number densities, or (LANDs), its formulation. By using LAND’s formulation, we allow double passing of the pump beam and the flexibility to include intra-cavity losses. The pump and laser beams are single frequency and centered at the peaks of the absorption and emission cross-sections. The pressure broad ended absorption and emission lines are assumed to be especially homogeneous. We specify in the model the intensity and number densities [6]. The diode pumped Alkali metal vapor laser (DPAL) model and pump and laser beam geometry includes two mirrors (M1 and M2) and laser Gain cell. We consider that the collision induced mixing or relaxation between 2 P3/2 and 2 P1/2 states is much faster by comparison with any other process in the system including stimulated emission. The mathematical description is of a quasi-two level system from which an analytical solution is extracted. The optical schematics for DPAL and the geometry for single or double pass longitudinal pumping are described in the next figure (Fig. 6.4). It includes the right and left running laser waves. Where Ip is the intensity of the input pump beam, I+ and I- are right and left running intra-cavity laser intensities. n1 is the lower level and n2 , n3 are the two upper levels. 1

; n 1 = n L ; n 2 = f · n U ; n 3 = (1 − f ) · n U

E −( 32 ) 1 + 2 · e k B ·T n U = n 2 + n 3 ; n tot = n U + n L f =

6.2 A Quasi-Two Level Analytic Model …

641

Fig. 6.4 Optical schematic for diode pumped Alkali metal vapor laser (DPAL)

Laser Gain Cell

I-

Pump power In single or Double pass

Ip I+

M1

lg

M2

where T is the temperature in degrees Kelvin, kB is the Boltzmann constant and

E 3,2 is the energy different between level 3 and level 2.n tot is the total Alkali atom number density.

E n3 −[ 32 ] = 2 · e k B ·T n2

The two upper levels n 2 and n 3 are decayed with a common life time since by assumption they are coupled by infinity collision induced processes.n 1 , n L —Density of lower level population (2 S1/2 ground state), n 2 —Density of the mid-level population (2 P1/2 state),n 3 —Density of the upper level population (2 P3/2 state), n U —Total density of the mid and upper states, f - Scaling parameter for the density of population in the mid  and upper state (0 ≤ f ≤ 1), n tot —Total density of all mode states (n tot = 3k=1 n k ). Parameters σ32 and σ21 are the line center simulated emission/absorption cross sections, h is Planck’s constant, v p and v L are the line center frequencies of the pump and laser transitions, and τ is the effective lifetime of the upper level manifold. The system of coupled differential equations is as follow. They describe the kinetics and radiation processes in the system at steady state [6]. Ip dn L = · σ31 · [(1 − f ) · n U − 2 · n L ] dt h · vp (I+ + I− ) nU + · σ21 · ( f · n U − n L ) + h · vL τ dn L dn U =− ; n tot = n U + n L dt dt At any plane z, of the laser cell, the number densities of the upper and lower levels are in steady state with the local pump and laser radiation fields present. Solving the above differential equations for n U and n L , and the expressions we get are n U = ξ1 (I p , I+ , I− )

642

6 Gas Laser Systems Stability Analysis …

= −(

τ · n tot · (2 · σ31 · I p · v L + σ21 · (I+ + I− ) · v p ) ) ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p

n L = ξ2 (I p , I+ , I− ) n tot · [( f − 1) · σ31 · τ · I p · v L − ( f · σ21 · τ · (I+ + I− ) + h · v L ) · v p ] = ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p The three rate equations for Ip (intensity of the input pump beam), I+ and I- (right and left running intra-cavity laser intensities) are as follow: d Ip = σ31 · [(1 − f ) · n U − 2 · n L ] · I p dz d I+ d I− = σ21 · ( f · n U − n L ) · I+ ; = −σ21 · ( f · n U − n L ) · I− dz dz And we can write the above rate equations by using ξ1 (I p , I+ , I− ) and ξ2 (I p , I+ , I− ) functions. d Ip = σ31 · [(1 − f ) · ξ1 (I p , I+ , I− ) − 2 · ξ2 (I p , I+ , I− )] dz ξ1 =ξ1 (I p , I+ , I− ); ξ2 =ξ2 (I p , I+ , I− ) d I+ = σ21 · ( f · ξ1 (I p , I+ , I− ) − ξ2 (I p , I+ , I− )) · I+ dz d I− = −σ21 · ( f · ξ1 (I p , I+ , I− ) − ξ2 (I p , I+ , I− )) · I− dz At fixed points (equilibrium points):

d Ip dz

= 0; ddzI+ = 0; ddzI− = 0

d Ip = 0 ⇒ σ31 · [(1 − f ) · ξ1 − 2 · ξ2 ] = 0 ⇒ σ31 = 0 dz ξ∗ 2 (1 − f ) · ξ1∗ = 2 · ξ2∗ ⇒ 1∗ = ξ2 (1 − f ) τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p ) ξ1∗ ) = −( ξ2∗ n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] 2 = (1 − f ) τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p )

n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] 2 = ( f − 1)

6.2 A Quasi-Two Level Analytic Model …

643

τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p ) · ( f − 1)

n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] =2 τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p ) · ( f − 1) = 2 · n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · vL ) · v p ] τ · n tot · 2 · σ31 · I p∗ · v L · ( f − 1) + τ · n tot · σ21 · v p · (I+∗ + I−∗ ) · ( f − 1) = 2 · n tot · ( f − 1) · σ31 · τ · I p∗ · v L − 2 · n tot · f · σ21 · τ · (I+∗ + I−∗ ) · v p − 2 · n tot · h · v L · v p [2 · τ · n tot · σ31 − 2 · τ · n tot · σ31 ] · ( f − 1) · v L · I p∗ + (3 · f − 1) · (I+∗ + I−∗ ) · v p · τ · n tot · σ21 = −2 · n tot · h · v L · v p [2 · τ · n tot · σ31 − 2 · τ · n tot · σ31 ] · ( f − 1) · v L · I p∗ = 0 ⇒ (3 · f − 1) · (I+∗ + I−∗ ) · v p · τ · n tot · σ21 = −2 · n tot · h · v L · v p I+∗ + I−∗ = −

2 · n tot · h · v L · v p (3 · f − 1) · v p · τ · n tot · σ21

d I+ = 0 ⇒ σ21 · ( f · ξ1∗ − ξ2∗ ) · I+∗ = 0; σ21 = 0 dz ( f · ξ1∗ − ξ2∗ ) · I+∗ = 0; Case 1 f · ξ1∗ − ξ2∗ = 0 ⇒ Case 1

ξ1∗ ξ2∗

=

f =

2 ; (1− f )

ξ1∗ ξ2∗

=

1 1+2·e

=3⇒e

E 32 −( k ·T B

E 32 B ·T

−( k

Meaning: E 32 = 0 ⇒ (I p(0) , I+(0) , I−(0) )

1 ; 2 f (1− f )

)

)

=

1 f

⇒ f =

=

1 ; f

Case 2 I+∗ = 0

1 3

E 1 −( 32 ) ⇒ 1 + 2 · e k B ·T 3

=1⇒

n3 n2

=

ξ1∗ ξ2∗

E 32 = 0; k B · T = 0; E 32 = 0 kB · T

= 2 ⇒ n 3 = 2 · n 2 ; First fixed point: E (0) =

644

6 Gas Laser Systems Stability Analysis … ·h·v ·v

tot L p Case 2 I+∗ = 0 ⇒ I−∗ = − (3· f −1)·v p ·τ ·n tot ·σ21

2·n

·h·v ·v

tot L p Second Fixed point: E (1) = (I p(1) , I+(1) , I−(1) ) = (I p(1) , 0, − (3· f −1)·v ) p ·τ ·n tot ·σ21

2·n

d I− dz

ξ2∗ ⇒

= 0 ⇒ −σ21 · ( f · ξ1∗ − ξ2∗ ) · I− ; σ21 = 0; Case 3 f · ξ1∗ − ξ2∗ = 0 ⇒ f · ξ1∗ =

ξ1∗ ξ2∗

=

1 f

τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p ) ξ1∗ ) ∗ = −( ξ2 n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] 1 = f f · τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p )

n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ]

= −1

f · τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p ) = −n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] f · τ · n tot · 2 · σ31 · I p∗ · v L + f · τ · n tot · σ21 · (I+∗ + I−∗ ) · v p = −n tot · [( f − 1) · σ31 · τ · I p∗ · v L − f · σ21 · τ · (I+∗ + I−∗ ) · v p − h · vL · v p ] f · τ · n tot · 2 · σ31 · I p∗ · v L + f · τ · n tot · σ21 · (I+∗ + I−∗ ) · v p = −n tot · ( f − 1) · σ31 · τ · I p∗ · v L + n tot · f · σ21 · τ · (I+∗ + I−∗ ) · v p + n tot · h · v L · v p f · τ · n tot · 2 · σ31 · I p∗ · v L + n tot · ( f − 1) · σ31 · τ · I p∗ · v L = n tot · f · σ21 · τ · (I+∗ + I−∗ ) · v p − f · τ · n tot · σ21 · (I+∗ + I−∗ ) · v p + n tot · h · v L · v p n tot · f · τ · σ21 · (I+∗ + I−∗ ) · v p − n tot · f · τ · σ21 · (I+∗ + I−∗ ) · v p = 0 then [3 · f − 1] · I p∗ · τ · v L · σ31 · n tot = n tot · h · v L · v p ⇒ I p∗ h · vp = [3 · f − 1] · τ · σ31 d Ip = 0 ⇒ σ31 · [(1 − f ) · ξ1 − 2 · ξ2 ] = 0 ⇒ σ31 = 0 dz

6.2 A Quasi-Two Level Analytic Model …

645

(1 − f ) · ξ1∗ = 2 · ξ2∗ ⇒

ξ1∗ 2 ∗ = ξ2 (1 − f )

d I− ξ∗ 2 1 d Ip = 0 ⇒ 1∗ = ; =0⇒ dz ξ2 f dz (1 − f ) 2 1 1 = ⇒1=3· f ⇒ f = f (1 − f ) 3 1

f =

1+2·e

=3⇒e

E 32 −( k ·T B

E 32 B ·T

−( k

Meaning: E 32 = 0 ⇒ ξ1∗ ξ2∗

=

)

)

=

E 1 −( 32 ) ⇒ 1 + 2 · e k B ·T 3

=1⇒

n3 n2

E 32 = 0; k B · T = 0; E 32 = 0 kB · T

= 2 ⇒ n 3 = 2 · n 2 ; I p∗ =

h·v p dI ; p [3· f −1]·τ ·σ31 dz

=0⇒

2 (1− f )

ξ1∗ ξ2∗

τ · n tot · (2 · σ31 · I p∗ · v L + σ21 · (I+∗ + I−∗ ) · v p )

= −(

n tot · [( f − 1) · σ31 · τ · I p∗ · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ] 2 = (1 − f )

)

ξ1∗ (I p∗ ) ξ2∗ (I p∗ )

p ∗ ∗ τ · n tot · (2 · σ31 · [ [3· f −1]·τ ·σ31 ] · v L + σ21 · (I+ + I− ) · v p )

h·v

= −( =

p ∗ ∗ n tot · [( f − 1) · σ31 · τ · [ [3· f −1]·τ ·σ31 ] · v L − ( f · σ21 · τ · (I+ + I− ) + h · v L ) · v p ]

h·v

2 (1 − f ) p τ · n tot · (2 · [ [3· f −1]·τ ] · v L + σ21 · (I+∗ + I−∗ ) · v p )

h·v

p n tot · [( f − 1) · τ · [ [3· f −1]·τ ] · v L − ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ]

h·v

=

2 ( f − 1) p ] · v L + τ · n tot · σ21 · (I+∗ + I−∗ ) · v p n tot · 2 · [ [3· f −1]

h·v

p [n tot · ( f − 1) · [ [3· f −1] ] · v L − n tot · ( f · σ21 · τ · (I+∗ + I−∗ ) + h · v L ) · v p ]

h·v

=

2 ( f − 1) n tot · 2 · [

h · vp ] · v L + τ · n tot · σ21 · (I+∗ + I−∗ ) · v p [3 · f − 1]

)

646

6 Gas Laser Systems Stability Analysis …

h · vp ] · v L − n tot · ( f · σ21 · τ [3 · f − 1] 2 · (I+∗ + I−∗ ) + h · v L ) · v p ] · ( f − 1)

= [n tot · ( f − 1) · [

h · vp ] · v L + τ · n tot · σ21 · (I+∗ + I−∗ ) · v p [3 · f − 1] h · vp ] · v L · 2 − n tot · f · σ21 · τ · (I+∗ + I−∗ ) = n tot · [ [3 · f − 1] 2 2 − n tot · h · v L · v p · · vp · ( f − 1) ( f − 1)

n tot · 2 · [

τ · n tot · σ21 · (I+∗ + I−∗ ) · v p + n tot · f · σ21 · τ 2 · (I+∗ + I−∗ ) · v p · ( f − 1) h · vp 2 ] · v L · 2 − n tot · h · v L · v p · = n tot · [ [3 · f − 1] ( f − 1) h · vp ] · vL − n tot · 2 · [ [3 · f − 1] 2 ] · (I+∗ + I−∗ ) · v p · n tot · σ21 · τ ( f − 1) h · vp h · vp ] · v L · 2 − n tot · 2 · [ ] = n tot · [ [3 · f − 1] [3 · f − 1] 2 · v L − n tot · h · v L · v p · ( f − 1)

[1 + f ·

2 ] · (I+∗ + I−∗ ) · v p · n tot · σ21 · τ ( f − 1) 2 = −n tot · h · v L · v p · ( f − 1)

[1 + f ·

(I+∗ + I−∗ ) = −[ (I+∗ + I−∗ ) = −[

n tot · h · v L · v p · [1 + f ·

2 ] ( f −1)

· v p · n tot · σ21 · τ

h · vL · v p · [1 + f ·

(I+∗ + I−∗ ) =

2 ] ( f −1)

2 ( f −1)

2 ( f −1)

· v p · σ21 · τ

]

]

h · vL · 2 [1 − 3 · f ] · σ21 · τ

h·v L ·2 p Third Fixed point: E (2) = (I p(2) , I+(2) , I−(2) ) = ( [3· f −1]·τ , I+∗ , [1−3· − I+∗ ) ·σ31 f ]·σ21 ·τ h·v

We can summary our system three fixed points: E (0) = (I p(0) , I+(0) , I−(0) )

6.2 A Quasi-Two Level Analytic Model …

647

E (1) = (I p(1) , I+(1) , I−(1) ) = (I p(1) , 0, −

2 · n tot · h · v L · v p ) (3 · f − 1) · v p · τ · n tot · σ21

E (2) = (I p(2) , I+(2) , I−(2) ) h · vp h · vL · 2 − I+∗ ) , I∗, =( [3 · f − 1] · τ · σ31 + [1 − 3 · f ] · σ21 · τ Stability analysis, we define three functions: g1 (I p , I+ , I− ) = σ31 · [(1 − f ) · ξ1 (I p , I+ , I− ) − 2 · ξ2 (I p , I+ , I− )] = σ31 · [(1 − f ) · ξ1 − 2 · ξ2 ] g2 (I p , I+ , I− ) = σ21 · ( f · ξ1 (I p , I+ , I− ) − ξ2 (I p , I+ , I− )) · I+ = σ21 · ( f · ξ1 − ξ2 ) · I+ g3 (I p , I+ , I− ) = −σ21 · ( f · ξ1 (I p , I+ , I− ) − ξ2 (I p , I+ , I− )) · I− = −σ21 · ( f · ξ1 − ξ2 ) · I− d Ip = g1 (I p , I+ , I− ) dz d I+ Our system set of differential equations: = g2 (I p , I+ , I− ) dz d I− = g3 (I p , I+ , I− ) dz Linearization: suppose that (I p∗ , I+∗ , I−∗ ) is a fixed point, g1 (I p∗ , I+∗ , I−∗ ) = 0; g2 (I p∗ , I+∗ , I−∗ ) = 0 g3 (I p∗ , I+∗ , I−∗ ) = 0 and let u 1 = I p − I p∗ ; u 2 = I+ − I+∗ ; u 3 = I− − I−∗ denote the components of a small disturbance from the fixed point. To see whether the disturbance grows or decays, we need to derive differential equations dI 1 for u 1 , u 2 and u 3 . Let’s do the u 1 -equation first: du = dtp since I p∗ is constant and dt du 1 dt

=

d Ip dt

= g1 (I p∗ + u 1 , I+∗ + u 2 , I−∗ + u 3 ) by substitution, then by using Taylor

1 1 series expansion du = dtp = g1 (I p∗ , I+∗ , I−∗ ) + . . . du = dtp = g1 (I p∗ , I+∗ , I−∗ ) + dt dt ∂g1 ∂g1 ∂g1 2 2 2 u 1 · ∂ I p + u 2 · ∂ I+ + u 3 · ∂ I− + O(u 1 , u 2 , u 3 , u 1 · u 2 · u 3 ) and since g1 (I p∗ , I+∗ , I−∗ ) = 0

d Ip dt

du 1 dt

= u1 ·

∂g1 ∂ Ip

dI

+ u2 ·

∂g1 ∂ I+

+ u3 ·

∂g1 ∂ I−

+ O(u 21 , u 22 , u 23 , u 1 · u 2 · u 3 ). To ∂g1 ∂g1 simplify the notation, we have written ∂ I p , ∂ I+ and ∂∂gI−1 , these partial derivatives are to be evaluated at the fixed point (I p∗ , I+∗ , I−∗ ); thus they are numbers, not functions. The shorthand notation, O(u 21 , u 22 , u 23 , u 1 · u 2 · u 3 ) denotes quadratic terms in u 1 , u 2 then

=

dI

648

6 Gas Laser Systems Stability Analysis …

and u 3 are small, these quadratic terms are extremely small [2, 3]. Similarly we find d I+ ∂g2 ∂g2 ∂g2 du 2 = = u1 · + u2 · + u3 · dt dt ∂ Ip ∂ I+ ∂ I− + O(u 21 , u 22 , u 23 , u 1 · u 2 · u 3 ) du 3 d I− ∂g3 ∂g3 ∂g3 = = u1 · + u2 · + u3 · dt dt ∂ Ip ∂ I+ ∂ I− + O(u 21 , u 22 , u 23 , u 1 · u 2 · u 3 ) Hence the disturbance (u 1 , u 2 , u 3 ) evolves according to ⎛ du ⎞ 1 ⎛ ∂g1 ∂g1 ∂g1 ⎞ ⎛ ⎞ ⎜ dt ⎟ u1 ∂ I p ∂ I+ ∂ I− ⎜ ⎟ ⎜ du 2 ⎟ ⎜ ∂g2 ∂g2 ∂g2 ⎟ ⎜ ⎟ ⎜ ⎟ = ⎝ ∂ I p ∂ I+ ∂ I− ⎠ · ⎝ u 2 ⎠ + quadratic term and the matrix ⎜ dt ⎟ ∂g3 ∂g3 ∂g3 ⎝ ⎠ u3 ∂ I p ∂ I+ ∂ I− du 3 dt ⎛ ⎞ ∂g1 ∂g1 ∂ I p ∂ I+ ∂g2 ∂ I+ ∂g3 ∂g3 ∂ I p ∂ I+

⎜ ∂g A = ⎝ ∂ I p2 (I p∗ , I+∗ , I−∗ ).

∂g1 ∂ I− ∂g2 ∂ I− ∂g3 ∂ I−

⎟ ⎠

is called the Jacobian matrix at the fixed point (I p∗ ,I+∗ ,I−∗ )

Since the quadratic terms are tiny, it’s tempting to neglect them altogether. If we do that, we obtain the linearized system. ⎛ du ⎞ 1

⎛ ∂g1 ⎜ dt ⎟ ∂ Ip ⎜ ⎟ ⎜ du 2 ⎟ ⎜ ∂g2 ⎜ ⎟ = ⎝ ∂ Ip ⎜ dt ⎟ ∂g3 ⎝ ⎠ ∂ Ip du 3 dt

∂g1 ∂ I+ ∂g2 ∂ I+ ∂g3 ∂ I+

∂g1 ∂ I− ∂g2 ∂ I− ∂g3 ∂ I−

⎞ ⎛

u1

⎞

⎟ ⎜ ⎟ ⎠ · ⎝ u2⎠ u3

Remark The linearized system gives a qualitatively correct picture of the phase portrait near (I p∗ , I+∗ , I−∗ ) as long as the fixed point for the linearized system is not one of the borderline cases. The linearized system predicts a saddle, node, or a spiral, and then it is the real fixed point classification for the original nonlinear system. g1 (I p , I+ , I− ) = σ31 · [(1 − f ) · ξ1 (I p , I+ , I− ) − 2 · ξ2 (I p , I+ , I− )] = σ31 · [(1 − f ) · ξ1 − 2 · ξ2 ] ∂g1 ∂ξ1 ξ2 = σ31 · [(1 − f ) · −2· ] ∂ Ip ∂ Ip Ip

6.2 A Quasi-Two Level Analytic Model …

649

∂ξ1 (I p , I+ , I− ) ∂ξ1 = ∂ Ip ∂ Ip ∂ξ2 (I p , I+ , I− ) ∂n U ∂ξ2 ∂n L = ; = = ∂ Ip ∂ Ip ∂ Ip ∂ Ip n U = ξ1 (I p , I+ , I− )   τ · n tot · (2 · σ31 · I p · v L + σ21 · (I+ + I− ) · v p ) =− ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p n L = ξ2 (I p , I+ , I− ) n tot · [( f − 1) · σ31 · τ · I p · v L − ( f · σ21 · τ · (I+ + I− ) + h · v L ) · v p ] = ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p We define new variables: ψ1 = ψ1 (I p , I+ , I− ); ψ2 = ψ2 (I p , I+ , I− ); ψ3 = ψ3 (I p , I+ , I− ) ψ1 = ψ1 (I p , I+ , I− ) = ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p ψ2 = ψ2 (I p , I+ , I− ) = τ · n tot · (2 · σ31 · I p · v L + σ21 · (I+ + I− ) · v p ) ψ2 n U = ξ1 (I p , I+ , I− ) = − ψ1 ψ3 = ψ3 (I p , I+ , I− ) = n tot · [( f − 1) · σ31 · τ · I p · v L − ( f · σ21 · τ · (I+ + I− ) + h · v L ) · v p ] ∂ψ2 1 · ψ1 − ∂ψ · ψ2 ∂ξ1 ∂n U ∂ Ip ∂ Ip = =− ∂ Ip ∂ Ip ψ12 ∂ψ2 ∂ψ1 = τ · n tot · 2 · σ31 · v L ; = ( f − 3) · σ31 · τ · v L ∂ Ip ∂ Ip

n L = ξ2 (I p , I+ , I− ) ψ3 ∂ξ2 ∂n L ; = = ψ1 ∂ I p ∂ I p ∂ξ2 (I p , I+ , I− ) = = ∂ Ip

∂ψ3 ∂ Ip

· ψ1 − ψ12

∂ψ1 ∂ Ip

· ψ3

650

6 Gas Laser Systems Stability Analysis …

∂ψ3 = n tot · ( f − 1) · σ31 · τ · v L ∂ Ip ∂g1 ∂ξ1 ξ2 = σ31 · [(1 − f ) · −2· ] ∂ Ip ∂ Ip Ip = σ31 · [−(1 − f ) · ( −2·(

∂ψ3 ∂ Ip

· ψ1 −

∂ψ2 ∂ Ip

· ψ1 −

∂ψ1 ∂ Ip

· ψ2

ψ12

∂ψ1 ∂ Ip

· ψ3

ψ12

)

)]

∂g1 ∂ξ1 ∂ξ2 = σ31 · [(1 − f ) · −2· ] ∂ I+ ∂ I+ ∂ I+ ∂ξ1 (I p , I+ , I− ) ∂ξ1 ∂n U = = ∂ I+ ∂ I+ ∂ I+ ∂ξ2 (I p , I+ , I− ) ∂ξ2 ∂n L = = ∂ I+ ∂ I+ ∂ I+ ∂ψ2 1 · ψ1 − ∂ψ · ψ2 ∂ξ1 ∂n U ∂ I+ ∂ I+ = =− ∂ I+ ∂ I+ ψ12 ∂ψ2 ∂ψ1 = τ · n tot · σ21 · v p ; = −( f + 1) · σ21 · τ · v p ∂ I+ ∂ I+

∂ψ3 = −n tot · f · σ21 · τ · v p ∂ I+ ∂ξ2 (I p , I+ , I− ) ∂ξ2 ∂n L = = ∂ I+ ∂ I+ ∂ I+ =

∂ψ3 ∂ I+

· ψ1 −

∂ψ1 ∂ I+

· ψ3

ψ12

∂g1 ∂ξ1 ∂ξ2 = σ31 · [(1 − f ) · −2· ] ∂ I− ∂ I− ∂ I− ∂ψ2 1 · ψ1 − ∂ψ · ψ2 ∂ξ1 ∂n U ∂I ∂ I− = =− − 2 ∂ I− ∂ I− ψ1

∂ψ1 = −( f + 1) · σ21 · τ · v p ∂ I− ∂ψ2 ∂ψ3 = τ · n tot · σ21 · v p ; = −n tot · f · σ21 · τ · v p ∂ I− ∂ I− ∂ξ2 (I p , I+ , I− ) ∂ξ2 ∂n L = = ∂ I− ∂ I− ∂ I− =

∂ψ3 ∂ I−

· ψ1 − ψ12

∂ψ1 ∂ I−

· ψ3

6.2 A Quasi-Two Level Analytic Model …

651

Table 6.7 Quasi-two level analytic model for metal vapor laser system’s Jacobian ∂ψ1 ∂ Ip

( f − 3) · σ31 · τ · v L

∂ψ1 ∂ I+

−( f + 1) · σ21 · τ · v p

∂ψ1 ∂ I−

∂ψ2 ∂ Ip

−( f + 1) · σ21 · τ · v p

τ · n tot · 2 · σ31 · v L

∂ψ2 ∂ I+

τ · n tot · σ21 · v p

∂ψ2 ∂ I−

τ · n tot · σ21 · v p

∂ψ3 ∂ Ip

n tot · ( f − 1) · σ31 · τ · v L

∂ψ3 ∂ I+

−n tot · f · σ21 · τ · v p

∂ψ3 ∂ I−

−n tot · f · σ21 · τ · v p

We can summary our last results in the following table (Table 6.7):

∂ψ1 ∂ψ1 ∂ψ1 ∂ψ2 ∂ψ2 ∂ψ2 ∂ψ3 ∂ψ3 ∂ψ3 Elements: ∂ I p , ∂ I+ , ∂ I− , ∂ I p , ∂ I+ , ∂ I− , ∂ I p , ∂ I+ , ∂ I− . g2 = g2 (I p , I+ , I− ) = σ21 · ( f · ξ1 (I p , I+ , I− ) − ξ2 (I p , I+ , I− )) · I+ = σ21 · ( f · ξ1 − ξ2 ) · I+ ∂g2 ∂ξ1 ∂ξ2 = σ21 · ( f · − ) · I+ ∂ Ip ∂ Ip ∂ Ip ∂g2 ∂ξ1 ∂ξ2 = σ21 · ( f · − ) · I+ + σ21 · ( f · ξ1 − ξ2 ) ∂ I+ ∂ I+ ∂ I+ ∂g2 ∂ξ1 ∂ξ2 = σ21 · ( f · − ) · I+ ∂ I− ∂ I− ∂ I− ∂g3 ξ1 ξ2 = −σ21 · ( f · − ) · I− ∂ Ip ∂ Ip ∂ Ip ∂g3 ∂ξ1 ∂ξ2 = −σ21 · ( f · − ) · I− ∂ I+ ∂ I+ ∂ I+ ∂g3 ∂ξ1 ∂ξ2 = −σ21 · ( f · − ) · I− − σ21 · ( f · ξ1 − ξ2 ) ∂ I− ∂ I− ∂ I− We can summary our last results in the following table (Table 6.8): The system Jacobian matrix at the fixed point (I p∗ , I+∗ , I−∗ ) is Table 6.8 Quasi-two level analytic model system’s Jacobian elements: ∂ξi /∂ I p,+,− ; i = 1, 2 ∂ξ1 ∂ Ip ∂ξ1 ∂ I+

−

∂ψ2 ∂Ip

∂ψ

·ψ1 − ∂ I p1 ·ψ2

∂ξ1 ∂ I−

ψ12 ∂ψ2

− ∂ I+

∂ψ +

·ψ1 − ∂ I 1 ·ψ2 ψ12

∂ξ2 ∂ Ip

∂ψ2

− ∂ I− ∂ψ3 ∂Ip

∂ψ −

·ψ1 − ∂ I 1 ·ψ2 ψ12 ∂ψ

·ψ1 − ∂ I p1 ·ψ3 ψ12

∂ξ2 ∂ I+

∂ψ3 ∂ I+

·ψ1 − ∂ I 1 ·ψ3

∂ξ2 ∂ I−

∂ψ3 ∂ I−

·ψ1 − ∂ I 1 ·ψ3

ψ12

ψ12

∂ψ +

∂ψ −

652

6 Gas Laser Systems Stability Analysis …

A−λ· I ⎛ σ31 · [(1 − f ) · ∂∂ξI 1p − 2 · ξI 2p ] − λ σ31 · [(1 − ⎜ ⎜ σ21 · ( f · ⎜ σ21 · ( f · ∂∂ξI 1p − ∂∂ξI 2p ) · I+ ⎜ =⎜ +σ21 · ( f ⎜ ⎜ ⎜ ⎝ −σ · ( f · ξ1 − ξ2 ) · I −σ21 · ( f 21 − ∂ Ip ∂ Ip

⎞ f ) · ∂∂ξI+1 − 2 · ∂∂ξI+2 ] σ31 · [(1 − f ) · ∂∂ξI−1 − 2 · ∂∂ξI−2 ] ⎟ ∂ξ1 ∂ξ2 ⎟ − ) · I+ σ21 · ( f · ∂∂ξI−1 − ∂∂ξI−2 ) · I+ ⎟ ∂ I+ ∂ I+ ⎟ ⎟ · ξ1 − ξ2 ) − λ ⎟ ⎟ ∂ξ1 ∂ξ2 ⎟ −σ · ( f · − ) · I 21 − ⎠ · ∂∂ξI+1 − ∂∂ξI+2 ) · I− ∂ I− ∂ I− −σ21 · ( f · ξ1 − ξ2 ) − λ (I ∗ ,I ∗ ,I ∗ ) p

For simplicity we define the following global Jacobian elements: ∂ξ1 ξ2 − 2 · ]|(I p∗ ,I+∗ ,I−∗ ) ∂ Ip Ip ∂ξ1 ∂ξ2 = σ31 · [(1 − f ) · −2· ]|(I ∗ ,I ∗ ,I ∗ ) ∂ I+ ∂ I+ p + −

ϒ11 = σ31 · [(1 − f ) · ϒ12

∂ξ1 ∂ξ2 −2· ]|(I ∗ ,I ∗ ,I ∗ ) ∂ I− ∂ I− p + − ∂ξ1 ∂ξ2 = σ21 · ( f · − ) · I+ |(I p∗ ,I+∗ ,I−∗ ) ∂ Ip ∂ Ip

ϒ13 = σ31 · [(1 − f ) · ϒ21

∂ξ1 ∂ξ2 − ) · I+ + σ21 · ( f · ξ1 − ξ2 )|(I p∗ ,I+∗ ,I−∗ ) ∂ I+ ∂ I+ ∂ξ1 ∂ξ2 = σ21 · ( f · − ) · I+(I p∗ ,I+∗ ,I−∗ ) ∂ I− ∂ I−

ϒ22 = σ21 · ( f · ϒ23

ξ1 ξ2 − ) · I− |(I p∗ ,I+∗ ,I−∗ ) ∂ Ip ∂ Ip ∂ξ1 ∂ξ2 = −σ21 · ( f · − ) · I− |(I p∗ ,I+∗ ,I−∗ ) ∂ I+ ∂ I+

ϒ31 = −σ21 · ( f · ϒ32

ϒ33 = −σ21 · ( f ·

∂ξ1 ∂ξ2 − ) · I− − σ21 · ( f · ξ1 − ξ2 )|(I p∗ ,I+∗ ,I−∗ ) ∂ I− ∂ I−

+

−

6.2 A Quasi-Two Level Analytic Model …

653

⎛

⎞ ϒ13 ϒ11 − λ ϒ12 A − λ · I = ⎝ ϒ21 ϒ22 − λ ϒ23 ⎠ ϒ31 ϒ32 ϒ33 − λ (I ∗ ,I ∗ ,I ∗ ) p + − Expanding the determinant yields: 

ϒ22 − λ ϒ23 det(A − λ · I ) = (ϒ11 − λ) · det ϒ32 ϒ33 − λ   ϒ21 ϒ23 − ϒ12 · det ϒ31 ϒ33 − λ   ϒ21 ϒ22 − λ + ϒ13 · det ϒ31 ϒ32



det(A − λ · I ) = (ϒ11 − λ) · [(ϒ22 − λ) · (ϒ33 − λ) − ϒ32 · ϒ23 ] − ϒ12 · [ϒ21 · (ϒ33 − λ) − ϒ31 · ϒ23 ] + ϒ13 · [ϒ21 · ϒ32 − ϒ31 · (ϒ22 − λ)] det(A − λ · I ) = (ϒ11 − λ) · [(ϒ22 · ϒ33 − ϒ32 · ϒ23 ) − λ · (ϒ22 + ϒ33 ) + λ2 ] − ϒ12 · [(ϒ21 · ϒ33 − ϒ31 · ϒ23 ) − ϒ21 · λ] + ϒ13 · [(ϒ21 · ϒ32 − ϒ31 · ϒ22 ) + ϒ31 · λ] det(A − λ · I ) = ϒ11 · (ϒ22 · ϒ33 − ϒ32 · ϒ23 ) − λ · ϒ11 · (ϒ22 + ϒ33 ) + ϒ11 · λ2 − λ · (ϒ22 · ϒ33 − ϒ32 · ϒ23 ) + λ2 · (ϒ22 + ϒ33 ) − λ3 − ϒ12 · (ϒ21 · ϒ33 − ϒ31 · ϒ23 ) + ϒ12 · ϒ21 · λ + ϒ13 · (ϒ21 · ϒ32 − ϒ31 · ϒ22 ) + ϒ13 · ϒ31 · λ det(A − λ · I ) = {ϒ11 · (ϒ22 · ϒ33 − ϒ32 · ϒ23 ) − ϒ12 · (ϒ21 · ϒ33 − ϒ31 · ϒ23 ) + ϒ13 · (ϒ21 · ϒ32 − ϒ31 · ϒ22 )} + λ · {ϒ12 · ϒ21 + ϒ13 · ϒ31 − ϒ11 · (ϒ22 + ϒ33 ) − (ϒ22 · ϒ33 − ϒ32 · ϒ23 )} + λ2 · (ϒ11 + ϒ22 + ϒ33 ) − λ3 We define for simplicity the following global parameters: 0 = ϒ11 · (ϒ22 · ϒ33 − ϒ32 · ϒ23 ) − ϒ12 · (ϒ21 · ϒ33 − ϒ31 · ϒ23 ) + ϒ13 · (ϒ21 · ϒ32 − ϒ31 · ϒ22 )

654

6 Gas Laser Systems Stability Analysis …

1 = ϒ12 · ϒ21 + ϒ13 · ϒ31 − ϒ11 · (ϒ22 + ϒ33 ) − (ϒ22 · ϒ33 − ϒ32 · ϒ23 ) 2 = ϒ11 + ϒ22 + ϒ33 ; 3 = −1 det(A − λ · I ) = 0 + λ · 1 + λ2 · 2 + λ3 · 3 det(A − λ · I ) =

3 

k · λk

k=0

det(A − λ · I ) = 0 ⇒

3 

k · λk = 0

k=0 3

0 + λ · 1 + λ · 2 + λ · 3 = 0 2

Eigenvalue stability discussion: Our Quasi-two level analytic model system involving N variables (N > 2, N = 3, arbitrarily small increments), the characteristic equation is of degree N = 3 and must often be solved numerically. Expect in some particular cases, such an equation has (N = 3) distinct roots that can be real or complex. These values are eigenvalues of the (3 × 3) Jacobian matrix (A). The general rule is that the Quasi-two level analytic model system is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our three variables (I p , I+ , I− ) system has three eigenvalues (three system’s arbitrarily small increments). The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five non-generated cases can be distinguished: (1) the three eigenvalues are real and negative (stable steady state), (2) the three eigenvalues are real, at least one of them is positive (unstable steady state), (3) and (4) two eigenvalues are complex conjugates with a negative real part and other eigenvalue real is negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) two eigenvalues are complex conjugates with a negative real part and other eigenvalue real is positive (unstable steady state) [7, 8]. det(A − λ · I ) = 0 ⇒

3 

k · λk = 0

k=0

0 + λ · 1 + λ2 · 2 + λ3 · 3 = 0

6.3 A Self-consistent Model Copper Vapor …

655

6.3 A Self-consistent Model Copper Vapor Laser (CVL) Circuity Stability Analysis Under Parameters Variation A self-consistent model copper vapor laser (CVL) is very helpful for characterization of the copper laser and its implementation. It uses vapors of copper as the lasing medium in a 3-level laser. It can produce green laser light (510.6 nm) and yellow laser light (578.2 nm). The peak power of the laser pulse can be between 50kw and 5 × 103 kw. The pulse width is typically from 5 nsec to 60 nsec and the pulse repetition frequencies are typically between 2 kHz to 100 kHz. We can characterize the average power of copper vapor laser in the range of 25 w to 2 kw. To a create vapor it is necessary to get extremely high temperature, about 1500 °C. Coper compound vapors increase the complexity of the pump signal applied to the device. Typically, two engineering pulses are required, the first to dissociate vapor molecules, and the second to cause the dissociated ions to lase. We can reduce the operating temperature by the use of copper nitrate or copper acetylacetonate. By using those coppers we get vapors with peak laser output power at 180 °C and 40 °C, respectively. If we want to use a copper vapor lasers that are cw (continuous wave) we need to use singly ionized species of Cu. These lasers can provide average UV powers of several mW mainly for spectroscopy applications. The copper vapor laser makes use of copper vapors as the lasing medium. Generally we use copper bromide, copper chloride, and copper iodide to implement copper vapor laser (CVL). The laser head of the copper vapor laser consists of a refractory ceramic tube that contains a low-pressure buffer gas, neon, and copper pellets. A pulsed electrical discharge is between electrodes at the tube ends, and the temperature of the laser head increases to ≈ 1450 C degree, and producing copper vapor at low temperature. The excited electrons collide with vaporized copper atoms and excite these atoms to the upper 2 P3/2 and 2 P1/2 laser levels. The electrons in the 2 P3/2 level decay to the lower 2 D5/2 laser level to produce green light, and the 2 P1/2 decay to 2 D3/2 level to produce yellow laser light. Copper vapor laser has good beam quality, excellent power stability and along operational life time. The main applications are laser cutting and precision micromachining [9]. When producing copper vapor laser it is recommended to use several mbars of neon which added as a buffer gas to prevent window contamination and loss of copper. The overall well plug efficiency of copper vapor laser is ~1% which is highest for visible gas lasers. The copper vapor laser tube is sealed with flat glass windows; the rear mirror is a total reflector, with 90% transmission chosen for the output windows. Beams from copper vapor lasers can be from 10 mm diameter to ~50 mm diameter, beam divergence for a stable type resonator is 3 mradians to 5 mradians. In the visible part of the spectrum, beam focusing is recommended to use glass optics. The copper vapor laser structure is present below (Fig. 6.5). We describe the equations for the discharge pulse, laser pulse, and inter pulse afterglow period. The simulation analysis is over many discharge cycles until a consistent solution is obtained. We present a consistent initial conditions and scaling characteristics of the copper laser. Ideal and real laser are simulate and a conclusion is accepted that the peak electron temperature and the initial density of metastable

656

6 Gas Laser Systems Stability Analysis …

Fig. 6.5 Copper vapor laser (CVL) structure

copper are the factors which dominate the performance of the laser. These processes affect these two quantities. We produce a high peak and high average power in the visible by using high repetition rate (HRR) electric discharge metal vapor laser. The most sophisticated HRR copper lasers are resonantly charged electric discharge devices. These lasers are self-heated and the heating energy of the laser tube and vaporize the copper comes from the discharge itself. The copper laser is a cyclic or self-terminating laser. The upper laser level is resonant to the ground state while the lower laser level is metastable. Atoms which making the laser transition, accumulate in the lower level and caused to gain to become negative. The choice of repetition rate immediately, sets threshold pumping requirements. The choice of repetition rate determined the initial conditions for the discharge pulse. The initial electron density, plasma tube impedance, and electron temperature are a function of inter pulse processes. In an ideal HRR copper vapor laser, quantities such as charging voltage, capacitance, metal density, and repetition rate are independent be uniquely specified. In an ideal laser, we specify thermodynamic quantities, access any tube temperature, and uniformly deposit energy throughout the active volume. The discharge circuit of an ideal laser is free of inductance. In real HRR laser, the tube temperature and metal density are functions of the discharge input power, and then the charging voltage, capacitance, repetition rate, and metal density are not independent. The buffer gas pressure can be specified exterior to the tube but not accurately within the tube due to transient discharge heating. The laser’s materials define an upper limit on the tube temperature, and inductance in the discharge circuit limits the rate at which current and voltage appear across the tube. These limitations mask the physical process which occur in a real HRR laser and make it difficult to de convolve the effect of changing a single variable. The analysis is divided into two sections. The first section is the modeling of an ideal laser and the second section examines and models the real

6.3 A Self-consistent Model Copper Vapor …

657

Fig. 6.6 Ideal copper vapor laser (CVL) discharge circuit

laser. The experiment is related to VENUS laser (15w device, 6hHz). The VENUS laser discharge tube is made of alumina and heat region. The laser power on metal density (tube temperature) is a function of the peak electron temperature. The peak electron temperature is a function of the peak voltage across the discharge tube, and it determines the optimum tube temperature and maximum laser power. The dependence of laser power on repetition rate is a function of the initial metastable density, thus the value is a function of the dominant mode of relaxation which is electron collision or diffusion. The model simulates a discharge pulse, laser pulse, and inter-pulse afterglow in a high repetition rate copper laser. The initial conditions are chosen and a number of cycles are computed, the outcome is consistent pulses. The voltage across the laser tube, and current through laser tube are a function of the plasma impedance and the particulars of the discharge circuit. Detailed discharge circuitry is an integral part of the model. There are two groups of discharge circuits which describe the ideal and real lasers [9]. Ideal laser discharge circuit: The ideal laser discharge circuit consists of a capacitor initially charged to a voltage V0 and a Thyratron with a switching time τs and discharge tube (Fig. 6.6). A thyratron is a type of gas-filled tube which used as a highpower electrical switch and controlled rectifier. The Thyratrons handle currents are much greater than hard-vacuum tube’s currents. The working details of Thyratron are as follow: An electron multiplication occurs when the gas becomes ionized, and Townsend discharge phenomenon is happened. The Townsend discharge or Townsend avalanche is a gas ionization process where free electrons are accelerated by an electric field, collide with gas molecules, and finally free additional electron. Then these electrons are in turn accelerated and free additional electrons. We get avalanche multiplication and electrical conduction through the gas is happened. The phenomenon occurs when there is a source of free electrons and a significant electric field. Thyratron, gas-filled discharge chamber grid controls the starting of a current and thus provides a trigger effect. The grid potential is negative with respect to the cathode and prevents electrons from flowing to the plate and exciting a discharge. The grid potential is raised enough to start electrons flowing from the cathode then

658

6 Gas Laser Systems Stability Analysis …

discharge happened. As free electrons stream toward the plate, they collide with gas molecules, freeing other electrons and ionizing the gas within the discharge chamber. When a sufficient number of ions and electrons are present, a “short” occurs, and a large current flows from the cathode to the plate, causing discharge. The discharge stops when the anode voltage has been sufficiently lowered. A discharge tube is an electron tube which contains gas or vapor at low pressure and through which conduction takes place when a high voltage is applied. The ionization is induced by an electric field and the gas molecules emit light as they return to the ground state. The time rate of change of voltage across the tube is ddtV = V0 · f (τs ) − RdV·C ,

e ·v where f (τs ) is the Thyratron voltage turn on function, and Rd , Rd = nl·m 2 , is the ·A·e e instantaneous discharge impedance. Where, m e is the mass of the electron, v is the electron collision frequency, and A is the effective cross-section area of the discharge tube. Real laser discharge circuit: The discharge circuit for the real laser simulates the actual circuit which use in VENUS laser (Fig. 6.7). The model discharge circuit elements are capacitor C1 which is the storage capacitor resonantly charged to voltage V0 by a DC power supply and inductanceL c . Capacitor C2 is the peaking capacitor. The Thyratron switch (S1 ) is modeled by a reverse bias voltage and removed with a switching time τs . Rc is the parallel charging resistance. The Thyratron behaves like a diode with current flowing through it in only one direction. The inductances L 1 and L 2 represent the lumped circuit values [9]. Where current I1 flows through L 1 and current I2 flows through L 2 , voltage V1 is the voltage across capacitor C1 and V 2 is the voltage across capacitor C2 . The

Rc ·Rd voltage across the discharge tube is Vd , Vd = I2 (t − τ2 ) · Rc +Rd , and the current c that flows through the discharge tube is Id , Id = I2 (t − τ2 ) · Rc R+R . The real model d differential equations circuitry which represents the actual behavior of our system is as follow:

(V1 − V2 ) + V0 · f (τs ) d I1 = dt L1

Fig. 6.7 Real copper vapor laser (CVL) discharge circuit

6.3 A Self-consistent Model Copper Vapor …

659

V2 − Vd d V1 I1 (t − τ1 ) d I2 = =− ; dt L2 dt C1 I1 (t − τ1 ) − I2 (t − τ2 ) d V2 = dt C2 Remark The real laser discharge circuit lumped inductances L1 and L2 are not ideal and there is time delays τ1 , τ2 of currents I1 and I2 that flow through them, respectively. τ1 and τ2 are circuit lumped inductances (L 1 , L 2 ) delay in time which causes by electromagnetic interferences. Equilibrium points (fixed points): lim I1 (t − τ1 )|t τ1 = I1 (t); lim I2 (t − τ2 ) = t→∞

I2 (t)

t→∞

d I2 d V1 d V2 d I1 = 0; = 0; = 0; =0 dt dt dt dt ∗ ∗ ∗ (V1 − V2 ) + V0 · f (τs ) V − Vd = 0; 2 = 0 ⇒ V2∗ = Vd L1 L2 (V1∗ − V2∗ ) + V0 · f (τs ) (V ∗ − Vd ) + V0 · f (τs ) =0⇒ 1 =0 L1 L1 (V1∗ − Vd ) + V0 · f (τs ) = 0 V1∗ = Vd − V0 · f (τs ); −

I1∗ = 0 ⇒ I1∗ = 0 C1

I1∗ − I2∗ = 0 ⇒ I1∗ = I2∗ = 0 C2

System fixed points: E ∗ (I1∗ , I2∗ , V1∗ , V2∗ ) = (0, 0, Vd − V0 · f (τs ), Vd ) Stability analysis: The standard local stability analysis about any one of the equilibrium point of the copper vapor laser (CVL) circuity consists in adding to coordinate [I1 , I2 , V1 , V2 ] arbitrarily small increments of exponential form [i 1 , i 2 , v1 , v2 ] · eλ·t and retaining the first order terms in I1 , I2 , V1 , V2 . The system of four homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into copper vapor laser (CVL) circuity equations. The copper vapor laser (CVL) circuity fixed values with arbitrarily small increments of exponential form [i 1 , i 2 , v1 , v2 ] · eλ·t are: j = 0(first fixed point), j = 1(second fixed point), j = 2(third fixed point), etc. [4, 5]. ( j)

( j)

I1 (t) = I1 + i 1 · eλ·t ; I2 (t) = I2 + i 2 · eλ·t ( j)

V1 (t) = V1

( j)

+ v1 · eλ·t ; V2 (t) = V2

+ v2 · eλ·t

660

6 Gas Laser Systems Stability Analysis … ( j)

I1 (t − τ1 ) = I1 + i 1 · eλ·(t−τ1 ) ; I2 (t − τ2 ) ( j)

= I2 + i 2 · eλ·(t−τ2 ) d I1 (t) d I2 (t) = i 1 · λ · eλ·t ; = i 2 · λ · eλ·t dt dt d V1 (t) d V2 (t) = v1 · λ · eλ·t ; = v2 · λ · eλ·t dt dt We choose these expressions for ourselves I1 (t), I2 (t), V1 (t), V2 (t) as a small displacement [i 1 , i 2 , v1 , v2 ] from the copper vapor laser (CVL) circuity fixed points in time t = 0. ( j)

( j)

I1 (t = 0) = I1 + i 1 ; I2 (t = 0) = I2 + i 2 ( j)

V1 (t = 0) = V1

( j)

+ v1 ; V2 (t = 0) = V2

+ v2

d I1 (V1 − V2 ) + V0 · f (τs ) = dt L1 i 1 · λ · eλ·t = i 1 · λ · eλ·t = At fixed point: v2 · eλ·t

d I1 dt

( j)

( j)

+ v1 · eλ·t − V2

(V1

( j)

(V1

=0⇒

− i1 · λ +

− v2 · eλ·t ) + V0 · f (τs ) L1

( j)

− V2 ) + V0 · f (τs ) (v1 · eλ·t − v2 · eλ·t ) + L1 L1 ( j)

( j)

(V1 −V2 )+V0 · f (τs ) L1

= 0; i 1 · λ · eλ·t =

1 L1

· v1 · eλ·t −

1 L1

·

V2 − Vd 1 1 d I2 = · v1 − · v2 = 0; L1 L1 dt L2 ( j)

i 2 · λ · eλ·t =

V2

i 2 · λ · eλ·t =

V2

+ v2 · eλ·t − Vd L2

( j)

( j)

− Vd v2 · eλ·t + L2 L2

−V

At fixed point: ddtI2 = 0 ⇒ 2 L 2 d = 0; i 2 ·λ·eλ·t = L12 ·v2 ·eλ·t ; −i 2 ·λ+ L12 ·v2 = 0 We have four sub cases for (1) τ1 = 0; τ2 = 0(2) τ1 = τ ; τ2 = 0(3) τ1 = 0; τ2 = τ (4) τ1 = τ ; τ2 = τ . We choose to analyze the second case τ1 = τ ; τ2 = 0. V

( j)

d V1 I + i 1 · eλ·(t−τ ) I1 (t − τ ) ; v1 · λ · eλ·t = − 1 =− dt C1 C1 v1 · λ · eλ·t = −

( j)

I1 i 1 · eλ·t · e−λ·τ − C1 C1

6.3 A Self-consistent Model Copper Vapor …

At fixed point:

d V1 dt

661

( j)

−λ·τ

−λ·τ

= 0 ⇒ − C1 1 = 0; v1 · λ = − i1 ·eC1 ; − i1 ·eC1 I

− v1 · λ = 0

d V2 I1 (t − τ ) − I2 (t) = dt C2 v2 · λ · eλ·t = v2 · λ · eλ·t =

( j)

( j)

I1 −I2 C2

+

( j)

( j)

I1 + i 1 · eλ·(t−τ ) − I2 − i 2 · eλ·t C2

i 1 ·eλ·(t−τ ) −i 2 ·eλ·t C2

, At fixed point:

d V2 dt

=0⇒

( j)

( j)

I1 −I2 C2

=0

i 1 · eλ·t · e−λ·τ − i 2 · eλ·t C2 1 · i1 − · i 2 − v2 · λ = 0 C2

v2 · λ · eλ·t = ⇒

e−λ·τ C2

We can summary our copper vapor laser (CVL) circuity four arbitrarily small increments equations: 1 1 1 · v1 − · v2 = 0; −i 2 · λ + · v2 = 0 L1 L1 L2 i 1 · e−λ·τ e−λ·τ 1 − − v1 · λ = 0; · i1 − · i 2 − v2 · λ = 0 C1 C2 C2 − i1 · λ +

The small increments Jacobian of our copper vapor laser (CVL) circuity is as follow: ⎛ ⎞ ⎛ ⎞ i1 ϒ11 . . . ϒ14 ⎜i ⎟ ⎜ .. . . .. ⎟ ⎜ 2 ⎟ ⎟=0 ⎝ . . . ⎠·⎜ ⎝ v1 ⎠ ϒ41 · · · ϒ44 v2 1 1 ϒ11 = −λ; ϒ12 = 0; ϒ13 = ; ϒ14 = − ; ϒ21 = 0; ϒ22 = −λ L1 L1 ϒ23 = 0; ϒ24 = ϒ33 = −λ; ϒ34

1 1 ; ϒ31 = − · e−λ·τ ; ϒ32 = 0 L2 C1 1 1 = 0; ϒ41 = · e−λ·τ ; ϒ42 = − C2 C2 ϒ43 = 0; ϒ44 = −λ

662

6 Gas Laser Systems Stability Analysis …

⎛

ϒ11 . . . ⎜ .. . . A−λ· I =⎝ . .

⎞ ϒ14 .. ⎟ . ⎠

ϒ41 · · · ϒ44 ⎛ ϒ11 . . . ⎜ .. . . det(A − λ · I ) = det ⎝ . . ϒ41 · · ·

⎞ ϒ14 .. ⎟ . ⎠

ϒ44

det(A − λ · I ) = 0 ⎛

⎞ ϒ22 ϒ23 ϒ24 det(A − λ · I ) = ϒ11 · det ⎝ ϒ32 ϒ33 ϒ34 ⎠ ϒ42 ϒ43 ϒ44 ⎛ ⎞ ϒ21 ϒ23 ϒ24 − ϒ12 · det ⎝ ϒ31 ϒ33 ϒ34 ⎠ ϒ41 ϒ43 ϒ44 ⎞ ⎛ ϒ21 ϒ22 ϒ24 + ϒ13 · det ⎝ ϒ31 ϒ32 ϒ34 ⎠ ϒ41 ϒ42 ϒ44 ⎛ ⎞ ϒ21 ϒ22 ϒ23 − ϒ14 · det ⎝ ϒ31 ϒ32 ϒ33 ⎠

ϒ12

ϒ41 ϒ42 ϒ43 ⎛ ⎞ ϒ22 ϒ23 ϒ24 = 0 ⇒ det(A − λ · I ) = ϒ11 · det ⎝ ϒ32 ϒ33 ϒ34 ⎠ ϒ42 ϒ43 ϒ44 ⎛ ⎞ ϒ21 ϒ22 ϒ24 + ϒ13 · det ⎝ ϒ31 ϒ32 ϒ34 ⎠ ϒ41 ϒ42 ϒ44 ⎛ ⎞ ϒ21 ϒ22 ϒ23 − ϒ14 · det ⎝ ϒ31 ϒ32 ϒ33 ⎠ ϒ41 ϒ42 ϒ43 ⎛ ⎞   ϒ22 ϒ23 ϒ24 ϒ33 ϒ34 det ⎝ ϒ32 ϒ33 ϒ34 ⎠ = ϒ22 · det ϒ43 ϒ44 ϒ42 ϒ43 ϒ44   ϒ32 ϒ34 − ϒ23 · det ϒ42 ϒ44   ϒ32 ϒ33 + ϒ24 · det ϒ42 ϒ43

6.3 A Self-consistent Model Copper Vapor …

663

⎛

⎞ ϒ22 ϒ23 ϒ24 ϒ23 = 0 ⇒ det ⎝ ϒ32 ϒ33 ϒ34 ⎠ ϒ42 ϒ43 ϒ44     ϒ33 ϒ34 ϒ32 ϒ33 + ϒ24 · det = ϒ22 · det ϒ43 ϒ44 ϒ42 ϒ43 ⎞ ⎛   ϒ22 ϒ23 ϒ24 −λ 0 ⎠ ⎝ det ϒ32 ϒ33 ϒ34 = −λ · det 0 −λ ϒ42 ϒ43 ϒ44   1 0 −λ · det + − C12 0 L2 = −λ3 −

1 ·λ C2 · L 2

⎛

⎞   ϒ21 ϒ22 ϒ24 ϒ32 ϒ34 det ⎝ ϒ31 ϒ32 ϒ34 ⎠ = ϒ21 · det ϒ42 ϒ44 ϒ41 ϒ42 ϒ44   ϒ31 ϒ34 − ϒ22 · det ϒ41 ϒ44   ϒ31 ϒ32 + ϒ24 · det ϒ41 ϒ42 ⎛ ⎞ ϒ21 ϒ22 ϒ24 ϒ21 = 0 ⇒ det ⎝ ϒ31 ϒ32 ϒ34 ⎠ ϒ41 ϒ42 ϒ44     ϒ31 ϒ34 ϒ31 ϒ32 + ϒ24 · det = −ϒ22 · det ϒ41 ϒ44 ϒ41 ϒ42 ⎞ ⎛   ϒ21 ϒ22 ϒ24 − C11 · e−λ·τ 0 det ⎝ ϒ31 ϒ32 ϒ34 ⎠ = λ · det 1 · e−λ·τ −λ C2 ϒ41 ϒ42 ϒ44   − C11 · e−λ·τ 0 1 + · det 1 · e−λ·τ − C12 L2 C2 ⎛

⎞ ϒ21 ϒ22 ϒ24 1 1 det ⎝ ϒ31 ϒ32 ϒ34 ⎠ = λ2 · · e−λ·τ + C1 C1 · C2 · L 2 ϒ41 ϒ42 ϒ44 1 1 · e−λ·τ = (λ2 · + ) · e−λ·τ C1 C1 · C2 · L 2

664

6 Gas Laser Systems Stability Analysis …

⎛

⎞   ϒ21 ϒ22 ϒ23 ϒ32 ϒ33 det ⎝ ϒ31 ϒ32 ϒ33 ⎠ = ϒ21 · det ϒ42 ϒ43 ϒ41 ϒ42 ϒ43   ϒ31 ϒ33 − ϒ22 · det ϒ41 ϒ43   ϒ31 ϒ32 + ϒ23 · det ϒ41 ϒ42 ⎛ ⎞ ϒ21 ϒ22 ϒ23 ϒ21 = 0; ϒ23 = 0 ⇒ det ⎝ ϒ31 ϒ32 ϒ33 ⎠ ϒ41 ϒ42 ϒ43   ϒ31 ϒ33 = −ϒ22 · det ϒ41 ϒ43   1 − C1 · e−λ·τ −λ = λ · det 1 · e−λ·τ 0 C2

⎛

⎞ ϒ21 ϒ22 ϒ23 1 det ⎝ ϒ31 ϒ32 ϒ33 ⎠ = · λ2 · e−λ·τ C2 ϒ41 ϒ42 ϒ43 ⎛ ⎞ ϒ22 ϒ23 ϒ24 det(A − λ · I ) = ϒ11 · det ⎝ ϒ32 ϒ33 ϒ34 ⎠ ϒ42 ϒ43 ϒ44 ⎛ ⎞ ϒ21 ϒ22 ϒ24 + ϒ13 · det ⎝ ϒ31 ϒ32 ϒ34 ⎠ ⎛

ϒ41 ϒ42 ϒ44

⎞ ϒ21 ϒ22 ϒ23 − ϒ14 · det ⎝ ϒ31 ϒ32 ϒ33 ⎠ ϒ41 ϒ42 ϒ43   1 3 det(A − λ · I ) = −λ · −λ − ·λ C2 · L 2   1 1 1 2 · λ · + + L1 C1 C1 · C2 · L 2 1 1 · · λ2 · e−λ·τ · e−λ·τ + L 1 C2 1 det(A − λ · I ) = λ4 + · λ2 C2 · L 2   1 1 + λ2 · + C1 · L 1 C1 · C2 · L 2 · L 1

6.3 A Self-consistent Model Copper Vapor …

· e−λ·τ +

665

1 1 · · λ2 · e−λ·τ L 1 C2

1 det(A − λ · I ) = λ4 + · λ2 C2 · L 2    1 1 1 · λ2 + · + L1 C1 C2  1 · e−λ·τ + C1 · C2 · L 2 · L 1 D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ n, m ∈ N0 ; n = 4; m = 2; n > m 1 · λ2 ; n = 4 C2 · L 2   1 1 1 1 · λ2 + Q m (λ, τ ) = · + ;m = 2 L1 C1 C2 C1 · C2 · L 2 · L 1 Pn (λ, τ ) = λ4 +

We can summary our copper vapor laser (CVL) circuity determinants small increments Jacobian results in the next table (Table 6.9) Pn (λ, τ ) =

n=4 

pk (τ ) · λk

k=0

= p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3 + p4 (τ ) · λ4 Table 6.9 Copper vapor laser (CVL) circuity determinants (small increments Jacobian results) Copper vapor laser (CVL) circuity determinants ⎞ ⎛ ϒ22 ϒ23 ϒ24 ⎟ ⎜ ⎟ det ⎜ ⎝ ϒ32 ϒ33 ϒ34 ⎠

Expression −λ3 −

1 C2 ·L 2

·λ

ϒ42 ϒ43 ϒ44

⎛

ϒ21 ϒ22 ϒ24

⎜ det ⎜ ⎝ ϒ31 ϒ41 ⎛ ϒ ⎜ 21 det ⎜ ⎝ ϒ31 ϒ41

ϒ32 ϒ42 ϒ22 ϒ32 ϒ42

⎞

⎟ ϒ34 ⎟ ⎠ ϒ44 ⎞ ϒ23 ⎟ ϒ33 ⎟ ⎠ ϒ43

(λ2 ·

1 C2

1 C1

+

1 −λ·τ C1 ·C2 ·L 2 ) · e

· λ2 · e−λ·τ

666

6 Gas Laser Systems Stability Analysis …

p0 = p0 (τ ); p1 = p1 (τ ); p2 = p2 (τ ); p3 = p3 (τ ) 1 p4 = p4 (τ ); p0 = 0; p1 =0; p2 = ; p3 =0; p4 = 1 C2 · L 2 Q m (λ, τ ) =

m=2 

qk (τ ) · λk

k=0

= q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 q0 = q0 (τ ); q1 =q1 (τ ); q2 = q2 (τ ) 1 ; q1 = 0 C1 · C2 · L 2 · L 1   1 1 1 q2 = · + L1 C1 C2

q0 =

The homogeneous system for I1 , I2 , V1 , V2 leads to a characteristic equation for the eigenvalue λhaving the form D(λ, τ ) + Q(λ, τ ) · τ2) = P(λ, 4 j j e−λ·τ = 0; and P(λ) = j=0 a j · λ ; Q(λ) = j=0 c j · λ . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delayτ . qi , qk are any Copper vapor laser (CVL) circuity’s parameters, other parameters kept as a constant [4, 5]. P(λ) =

4 

aj · λj

j=0

= a 0 + a 1 · λ + a 2 · λ2 + a 3 · λ3 + a 4 · λ4 Q(λ) =

2 

c j · λ j = c0 + c1 · λ + c2 · λ2

j=0

1 ; a3 = 0 C2 · L 2   1 1 1 1 a4 = 1; c0 = ; c1 = 0; c2 = · + C1 · C2 · L 2 · L 1 L1 C1 C2

a0 = 0; a1 = 0; a2 =

Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang J and Cong Y 2005; Kuang Y 1993) can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) If | Q(λ) P(λ)

6.3 A Self-consistent Model Copper Vapor …

667

(c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real numberω: Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and Pn (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + p4 (τ ) · ω4 + [ p1 (τ ) · ω − p3 (τ ) · ω3 ] · i 1 · ω2 + ω4 C2 · L 2 Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω − q2 (τ ) · ω2 Pn (λ = i · ω, τ ) = −

Q m (λ = i · ω, τ ) =

1 1 − · C1 · C2 · L 2 · L 1 L1



1 1 + C1 C2

 · ω2

Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) 1 1 =− · ω2 + ω4 + C2 · L 2 C1 · C2 · L 2 · L 1 1 1 1 − ·( + ) · ω2  = 0 L 1 C1 C2 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ )    1 1 1 1 1 − + · + = C1 · C2 · L 2 · L 1 C2 · L 2 L1 C1 C2 · ω2 + ω4  = 0 1 · ω 2 + ω 4 ]2 C2 · L 2 1 1 = · ω4 − 2 · · ω6 + ω8 2 (C2 · L 2 ) C2 · L 2

|P(i · ω, τ )|2 = [−

1 1 1 1 − ·( + ) · ω 2 ]2 C1 · C2 · L 2 · L 1 L 1 C1 C2 1 1 = −2· (C1 · C2 · L 2 · L 1 )2 (C1 · C2 · L 2 · L 1 ) 1 1 1 1 1 1 2 4 · ·( + ) · ω2 + 2 · ( + ) ·ω L 1 C1 C2 C2 L 1 C1

|Q(i · ω, τ )|2 = [

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2

668

6 Gas Laser Systems Stability Analysis …

1 1 · ω4 − 2 · · ω6 + ω8 2 (C2 · L 2 ) C2 · L 2 1 1 1 · − +2· (C1 · C2 · L 2 · L 1 )2 (C1 · C2 · L 2 · L 1 ) L 1     1 1 1 1 1 2 4 2 ·ω − 2 · · + + ·ω C1 C2 C1 C2 L1

=

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 1 1 =− +2· 2 (C1 · C2 · L 2 · L 1 ) (C1 · C2 · L 2 · L 1 )     1 1 1 1 1 1 2 1 2 ·( + )·ω + − 2· + · L 1 C1 C2 (C2 · L 2 )2 C1 C2 L1 · ω4 − 2 ·

1 · ω6 + ω8 C2 · L 2

We define the following parameters for simplicity: 0 , 2 , 4 , 6 .8 1 (C1 · C2 · L 2 · L 1 )2   1 1 1 1 · 2 = 2 · · + (C1 · C2 · L 2 · L 1 ) L 1 C1 C2 2  1 1 1 1 4 = − 2· + (C2 · L 2 )2 C1 C2 L1 0 = −

6 = −2 · Hence F(ω, τ ) = 0 implies solving the above polynomial. PR (iω, τ ) = −

4 k=0

1 ; 8 = 1 C2 · L 2 2·k · ω2·k = 0 and its roots are given by

1 · ω2 + ω4 C2 · L 2

PI (iω, τ ) = 0; Q R (iω, τ ) =

1 1 1 1 − ·( + ) · ω2 C1 · C2 · L 2 · L 1 L 1 C1 C2

Q I (iω, τ ) = 0 −PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) sin θ (τ ) = |Q(iω, τ )|2 cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

6.3 A Self-consistent Model Copper Vapor …

669

We use different terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j n, m ∈ N0 ; n = 4; m = 2; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

4 

a j · λ j ; Q(λ) =

j=0

2 

cj · λj

j=0

P(λ) = a0 + a1 · λ + a2 · λ2 + a3 · λ3 + a4 · λ4 Q(λ) = c0 + c1 · λ + c2 · λ2 n, m ∈ N0 ; n > m and a j , c j : R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denoted complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ . The coefficientsa j (L 1 , L 2 , C1 , C2 , V0 , τ, . . .) ∈ R and c j (L 1 , L 2 , C1 , C2 , V0 , τ, . . .) ∈ R depend on Copper vapor laser (CVL) circuity’s parameters, L 1 , L 2 , C1 , C2 , V0 , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: L 1 , L 2 , C1 , C2 , V0 , τ, . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0[4, 5]. 1 C1 · C2 · L 2 · L 1 1 a0 + c0 = = 0 C1 · C2 · L 2 · L 1 a0 = 0; c0 =

∀ L 1 , L 2 , C1 , C2 , V0 , τ, . . . ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang, 1993, Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire L 1 , L 2 , C1 , C2 , V0 , τ, . . . domain of interest. P(λ) | is bounded for |λ| → ∞ ; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, (b) | Q(λ) c0 +c1 ·λ+c2 ·λ in the limit: | Q(λ) | = | a0 +a1 ·λ+a 2 3 4 |. P(λ) 2 ·λ +a3 ·λ +a4 ·λ 2 (c) The following expressions exist: F(ω) = |P(i · ω)|2 4· ω)| − |Q(i 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω8 ). (d) Each positive root ω( L 1 , L 2 , C1 , C2 , V0 , τ, . . .) of F(ω) = 0 is continuous and differentiable with respect to L 1 , L 2 , C1 , C2 , V0 , τ, . . . and the condition can only be assessed numerically. 2

670

6 Gas Laser Systems Stability Analysis …

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: L 1 , L 2 , C1 , C2 , V0 , τ, . . ., Reλ may, at the crossing, change its sign from (−) to (+). i.e. from a stable focus E ∗ (I1∗ , I2∗ , V1∗ , V2∗ ) = (0, 0, Vd − V0 · f (τs ), Vd ) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to L 1 , L 2 , C1 , C2 , V0 , τ, . . . and any system parameters. −1 (L 1 ) = (

∂Reλ )λ=i·ω ; L 2 , C1 , C2 , V0 , Vd , τ, . . . = const ∂ L1

−1 (L 2 ) = (

∂Reλ )λ=i·ω ; L 1 , C1 , C2 , V0 , Vd , τ, . . . = const ∂ L2

−1 (C1 ) = (

∂Reλ )λ=i·ω ; L 1 , L 2 , C2 , V0 , Vd , τ, . . . = const ∂C1

−1 (C2 ) = (

∂Reλ )λ=i·ω ; L 1 , L 2 , C1 , V0 , Vd , τ, . . . = const ∂C2

−1 (V0 ) = (

∂Reλ )λ=i·ω ; L 1 , L 2 , C1 , C2 , Vd , τ, . . . = const ∂ V0

−1 (τ ) = (

∂Reλ )λ=i·ω ; L 1 , L 2 , C1 , C2 , V0 , Vd , . . . = const ∂τ

P(λ) = PR (λ)+i · PI (λ); Q(λ) = Q R (λ)+i · Q I (λ), When writing and inserting λ = i · ω into Copper vapor laser (CVL) circuity’s characteristic equation ω must satisfy the following equations: sin(ω · τ ) = g(ω) −PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) = |Q(iω, τ )|2 cos(ω · τ ) = h(ω) PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) =− |Q(iω, τ )|2 where |Q(iω, τ )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ . Now it is important to notice that if τ ∈ / I (assume that / I , ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are

6.3 A Self-consistent Model Copper Vapor …

671

no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ ·ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) + 2 · n ·  ∀ n ∈ N0 Hence we can define the maps: ; n ∈ N0 ; τ ∈ I . Let us introduce the τn : I → R+0 , is given by τn (τ ) = θ(τ )+2·n· ω(τ ) function I → R ; Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ . In the following, the subscripts λ, ω, L 1 , L 2 , C1 , C2 , V0 , Vd , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(L 1 , L 2 , C1 , C2 , V0 , Vd , . . .) and ω(L 1 , L 2 , C1 , C2 , V0 , Vd , . . .), and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Copper vapor laser (CVL) circuity’s characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = L 1 , L 2 , C1 , C2 , V0 , Vd , τ, . . . ∂λ −1 ) ∂x −Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) = Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

(

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω = λ (i·ω,x)·Q(λ,x)−τ ·|P(i·ω,x)| ( i·Pω (i·ω,x)·P(i·ω,x)+i·Q ) Upon separating into real and imagiPx (i·ω,x)·P(i·ω,x)−Q x (i·ω,x)·Q(i·ω,x) nary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω +i · Q I ω ; Px = PRx +i · PI x ; Q x = Q Rx +i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any Copper vapor laser (CVL) circuity’s parameters L 1 , L 2 , C1 , C2 , V0 , Vd , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i·ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] And ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delayx = τ ; PR = − C21·L 2 · ω2 + ω4 ; PI = 0; Q R = 1 − L11 · ( C11 + C12 ) · ω2 ; Q I = 0: C1 ·C2 ·L 2 ·L 1 2

1 1 1 ·( + ) · ω; Q I ω = 0 L 1 C1 C2 1 = −2 · · ω + 4 · ω3 ; PI ω = 0 C2 · L 2

Q Rω = −2 · PRω

PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; Fτ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0

672

6 Gas Laser Systems Stability Analysis …

Fω = . . . . Elements: PRω · PR = [−2 ·

1 C2 ·L 2

· ω + 4 · ω3 ] · [− C21·L 2 · ω2 + ω4 ]

1 1 · ω + 4 · ω3 ] · [− · ω2 + ω4 ] C2 · L 2 C2 · L 2 1 1 =2· · ω3 − 6 · · ω5 + 4 · ω7 (C2 · L 2 )2 C2 · L 2

PRω · PR = [−2 ·

We define the following parameters for simplicity: A3 = 2 · 1 ; A7 = 4 C2 ·L 2 PRω · PR =

3 

1 ; (C2 ·L 2 )2

A5 = −6 ·

A1+2·k · ω1+2·k ; PI ω · PI = 0; Q I ω · Q I = 0

k=1

    1 1 1 ·ω Q Rω · Q R = −2 · · + L1 C1 C2     1 1 1 1 2 ·ω · − · + C1 · C2 · L 2 · L 1 L1 C1 C2   1 1 1 1 · · + ·ω Q Rω · Q R = −2 · L1 C1 C2 C1 · C2 · L 2 · L 1 1 1 1 2 3 +2· 2 ·( + ) ·ω C2 L 1 C1 We define the following parameters for simplicity: B1 = −2 · 1 C1 ·C2 ·L 2 ·L 1

1 L1

· ( C11 +

1 ) C2

·

 1 1 1 2 ·( + ) ; Q Rω · Q R = B2·k−1 · ω2·k−1 2 C2 L 1 C1 k=1 2

B3 = 2 ·

U = . . . . Elements: PR · PI ω = 0; PI · PRω = 0; Q R · Q I ω = 0; Q I · Q Rω = 0 + Fτ = 0; τ ∈ F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · ∂ω ∂τ Fτ ∂ω I ⇒ ∂τ = − Fω ∂Reλ ∂ω Fτ )λ=iω ; = ωτ = − ∂τ ∂τ Fω −2 · [U + τ · |P|2 + i · Fω −1 (τ ) = Re{ } Fτ + i · 2 · [V + ω · |P|2 ] −1 (τ ) = (

∂Reλ )λ=iω } ∂τ V + ∂ω ·U ∂ω ∂τ sign{−1 (τ )} = sign{Fω } · sign{ · τ} +ω+ |P|2 ∂τ

sign{−1 (τ )} = sign{(

6.3 A Self-consistent Model Copper Vapor …

673

We shall presently examine the possibility of stability transitions (bifurcations) Copper vapor laser (CVL) circuity, about the equilibrium point E ∗ (I1∗ , I2∗ , V1∗ , V2∗ ) = (0, 0, Vd − V0 · f (τs ), Vd ). The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameter τ . Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [4, 5]. −1 (τ ) = (

∂Reλ )λ=i·ω ; L 1 , L 2 , C1 , C2 , V0 , Vd , . . . = const ∂τ

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 Fω = 2 · [(

3 

A1+2·k · ω1+2·k ) − (

k=1

2 

B2·k−1 · ω2·k−1 )]

k=1

∂ω Fτ Fτ = ωτ = − | Fτ =0,Fω =0 = − =0 ∂τ Fω Fω Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0 sign{−1 (τ )} = sign{Fω } · sign{ +ω+ −1

∂ω · τ} ∂τ

sign{ (τ )} = sign{2 · [(

3 

V + ∂ω ·U ∂τ |P|2

A1+2·k · ω1+2·k )

k=1 2  B2·k−1 · ω2·k−1 )]} · sign{ω} −( k=1

sign{−1 (τ )} = sign{[(

3 

A1+2·k · ω1+2·k )

k=1

−(

2 

B2·k−1 · ω2·k−1 )]} · sign{ω}

k=1

  We get the expression for Fω = 2·[( 3k=1 A1+2·k ·ω1+2·k )−( 2k=1 B2·k−1 ·ω2·k−1 )] Copper vapor laser (CVL) circuity parameter values. We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for

674

6 Gas Laser Systems Stability Analysis …

specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Copper vapor laser (CVL) circuity. ∂Reλ )λ=iω ∂τ −2 · [U + τ · |P|2 ] + i · Fω } = Re{ Fτ + 2 · i · [V + ω · |P|2 ]

−1 (τ ) = (

∂Reλ )λ=iω ∂τ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = Fτ2 + 4 · (V + ω · P 2 )2

−1 (τ ) = (

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all Copper vapor laser (CVL) circuity’s parameters are known and the stability switch due to various time delay values τ is described in the following expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ ) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + } |P(ω(τ ))|2 Remark we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the paramester F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = . . . . and cos θ (τ ) = . . . .; sin(ω · τ ) = −PR ·Q I +PI ·Q R R +PI ·Q I ) ; cos(ω · τ ) = − (PR ·Q|Q| |Q|2 = Q 2R + Q 2I . Finally we plot the 2 |Q|2 ) stability switch diagram g(τ ) = −1 (τ ) = ( ∂Reλ ∂τ λ=iω g(τ ) = −1 (τ )   ∂Reλ = ∂τ λ=iω =

2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2

sign[g(τ )] = sign[−1 (τ )]

6.3 A Self-consistent Model Copper Vapor …

675

∂Reλ )λ=iω ] ∂τ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} ] = sign[ Fτ2 + 4 · (V + ω · P 2 )2

= sign[(

Fτ2 + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ )] = sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )] sign[−1 (τ )] = sign{[Fω ] · [(V + ω · P 2 ) Fτ · (U + τ · P 2 )]} − Fω Fτ ∂ω ∂ F/∂ω ωτ = − ; ωτ = ( )−1 = − Fω ∂τ ∂ F/∂τ V + ωτ · U P2 2 + ω + ωτ · τ ]} ; sign[P ] > 0

sign[−1 (τ )] = sign{[Fω ] · [P 2 ] · [

sign[−1 (τ )] = sign{[Fω ] · [

V + ωτ · U + ω + ωτ · τ ]} P2

sign[−1 (τ )] = sign[Fω ] · sign[

V + ωτ · U + ω + ωτ · τ ] P2

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [(

3 

A1+2·k · ω1+2·k ) − (

k=1

2 

B2·k−1 · ω2·k−1 )]

k=1

We check the sign of −1 (τ ) according to the following rule: (Table 6.10) If, sign[−1 (τ )] > 0, then the crossing proceeds from (-) to (+) respectively (stable to unstable). If, sign[−1 (τ )] < 0, then the crossing proceeds from (+) to (-) respectively (unstable to stable). Table 6.10 Copper vapor laser (CVL) circuity sign of sign[−1 (τ )] sign[Fω ]

τ ·U sign[ V +ω + ω + ωτ · τ ] P2

sign[−1 (τ )]

±

±

+

±

∓

–

676

6 Gas Laser Systems Stability Analysis …

6.4 A Self-consistent Model Copper Vapor Laser (CVL) Electron Density Upper and Lower Laser Levels Stability Analysis Under Parameters Variation A copper vapor laser makes use of copper vapor as a lasing medium. It emits yellow and green laser light at 578.2 nm and 510.6 nm of the electromagnetic spectrum, respectively. Laser is made using pure metal vapor produced from elemental copper is hard to make because of the high temperature required to form vapor. This is the reason why compounds like chloride, copper iodide, and copper bromide are used and they form vapors at low temperatures. The laser head of copper vapor laser consists of a refractory ceramic tube that contains a low-pressure buffer gas, neon, and copper pellets. A pulsed electrical discharge between electrodes at the tube ends is happened. The temperature of the laser head increased to about 1450 C degree, thereby producing copper vapor at low pressure. The excited electrons collide with vaporized copper atoms and excite these atoms to the upper 2 P3/2 and 2 P1/2 laser level. The electrons in the 2 P3/2 level decay to the lower 2 D5/2 laser level to produce green light, and the 2 P1/2 decay to 2 D3/2 level to produce yellow laser light. The features of the laser are good power stability, good beam quality, and a long operational lifetime. The copper laser has a wide range of applications due to his very good beam characteristics. Some of the applications are high speed photography, dermatological applications, and pump for dye lasers. The copper vapor laser (CVL) is very applicable to engineering lasers. Diverse applications exist because of an inherent beam characteristic and a natural choice for high precision micromachining of features down to 1 µm. Laser is an acronym for Light Amplification by the Stimulated Emission of Radiation. The emission of radiation is stimulated when atoms are excited to an energy state above their ground state. The copper vapor laser (CVL) uses vaporized copper atoms as the lasing medium. The self- consistent model for high repetition rate copper vapor laser (CVL) simulates a discharge pulse, laser pulse, an inter-pulse afterglow in a high repetition rate. We need to choose initial conditions and number of cycles for computing until consistent pulses are obtained [9]. The model includes various species as follow (Table 6.1). The helium metastable state is the 23 S level. Lamped radiative state (H e∗ , Cu ∗∗ ) is induced for both helium and copper. These states act as a buffer for recombining ions. Result atom in an excited state is happened when an electron and ion recombine. This state is H e∗ or Cu ∗∗ . Subsequent processes determine what distribution of excited states results from electron-ion recombination. The subsequent processes which involving these states are spontaneous emission, electron excitation or relaxation, etc. If electron-ion recombination preferentially populates the metastable 2 D lower laser level, then pumping requirements to reach threshold are increased for the next laser pulse. System model thermodynamic quantities are Tg , PCu , and Pg . During the operation of the laser, the temperature of the discharge tube, which determines the copper vapor pressure, can be measured by an optical pyrometer. The optical pyrometer is a non-contact type temperature measuring device. It works on

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

677

Table 6.11 Copper vapor laser (CVL) model species Copper vapor laser (CVL) model species

Specie meaning

ne

Electrons

Cu

2S

Cu m

2D

metastable copper (lower laser level)

Cu ∗

2P

copper (upper laser level)

Cu ∗∗

Lumped radiative state in copper

Cu +

Atomic copper ion

He

Ground state helium

Hem

Metastable state helium

H e∗

Lumped radiative state in helium

H e+

Atomic helium ion

H e2+

Molecular helium ion

Te

Electron temperature

Tg

Gas temperature

Pg

Buffer gas pressure

PCu

Copper vapor pressure

Ii , Vi

Discharge current and voltage for circuit elements i

ground state copper

R

Copper vapor laser (CVL) mirror reflectivity



Laser intensity

the principle of matching the brightness of an object to the brightness of the filament which is placed inside the pyrometer. Mainly the optical pyrometer is used for measuring the temperature of the furnaces, molten metals, and other overhead material or liquids. Furnace is a device used for high-temperature heating. The heat energy to fuel a furnace is supplied directly by fuel combustion, by electricity such as the electric arc furnace, or through induction heating in induction furnaces. The pyrometer is cylindrical inside which the lens is placed on one end and the eyepiece on the other end. An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. The objective lens or mirror collects light and brings it to focus creating an image. The eyepiece is placed near the focal point of the objective to magnify this image. The amount of magnification depends on the focal length of the eyepiece. The lamp is kept between the eyepiece and the lens. The filter is placed in front of the eyepiece. By using filter we get monochromatic light. The lamp has the filament which is connected to the battery, ammeter and rheostat. A rheostat is a variable resistor which is used to control current. It is able to vary the resistance in a circuit without interruption. The rheostats have to carry a significant current therefore they mostly constructed as a wire wound resistors. Resistive wire is wound around an insulating ceramic core and the wiper slides over the windings. There are some types of rheostats, rotary type, multi-gang types, slide rheostats, and linear rheostats [9]. The optical pyrometer

678

6 Gas Laser Systems Stability Analysis …

Objective lens Eyepiece

Filament

Heated source

Filter

Absorption screen

Rheostat

Meter

Battery

Fig. 6.8 Optical pyrometer structure

(Fig. 6.8) consists the lens which are focused the radiated energy from the heated object and target it on the electric filament lamp. The intensity of the filament depends on the current passes through it. The adjustable current is passed through the lamp. The magnitude of the current is adjusted until the brightness of the filament is similar to the brightness of the object. When the brightness of the filament and the brightness of the object are same, then the outline of the filament is completely disappeared. The optical pyrometer has advantages of high accuracy, temperature measurement without contacting the heated body. Typically pyrometer is used for measuring the temperature above 700 °C. The pyrometer accuracy depends on the adjustment of the filament current (Table 6.11). In our system the buffer gas pressure is measured external to the laser tube. The temperature of the discharge tube remains constant, as does the external buffer gas pressure. The gas temperature within the tube is changed by many hundreds of degrees during the discharge pulse. An increase is gas temperature increases the local gas pressures Pg and PCu at the same rate, but the relaxation of Pg and PCu proceeds at different rates. The equilibrium vapor pressure for PCu is the pressure determined by the wall temperature. The changes in the gas temperature, the time constant for relaxation of PCu to its equilibrium value is vd , where the radial diffusion length of the discharge tube is  and vd is the speed at which copper atoms diffuse to the wall. The equilibrium pressure for the buffer gas is the pressure measured external to the tube. Exclusive of changes in the gas temperature, the time constant for relaxation of Pg to its equilibrium value is 2·vl s ; where the length of the discharge tube is L and vs is the sound speed. The gas temperature relaxes to the wall temperature with a time constant proportional to the radius of the discharge tube. We can express the time rate of change of Pg and PCu :

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

679

d Pg dTg Pg (t − τ ) 2 · v2 − [Pg (t − τ ) − Pex ] · = · dt dt Tg l dTg Pcu d Pcu vd = · − [Pcu − Pvp ] · dt dt Tg R Pcu = Pcu (t); Pg = Pg (t); Tg = Tg (t) Pg (t−τ ) dTg , dt · PTcug . Tg elements: [Pg (t − τ ) − Pex ] · 2·vl 2 , [Pcu − Pvp ] · vRd . gas pressure, Tg - Gas temperature, Pcu - Copper vapor

Temperature increasing elements:

dTg dt

·

Relaxation pressure, Pex Pg - Buffer External buffer gas pressure, Pvp - Copper vapor pressure based on the wall temperature, R- Mirror reflectivity [9]. Time delay τ is related to the operational interferences of system discharge tube which affect the thermodynamic quantity Pg . The time delay τ is related to buffer gas pressure. Remark The time delay

parameter τ d Pg does not affect the derivative of the buffer gas pressure in time dt . We assume that the gas density stays in equilibrium with the local temperature and pressure, and we express the changes in gas density due to changes in temperature and pressure by the following differential equations: n g dTg ng d Pg ∂n g =− · + · ∂t Tg dt Pg (t − τ ) dt n g · 2 · vs = −[Pg (t − τ ) − Pex ] · l · Pg (t − τ ) n cu dTg n cu d Pcu ∂n cu =− + · · ∂t Tg dt Pcu dt n cu · vd = −[Pcu − Pvp ] ·  · Pcu where n g and n cu are the densities of the buffer gas and copper atoms, respectively. ∂n dn We can consider ∂tg ↔ dtg ; ∂n∂tcu ↔ dndtcu , and then we can claim that the above equations: n g · 2 · vs dn g = −[Pg (t − τ ) − Pex ] · dt l · Pg (t − τ ) dn cu n cu · vd = −[Pcu − Pvp ] · dt  · Pcu We assume in our system that Pex (external buffer gas pressure), and Pcu (copper Pex Pcu → 0; ddt → 0. vapor pressure) are constant in time, then ddt dTg Pcu dTg vd · Tg d Pcu vd →0⇒ · = 0; = [Pcu − Pvp ] · − [Pcu − Pvp ] · dt dt Tg R dt R · Pcu

680

6 Gas Laser Systems Stability Analysis …

Submitting

dTg dt

= [Pcu − Pvp ] ·

vd ·Tg R·Pcu

in

d Pg dt

= . . . equation gives:

d Pg vd · Pg (t − τ ) = [Pcu − Pvp ] · dt R · Pcu 2 · v2 − [Pg (t − τ ) − Pex ] · l We can summary our system delay differential equations: d Pg vd = [Pcu − Pvp ] · · Pg (t − τ ) dt R · Pcu 2 · v2 − [Pg (t − τ ) − Pex ] · l n g · 2 · vs dn g = −[Pg (t − τ ) − Pex ] · dt l · Pg (t − τ ) dn cu n cu · vd = −[Pcu − Pvp ] · dt  · Pcu Our system variables are Pg , n g , n cu and all other parameters are constant. dP dn At fixed point dtg = 0; dtg = 0; dndtcu = 0; lim Pg (t − τ ) = Pg (t); t τ t→∞

d Pg vd 2 · v2 = 0 ⇒ [Pcu − Pvp ] · =0 · Pg∗ − [Pg∗ − Pex ] · dt R · Pcu l n ∗g · 2 · vs dn g = 0 ⇒ −[Pg∗ − Pex ] · =0 dt l · Pg∗ dn cu n ∗ · vd = 0 ⇒ −[Pcu − Pvp ] · cu =0 dt  · Pcu Pcu = Pvp ⇒ n ∗cu = 0 vd 2 · v2 =0 · Pg∗ − [Pg∗ − Pex ] · R · Pcu l vd 2 · v2 2 · v2 ⇒ [Pcu − Pvp ] · − Pex · · Pg∗ = Pg∗ · R · Pcu l l

[Pcu − Pvp ] ·

2 · v2 2 · v2 vd =( − [Pcu − Pvp ] · ) · Pg∗ l l R · Pcu Pex · 2·vl 2 ⇒ Pg∗ = 2·v2 vd − [Pcu − Pvp ] · R·P l cu

Pex ·

−[Pg∗ − Pex ] ·

n ∗g ·2·vs l·Pg∗

= 0 then we have two cases:

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

Pg∗ = Pex ⇒ (a)

Pex · 2·v2 l

2·v2 l

− [Pcu − Pvp ] ·

vd R·Pcu

681

= Pex

2 · v2 vd 2 · v2 ⇒ Pcu = Pvp = − [Pcu − Pvp ] · l l R · Pcu (b) Pg∗ = Pex ; n ∗g = 0 We can summary our system fixed points: (0) E (0) = (Pg(0) , n (0) g , n cu )  Pex · 2·vl 2 = 2·v2 − [Pcu − Pvp ] · l

 vd R·Pcu

, 0, 0

(1) ∗ E (1) = (Pg(1) , n (1) g , n cu ) = (Pex , n g , 0) ∀ Pcu = Pvp

Stability analysis: The standard local stability analysis about any one of the equilibrium point of the copper vapor laser (CVL) rate equations consists in adding to coordinate [Pg , n g , n cu ] arbitrarily small increments of exponential form [ pg , n g , n cu ] · eλ·t and retaining the first order terms in Pg , n g , n cu . The system of three homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into copper vapor laser (CVL) rate equations. The copper vapor laser (CVL) rate equations fixed values with arbitrarily small increments of exponential form [ pg , n g , n cu ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2(third fixed point), etc. [4, 5]. Pg (t) = Pg( j) + pg · eλ·t ; n g (t) = n (gj) + n g · eλ·t n cu (t) = n (cuj) + n cu · eλ·t ; Pg (t − τ ) = Pg( j) + pg · eλ·(t−τ ) d Pg (t) = pg · λ · eλ·t dt dn g (t) dn cu (t) = n g · λ · eλ·t ; = n cu · λ · eλ·t dt dt We choose these expressions for ourselves Pg (t), n g (t), V1 (t), n cu (t) as a small displacement [ pg , n g , n cu ] from the copper vapor laser (CVL) rate equations fixed points in time t = 0. Pg (t = 0) = Pg( j) + pg ; n g (t = 0) = n (gj) + n g n cu (t = 0) = n (cuj) + n cu d Pg vd · Pg (t − τ ) = [Pcu − Pvp ] · dt R · Pcu

682

6 Gas Laser Systems Stability Analysis …

− [Pg (t − τ ) − Pex ] ·

2 · v2 l

vd · [Pg( j) + pg · eλ·(t−τ ) ] R · Pcu 2 · v2 − [Pg( j) + pg · eλ·(t−τ ) − Pex ] · l

pg · λ · eλ·t = [Pcu − Pvp ] ·

vd · Pg( j) R · Pcu vd + [Pcu − Pvp ] · · pg · eλ·(t−τ ) R · Pcu 2 · v2 2 · v2 − [Pg( j) − Pex ] · − pg · eλ·(t−τ ) · l l

pg · λ · eλ·t = [Pcu − Pvp ] ·

vd · Pg( j) − [Pg( j) − Pex ] R · Pcu 2 · v2 vd · + [Pcu − Pvp ] · · pg · eλ·(t−τ ) l R · Pcu 2 · v2 − pg · eλ·(t−τ ) · l

pg · λ · eλ·t = [Pcu − Pvp ] ·

At fixed points: [Pcu − Pvp ] · ([Pcu − Pvp ] ·

vd R·Pcu

( j)

( j)

· Pg − [Pg − Pex ] ·

2·v2 l

=0

vd 2 · v2 − ) · pg · e−λ·τ − pg · λ = 0 R · Pcu l

dn g n g · 2 · vs = −[Pg (t − τ ) − Pex ] · dt l · Pg (t − τ ) n g · λ · eλ·t = −[Pg( j) + pg · eλ·(t−τ ) − Pex ] ·

( j)

[n g + n g · eλ·t ] · 2 · vs ( j)

l · [Pg + pg · eλ·(t−τ ) ]

n g · λ · eλ·t = −[Pg( j) + pg · eλ·(t−τ ) − Pex ] ( j)

·

[n g + n g · eλ·t ] · 2 · vs ( j)

l · [Pg + pg · eλ·(t−τ ) ] ( j)

First we analyze the expression:

[n g +n g ·eλ·t ]·2·vs ( j) l·[Pg + pg ·eλ·(t−τ ) ]

( j)

[n g + n g · eλ·t ] · 2 · vs ( j)

( j)

·

[Pg − pg · eλ·(t−τ ) ] ( j)

l · [Pg + pg · eλ·(t−τ ) ] [Pg − pg · eλ·(t−τ ) ]

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

683

2 · vs · [n (gj) · Pg( j) − n (gj) · pg · eλ·(t−τ ) + n g · eλ·t · Pg( j) − n g · pg · eλ·(t−τ ) · eλ·t ]

=

( j)

l · [(Pg )2 − pg2 · e2·λ·(t−τ ) ]

Assumptions n g · pg ≈ 0; pg2 ≈ 0 ( j)

( j)

[n g + n g · eλ·t ] · 2 · vs

·

( j)

[Pg − pg · eλ·(t−τ ) ] ( j)

l · [Pg + pg · eλ·(t−τ ) ] [Pg − pg · eλ·(t−τ ) ] ( j)

=

( j)

( j)

( j)

2 · vs · [n g · Pg − n g · pg · eλ·(t−τ ) + n g · eλ·t · Pg ] ( j)

l · (Pg )2

( j)

[n g + n g · eλ·t ] · 2 · vs ( j)

( j)

·

[Pg − pg · eλ·(t−τ ) ] ( j)

l · [Pg + pg · eλ·(t−τ ) ] [Pg − pg · eλ·(t−τ ) ] ( j)

=

( j)

( j)

( j)

n g · pg · eλ·(t−τ ) n g · eλ·t · Pg 2 · vs n g · Pg ·[ − + ] ( j) ( j) ( j) l (Pg )2 (Pg )2 (Pg )2 ( j)

[n g + n g · eλ·t ] · 2 · vs ( j)

( j)

·

[Pg − pg · eλ·(t−τ ) ] ( j)

l · [Pg + pg · eλ·(t−τ ) ] [Pg − pg · eλ·(t−τ ) ] ( j)

( j)

=

n g · pg · eλ·(t−τ ) n g · eλ·t ng 2 · vs · [ ( j) − + ] ( j) ( j) l Pg (Pg )2 Pg

( j)

[n g + n g · eλ·t ] · 2 · vs ( j)

l · [Pg + pg · eλ·(t−τ ) ] ( j)

( j)

=

n g · pg · eλ·(t−τ ) n g · eλ·t ng 2 · vs · [ ( j) − + ] ( j) ( j) l Pg (Pg )2 Pg

n g · λ · eλ·t = −[Pg( j) + pg · eλ·(t−τ ) − Pex ] ( j)

( j)

n g · pg · eλ·(t−τ ) n g · eλ·t 2 · vs ng · [ ( j) − + ] ( j) ( j) l Pg (Pg )2 Pg

·

n g · λ · eλ·t = −[(Pg( j) − Pex ) + pg · eλ·(t−τ ) ] ( j)

( j)

n g · pg · eλ·(t−τ ) n g · eλ·t ng 2 · vs · [ ( j) − + ] ( j) ( j) l Pg (Pg )2 Pg

·

n g · λ · eλ·t = −(Pg( j) − Pex ) · ( j)

·

( j)

2 · vs n g · ( j) + (Pg( j) − Pex ) l Pg

2 · vs n g · pg · eλ·(t−τ ) · − (Pg( j) − Pex ) ( j) l (Pg )2

684

6 Gas Laser Systems Stability Analysis … ( j)

·

2 · vs n g · eλ·t 2 · vs n g · · ( j) − pg · eλ·(t−τ ) · ( j) l l Pg Pg ( j)

+ pg · eλ·(t−τ ) ·

2 · vs n g · pg · eλ·(t−τ ) · ( j) l (Pg )2

− pg · eλ·(t−τ ) ·

2 · vs n g · eλ·t · ( j) l Pg ( j)

2 · vs n g · ( j) + (Pg( j) − Pex ) l Pg

n g · λ · eλ·t = −(Pg( j) − Pex ) · ( j)

·

2 · vs n g · pg · eλ·(t−τ ) · − (Pg( j) − Pex ) ( j) l (Pg )2

·

2 · vs n g · eλ·t 2 · vs n g − pg · eλ·(t−τ ) · · · ( j) ( j) l l Pg Pg

( j)

( j)

+ pg · pg · eλ·(t−τ ) ·

2 · vs n g · eλ·(t−τ ) · ( j) l (Pg )2

− pg · n g · eλ·(t−τ ) ·

2 · vs eλ·t · ( j) l Pg

Assumption pg · pg ≈ 0; pg · n g ≈ 0 ( j)

2 · vs n g · ( j) + (Pg( j) − Pex ) l Pg

n g · λ · eλ·t = −(Pg( j) − Pex ) · ( j)

·

ng 2 · vs 2 · vs · ( j) · pg · eλ·(t−τ ) − l l (Pg )2 ( j)

· · ng · λ · e

λ·t

= ·

ng

( j) Pg

1 ( j)

Pg

· pg · eλ·(t−τ ) − (Pg( j) − Pex ) · · n g · eλ·t

−(Pg( j) 1 ( j) Pg

2 · vs l

( j)

2 · vs n g · ( j) + [(Pg( j) − Pex ) − Pex ) · l Pg ( j)

− 1] ·

ng

( j) Pg

− (Pg( j) − Pex ) ·

·

2 · vs · pg · eλ·(t−τ ) l

1 2 · vs · ( j) · n g · eλ·t l Pg

6.4 A Self-consistent Model Copper Vapor Laser (CVL) … ( j)

At fixed point: −(Pg − Pex ) ·

2·vs l

( j)

ng ( j) Pg

·

n g · λ · eλ·t = [(Pg( j) − Pex ) ·

=0 ( j)

1 ( j) Pg

− 1] ·

· eλ·(t−τ ) − (Pg( j) − Pex ) · n g · λ = [(Pg( j) − Pex ) ·

( j) Pg

( j) Pg

− (Pg( j) − Pex ) ·

− 1] ·

( j) Pg

·

2 · vs · pg l

1 2 · vs · ( j) · n g · eλ·t l Pg

− 1] ·

ng

( j) Pg

·

2 · vs · pg l

1 2 · vs · ( j) · n g l Pg

( j)

1

ng

( j)

1

· e−λ·τ − (Pg( j) − Pex ) · [(Pg( j) − Pex ) ·

685

ng

( j) Pg

·

2 · vs · pg · e−λ·τ l

1 2 · vs · ( j) · n g − n g · λ = 0 l Pg

dn cu n cu · vd = −[Pcu − Pvp ] · ; n cu · λ · eλ·t dt  · Pcu ( j)

= −[Pcu − Pvp ] ·

(n cu + n cu · eλ·t ) · vd  · Pcu

n cu · λ · eλ·t = −[Pcu − Pvp ] · n (cuj) · − [Pcu − Pvp ] ·

vd  · Pcu

vd · n cu · eλ·t  · Pcu

( j)

vd vd At fixed point: −[Pcu − Pvp ]·n cu · ·P = 0; −[Pcu − Pvp ]· ·P ·n cu −n cu ·λ = 0 cu cu We can summary our copper vapor laser (CVL) rate equations three arbitrarily small increments equations:

([Pcu − Pvp ] ·

vd 2 · v2 ) · pg · e−λ·τ − pg · λ = 0 − R · Pcu l

[(Pg( j) − Pex ) ·

1 ( j) Pg

( j)

− 1] ·

ng

( j) Pg

·

2 · vs · pg · e−λ·τ l

1 2 · vs · ( j) · n g − n g · λ = 0 l Pg vd −[Pcu − Pvp ] · · n cu − n cu · λ = 0  · Pcu

− (Pg( j) − Pex ) ·

686

6 Gas Laser Systems Stability Analysis …

The small increments Jacobian of our copper vapor laser (CVL) rate equations is as follow: ⎛

vd − 2·vl 2 ) · e−λ·τ − λ 0 ([Pcu − Pvp ] · R·P cu ( j) ⎜ ⎜ [(Pg( j) − Pex ) · 1( j) − 1] · n g( j) · 2·vs · e−λ·τ −(Pg( j) − Pex ) · l ⎝ Pg Pg

0

2·vs l

·

1 ( j) Pg

−λ

⎟ ⎟ ⎠

0 −[Pcu − Pvp ] ·

0

⎞

⎛

⎞

0 vd ·Pcu

−λ

pg ⎟ ⎜ · ⎝ ng ⎠ = 0 n cu

A−λ· I = ⎛ vd ([Pcu − Pvp ] · ⎜ R · Pcu 0 ⎜ 2 · v2 −λ·τ ⎜− )·e −λ ⎜ l ⎜ ⎜ n (gj) 2 · vs ( j) ⎜ · · e−λ·τ −(Pg − Pex ) · · ⎜ ( j) l ⎜ Pg ⎜ ⎜ ⎝ 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −λ 0 ⎟ ⎟ vd ⎟ ⎟ −[Pcu − Pvp ] ·  · Pcu ⎠ −λ 0

2·vs l

det(A − λ · I ) = ⎛ vd ([Pcu − Pvp ] · R · Pcu ⎜ 0 ⎜ 2 · v2 ⎜− ) · e−λ·τ − λ ⎜ l ⎜ ( j) ⎜ det ⎜ · n g · 2 · vs · e−λ·τ −(P ( j) − P ) · ex g ⎜ l ⎜ Pg( j) ⎜ ⎜ ⎝ 0 0

·

1 ( j) Pg

⎞ 0 2·vs l

·

1 ( j) Pg

−λ

0 −[Pcu − Pvp ] · −λ

vd  · Pcu

det(A − λ · I ) = 0 det(A − λ · I ) vd 2 · v2 = {([Pcu − Pvp ] · ) · e−λ·τ − λ} − R · Pcu l  ( j) −(Pg − Pex ) · 2·vl s · 1( j) − λ 0 Pg · det 0 −[Pcu − Pvp ] ·

 vd ·Pcu

−λ

vd 2 · v2 ) · e−λ·τ − λ} − R · Pcu l 1 2 · vs · {[−(Pg( j) − Pex ) · · ( j) − λ] l Pg

det(A − λ · I ) = {([Pcu − Pvp ] ·

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

· [−[Pcu − Pvp ] ·

687

vd − λ]}  · Pcu

We define for simplicity global parameters: ϒ1 = [Pcu − Pvp ] ·

vd R·Pcu

−

2·v2 l

2 · vs 1 · ( j) l Pg vd ϒ3 = −[Pcu − Pvp ] ·  · Pcu ϒ2 = −(Pg( j) − Pex ) ·

det(A − λ · I ) = (ϒ1 · e−λ·τ − λ) · ([ϒ2 − λ] · [ϒ3 − λ]) = (ϒ1 · e−λ·τ − λ) · (ϒ2 · ϒ3 − [ϒ2 + ϒ3 ] · λ + λ2 ) det(A − λ · I ) = (ϒ1 · e−λ·τ − λ) · (ϒ2 · ϒ3 − [ϒ2 + ϒ3 ] · λ + λ2 ) = ϒ2 · ϒ3 · ϒ1 · e−λ·τ − [ϒ2 + ϒ3 ] · ϒ1 · e−λ·τ · λ + λ2 · ϒ1 · e−λ·τ − ϒ2 · ϒ3 · λ + [ϒ2 + ϒ3 ] · λ2 − λ3 det(A − λ · I ) = −ϒ2 · ϒ3 · λ + [ϒ2 + ϒ3 ] · λ2 − λ3 + (ϒ2 · ϒ3 · ϒ1 − [ϒ2 + ϒ3 ] · ϒ1 · λ + λ2 · ϒ1 ) · e−λ·τ D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ n, m ∈ N0 ; n = 3; m = 2; n > m Pn (λ, τ ) = −ϒ2 · ϒ3 · λ + [ϒ2 + ϒ3 ] · λ2 − λ3 n = 3; Q m (λ, τ ) = ϒ2 · ϒ3 · ϒ1 − [ϒ2 + ϒ3 ] · ϒ1 · λ + λ2 · ϒ1 ; m = 2 Pn (λ, τ ) =

n=3 

pk (τ ) · λk

k=0

= p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3 p0 = p0 (τ ); p1 = p1 (τ ); p2 = p2 (τ ) p3 = p3 (τ ); p0 = 0; p1 = − ϒ2 · ϒ3 p2 =ϒ2 + ϒ3 ; p3 = − 1 Q m (λ, τ ) =

m=2 

qk (τ ) · λk

k=0

= q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2

688

6 Gas Laser Systems Stability Analysis …

q0 = q0 (τ ); q1 =q1 (τ ); q2 = q2 (τ ) Q m (λ, τ ) =

m=2 

qk (τ ) · λk

k=0

= q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 q0 = q0 (τ ); q1 =q1 (τ ); q2 = q2 (τ ) q0 = ϒ2 · ϒ3 · ϒ1 ; q1 = − [ϒ2 + ϒ3 ] · ϒ1 ; q2 = ϒ1 The homogeneous system for Pg , n g , V1 , n cu leads to a characteristic equation for the eigenvalue λhaving the form D(λ, τ ) + Q(λ, τ ) · τ2) = P(λ, 3 j j a · λ ; Q(λ) = c · λ . The coefficients e−λ·τ = 0; and P(λ) = j=0 j j=0 j {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delayτ . qi , qk are any copper vapor laser (CVL) rate equations parameters, other parameters kept as a constant [4, 5]. P(λ) =

3 

a j · λ j = a 0 + a 1 · λ + a 2 · λ2 + a 3 · λ3

j=0

Q(λ) =

2 

c j · λ j = c0 + c1 · λ + c2 · λ2

j=0

a0 = 0; a1 = − ϒ2 · ϒ3 ; a2 =ϒ2 + ϒ3 a3 = − 1; c0 = ϒ2 · ϒ3 · ϒ1 ; c1 = − [ϒ2 + ϒ3 ] · ϒ1 ; c2 = ϒ1 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang J and Cong Y 2005; Kuang Y 1993) can also be verified in the present case: (a) If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. (b) If | Q(λ) P(λ) (c) F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. (d) Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real numberω:Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

Pn (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + [ p1 (τ ) · ω − p3 (τ ) · ω3 ] · i Pn (λ = i · ω, τ ) = −[ϒ2 + ϒ3 ] · ω2 + [ω3 − ϒ2 · ϒ3 · ω] · i Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω − q2 (τ ) · ω2 Q m (λ = i · ω, τ ) = ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 − [ϒ2 + ϒ3 ] · ϒ1 · ω · i Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = −[ϒ2 + ϒ3 ] · ω2 + [ω3 − ϒ2 · ϒ3 · ω] · i + ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 − [ϒ2 + ϒ3 ] · ϒ1 · i · ω = 0 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = −[ϒ2 + ϒ3 ] · ω2 + ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 − [ϒ2 + ϒ3 ] · ϒ1 · i · ω + [ω3 − ϒ2 · ϒ3 · ω] · i = 0 |P(i · ω, τ )|2 = [ϒ2 + ϒ3 ]2 · ω4 + [ω3 − ϒ2 · ϒ3 · ω]2 = [ϒ2 + ϒ3 ]2 · ω4 + ω6 − 2 · ϒ2 · ϒ3 · ω4 + (ϒ2 · ϒ3 )2 · ω2 |P(i · ω, τ )|2 = [ϒ2 + ϒ3 ]2 · ω4 + [ω3 − ϒ2 · ϒ3 · ω]2 = ω6 + ([ϒ2 + ϒ3 ]2 − 2 · ϒ2 · ϒ3 ) · ω4 + (ϒ2 · ϒ3 )2 · ω2 |Q(i · ω, τ )|2 = [ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ]2 + [ϒ2 + ϒ3 ]2 · ϒ12 · ω2 |Q(i · ω, τ )|2 = [ϒ2 · ϒ3 · ϒ1 ]2 + ([ϒ2 + ϒ3 ]2 · ϒ12 − 2 · ϒ2 · ϒ3 · ϒ12 ) · ω2 + ϒ12 · ω4 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ω6 + ([ϒ2 + ϒ3 ]2 − 2 · ϒ2 · ϒ3 ) · ω4 + (ϒ2 · ϒ3 )2 · ω2 − [ϒ2 · ϒ3 · ϒ1 ]2 − ([ϒ2 + ϒ3 ]2 · ϒ12 − 2 · ϒ2 · ϒ3 · ϒ12 ) · ω2 − ϒ12 · ω4 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2

689

690

6 Gas Laser Systems Stability Analysis …

= −[ϒ2 · ϒ3 · ϒ1 ]2 + {(ϒ2 · ϒ3 )2 − ([ϒ2 + ϒ3 ]2 · ϒ12 − 2 · ϒ2 · ϒ3 · ϒ12 )} · ω2 + ([ϒ2 + ϒ3 ]2 − 2 · ϒ2 · ϒ3 − ϒ12 ) · ω4 + ω6 We define the following parameters for simplicity: 0 , 2 , 4 , 6 0 = −[ϒ2 · ϒ3 · ϒ1 ]2 2 = (ϒ2 · ϒ3 )2 − ([ϒ2 + ϒ3 ]2 · ϒ12 − 2 · ϒ2 · ϒ3 · ϒ12 ) 4 = [ϒ2 + ϒ3 ]2 − 2 · ϒ2 · ϒ3 − ϒ12 ; 6 = 1 Hence F(ω, τ ) = 0 implies solving the above polynomial.

3 k=0

2·k · ω2·k = 0 and its roots are given by

PR (iω, τ ) = −[ϒ2 + ϒ3 ] · ω2 PI (iω, τ ) = ω3 − ϒ2 · ϒ3 · ω Q R (iω, τ ) = ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 Q I (iω, τ ) = −[ϒ2 + ϒ3 ] · ϒ1 · ω sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

We use different terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j n, m ∈ N0 ; n = 3; m = 2; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

3 

a j · λ j ; Q(λ) =

j=0

2 

cj · λj

j=0

P(λ) = a0 + a1 · λ + a2 · λ2 + a3 · λ3 Q(λ) = c0 + c1 · λ + c2 · λ2 n, m ∈ N0 ; n > m and a j , c j : R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denoted

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

691

complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ . The coefficientsa j (Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ ) ∈ R and c j (Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ ) ∈ R depend on copper vapor laser (CVL) rate equations parameters, Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ values. Unless strictly necessary, the designation of the varied arguments: Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 [4, 5]. a0 = 0; c0 = ϒ2 · ϒ3 · ϒ1 3

a0 + c0 = ϒ2 · ϒ3 · ϒ1 =

ϒk = 0 k=1

∀ Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang, 1993, Sect. 3.4) can also be verified in the present case. (a) If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ domain of interest. P(λ) (b) | Q(λ) | is bounded for |λ| → ∞ ; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, 1 ·λ+c2 ·λ in the limit: | Q(λ) | = | a0 +ac01+c |. P(λ) ·λ+a2 ·λ2 +a3 ·λ3 2 (c) The following expressions exist: F(ω) = |P(i · ω)|2 3· ω)| − |Q(i 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω6 ). of (d) Each positive root ω(Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ ) F(ω) = 0 is continuous and differentiable with respect to Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ and the condition can only be assessed numerically. 2

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ , Reλ may, at the crossing, change its (∗) sign from (-) to (+). i.e. from a stable focus E (∗) = (Pg(∗) , n (∗) g , n cu ) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to Pcu , Pvp , vd , R, v2 , l, Pex , vs , , τ and any system parameters. −1

 (Pcu ) =



∂Reλ ∂ Pcu

 λ=i·ω

; Pvp , vd , R, v2 , l, Pex , vs , , τ = const

692

6 Gas Laser Systems Stability Analysis …

 ∂Reλ ; Pcu , vd , R, v2 , l, Pex , vs , , τ = const ∂ Pvp λ=i·ω   ∂Reλ −1  (vd ) = ; Pcu , Pvp , R, v2 , l, Pex , vs , , τ = const ∂vd λ=i·ω   ∂Reλ −1 (R) = ; Pcu , Pvp , vd , v2 , l, Pex , vs , , τ = const ∂ R λ=i·ω   ∂Reλ −1 (v2 ) = ; Pcu , Pvp , vd , R, l, Pex , vs , , τ = const ∂v2 λ=i·ω   ∂Reλ −1  (τ ) = ; Pcu , Pvp , vd , R, v2 , l, Pex , vs ,  = const ∂τ λ=i·ω

−1 (Pvp ) =



P(λ) = PR (λ)+i · PI (λ); Q(λ) = Q R (λ)+i · Q I (λ), When writing and inserting λ = i ·ω into copper vapor laser (CVL) rate equations characteristic equation ω must satisfy the following equations: sin(ω · τ ) = g(ω) −PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) = |Q(iω, τ )|2 cos(ω · τ ) = h(ω) PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) =− |Q(iω, τ )|2 where |Q(iω, τ )|2 = 0 in view of requirement (a) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ . Now it is important to notice that if τ ∈ / I (assume that / I , ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ ·ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) + 2 · n ·  ∀ n ∈ N0 Hence we can define the maps: ; n ∈ N0 ; τ ∈ I . Let us introduce the τn : I → R+0 , is given by τn (τ ) = θ(τ )+2·n· ω(τ ) function I → R ; Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ . In the following, the subscripts λ, ω, Pcu , Pvp , vd , R, v2 , l, Pex , vs ,  indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(Pcu , Pvp , vd , R, v2 , l, Pex , vs , , . . .) and ω(Pcu , Pvp , vd , R, v2 , l, Pex , vs , , . . .), and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating copper

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

693

vapor laser (CVL) rate equations characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = Pcu , Pvp , vd , R, v2 , l, , τ, . . . 

 ∂λ −1 ∂x −Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) = Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω =

λ (i·ω,x)·Q(λ,x)−τ ·|P(i·ω,x)| ( i·Pω (i·ω,x)·P(i·ω,x)+i·Q ) Px (i·ω,x)·P(i·ω,x)−Q x (i·ω,x)·Q(i·ω,x) Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any copper vapor laser (CVL) rate equations parameters Pcu , Pvp , vd , R, v2 , l, , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delayx = τ ; PR = −[ϒ2 + ϒ3 ] · ω2 ; PI = ω3 − ϒ2 · ϒ3 · ω; Q R = ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ; Q I = −[ϒ2 + ϒ3 ] · ϒ1 · ω: 2

PRω = −2 · [ϒ2 + ϒ3 ] · ω; PI ω = 3 · ω2 − ϒ2 · ϒ3 Q Rω = −2 · ϒ1 · ω; Q I ω = −[ϒ2 + ϒ3 ] · ϒ1 PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; Fτ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 Fω = . . . . Elements: PRω · PR = 2 · [ϒ2 + ϒ3 ]2 · ω3 ; PI ω · PI = (3 · ω2 − ϒ2 · ϒ3 ) · (ω3 − ϒ2 · ϒ3 · ω) Q I ω · Q I = [ϒ2 + ϒ3 ]2 · ϒ12 · ω Q Rω · Q R = −2 · ϒ1 · ω · (ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ) U = . . . . Elements: PR · PI ω = −(ϒ2 + ϒ3 ) · ω2 · (3 · ω2 − ϒ2 · ϒ3 ); PI · PRω = −2 · (ω3 − ϒ2 · ϒ3 · ω) · (ϒ2 + ϒ3 ) · ω Q R · Q I ω = −(ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ) · (ϒ2 + ϒ3 ) · ϒ1 ; Q I · Q Rω = (ϒ2 + ϒ3 ) · 2 · ϒ12 · ω2 F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · ∂ω + Fτ = 0; τ ∈ ∂τ Fτ ∂ω I ⇒ ∂τ = − Fω

694

6 Gas Laser Systems Stability Analysis …

∂Reλ ∂ω Fτ )λ=iω ; = ωτ = − ∂τ ∂τ Fω 2 + i · F −2 · [U + τ · |P| ω −1 (τ ) = Re{ } Fτ + i · 2 · [V + ω · |P|2 ] −1 (τ ) = (

∂Reλ )λ=iω } ∂τ V + ∂ω ·U ∂ω ∂τ sign{−1 (τ )} = sign{Fω } · sign{ · τ} +ω+ 2 |P| ∂τ

sign{−1 (τ )} = sign{(

We shall presently examine the possibility of stability transitions (bifurcations) copper vapor laser (CVL) rate equations, about the equilibrium point E (∗) = (∗) (Pg(∗) , n (∗) g , n cu ). The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameter τ . Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [4, 5]. −1 (τ ) = (

∂Reλ )λ=i·ω ; Pcu , Pvp , vd , R, v2 , l, Pex , vs ,  = const ∂τ

V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = −(ϒ2 + ϒ3 ) · ω2 · (3 · ω2 − ϒ2 · ϒ3 ) + 2 · (ω3 − ϒ2 · ϒ3 · ω) · (ϒ2 + ϒ3 ) · ω + (ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ) · (ϒ2 + ϒ3 ) · ϒ1 + (ϒ2 + ϒ3 ) · 2 · ϒ12 · ω2 Fω = 2 · [2 · [ϒ2 + ϒ3 ]2 · ω3 + (3 · ω2 − ϒ2 · ϒ3 ) · (ω3 − ϒ2 · ϒ3 · ω) + 2 · ϒ1 · ω · (ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ) ∂ω Fτ Fτ =0 − [ϒ2 + ϒ3 ]2 · ϒ12 · ω]; = ωτ = − | Fτ =0,Fω =0 = − ∂τ Fω Fω Fτ = 2 · [(PRτ · PR + PI τ · PI ) − (Q Rτ · Q R + Q I τ · Q I )] = 0 sign{−1 (τ )} = sign{Fω } · sign{ +ω+

∂ω · τ} ∂τ

V + ∂ω ·U ∂τ |P|2

6.4 A Self-consistent Model Copper Vapor Laser (CVL) …

695

We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Copper vapor laser (CVL) circuity. −1



 (τ ) =

∂Reλ ∂τ





λ=iω

−2 · [U + τ · |P|2 ] + i · Fω = Re Fτ + 2 · i · [V + ω · |P|2 ]



∂Reλ )λ=iω ∂τ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = Fτ2 + 4 · (V + ω · P 2 )2

−1 (τ ) = (

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all copper vapor laser (CVL) rate equations parameters are known and the stability switch due to various time delay values τ is described in the following expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) +ω(τ ) +

U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) |P(ω(τ ))|2



Remark we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = . . . . and cos θ (τ ) = . . . .; sin(ω · τ ) = −PR ·Q I +PI ·Q R R +PI ·Q I ) ; cos(ω · τ ) = − (PR ·Q|Q| |Q|2 = Q 2R + Q 2I . Finally we plot the 2 |Q|2 ) stability switch diagram g(τ ) = −1 (τ ) = ( ∂Reλ ∂τ λ=iω g(τ ) = −1 (τ ) =



∂Reλ ∂τ

 λ=iω

2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = Fτ2 + 4 · (V + ω · P 2 )2    ∂Reλ sign[g(τ )] = sign[−1 (τ )] = sign ∂τ λ=iω   2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2 Fτ2 + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ )]

696

6 Gas Laser Systems Stability Analysis …

Table 6.12 Copper vapor laser (CVL) rate equations sign of sign[−1 (τ )] sign[Fω ]

τ ·U sign[ V +ω + ω + ωτ · τ ] P2

sign[−1 (τ )]

±

±

+

±

∓

–

= sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )]    Fτ sign[−1 (τ )] = sign [Fω ] · (V + ω · P 2 ) − · (U + τ · P 2 ) Fω  −1 Fτ ∂ω ∂ F/∂ω ωτ = − ; ωτ = =− Fω ∂τ ∂ F/∂τ   V + ωτ · U sign[−1 (τ )] = sign [Fω ] · [P 2 ] · P2 + ω + ωτ · τ ]}; sign[P 2 ] > 0    V + ωτ · U sign[−1 (τ )] = sign [Fω ] · + ω + ω · τ τ P2   V + ωτ · U + ω + ω · τ sign[−1 (τ )] = sign[Fω ] · sign τ P2 Fω = 2 · [2 · [ϒ2 + ϒ3 ]2 · ω3 + (3 · ω2 − ϒ2 · ϒ3 ) · (ω3 − ϒ2 · ϒ3 · ω) + 2 · ϒ1 · ω · (ϒ2 · ϒ3 · ϒ1 − ϒ1 · ω2 ) − [ϒ2 + ϒ3 ]2 · ϒ12 · ω] We check the sign of −1 (τ ) according to the following rule: (Table 6.12) If, sign[−1 (τ )] > 0, then the crossing proceeds from (-) to (+) respectively (stable to unstable). If, sign[−1 (τ )] < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

6.5 Questions 1. Optical system with diode pumped Alkali metal vapor laser (DPAL), Where Ip is the intensity of the input pump beam, I+ and I- are right and left running intra-cavity laser intensities. n1 is the lower level and n2 , n3 are the two upper levels. f =

1 1+2·e

E 32 B ·T

−( k

)

; n 1 = n L ; n 2 = f · nU

6.5 Questions

697

n 3 = (1 − f ) · n U ; n U = n 2 + n 3 ; n tot = n U + n L Where T is the temperature in degrees Kelvin, kB is the Boltzmann constant and

E 3,2 is the energy different between level 3 and level 2. n tot is the total Alkali atom −[

E 32

]

number density, nn 23 = 2·e k B ·T . The diode pumped Alkali metal vapor laser (DPAL) model and pump and laser beam geometry includes two mirrors (M1 and M2) and laser Gain cell. We consider that the collision induced mixing or relaxation between 2 P3/2 and 2 P1/2 states is much faster by comparison with any other process in the system including stimulated emission. The mathematical description is of a quasi-two level system from which an analytical solution is extracted. The optical schematics for DPAL and the geometry for single or double pass longitudinal pumping are described our system. It includes the right and left running laser waves. Due to nonideal mirrors (M1 and M2) in Optical schematic for diode pumped Alkali metal vapor laser (DPAL), there are shifting in z, ( z1 , z2 ) for the right and left running intra-cavity laser intensities (I+ , I− ), respectively. The shifts in z do not affect the derivatives of I+ , I− in z ( ddzI+ , ddzI− ). The non-ideal mirrors M1 and M2 is caused by mirror’s surfaces and non-uniformity reflection (all other parameters as describe in subchapter 6.2). d I p (z) = σ31 · [(1 − f ) · n U − 2 · n L ] · I p (z) dz d I+ (z) = σ21 · ( f · n U − n L ) · I+ (z − z1 ) dz d I− (z) = −σ21 · ( f · n U − n L ) · I− (z − z 2 ); n U = ξ1 (I p , I+ (z − z 1 ), I− (z − z 2 )) dz

n L = ξ2 (I p , I+ (z − z1 ), I− (z − z2 )); n U = ξ1 (I p , I+ (z − z1 ), I− (z − z2 )) n U = ξ1 (I p , I+ , I− )   τ · n tot · (2 · σ31 · I p · v L + σ21 · (I+ + I− ) · v p ) =− ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p n L = ξ2 (I p , I+ , I− ) n tot · [( f − 1) · σ31 · τ · I p · v L − ( f · σ21 · τ · (I+ + I− ) + h · v L ) · v p ] = ( f − 3) · σ31 · τ · I p · v L − [( f + 1) · σ21 · τ · (I+ + I− ) + h · v L ] · v p 1.1 Find system fixed points. Draw fixed points coordinates as a function of f (Scaling parameter for the density of population in the mid and upper state, 0 ≤ f ≤ 1), and σ31 - 3D graph. 1.2 Discuss stability and stability switching for different values of z1 , z2 → 0. 1.3 Discuss stability and stability switching for different values of z2 , z1 → 0. 1.4 Parameter σ21 (line center simulated emission/absorption cross sections) is Very small, σ21 → ε, return (1.1) and (1.2). Explain how our results change.

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6 Gas Laser Systems Stability Analysis …

1.5 Parameter σ21 (line center simulated emission/absorption cross sections) doubles his value, σ21−new = 2 · σ21 , Explain how our results change? 2. A three-level analytic DPAL model is applied to pulsed potassium and quasi cw rubidium laser demonstrations. This model is adequate to describe the entire output laser dynamics observed for the broad range of gain cell conditions. Application of the model to gain cells with helium buffer gases requires a modification to the alkali concentration and is further interpreted as a thermal effect. The model uses longitudinally averaged number densities to predict cw or temporally evolving output intensities, including conditions where the spin-orbit relaxation rate limits performance. Determining the number of absorbed pump photons via the longitudinally averaged pump intensity, (t), requires the solution of a transcendental equation, we take it in our analysis as a constant (t) = . Where the populations in the ground 2 S1/2 state, n 1 (t), pumped 2 P3/2 state, n 3 (t), and upper laser 2 P1/2 , n 2 (t), are implicitly a function of the pump intensity and specified by the standard laser rate equations: n 1 = n 1 (t); n 2 = n 2 (t); n 3 = n 3 (t) dn 1  ψ = σ31 · (n 3 − 2 · n 1 ) · + σ21 · (n 2 − n 1 ) · + n 2 · 21 + n 3 · 31 dt h · vp h · vL dn 2 ψ = −σ21 · (n 2 − n 1 ) · + γmi x · (n 3 − 2 · e−θ · n 2 ) − n 2 · 21 dt h · vL dn 3  = −σ31 · (n 3 − 2 · n 1 ) · + γmi x · (n 3 − 2 · e−θ · n 2 ) − n 3 · 31 dt h · vp The material and cavity parameters are defined as: σ31 is the stimulated emission cross-section for the pump transition, σ21 is the stimulated emission cross-section for the lasing transition, 21 = 3.61 × 107 s−1 (assumes no quenching), 31 = 21 = 0.91; T = 376 K, Rubidium 3.81 × 107 s−1 (assumes no quenching), θ = E 3 k·T , γmi x concentration, n, n = n 1 + n 2 + n 3 = i=1 n i = 0.15 − 2.01 × 1013 atoms cm3 is spin-orbit mixing rate, typically is equal to 4.56 − 9.14 × 109 s−1 (depending on gas mixture). The average intra-cavity DPAL laser intensity, ψ, is amplified via stimulated emission from the spontaneous noise and limited by output coupling and cavity losses: 2 dψ = 0 = {r · T 4 · e(σ21 ·(n 2 −n 1 )·2·lg − 1} · τψRT + n 2 ·c ·σlg21 ·h·vL , where r is the output dt coupler reflectivity, l g is the gain length (12.7cm), τ RT = speed of light in vacuum.

2·l g c

= 2.5nsec, c is the

2.1 Find system fixed points (equilibrium points), Plot system fixed points (1D) as a function of σ31 (n 1 , n 2 and n 3 are system variables). 2.2 Discuss system stability and stability switching for different values of σ31 and σ21 parameters. 2.3 Discuss stability and stability switching, where n 3 (t) → n 3 (t − ) is delay in 3 ) for different values of , delay parameter. time (not effect dn dt

6.5 Questions

699

21 2.4 Parameter θ = E is very small, E 21 → 0 ⇒ θ → 0, find system fixed k·T points and discuss stability and stability switching for different values of σ31 parameter. 2.5 The system gain length (l g ) triple his value, l g → 3·l g , How the system dynamic is changed? Discuss stability.

3. We have a Nitrogen gas laser filament plasma kinetic system. The set of plasma kinetic equations is associated with the relevant rate equations for the population of the lasing levels and number of emitted photons. The change of concentration of different neutral and ionic atomic and molecular s] = species in the filament plasma is described by set of rate equations d[N dt + + + + + [G s ] − [L s ]. Where the subscript s stands for e, Ar , Ar2 , N2 , N4 , N3 and N + , N , N2 (A3 u ), N2 (B 3 g ), N2 (C 3 u ), Ar ∗ (43 P2 ); Ns is the density of species of type- s; and G s ,L s are the relevant generation and loss rates. The rate equations are solved jointly with the equations for the electron temperature and the vibrational temperature of ground-state nitrogen molecules. 3 dTe 3 · k · Ne · = − · Ne · k · (Te − Tvibr ) · v∗ 2 dt 2 3 1 − · Ne · k · (Te − T ) · [δ N2 · (v N2 + v N2+ + · v N4+ ) 2 2 1 + δ Ar · (v Ar + v Ar+ + · v Ar + )] + ke · Ne · N Ar ∗ · Iexc 2 2 Ne dTvibr 1 NA =( ) · (Te − Tvibr ) · v∗ − (Tvibr − T ) × [ + ( r )/τV T,Ar ] dt N N2 τV T,N2 N2 All system parameters and variables as describe in subchapter (6.1), T, Te , Tvibr are gas, electronic, and vibrational temperatures, respectively. The concentration of system gas is not balanced and T√ e ≈ Tvibr , and additionally there is a new behavior for T parameter, T → T +  · T ; T ∈ N;  ∈ R. 3.1 Find system fixed points and discuss how the fixed points coordinate are changed for different values of T and  parameters. 3.2 Do linearization to system f 1 (Te , Tvibr ); f 2 (Te , Tvibr ) equations and get the system eigenvalues of the matrix A. 3.3 Discuss stability and stability switching for different values of T and  parameters. √ 3.4 T → T +  · T ; T ∈ N;  ∈ R ; T = 0, How the dynamic of the system is changed? Find fixed points and discuss stability. 3.5 How our system dynamic is changed for T , discuss stability. 4. A laser initiation RF technique create and sustain large-volume, high-pressure air and nitrogen plasmas. The technique utilizes a laser-initiated, 15 mTorr partial tetrakis ethylene seed plasma with a 75 Torr background gas pressure to achieve high-pressure air/nitrogen plasma breakdown and reduce the RF power requirement needed to sustain the plasma. The target is to create a power-efficient, large-volume

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6 Gas Laser Systems Stability Analysis …

atmospheric plasma, breakdown of atmospheric-pressure air is assisted by a laserproduced seed plasma. The phase and attenuation constants of the plasma is βp = αp =

ω2p ω2p ω2p ω 1 1 v 1 1 2 · { · (1 − 2 · [(1 − ) + ) + ( · )2 ] 2 } 2 2 2 2 2 2 c 2 ω +v 2 ω +v ω +v ω ω2p ω2p ω2p 1 ω 1 v 1 1 2 · {− · (1 − 2 · [(1 − ) + ) + ( · )2 ] 2 } 2 2 2 2 2 2 c 2 ω +v 2 ω +v ω +v ω

where, ω is the frequency of the electromagnetic wave, c!is the speed of light in the

free space, and the electron plasma frequency is ω p = nε0e ·e·m textrad . The plasma sec interferometry objective is to accurately diagnose the line-averaged v and n e during the entire plasma pulse. The plasma characteristics are v, ω p , n e and Te . The phase constant is β p and the attenuation constant is α p of the wave. We know that due to system non linearity there is a dependent of β p parameter on ω (frequency of the electromagnetic wave), β p (ω) and α p on ω, α p (ω). We define our system by two dβ dα differential equations: dωp = f 1 (β p , α p ); dωp = f 2 (β p , α p ). 2

dβ

dα

4.1 Find system fixed points ( dωp = 0; dωp = 0). 4.2 Discuss stability and stability switching for different values of ω p , plasma parameter. 4.3 How the system dynamic is changed if the plasma frequency (ω p ) is equal to the frequency of the electromagnetic wave (ω p ≈ ω)? 4.4 Find system fixed points and discuss stability if ω p =  + ω;  ∈ R, Hint:  parameter can be negative or positive and fulfil, ω p > 0; ω > 0. 4.5 How the system dynamics is changed if ω p ω and if ω p  ω? 5. We have a self-consistent model copper vapor laser (CVL) system. Our system is represented by a related CVL discharge circuit (see subchapter 6.3, Fig. 6.7). The model discharge circuit elements are capacitor C1 which is the storage capacitor resonantly charged to voltage V0 by a DC power supply and inductance L c . Capacitor C2 is the peaking capacitor. The Thyratron switch (S1 ) is modeled by a reverse bias voltage and removed with a switching time τs . Rc is the parallel charging resistance. The Thyratron behaves like a diode with current flowing through it in only one direction. The inductances L 1 and L 2 represent the lumped circuit values. Where current I1 flows through L 1 and current I2 flows through L 2 , voltage V1 is the voltage across capacitor C1 and V2 is the voltage across capacitor C2 . The voltage across the discharge tube is Vd , and the current that flows through the discharge tube is c . The real model differential equations circuitry which Id , Id = I2 (t − τ2 ) · Rc R+R d represents the actual behavior of our system is as follow: d I1 (V1 − V2 ) + V0 · f (τs ) d I2 V2 − Vd = = ; dt L1 dt L2 I1 (t − τ1 ) d V2 I1 (t − τ1 ) − I2 (t − τ2 ) d V1 =− = ; dt C1 dt C2

6.5 Questions

701

voltage turn on function, function 2f (τs ) is the dThyratron I2 I (t − τ )+ . k k=1 k dt

f (τs )

=

5.1 Find system (CVL discharge circuit) fixed points. How the value of ddtI2 influences the fixed points coordinates? 5.2 Discuss stability and stability switching for τ1 = τ ; τ2 = 0, x = τ . 5.3 Discuss stability and stability switching for τ1 = 0; τ2 = τ , x = τ . 5.4 Return 5.1 – 5.3 for I2 (t − τ2 ) → 0, at fixed point lim I2 (t − τ2 )|t τ2 = I2 (t). t→∞

5.5 Return 5.1 – 5.3 for I1 (t − τ1 ) → 0, at fixed point lim I1 (t − τ2 )|t τ1 = I1 (t). t→∞

6. We have a self-consistent model copper vapor laser (CVL) system. Our system is represented by a related CVL discharge circuit (see subchapter 6.3, Fig. 6.7). The model discharge circuit elements are capacitor C1 which is the storage capacitor resonantly charged to voltage V0 by a DC power supply and inductance L c . Capacitor C2 is the peaking capacitor. The Thyratron switch (S1 ) is modeled by a reverse bias voltage and removed with a switching time τs . Rc is the parallel charging resistance. The Thyratron behaves like a diode with current flowing through it in only one direction. The inductances L 1 and L 2 represent the lumped circuit values. Where current I1 flows through L 1 and current I2 flows through L 2 , voltage V1 is the voltage across capacitor C1 and V2 is the voltage across capacitor C2 . The voltage across the discharge tube is Vd , and the current that flows through the discharge tube is c . The real model differential equations circuitry which Id , Id = I2 (t − τ2 ) · Rc R+R d represents the actual behavior of our system is as follow: d I1 (V1 − V2 ) + V0 · f (τs ) d I2 V2 − Vd = = ; dt L1 dt L2 d V1 I1 (t − τ1 ) d V2 I1 (t − τ1 ) − I2 (t − τ2 ) ; =− = dt C1 dt C2 Thyratron voltage turn on function, function 2f (τs 2) is the √  k=1 Ik (t − τk )+ I2 ;  ∈ N.

f (τs )

=

6.1 Find system (CVL discharge circuit) fixed points. Plot fixed points coordinates. 6.2 Discuss stability and stability switching for different values of  parameter (τ1 = 0, τ2 = 0). 6.3 Discuss stability and stability switching for different values of τ parameter (τ1 = τ , τ2 = 0). 6.4 Discuss stability and stability switching for different values of τ parameter (τ1 = 0, τ2 = τ ). 6.5 Discuss √ stability and stability switching for different values of τ parameter (τ1 = τ , τ2 = τ ). 7. We have a heat and mass transfer in pulsed laser alloying system. Typical process monitoring is based on the detection of electromagnetic radiation from the zone of laser action. We get information on the various surface phenomena and the

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material removal (vapor and plasma flows, droplets of melt, etc.), but no information from the inside of irradiated metal is available. By analysis of the treatment results after the end of energy action, we reconstruct mainly integral parameters, but not the dynamics of the process. The calculation of temperature fields on the base of linear mathematical models is crucial. The main difficulties appear during the simulation of phase transformations of melting and evaporation. The mathematical models are nonlinear. The heat transfer phenomena are described by the dynamics of phase boundaries in pulsed laser action. The two phase fronts model: movement of two phase fronts, melting and evaporation. The evaporation phenomenon into the model prevents the rise of temperature up to nonrealistic values and makes it possible to consider comparatively high values of energy density flows. We assume that the energy flow is absorbed on the irradiated surface; convection and radiation mechanisms of heat losses from both the sides of the slab are taking care and the melting is determined. The model includes both melting and evaporation phenomena. There is a wide range of energy density flow and pulse duration values. The mathematical model is describes by the following equations: T (x, t = 0) = T0 ∂ 2 T1 1 ∂ T1 = · ; S1 (t) < x < S2 (t); tm < t < 0 ∂x2 a1 ∂t q0 (t) − αg · [T1 (x, t) − Tg ] − σ · [ε1 · T14 (x, t) − εg · Tg4 ] = −λ1 ·

∂ T1 d S1 + ρ1 · L v · ∂x dt

T# V# d S1 [− ] =√ · e T1 (S1 (t),t) ; x = S1 (t) dt T1 (S1 (t), t)

λ1 ·

∂ T1 ∂ T2 d S2 = λ2 · − ρ2 · L m · ; T1 = T2 = Tm ; x = S2 (t); S2 (Tm ) = 0 ∂x ∂x dt ∂ 2 T2 1 ∂ T2 ; S2 (t) < x < L = · 2 ∂x a2 ∂t −λ2 ·

∂ T2 = α f · (T2 − T f ) + σ · (ε2 · T24 − ε f · T f4 ); x = L ∂x

Where T1 (x, t) temperature of liquid phase; T2 (x, t) temperature of solid phase; x, t are distance and time; S1 (t), S2 (t) positions of evaporation and melting phase boundaries, respectively; q0 (t) is the absorbed energy density flux; a1 , a2 are thermal diffusivity of liquid and solid phase, respectively; λ1 , λ2 are thermal conductivity of liquid and solid phase; ρ1 , ρ2 are densities of liquid and solid phase, respectively; L v is specific heat of evaporation; L m is latent heat of melting; αg , α f are convection heat losses coefficient of irradiated and rear surfaces of the slab; ε1 , ε2 are emissivity of irradiated and rear surfaces of the slab; εg , ε f are emissivity of the environment near the irradiated and rear surfaces of the slab; Tm is the temperature of melting; tm is the starting time for melting; L is the thickness of the slab; T0 is the initial temperature. V# , T# are constants which determined by the Herz-Knudsen law of evaporation:

6.5 Questions

703

V# =

[ L vk ] Pv Lv ! · e Tv ·( m ) ; T# = (k/m) 2 · ρ1 · 2·π·k m

Where k = 1.38×10−23 [ KJ ], Boltzmann constant; m atomic mass of slab material; Tv is the boiling temperature corresponding to the pressure Pv . Hint: we can replace the partial derivative by regular derivative, ∂∂x ↔ ddx ; ∂t∂ ↔ dtd . The variables of our system are T1 , T2 , S1 , S2 and all other parameters. 7.1 Try to find system fixed points in each distance interval which is characterize by nonlinear partial differential equations. 7.2 Discuss stability and stability switching for values of αg parameter. 7.3 Discuss stability and√stability switching for values of α f parameter. √ 7.4 Assume that ∂∂Tx1 = S2 ; ∂∂Tx2 =  S2 ;  ∈ N, how the dynamic of the system is changed? Discuss stability and stability switching for different values of  parameter. 7.5 Variable T1 (x, t) is delayed in time, T1 (x, t − τ ), discuss stability and stability switching for different values of τ parameter. Remark T1 (x, t) derivative in distance, ∂∂Tx1 do not effected by τ parameter. 8. Laser cladding is a technique to coat structural materials with a wear and/or corrosion resistance material without altering the mechanical properties of the bulk. This process is suited to coat medium to small areas with a moderate thickness coating. The advantages include easy control of the coating thickness, minimal heat load to the substrate, low roughness, excellent adhesion between the coating and substrate, minimal dilution of the coating’s material by the substrate’s material, and fine grained micro structures with superior hardness, wear and corrosion properties which result from the high cooling rates involved. Laser cladding is performed by either by melting a power preplaced on the substrate or by blowing the powder into the molten pool produced by the laser beam. There is an influence of the process parameters on the cladding quality and concluded that dilution depends on the specific energy and on the injected powder mass flow (q). The angel α depends on the dilution, the spot diameter, the scanning speed (v), the powder feed rate, the catchment efficiency and the powder’s material density (ρ). A simple theoretical correlation between the cladding parameters and the dimensions and shape of the clad track that is valid for low dilution. The model: we consider the cladding process is stationary, the area q · η, (A) of the cross section of the clad track is given by the mass balance: A = ρ·v where q is the powder feed rate, ρ is the density of the powder’s material, v is the scanning speed and η is the catchment efficiency of the powder. The variation of the track width (w) with the scanning speed is expressed by w = a − b · v, where, a, b are constants that depend on the laser power, the beam diameter, the powder mass flow rate and the powder particle velocity. The profile of the transverse cross section of the liquid track (S) is S : y = f (x) in two dimensional frames. The position of the contact line between three different single-phase regions, which corresponds to the points ( w2 ) and (− w2 ), and we can use the width values with no powder injection. The shape of the interface at x = ± w2 determines the contact

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6 Gas Laser Systems Stability Analysis …

angle between the solid and the liquid tan(α) = dd xf |x= w2 ). This shape is determined by the normal component of the force balance over the interface which is given by 2 3

p(x) + (μS · n · n) = γ · [1 + ( dd xf )2 ]− 2 · dd x 2f Where, Q is the jump at the interface of a property Q, p is the pressure, m is the viscosity, μS is the viscous stress tensor (constant) and n is the unit normal pointing out of liquid. g is the surface tension times the mean curvature of S. The gravitational forces are p(x) = p and the variation of γ with temperature (T) and the concentration (C) which lead to the Marangoni effect. If S = 0 s (static case), the balance nonlinear differential equation states that the interface S has a constant 1 and all curvature radius. We define two variables for our system y1 = dd xf ; y2 = dy dx other balance nonlinear differential equations are constant. 8.1 Find system fixed points, and plot as a function of viscous stress tensor μS. 8.2 Discuss stability and stability switching for different values of gravitational forces parameter p. 8.3 We have y1 variable shifting in x coordinate, y1 (x) → y1 (x − x), Investigate stability and stability switching for different values of x parameter. 8.4 We know ! that the gravitational forces parameter p is a function of f function,

p = f + dd xf , find system fixed points and discuss stability. 8.5 We know ! that the gravitational forces parameter p is a function of f function,

p = f 2 + ( dd xf ) ;  ∈ N, find system fixed points and discuss stability and stability switching for different values of  parameter.

9. We have a copper vapor laser (CVL) which characterize by set of delay differential equations (DDEs), rate equations. The self- consistent model for high repetition rate copper vapor laser (CVL) simulates a discharge pulse, laser pulse, an inter-pulse afterglow in a high repetition rate. System model thermodynamic quantities are Tg , PCu , and Pg . The radial diffusion length of the discharge tube is  and vd is the speed at which copper atoms diffuse to the wall. Exclusive of changes in the gas temperature, the time constant for relaxation of Pg to its equilibrium value is 2·vl s ; where the length of the discharge tube is land vs is the sound speed. Pg —Buffer gas pressure, Tg —Gas temperature, Pcu —Copper vapor pressure, Pex —External buffer gas pressure, Pvp —Copper vapor pressure based on the wall temperature, R—Mirror reflectivity. The time delays τ1 , τ2 and τ3 are related to buffer gas pressure, and densities of the buffer gas and copper atoms, respectively. Remark The time delays d P dn parameters τ1 , τ2 and τ3 do not affect the derivatives dtg , dtg , dndtcu . d Pg vd 2 · v2 = [Pcu − Pvp ] · · Pg (t − τ1 ) − [Pg (t − τ1 ) − Pex ] · dt R · Pcu l ! n g (t − τ2 ) · 2 · vs dn cu dn g n cu (t − τ3 ) · vd = − [Pg (t − τ1 ) − Pex ] · ; = −[Pcu − Pvp ] · dt l · Pg (t − τ1 ) dt  · Pcu

9.1 Find system fixed points, and draw coordinates as a function of Pex parameter. 9.2 Discuss stability and stability switching (for different values of τ parameter) for τ1 = 0, τ2 = τ and τ3 = 0.

6.5 Questions

705

9.3 9.3 Discuss √ stability and stability switching (for different values of τ parameter) for τ1 = τ , τ2 = 0 and τ3 = τ . 9.4 Discuss stability and (for different values of τ parameter) √ √ stability switching √ for τ1 = τ , τ2 = 3 τ and τ3 = 4 τ . 9.5 The external buffer gas pressure Pex is very small, Pex → 0, How it influence the dynamic of the system? Find fixed points and discuss stability for τ1 = 0, τ2 = τ and τ3 = 0. 10. We have a copper vapor laser (CVL) which characterize by set of delay differential equations (DDEs), rate equations. The self- consistent model for high repetition rate copper vapor laser (CVL) simulates a discharge pulse, laser pulse, an inter-pulse afterglow in a high repetition rate. System model thermodynamic quantities are Tg , PCu , and Pg . The radial diffusion length of the discharge tube is and vd is the speed at which copper atoms diffuse to the wall. Exclusive of changes in the gas temperature, the time constant for relaxation of Pg to its equilibrium value is l ; where the length of the discharge tube is l and vs is the sound speed. Pg —Buffer 2·vs gas pressure, Tg —Gas temperature, Pcu —Copper vapor pressure, Pex —External buffer gas pressure, Pvp —Copper vapor pressure based on the wall temperature, R— Mirror reflectivity. The time delays τ1 , τ2 and τ3 are related to buffer gas pressure, and densities of the buffer gas and copper atoms, respectively,  parameter 1 >  > 0;  ∈ R. Remark: The time delays parameters τ1 , τ2 and τ3 do not affect the d P dn derivatives dtg , dtg , dndtcu . d Pg vd 2 · v2 = [Pcu − Pvp ] · · Pg (t − τ1 ) − [Pg (t − τ1 ) − Pex ] · dt R · Pcu l #

" n g (t − τ2 ) · 2 · vs dn cu dn g = −[Pg (t − τ1 ) − Pex ] · ; = −[Pcu − Pvp ] · dt l · Pg (t − τ1 ) dt

(1+)

n cu (t − τ3 ) · vd  · Pcu

10.1 Find system fixed points, and draw coordinates as a function of  parameter. 10.2 Discuss stability and stability switching (for different values of τ parameter) for τ1 = 0, τ2 = 0 and τ3 = τ . 10.3 Discuss stability and √ stability switching (for different values of τ parameter) for τ1 = 0, τ2 = τ and τ3 = 0. 10.4 Discuss stability and√stability switching (for different √ √values of  Parameter) for τ1 = τ , τ2 =  τ ; 1 >> 0; ∈ R and τ3 = 4 τ . 10.5 The external buffer gas pressure Pex is very small, Pex → 0, How it influence the dynamic of the system? Find fixed points and discuss stability for τ1 = τ, τ2 = 0 and τ3 = 0.

References 1. K. Daniel, A. Skimantas, et al., Free-space nitrogen gas laser driven by a femtosecond filament. Phys. Rev. A 86, 033831 (2012)

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2. J.K. Hale, Dynamics and Bifurcations. Texts in Applied Mathematics, vol. 3 3. S.H. Strogatz, Nonlinear Dynamics and Chaos. Westview Press 4. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993) 5. E. Betta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002) 6. G. Hager, J. McIver, D. Hostutler, G. Pitz, G. Perram, A Quasi-two level analytic model for end pumped Alkali metal vapor laser, in Proceedings of SPIE, vol. 7005 700528-1 7. J. Guckenheimer, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, in Applied Mathematical Sciences, vol. 42 8. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, in Text in Applied Mathematics (Hardcover) 9. M.J. Kushner, A self-consistent model for high repetition rate copper vapor lasers. IEEE J. Quantum Electronics QE-17(8) (1981)

Chapter 7

Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (I)

Dual wavelength lasers are not common in scientific facilities but some applications need to use them. Dual wave lasers demonstrate simultaneous dual wavelength lasing and outcome of operating in multiple wavelength is possible. The related sub topics to dual wavelength lasers are dual wavelength laser Raman, simultaneous dual wavelength laser, dual wavelength laser tunable, dual wavelength laser speckle contrast imaging, dual wavelength laser operation, dual wavelength laser diode, dual wavelength laser interference, and dual wavelength laser output power. Dual wavelength Ti:Sapphire laser is used in many scientific and industrial applications. Ti:Sapphire laser system simultaneously generates sequence of femtosecond pulses at two independent wavelength regions. Rate-equation characterize the four energy levels of dual wavelength Ti:Sapphire laser. There is a separation of the wavelengths that increased with the pump power which is thermal lensing in Ti:Sapphire laser. Ti:Sapphire laser can generates a laser beam at more than one wavelength simultaneously. It is non-linear frequency changing via dual-wavelength laser operation of Ti:Sapphire. The dual wavelength tunable laser is characterized by rate equations for dynamical behavior over time and stability analysis. We split the Ti:Sapphire laser cavity to two sub resonator and one wavelength is established by both resonator. Ti:Sapphire sub resonators (A and B) are inspected for their operation and stability. We can get dual wavelength emission laser from Vertical External Cavity Surface Emitting Laser (VECSEL). It is simultaneously dual-wavelength emission, an inhomogeneous optical pumping of the non-identical quantum wells of the active region. The VECSEL rate equations include a strong time-delay feedback. The dynamic of dual-wavelength lasing VECSEL is analyzed and steady-state stability conditions is inspected. There are many applications that use laser sources which emit coaxial beams at two different wavelengths. The dual-wavelength VECSEL model is characterized and a stability analysis for the factors that impede simultaneous steady-state dual-wavelength emission. The VECSEL laser outcome can be stable or self-modulated optical output. Erbium-doped fiber laser is Er-doped fiber laser (EDFL) which operates in a particular regime where coherent oscillation of ASE occurs by feedback. The Erbium doped fiber laser (EDFLs) are used as sources for © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_7

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708

7 Dual-Wavelength Laser Systems Stability Analysis …

coherent light signal generation, while EDFAs are used as wave-wave amplifiers for coherent light signal generation. Erbium-Doped fiber amplifier (EDFA) is an optical amplifier used in the C-band and L-band, where loss of telecom optical fibers becomes lowest in the entire optical communication wavelength bands. An EDFA is used to compensate the loss of an optical fiber in long-distance optical communication. An EDFA can amplify multiple optical signals simultaneously and can be easily integrated with WDM solution. Erbium-doped fiber laser exhibits quasiperiodic route to chaos and operates simultaneously at two wavelengths. The EDFL start from a CW state for high pumping rates and the system becomes T, 2T, 3T, …, nT periodics and chaotic for decreasing pumping ratios. There are two fundamental frequencies and the nT- periodic (n = 1, 2, 3, …) regimes are related to a frequency locking of the low frequency on a subharmonic of the high frequency. The model is based on the rate equations and the dynamic is analyzed and simulate. An intra cavity sum-frequency mixing in a diode and pump Q-switched Nd:YVO4 dual-wavelength laser emits a compact high power yellow pulses laser beam. The gain match for simultaneous dual-wavelength emission Q-switched is optimized by a three mirror configuration forming two separate laser cavities. The Nd:YVO4 crystal is good for diode-pumped lasers with dual-wavelength emission. It has high absorption over a wide pump-wavelength bandwidth. The model exhibits the dynamical behavior of a dual-wavelength Q-switched laser. The dynamic and stability of the laser model is inspected and analyze.

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser Stability Analysis Under Delay Variables in Time A four-energy-level system of dual-wavelength Ti:Sapphire laser is characterized by rate-equation. Two sub-resonators structure gives dual wavelength laser. The Ti:Sapphire laser system simultaneously generates sequence of femtosecond pulses at two independent wavelength regions. It shows a high degree of synchronization and reduced relative timing jitter. Laser beam which oscillates at dual wavelength shows separation of the wavelengths that increased with the pump power (thermal lensing in Ti:Sapphire). The dual wavelengths laser source is very applicable in differential absorption lidar (DIAL), laser probing of the atmosphere, non-linear frequency conversion, and laser multistage ionization of atoms and molecules. Differential absorption lidar (DIAL) is a laser remote-sensing technique used for rangeresolved measurements of atmospheric gas concentrations. It’s provides species specific concentrations of gases with range resolution. The LIDAR technique also uses as an optical tool for distance and speed measurement, or for the determination of gas parameters. Differential LIDAR uses two different optical wavelengths and provides a self-calibrating method for the depth profile of gas concentration. A ground-based differential absorption LIDAR (light detection and ranging) system

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

709

(DIAL) uses an optical parametric oscillator (OPO). The optical parametric oscillator (OPO) is injection seeded with the output of a confocal filter cavity at frequencies generated by an electro-optic phase modulator (EOM) from a fixed-frequency external cavity diode laser (ECDL). Laser probing of the gas used in optical interferometry to study the propagation of femtosecond laser pulses in gases. The propagation in a gas jet gives details on the temporal structure of the laser pulse and the phase and resulting electron-density map. We get the gas ionization dynamic profile and characteristics. If we want to know the behavior of gas density, we use an array of laser beams, which are used to probe discrete points within the volume. The fluctuations in gas density are inspected, and therefore refractive index, and resulting beam deflection, which is detected by a neutral density wedge. Nonlinear frequency conversion is the conversion of input light to light of other frequencies by using optical nonlinearities. We get an efficient frequency conversion at sufficiently high optical intensities. By using non-linear crystal we can modify and change the laser’s wavelength to that required for our system [1]. The principles of nonlinear frequency conversion is based on the fact that light travels through a material interacts with it and the electric field from the incident light drives material dipoles and causing them to oscillate at it travels through the material. The light forces dipoles to oscillate with a nonlinear response; reemitted light contains additional frequencies, harmonics. The type of nonlinear response is dependent on the structure of the material. The multistage ionization limits the increase in the efficiency and emission power of self-mode-locking lasers. The lasers which have outcome of two different wavelengths simultaneously are using dyes, alexandrite, and crystals containing F color centers. Ti:Sapphire laser generates a laser beam at more than one wavelength simultaneously. Non-linear frequency changing via dual-wavelength laser operation of Ti:Sapphire is used in many scientific experimental facilities. The unique of a pulsed dual-wavelength Ti:Sapphire laser is the pump source which is a Q-switched nanosecond laser pulse, the changeable operation wavelengths, and the different gains for different wavelengths. We characterize the dual wavelength tunable laser by the rate equations, and inspect the Ti:Sapphire dual wavelength results and nonlinear dynamic. The lower energy level of a Ti:Sapphire crystal is split into an energy band with closely spaced energy levels. The Ti:Sapphire laser can operate at two or more different-wavelengths. A single resonator with specific wavelength of the laser, the laser oscillates at one wavelength. By splitting the laser cavity to two sub-resonators, the one wavelength is established by both resonators. Outcome of two laser wavelengths is result of balance between the gain and loss on specific wavelength. The energy levels scheme of a Ti:Sapphire crystal contains an upper state (E3 ) and two B states (EA 2 and E2 ) which are the lower energy band (Fig. 7.1). There are two sub-resonators, A and B. The sub-resonator A causes a transition of a part of the population inversion from E3 state to EA 2 state and the laser oscillates at wavelength λ A . Simultaneously, the sub-resonator B causes a transition of the other part of the population inversion from the E3 state to the E2B state and the laser oscillates at wavelength λ B . The lasers at two wavelengths deplete the population inversion at the same time (homogeneous broadening). The rate equations of tunable solid-state lasers, a quasi-four energy level system simulate the dual-wavelength operation of

710

7 Dual-Wavelength Laser Systems Stability Analysis …

E4

E3 A41

W14

W23A

S41

A S32

A A32

W23B

B S32

A 32

W

W32B A

B 32

E2A E2B E1 Fig. 7.1 Energy levels in the dual-wavelength operation of Ti:Sapphire laser

a Ti:Sapphire laser. The subscripts A and B refer to the sub-resonators A and B, respectively [1]. We define the following system parameters list (Table 7.1). The system rate equations for the two separate lower energy levels are Table 7.1 Dual-wavelength Ti:Sapphire laser system parameters list Dual-wavelength Ti:Sapphire laser system parameter

Description



Photon density

ν

Center frequency of laser pulse in sub-resonator

σ

Stimulated-emission cross-section

g

Small-signal gain coefficient

Amn

Rate of spontaneous transition between energy levels Em and En

Smn

Non-radiative emission rate between energy levels Em and En

Wmn

Stimulate transition probability between energy levels Em and En

l

Length of Ti:Sapphire crystal

L

Length of sub-resonators

V

Velocity of light in Ti:Sapphire crystal

δ τR =

Single-pass loss in sub-resonator L δc

Photon lifetime in sub-resonator

ni

Population density at every energy level Ei i = 1, 2, 3, 4

n tot

Total population density

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

711

  AA d A g3 l  A (t − τ A ) · n 2 · 32 · g(v, v A ) ·  A (t − τ A ) · − = n3 − dt g2 nvA LA τR A   AB d B g3 l  B (t − τ B ) = n3 − · n 2 · 32 · g(v, v B ) ·  B (t − τ B ) · − dt g2 nvB LB τR B dn 4 = n 1 · W14 − n 4 · (S43 + S41 + A41 ) dt   dn 3 AA g3 = − n3 − · n 2 · 32 · g(v, v A ) ·  A (t − τ A ) dt g2 nvA   AB g3 − n3 − · n 2 · 32 · g(v, v B ) ·  B (t − τ B ) g2 nvB A A B B − n 3 · (A32 + S32 ) − n 3 · (A32 + S32 ) + n 4 · S43 4  dn 1 A B n k = n tot = n 3 · (S21 + S21 ) − n 1 · W14 ; dt k=1

Our system variables are  A ,  B , n 3 , n 4 , n 1 ;  A =  A (t),  B =  B (t), n 3 = n 3 (t), n 4 = n 4 (t), n 1 = n 1 (t). We consider that all other parameters in our system are constant. 4 

n k = n tot ⇒ n 2 = n tot − n 1 − n 3 − n 4 = n tot −

k=1

4 

nk

k=1 k=2

We can reorganize our system DDEs as ⎛

⎤⎞

⎡ 4 

A d A g3 ⎢ ⎜ ⎥⎟ A = ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t dt g2 nvA k=1 k=2

l  A (t − τ A ) − τA) · − LA τR A ⎛ ⎤⎞ ⎡ 4 B  d B g3 ⎢ ⎜ ⎥⎟ A = ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t dt g2 nvB k=1 k=2

− τB ) ·

l  B (t − τ B ) − LB τR B

dn 4 dn 1 A B + S21 ) − n 1 · W14 = n 1 · W14 − n 4 · (S43 + S41 + A41 ); = n 3 · (S21 dt dt

712

7 Dual-Wavelength Laser Systems Stability Analysis …

⎛

⎤⎞

⎡ 4 

A dn 3 g3 ⎢ ⎜ ⎥⎟ A = −⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t − τ A ) dt g2 nvA k=1

⎛

k=2

⎡

⎤⎞

4 

B g3 ⎢ ⎜ ⎥⎟ A − ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t − τ B ) g2 nvB k=1 k=2

− n3 ·

A [A32

+

A S32

+

B A32

B + S32 ] + n 4 · S43

Parameters τ A , τ B are photon densities ( A ,  B ) delays in time, respectively. The photon densities ( A ,  B ) delays in time is caused by Dual-wavelength Ti:Sapphire A B , d are not affected by τ A , τ B time delays. optical interferences. The derivatives d dt dt d A d B dn 4 1 3 = 0; dn = 0; limt→∞  A (t − At fixed points, dt = 0; dt = 0; dt = 0; dn dt dt τ A ) =  A (t) ∀ t  τ A lim  B (t − τ B ) =  B (t) ∀ t  τ B . We define for t→∞  simplicity  ∗ = n ∗3 − gg23 · [n tot − 4k=1 n ∗k ], where  ∗ is fixed global variable. k=2   ∗ = n ∗3 − gg23 · [n tot − 4k=1 n ∗k ] = n ∗3 − gg23 · [n tot − (n ∗1 + n ∗3 + n ∗4 )]. k=2

∗

 =

∗

(n ∗1 , n ∗3 , n ∗4 )

  g3 g3 g3 ∗ g3 ∗ · n ∗3 − = 1+ · n tot + ·n + ·n g2 g2 g2 1 g2 4

d A AA l ∗ = 0 ⇒  ∗ · 32 · g(v, v A ) · ∗A · − A =0 dt nvA LA τR A   A A l 1  ∗ · 32 · g(v, v A ) · · ∗A = 0 − nvA LA τR A Case A: ∗A = 0   d B AB l 1 · ∗B = 0, Case A1 : ∗B = 0 = 0 ⇒  ∗ · 32 · g(v, v B ) · − dt nvB LB τR B dn 4 = 0 ⇒ n ∗1 · W14 − n ∗4 · (S43 + S41 + A41 ) = 0 dt dn 1 A B = 0 ⇒ n ∗3 · (S21 + S21 ) − n ∗1 · W14 = 0 dt 1 1 A B · (S43 + S41 + A41 ); n ∗1 = n ∗3 · · (S21 + S21 ) W14 W14 (S43 + S41 + A41 ) n ∗3 = n ∗4 · A B (S21 + S21 ) n ∗1 = n ∗4 ·

dn 3 AA AB = 0 ⇒ − ∗ · 32 · g(v, v A ) · ∗A −  ∗ · 32 · g(v, v B ) · ∗B dt nvA nvB

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

713

A A B B − n ∗3 · [A32 + S32 + A32 + S32 ] + n ∗4 · S43 = 0

Case A, A1: ∗A = 0; ∗B = 0; n ∗3 = n ∗4 ·

S43 A A B B A32 +S32 +A32 +S32

S43 B B + + A32 + S32 (S43 + S41 + A41 ) S43 n ∗3 = n ∗4 · ⇒ A A B A B B (S21 + S21 ) A32 + S32 + A32 + S32 (S43 + S41 + A41 ) = A B (S21 + S21 ) n ∗3 = n ∗4 ·

A A32

A S32

First fixed point: (0) (0) (0) (0) E (0) = ((0) A , B , n1 , n3 , n4 )   1 S43 ∗ ∗ ∗ · (S43 + S41 + A41 ), n 4 · A , n4 = 0, 0, n 4 · A B B W14 A32 + S32 + A32 + S32

1 · (S43 + S41 + A41 ) W14 S43 = f 2 (n ∗4 ); f 2 (n ∗4 ) = n ∗4 · A A B B A32 + S32 + A32 + S32

∗ ∗ ∗ n (0) 1 = f 1 (n 4 ); f 1 (n 4 ) = n 4 ·

n (0) 3

(0) (0) (0) (0) ∗ ∗ ∗ E (0) = ((0) A ,  B , n 1 , n 3 , n 4 ) = (0, 0, f 1 (n 4 ), f 2 (n 4 ), n 4 )

Case A, A2:  ∗ · 1

τR B

∗

; =

B A32 nvB

L B ·n v B B τ R B ·g(v,v B )·A32 ·l

· g(v, v B ) ·

l LB

−

1 τR B

= 0 ⇒ ∗ ·

B A32 nvB

· g(v, v B ) ·

∗A = 0; ∗B = 0   g3 g3 g3 ∗ g3 ∗ · n ∗3 − · n tot + ·n + ·n ∗ = 1 + g2 g2 g2 1 g2 4 L B · nvB = B τ R B · g(v, v B ) · A32 ·l dn 3 AB = 0 ⇒ − ∗ · 32 · g(v, v B ) · ∗B dt nvB A A B B − n ∗3 · [A32 + S32 + A32 + S32 ] + n ∗4 · S43 = 0

−

τR B

B L B · nvB A32 · · g(v, v B ) · ∗B B · g(v, v B ) · A32 · l nvB

A A B B − n ∗3 · [A32 + S32 + A32 + S32 ] + n ∗4 · S43 = 0

l LB

=

714

7 Dual-Wavelength Laser Systems Stability Analysis …

∗B =

A A B B + S32 + A32 + S32 ] n ∗4 · S43 − n ∗3 · [A32 L B ·n v B B τ R B ·g(v,v B )·A32 ·l

·

B A32 nvB

· g(v, v B )

We get three fixed points equations:  1+

g3 g2



· n ∗3 −

g3 g3 ∗ g3 ∗ L B · nvB · n tot + · n1 + · n4 = B g2 g2 g2 τ R B · g(v, v B ) · A32 ·l

dn 4 1 = 0 ⇒ n ∗1 = n ∗4 · · (S43 + S41 + A41 ) dt W14 dn 1 1 (S43 + S41 + A41 ) A B = 0 ⇒ n ∗1 = n ∗3 · · (S21 + S21 ); n ∗3 = n ∗4 · A B dt W14 (S21 + S21 )   (S43 + S41 + A41 ) g3 g3 ∗ 1 g3 − · n tot + ·n · · (S43 + S41 + A41 ) · n ∗4 · 1+ A B g2 g2 g2 4 W14 (S21 + S21 ) g3 ∗ L B · nvB + · n4 = B g2 τ R B · g(v, v B ) · A32 ·l    (S43 + S41 + A41 ) g3 g3 1 g3 · · n ∗4 1+ + · · (S43 + S41 + A41 ) + A B g2 g2 W14 g2 (S21 + S21 ) g3 L B · nvB + = · n tot B g2 τ R B · g(v, v B ) · A32 ·l n (1) 4

=

1+

⎡ ⎣ n (1) 3 = 

1+

g3 g2

g3 g2





L B ·n v B B τ R B ·g(v,v B )·A32 ·l

·

(S43 +S41 +A41 ) A B (S21 +S21 )

= ⎣  ·

1+

g3 g2



g3 g2

·

L B ·n v B B τ R B ·g(v,v B )·A32 ·l

·

(S43 +S41 +A41 ) A B (S21 +S21 )

(S43 + S41 + A41 ) · A B (S21 + S21 ) ⎡ n (1) 1

+



+

g3 g2

·

L B ·n v B B τ R B ·g(v,v B )·A32 ·l

·

(S43 +S41 +A41 ) A B (S21 +S21 )

(S43 + S41 + A41 ) W14



+

g3 g2

·

+ 1 W14

+ 1 W14

+ 1 W14

g3 g2

· n tot

· (S43 + S41 + A41 ) + g3 g2

· n tot

· (S43 + S41 + A41 ) +

g3 g2

g3 g2

g3 g2

⎦

⎤

· n tot

· (S43 + S41 + A41 ) +

⎤

g3 g2

⎦

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎧⎡ ⎨ ⎣ ⎩ (1 + ⎡

(1) B =

⎤ S41 + A41 ) +

g3 g2

S41 + A41 ) +     (S43 + S41 + A41 )  A A B B · + S + A + S · A 32 32 32 32 A B (S21 + S21 )

g3 g2

−⎣

(1 +

g3 ) g2

L B ·n v B + gg23 · n tot B τ R B ·g(v,v B )·A32 ·l (S43 +S41 +A41 ) + gg23 · W114 · (S43 + A B (S21 +S21 )

715

g3 ) g2

·

·

L B ·n v B + gg23 · n tot B τ R B ·g(v,v B )·A32 ·l (S43 +S41 +A41 ) + gg23 · W114 · (S43 + A B (S21 +S21 )

L B ·n v B B τ R B ·g(v,v B )·A32 ·l

·

B A32 nvB

⎦ · S43 ⎤ ⎦

· g(v, v B )

(1) (1) (1) (1) (1) (1) (1) (1) Second fixed point: E (1) = ((1) A ,  B , n 1 , n 3 , n 4 ) = (0,  B , n 1 , n 3 , n 4 ). B A B Case B: ∗B = 0; d = 0 ⇒ [ ∗ · n v32 · g(v, v B ) · LlB − τ R1B ] · ∗B = 0 dt B

AA l ∗ d A = 0 ⇒  ∗ · 32 · g(v, v A ) · ∗A · − A =0 dt nvA LA τR A   A A l 1  ∗ · 32 · g(v, v A ) · · ∗A = 0 − nvA LA τR A 43 Case B, B1: ∗A = 0 then ∗A = 0; ∗B = 0; n ∗3 = n ∗4 · A A +S AS+A B B and the 32 32 32 +S32 fixed point is identical as in Case A, A1, and we don’t count it as a new fixed point.

(0) (0) (0) (0) E (0) = ((0) A , B , n1 , n3 , n4 )   1 S43 ∗ ∗ ∗ · (S43 + S41 + A41 ), n 4 · A , n4 = 0, 0, n 4 · A B B W14 A32 + S32 + A32 + S32

1 · (S43 + S41 + A41 ) W14 S43 = f 2 (n ∗4 ); f 2 (n ∗4 ) = n ∗4 · A A B B A32 + S32 + A32 + S32

∗ ∗ ∗ n (0) 1 = f 1 (n 4 ); f 1 (n 4 ) = n 4 ·

n (0) 3

(0) (0) (0) (0) ∗ ∗ ∗ E (0) = ((0) A ,  B , n 1 , n 3 , n 4 ) = (0, 0, f 1 (n 4 ), f 2 (n 4 ), n 4 )

Case B, B2:  ∗ ·

A A32 nv A

· g(v, v A ) ·

l LA

−

1 τR A

= 0 ⇒ ∗ =

n v A ·L A B τ R A ·A32 ·g(v,v B )·l

∗A = 0; ∗B = 0   g3 g3 g3 ∗ g3 ∗ ∗ · n ∗3 − · n tot + ·n + ·n  = 1+ g2 g2 g2 1 g2 4

1 τR A A A32 nv A

·g(v,v A )· Ll

A

=

716

7 Dual-Wavelength Laser Systems Stability Analysis …

=

nvA · L A B τ R A · A32 · g(v, v B ) · l

dn 3 AA A A B B = 0 ⇒ − ∗ · 32 · g(v, v A ) · ∗A − n ∗3 · [A32 + S32 + A32 + S32 ] dt nvA + n ∗4 · S43 = 0  −

nvA · L A B τ R A · A32 · g(v, v B ) · l

 ·

A A32 · g(v, v A ) · ∗A nvA

A A B B − n ∗3 · [A32 + S32 + A32 + S32 ] + n ∗4 · S43 = 0



nvA · L A B τ R A · A32 · g(v, v B ) · l

 ·

A A32 · g(v, v A ) · ∗A nvA

A A B B = −n ∗3 · [A32 + S32 + A32 + S32 ] + n ∗4 · S43

∗A =

A A B + S32 + A32 + S B ] + n ∗4 · S43 −n ∗3 · [A32   A 32 n v A ·L A A · n v32 · g(v, v A ) τ ·A B ·g(v,v )·l RA

32

B

A

We get three fixed points equations:   g3 g3 g3 ∗ g3 ∗ nvA · L A 1+ · n ∗3 − · n tot + · n1 + · n4 = B g2 g2 g2 g2 τ R A · A32 · g(v, v B ) · l dn 4 1 = 0 ⇒ n ∗1 = n ∗4 · · (S43 + S41 + A41 ) dt W14 dn 1 1 (S43 + S41 + A41 ) A B = 0 ⇒ n ∗1 = n ∗3 · · (S21 + S21 ); n ∗3 = n ∗4 · A B dt W14 (S21 + S21 )   g3 (S43 + S41 + A41 ) g3 · n ∗4 · 1+ − · n tot A B g2 g2 (S21 + S21 ) g3 ∗ 1 g3 ∗ nvA · L A + · n4 · · (S43 + S41 + A41 ) + · n4 = B g2 W14 g2 τ R A · A32 · g(v, v B ) · l    (S43 + S41 + A41 ) g3 g3 1 g3 · · n ∗4 1+ + · · (S + S + A ) + 43 41 41 A B g2 g2 W14 g2 (S21 + S21 ) nvA · L A g3 = · n tot + B g2 τ R A · A32 · g(v, v B ) · l n (2) 4

=

(1 +

g3 ) g2

·

n v A ·L A + gg23 · n tot B τ R A ·A32 ·g(v,v B )·l (S43 +S41 +A41 ) + gg23 · W114 · (S43 + A B (S21 +S21 )

S41 + A41 ) +

g3 g2

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

n (2) 3

= ·

n (2) 1

g3 g2

·

(S43 +S41 +A41 ) A B (S21 +S21 )

+

g3 g2

·

+ 1 W14

g3 g2

· n tot

· (S43 + S41 + A41 ) +

g3 g2

(S43 + S41 + A41 ) A B (S21 + S21 )

= ·

1+



n v A ·L A B τ R A ·A32 ·g(v,v B )·l

717

1+

g3 g2



n v A ·L A B τ R A ·A32 ·g(v,v B )·l

·

(S43 +S41 +A41 ) A B (S21 +S21 )

+

g3 g2

·

+ 1 W14

g3 g2

· n tot

· (S43 + S41 + A41 ) +

g3 g2

(S43 + S41 + A41 ) W14

⎧ ⎡ ⎨ −⎣  ⎩ 1+

g3 g2



n v A ·L A B τ R A ·A32 ·g(v,v B )·l

·

(S43 +S41 +A41 ) A B (S21 +S21 )



+

g3 g2

·

+ 1 W14

g3 g2

· n tot

· (S43 + S41 + A41 ) +

(S43 + S41 + A41 ) A A B B · [A32 + S32 + A32 + S32 ] A B (S21 + S21 ) ⎡ n v A ·L A + gg23 · n tot B τ R A ·A32 ·g(v,v B )·l ⎣  +  41 +A41 ) 1 + gg23 · (S43(S+S + gg23 · W114 · (S43 + S41 + A41 ) + A B 21 +S21 )   A = n v A ·L A A · n v32 · g(v, v A ) B τ ·A ·g(v,v )·l

g3 g2

·

(2) A

RA

32

B

⎤ g3 g2

⎦ · S43

⎫ ⎬ ⎭

A

(2) (2) (2) (2) (2) (2) (2) (2) Third fixed point: E (2) = ((2) A ,  B , n 1 , n 3 , n 4 ) = ( A , 0, n 1 , n 3 , n 4 ). We can summary our system fixed points (Fig. 7.2). (0) (0) (0) (0) E (0) = ((0) A , B , n1 , n3 , n4 )   1 S43 ∗ · (S43 + S41 + A41 ), n ∗4 · A , n = 0, 0, n ∗4 · 4 A B B W14 A32 + S32 + A32 + S32

Fig. 7.2 Dual-wavelength operation of Ti:Sapphire laser system fixed points

718

7 Dual-Wavelength Laser Systems Stability Analysis … (1) (1) (1) (1) (1) (1) (1) (1) E (1) = ((1) A ,  B , n 1 , n 3 , n 4 ) = (0,  B , n 1 , n 3 , n 4 ) (2) (2) (2) (2) (2) (2) (2) (2) E (2) = ((2) A ,  B , n 1 , n 3 , n 4 ) = ( A , 0, n 1 , n 3 , n 4 )

Stability analysis: The standard local stability analysis about any one of the equilibrium point of the Dual-wavelength operation of Ti:Sapphire laser system consists in adding to coordinate [ A ,  B , n 1 , n 3 , n 4 ] arbitrarily small increments of exponential form [φ A , φ B , n 1 , n 3 , n 4 ] · eλ·t and retaining the first order terms in  A ,  B , n 1 , n 3 , n 4 . The system of five homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into Dual-wavelength system equations. The Dual-wavelength system fixed values with arbitrarily small increments of exponential form [φ A , φ B , n 1 , n 3 , n 4 ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. [2, 3]. ( j)

( j)

 A (t) =  A + φ A · eλ·t ;  B (t) =  B + φ B · eλ·t ( j)

( j)

n 1 (t) = n 1 + n 1 · eλ·t ; n 3 (t) = n 3 + n 3 · eλ·t d A (t) = φ A · λ · eλ·t dt d B (t) dn 1 (t) = φ B · λ · eλ·t ; = n 1 · λ · eλ·t dt dt ( j)

n 4 (t) = n 4 + n 4 · eλ·t ;

dn 3 (t) dn 4 (t) = n 3 · λ · eλ·t ; = n 4 · λ · eλ·t dt dt ( j) ( j)  A (t − τ A ) =  A + φ A · eλ·(t−τ A ) ;  B (t − τ B ) =  B + φ B · eλ·(t−τ B ) We choose these expressions for ourselves  A (t),  B (t), n 1 (t), n 3 (t), n 4 (t) as a small displacement [φ A , φ B , n 1 , n 3 , n 4 ] from the Dual-wavelength operation of ( j) Ti:Sapphire laser system fixed points in time t = 0,  A (t = 0) =  A +φ A ;  B (t = ( j) ( j) 0) =  B + φ B ; n 1 (t = 0) = n 1 + n 1 . ⎛ 1.

⎤⎞

⎡ 4 

A d A g3 ⎢ l ⎜ ⎥⎟ A = ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t − τ A ) · dt g2 n L v A A k=1 k=2

 A (t − τ A ) − τR A

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎛

719

⎡

⎤⎞ 4 

g3 ⎢ ⎜ ( j) ⎥⎟ ( j) φ A · λ · eλ·t = ⎝n 3 + n 3 · eλ·t − · ⎣n tot − (n k + n k · eλ·t )⎦⎠ g2 k=1 k=2

l ( j) · g(v, v A ) · ( A + φ A · eλ·(t−τ A ) ) · nvA LA 1 ( j) − · ( A + φ A · eλ·(t−τ A ) ) τR A ⎞⎤ ⎡ ⎤ ⎞ ⎛⎡ ⎛ 4 4   ⎢ g3 ⎥ ⎟ ⎜⎢ ( j) g3 ⎜ ( j) ⎟⎥ = ⎝⎣n 3 − · ⎝n tot − n k ⎠⎦ + ⎣ · n k + n 3 ⎦ · eλ·t ⎠ g2 g2 k=1 k=1 ·

φ A · λ · eλ·t

A A32

k=2

l ( j) · g(v, v A ) · ( A + φ A · eλ·t · e−λ·τ A ) · nvA LA 1 1 ( j) − · A − · φ A · eλ·t · e−λ·τ A τR A τR A ⎞⎤ ⎡ ⎛ 4  ( j) ⎟⎥ A A ⎢ ( j) g3 ⎜ ( j) = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · ( A g2 n v A k=1 ·

φ A · λ · eλ·t

k=2

A A32

k=2

l + φ A · eλ·t · e−λ·τ A ) · LA ⎤ ⎡ 4  AA ⎥ ⎢ g3 ( j) +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v A ) · ( A g2 k=1 nvA k=2

l 1 1 ( j) − · A − · φ A · eλ·t · e−λ·τ A LA τR A τR A ⎞⎤ 4 A  l ( j) ⎟⎥ A ( j) − n k ⎠⎦ · 32 · g(v, v A ) ·  A · n L vA A k=1

+ φ A · eλ·t · e−λ·τ A ) · ⎡

⎛

⎢ ( j) g3 ⎜ φ A · λ · eλ·t = ⎣n 3 − · ⎝n tot g2 ⎡

⎛

k=2

⎞⎤

4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · eλ·t · e−λ·τ A g2 nvA k=1 k=2

⎤

⎡ ·

4 

l AA l ⎥ ⎢ g3 ( j) +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v A ) ·  A · LA g2 k=1 nvA LA k=2

720

7 Dual-Wavelength Laser Systems Stability Analysis …

⎤

⎡ 4 

AA l ⎥ ⎢ g3 +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v A ) · φ A · eλ·t · e−λ·τ A · g2 k=1 nvA LA k=2

−

1

1

( j)

· A −

τR A ⎡

τR A ⎛

· φ A · eλ·t · e−λ·τ A ⎞⎤ 4 

A l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) φ A · λ · eλ·t = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) ·  A · g2 n L vA A k=1 k=2

⎡ −

⎞⎤

⎛ 4 

1 τR A

A ⎢ ( j) g3 ⎜ ( j) ( j) ⎟⎥ A ·  A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) g2 nvA k=1 k=2

⎡ · φ A · eλ·t · e−λ·τ A ·

⎤

4 

l ⎥ ⎢ g3 +⎣ · n k + n 3 ⎦ · eλ·t LA g2 k=1 k=2

AA l ( j) · 32 · g(v, v A ) ·  A · nvA LA ⎤ ⎡ 4 AA ⎥ ⎢ g3  +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v A ) g2 k=1 nvA k=2

· φ A · eλ·t · e−λ·τ A · ( j)

At fixed point [n 3 −

g3 g2

l 1 − · φ A · eλ·t · e−λ·τ A LA τR A

· (n tot −

4

0 ⎡

k=1 k=2

( j)

AA

( j)

( j)

n k )]· n v32 ·g(v, v A )· A · LlA − τ R1A · A = A

⎞⎤

⎛ 4 

A l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A φ A · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · e−λ·τ A · g2 n L v A A k=1

⎤

⎡

k=2

4 A l ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v A ) ·  A · g2 k=1 nvA LA

⎡

k=2

⎤

4 

A ⎥ A ⎢ g3 +⎣ · n k + n 3 ⎦ · 32 · g(v, v A ) g2 k=1 nvA k=2

· φ A · eλ·t · e−λ·τ A ·

l 1 − · φ A · e−λ·τ A LA τR A

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎡

721

⎞⎤

⎛ 4 

A l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A φ A · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · e−λ·τ A · g2 nvA LA k=1

⎤

⎡

k=2

4 A l ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v A ) ·  A · g2 k=1 nvA LA

⎡

k=2

⎤

4 

A ⎥ A ⎢ g3 +⎣ · n k · φ A + n 3 · φ A ⎦ · 32 · g(v, v A ) g2 k=1 nvA k=2

· eλ·t · e−λ·τ A ·

l 1 − · φ A · e−λ·τ A LA τR A

Assumption: n k · φ A ≈ 0; n 3 · φ A ≈ 0 ⎡

⎞⎤

⎛ 4 

A l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A φ A · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · e−λ·τ A · g2 nvA LA k=1

⎤

⎡

k=2

4 A l 1 ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v A ) ·  A · − · φ A · e−λ·τ A g2 k=1 nvA LA τR A k=2

(

consider that n tot is slightly changes and take it as a constant We 4 k=1 n k = n 1 + n 3 + n 4 ). k=2

⎡

⎞⎤

⎛ 4 

A l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · e−λ·τ A · φ A · λ = ⎣n 3 − g2 nvA LA k=1 k=2

 AA g3 l ( j) + · (n 1 + n 3 + n 4 ) + n 3 · 32 · g(v, v A ) ·  A · g2 nvA LA 1 − · φ A · e−λ·τ A τR A ⎧⎡ ⎫ ⎞⎤ ⎛ ⎪ ⎪ 4 ⎨  ( j) ⎟⎥ A A l 1 ⎬ ⎢ ( j) g3 ⎜ · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · − · φ A · e−λ·τ A ⎣n 3 − ⎪ ⎪ g n L τ 2 v A R A ⎩ ⎭ A k=1 

k=2

A g3 A32 l ( j) − φA · λ + · · g(v, v A ) ·  A · · n1 g2 n v A LA

722

7 Dual-Wavelength Laser Systems Stability Analysis …

 AA g3 l ( j) · 32 · g(v, v A ) ·  A · + 1+ · n3 g2 nvA LA A g3 A32 l ( j) + · · g(v, v A ) ·  A · · n4 = 0 g2 n v A LA 

We define for simplicity some global parameters: ⎡

⎞⎤

⎛ 4 

A l 1 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A

1 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · − g2 n L τ vA A RA k=1 k=2

A A32

g3 l ( j) · · g(v, v A ) ·  A · g2 n v A LA   AA g3 l ( j) · 32 · g(v, v A ) ·  A ·

3 = 1 + g2 nvA LA

2 =

1 · φ A · e−λ·τ A − φ A · λ + 2 · n 1 + 3 · n 3 + 2 · n 4 = 0 First arbitrarily small increments equation: 1 · φ A · e−λ·τ A − φ A · λ + 2 · n 1 +

3 · n 3 + 2 · n 4 = 0. ⎛ 2.

⎤⎞

⎡ 4 

B g3 ⎢ l d B ⎜ ⎥⎟ A = ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t − τ B ) · dt g2 n L v B B k=1 k=2

 B (t − τ B ) − τR B ⎛

⎡

⎤⎞ 4 

B g3 ⎢ ⎜ ( j) ⎥⎟ A ( j) φ B · λ · eλ·t = ⎝n 3 + n 3 · eλ·t − · ⎣n tot − (n k + n k · eλ·t )⎦⎠ · 32 g2 nvB k=1 k=2

( j)

· g(v, v B ) · ( B + φ B · eλ·(t−τ B ) ) · −

1

τR B ⎛⎡

l LB

( j)

· ( B + φ B · eλ·(t−τ B ) ) ⎞⎤

⎛

⎡

⎤

4 

4 

k=2

k=2

⎞

⎢ g3 ⎥ ⎟ ⎜⎢ ( j) g3 ⎜ ( j) ⎟⎥ φ B · λ · eλ·t = ⎝⎣n 3 − · ⎝n tot − n k ⎠⎦ + ⎣ · n k + n 3 ⎦ · eλ·t ⎠ g2 g 2 k=1 k=1 ·

B A32

nvB

( j)

· g(v, v B ) · ( B + φ B · eλ·t · e−λ·τ B ) ·

l LB

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

−

1

1

( j)

τR B ⎡

· B −

τR B ⎛

723

· φ B · eλ·t · e−λ·τ B ⎞⎤ 4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) φ B · λ · eλ·t = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · ( B g2 n vB k=1 k=2

l + φ B · eλ·t · e−λ·τ B ) · L ⎤B ⎡ 4 AA ⎥ ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v B ) · ( B g2 k=1 nvB k=2

l 1 1 ( j) − · B − · φ B · eλ·t · e−λ·τ B LB τR B τR B ⎞⎤ 4  ( j) ⎟⎥ A B l ( j) − n k ⎠⎦ · 32 · g(v, v B ) ·  B · nvB LB k=1

+ φ B · eλ·t · e−λ·τ B ) · ⎡

⎛

⎢ ( j) g3 ⎜ φ B · λ · eλ·t = ⎣n 3 − · ⎝n tot g2 ⎡

k=2

⎛

⎞⎤

4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B g2 nvB k=1 k=2

l · eλ·t · e−λ·τ B · LB ⎤ ⎡ 4  AB l ⎥ ⎢ g3 ( j) +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v B ) ·  B · g2 k=1 nvB LB k=2

⎡

⎤

4 

AB l ⎥ ⎢ g3 +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v B ) · φ B · eλ·t · e−λ·τ B · g2 k=1 nvB LB k=2

− ⎡

1

1

( j)

τR B

· B − ⎛

τR B

· φ B · eλ·t · e−λ·τ B ⎞⎤ 4 

B l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) φ B · λ · eλ·t = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) ·  B · g2 n L vB B k=1

⎡ −

1 τR B

k=2

⎞⎤

⎛ 4 

B ⎢ ( j) g3 ⎜ ( j) ( j) ⎟⎥ A ·  B + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B g2 nvB k=1 k=2

724

7 Dual-Wavelength Laser Systems Stability Analysis …

⎤

⎡ · eλ·t · e−λ·τ B ·

4 

l AB ⎥ ⎢ g3 ( j) +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v B ) ·  B LB g2 k=1 nvB k=2

⎤

⎡ ·

4 l AB ⎥ ⎢ g3  +⎣ · n k + n 3 ⎦ · eλ·t · 32 · g(v, v B ) · φ B · eλ·t LB g2 k=1 nvB k=2

l 1 · − · φ B · eλ·t · e−λ·τ B LB τR B

· e−λ·τ B ( j)

At fixed point [n 3 −

g3 g2

· (n tot −

4

0 ⎡

k=1 k=2

( j)

AB

( j)

( j)

n k )]· n v32 ·g(v, v B )· B · LlB − τ R1B · B = B

⎞⎤

⎛ 4 

B l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A φ B · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B · g2 n L v B B k=1

⎤

⎡

k=2

4 B l ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v B ) ·  B · g2 k=1 nvB LB

⎡

k=2

⎤

4 

B ⎥ A ⎢ g3 +⎣ · n k + n 3 ⎦ · 32 · g(v, v B ) · φ B g2 k=1 nvB k=2

l 1 − · φ B · e−λ·τ B LB τR B ⎞⎤ ⎡ ⎛ 4  ( j) ⎟⎥ A B l ⎢ ( j) g3 ⎜ φ B · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B · g2 nvB LB k=1 · eλ·t · e−λ·τ B ·

⎤

⎡

k=2

4 B l ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v B ) ·  B · g2 k=1 nvB LB

⎡

k=2

⎤

4 

B ⎥ A ⎢ g3 +⎣ · n k · φ B + n 3 · φ B ⎦ · 32 · g(v, v B ) g2 k=1 nvB k=2

· eλ·t · e−λ·τ B ·

l 1 − · φ B · e−λ·τ B LB τR B

Assumption: n k · φ B ≈ 0; n 3 · φ B ≈ 0

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎡

725

⎞⎤

⎛ 4 

B l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A φ B · λ = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B · g2 n L vB B k=1

⎤

⎡

k=2

4 B l 1 ⎥ A ⎢ g3  ( j) +⎣ · n k + n 3 ⎦ · 32 · g(v, v B ) ·  B · − · φ B · e−λ·τ B g2 k=1 nvB LB τR B k=2

(

consider that n tot is slightly changes and take it as a constant We 4 n k=1 k = n 1 + n 3 + n 4 ). k=2

⎡

⎞⎤

⎛ 4 

B l ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B · φ B · λ = ⎣n 3 − g2 n L vB B k=1 k=2

 AB g3 l ( j) + · (n 1 + n 3 + n 4 ) + n 3 · 32 · g(v, v B ) ·  B · g2 nvB LB 1 − · φ B · e−λ·τ B τR B ⎧⎡ ⎫ ⎞⎤ ⎛ ⎪ ⎪ 4 ⎨ B  ( j) ⎟⎥ A l 1 ⎬ ⎢ ( j) g3 ⎜ 32 · φ B · e−λ·τ B · ⎝n tot − n k ⎠⎦ · · g(v, v B ) · − ⎣n 3 − ⎪ ⎪ g2 n L τ v B R B ⎩ ⎭ B k=1 

k=2

  g3 l g3 ( j) − φB · λ + · · g(v, v B ) ·  B · · n1 + 1 + g2 n v B LB g2 B A32 ( j) · · g(v, v B ) ·  B nvB B l g3 A32 l ( j) · · n3 + · · g(v, v B ) ·  B · · n4 = 0 LB g2 n v B LB B A32

We define for simplicity some global parameters: ⎡

⎞⎤

⎛ 4 

B l 1 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A

4 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · − g2 nvB LB τR B k=1 k=2

B g3 A32 l ( j)

5 = · · g(v, v B ) ·  B · g2 n v B LB   AB g3 l ( j) · 32 · g(v, v B ) ·  B ·

6 = 1 + g2 nvB LB

726

7 Dual-Wavelength Laser Systems Stability Analysis …

4 · φ B · e−λ·τ B − φ B · λ + 5 · n 1 + 6 · n 3 + 5 · n 4 = 0 Second arbitrarily small increments equation: 4 · φ B · e−λ·τ B − φ B · λ + 5 · n 1 +

6 · n 3 + 5 · n 4 = 0. 3.

dn 1 A B = n 3 · (S21 + S21 ) − n 1 · W14 ⇒ n 1 · λ · eλ·t dt ( j) ( j) A B = (n 3 + n 3 · eλ·t ) · (S21 + S21 ) − (n 1 + n 1 · eλ·t ) · W14

( j)

( j)

A B A B n 3 · (S21 + S21 ) − n 1 · W14 + n 3 · (S21 + S21 ) · eλ·t

− n 1 · W14 · eλ·t − n 1 · λ · eλ·t = 0 ( j)

( j)

A B A B At fixed point: n 3 ·(S21 +S21 )−n 1 ·W14 = 0; −n 1 ·W14 −n 1 ·λ+n 3 ·(S21 +S21 )=

0. A B +S21 ) = 0. Third arbitrarily small increments equation: −n 1 ·W14 −n 1 ·λ+n 3 ·(S21

⎛ 4.

⎤⎞

⎡ 4 

A dn 3 g3 ⎢ ⎜ ⎥⎟ A = −⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t − τ A ) dt g2 nvA k=1

⎛

k=2

⎡

⎤⎞

4 

B g3 ⎢ ⎜ ⎥⎟ A − ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t − τ B ) g2 nvB k=1 k=2

− n3 ·

A [A32

+

⎛

A S32

+

B A32

B + S32 ] + n 4 · S43

⎤⎞

⎡ 4 

A g3 ⎢ ⎥⎟ A ⎜ n 3 · λ · eλ·t = −⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t − τ A ) g2 nvA k=1

⎛

⎡

k=2

⎤⎞

4 

B g3 ⎢ ⎜ ⎥⎟ A − ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t − τ B ) g2 nvB k=1 k=2

−

( j) (n 3

+ n3 · e

λ·t

B B + A32 + S32 ]+

A A ) · [A32 + S32 ( j) (n 4 + n 4 · eλ·t )

 First step, arbitrarily expression: n 3 −  A (t − τ A )

g3 g2

· S43

   · · n tot − 4k=1 n k k=2

A A32 nv A

· g(v, v A ) ·

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎛

727

⎡

⎤⎞ 4 

A g3 ⎢ ⎜ ( j) ⎥⎟ A ( j) λ·t · ⎣n tot − (n k + n k · eλ·t )⎦⎠ · 32 ⎝n 3 + n 3 · e − g2 nvA k=1 k=2

· g(v, v A ) ·

( j) ( A

+ φA · e

λ·(t−τ A )

) ⎞

⎛

⎡

⎤

4 

4 

k=2

k=2

A g3 g3 ⎜ ⎥ A ⎢ ( j) ( j) ⎟ λ·t · ⎝n tot − nk ⎠ + · n k · eλ·t ⎦ · 32 ⎣n 3 + n 3 · e − g2 g2 k=1 nvA k=1

· g(v, v A ) ·

( j) ( A

+ φA · e

λ·t

·e

−λ·τ A

)

⎫ ⎧⎡ ⎞⎤ ⎛ ⎪ ⎪ 4 4 ⎬ AA ⎨   g3 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ λ·t λ·t · 32 · ⎝n tot − n k ⎠⎦ + · nk · e + n3 · e ⎣n 3 − ⎪ ⎪ g2 g nvA 2 ⎭ ⎩ k=1 k=1 k=2

· g(v, v A ) ·

( j) ( A

+ φA · e

k=2

λ·t

·e

−λ·τ A

)

⎧⎡ ⎞⎤ ⎛ ⎪ 4 ⎨  ⎢ ( j) g3 ⎜ ( j) ⎟⎥ · ⎝n tot − n k ⎠⎦ ⎣n 3 − ⎪ g2 ⎩ k=1 k=2 ⎫ ⎪ 4 ⎬ AA  g3 ( j) λ·t λ·t nk · e + n3 · e · 32 · g(v, v A ) ·  A + · ⎪ g2 k=1 ⎭ nvA k=2

⎧⎡ ⎞⎤ ⎛ ⎪ 4 ⎨  ⎢ ( j) g3 ⎜ ( j) ⎟⎥ + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · φ A ⎪ g 2 ⎩ k=1 k=2 ⎫ ⎪ 4 ⎬ AA  g3 λ·t λ·t + · nk · φ A · e + n3 · φ A · e · 32 · g(v, v A ) · eλ·t · e−λ·τ A ⎪ g2 k=1 ⎭ nvA k=2

Assumption: n k · φ A ≈ 0; n 3 · φ A ≈ 0 ⎡

⎞⎤

⎛ 4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) ·  A ⎣n 3 − g2 n v A k=1 k=2

728

7 Dual-Wavelength Laser Systems Stability Analysis …

⎤

⎡ 4 

A ⎥ A ⎢ g3 ( j) +⎣ · n k · eλ·t + n 3 · eλ·t ⎦ · 32 · g(v, v A ) ·  A g2 k=1 nvA k=2

⎡

⎞⎤

⎛ 4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · eλ·t · e−λ·τ A g2 nvA k=1 k=2

(

consider that n tot is slightly changes and take it as a constant We 4 n k=1 k = n 1 + n 3 + n 4 ). ⎡

k=2

⎞⎤

⎛

4  ⎢ ( j) g3 ⎜ AA ( j) ⎟⎥ ⎢n − ⎥ · 32 · g(v, v A ) · ( j) ·⎜ n tot − nk ⎟ A ⎣ 3 ⎠ ⎦ ⎝ g2 nv A

⎡

⎛

k=1 k=2

⎞⎤

4 A  ⎢ ( j) g3 ⎜ A32 ( j) ⎟⎥ λ·t −λ·τ A ⎥ ⎜ +⎢ nk ⎟ ⎠⎦ · n v · g(v, v A ) · φ A · e · e ⎣n 3 − g · ⎝n tot − 2 A k=1 k=2

A g3 A32 ( j) · · g(v, v A ) ·  A · eλ·t · n 1 g2 n v A   A AA g3 g3 A32 ( j) ( j) + + 1 · eλ·t · 32 · g(v, v A ) ·  A · n 3 + · · g(v, v A ) ·  A · eλ·t · n 4 g2 nv A g2 n v A

+

( j)

At fixed points: [n 3 −

g3 g2

· (n tot −

k=1 k=2

( j)

n k )] ·

A A32 nv A

( j)

· g(v, v A ) ·  A = 0.

⎞⎤

⎛

⎡

4

4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · eλ·t · e−λ·τ A ⎣n 3 − g2 nvA k=1 k=2

g3 + · g2  g3 + g2 g3 + · g2

A A32 ( j) · g(v, v A ) ·  A · eλ·t · n 1 nvA  AA ( j) + 1 · eλ·t · 32 · g(v, v A ) ·  A · n 3 nvA A A32 ( j) · g(v, v A ) · eλ·t ·  A · n 4 nvA

We divide it by eλ·t and get

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎡

729

⎞⎤

⎛ 4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A · e−λ·τ A ⎣n 3 − g2 nvA k=1 k=2

A A32

g3 ( j) · · g(v, v A ) ·  A · n 1 g2 n v A   A g3 AA g3 A32 ( j) ( j) + + 1 · 32 · g(v, v A ) ·  A · n 3 + · · g(v, v A ) ·  A · n 4 g2 nvA g2 n v A +

 Second step, arbitrarily expression: n 3 −

g3 g2

  4 AB · n tot − k=1 n k · n v32 ·g(v, v B )· k=2

 B (t − τ B ). ⎛

B

⎤⎞

⎡ 4 

B g3 ⎢ ⎜ ( j) ⎥⎟ A ( j) λ·t · ⎣n tot − (n k + n k · eλ·t )⎦⎠ · 32 ⎝n 3 + n 3 · e − g2 nvB k=1 k=2

· g(v, v B ) ·

( j) ( B

+ φB · e

λ·(t−τ B )

) ⎞

⎛

⎡

⎤

4 

4 

k=2

k=2

B g3 g3 ⎜ ⎥ A ⎢ ( j) ( j) ⎟ λ·t · ⎝n tot − nk ⎠ + · n k · eλ·t ⎦ · 32 ⎣n 3 + n 3 · e − g2 g2 k=1 nvB k=1

· g(v, v B ) ·

( j) ( B

+ φB · e

λ·t

·e

−λ·τ B

)

⎫ ⎧⎡ ⎞⎤ ⎛ ⎪ ⎪ 4 4 ⎬ ⎨  ( j) ⎟⎥ g3  ⎢ ( j) g3 ⎜ λ·t λ·t · ⎝n tot − n k ⎠⎦ + · nk · e + n3 · e ⎣n 3 − ⎪ ⎪ g2 g2 k=1 ⎭ ⎩ k=1 k=2

·

B A32

nvB

k=2

( j)

· g(v, v B ) · ( B + φ B · eλ·t · e−λ·τ B )

⎧⎡ ⎫ ⎞⎤ ⎛ ⎪ ⎪ 4 4 ⎨ ⎬   g3 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ λ·t λ·t n + − · n − n · n · e + n · e ⎣ 3 ⎝ tot k 3 k ⎠⎦ ⎪ ⎪ g2 g2 k=1 ⎩ ⎭ k=1 k=2 k=2 ⎧⎡ ⎞⎤ ⎛ ⎪ 4 ⎨ B  A ⎢ ( j) g3 ⎜ ( j) ( j) ⎟⎥ · 32 · g(v, v B ) ·  B + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · φ B ⎪ nvB g 2 ⎩ k=1 k=2 ⎫ ⎪ 4 ⎬ AB g3  + · n k · φ B · eλ·t + n 3 · φ B · eλ·t · 32 ⎪ g2 k=1 ⎭ nvB k=2

730

7 Dual-Wavelength Laser Systems Stability Analysis …

· g(v, v B ) · eλ·t · e−λ·τ B Assumption: n k · φ B ≈ 0; n 3 · φ B ≈ 0 ⎡

⎞⎤

⎛ 4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) ·  B ⎣n 3 − g2 n v B k=1 k=2

⎡

⎤

4 

B ⎥ A ⎢ g3 ( j) +⎣ · n k · eλ·t + n 3 · eλ·t ⎦ · 32 · g(v, v B ) ·  B g2 k=1 nvB

⎡

k=2

⎞⎤

⎛ 4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · eλ·t · e−λ·τ B g2 nvB k=1 k=2

(

consider that n tot is slightly changes and take it as a constant We 4 k=1 n k = n 1 + n 3 + n 4 ). k=2

⎞⎤

⎛

⎡

4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A ( j) · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) ·  B ⎣n 3 − g2 n v B k=1

⎡

⎛

k=2

⎞⎤ 4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A + ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · eλ·t · e−λ·τ B g2 nvB k=1 k=2

  g3 g3 AB ( j) λ·t + · · g(v, v B ) ·  B · e · n 1 + + 1 · eλ·t · 32 · g(v, v B ) g2 n v B g2 nvB B g3 A32 ( j) ( j) · B · n3 + · · g(v, v B ) ·  B · eλ·t · n 4 g2 n v B B A32

( j)

At fixed points: [n 3 −

g3 g2

· (n tot −

k=1 k=2

( j)

n k )] ·

B A32 nvB

( j)

· g(v, v B ) ·  B = 0.

⎞⎤

⎛

⎡

4

4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · eλ·t · e−λ·τ B ⎣n 3 − g2 nvB k=1 k=2

B A32

g3 ( j) · · g(v, v B ) ·  B · eλ·t · n 1 g2 n v B   g3 AB ( j) + + 1 · eλ·t · 32 · g(v, v B ) ·  B · n 3 g2 nvB +

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

731

B g3 A32 ( j) · · g(v, v B ) ·  B · eλ·t · n 4 g2 n v B

+

We divide it by eλ·t and get ⎡

⎞⎤

⎛ 4 

B ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B ⎣n 3 − g2 nvB k=1 k=2

B A32

g3 ( j) · · g(v, v B ) ·  B · n 1 g2 n v B   B g3 AB g3 A32 ( j) ( j) + + 1 · 32 · g(v, v B ) ·  B · n 3 + · · g(v, v B ) ·  B · n 4 g2 nvB g2 n v B +

( j)

A A B B Third step, arbitrarily expression: −(n 3 + n 3 · eλ·t ) · [A32 + S32 + A32 + S32 ] ( j) +(n 4 + n 4 · eλ·t ) · S43 . ( j)

A A B B A A B B − n 3 · [A32 + S32 + A32 + S32 ] − [A32 + S32 + A32 + S32 ] · n 3 · eλ·t ( j)

+ n 4 · S43 + n 4 · S43 · eλ·t ( j)

( j)

A A B B A A B − n 3 · [A32 + S32 + A32 + S32 ] + n 4 · S43 − [A32 + S32 + A32 B + S32 ] · n 3 · eλ·t + n 4 · S43 · eλ·t ( j)

( j)

A A B B At fixed point: −n 3 · [A32 + S32 + A32 + S32 ] + n 4 · S43 = 0. A A B B + S32 + A32 + S32 ] · n 3 · eλ·t + n 4 · S43 · eλ·t −[A32 A A B B We divide it by eλ·t and get −[A32 + S32 + A32 + S32 ] · n 3 + n 4 · S43 . We can summary our third arbitrarily equation’s three steps:

⎛

⎤⎞

⎡ 4 

A dn 3 g3 ⎢ ⎜ ⎥⎟ A = −⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v A ) ·  A (t − τ A ) dt g2 nvA k=1

⎛

k=2

⎡

⎤⎞

4 

B g3 ⎢ ⎜ ⎥⎟ A − ⎝n 3 − · ⎣n tot − n k ⎦⎠ · 32 · g(v, v B ) ·  B (t − τ B ) g2 nvB k=1 k=2

− n3 ·

A [A32

+

A S32

+

B A32

B + S32 ] + n 4 · S43

732

7 Dual-Wavelength Laser Systems Stability Analysis …

⎧⎡ ⎞⎤ ⎛ ⎪ 4 ⎨ A  ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A n 3 · λ = − ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · φ A ⎪ g2 nvA ⎩ k=1 k=2

A A32

g3 ( j) · e−λ·τ A + · · g(v, v A ) ·  A · n 1 g2 n v A   g3 AA ( j) + + 1 · 32 · g(v, v A ) ·  A · n 3 g2 nvA  A g3 A32 ( j) + · · g(v, v A ) ·  A · n 4 g2 n v A ⎧⎡ ⎞⎤ ⎛ ⎪ 4 ⎨  ( j) ⎟⎥ A B ⎢ ( j) g3 ⎜ − ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · φ B · e−λ·τ B ⎪ g nvB 2 ⎩ k=1 k=2

B g3 A32 ( j) + · · g(v, v B ) ·  B · n 1 g2 n v B   g3 AB ( j) + + 1 · 32 · g(v, v B ) ·  B · n 3 g2 nvB  g3 A B ( j) + · 32 · g(v, v B ) ·  B · n 4 g2 n v B A A B B − [A32 + S32 + A32 + S32 ] · n 3 + n 4 · S43

For simplicity we define some global parameters: ⎡

⎞⎤

⎛ 4 

A ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) 1 = ⎣n 3 − g2 nvA k=1 k=2

A A32

g3 ( j) · · g(v, v A ) ·  A g2 n v A   g3 AA ( j) 3 = + 1 · 32 · g(v, v A ) ·  A g2 nvA ⎞⎤ ⎡ ⎛ 4 B  ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A 4 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) g2 nvB k=1 2 =

k=2

B g3 A32 ( j) · · g(v, v B ) ·  B g2 n v B   g3 AB ( j) 6 = + 1 · 32 · g(v, v B ) ·  B g2 nvB

5 =

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

733

A A B B 7 = A32 + S32 + A32 + S32

n 3 · λ = −(1 · φ A · e−λ·τ A + 2 · n 1 + 3 · n 3 + 2 · n 4 ) − (4 · φ B · e−λ·τ B + 5 · n 1 + 6 · n 3 + 5 · n 4 ) − 7 · n 3 + n 4 · S43 Forth arbitrarily small increments equation: − 1 · φ A · e−λ·τ A − 4 · φ B · e−λ·τ B − (2 + 5 ) · n 1 − (3 + 6 + 7 ) · n 3 − n 3 · λ + (S43 − 2 − 5 ) · n 4 = 0 5.

dn 4 = n 1 · W14 − n 4 · (S43 + S41 + A41 ) dt

( j)

( j)

n 4 · λ · eλ·t = (n 1 + n 1 · eλ·t ) · W14 − (n 4 + n 4 · eλ·t ) · (S43 + S41 + A41 ) ( j)

( j)

n 4 · λ · eλ·t = (n 1 · W14 + W14 · n 1 · eλ·t ) − [n 4 · (S43 + S41 + A41 ) + (S43 + S41 + A41 ) · n 4 · eλ·t ] ( j)

( j)

n 4 · λ · eλ·t = n 1 · W14 − n 4 · (S43 + S41 + A41 ) + W14 · n 1 · eλ·t − (S43 + S41 + A41 ) · n 4 · eλ·t ( j)

( j)

At fixed points: n 1 · W14 − n 4 · (S43 + S41 + A41 ) = 0. W14 · n 1 − (S43 + S41 + A41 ) · n 4 − n 4 · λ = 0; 8 = S43 + S41 + A41 Fifth arbitrarily small increments equation: W14 · n 1 − 8 ·n 4 − n 4 · λ = 0. We can summary our system global parameters list: 6 = gg23 + 1 · 5 · gg23 ⎡

⎞⎤

⎛ 4 

A l 1 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A

1 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) · − g2 n L τ v A R A A k=1 k=2

A g3 A32 l ( j)

2 = · · g(v, v A ) ·  A · g2 n v A LA   AA g3 l ( j) · 32 · g(v, v A ) ·  A ·

3 = 1 + g2 nvA LA

734

7 Dual-Wavelength Laser Systems Stability Analysis …

⎡

⎞⎤

⎛ 4 

B l 1 ⎢ ( j) g3 ⎜ ( j) ⎟⎥ A

4 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) · − g2 n L τ vB B RB k=1 k=2

B g3 A32 l ( j) · · g(v, v B ) ·  B · g2 n v B LB   AB g3 l ( j) · 32 · g(v, v B ) ·  B ·

6 = 1 + g2 nvB LB ⎞⎤ ⎡ ⎛ 4  ( j) ⎟⎥ A A ⎢ ( j) g3 ⎜ 1 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v A ) g2 nvA k=1

5 =

k=2

A g3 A32 ( j) 2 = · · g(v, v A ) ·  A g2 n v A   g3 AA ( j) 3 = + 1 · 32 · g(v, v A ) ·  A g2 nvA ⎞⎤ ⎡ ⎛ 4  ( j) ⎟⎥ A B ⎢ ( j) g3 ⎜ 4 = ⎣n 3 − · ⎝n tot − n k ⎠⎦ · 32 · g(v, v B ) g2 nvB k=1 k=2

B g3 A32 ( j) · · g(v, v B ) ·  B g2 n v B   g3 AB ( j) 6 = + 1 · 32 · g(v, v B ) ·  B g2 nvB

5 =

A A B B 7 = A32 + S32 + A32 + S32



 g3 g2

3 = + 1 · 2 · g2 g3   g3 g2

6 = + 1 · 5 · g2 g3   g3 g2 3 = + 1 · 2 · g2 g3 The next table presents the System arbitrarily small increments equations (Table 7.2). The small increments Jacobian of our Dual-wavelength Ti:Sapphire laser system is as follow:

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

735

Table 7.2 Dual-wavelength Ti:Sapphire laser system arbitrarily small increments equations First arbitrarily small increments equation

1 · φ A · e−λ·τ A − φ A · λ + 2 · n 1 + 3 · n 3 + 2 · n 4 = 0

Second arbitrarily small increments equation

4 · φ B · e−λ·τ B − φ B · λ + 5 · n 1 + 6 · n 3 + 5 · n 4 = 0 A + SB ) = 0 −n 1 · W14 − n 1 · λ + n 3 · (S21 21

Third arbitrarily small increments equation

− 1 · φ A · e−λ·τ A − 4 · φ B · e−λ·τ B − (2 + 5 ) · n 1

Forth arbitrarily small increments equation

− (3 + 6 + 7 ) · n 3 − n 3 · λ + (S43 − 2 − 5 ) · n 4 = 0 W14 · n 1 − 8 · n 4 − n 4 · λ = 0

Fifth arbitrarily small increments equation

⎞ φA ⎛ ⎞ ⎜ ⎟ ϒ11 . . . ϒ15 ⎜ φB ⎟ ⎟ ⎜ .. . . .. ⎟ ⎜ −λ·τ A ⎟ −λ ⎝ . . . ⎠·⎜ ⎜ n 1 ⎟ = 0; ϒ11 = 1 · e ⎜ ⎟ n ⎝ 3⎠ ϒ51 · · · ϒ55 n4 ⎛

ϒ12 = 0; ϒ13 = 2 ; ϒ14 = 3 ; ϒ15 = 2 ϒ21 = 0; ϒ22 = 4 · e−λ·τ B − λ; ϒ23 = 5 ; ϒ24 = 6 ϒ25 = 5 ; ϒ31 = 0; ϒ32 = 0 A B ϒ33 = −W14 − λ; ϒ34 = S21 + S21 ; ϒ35 = 0

ϒ41 = −1 · e−λ·τ A ; ϒ42 = −4 · e−λ·τ B ϒ43 = −(2 + 5 ); ϒ44 = −(3 + 6 + 7 ) − λ ϒ45 = S43 − 2 − 5 ; ϒ51 = 0; ϒ52 = 0 ϒ53 = W14 ; ϒ54 = 0; ϒ55 = −8 − λ ⎛

ϒ11 . . . ⎜ .. . . A−λ· I =⎝ . . ϒ51 · · · det(A − λ · I ) = 0

⎞ ⎛ ϒ15 ϒ11 . . . .. ⎟; det(A − λ · I ) = det ⎜ .. . . ⎝ . . . ⎠ ϒ55 ϒ51 · · ·

⎞ ϒ15 .. ⎟ . ⎠

ϒ55

736

7 Dual-Wavelength Laser Systems Stability Analysis …

⎞ ⎞ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ϒ21 ϒ23 ϒ24 ϒ25 ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ⎟ ⎜ det(A − λ · I ) = ϒ11 · det ⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ − ϒ12 · det ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎞ ⎛ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ⎟ ⎜ + ϒ13 · det ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ − ϒ14 · det ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55 ϒ51 ϒ52 ϒ53 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ + ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ⎛

ϒ51 ϒ52 ϒ53 ϒ54 ⎛

ϒ22 ⎜ ϒ32 Step 1: Find the expression for ϒ11 · det ⎜ ⎝ ϒ42 ϒ52 ⎛

( 1 · e−λ·τ A

ϒ22 ⎜ ϒ32 − λ) · det⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟. ϒ45 ⎠ ϒ55

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= ( 1 · e−λ·τ A − λ) ⎞ ⎛

5

6

5

4 · e−λ·τ B − λ A B ⎟ ⎜ 0 −W14 − λ S21 + S21 0 ⎟ · det ⎜ ⎝ −4 · e−λ·τ B −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ 0 W14 0 −8 − λ ⎞ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ( 1 · e−λ·τ A − λ) · det⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 = ( 1 · e−λ·τ A − λ) ⎞ ⎛

5

6

5

4 · e−λ·τ B − λ A B ⎟ ⎜ 0 −W14 − λ S21 + S21 0 ⎟ · det ⎜ ⎝ −4 · e−λ·τ B −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ 0 W14 0 −8 − λ = ( 1 · e−λ·τ A − λ) · ( 4 · e−λ·τ B − λ)

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎛

⎞ A B (S21 + S21 ) 0 −(W14 + λ) · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ) ⎛ ⎞⎫

6

5

5 ⎬ A B ⎠ −4 · e−λ·τ B · det ⎝ −(W14 + λ) (S21 + S21 ) 0 ⎭ 0 −(8 + λ) W14 Step 1.1: ( 4 · e−λ·τ B − λ) ⎞ ⎛ A B (S21 + S21 ) 0 −(W14 + λ) . · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ) ( 4 · e−λ·τ B − λ) ⎞ ⎛ A B (S21 + S21 ) 0 −(W14 + λ) · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ) = ( 4 · e−λ·τ B − λ) · {−(W14 + λ)   −(3 + 6 + 7 ) − λ (S43 − 2 − 5 ) · det 0 −(8 + λ)   −(2 + 5 ) (S43 − 2 − 5 ) A B + S21 ) · det −(S21 −(8 + λ) W14 = ( 4 · e−λ·τ B − λ) · {−(W14 + λ) · ([3 + 6 + 7 ] + λ) · (8 + λ) A B − (S21 + S21 ) · [(2 + 5 ) · (8 + λ) − W14 · (S43 − 2 − 5 )]}

Step 1.1.1: −(W14 + λ) · ([3 + 6 + 7 ] + λ) · (8 + λ).

− (W14 + λ) · ([3 + 6 + 7 ] + λ) · (8 + λ) = −{W14 · [3 + 6 + 7 ] · 8 + λ · (W14 · [3 + 6 + 7 ] + 8 · [W14 + 3 + 6 + 7 ]) + λ2 · (W14 + 3 + 6 + 7 + 8 ) + λ 3 } We define some parameters for simplicity: 1 = W14 · [3 + 6 + 7 ] · 8

737

738

7 Dual-Wavelength Laser Systems Stability Analysis …

2 = W14 · [3 + 6 + 7 ] + 8 · [W14 + 3 + 6 + 7 ] 3 = W14 + 3 + 6 + 7 + 8 −(W14 + λ) · ([3 + 6 + 7 ] + λ) · (8 + λ) = −{ 1 + λ · 2 + λ2 · 3 + λ3 } A B + S21 ) · [(2 + 5 ) · (8 + λ) − W14 · (S43 − 2 − 5 )]. Step 1.1.2: (S21

A B (S21 + S21 ) · [(2 + 5 ) · (8 + λ) − W14 · (S43 − 2 − 5 )] A B = (S21 + S21 ) · [{(2 + 5 ) · 8

− W14 · (S43 − 2 − 5 )} + (2 + 5 ) · λ] A B = (S21 + S21 ) · {(2 + 5 ) · 8 − W14 · (S43 − 2 − 5 )} A B + (S21 + S21 ) · (2 + 5 ) · λ A B + S21 ) · {(2 + 5 ) · 8 − We define some parameters for simplicity: 4 = (S21 W14 · (S43 − 2 − 5 )} A B + S21 ) · (2 + 5 ) 5 = (S21 A B (S21 + S21 ) · [(2 + 5 ) · (8 + λ) − W14 · (S43 − 2 − 5 )] = 4 + 5 · λ

⎛

( 4 · e−λ·τ B

⎞ A B (S21 + S21 ) 0 −(W14 + λ) − λ) · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ)

= ( 4 · e−λ·τ B − λ) · {−(W14 + λ) · ([3 + 6 + 7 ] + λ) · (8 + λ) A B − (S21 + S21 ) · [(2 + 5 ) · (8 + λ) − W14 · (S43 − 2 − 5 )]}

= ( 4 · e−λ·τ B − λ) · {−( 1 + λ · 2 + λ2 · 3 + λ3 ) − ( 4 + 5 · λ)} = ( 4 · e−λ·τ B − λ) · {−( 1 + 4 ) − ( 2 + 5 ) · λ − λ2 · 3 − λ3 } ⎛

( 4 · e−λ·τ B

⎞ A B (S21 + S21 ) 0 −(W14 + λ) − λ) · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ)

= λ · ( 1 + 4 ) + λ2 · ( 2 + 5 ) + λ3 · 3 + λ4 − [( 1 + 4 ) · 4 + ( 2 + 5 ) · 4 · λ + 3 · 4 · λ2 + 4 · λ3 ] · e−λ·τ B

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

739

⎛

Step 1.2: 4 · e−λ·τ B

⎞

6

5

5 A B ⎠. · det ⎝ −(W14 + λ) (S21 + S21 ) 0 0 −(8 + λ) W14 ⎛

⎞

6

5

5 A B ⎠ 4 · e−λ·τ B · det ⎝ −(W14 + λ) (S21 + S21 ) 0 0 −(8 + λ) W14  !  A B 0 (S21 + S21 ) = 4 · e−λ·τ B · 5 · det 0 −(8 + λ)   0 −(W14 + λ) − 6 · det −(8 + λ) W14   A B + S21 ) −(W14 + λ) (S21 + 5 · det 0 W14 ⎛

⎞

5

6

5 A + SB ) ⎠ 4 · e−λ·τ B · det ⎝ −(W14 + λ) (S21 0 21 W14 0 −(8 + λ)  !  A B) 0 (S21 + S21 = 4 · e−λ·τ B · 5 · det 0 −(8 + λ) 

− 6 · det

0 −(W14 + λ) −(8 + λ) W14

 Step 1.2.1: 5 · det 

5 · det





+ 5 · det

A + SB ) −(W14 + λ) (S21 21 W14 0



 A B + S21 ) 0 (S21 . 0 −(8 + λ)

A B + S21 ) 0 (S21 0 −(8 + λ)

 A B = − 5 · (S21 + S21 ) · (8 + λ) A B = − 5 · (S21 + S21 ) · 8 A B − 5 · (S21 + S21 )·λ

 Step 1.2.2: 6 · det 

6 · det

 0 −(W14 + λ) . −(8 + λ) W14

0 −(W14 + λ) −(8 + λ) W14

 = 6 · (W14 + λ) · (8 + λ) = 6 · W14 · 8 + 6 · (W14 + 8 ) · λ + 6 · λ2

740

7 Dual-Wavelength Laser Systems Stability Analysis …



A B + S21 ) −(W14 + λ) (S21 Step 1.2.3: 5 · det 0 W14

 A B = − 5 · W14 · (S21 + S21 ).

⎛

4 · e−λ·τ B

⎞

6

5

5 A B ⎠ · det ⎝ −(W14 + λ) (S21 + S21 ) 0 0 −(8 + λ) W14

A B A B = 4 · e−λ·τ B · {− 5 · (S21 + S21 ) · 8 − 5 · (S21 + S21 )·λ A B − [ 6 · W14 · 8 + 6 · (W14 + 8 ) · λ + 6 · λ2 ] − 5 · W14 · (S21 + S21 )} A B = 4 · e−λ·τ B · {− 5 · (S21 + S21 ) · 8 A B − 5 · (S21 + S21 ) · λ − 6 · W14 · 8 A B − 6 · (W14 + 8 ) · λ − 6 · λ2 − 5 · W14 · (S21 + S21 )}

⎛

4 · e−λ·τ B

⎞

6

5

5 A B ⎠ · det ⎝ −(W14 + λ) (S21 + S21 ) 0 0 −(8 + λ) W14

A B = 4 · e−λ·τ B · {− 5 · (S21 + S21 ) · 8 − 6 · W14 · 8 A B A B − 5 · W14 · (S21 + S21 ) − [ 5 · (S21 + S21 )

+ 6 · (W14 + 8 )] · λ − 6 · λ2 } A B +S21 )+ 6 ·(W14 +8 )] We define some parameters for simplicity: 7 = [ 5 ·(S21 A B A B + S21 ) · 8 − 6 · W14 · 8 − 5 · W14 · (S21 + S21 ) 6 = − 5 · (S21

⎛

4 · e−λ·τ B

⎞

6

5

5 A B ⎠ · det ⎝ −(W14 + λ) (S21 + S21 ) 0 0 −(8 + λ) W14

= 4 · e−λ·τ B · ( 6 − 7 · λ − 6 · λ2 ) We can implement 1.1 and 1.2 into step 1 expression: ⎛

( 1 · e−λ·τ A

ϒ22 ⎜ ϒ32 − λ) · det⎜ ⎝ ϒ42 ϒ52

= ( 1 · e−λ·τ A − λ)

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

741

⎞

5

6

5

4 · e−λ·τ B − λ A B ⎟ ⎜ 0 −W14 − λ S21 + S21 0 ⎟ · det ⎜ ⎝ −4 · e−λ·τ B −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ 0 W14 0 −8 − λ ⎛

= ( 1 · e−λ·τ A − λ) · ( 4 · e−λ·τ B − λ) ⎞ ⎛ A B (S21 + S21 ) 0 −(W14 + λ) · det ⎝ −(2 + 5 ) −(3 + 6 + 7 ) − λ S43 − 2 − 5 ⎠ W14 0 −(8 + λ) ⎛ ⎞⎫

6

5

5 ⎬ A B ⎠ −4 · e−λ·τ B · det ⎝ −(W14 + λ) (S21 + S21 ) 0 ⎭ 0 −(8 + λ) W14 = ( 1 · e−λ·τ A − λ) · {λ · ( 1 + 4 ) + λ2 · ( 2 + 5 ) + λ3 · 3 + λ4 − [( 1 + 4 ) · 4 + ( 2 + 5 ) · 4 · λ + 3 · 4 · λ2 + 4 · λ3 ] · e−λ·τ B − 4 · e−λ·τ B · ( 6 − 7 · λ − 6 · λ2 )} ⎛

( 1 · e−λ·τ A

ϒ22 ⎜ ϒ32 − λ) · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= ( 1 · e−λ·τ A − λ) · {λ · ( 1 + 4 ) + λ2 · ( 2 + 5 ) + λ3 · 3 + λ4 − {[( 1 + 4 ) · 4 + 6 · 4 ] + [( 2 + 5 ) · 4 − 7 · 4 ] · λ + [ 3 · 4 − 6 · 4 ] · λ2 + 4 · λ3 } · e−λ·τ B } We choose the case: τ A = τ ; τ B = 0 ⇒ e−λ·τ B = 1 then ⎛

( 1 · e−λ·τ

ϒ22 ⎜ ϒ32 − λ) · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= ( 1 · e−λ·τ − λ) · {λ · ( 1 + 4 ) + λ2 · ( 2 + 5 ) + λ3 · 3 + λ4 − [( 1 + 4 ) · 4 + 6 · 4 ] − [( 2 + 5 ) · 4 − 7 · 4 ] · λ − [ 3 · 4 − 6 · 4 ] · λ2 − 4 · λ3 } ⎛

( 1 · e−λ·τ

ϒ22 ⎜ ϒ32 − λ) · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

742

7 Dual-Wavelength Laser Systems Stability Analysis …

= ( 1 · e−λ·τ − λ) · {−[( 1 + 4 ) · 4 + 6 · 4 ] + [( 1 + 4 ) − ( 2 + 5 ) · 4 + 7 · 4 ] · λ + [ 2 + 5 − 3 · 4 + 6 · 4 ] · λ2 + [ 3 − 4 ] · λ3 + λ4 } We define some parameters for simplicity: 8 = −[( 1 + 4 ) · 4 + 6 · 4 ] 9 = ( 1 + 4 ) − ( 2 + 5 ) · 4 + 7 · 4 10 = 2 + 5 − 3 · 4 + 6 · 4 11 = 3 − 4 ⎛

( 1 · e−λ·τ

ϒ22 ⎜ ϒ32 − λ) · det ⎜ ⎝ ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= ( 1 · e−λ·τ − λ) · { 8 + 9 · λ + 10 · λ2 + 11 · λ3 + λ4 } = −( 8 · λ + 9 · λ2 + 10 · λ3 + 11 · λ4 + λ5 ) + ( 8 · 1 + 9 · 1 · λ + 10 · 1 · λ2 + 11 · 1 · λ3 + 1 · λ4 ) · e−λ·τ ⎛

⎞ ϒ25 ϒ35 ⎟ ⎟. ϒ45 ⎠

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

ϒ23 ϒ33 ϒ43 ϒ53

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟=0 ϒ45 ⎠ ϒ55

ϒ21 ⎜ ϒ31 Step 3: Find the expression for ϒ13 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

ϒ21 ⎜ ϒ31 Step 2: Find the expression for ϒ12 · det ⎜ ⎝ ϒ41 ϒ51 ⎛

ϒ12

ϒ21 ⎜ ϒ31 = 0 ⇒ ϒ12 · det ⎜ ⎝ ϒ41 ϒ51 ⎛

⎛

ϒ21 ⎜ ϒ31

2 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

ϒ55

⎞ ϒ25 ϒ35 ⎟ ⎟; ϒ13 = 2 . ϒ45 ⎠ ϒ55

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎛

0 0

⎜ ⎜ = 2 · det ⎜ ⎜ −1 · e−λ·τ A ⎝ 0 ⎛

ϒ21 ϒ22 ⎜ ϒ31 ϒ32

2 · det ⎜ ⎝ ϒ41 ϒ42 ϒ51 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

743

⎞ ( 4 · e−λ·τ B − λ)

6

5 A B ⎟ 0 (S21 + S21 ) 0 ⎟ ⎟ −( +  3 6 −λ·τ B −4 · e S − 2 − 5 ⎟ ⎠ +7 ) − λ 43 0 0 −8 − λ ⎞

ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55 ⎛

⎞

6

5 ( 4 · e−λ·τ B − λ) A B ⎠ = 2 · (−1 · e−λ·τ A ) · det ⎝ 0 (S21 + S21 ) 0 0 0 −8 − λ ⎞ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟

2 · det ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55   A B ) 0 (S21 + S21 −λ·τ A −λ·τ B = 2 · (−1 · e ) · ( 4 · e − λ) · det 0 −(8 + λ) We choose the case: τ A = τ ; τ B = 0 ⇒ e−λ·τ B = 1 then ⎛

ϒ21 ⎜ ϒ31

2 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= 2 · (−1 · e−λ·τ ) · ( 4 − λ) · det



A B (S21 + S21 ) 0 0 −(8 + λ)



A B + S21 ) · (8 + λ) = 2 · 1 · e−λ·τ · ( 4 − λ) · (S21 A B = (S21 + S21 ) · 2 · 1 · ( 4 − λ) · (8 + λ) · e−λ·τ A B A B = [(S21 + S21 ) · 2 · 1 · 4 · 8 + (S21 + S21 ) · 2 · 1 · ( 4 − 8 ) · λ A B − (S21 + S21 ) · 2 · 1 · λ2 ] · e−λ·τ A B We define some parameters for simplicity: 12 = (S21 + S21 ) · 2 · 1 · 4 · 8 A B A B 13 = (S21 + S21 ) · 2 · 1 · ( 4 − 8 ); 14 = (S21 + S21 ) · 2 · 1

744

7 Dual-Wavelength Laser Systems Stability Analysis …

⎛

ϒ22 ϒ32 ϒ42 ϒ52

ϒ21 ⎜ ϒ31

2 · det ⎜ ⎝ ϒ41 ϒ51

ϒ24 ϒ34 ϒ44 ϒ54

⎞ ϒ25 ϒ35 ⎟ ⎟ = [ 12 + 13 · λ − 14 · λ2 ] · e−λ·τ ϒ45 ⎠ ϒ55 ⎛

ϒ21 ⎜ ϒ31 Step 4: Find the expression for ϒ14 · det ⎜ ⎝ ϒ41 ϒ51 ⎛

ϒ21 ϒ22 ⎜ ϒ31 ϒ32 ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ51 ϒ52 ⎛

ϒ23 ϒ33 ϒ43 ϒ53

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟; ϒ14 = 3 . ϒ45 ⎠ ϒ55

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

⎞

5

5 0 ( 4 · e−λ·τ B − λ) ⎟ ⎜ 0 0 −(W14 + λ) 0 ⎟ = 3 · det ⎜ ⎝ −1 · e−λ·τ A −4 · e−λ·τ B −(2 + 5 ) S43 − 2 − 5 ⎠ 0 0 W14 −(8 + λ) ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55 ⎛ ⎞

5

5 ( 4 · e−λ·τ B − λ) ⎠ = 3 · (−1 · e−λ·τ A ) · det ⎝ 0 −(W14 + λ) 0 −(8 + λ) 0 W14 ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55

= 3 · (−1 · e ⎛

−λ·τ A

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

) · ( 4 · e

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

−λ·τ B



0 −(W14 + λ) − λ) · det −(8 + λ) W14

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= 3 · (−1 · e−λ·τ A ) · ( 4 · e−λ·τ B − λ) · (W14 + λ) · (8 + λ) We choose the case: τ A = τ ; τ B = 0 ⇒ e−λ·τ B = 1 then



7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

745

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= − 3 · 1 · e−λ·τ · ( 4 − λ) · (W14 + λ) · (8 + λ) ⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

= − 3 · 1 · e

ϒ23 ϒ33 ϒ43 ϒ53

−λ·τ

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

· {8 4 W14 + (8 · [ 4 − W14 ] + 4 W14 ) · λ

+ ( 4 − W14 − 8 ) · λ2 − λ3 } ⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟ ϒ45 ⎠ ϒ55

= {− 3 · 1 · 8 · 4 · W14 − 3 · 1 · (8 · [ 4 − W14 ] + 4 · W14 ) · λ − 3 · 1 · ( 4 − W14 − 8 ) · λ2 + 3 · 1 · λ3 } · e−λ·τ We define some parameters for simplicity: 15 = − 3 · 1 · 8 · 4 · W14 16 = 3 · 1 · (8 · [ 4 − W14 ] + 4 · W14 ) 17 = 3 · 1 · ( 4 − W14 − 8 ); 18 = 3 · 1 ⎛

ϒ21 ⎜ ϒ31 ϒ14 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ25 ϒ35 ⎟ ⎟ = ( 15 − 16 · λ − 17 · λ2 + 18 · λ3 ) · e−λ·τ ϒ45 ⎠ ϒ55 ⎛

ϒ21 ⎜ ϒ31 Step 5: Find the expression for ϒ15 · det ⎜ ⎝ ϒ41 ϒ51 ⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟. ϒ44 ⎠ ϒ54

746

7 Dual-Wavelength Laser Systems Stability Analysis …

⎛

0 0

⎜ ⎜ = 2 · det ⎜ ⎜ −1 · e−λ·τ A ⎝

−4 · e−λ·τ B

0 ⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

0 ϒ23 ϒ33 ϒ43 ϒ53

⎞

6 A B ⎟ (S21 + S21 ) ⎟ −[(3 + 6 + 7 ) ⎟ ⎟ −(2 + 5 ) +λ] ⎠ 0 W14

( 4 · e−λ·τ B − λ)

5 0 −(W14 + λ)

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54 ⎛

⎞

5

6 ( 4 · e−λ·τ B − λ) A B ⎠ = 2 · (−1 · e−λ·τ A ) · det ⎝ 0 −(W14 + λ) (S21 + S21 ) 0 0 W14 ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54   A B + S21 ) −(W14 + λ) (S21 −λ·τ A −λ·τ B = 2 · (−1 · e ) · ( 4 · e − λ) · det 0 W14 We choose the case: τ A = τ ; τ B = 0 ⇒ e−λ·τ B = 1 then ⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

= 2 · (−1 · e−λ·τ ) · ( 4 − λ) · det ⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53



A B −(W14 + λ) (S21 + S21 ) 0 W14



⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

A B = 2 · 1 · W14 · (S21 + S21 ) · ( 4 − λ) · e−λ·τ

⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

A B A B = [ 2 · 1 · W14 · (S21 + S21 ) · 4 − 2 · 1 · W14 · (S21 + S21 ) · λ] · e−λ·τ

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

747

A B We define some parameters for simplicity: 19 = 2 · 1 · W14 · (S21 + S21 ) · 4 A B 20 = 2 · 1 · W14 · (S21 + S21 ); 19 = 4 · 20

⎛

ϒ21 ⎜ ϒ31 ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ = [ 19 − 20 · λ] · e−λ·τ ϒ44 ⎠ ϒ54

We can summary our system additional parameters in the next table (Table 7.3). We can summary steps 1 to 5 in the next table (Table 7.4). ⎞ ⎞ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ϒ21 ϒ23 ϒ24 ϒ25 ⎟ ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎜ ⎟ − ϒ12 · det ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ det(A − λ · I ) = ϒ11 · det ⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎞ ⎛ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ϒ21 ϒ22 ϒ23 ϒ25 ⎟ ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎜ ⎟ − ϒ14 · det ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ + ϒ13 · det ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55 ϒ51 ϒ52 ϒ53 ϒ55 ⎛

Table 7.3 Dual-wavelength Ti:Sapphire laser system additional parameters Parameter

Expression

Parameter

1

W14 · [3 + 6 + 7 ] · 8

11

3 − 4

2

W14 · [3 + 6 + 7 ]

12

A + SB ) · ·  · (S21 2 1 21

4 · 8

+ 8 · [W14 + 3 + 6 + 7 ] 3

W14 + 3 + 6 + 7 + 8

13

A + SB ) · ·  · (S21 2 1 21 ( 4 − 8 )

4

A B (S21 + S21 ) · {(2 + 5 ) · 8

14

A + SB ) · ·  (S21 2 1 21

15

− 3 · 1 · 8 · 4 · W14

16

3 · 1 · (8 · [ 4 − W14 ] + 4 · W14 )

− W14 · (S43 − 2 − 5 )} 5 6

A + S B ) · ( +  ) (S21 2 5 21 A − 5 · (S21

− 5 ·

B + S21 ) · 8 − 6 A B W14 · (S21 + S21 )

· W14 · 8

7

A + S B ) + · (W +  )] [ 5 · (S21 6 14 8 21

17

3 ·1 ·( 4 − W14 −8 )

8

−[( 1 + 4 ) · 4 + 6 · 4 ]

18

3 · 1

9

( 1 + 4 ) − ( 2 + 5 ) · 4 + 7 · 4

19

A +

2 · 1 · W14 · (S21 B S21 ) · 4

10

2 + 5 − 3 · 4 + 6 · 4

20

A + SB )

2 · 1 · W14 · (S21 21

Matrix’s determinant element ⎞ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ϒ11 · det ⎜ ⎟ ⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ23 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ31 ϒ33 ϒ34 ϒ35 ⎟ ⎟ ϒ12 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ43 ϒ44 ϒ45 ⎠ ϒ51 ϒ53 ϒ54 ϒ55 ⎞ ⎛ ϒ21 ϒ22 ϒ24 ϒ25 ⎟ ⎜ ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟ ϒ13 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ϒ51 ϒ52 ϒ54 ϒ55 ⎛ ⎞ ϒ21 ϒ22 ϒ23 ϒ25 ⎜ ⎟ ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎟ ϒ14 · det ⎜ ⎜ ⎟ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ ϒ51 ϒ52 ϒ53 ϒ55 ( 15 − 16 · λ − 17 · λ2 + 18 · λ3 ) · e−λ·τ

[ 12 + 13 · λ − 14 · λ2 ] · e−λ·τ

ϒ12 = 0 ⇒ 0

+ ( 8 · 1 + 9 · 1 · λ + 10 · 1 · λ2 + 11 · 1 · λ3 + 1 · λ4 ) · e−λ·τ

− ( 8 · λ + 9 · λ2 + 10 · λ3 + 11 · λ4 + λ5 )

Expression

Table 7.4 Dual-wavelength Ti:Sapphire laser system small increments Jacobian Matrix’s determinant elements

(continued)

748 7 Dual-Wavelength Laser Systems Stability Analysis …

Matrix’s determinant element ⎞ ⎛ ϒ21 ϒ22 ϒ23 ϒ24 ⎟ ⎜ ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ϒ15 · det ⎜ ⎟ ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ54

Table 7.4 (continued) [ 19 − 20 · λ] · e−λ·τ

Expression

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser … 749

750

7 Dual-Wavelength Laser Systems Stability Analysis …

⎛

ϒ21 ⎜ ϒ31 + ϒ15 · det ⎜ ⎝ ϒ41 ϒ51

ϒ22 ϒ32 ϒ42 ϒ52

ϒ23 ϒ33 ϒ43 ϒ53

⎞ ϒ24 ϒ34 ⎟ ⎟ ϒ44 ⎠ ϒ54

ϒ12 = 0 ⇒ det(A − λ · I ) ⎞ ⎞ ⎛ ⎛ ϒ22 ϒ23 ϒ24 ϒ25 ϒ21 ϒ22 ϒ24 ϒ25 ⎜ ϒ32 ϒ33 ϒ34 ϒ35 ⎟ ⎜ ϒ31 ϒ32 ϒ34 ϒ35 ⎟ ⎟ ⎟ ⎜ = ϒ11 · det ⎜ ⎝ ϒ42 ϒ43 ϒ44 ϒ45 ⎠ + ϒ13 · det ⎝ ϒ41 ϒ42 ϒ44 ϒ45 ⎠ ϒ52 ϒ53 ϒ54 ϒ55 ϒ51 ϒ52 ϒ54 ϒ55 ⎞ ⎞ ⎛ ⎛ ϒ21 ϒ22 ϒ23 ϒ25 ϒ21 ϒ22 ϒ23 ϒ24 ⎜ ϒ31 ϒ32 ϒ33 ϒ35 ⎟ ⎜ ϒ31 ϒ32 ϒ33 ϒ34 ⎟ ⎟ ⎟ ⎜ − ϒ14 · det ⎜ ⎝ ϒ41 ϒ42 ϒ43 ϒ45 ⎠ + ϒ15 · det ⎝ ϒ41 ϒ42 ϒ43 ϒ44 ⎠ ϒ51 ϒ52 ϒ53 ϒ55 ϒ51 ϒ52 ϒ53 ϒ54 We choose the case: τ A = τ ; τ B = 0 ⇒ e−λ·τ B = 1 then det(A − λ · I ) = −( 8 · λ + 9 · λ2 + 10 · λ3 + 11 · λ4 + λ5 ) + ( 8 · 1 + 9 · 1 · λ + 10 · 1 · λ2 + 11 · 1 · λ3 + 1 · λ4 ) · e−λ·τ + [ 12 + 13 · λ − 14 · λ2 ] · e−λ·τ − ( 15 − 16 · λ − 17 · λ2 + 18 · λ3 ) · e−λ·τ + [ 19 − 20 · λ] · e−λ·τ det(A − λ · I ) = − 8 · λ − 9 · λ2 − 10 · λ3 − 11 · λ4 − λ5 + { 8 · 1 + 12 − 15 + 19 + ( 9 · 1 + 13 + 16 − 20 ) · λ + ( 10 · 1 − 14 + 17 ) · λ2 + ( 11 · 1 − 18 ) · λ3 + 1 · λ4 } · e−λ·τ We define some parameters for simplicity: 21 = 8 · 1 + 12 − 15 + 19 22 = 9 · 1 + 13 + 16 − 20 23 = 10 · 1 − 14 + 17 24 = 11 · 1 − 18 det(A − λ · I ) = − 8 · λ − 9 · λ2 − 10 · λ3 − 11 · λ4 − λ5 + ( 21 + 22 · λ + 23 · λ2 + 24 · λ3 + 1 · λ4 ) · e−λ·τ

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

751

D(λ, τ ) = − 8 · λ − 9 · λ2 − 10 · λ3 − 11 · λ4 − λ5 + ( 21 + 22 · λ + 23 · λ2 + 24 · λ3 + 1 · λ4 ) · e−λ·τ D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n = 5; m = 4; n > m Pn (λ, τ ) = − 8 · λ − 9 · λ2 − 10 · λ3 − 11 · λ4 − λ5 ; n = 5 Q m (λ, τ ) = 21 + 22 · λ + 23 · λ2 + 24 · λ3 + 1 · λ4 ; m = 4 Pn (λ, τ ) =

n=5 

pk (τ ) · λk

k=0

= p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3 + p4 (τ ) · λ4 + p5 (τ ) · λ5 p0 (τ ) = 0; p1 (τ ) = − 8 ; p2 (τ ) = − 9 p3 (τ ) = − 10 ; p4 (τ ) = − 11 ; p5 (τ ) = −1 Q m (λ, τ ) =

m=4 

qk (τ ) · λk

k=0

= q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 + q3 (τ ) · λ3 + q4 (τ ) · λ4 q0 (τ ) = 21 ; q1 (τ ) = 22 ; q2 (τ ) = 23 ; q3 (τ ) = 24 ; q4 (τ ) = 1 The homogeneous system for  A ,  B , n 1 .n 3 , n 4 leads to a characteristic equation for the eigenvalue λ  having the form D(λ,  τ ) = P(λ, τ ) +Q(λ, τ ) · 5 4 j j e−λ·τ = 0; and P(λ) = j=0 a j · λ ; Q(λ) = j=0 c j · λ . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delay τ . qi , qk are any Dual-wavelength Ti:Sapphire laser system’s parameters, other parameters kept as a constant [2, 3]. a0 = 0; a1 = − 8 ; a2 = − 9 ; a3 = − 10 ; a4 = − 11 ; a5 = −1 c0 = 21 ; c1 = 22 ; c2 = 23 ; c3 = 24 ; c4 = 1 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous

752

7 Dual-Wavelength Laser Systems Stability Analysis …

and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis [2] can also be verified in the present case: 1. If λ " = i" · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. " " is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. 2. If " Q(λ) P(λ) " 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω:Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and Pn (λ = i · ω, τ ) = p0 (τ ) + p1 (τ ) · i · ω − p2 (τ ) · ω2 − p3 (τ ) · i · ω3 + p4 (τ ) · ω4 + p5 (τ ) · i · ω5 Pn (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + p4 (τ ) · ω4 + [ p1 (τ ) · ω − p3 (τ ) · ω3 + p5 (τ ) · ω5 ] · i Pn (λ = i · ω, τ ) = 9 · ω2 − 11 · ω4 + [− 8 · ω + 10 · ω3 − ω5 ] · i Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω − q2 (τ ) · ω2 − q3 (τ ) · i · ω3 + q4 (τ ) · ω4 Q m (λ = i · ω, τ ) = q0 (τ ) − q2 (τ ) · ω2 + q4 (τ ) · ω4 + [q1 (τ ) · ω − q3 (τ ) · ω3 ] · i Q m (λ = i · ω, τ ) = 21 − 23 · ω2 + 1 · ω4 + [ 22 · ω − 24 · ω3 ] · i Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 9 · ω2 − 11 · ω4 + [− 8 · ω + 10 · ω3 − ω5 ] · i + 21 − 23 · ω2 + 1 · ω4 + [ 22 · ω − 24 · ω3 ] · i = 0 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 21 + ( 9 − 23 ) · ω2 + ( 1 − 11 ) · ω4 + [( 22 − 8 ) · ω + ( 10 − 24 ) · ω3 − ω5 ] · i = 0

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

753

Table 7.5 Dual-wavelength Ti:Sapphire laser system’s stability analysis F(ω, τ ) elements Expression with ω

Equivalent expression

( 9 · ω2 − 11 · ω4 )2 (− 8 · ω + 10

· ω3

29 · ω4 − 2 · 9 · 11 · ω6 + 211 · ω8

− ω 5 )2

28 · ω2 − 2 · 10 · 8 · ω4 + (2 · 8 + 210 ) · ω6 − 2 · 10 · ω8 + ω10

( 21 − 23 · ω2 + 1 · ω4 )2

221 − 2 · 21 · 23 · ω2 + (2 · 21 · 1 + 223 ) · ω4 − 2 · 23 · 1 · ω6 + 21 · ω8

( 22 · ω − 24 · ω3 )2

222 · ω2 − 2 · 22 · 24 · ω4 + 224 · ω6

|P(i · ω, τ )|2 = ( 9 · ω2 − 11 · ω4 )2 + (− 8 · ω + 10 · ω3 − ω5 )2 |Q(i · ω, τ )|2 = ( 21 − 23 · ω2 + 1 · ω4 )2 + ( 22 · ω − 24 · ω3 )2 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = ( 9 · ω2 − 11 · ω4 )2 + (− 8 · ω + 10 · ω3 − ω5 )2 − ( 21 − 23 · ω2 + 1 · ω4 )2 − ( 22 · ω − 24 · ω3 )2 The Dual-wavelength Ti:Sapphire laser system’s stability analysis F(ω, τ ) elements are (Table 7.5). F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = 29 · ω4 − 2 · 9 · 11 · ω6 + 211 · ω8 + 28 · ω2 − 2 · 10 · 8 · ω4 + (2 · 8 + 210 ) · ω6 − 2 · 10 · ω8 + ω10 − [ 221 − 2 · 21 · 23 · ω2 + (2 · 21 · 1 + 223 ) · ω4 − 2 · 23 · 1 · ω6 + 21 · ω8 ] − [ 222 · ω2 − 2 · 22 · 24 · ω4 + 224 · ω6 ] F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = 29 · ω4 − 2 · 9 · 11 · ω6 + 211 · ω8 + 28 · ω2 − 2 · 10 · 8 · ω4 + (2 · 8 + 210 ) · ω6 − 2 · 10 · ω8 + ω10 − 221 + 2 · 21 · 23 · ω2 − (2 · 21 · 1 + 223 ) · ω4 + 2 · 23 · 1 · ω6 − 21 · ω8 − 222 · ω2 + 2 · 22 · 24 · ω4 − 224 · ω6

754

7 Dual-Wavelength Laser Systems Stability Analysis …

F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = − 221 + [ 28 + 2 · 21 · 23 − 222 ] · ω2 + [ 29 − 2 · 10 · 8 − (2 · 21 · 1 + 223 ) + 2 · 22 · 24 ] · ω4 + [(2 · 8 + 210 ) − 2 · 9 · 11 + 2 · 23 · 1 − 224 ] · ω6 + [ 211 − 2 · 10 − 21 ] · ω8 + ω10 We define the following parameters for simplicity: 0 , 2 , 4 , 6 .8 , 10 0 = − 221 ; 2 = 28 + 2 · 21 · 23 − 222 4 = 29 − 2 · 10 · 8 − (2 · 21 · 1 + 223 ) + 2 · 22 · 24 6 = (2 · 8 + 210 ) − 2 · 9 · 11 + 2 · 23 · 1 − 224 8 = 211 − 2 · 10 − 21 ; 10 = 1 Hence F(ω, τ ) = 0 implies solving the above polynomial.

5 k=0

2·k · ω2·k = 0 and its roots are given by

PR (iω, τ ) = 9 · ω2 − 11 · ω4 ; PI (iω, τ ) = − 8 · ω + 10 · ω3 − ω5 Q R (iω, τ ) = 21 − 23 · ω2 + 1 · ω4 ; Q I (iω, τ ) = 22 · ω − 24 · ω3 sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

We use different terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n, m ∈ N0 ; n = 5; m = 4; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ) P(λ) =

5  j=0

a j · λ j ; Q(λ) =

4 

cj · λj

j=0

P(λ) = a0 + a1 · λ + a2 · λ2 + a3 · λ3 + a4 · λ4 + a5 · λ5 Q(λ) = c0 + c1 · λ + c2 · λ2 + c3 · λ3 + c4 · λ4

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

755

n, m ∈ N0 ; n > m and a j , c j :R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “—” denoted complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differenA , L A , l, W14 , n v A , n v B , τ , . . .) ∈ R and tiable in τ . The coefficients a j (A32 A B c j (A32 , A32 , L A , L B , l, W14 , n v A , n v B , τ , . . .) ∈ R depend on Dual-wavelength A B , A32 , L A , L B , l, W14 , n v A , n v B , τ , . . . Ti:Sapphire laser system’s parameters, A32 values. Unless strictly necessary, the designation of the varied arguments: A B , A32 , L A , L B , l, W14 , n v A , n v B , τ , . . . will subsequently be omitted from A32 P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 [2, 3]. a0 = 0; c0 = 21 = 8 · 1 + 12 − 15 + 19 8 = −[( 1 + 4 ) · 4 + 6 · 4 ] A B 12 = (S21 + S21 ) · 2 · 1 · 4 · 8 ; 15 = − 3 · 1 · 8 · 4 · W14 A B 19 = 2 · 1 · W14 · (S21 + S21 ) · 4 A B c0 = (−[( 1 + 4 ) · 4 + 6 · 4 ]) · 1 + (S21 + S21 ) · 2 · 1 · 4 · 8 A B + 3 · 1 · 8 · 4 · W14 + 2 · 1 · W14 · (S21 + S21 ) · 4

1 = W14 · [3 + 6 + 7 ] · 8 ; 4 A B = (S21 + S21 ) · {(2 + 5 ) · 8 − W14 · (S43 − 2 − 5 )} A B A B 6 = − 5 · (S21 + S21 ) · 8 − 6 · W14 · 8 − 5 · W14 · (S21 + S21 ) A B c0 = (−[(W14 · [3 + 6 + 7 ] · 8 + (S21 + S21 ) · {(2 + 5 ) · 8 A B − W14 · (S43 − 2 − 5 )}) · 4 + (− 5 · (S21 + S21 ) · 8 − 6 · W14 · 8 A B A B − 5 · W14 · (S21 + S21 )) · 4 ]) · 1 + (S21 + S21 ) · 2 · 1 · 4 · 8 A B + 3 · 1 · 8 · 4 · W14 + 2 · 1 · W14 · (S21 + S21 ) · 4 A B a0 + c0 = (−[(W14 · [3 + 6 + 7 ] · 8 + (S21 + S21 ) · {(2 + 5 ) · 8 A B − W14 · (S43 − 2 − 5 )}) · 4 + (− 5 · (S21 + S21 ) · 8 − 6 · W14 · 8 A B A B − 5 · W14 · (S21 + S21 )) · 4 ]) · 1 + (S21 + S21 ) · 2 · 1 · 4 · 8 A B + 3 · 1 · 8 · 4 · W14 + 2 · 1 · W14 · (S21 + S21 ) · 4 = 0 A B , A32 , L A , L B , l, W14 , n v A , n v B , τ , . . . ∈ R+ I.e. λ = 0 is not a root of ∀ A32 the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for

756

7 Dual-Wavelength Laser Systems Stability Analysis …

which the following requirements of the analysis (see [2], Sect. 3.4) can also be verified in the present case. 1. If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire A "A32 , L " A , l, W14 , n v A , n v B , τ , . . . domain of interest. " P(λ) " 2. " Q(λ) " is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. Indeed, " " " " " " " c0 +c1 ·λ+c2 ·λ2 +c3 ·λ3 +c4 ·λ4 " in the limit: " Q(λ) = " " 2 3 4 5 P(λ) a0 +a1 ·λ+a2 ·λ +a3 ·λ +a4 ·λ +a5 ·λ ". 2 2 3. The following expressions exist: F(ω) = |P(i · ω)| . 5 − |Q(i · ω)| 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω10 ). A , L A , l, W14 , n v A , n v B , τ , . . .) of F(ω) = 0 is contin4. Each positive root ω(A32 A , L A , l, W14 , n v A , n v B , τ , . . . and the uous and differentiable with respect to A32 condition can only be assessed numerically.

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: A , L A , l, W14 , n v A , n v B , τ , . . ., Reλ may, at the crossing, change its sign from (−) A32 ( j) ( j) ( j) ( j) ( j) to (+). i.e. from a stable focus E ( j) = ( A ,  B , n 1 , n 3 , n 4 ); j = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the A , L A , l, W14 , n v A , n v B , τ , . . . and sign of the partial derivatives with respect to A32 any system parameters.  ∂Reλ B = ; A32 , L A , L B , l, W14 , n v A , n v B , τ , . . . = const  A ∂ A32 λ=i·ω   ∂Reλ −1  (L A ) = ; A A , A B , L B , l, W14 , n v A , n v B , τ , . . . = const ∂ L A λ=i·ω 32 32   ∂Reλ −1 (W14 ) = ; A A , A B , L A , L B , l, n v A , n v B , τ , . . . = const ∂ W14 λ=i·ω 32 32   ∂Reλ −1 nvA ) = (  ; A A , A B , L A , L B , l, W14 , n v B , τ , . . . = const ∂n v A λ=i·ω 32 32   ∂Reλ −1  (n v B ) = ; A A , A B , L A , L B , l, W14 , n v A , τ , . . . = const ∂n v B λ=i·ω 32 32   ∂Reλ −1 (τ ) = ; A A , A B , L A , L B , l, W14 , n v A , n v B , . . . = const ∂τ λ=i·ω 32 32 −1



A (A32 )

P(λ) = PR (λ)+i · PI (λ); Q(λ) = Q R (λ)+i · Q I (λ), When writing and inserting λ = i · ω into Dual-wavelength Ti:Sapphire laser system’s characteristic equation ω must satisfy the following equations:

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

sin(ω · τ ) = g(ω) =

757

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos(ω · τ ) = h(ω) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

where |Q(iω, τ )|2 = 0 in view of requirement (1) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω·τ ) and cos(ω·τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: / I (assume that F(ω) is independent on τ . Now it is important to notice that if τ ∈ / I , ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ · ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) +2 · n · π ∀ n ∈ N0 Hence we can define the maps: )+2·n·π ; n ∈ N0 ; τ ∈ I . τ (n) : I → R+0 , is given by τ (n) (τ ) = θ(τ ω(τ

) Let us introduce the function I → R; Sn (τ ) = τ − τ (n) (τ ) ; τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ . In the following, the subscripts A , L A , l, W14 , n v A , n v B , . . . indicate the corresponding partial derivatives. λ, ω, A32 A , L A , l, W14 , n v A , n v B , . . .) and Let us first concentrate on (x), remember in λ(A32 A ω(A32 , L A , l, W14 , n v A , n v B , . . .), and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Dualwavelength Ti:Sapphire laser system’s characteristic equation P(λ)+ Q(λ)·e−λ·τ D = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, A , L A , l, W14 , n v A , n v B , τ , . . . one calculates: x = A32 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω) ; i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω = λ (i·ω,x)·Q(λ,x)−τ ·|P(i·ω,x)| ( i·Pω (i·ω,x)·P(i·ω,x)+i·Q ). Px (i·ω,x)·P(i·ω,x)−Q x (i·ω,x)·Q(i·ω,x) Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any Dual-wavelength A Ti:Sapphire laser system’s parameters A32 , L A , l, W14 , n v A , n v B , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2·[(PRω · PR + PI ω · PI ) −(Q Rω · Q R + Q I ω · Q I )]; Fx = 2·[(PRx · PR + PI x · PI ) −(Q Rx ·Q R +Q I x ·Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) −(Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) −(Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delay x = τ : PR = 9 · ω2 − 11 · ω4 ; PI = 2

758

7 Dual-Wavelength Laser Systems Stability Analysis …

− 8 · ω + 10 · ω3 − ω5 ; Q R = 21 − 23 · ω2 + 1 · ω4 ; Q I = 22 · ω − 24 · ω3 ; PRω = 2· 9 ·ω−4· 11 ·ω3 ; PI ω = − 8 +3· 10 ·ω2 −5·ω4 ; Q I ω = 22 −3· 24 ·ω2 ; Q Rω = −2 · 23 · ω + 4 · 1 · ω3 ; PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; Fτ = 0; PR · PI τ − PI · PRτ = 0; Q R · Q I τ − Q I · Q Rτ = 0; V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0. Fω = . . . Elements: PRω · PR = (2 · 9 · ω − 4 · 11 · ω3 ) · ( 9 · ω2 − 11 · ω4 ) = 2 · 29 · ω3 − 6 · 11 · 9 · ω5 + 4 · 211 · ω7 PI ω · PI = (− 8 + 3 · 10 · ω2 − 5 · ω4 ) · (− 8 · ω + 10 · ω3 − ω5 ) = 28 · ω − 4 · 10 · 8 · ω3 + 3 · ( 210 + 2 · 8 ) · ω5 − 8 · 10 · ω7 + 5 · ω9 Q Rω · Q R = (−2 · 23 · ω + 4 · 1 · ω3 ) · ( 21 − 23 · ω2 + 1 · ω4 ) = −2 · 23 21 · ω + 2 · ( 223 + 2 · 1 · 21 ) · ω3 − 2 · ( 23 · 1 + 2 · 1 · 23 ) · ω5 + 4 · 21 · ω7 Q I ω · Q I = ( 22 − 3 · 24 · ω2 ) · ( 22 · ω − 24 · ω3 ) = 222 · ω − 4 · 24 · 22 · ω3 + 3 · 224 · ω5 U = . . . Elements: PR · PI ω = ( 9 · ω2 − 11 · ω4 ) · (− 8 + 3 · 10 · ω2 − 5 · ω4 ) = − 9 · 8 · ω2 + (3 · 9 · 10 + 11 · 8 ) · ω4 − (3 · 11 · 10 + 5 · 9 ) · ω6 + 5 · 11 · ω8 PI · PRω = (− 8 · ω + 10 · ω3 − ω5 ) · (2 · 9 · ω − 4 · 11 · ω3 ) = −2 · 8 · 9 · ω2 + 2 · (2 · 8 · 11 + 10 · 9 ) · ω4 − 2 · (2 · 10 · 11 + 9 ) · ω6 + 4 · 11 · ω8 Q R · Q I ω = ( 21 − 23 · ω2 + 1 · ω4 ) · ( 22 − 3 · 24 · ω2 ) = 21 · 22 − (3 · 21 · 24 + 23 · 22 ) · ω2 + (3 · 23 · 24 + 1 · 22 ) · ω4 − 3 · 1 · 24 · ω6

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

759

Table 7.6 Dual-wavelength Ti:Sapphire laser system’s stability analysis Fω , U elements Fω , U elements

Expression

PRω · PR

2 · 29 · ω3 − 6 · 11 · 9 · ω5 + 4 · 211 · ω7

PI ω · PI

28 · ω − 4 · 10 · 8 · ω3 + 3 · ( 210 + 2 · 8 ) · ω5 − 8 · 10 · ω7 + 5 · ω9

Q Rω · Q R

− 2 · 23 21 · ω + 2 · ( 223 + 2 · 1 · 21 ) · ω3 − 2 · ( 23 · 1 + 2 · 1 · 23 ) · ω5 + 4 · 21 · ω7

QIω · QI

222 · ω − 4 · 24 · 22 · ω3 + 3 · 224 · ω5

PR · PI ω

− 9 · 8 · ω2 + (3 · 9 · 10 + 11 · 8 ) · ω4 − (3 · 11 · 10 + 5 · 9 ) · ω6 + 5 · 11 · ω8

PI · PRω

− 2 · 8 · 9 · ω2 + 2 · (2 · 8 · 11 + 10 · 9 ) · ω4 − 2 · (2 · 10 · 11 + 9 ) · ω6 + 4 · 11 · ω8

QR · QIω

21 · 22 − (3 · 21 · 24 + 23 · 22 ) · ω2 + (3 · 23 · 24 + 1 · 22 ) · ω4 − 3 · 1 · 24 · ω6

Q I · Q Rω

−2 · 22 · 23 · ω2 + 2 · (2 · 22 · 1 + 24 · 23 ) · ω4 − 4 · 24 · 1 · ω6

Q I · Q Rω = ( 22 · ω − 24 · ω3 ) · (−2 · 23 · ω + 4 · 1 · ω3 ) = −2 · 22 · 23 · ω2 + 2 · (2 · 22 · 1 + 24 · 23 ) · ω4 − 4 · 24 · 1 · ω6 We can summary our last results in the next table (Table 7.6). F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · 0; τ ∈ I ⇒

∂ω ∂τ

=

F − Fτω

∂ω ∂τ

+ Fτ =

 ∂Reλ ∂ω Fτ  (τ ) = ; = ωτ = − ∂τ λ=iω ∂τ Fω  ! 2 −2 · [U + τ · |P| + i · Fω −1  (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ]   ! ∂Reλ sign{−1 (τ )} = sign ∂τ λ=iω −1



∂ω V + ∂τ ·U # ∂ω

+ω+ · τ } sign −1 (τ ) = sign{Fω } · sign{ 2 |P| ∂τ

We shall presently examine the possibility of stability transitions (bifurcations) Dual-wavelength Ti:Sapphire laser system, about the equilibrium points, E ( j) ; j = 0, 1, 2, . . .,

760

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

( j)

( j)

( j)

( j)

E ( j) = ( A ,  B , n 1 , n 3 , n 4 ); j = 0, 1, 2, . . .. The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameter τ . Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [2, 3]. −1



 (τ ) =

∂Reλ ∂τ

 λ=i·ω

A B ; A32 , A32 , L A , L B , l, W14 , n v A , n v B , . . . = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = − 21 · 22 + [2 · 8 · 9 + (3 · 21 · 24 + 23 · 22 ) − 2 · 22 · 23 − 9 · 8 ] · ω2 + [(3 · 9 · 10 + 11 · 8 ) − 2 · (2 · 8 · 11 + 10 · 9 ) − (3 · 23 · 24 + 1 · 22 ) + 2 · (2 · 22 · 1 + 24 · 23 )] · ω4 + [2 · (2 · 10 · 11 + 9 ) + 3 · 1 · 24 − 4 · 24 · 1 − (3 · 11 · 10 + 5 · 9 )] · ω6 + [5 · 11 − 4 · 11 ] · ω8 We define for simplicity the following global parameters: A0 = − 21 · 22 A2 = 2 · 8 · 9 + (3 · 21 · 24 + 23 · 22 ) − 2 · 22 · 23 − 9 · 8 A4 = (3 · 9 · 10 + 11 · 8 ) − 2 · (2 · 8 · 11 + 10 · 9 ) − (3 · 23 · 24 + 1 · 22 ) + 2 · (2 · 22 · 1 + 24 · 23 ) A6 = 2 · (2 · 10 · 11 + 9 ) + 3 · 1 · 24 − 4 · 24 · 1 − (3 · 11 · 10 + 5 · 9 ); A8 = 5 · 11 − 4 · 11 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) =

4 

A2k · ω2·k

k=0

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [ 28 · ω + (2 · 29 − 4 · 10 · 8 ) · ω3 + (3 · ( 210 + 2 · 8 ) − 6 · 11 · 9 ) · ω5 + (4 · 211 − 8 · 10 ) · ω7 + 5 · ω9 + (2 · 23 21 − 222 ) · ω + (4 · 24 · 22 − 2 · ( 223 + 2 · 1 · 21 )) · ω3 + (2 · ( 23 · 1 + 2 · 1 · 23 ) − 3 · 224 ) · ω5 − 4 · 21 · ω7 ]

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

761

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [( 28 + 2 · 23 21 − 222 ) · ω + {2 · 29 − 4 · 10 · 8 + 4 · 24 · 22 − 2 · ( 223 + 2 · 1 · 21 )} · ω3 + {3 · ( 210 + 2 · 8 ) − 6 · 11 · 9 + 2 · ( 23 · 1 + 2 · 1 · 23 ) − 3 · 224 } · ω5 + {4 · 211 − 8 · 10 − 4 · 21 } · ω7 + 5 · ω9 ] We define for simplicity the following global parameters: B1 = 2 · ( 28 + 2 · 23 21 − 222 ) B3 = 2 · {2 · 29 − 4 · 10 · 8 + 4 · 24 · 22 − 2 · ( 223 + 2 · 1 · 21 )} B5 = {3 · ( 210 + 2 · 8 ) − 6 · 11 · 9 + 2 · ( 23 · 1 + 2 · 1 · 23 ) − 3 · 224 } B7 = 2{4 · 211 − 8 · 10 − 4 · 21 }; B9 = 10 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

4 

B2·k+1 · ω2·k+1

k=0

4 2·k+1 Then we get the expression for Fω = , Dual-wavelength k=0 B2·k+1 · ω Ti:Sapphire laser system parameter values. We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Dualwavelength Ti:Sapphire laser system.   ! ∂Reλ −2 · [U + τ · |P|2 ] + i · Fω  (τ ) = = Re ∂τ λ=iω Fτ + 2i · [V + ω · |P|2 ]   ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = −1 (τ ) = ∂τ λ=iω Fτ2 + 4 · (V + ω · P 2 )2 −1



The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ

)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all Dual-wavelength Ti:Sapphire laser system’s parameters are known and the stability switch due to various time delay values τ is described in the following expression:

762

7 Dual-Wavelength Laser Systems Stability Analysis …

sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign τ · ωτ (ω(τ )) U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) +ω(τ ) + |P(ω(τ ))|2



Remark: we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = . . . . and cos θ (τ ) = . . . .; I +PI ·Q R R +PI ·Q I ) |Q|2 = Q 2R + Q 2I . Finally ; cos(ω ·τ ) = − (PR ·Q|Q| sin(ω ·τ ) = −PR ·Q|Q| 2 2 we plot the stability switch diagram g(τ ) = −1 (τ ) = ( ∂Reλ ) . ∂τ λ=iω

g(τ ) = −1 (τ )   ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = = ∂τ λ=iω Fτ2 + 4 · (V + ω · P 2 )2   ∂Reλ sign[g(τ )] = sign[−1 (τ )] = sign[ ] ∂τ λ=iω % $ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2 Fτ2 + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ )] = sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )] !   Fτ 2 2 sign[ (τ )] = sign [Fω ] · (V + ω · P ) − · (U + τ · P ) Fω   Fτ ∂ω −1 ∂ F/∂ω ωτ = − ; ωτ = =− Fω ∂τ ∂ F/∂τ  !  V + ωτ · U sign[−1 (τ )] = sign [Fω ] · [P 2 ] · + ω + ω · τ τ

P2 −1

sign[P 2 ] > 0  !  V + ωτ · U sign[−1 (τ )] = sign [Fω ] · + ω + ω · τ τ

P2   V + ωτ · U −1 + ω + ωτ · τ sign[ (τ )] = sign[Fω ] · sign P2

7.1 Dual-Wavelength Operation of a Ti:Sapphire Laser …

763

Table 7.7 Dual-wavelength Ti:Sapphire laser system sign of sign[−1 (τ )] ' & V +ω ·U τ sign[Fω ] sign[−1 (τ )] sign + ω + ω · τ τ

2

P ±

±

+

±

∓

–

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

4 

B2·k+1 · ω2·k+1

k=0

We check the sign of −1 (τ ) according to the following rule (Table 7.7). If, sign[−1 (τ )] > 0, then the crossing proceeds from (−) to (+) respectively (stable to unstable). If, sign[−1 (τ )] < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

7.2 Dual-Wavelength Emission from Vertical External-Cavity Surface-Emitting Laser Stability Analysis Under Delay Parameters in Time A vertical external-cavity surface-emitting laser (VECSEL) is a semiconductor laser based on a surface emitting semiconductor gain chip and a laser resonator. VECSELs generate high optical power in diffraction limited beams and have wavelength versatility. VECSEL unit is constructed from a semiconductor gain chip and an external laser resonator and additionally there are arrangements for pumping and cooling. The semiconductor device, gain chip contains a single semiconductor Bragg mirror and the active/gain region with several quantum wells (QWs). The semiconductor structure total thickness is around few micrometers and it is amounted on heat sink substrate. The laser resonator and external mirror is one functional VECSEL unit. The semiconductor chip laser mode size is defined by the external resonator. The external resonator is folded with flat or curved mirrors and additional optical elements. The additional optical elements can be optical filters (single frequency operation, wavelength tuning), a nonlinear crystal (intra-cavity frequency doubling), saturable absorber (passive mode locking). The VECSEL resonator can be monolithic with a micro lens. The monolithic resonator is being contacted with the gain chip on one side and has an output coupler mirror coating on the other surface [4]. The schematic structure of a VECSEL with a semiconductor gain chip and an external laser resonator is described (Fig. 7.3).

764

7 Dual-Wavelength Laser Systems Stability Analysis … Emied Laser beam

Metal contact p-type

Output light beam

Upper Bragg reflector Quantum well

n-type n-substrate

Acon region With quantum wells

Lower Bragg reflector

Bragg reflector

Metal contact

Fig. 7.3 Schematic structure of a VECSEL with a semiconductor gain chip and an external laser resonator

A VECSEL thin film Bragg reflector (Bragg mirror) is constructed from a multilayer-stack of alternate high-index and low-index films, which characterized by one quarter wavelength thick. The geometrical structure thickness of the high index and low index films are t H = 4·nλ H ; t L = 4·nλ L , respectively. n H and n L are the indices of refraction of high and low index films, respectively and λ is the center wavelength of the Bragg mirror. On every interface in the stack a part of the incident beam is reflected. There is a phase shift of 180° which related to reflected parts if the incident light goes from low index medium in a high index medium. The relative phase difference of all reflected beams is zero or a multiple of 360° then they interfere constructively. The intensity of the incident light beam is decreases during his travel through the quarter-wave stack and at the same time the reflected light increases, if the absorptance A of the stack is negligible (Fig. 7.4). Incident light=Reflected light, combinaon of N beams n=1 in air nH substrate – high index nL substrate – low index nH substrate – high index nL substrate – low index nH substrate – high index ns substrate Transmied light Fig. 7.4 VECSEL thin film Bragg reflector with multilayer-stack

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

765

The lasing occurs in VECSEL quantum well. Quantum well is hetero structure in which a thin layer of one semiconductor is sandwiched between two layers of a different semiconductor material by forming a heterojunction. The demand is that the two semiconductors have different energy gaps and different refractive indices (VECSEL). We choose the material those electrons available for conduction in the middle layer has lower energy than those in the outer layers, creating energy well (dip) that confines the electrons in the middle layer. Differences in the energy gap permit spatial confinement of electrons and holes injected into the middle layer. A quantum well laser is a laser diode unit in which the active region of the device is narrow that quantum confinement occurs. The wavelength of the light emitted by a quantum well laser is determined by the thickness of the active region rather than the bandgap of the material. Shorter wavelengths are obtained from quantum well lasers compare to conventional laser diode by using a particular semiconductor material. The efficiency of a quantum well laser is greater than a conventional laser diode due to the stepwise form of its density of states function. A VECSEL can be designed for simultaneous dual wavelength emission. The operation mechanism is inhomogeneous optical pumping of the non-identical quantum wells of the active region. The simultaneous dual-wavelength emission VECSEL is characterized by rate equations and exhibits a strong time-delayed feedback. VECSEL tendency to self-pulsation due to Q-switching at high pump power is avoidable by accurate modeling and epitaxial growth of the active region of the laser. VECSELs have features as circular nondiffraction limited beams, high output power, and intra cavity nonlinear frequency conversion. Many industrial applications need laser source which emits coaxial beam at two different wavelengths. The two-color laser is applied to dual-wavelength interferometry, and generation of middle infrared and far-infrared radiation. Dualwavelength lasing source systems utilize an array of lasers with physically separated gain media for each wavelength. We can get dual-wavelength emission with a wideband semiconductor amplifier and with a super luminescent diode located inside an external cavity containing optical grating and a slit mirror. The term “optical grating” is synonym with diffraction grating or multiple slit. Optical gratings are periodical structures for diffraction of light. Other relevant devices are characterized by vertical coupled-cavity geometry. Simultaneous dual-wavelength emissions from optically pumped VECSEL can gives a single transverse mode continues wave (CW) at specific two different wavelengths. The simultaneous dual-wavelength VECSEL demonstrated an inhomogeneous pumping of the active region containing non-identical quantum wells (QWs). A single-cavity quantum well (QW) hetero structure is designed for dual-wavelength emission. The active region is divided into three sections by blocking layers. We enhance the achievable gain coefficient by placing the QWs at the antinodes of their own standing waves. Quantum well L and S, QWL,S are located corresponds to the standing wave at λ L ,S . QWL,S are near the nodes of the cavity mode of λ S to minimize optical absorption of short-wavelength emission in the deeper QWs. The blocking layers are constructed from wide bandgap AlAs, are transparent to the pump light and laser emission, but impenetrable for

766

7 Dual-Wavelength Laser Systems Stability Analysis …

charge carriers. The distributed Bragg reflector (DBR), made of alternating GaAs and AlAs layers, and the window layer is the same of a conventional VECSEL at these wavelengths. The field distribution and geometry of the device are accepted by solving the boundary eigenvalue problem for the Helmholts equation within the onedimensional transfer matrix. The VECSEL pump beam is incident to the absorber region through the output window layer. A one-dimensional diffusion equation is solved with the assumption of no electric bias applied to VECSEL, and then there is no carrier drift contribution. The carrier distribution in the absorber region is charac2 terized by the second order differential equation: n = n(x), Da · dd xn2 − τn + G = 0; where n is the carrier density, Da is the ambipolar diffusion coefficient,√and τ is the ambipolar carrier lifetime. It is related to the diffusion length by L a = Da · τ , due Pin α·λ p · π·r 2 · e−α·x = to spontaneous recombination. The carrier generation rate is G = h·c Pin α·λ p −α·x G0 · e ; G 0 = h·c · π·r 2 ; where Pin is the pump power at the onset (x = 0) of the absorber region, r is the spot radius, λ p is the pump wavelength, α is the absorption coefficient, h is Planck’s constant, and c is the speed of light in vacuum. The general solution we get is n(X ) = C1 · sinh(X )+ C2 · cosh(X ) + n G · e−a·X , where X = Lxa ; a = α · L a . And The general solution is consistent within a barrier region between two quantum wells, QWs or between a QW and a blocking layer [4]. We can characterize dual-wavelength VECSEL by rate equations. The laser consists of two cavities, a sub-cavity with a low reflectivity mirror (close to zero) and an external cavity with a high feedback mirror (close to one). The sub-cavity is formed between the semiconductor DBR (Distributed Bragg Reflector) and the top of the diamond heat spreader in spite of its Anti-Reflect (AR) coated surface. By inspecting the rate equations for dual-wavelength emission, we reduce the number of rate equations. It is done by the assumption that the gain coefficient of the quantum wells (QWs) of each section is lumped to a single “equivalent” quantum well (QW) per section. Parameter n QW in the QWs within each section is constant with high probably. The outcome is that the optical field confinement factor of the equivalent QW equals the number of the QWs in that section times the confinement factor of one QW. The whole active region contains three equivalents QWs. Minority (very small) influence is related to the contributions of spontaneous emission to the lasing modes and effects or random generation-recombination process. The rate equations for dual-wavelength emission are: ⎤ ⎡ 3  " " 1 d S1 = vg · ⎣ 1 j · g1 j − αin + · ln(r D B R (λ S ) · "re f (λ S )")⎦ · S1 dt L in j=1 ⎤ ⎡ 3  " " d S2 1 = vg · ⎣ 2 j · g2 j − αin + · ln(r D B R (λ L ) · "re f (λ L )")⎦ · S2 dt L in j=1 S1 and S2 are the photon densities at wavelength λ S and λ L , respectively, i j is the confinement factor of the ith optical field (i = 1 for λ S and i = 2 for λ L ) in the jth equivalent QW, gi j is the gain coefficient for the jth QW at λ S and λ L , αin is the loss

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

767

factor inside the sub cavity, vg is the group velocity of optical fields, L in is the sub cavity length, r D B R (r D B R (λ S ), r D B R (λ L )) is the reflectivity of the DBR mirror, and re f (re f (λ S ), re f (λ L )) is the effective reflectivity of the external mirror and defined as ( ∞  S(t − m · τext ) − j·θm 2 m−1 ·e (r f · rext ) · re f = r f − rext · (1 − r f ) · S(t) m=1 r f is the sub cavity facet reflectivity and rext is the external mirror reflectivity, τext is the round-trip time in the external cavity, and θm , θm = m · ω · τext + ϕ(t) −ϕ(t − m · τext ), where ω is the angular frequency and ϕ is the time-varying phase deviation from the steady state of the solitary laser [4].  The rate equations for  dual-wavelength

emission with high feedback reflectivity χ = rext ·

(1−r 2f ) rf

 1 , are

⎡ ⎤   3  d S1 (t − τ ) 1 S 1 ext ⎦ · S1 (t); S1 = S1 (t) = vg · ⎣ 1 j · g1 j − αs1 + · ln dt 2 · L S (t) in 1 j=1 ⎡ ⎤   3  d S2 (t − τ ) 1 S 2 ext ⎦ · S2 (t); S2 = S2 (t) = vg · ⎣ 2 j · g2 j − αs2 + · ln dt 2 · L S (t) in 2 j=1 αs1 and αs2 are the total loss coefficients, αs1,s2 = αin − ( L1in ) · ln[rext · (1 − r 2f ) · r D B R (λ S,L )]. At fixed points ddtS1 = 0; ddtS2 = 0; lim t→∞ S1 (t − τext ) = S1 (t) ; limt→∞ S2 (t − τext ) = S2 (t) ∀ t  τext . ⎡ 3  1 j · g1 j − αs1 + vg · ⎣ j=1



vg = 0; ln

S1∗ S1∗



= ln(1) = 0

⎡ 3  vg · ⎣ 2 j · g2 j − αs2 + j=1



S∗ vg = 0; ln 2∗ S2

⎤  ∗ 1 S1 ⎦ ∗ · ln ∗ · S1 = 0 2 · L in S1



1 · ln 2 · L in



⎤

 S2∗ ⎦ S2∗

· S2∗ = 0

= ln(1) = 0

We get the system fixed points equations: ⎡ ⎤ 3 3   ⎣ 1 j · g1 j − αs1 ⎦ · S1∗ = 0 ⇒ 1 j · g1 j = αs1 ; S1∗ = 0 j=1

j=1

768

7 Dual-Wavelength Laser Systems Stability Analysis …

⎡ ⎣

3 

⎤ 2 j · g2 j − αs2 ⎦ · S2∗ = 0 ⇒

j=1

3 

2 j · g2 j = αs2 ; S2∗ = 0

j=1

 Remark: Special case is when 3j=1 1 j · g1 j = αs1 then S1∗ can be any value,  S1∗ > 0; S1∗ ∈ R or/and 3j=1 2 j · g2 j = αs2 then S2∗ can be any value, S2∗ > 0; S2∗ ∈ R. Dual-wavelength emission VECSEL system fixed point: E (∗) = (S1∗ , S2∗ ) = (0, 0). Assumption: τext is the round-trip time in the external cavity is not the same for the photon densities S1 and S2 . We consider τext−1 and τext−2 for photon densities emission with high S1 and S2 , respectively.  The rate equations for dual-wavelength 

feedback reflectivity χ = rext−1,2 ·

(1−r 2f ) rf

 1 , are

⎡ ⎤   3  d S1 (t − τ ) 1 S 1 ext−1 ⎦ · S1 (t) = vg · ⎣ 1 j · g1 j − αs1 + · ln dt 2 · L S (t) in 1 j=1 S1 = S1 (t) ⎡ ⎤   3  d S2 (t − τ ) 1 S 2 ext−2 ⎦ · S2 (t) = vg · ⎣ 2 j · g2 j − αs2 + · ln dt 2 · L S (t) in 2 j=1 S2 = S2 (t) Stability analysis: The standard local stability analysis about any one of the equilibrium point of the dual-wavelength emission VECSEL system consists in adding to coordinate [S1 , S2 ] arbitrarily small increments of exponential form [s1 , s2 ] · eλ·t and retaining the first order terms in S1 , S2 . The system of two homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into dual-wavelength emission VECSEL system rate equations (DDEs). The dual-wavelength emission VECSEL system equations fixed values with arbitrarily small increments of exponential form [s1 , s2 ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. [2, 3]. ( j)

( j)

S1 (t) = S1 + s1 · eλ·t ; S2 (t) = S2 + s2 · eλ·t ( j)

S1 (t − τext−1 ) = S1 + s1 · eλ·(t−τext−1 )

7.2 Dual-Wavelength Emission from Vertical External-Cavity … ( j)

S2 (t − τext−2 ) = S2 + s2 · eλ·(t−τext−2 ) ; d S2 (t) = s2 · λ · eλ·t dt

769

d S1 (t) = s1 · λ · eλ·t dt

We choose these expressions for ourselves S1 (t), S2 (t) as a small displacement [s1 , s2 ] from the dual-wavelength emission VECSEL system rate equations fixed ( j) ( j) points in time t = 0, S1 (t = 0) = S1 + s1 ; S2 (t = 0) = S2 + s2 . There are four possible cases: (1) τext−1 = 0, τext−2 = 0; (2) τext−1 = τ, τext−2 = 0; (3) τext−1 = 0, τext−2 = τ , (4) τext−1 = τ, τext−2 = τ . We analyze the second case, where τext−1 = τ, τext−2 = 0. emission with high feedback reflectivity  The rate equations for dual-wavelength  χ = rext−1,2 ·

(1−r 2f ) rf

 1 , are

⎡ ⎤   3  d S1 1 S1 (t − τ ) ⎦ = vg · ⎣ 1 j · g1 j − αs1 + · ln · S1 (t); S1 = S1 (t) dt 2 · L S1 (t) in j=1 ⎡ ⎤ 3  d S2 = vg · ⎣ 2 j · g2 j − αs2 ⎦ · S2 (t); S2 = S2 (t) dt j=1 Stability analysis: ⎡ ⎤   3  d S1 (t − τ ) 1 S 1 ⎦ · S1 (t) = vg · ⎣ 1 j · g1 j − αs1 + · ln dt 2 · L S (t) in 1 j=1 s1 · λ · eλ·t

⎡ 3  = vg · ⎣ 1 j · g1 j − αs1 j=1

) ( j) *% S1 + s1 · eλ·(t−τ ) 1 ( j) + · ln · (S1 + s1 · eλ·t ) ( j) 2 · L in S1 + s1 · eλ·t ) ( j) * ) ( j) * ( j) S1 + s1 · eλ·(t−τ ) S1 − s1 · eλ·t S1 + s1 · eλ·(t−τ ) = · ( j) ( j) ( j) S1 + s1 · eλ·t S1 + s1 · eλ·t S1 − s1 · eλ·t ( j)

= Assumption: s12 ≈ 0 then

( j)

( j)

{[S1 ]2 − S1 · s1 · eλ·t + s1 · eλ·(t−τ ) · S1 −s12 · eλ·(t−τ ) · eλ·t } ( j)

[S1 ]2 − s12 · e2·λ·t ( j)

S1 +s1 ·eλ·(t−τ ) ( j) S1 +s1 ·eλ·t

=

( j)

( j)

( j)

[S1 ]2 −S1 ·s1 ·eλ·t +s1 ·eλ·(t−τ ) ·S1 ( j) [S1 ]2

770

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

S1 + s1 · eλ·(t−τ ) ( j)

S1 + s1 · eλ·t

s1 · λ · eλ·t

( j)

=

( j)

( j)

[S1 ]2 − S1 · s1 · eλ·t + s1 · eλ·(t−τ ) · S1 ( j)

[S1 ]2 1 1 = 1 − ( j) · s1 · eλ·t + ( j) · s1 · eλ·(t−τ ) S1 S1 ⎡ 3  1 = vg · ⎣ 1 j · g1 j − αs1 + 2 · L in j=1 ) *% 1 1 λ·t λ·(t−τ ) · ln 1 − ( j) · s1 · e + ( j) · s1 · e S1 S1 ( j)

⎡ s1 · λ · eλ·t = vg · ⎣ $

· (S1 + s1 · eλ·t ) 3 

⎤

( j)

1 j · g1 j − αs1 ⎦ · S1

j=1

) *% 1 1 1 ( j) λ·t λ·(t−τ ) + vg · · ln 1 − ( j) · s1 · e + ( j) · s1 · e · S1 2 · L in S1 S1 ⎡ ⎤ 3  + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 · eλ·t $

j=1

) *% 1 1 1 λ·t λ·(t−τ ) + vg · · ln 1 − ( j) · s1 · e + ( j) · s1 · e · s1 · eλ·t 2 · L in S1 S1

At fixed points:  ∗ S 0; ln S1∗ = 0.

d S1 dt

= 0 ⇒ vg · [

3

j=1 1 j · g1 j − αs1 +

1 2·L in

· ln

 ∗ S1 S1∗

] ·S1∗ =

1

⎡ ⎡ ⎤ ⎤ 3 3   d S1 ( j) 1 j · g1 j − αs1 ⎦ · S1∗ = 0 ⇒ vg · ⎣ 1 j · g1 j − αs1 ⎦ · S1 = 0 ⇒ vg · ⎣ dt j=1 j=1 $ ) *% 1 1 1 ( j) λ·t λ·t λ·(t−τ ) s1 · λ · e = v g · · ln 1 − ( j) · s1 · e + ( j) · s1 · e · S1 2 · L in S1 S1 ⎡ ⎤ 3  + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 · eλ·t $

j=1

) *% 1 1 1 λ·t λ·(t−τ ) + vg · · ln 1 − ( j) · s1 · e + ( j) · s1 · e · s1 · eλ·t 2 · L in S1 S1

A=

1 ( j) S1

· s1 · eλ·t ; B =

1 ( j) S1

· s1 · eλ·(t−τ )

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

) ln 1 −

1 ( j)

S1

· s1 · e

λ·t

+

771

*

1 ( j)

S1

· s1 · e

λ·(t−τ )

= ln([1 − A] + B)

  B ln([1 − A] + B) = ln[1 − A] + ln 1 + 1− A $ % 1 ln[1 − A] = ln 1 − ( j) · s1 · eλ·t S1 ⎡ ⎤ 1 λ·(t−τ )   ( j) · s1 · e B S1 ⎦ = ln⎣1 + ln 1 + 1− A 1 − 1( j) · s1 · eλ·t S1

∞  A2 A3 An ln[1 − A] = − = −A − − − ··· n 2 3 n=1

ln[1 − A] = − =−

∞  An n n=1

1 ( j)

S1

· s1 · e

λ·t

−

[

1 ( j) S1

· s1 · eλ·t ]2 2

−

[

1 ( j) S1

· s1 · eλ·t ]3 3

− ···

Assumption: ∀ k ≥ 2; k ∈ N ∃ s1k → 0 ∞  1 An ≈ − ( j) · s1 · eλ·t n S1 n=1    n  ∞ B B 1 = ln 1 + (−1)n+1 · · 1− A n 1− A n=1  2  3 1 B B B 1 − · = + · − ··· 1− A 2 1− A 3 1− A ⎡ 1 ⎤ ⎡ ⎤ 1 λ·(t−τ ) λ·(t−τ ) 1 + 1( j) · s1 · eλ·t ( j) · s1 · e ( j) · s1 · e B S1 S1 S1 ⎦·⎣ ⎦ = =⎣ 1− A 1 − 1( j) · s1 · eλ·t 1 − 1( j) · s1 · eλ·t 1 + 1( j) · s1 · eλ·t S1 S1 S1  2 1 λ·(t−τ ) + 1( j) · s12 · eλ·(t−τ ) · eλ·t ( j) · s1 · e S1 S1 =  2 1 − 1( j) · s12 · e2·λ·t

ln[1 − A] = −

S1

Assumption: s12 → 0;  B k →0 1−A

B 1−A

≈

1 ( j) S1

· s1 · eλ·(t−τ ) ; ∀ k ≥ 2; k ∈ N ∃ s1k → 0 then

772

7 Dual-Wavelength Laser Systems Stability Analysis …

 2  3 B B 1 1 − · + · − ··· → 0 2 1− A 3 1− A   1 B ≈ ( j) · s1 · eλ·(t−τ ) ln 1 + 1− A S1 ln[1 − A] ≈ −

1 ( j)

S1

· s1 · eλ·t

) * 1 1 ln 1 − ( j) · s1 · eλ·t + ( j) · s1 · eλ·(t−τ ) = ln([1 − A] + B) S1 S1  = ln[1 − A] + ln 1 +

 B 1− A 1 1 ≈ − ( j) · s1 · eλ·t + ( j) · s1 · eλ·(t−τ ) S1 S1 1 ≈ ( j) · s1 · eλ·t · (e−λ·τ − 1) S1

Back to the original expression: $

) 1 1 · ln 1 − ( j) · s1 · eλ·t + 2 · L in S1 ⎡ ⎤ 3  + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 · eλ·t

s1 · λ · eλ·t = vg ·

$

1

*%

· s1 · eλ·(t−τ ) ( j)

S1

j=1

( j)

· S1

) *% 1 1 1 λ·t λ·(t−τ ) + vg · · ln 1 − ( j) · s1 · e + ( j) · s1 · e · s1 · eλ·t 2 · L in S1 S1

1 1 ( j) · ( j) · s1 · eλ·t · (e−λ·τ − 1) · S1 2 · L in S1 ⎡ ⎤ 3  + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 · eλ·t

s1 · λ · eλ·t = vg ·

j=1

+ vg ·

1 1 · · eλ·t · (e−λ·τ − 1) · s12 · eλ·t 2 · L in S1( j) 1 1 ( j) · · s1 · (e−λ·τ − 1) · S1 2 · L in S1( j) ⎡ ⎤ 3  + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1

s12 → 0 ⇒ s1 · λ = vg ·

j=1

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

773

⎤⎫ ⎡ 3 ⎬  1 · (e−λ·τ − 1) + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 − s1 · λ = 0 vg · ⎭ ⎩ 2 · L in j=1 ⎧ ⎨

⎤ ⎡ 3  d S2 = vg · ⎣ 2 j · g2 j − αs2 ⎦ · S2 (t); S2 = S2 (t) dt j=1 ⎡ ⎤ 3  ( j) s2 · λ · eλ·t = vg · ⎣ 2 j · g2 j − αs2 ⎦ · (S2 + s2 · eλ·t ) j=1

s2 · λ · eλ·t

⎡ ⎤ 3  ( j) = vg · ⎣ 2 j · g2 j − αs2 ⎦ · S2 j=1

⎡ ⎤ 3  + vg · ⎣ 2 j · g2 j − αs2 ⎦ · s2 · eλ·t j=1

At fixed points: vg · [

3 j=1

( j)

2 j · g2 j − αs2 ] · S2 = 0.

⎡ ⎡ ⎤ ⎤ 3 3   s2 · λ = v g · ⎣ 2 j · g2 j − αs2 ⎦ · s2 ⇒ vg · ⎣ 2 j · g2 j − αs2 ⎦ · s2 j=1

j=1

− s2 · λ = 0 The System arbitrarily small increments equations: ⎧ ⎨ ⎩

vg ·

1 · (e−λ·τ 2 · L in

⎤⎫ ⎡ 3 ⎬  − 1) + vg · ⎣ 1 j · g1 j − αs1 ⎦ · s1 − s1 · λ = 0 ⎭ j=1 ⎤

⎡ 3  vg · ⎣ 2 j · g2 j − αs2 ⎦ · s2 − s2 · λ = 0 j=1

The small increments Jacobian of our dual-wavelength emission VECSEL system is as follow:

774

7 Dual-Wavelength Laser Systems Stability Analysis …

! 1 det(A − λ · I ) = 0 ⇒ vg · · (e−λ·τ − 1) 2 · L in ⎫ ⎤ ⎡ 3 ⎬  +vg · ⎣ 1 j · g1 j − αs1 ⎦ − λ ⎭ j=1 ⎧ ⎫ ⎡ ⎤ 3 ⎨ ⎬  · vg · ⎣ 2 j · g2 j − αs2 ⎦ − λ = 0 ⎩ ⎭ j=1

⎫ ⎧ ⎡ ⎤ 3 ⎬ ⎨  1 · (e−λ·τ − 1) + vg · ⎣ 1 j · g1 j − αs1 ⎦ − λ vg · ⎭ ⎩ 2 · L in j=1 ⎧ ⎫ ⎤ ⎡ 3 ⎨ ⎬  · vg · ⎣ 2 j · g2 j − αs2 ⎦ − λ ⎩ ⎭ j=1 ⎡ ⎤ 3  1 = vg · · (e−λ·τ − 1) · vg · ⎣ 2 j · g2 j − αs2 ⎦ 2 · L in j=1 − vg ·

1 · (e−λ·τ − 1) · λ 2 · L in

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

775

⎤ ⎤ ⎡ ⎡ 3 3   + vg · ⎣ 1 j · g1 j − αs1 ⎦ · vg · ⎣ 2 j · g2 j − αs2 ⎦ j=1

j=1

⎡ ⎤ 3  − vg · ⎣ 1 j · g1 j − αs1 ⎦ · λ j=1

⎡ ⎤ 3  − vg · ⎣ 2 j · g2 j − αs2 ⎦ · λ + λ2 = 0 j=1

⎧ ⎨

⎫ ⎬ 1 · λ · e−λ·τ 2 · L in ⎭ ⎤ ⎡ ⎤ 3  ⎦·⎣ 2 j · g2 j − αs2 ⎦

⎡ ⎤ 3  ·⎣ 2 j · g2 j − αs2 ⎦ − vg ·

1 v2 · ⎩ g 2 · L in j=1 ⎡ 3  + vg2 · ⎣ 1 j · g1 j − αs1 −

+ vg · ⎡ −⎣

⎧ ⎨

j=1

⎡

1 −⎣ ⎩ 2 · L in

3 

2 j · g2 j

j=1

3 

1 2 · L in

⎤

j=1

1 j · g1 j − αs1 ⎦

j=1

⎤⎫ ⎬ − αs2 ⎦ · λ + λ2 = 0 ⎭

We define global parameters for simplicity: 0 = vg2 · 2·L1 in ·[ 1 1 = −vg · 2 · L in ⎡ 3  2 = vg2 · ⎣ 1 j · g1 j − αs1 − j=1

3 j=1

2 j · g2 j − αs2 ]

⎤ ⎡ ⎤ 3 1 ⎦ ⎣ 2 j · g2 j − αs2 ⎦ · 2 · L in j=1

⎧ ⎨

⎡ ⎤ ⎡ ⎤⎫ 3 3 ⎬   1 3 = v g · −⎣ 1 j · g1 j − αs1 ⎦ − ⎣ 2 j · g2 j − αs2 ⎦ ⎩ 2 · L in ⎭ j=1 j=1 2 + 3 · λ + λ2 + ( 0 + 1 · λ) · e−λ·τ = 0 det(A − λ · I ) = 2 + 3 · λ + λ2 + ( 0 + 1 · λ) · e−λ·τ D(λ, τ ) = 2 + 3 · λ + λ2 + ( 0 + 1 · λ) · e−λ·τ D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ n, m ∈ N0 ; n = 2; m = 1; n > m

776

7 Dual-Wavelength Laser Systems Stability Analysis …

Pn (λ, τ ) = 2 + 3 · λ + λ2 ; n = 2; Q m (λ, τ ) = 0 + 1 · λ; m = 1 Pn (λ, τ ) =

n=2 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2

k=0

⎡ 3  p0 (τ ) = 2 = vg2 · ⎣ 1 j · g1 j − αs1 − j=1

1 2 · L in

⎤ ⎡ ⎤ 3  ⎦·⎣ 2 j · g2 j − αs2 ⎦ j=1

p1 (τ ) = 3 ⎧ ⎡ ⎤ ⎡ ⎤⎫ 3 3 ⎨ 1 ⎬   = vg · −⎣ 1 j · g1 j − αs1 ⎦ − ⎣ 2 j · g2 j − αs2 ⎦ ⎩ 2 · L in ⎭ j=1

j=1

p2 (τ ) = 1 Q m (λ, τ ) =

m=1 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ

k=0

q0 (τ ) = 0 = vg2 ·

1 2 · L in

⎡ ⎤ 3  ·⎣ 2 j · g2 j − αs2 ⎦; q1 (τ ) = 1 = −vg · j=1

1 2 · L in

The homogeneous system for S1 , S2 leads to a characteristic equation for the eigenvalue λ having the form D(λ, τ ) = P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; and P(λ) = 2 1 j j j=0 a j · λ ; Q(λ) = j=0 c j · λ . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delay τ. qi , qk are any Dual-wavelength emission VECSEL system’s parameters, other parameters kept as a constant [2, 3]. ⎡ 3  a0 = 2 = vg2 · ⎣ 1 j · g1 j − αs1 − j=1

⎤ ⎡ ⎤ 3 1 ⎦ ⎣ · 2 j · g2 j − αs2 ⎦ 2 · L in j=1

⎧ ⎨

⎡ ⎤ ⎡ ⎤⎫ 3 3 ⎬   1 a 1 = 3 = v g · −⎣ 1 j · g1 j − αs1 ⎦ − ⎣ 2 j · g2 j − αs2 ⎦ ⎩ 2 · L in ⎭ j=1 j=1 a2 = 1 ⎤ ⎡ 3  1 1 ·⎣ 2 j · g2 j − αs2 ⎦; c1 = 1 = −vg · c0 = 0 = vg2 · 2 · L in 2 · L in j=1

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

777

Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang and Cong 2005) [2] can also be verified in the present case: 1. If λ " = i" · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. " " is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. 2. If " Q(λ) P(λ) " 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω:Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and Pn (λ = i · ω, τ ) = p0 (τ ) + p1 (τ ) · i · ω − p2 (τ ) · ω2 = p0 (τ ) − p2 (τ ) · ω2 + p1 (τ ) · i · ω Pn (λ = i · ω, τ ) = 2 − ω2 + 3 · ω · i Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω Q m (λ = i · ω, τ ) = 0 + 1 · ω · i Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = q0 (τ ) + p0 (τ ) − p2 (τ ) · ω2 + [ p1 (τ ) + q1 (τ )] · i · ω = 0 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 + 2 − ω2 + [ 3 + 1 ] · i · ω = 0 |P(i · ω, τ )|2 = ( 2 − ω2 )2 + ( 3 · ω)2 = 22 + ( 23 − 2 · 2 ) · ω2 + ω4 ; |Q(i · ω, τ )|2 = 20 + 21 · ω2 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = 22 − 20 + ( 23 − 2 · 2 − 21 ) · ω2 + ω4 We define the following parameters for simplicity: 0 , 2 , 4 0 = 22 − 20 ; 2 = 23 − 2 · 2 − 21 ; 4 = 1

778

7 Dual-Wavelength Laser Systems Stability Analysis …

 Hence F(ω, τ ) = 0 implies 2k=0 2·k ·ω2·k = 0 and its roots are given by solving the above polynomial. PR (iω, τ ) = 2 − ω2 ; PI (iω, τ ) = 3 · ω; Q R (iω, τ ) = 0 ; Q I (iω, τ ) = 1 · ω sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

We use different terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n, m ∈ N0 ; n = 2; m = 1; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ); P(λ) =

2 

a j · λ j ; Q(λ) =

j=0

1 

cj · λj

j=0

P(λ) = a0 + a1 · λ + a2 · λ2 ; Q(λ) = c0 + c1 · λ n, m ∈ N0 ; n > m and a j , c j :R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “—” denoted complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ. The coefficients a j (vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . .) ∈ R and c j (vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . .) ∈ R depend on Dual-wavelength emission VECSEL system’s parameters, vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 [2, 3]. a0 + c0 = 2 + 0 ⎡ 3  = vg2 · ⎣ 1 j · g1 j − αs1 − j=1

−αs2 ] + vg2 ·

1 2 · L in

1 2 · L in

1 + 2 · L in



j=1

⎤ 1 ⎦ 2 · L in ⎤

⎡ 3  ·⎣ 2 j · g2 j − αs2 ⎦ j=1

j=1

⎡ ⎤ 3  ·⎣ 2 j · g2 j − αs2 ⎦

⎛⎡ 3  = vg2 · ⎝⎣ 1 j · g1 j − αs1 − j=1

⎤ ⎡ 3  ⎦·⎣ 2 j · g2 j

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

779

⎤ ⎡ ⎤ ⎡ 3 3   = vg2 · ⎣ 1 j · g1 j − αs1 ⎦ · ⎣ 2 j · g2 j − αs2 ⎦ = 0 j=1

j=1

∀ vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang 2, Sect. 3.4) can also be verified in the present case. 1. If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire "∀ vg , "1 j , αs1 , L in , 2 j , αs2 , τ, . . . domain of interest. " P(λ) " 2. " Q(λ) " is bounded for |λ| → ∞ ; Reλ ≥ 0. No roots bifurcation from ∞. Indeed, " " " " " " " c0 +c1 ·λ " = in the limit: " Q(λ) " " 2 P(λ) a0 +a1 ·λ+a2 ·λ ". 2 3. The following expressions exist: F(ω) = |P(i · ω)|2 . ·2 ω)| − |Q(i 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω4 ). 4. Each positive root ω(vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . .) of F(ω) = 0 is continuous and differentiable with respect to vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . and the condition can only be assessed numerically.

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . ., Reλ may, at the crossing, change its ( j) ( j) sign from (−) to (+). i.e. from a stable focus E ( j) = (S1 , S2 ); j = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . and any system parameters.  ∂Reλ ; 1 j , αs1 , L in , 2 j , αs2 , τ, . . . = const  (vg ) = ∂vg λ=i·ω   ∂Reλ −1 (1 j ) = ; vg , αs1 , L in , 2 j , αs2 , τ, . . . = const ∂1 j λ=i·ω   ∂Reλ −1  (L in ) = ; vg , 1 j , αs1 , 2 j , αs2 , τ, . . . = const ∂ L in λ=i·ω   ∂Reλ −1  (2 j ) = ; vg , 1 j , αs1 , L in , αs2 , τ, . . . = const ∂2 j λ=i·ω   ∂Reλ −1 (αs2 ) = ; vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . = const ∂αs2 λ=i·ω −1



780

7 Dual-Wavelength Laser Systems Stability Analysis …

−1



 (τ ) =

∂Reλ ∂τ

 λ=i·ω

; vg , 1 j , αs1 , L in , 2 j , αs2 , . . . = const

P(λ) = PR (λ) + i · PI (λ) ; Q(λ) = Q R (λ) + i · Q I (λ), When writing and inserting λ = i · ω into Dual-wavelength emission VECSEL system’s characteristic equation ω must satisfy the following equations: sin(ω · τ ) = g(ω) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos(ω · τ ) = h(ω) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

where |Q(iω, τ )|2 = 0 in view of requirement (1) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ. Now it is important to notice that if τ ∈ / I (assume that / I , ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ · ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) + 2 · n · π ∀ n ∈ N0 Hence we can define the maps: )+2·n·π ; n ∈ N0 ; τ ∈ I . τn : I → R+0 , is given by τn (τ ) = θ(τ ω(τ ) Let us introduce the function I → R; Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ. In the following, the subscripts λ, ω, vg , 1 j , αs1 , L in , 2 j , αs2 , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(vg , 1 j , αs1 , L in , 2 j , αs2 , . . .) and ω(vg , 1 j , αs1 , L in , 2 j , αs2 , . . .), and keeping all parameters except one (x) and τ. The derivation closely follows that in reference [BK]. Differentiating Dualwavelength emission VECSEL system’s characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = vg , 1 j , αs1 , L in , 2 j , αs2 , τ, . . . 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω = λ (i·ω,x)·Q(λ,x)−τ ·|P(i·ω,x)| ( i·Pω (i·ω,x)·P(i·ω,x)+i·Q ). Px (i·ω,x)·P(i·ω,x)−Q x (i·ω,x)·Q(i·ω,x) Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R +i · Q I and Pω = PRω +i · PI ω ; Q ω = Q Rω +i · Q I ω ; Px = PRx +i · PI x ; Q x = 2

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

781

Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any Dual-wavelength emission VECSEL system’s parameters vg , 1 j , αs1 , L in , 2 j , αs2 , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delay x = τ : PR = 2 − ω2 ; PI = 3 · ω; Q R = 0 ; Q I = 1 · ω; PRω = −2 · ω; PI ω = 3 ; Q I ω = 1 . Q Rω = 0; PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0 Fτ = 0; PR · PI τ − PI · PRτ = 0 Q R · Q I τ − Q I · Q Rτ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 Fω = . . . Elememts: PRω · PR = −2 · ω · ( 2 − ω2 ) = 2 · ω3 − 2 · ω · 2 PI ω · PI = 23 · ω; Q Rω · Q R = 0 Q I ω · Q I = 21 · ω U = . . . Elememts:

PR · PI ω = ( 2 − ω2 ) · 3 ; PI · PRω = −2 · 3 · ω2 Q R · Q I ω = 0 · 1 ; Q I · Q Rω = 0 We can summary our last results in the next table (Table 7.8). F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · = − FFωτ I ⇒ ∂ω ∂τ Table 7.8 Dual-wavelength emission VECSEL system’s stability analyses Fω , U elements

∂ω ∂τ

+ Fτ = 0; τ ∈

Fω , U elements

Expression

Fω , U elements

Expression

PRω · PR

2 · ω 3 − 2 · ω · 2

PR · PI ω

( 2 − ω2 ) · 3

PI ω · PI

23

PI · PRω

−2 · 3 · ω2

Q Rω · Q R

0

QR · QIω

0 · 1

QIω · QI

21

Q I · Q Rω

0

·ω ·ω

782

7 Dual-Wavelength Laser Systems Stability Analysis …

 ∂Reλ ∂ω Fτ = ωτ = − ; ∂τ λ=iω ∂τ Fω  ! −2 · [U + τ · |P|2 + i · Fω −1  (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ]   ! ∂Reλ sign{−1 (τ )} = sign ∂τ λ=iω + , V + ∂ω ·U ∂ω ∂τ −1 sign{ (τ )} = sign{Fω } · sign +ω+ ·τ ∂τ |P|2 −1 (τ ) =



We shall presently examine the possibility of stability transitions (bifurcations) Dual-wavelength emission VECSEL system, about the equilibrium points, E ( j) ; j = 0, 1, 2, . . ., ( j) ( j) E ( j) = (S1 , S2 ); j = 0, 1, 2. The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λplane, where by increasing the delay parameter τ. Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [2, 3]. −1 (τ ) =



∂Reλ ∂τ

 λ=i·ω

; vg , 1 j , αs1 , L in , 2 j , αs2 , . . . = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = ( 2 − ω2 ) · 3 + 2 · 3 · ω2 − 0 · 1 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = 2 · 3 − 0 · 1 + 3 · ω 2 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) =

1 

A2k · ω2·k

k=0

A 0 = 2 · 3 − 0 · 1 ; A 2 = 3 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · ( 23 − 2 · 2 − 21 ) · ω + 4 · ω3 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

1  k=0

B2·k+1 · ω2·k+1 ; B1 = 2 · ( 23 − 2 · 2 − 21 ); B3 = 4

7.2 Dual-Wavelength Emission from Vertical External-Cavity …

783

1 2·k+1 Then we get the expression for Fω = , Dual-wavelength k=0 B2·k+1 · ω emission VECSEL system parameter values. We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Dual-wavelength emission VECSEL system.   ! ∂Reλ −2 · [U + τ · |P|2 ] + i · Fω  (τ ) = = Re ∂τ λ=iω Fτ + 2i · [V + ω · |P|2 ]   ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = −1 (τ ) = ∂τ λ=iω Fτ2 + 4 · (V + ω · P 2 )2 

−1

The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all Dual-wavelength emission VECSEL system’s parameters are known and the stability switch due to various time delay values τ is described in the following expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ )  U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + |P(ω(τ ))|2 Remark: we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω . Next is to find those ω, τ values which fulfill sin θ (τ ) = . . . . and cos θ (τ ) = . . . .; sin(ω · τ ) = −PR ·Q I +PI ·Q R R +PI ·Q I ) |Q|2 = Q 2R + Q 2I . Finally we plot the ; cos(ω · τ ) = − (PR ·Q|Q| 2 |Q|2 stability switch diagram g(τ ) = −1 (τ ) = ( ∂Reλ ) ∂τ λ=iω 



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2 λ=iω    ∂Reλ −1 sign[g(τ )] = sign[ (τ )] = sign ∂τ λ=iω   2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2

g(τ ) = −1 (τ ) =

∂Reλ ∂τ

=

Fτ2 + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ )] = sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )]

784

7 Dual-Wavelength Laser Systems Stability Analysis …

Table 7.9 Dual-wavelength emission VECSEL system sign of sign[−1 (τ )]

sign[Fω ]

τ ·U sign[ V +ω + ω + ωτ · τ ] P2

sign[−1 (τ )]

±

±

+

±

∓

–

!   Fτ 2 2 sign[ (τ )] = sign [Fω ] · (V + ω · P ) − · (U + τ · P ) Fω  −1 Fτ ∂ω ∂ F/∂ω =− ωτ = − ; ωτ = Fω ∂τ ∂ F/∂τ !   V + ωτ · U + ω + ω · τ ; sign[P 2 ] > 0 sign[−1 (τ )] = sign [Fω ] · [P 2 ] · τ P2 !   V + ωτ · U −1 sign[ (τ )] = sign [Fω ] · + ω + ωτ · τ P2   V + ωτ · U + ω + ω · τ sign[−1 (τ )] = sign[Fω ] · sign τ P2 −1

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

1 

B2·k+1 · ω2·k+1

k=0

We check the sign of −1 (τ ) according to the following rule (Table 7.9). If, sign[−1 (τ )] > 0, then the crossing proceeds from (−) to (+) respectively (stable to unstable). If, sign[−1 (τ )] < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser Nonlinearity and Stability Analysis Er-doped fiber lasers (EDFLs) is a EDFA (Erbium-doped fiber amplifier) which operate in the particular regime where coherent oscillation of ASE occurs due to feedbacks mean. EDFLs are used as sources for coherent light signal generation, while EDFAs are used as wave-wave amplifiers for coherent light signal generation. EDFLs can be pumped with compact, efficient laser diodes. They are compatible with differ fibers and fiber optic components used in communications; they have a negligible coupling losses. Fiber wave guiding and splicing alleviate any mechanical alignment of parts and provide environment stability. The tunable EDFL configuration uses an all-fiber ring laser cavity. Wavelength selectivity is achieved by using

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser …

785

a tunable transmission filter. Basically Erbium (chemical symbol: Er) is a chemical element which belong to the group of metals. It is used also in the form of the trivalent ion Er3+ as the laser-active dopant of gain media based on various host materials (crystal and glasses). Erbium-doped glasses are mostly of silicate and phosphate types, are used both for bulk lasers and fiber lasers and amplifiers. Erbium-Doped Fiber Amplifier (EDFA) is an optical amplifier used in the C-band and L-band, where loss of telecom optical fibers becomes lowest in the entire optical telecommunication wavelength bands. EDFA is commonly used to compensate the loss of an optical fiber in long-distance optical communication. EDFA can amplify multiple optical signals simultaneously, and thus can be easily combined with WDM (wavelength division multiplexing) systems. EDFAs are used as a booster, inline, and pre-amplifier in an optical transmission line. The booster amplifier is placed after the transmitter to increase the optical power launched to the transmission line. The inline amplifiers are placed in the transmission line, compensating the attenuation induced by the optical fiber. The pre-amplifier is placed before the receiver, such that sufficient optical power is launched to the receiver. A typical distance between each of the EDFAs is several tens of kilometers. When an EDFA is pumped at 1480 nm, Er ion doped in the fiber absorbs the pump light and is excited to an excited state. There is a quasiperiodic route to chaos in erbium-doped fiber laser operating simultaneously at 1.55 and 1.536 μm. It is starting from a CW state for high pumping rates the system become T-periodic, 2T-periodic, 3T-periodic and chaotic for decreasing pumping ratios. The low-frequency spectra show two fundamental frequencies. The nT-periodic regimes correspond to a frequency locking of the low frequency on a subharmonic of the high frequency. The group of rare-earth doped fibers is focused on their potential applications as amplifiers or optical sources for telecommunication networks [5]. Commercially optical amplifier are at 1.55 μm and they demonstrated a large variety of dynamical behaviors including static and dynamics polarization effects, antiphase and chaotic dynamics under autonomous or pump modulation conditions. They are characterized by highly multimode operation due to the large inhomogeneous linewidth, and the fiber lasers exhibit cooperative effects between longitudinal modes leading to a simple modeling of the dynamics by low dimensional equations. Erbium-doped fiber laser (EDFL) is preferred because of its spontaneous different dynamical behaviors. EDFL can operate in a CW, sinusoidal or self-pulsing regime, depending on some parameters: the ion pairs concentration (x), the pumping ration (r = PPth ), and the photon lifetime in the cavity. The ion pair’s concentration is responsible for the dynamics. For low x-values, EDFL is always CW, whereas for large x, the laser is self-pulsing for any pumping ratio achievable experimentally and for intermediate x-values, the EDFL behavior changes continuously from selfpulsing to CW operation via a Hopf bifurcation. The ion clusters (ion pairs) are able to explain qualitatively the whole dynamics observed when operating around 1.55 μm. There is a antiphase and chaotic dynamics in EDFL having an ion pair concentration of about ≈ 18%, when operating simultaneously at 1.55 and 1.536 μm. Experimentally it is approve that quasiperiodic route to chaos in erbium-doped fiber laser operate simultaneously at 1.55 and 1.536 μm (Fig. 7.5).

786

7 Dual-Wavelength Laser Systems Stability Analysis … M1 Prism

Erbium-doped Fiber

Source Al2TiO3 Laser

M1 lens Mono Chromator λ=1.536 um

Mono Chromator λ=1.55 um

Ge

Ge

Scope

Fig. 7.5 Experimental set up system for two-wavelength operation

The experimental system includes the following elements: A Titane Sapphire laser which is used to optically pump the erbium-doped fiber. The output power is around 810 nm and about 800 mW which allows pumping ratios up to 20 times the threshold. The geometrical characteristics of the fiber are core diameter c , cutoff wavelength λc , and length (l). Typical doping concentrations are: 100 ppm Er3+ , 1000 ppm Al3+ and 145,000 ppm Ge4+ . The ion pair concentration is 7.5% (x  7.5%). The ion pair concentration is responsible of the pulsed behavior of erbium-doped fiber lasers. We do not operate the laser pumping at 514.5 nm to avoid the photochromism resulting from the high Ge4+ concentration. The prism is inside the cavity in order to allow a spectral tune-ability together with a spatial separation between the pump beam and the laser beam. The spectral tune-ability is achieved with a tilt of mirror M1. We get an oscillation between the mirror M1 (R = 100%@1.5 μm) and the mirror M2 (R = 80%@1.5 μm). The output fiber end is directly butted on the output mirror. A collimating lens is placed after the output mirror. The laser beam is then split by a non-polarizing cube. The signals are analyzed through mono chromators adjusted respectively at 1.536 and 1.55 μm and detected with two germanium photodetectors. The output signal from the scope is analyzed by appropriate PC [5]. The erbium laser model simulation is for the case of laser operation around 1.55 μm. The model is based on existence of two atomic systems: isolated ions and ion pairs coupled via the laser field. The diagram of the energy levels adopted for two systems (an isolated ion and an ion pair) is described (Fig. 7.6). An isolated ion is modeled as a two-level system while ion pairs are assumed three level systems [6]. The system rate equations described by four differential nonlinear coupled equations. dd =  − a2 · (1 + d) − 2 · I · d dt

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser … Fig. 7.6 Schematic energy levels for an isolated ion and an ion pair

787

4

+Λ

+Λ I13/2

2

22 σl

σl

12 σl

4 I15/2

1 -Λ

11 -Λ

dd+ a22 = a12 · (1 − d+ ) − · (d+ + d− ) + y · I · (2 − 3 · d+ ) dt 2 dd− a22 =  − a12 · (1 − d+ ) − · (d+ + d− ) + y · I · d− dt 2 dI = −I + (1 − 2 · x) · A · I · d + x · y · A · I · d− dt d = d(t); d+ = d+ (t); d− = d− (t); I = I (t) where d = n 2 − n 1 ; d± = n 22 ± n 11 ; a2 = ττ21 ; ai j = ττi1j ; A = σl · N0 · τ1 ; I, is the normalized laser intensity.  is the pumping rate, x is the ion pair concentration, y is the ratio between the absorption cross-sections of an ion pair and an isolated ion, n k is the population of level |k for the isolated ions and n kl is the population of level |kl for the ion pairs, N0 is the erbium concentration, τ1 is the photon lifetime and τk is the lifetime of level |k. Equation dd = . . ., describe the evolution of the dt population inversion for isolated ions. Equations dddt+ = . . . . and dddt− = . . . are associated respectively with the sum (difference) of the populations of the upper and lower levels of ion pairs. Equation ddtI = . . ., describe the field dynamics. At fixed points (equilibrium points), dd = 0; dddt+ = 0. dt dd− dI = 0; = 0,  − a2 · (1 + d ∗ ) − 2 · I ∗ · d ∗ = 0 dt dt a22 · (d+∗ + d−∗ ) + y · I ∗ · (2 − 3 · d+∗ ) = 0 a12 · (1 − d+∗ ) − 2 a22 · (d+∗ + d−∗ ) + y · I ∗ · d−∗ = 0 2 − I ∗ + (1 − 2 · x) · A · I ∗ · d ∗ + x · y · A · I ∗ · d−∗ = 0

 − a12 · (1 − d+∗ ) −

− I ∗ + (1 − 2 · x) · A · I ∗ · d ∗ + x · y · A · I ∗ · d−∗ = 0 ⇒ [−1 + (1 − 2 · x) · A · d ∗ + x · y · A · d−∗ ] · I ∗ = 0 Case A: I ∗ = 0 then

788

7 Dual-Wavelength Laser Systems Stability Analysis …

 − a2 · (1 + d ∗ ) = 0; a12 · (1 − d+∗ ) −  − a12 · (1 − d+∗ ) −

a22 · (d+∗ + d−∗ ) = 0 2

a22 · (d+∗ + d−∗ ) = 0 2

 − a2 · (1 + d ∗ ) = 0 ⇒ d ∗ =

 a22 − 1; · (d+∗ + d−∗ ) = a12 · (1 − d+∗ ) a2 2

a22 · (d+∗ + d−∗ ) =  − a12 · (1 − d+∗ ) 2 a12 · (1 − d+∗ ) =  − a12 · (1 − d+∗ ) ⇒ d+∗ = 1 − a12 a22 a12 =2· a22 a12 =2· a22 a12 =2· a22

d−∗ = 2 ·

d−∗ = 2 ·

( a2

 a12 a12 − 2· a22 a22

 2 · a12

· (1 − d+∗ ) − d+∗

a12 ∗ −2· · d − d+∗ a22 +   a12 − 2· + 1 · d+∗ a22     a12  − 2· +1 · 1− a22 2 · a12      1  1 = ·−1 +1 · 1− + 2 · a12 a22 2 · a12

First fixed point: E (0)  1 1 − 1, 1 − 2·a12 , ( a22 + 2·a12 ) ·  − 1, 0).

=

(d (0) , d+(0) , d−(0) , I (0) )

Case B: −1 + (1 − 2 · x) · A · d ∗ + x · y · A · d−∗ = 0 ⇒ d ∗ =

1−x·y·A·d−∗ (1−2·x)·A

 − a2 · (1 + d ∗ ) 2 · d∗  − a2 − a2 · d ∗ = 2 · d∗ 1 a2 = · ( − a2 ) − 2 · d∗ 2

 − a2 · (1 + d ∗ ) − 2 · I ∗ · d ∗ = 0 ⇒ I ∗ =

a22 · (d+∗ + d−∗ ) 1. a12 · (1 − d+∗ ) − 2    − a2 · (1 + d ∗ ) · (2 − 3 · d+∗ ) = 0 +y· 2 · d∗

a12 − a12 · d+∗ −

a22 ∗ a22 ∗ · d+ − · d− 2 2

=

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser …

 +y·

 a2 a2 − − 2 · d∗ 2 · d∗ 2

789



· (2 − 3 · d+∗ ) = 0

  a22 ∗ a22 ∗ 3· · d+ − · d− + y · a12 − a12 · − − · d∗ 2 2 d∗ 2 · d∗ +  a2 3 · a2 3 · a2 ∗ ∗ − ∗+ · d − a2 + · d+ = 0 d 2 · d∗ + 2 d+∗

a22 ∗ a22 ∗ y· y·3· ∗ · d+ − · d− + ∗ − · d+ 2 2 d 2 · d∗ y · a2 y · 3 · a2 ∗ y · 3 · a2 ∗ · d+ = 0 − + · d+ − y · a2 + ∗ ∗ d 2·d 2  a22 ∗ a22 ∗ 1 y·3· ∗ · d+ − · d− + ∗ · y ·  − · d+ a12 − a12 · d+∗ − 2 2 d 2  y · 3 · a2 ∗ y · 3 · a2 ∗ −y · a2 + · d+ − y · a2 + · d+ = 0 2 2   1 y·3· ∗ y · 3 · a2 ∗ · d+ + · d+ · y · ( − a2 ) − d∗ 2 2 y · 3 · a2 ∗ a22 ∗ a22 ∗ · d+ + · d− + · d+ + a12 · d+∗ − a12 = y · a2 − 2 2 2 a12 − a12 · d+∗ −

2 y · a2 − y·3·a · d+∗ + a222 · d−∗ + a222 · d+∗ + a12 · d+∗ − a12 1 2 = 2 d∗ y · ( − a2 ) − y·3· · d+∗ + y·3·a · d+∗ 2 2

d∗ =

y · a2 −

y·3· · d+∗ 2 a22 · d−∗ + a222 2

y·3·a2 · d+∗ 2 d+∗ + a12 · d+∗

y · ( − a2 ) −

+

· d+∗ +

·

y·3·a2 2

− a12

2 y · ( − a2 ) − y·3· · d+∗ + y·3·a · d+∗ 1 − x · y · A · d−∗ 2 2 = 2 (1 − 2 · x) · A y · a2 − y·3·a · d+∗ + a222 · d−∗ + a222 · d+∗ + a12 · d+∗ − a12 2   a22  − a2 · (1 + d ∗ ) ∗ ∗ ∗ · d−∗ = 0 · (d+ + d− ) + y · 2.  − a12 · (1 − d+ ) − 2 2 · d∗

 − a12 + a12 · d+∗ −

a22 ∗ a22 ∗ · d+ − · d− + y · 2 2



 a2 a2 − − 2 · d∗ 2 · d∗ 2

a22 ∗ a22 ∗  · d+ − · d− + y · · d∗ 2 2 2 · d∗ − a2 a2 ∗ · d− = 0 −y· · d−∗ − y · ∗ 2·d 2

 − a12 + a12 · d+∗ −



· d−∗ = 0

790

7 Dual-Wavelength Laser Systems Stability Analysis …

 a y 1 22 ∗ ∗ · (y · a − a · d+∗ −  + a12 · ( − a ) · d = + a ) · d + 2 2 22 12 − − 2 · d∗ 2 2 

∗ = 2 · d∗ · y · ( − a2 ) · d−

d∗ =

  1 ∗ + a22 − a ∗ · (y · a2 + a22 ) · d− 12 · d+ −  + a12 2 2

y · ( − a2 ) · d−∗

2 · d∗ ·

[ 21

· (y · a2 + a22 ) · d−∗ + ( a222 − a12 ) · d+∗ −  + a12 ]

y · ( − a2 ) · d−∗ 1 − x · y · A · d−∗ = (1 − 2 · x) · A 2 · d ∗ · [ 21 · (y · a2 + a22 ) · d−∗ + ( a222 − a12 ) · d+∗ −  + a12 ] We can summary our last results for second, third… Fixed points d+∗ and d−∗ coordinate 2 y · ( − a2 ) − y·3· · d+∗ + y·3·a · d+∗ 1 − x · y · A · d−∗ 2 2 = 2 (1 − 2 · x) · A y · a2 − y·3·a · d+∗ + a222 · d−∗ + a222 · d+∗ + a12 · d+∗ − a12 2

y · ( − a2 ) · d−∗ 1 − x · y · A · d−∗ = (1 − 2 · x) · A 2 · d ∗ · [ 21 · (y · a2 + a22 ) · d−∗ + ( a222 − a12 ) · d+∗ −  + a12 ] It is recommended to solve the above two equations for d+∗ and d−∗ coordinate numerically rather than analytic. Then we can get system second, third… fixed points. E (k) = (d (k) , d+(k) , d−(k) , I (k) )  1 − x · y · A · d−∗ ∗ ∗ 1 , d+ , d− , · ( = (1 − 2 · x) · A 2 · d∗ a2  ; k > 0; k ∈ N; k = 1, 2, 3, . . . −a2 ) − 2 E (k) = (d (k) , d+(k) , d−(k) , I (k) )   1 − x · y · A · d−∗ ∗ ∗ (1 − 2 · x) · A · ( − a2 ) a2 , d+ , d− , − = (1 − 2 · x) · A 2 · (1 − x · y · A · d−∗ ) 2 ∀ k > 0; k ∈ N; k = 1, 2, 3, . . . Stability analysis: The standard local stability analysis about any one of the equilibrium point of the Er-doped fiber laser (EDFL) system consists in adding to coordinate [d, d+ , d− , I ] arbitrarily small increments of exponential form [d, d+ , d− , I ] · eλ·t and retaining the first order terms in d, d+ , d− , I . The system of four homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into Er-doped fiber laser (EDFL) system equations. The Er-doped fiber laser (EDFL) system fixed values with arbitrarily small increments of exponential form [d, d+ , d− , I ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. [2, 3].

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser …

791

( j)

( j)

d(t) = d ( j) + d · eλ·t ; d+ (t) = d+ + d+ · eλ·t ; d− (t) = d− + d− · eλ·t I (t) = I ( j) + I · eλ·t dd(t) dd+ (t) dd− (t) = λ · d · eλ·t ; = λ · d+ · eλ·t ; = λ · d− · eλ·t dt dt dt d I (t) = λ · I · eλ·t dt dd =  − a2 · (1 + d) − 2 · I · d dt λ · d · eλ·t =  − a2 · (1 + d ( j) + d · eλ·t ) − 2 · (I ( j) + I · eλ·t ) · (d ( j) + d · eλ·t ) λ · d · eλ·t =  − (a2 + a2 · d ( j) + a2 · d · eλ·t ) − 2 · (I ( j) · d ( j) + I ( j) · d · eλ·t + d ( j) · I · eλ·t + I · d · eλ·t · eλ·t ) Assumption: I · d ≈ 0 λ · d · eλ·t =  − (a2 + a2 · d ( j) + a2 · d · eλ·t ) − 2 · (I ( j) · d ( j) + I ( j) · d · eλ·t + d ( j) · I · eλ·t ) λ · d · eλ·t =  − a2 · (1 + d ( j) ) − 2 · I ( j) · d ( j) − a2 · d · eλ·t − 2 · I ( j) · d · eλ·t − 2 · d ( j) · I · eλ·t At fixed points:  − a2 · (1 + d ( j) ) − 2 · I ( j) · d ( j) = 0; −(a2 + 2 · I ( j) ) · d − λ · d − 2 · d ( j) · I = 0. a22 dd+ = a12 · (1 − d+ ) − · (d+ + d− ) + y · I · (2 − 3 · d+ ) dt 2 ( j)

λ · d+ · eλ·t = a12 · (1 − [d+ + d+ · eλ·t ]) a22 ( j) ( j) · (d+ + d+ · eλ·t + d− + d− · eλ·t ) − 2 ( j) + y · (I ( j) + I · eλ·t ) · (2 − 3 · [d+ + d+ · eλ·t ]) ( j)

λ · d+ · eλ·t = a12 · (1 − d+ ) − a12 · d+ · eλ·t −

a22 ( j) ( j) · (d+ + d− ) 2

a22 · (d+ · eλ·t + d− · eλ·t ) 2 ( j) + y · (2 · I ( j) − 3 · I ( j) · [d+ + d+ · eλ·t ] + 2 · I · eλ·t −

792

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

− I · eλ·t · 3 · [d+ + d+ · eλ·t ]) ( j)

λ · d+ · eλ·t = a12 · (1 − d+ ) − a12 · d+ · eλ·t −

a22 ( j) ( j) · (d+ + d− ) 2

a22 · (d+ · eλ·t + d− · eλ·t ) 2 ( j) + y · (2 · I ( j) − 3 · I ( j) · d+ − 3 · I ( j) · d+ · eλ·t + 2 · I · eλ·t −

( j)

− I · eλ·t · 3 · d+ − I · d+ · eλ·t · 3 · eλ·t ) a22 ( j) ( j) ( j) · (d+ + d− ) + y · I ( j) · (2 − 3 · d+ ) 2 a22 · (d+ · eλ·t + d− · eλ·t ) + y · (2 · I − a12 · d+ · eλ·t − 2 ( j) − I · 3 · d+ − I · d+ · eλ·t · 3 − 3 · I ( j) · d+ ) · eλ·t ( j)

λ · d+ · eλ·t = a12 · (1 − d+ ) −

Assumption: I · d+ ≈ 0 a22 ( j) ( j) ( j) · (d+ + d− ) + y · I ( j) · (2 − 3 · d+ ) 2 a22 ( j) · (d+ + d− ) · eλ·t + y · (2 · I − I · 3 · d+ − a12 · d+ · eλ·t − 2 − 3 · I ( j) · d+ ) · eλ·t ( j)

λ · d+ · eλ·t = a12 · (1 − d+ ) −

( j)

At fixed points: a12 · (1 − d+ ) −

a22 2

( j)

( j)

( j)

· (d+ + d− ) + y · I ( j) · (2 − 3 · d+ ) = 0.

a22 ( j) · (d+ + d− ) + y · 2 · I − I · y · 3 · d+ 2 − 3 · y · I ( j) · d+ − λ · d+ = 0

− a12 · d+ −

−(a12 +

a22 a22 ( j) + 3 · y · I ( j) ) · d+ − λ · d+ − · d− + y · (2 − 3 · d+ ) · I = 0 2 2 dd− a22 =  − a12 · (1 − d+ ) − · (d+ + d− ) + y · I · d− dt 2 ( j)

λ · d− · eλ·t =  − a12 · (1 − [d+ + d+ · eλ·t ]) a22 ( j) ( j) · (d+ + d+ · eλ·t + d− + d− · eλ·t ) − 2 ( j) + y · (I ( j) + I · eλ·t ) · (d− + d− · eλ·t ) ( j)

λ · d− · eλ·t =  − a12 · (1 − d+ ) + a12 · d+ · eλ·t − −

a22 · (d+ + d− ) · eλ·t 2

a22 ( j) ( j) · (d+ + d− ) 2

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser … ( j)

793 ( j)

+ y · (I ( j) · d− + I ( j) · d− · eλ·t + d− · I · eλ·t + I · d− · eλ·t · eλ·t ) Assumption: I · d− ≈ 0 ( j)

λ · d− · eλ·t =  − a12 · (1 − d+ ) + a12 · d+ · eλ·t −

a22 ( j) ( j) · (d+ + d− ) 2

a22 · (d+ + d− ) · eλ·t 2 ( j) ( j) + y · I ( j) · d− + y · (I ( j) · d− + d− · I ) · eλ·t −

a22 ( j) ( j) · (d+ + d− ) 2 ( j) + y · I ( j) · d− + a12 · d+ · eλ·t a22 ( j) · (d+ + d− ) · eλ·t + y · (I ( j) · d− + d− · I ) · eλ·t − 2 ( j)

λ · d− · eλ·t =  − a12 · (1 − d+ ) −

( j)

At fixed points:  − a12 · (1 − d+ ) −

a22 2

( j)

( j)

( j)

· (d+ + d− ) + y · I ( j) · d− = 0.

a22 ( j) · (d+ + d− ) + y · (I ( j) · d− + d− · I ) 2   a22  a22  ( j) · d+ + y · I ( j) − · d− − λ · d− + y · d− · I = 0 a12 − 2 2 λ · d− = a12 · d+ −

dI = −I + (1 − 2 · x) · A · I · d + x · y · A · I · d− dt λ · I · eλ·t = −(I ( j) + I · eλ·t ) + (1 − 2 · x) · A · (I ( j) + I · eλ·t ) · (d ( j) + d · eλ·t ) ( j)

+ x · y · A · (I ( j) + I · eλ·t ) · (d− + d− · eλ·t ) λ · I · eλ·t = −(I ( j) + I · eλ·t ) + (1 − 2 · x) · A · (I ( j) · d ( j) + I ( j) · d · eλ·t + d ( j) · I · eλ·t + I · d · eλ·t · eλ·t ) ( j)

( j)

+ x · y · A · (I ( j) · d− + I ( j) · d− · eλ·t + d− · I · eλ·t + I · d− · eλ·t · eλ·t ) At fixed points: I · d ≈ 0; I · d− ≈ 0. λ · I · eλ·t = −(I ( j) + I · eλ·t ) + (1 − 2 · x) · A · (I ( j) · d ( j) ) + (1 − 2 · x) · A · (I ( j) · d + d ( j) · I ) · eλ·t ( j)

( j)

+ x · y · A · (I ( j) · d− ) + x · y · A · (I ( j) · d− + d− · I ) · eλ·t

794

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

λ · I · eλ·t = −I ( j) + (1 − 2 · x) · A · I ( j) · d ( j) + x · y · A · I ( j) · d− − I · eλ·t + (1 − 2 · x) · A · (I ( j) · d + d ( j) · I ) · eλ·t ( j)

+ x · y · A · (I ( j) · d− + d− · I ) · eλ·t ( j)

At fixed points: −I ( j) + (1 − 2 · x) · A · I ( j) · d ( j) + x · y · A · I ( j) · d− = 0. λ · I = −I + (1 − 2 · x) · A · (I ( j) · d + d ( j) · I ) ( j)

+ x · y · A · (I ( j) · d− + d− · I ) (1 − 2 · x) · A · I ( j) · d + x · y · A · I ( j) · d− + [(1 − 2 · x) · A · d ( j) ( j)

+ x · y · A · d− − 1] · I − λ · I = 0 We can summary our system four arbitrarily equations: −(a2 + 2 · I ( j) ) · d − λ · d − 2 · d ( j) · I = 0   a22 a22 + 3 · y · I ( j) · d+ − λ · d+ − · d− − a12 + 2 2 ( j) + y · (2 − 3 · d+ ) · I = 0   a22  a22  ( j) · d+ + y · I ( j) − · d− − λ · d− + y · d− · I = 0 a12 − 2 2 (1 − 2 · x) · A · I ( j) · d + x · y · A · I ( j) · d− + [(1 − 2 · x) · A · d ( j) ( j)

+ x · y · A · d− − 1] · I − λ · I = 0 We define for simplicity the following global parameters: ϒ1 = −(a2 + 2 · I ( j) )   a22 ( j) + 3 · y · I ( j) ; ϒ3 = y · (2 − 3 · d+ ) ϒ2 = − a12 + 2 a22 a22 ; ϒ5 = y · I ( j) − ϒ4 = a12 − 2 2 ϒ6 = (1 − 2 · x) · A · I ( j) ; ϒ7 = x · y · A · I ( j) ( j)

ϒ8 = (1 − 2 · x) · A · d ( j) + x · y · A · d− − 1 We can summary our system four arbitrarily equations (ϒk ; k = 1, 2, . . . , 8): ϒ1 · d − λ · d − 2 · d ( j) · I = 0; ϒ2 · d+ − λ · d+ −

a22 · d− + ϒ3 · I = 0 2

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser …

795 ( j)

ϒ4 · d+ + ϒ5 · d− − λ · d− + y · d− · I = 0 ϒ6 · d + ϒ7 · d− + ϒ8 · I − λ · I = 0 Er-doped fiber laser (EDFL) system arbitrarily increment matrix is ⎤ ⎡ d⎤ (ϒ1 − λ) 0 0 −2 · d ( j) ⎢d ⎥ ⎢ 0 (ϒ2 − λ) − a222 ϒ3 ⎥ +⎥ ⎥·⎢ ⎢ ⎢ ⎥=0 ( j) ⎦ ⎣ ⎣ d− ⎦ 0 ϒ4 (ϒ5 − λ) y · d− ϒ6 0 ϒ7 (ϒ8 − λ) I ⎡

⎡

⎤ (ϒ1 − λ) 0 0 −2 · d ( j) ⎢ 0 (ϒ2 − λ) − a222 ϒ3 ⎥ ⎥ A−λ· I =⎢ ( j) ⎣ 0 ϒ4 (ϒ5 − λ) y · d− ⎦ ϒ6 0 ϒ7 (ϒ8 − λ) det(A − λ · I ) = 0 ⎛

⎞ ϒ3 (ϒ2 − λ) − a222 ( j) det(A − λ · I ) = (ϒ1 − λ) · det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠ 0 ϒ7 (ϒ8 − λ) ⎛ ⎞ a22 0 (ϒ2 − λ) − 2 + 2 · d ( j) · det ⎝ 0 ϒ4 (ϒ5 − λ) ⎠ 0 ϒ7 ϒ6 ⎛

⎞ (ϒ2 − λ) − a222 ϒ3 ( j) Step 1: det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠. 0 ϒ7 (ϒ8 − λ) ⎛

⎞ ϒ3 (ϒ2 − λ) − a222 ( j) det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠ 0 ϒ7 (ϒ8 − λ)  ( j)  (ϒ5 − λ) y · d− = (ϒ2 − λ) · det ϒ7 (ϒ8 − λ)   a22 ϒ3 −2 − ϒ4 · det ϒ7 (ϒ8 − λ) ⎞ ⎛ ϒ3 (ϒ2 − λ) − a222 ( j) det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠ 0 ϒ7 (ϒ8 − λ) = (ϒ2 − λ) · [(ϒ5 − λ) · (ϒ8 − λ)

796

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

&a

22

· (ϒ8 − λ) + ϒ7 · ϒ3 2  ( j) = −λ3 + λ2 · (ϒ8 + ϒ2 + ϒ5 ) + λ · ϒ7 · y · d− a22  −ϒ8 · ϒ5 − ϒ2 · ϒ5 − ϒ2 · ϒ8 − ϒ4 · 2 ( j) + ϒ2 · ϒ8 · ϒ5 − ϒ2 · ϒ7 · y · d− a22 · ϒ8 + ϒ4 · ϒ7 · ϒ3 + ϒ4 · 2 − ϒ7 · y · d− ] + ϒ4 ·

'

We define for simplicity global parameters: 1 = ϒ8 + ϒ2 + ϒ5 ( j)

2 = ϒ7 · y · d− − ϒ8 · ϒ5 − ϒ2 · ϒ5 − ϒ2 · ϒ8 − ϒ4 · ( j)

3 = ϒ2 · ϒ8 · ϒ5 − ϒ2 · ϒ7 · y · d− + ϒ4 ·

a22 2

a22 · ϒ8 + ϒ4 · ϒ7 · ϒ3 2

⎛

⎞ ϒ3 (ϒ2 − λ) − a222 ( j) det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠ = −λ3 + λ2 · 1 + λ · 2 + 3 0 ϒ7 (ϒ8 − λ) ⎛

⎞ 0 (ϒ2 − λ) − a222 Step 2: det ⎝ 0 ϒ4 (ϒ5 − λ) ⎠. 0 ϒ7 ϒ6 ⎛

⎞   0 (ϒ2 − λ) − a222 (ϒ2 − λ) − a222 ⎝ ⎠ det 0 ϒ4 (ϒ5 − λ) = ϒ6 · det ϒ4 (ϒ5 − λ) 0 ϒ7 ϒ6 & a22 ' = ϒ6 · (ϒ2 − λ) · (ϒ5 − λ) + ϒ4 · 2 ⎛ ⎞ 0 (ϒ2 − λ) − a222 det ⎝ 0 ϒ4 (ϒ5 − λ) ⎠ = ϒ6 · λ2 − ϒ6 · (ϒ2 + ϒ5 ) · λ 0 ϒ7 ϒ6 a22 + ϒ6 · ϒ2 · ϒ5 + ϒ6 · ϒ4 · 2 a22 2

We define for simplicity global parameters: 4 = ϒ6 · (ϒ2 + ϒ5 ); 5 = ϒ6 · ϒ4 · + ϒ6 · ϒ2 · ϒ5 ⎛

⎞ 0 (ϒ2 − λ) − a222 det ⎝ 0 ϒ4 (ϒ5 − λ) ⎠ = ϒ6 · λ2 − 4 · λ + 5 0 ϒ7 ϒ6

7.3 A Quasi-periodic in Erbium-Doped Fiber Laser …

797

We can summary steps 1 and 2: ⎛

⎞ ϒ3 (ϒ2 − λ) − a222 ( j) det ⎝ ϒ4 (ϒ5 − λ) y · d− ⎠ = −λ3 + λ2 · 1 + λ · 2 + 3 0 ϒ7 (ϒ8 − λ) ⎛ ⎞ 0 (ϒ2 − λ) − a222 det ⎝ 0 ϒ4 (ϒ5 − λ) ⎠ = ϒ6 · λ2 − 4 · λ + 5 0 ϒ7 ϒ6 det(A − λ · I ) = (ϒ1 − λ) · (−λ3 + λ2 · 1 + λ · 2 + 3 ) + 2 · d ( j) · (ϒ6 · λ2 − 4 · λ + 5 ) det(A − λ · I ) = −λ3 · ϒ1 + λ2 · 1 · ϒ1 + λ · 2 · ϒ1 + 3 · ϒ1 + λ4 − λ3 · 1 − λ2 · 2 − λ · 3 + 2 · d ( j) · ϒ6 · λ2 − 2 · d ( j) · 4 · λ + 2 · d ( j) · 5 det(A − λ · I ) = λ4 − λ3 · (ϒ1 + 1 ) + λ2 · ( 1 · ϒ1 − 2 + 2 · d ( j) · ϒ6 ) + λ · ( 2 · ϒ1 − 3 − 2 · d ( j) · 4 ) + 3 · ϒ1 + 2 · d ( j) · 5 We define for simplicity global parameters: 4 = 1; 3 = −(ϒ1 + 1 ) 2 = 1 · ϒ1 − 2 + 2 · d ( j) · ϒ6 ; 1 = 2 · ϒ1 − 3 − 2 · d ( j) · 4 0 = 3 · ϒ1 + 2 · d ( j) · 5 det(A − λ · I ) = λ4 · 4 + λ3 · 3 + λ2 · 2 + λ · 1 + 0 =

4 

λk · k

k=0

Eigenvalue stability discussion: Our Er-doped fiber laser (EDFL) system involving N variables (N > 2, N = 4, arbitrarily small increments), the characteristic equation is of degree N = 4 and must often be solved numerically. Expect in some particular cases, such an equation has (N = 4) distinct roots that can be real or complex. These values are eigenvalues of the (4 × 4) Jacobian matrix (A). The general rule is that the Er-doped fiber laser (EDFL) system is stable if there is no eigenvalue with positive real part. It is sufficient that one eigenvalue is positive for the steady state to be unstable. Our four variables (d, d+ , d− , I ) system has four eigenvalues (four system’s arbitrarily small increments). The type of behavior can be characterized as a function of the position of these eigenvalues in the Re/Im plane. Five nongenerated cases can be distinguished: (1) the four eigenvalues are real and negative

798

7 Dual-Wavelength Laser Systems Stability Analysis …

(stable steady state), (2) the four eigenvalues are real, at least one of them is positive (unstable steady state), (3) and (4) three eigenvalues are complex conjugates with a negative real part and other eigenvalue real is negative (stable steady state), two cases can be distinguished depending on the relative value of the real part of the complex eigenvalues and of the real one, (5) three eigenvalues are complex conjugates with a negative real part and other eigenvalue real is positive (unstable steady state) [7, 8].

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser Analysis Under Delay Variables in Time A compact high power yellow pulsed laser use an intra cavity sum-frequency mixing in a diode-end-pumped Q-switched Nd:YVO4 dual-wavelength laser. A threemirror configuration forming two separate laser cavities is used to optimize the gain match for simultaneous dual-wavelength emission in Q-switched operation. A diodepumped Nd-doped lasers is a compact all-solid-state sources in the blue, green, and red spectral regions by use of intra cavity frequency doubling. These sources cover the region from 550 to 650 nm. The laser sources in the yellow-orange spectrum are used mainly for medical applications. The system consists of pulse dye lasers, copper-vapor lasers, and kryptonion lasers, on all-solid-state laser is another possible source. Krypton laser belongs to the gas lasers family, which use rare gases as the lasing medium. It is referred to as a krypton ion laser. The stimulated emission process in ion lasers occurs between the two energy states of the ion. Argon is one of the most common gases to be used in gas lasers, as it generates white light. The unique feature of krypton lasers is that with the use of proper mirrors it will lease on four sharp spectral lines of red, blue, yellow and green color. The krypton laser is capable of emitting lights of a number of wavelengths (as many as 10), the most significant one being the ones in the visible spectrum of the electromagnetic spectrum. The system is based on solid-state laser techniques for generating yellow lasers include sumfrequency mixing (SFM) of a flash-pumped Q-switched Nd:YAG dual-wavelength laser and frequency doubling of an intra cavity Raman-shifted Nd:YAG laser. A sum and difference frequency generation is a nonlinear process generating beams with the sum or difference of the frequencies of the input beams. Crystal materials lacking inversion symmetry exhibits χ (2) nonlinearity. By using a nonlinear crystal we get the sum frequency generation (SFG) or difference frequency generation (DFG), where two pump beams generate another beam with the sum or difference of the optical frequencies of the pump beams. A possible model of the self-sum frequency mixing (self-SFM) laser is generated by a single crystal and characterized by the spatial distribution of the pump and circulating fundamental lasers with arbitrary beam waists. The beam waist (or beam focus) of a laser beam is the location along the propagation direction where the beam radius has a minimum. The waist radius is the beam radius at this location. There are several ways to improve the self-SFM laser output. A radiation with a high frequency density is generated by sum frequency

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

799

mixing the output of two frequency modulated lasers. The radiation is useful for efficient excitation of a Doppler broadened atomic vapor. The Nd:YVO4 crystal is the preferred material for diode-pumped lasers with dual-wavelength emission. It is characterized by high absorption over a wide pump-wavelength bandwidth and a large stimulated-emission cross section at both 4 F3/2 –4 I11/2 and the 4 F3/2 –4 I13/2 transitions [9]. Neodymium Doped Yttrium Aluminum Garnet—Nd:YAG laser crystal is the most popular lasing medial for solid-state lasers. It possesses a combination of properties uniquely favorable for laser operations. There are Nd:YAG crystals with diffusion bonded undoped YAG endcaps. Nd:YAG crystals have a good fluorescence lifetime and thermal conductivity, as well as a robust nature. It is suitable for high-power continuous wave (CW), high-intensity Q-switched and single-mode operations. In Nd:YAG crystal, the ratio of the stimulated-emission cross section between the 4 F3/2 –4 I11/2 and the 4 F3/2 –4 I13/2 transitions is approximately 5/5.1, whereas this ratio is approximately 2/2.1 for Nd:YVO4 crystal. There is a compact, efficient scheme for generating a 593 nm laser based on intra-cavity SFM (sumfrequency mixing) of a diode-pumped Q-switched Nd:YVO4 dual-wavelength laser. There is a spatial and temporal overlap of the two different wavelengths in SFM. It is an overlapping collinear cavities for simultaneous 1064 and 1342 nm emission from a Q-switched Nd:YVO4 laser and empty an intra cavity β-barium borate crystal to obtain SFM 593 nm output. There is an option for Y cavity, Q-switched dualwavelength Nd:YAG lasing and offset of the open times of the Q switches for the different wavelength emissions and the lasers are characterized by relatively good temporal and spatial overlap between the two wavelengths. The schematic of intra cavity SFM in the diode-end-pumped Q-switched Nd:YVO4 dual-wavelength laser at 1064 and 1342 nm is presented (Fig. 7.7). AO, acousto-optic; HR, highly reflective; HT, highly transmitting; BBO, β-barium borate. The Nd3+ concentration of the laser crystal is 0.5 and its length is 6 mm. The pump source is 15 W fiber-coupled laser diode with a core diameter of 0.8 mm and a numerical aperture of 0.18. The fiber output is focused into the crystal, and the pump spot size is ~0.35 mm. The input mirror, M1, is a 1-m radius-of-curvature concave mirror with anti-reflection coating M2 Nd:YVO4

M3

BBO

AO Q-switch Coupling Lens Laser Diode

M1

Highly reflective (HR) at 1064nm and 1342nm, Highly transmission (HT) At 808nm

HR at 1064nm HR at 1342nm HT at 1064nm

Fig. 7.7 Schematic of intra cavity SFM in the diode-end-pumped Q-switched Nd:YVO4 dualwavelength laser at 1064 and 1342 nm

800

7 Dual-Wavelength Laser Systems Stability Analysis …

at the pump wavelength on the entrance face (R < 0.2%), high-reflection coating at both lasing wavelengths (R > 99.8%), and high-transmission coating at the pump wavelength on the other surface (T > 90%). The mirror M1 is not optimum because of limited mirror availability. The optimum mirror M1 has a high-reflection coating at 593 nm. One side of flat mirror M2 is coated to be highly reflecting at 1342 nm (R > 99.8%) and highly transmitting at 1064 nm (T > 95%). The other side of mirror M2 is anti-reflective at 1064 nm (R < 0.2%). Flat mirror M3 has one side coated to be highly reflecting at 1064 nm and highly transmitting at 593 nm (T > 90%). The other side of mirror M3 is anti-reflective at 593 nm. The 20 mm long Q switcher has anti-reflection coatings at 1064 and 1342 nm on both faces and is driven at a 41 MHz center frequency with 3 W of RF power. The cavity length between M1 and M2 is 7 cm for 1342 nm oscillation. The cavity length between M1 and M3 is 15 cm for 1064 nm emission. The dynamical model of our system described the behavior of a dual-wavelength Q-switched laser. The model is presented by three delay differential equations (DDEs). N dN = R p − c · N · [σ1 · φ1 (t − τ1 ) + σ2 · φ2 (t − τ2 )] − dt τf N = N (t); φ1 = φ1 (t); φ2 = φ2 (t) dφ1 φ1 (t − τ1 ) lcr · c · σ1 · N − − η S F M · φ1 (t − τ1 ) · φ2 (t − τ2 ) = dt l1 τc1 lcr dφ2 1 = · c · σ2 · φ2 (t − τ2 ) · N − · φ2 (t − τ2 ) dt l2 τc2 − η S F M · φ1 (t − τ1 ) · φ2 (t − τ2 ) 2 

σk · φk (t − τk ) = σ1 · φ1 (t − τ1 ) + σ2 · φ2 (t − τ2 )

k=1 2 -

φk (t − τk ) = φ1 (t − τ1 ) · φ2 (t − τ2 )

k=1

We can write our Diode-pumped Q-switched Nd:YVO4 yellow laser system DDEs: $ 2 %  dN N = Rp − c · N · σk · φk (t − τk ) − dt τ f k=1 2 lcr dφ1 φ1 (t − τ1 ) = · c · σ1 · N − − ηS F M · φk (t − τk ) dt l1 τc1 k=1

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

801

Table 7.10 Diode-pumped Q-switched Nd:YVO4 yellow laser system parameters for λ1 = 1342 nm and λ2 = 1064 nm Parameter @ λ1 = 1342 nm

Parameter @ λ2 = 1064 nm

Description

φ1

φ2

Photon intensity Stimulated-emission cross section

σ1

σ2

l1

l2

Cavity length

N

N

Population inversion density

Rp

Rp

Average pump intensity

τf

τf

Emission lifetime

lcr

lcr

Crystal length

τc1

τc2

Effective photon decay time as a result of all the linear losses

ηS F M

ηS F M

Effective conversion rate as a result of the intra cavity SFM (sum-frequency mixing)

2 lcr dφ2 1 = · c · σ2 · φ2 (t − τ2 ) · N − · φ2 (t − τ2 ) − η S F M · φk (t − τk ) dt l2 τc2 k=1

The parameters of our system are described in the next table (Table 7.10), where the subscripts 1 and 2 for λ1 = 1342 nm and λ2 = 1064 nm, respectively. c is the speed of light in vacuum. The photon densities at time φ1 (t) and φ2 (t) for λ1 = 1342 nm and λ2 = 1064 nm, respectively is shifted in time φ1 (t) → φ1 (t −τ1 ), 1 2 , dφ . The shifting in φ2 (t) → φ2 (t − τ2 ) but it is not affect the derivatives in time dφ dt dt time (τ1 and τ2 for λ1 = 1342 nm and λ2 = 1064 nm, respectively) of these variables is due to parasitic effects in our system. The optimum temporal overlap between the 1 2 = dφ . When σl11 = σl22 and τc1 = τc2 then two wavelengths is happened when dφ dt dt dφ1 2 = dφ , parameter τc is varied by the fine adjustment of the cavity alignment. dt dt The cavity lengths satisfy the relationship σσ21 = ll21 for optimum temporal overlap. By increasing the cavity length we lower the average photon density. The result is that the competing ability of the transition with larger emission cross section (σ ) is reduced. 1 2 = 0; dφ = 0; lim φ1 (t − τ1 ) = φ1 (t) ∀ t  τ1 . At fixed points ddtN = 0; dφ dt dt t→∞

lim φ1 (t − τ2 ) = φ2 (t) ∀ t  τ2 ; N = N (t); φ1 = φ1 (t); φ2 = φ2 (t)

t→∞

dN N∗ = 0 ⇒ R p − c · N ∗ · [σ1 · φ1∗ + σ2 · φ2∗ ] − =0 dt τf   1 ⇒ R p − N ∗ · c · [σ1 · φ1∗ + σ2 · φ2∗ ] + =0 τf

802

7 Dual-Wavelength Laser Systems Stability Analysis …

  Rp 1 ∗ ∗ N · c · [σ1 · φ1 + σ2 · φ2 ] + = Rp ⇒ N ∗ = τf c · [σ1 · φ1∗ + σ2 · φ2∗ ] + ∗

1 τf

lcr dφ1 φ∗ =0⇒ · c · σ1 · N ∗ − 1 − η S F M · φ1∗ · φ2∗ = 0 dt l1 τc1 $ % Rp lcr φ∗ · c · σ1 · − 1 − η S F M · φ1∗ · φ2∗ = 0 1 ∗ ∗ l1 τc1 c · [σ1 · φ1 + σ2 · φ2 ] + τ f lcr dφ2 1 =0⇒ · c · σ2 · φ2∗ · N ∗ − · φ ∗ − η S F M · φ1∗ · φ2∗ = 0 dt l2 τc2 2 $ % Rp lcr 1 ∗ · c · σ2 · φ2 · · φ ∗ − η S F M · φ1∗ · φ2∗ = 0 − l2 τc2 2 c · [σ1 · φ1∗ + σ2 · φ2∗ ] + τ1f We get two fixed points coordinate equations: (φ1∗ , φ2∗ ). $ % Rp lcr φ1∗ 1. · c · σ1 · − η S F M · φ1∗ · φ2∗ = 0 − l1 τc1 c · [σ1 · φ1∗ + σ2 · φ2∗ ] + τ1f $ % Rp lcr ∗ 2. · c · σ2 · φ2 · l2 c · [σ1 · φ1∗ + σ2 · φ2∗ ] + τ1f −

1 · φ ∗ − η S F M · φ1∗ · φ2∗ = 0 τc2 2 )

φ2∗ ·

$ Rp lcr · c · σ2 · ∗ l2 c · [σ1 · φ1 + σ2 · φ2∗ ] +

% 1 τf

1 − η S F M · φ1∗ − τc2

* =0

Case A: φ2∗ = 0 $ % Rp lcr φ∗ · c · σ1 · − 1 − η S F M · φ1∗ · φ2∗ |φ2∗ =0 1 ∗ ∗ l1 τc1 c · [σ1 · φ1 + σ2 · φ2 ] + τ f $ % Rp lcr φ1∗ = · c · σ1 · =0 − l1 τc1 c · σ1 · φ1∗ + τ1f $ % Rp lcr φ1∗ · c · σ1 · =0 − l1 τc1 c · σ1 · φ1∗ + τ1f

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

803

  · c · σ1 · R p · τc1 − c · σ1 · φ1∗ + τ1f · φ1∗   ⇒ =0 c · σ1 · φ1∗ + τ1f · τc1   1 · τc1 = 0 ⇒ τc1 = 0 and c · σ1 · φ1∗ + τf 1 1 c · σ1 · φ1∗ + = 0 ⇒ φ1∗ = − τf c · σ1 · τ f   lcr 1 · φ1∗ = 0 · c · σ1 · R p · τc1 − c · σ1 · φ1∗ + l1 τf   lcr 1 · c · σ1 · R p · τc1 − c · σ1 · [φ1∗ ]2 + · φ1∗ = 0 l1 τf lcr l1

1 lcr c · σ1 · [φ1∗ ]2 + · φ1∗ − · c · σ1 · R p · τc1 = 0 τf l1 . − τ1f ± τ12 + 4 · llcr1 · c2 · σ12 · R p · τc1 f ∗ φ1 = 2 · c · σ1 Case

A1 : φ1∗

=

/ − τ1 + f

1 τ 2f

+4· llcr ·c2 ·σ12 ·R p ·τc1 1

;

2·c·σ1

N∗ = =

c · [σ1 ·

φ1∗

⎡ c · σ1 · ⎣

N ∗ |φ2∗ =0 = =

First fixed point: E (0) (N (0) , φ1(0) , φ2(0) )

− 2·τ1 f 1 τf

+

+ .

1 2

1 τ 2f

Rp + σ2 · φ2∗ ] +

Rp c · σ1 · φ1∗ +

=

.

− τ1 f

·

lcr l1

· c2 · σ12 · R p · τc1 >

| ∗ 1 φ2 =0 τf

1 τf / +

Rp 1 τ 2f

+4· llcr ·c2 ·σ12 ·R p ·τc1 1

2·c·σ1

.

+4·

⎤ ⎦+

Rp 1 τ 2f

+4·

lcr l1

· c2 · σ12 · R p · τc1 +

2 · Rp 1 τ 2f

1 τf

+4·

lcr l1

· c2 · σ12 · R p · τc1

1 τf

1 τf

804

7 Dual-Wavelength Laser Systems Stability Analysis … ⎛ ⎜ =⎜ ⎝

Case

1 τf

+

/

A2 : φ1∗

2 · Rp +4·

1 τ 2f

=

lcr l1

/ − τ1 − f

· c2 · σ12 · R p · τc1 1 τ 2f

,

+4· llcr ·c2 ·σ12 ·R p ·τc1 1

− τ1f +

/

1 τ 2f

+4·

lcr l1

· c2 · σ12 · R p · τc1

2 · c · σ1

⎞ ⎟ , 0⎟ ⎠

; φ1∗ < 0.

2·c·σ1

Remark: It is a mathematical fixed point, since Photon intensity cannot be negative. N ∗ |φ2∗ =0 =

− 2·τ1 f

=

1 τf

−

− .

1 2

·

.

Rp 1 τ 2f

+4·

lcr l1

· c2 · σ12 · R p · τc1 +

1 τf

2 · Rp 1 τ 2f

+4·

· c2 · σ12 · R p · τc1

lcr l1

Second fixed point: (1)

(1)

E (1) (N (1) , φ1 , φ2 ) ⎛ ⎜ =⎜ ⎝

1 τf

−

Case B:

lcr l2

/

2 · Rp 1 τ 2f

+4·

lcr l1

· c2 · σ12 · R p · τc1

/

1 τ 2f

+4·

lcr l1

· c2 · σ12 · R p · τc1

2 · c · σ1

⎞ ⎟ , 0⎟ ⎠



 · c · σ2 ·

,

− τ1f −

Rp c·[σ1 ·φ1∗ +σ2 ·φ2∗ ]+ τ1

f

−

1 τc2

− η S F M · φ1∗ = 0

Rp lcr 1  = η S F M · φ1∗ + · c · σ2 ·  l2 τc2 c · [σ1 · φ1∗ + σ2 · φ2∗ ] + τ1f     lcr 1 1 · c · [σ1 · φ1∗ + σ2 · φ2∗ ] + ⇒ · c · σ2 · R p = η S F M · φ1∗ + l2 τc2 τf      lcr 1 1 · c · σ1 · φ1∗ + + c · σ2 · φ2∗ · c · σ2 · R p = η S F M · φ1∗ + l2 τc2 τf     lcr 1 1 ∗ ∗ · c · σ1 · φ1 + · c · σ2 · R p = η S F M · φ1 + l2 τc2 τf   1 · c · σ2 · φ2∗ + η S F M · φ1∗ + τc2   1 lcr η S F M · φ1∗ + · c · σ2 · φ2∗ = · c · σ2 · R p τc2 l2

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

φ2∗ |CaseB

=

lcr l2

805

    1 1 · c · σ1 · φ1∗ + − η S F M · φ1∗ + τc2 τf     · c · σ2 · R p − η S F M · φ1∗ + τ1c2 · c · σ1 · φ1∗ + τ1f   η S F M · φ1∗ + τ1c2 · c · σ2

Rp dN = 0 ⇒ N∗ = ∗ dt c · [σ1 · φ1 + σ2 · φ2∗ ] + N∗ = N∗ =

c · [σ1 ·

φ1∗

Rp + σ2 · φ2∗ |CaseB ] +

1 τf

1 τf

Rp   ! lcr 1 1 ∗ ∗ ·c·σ ·R −(η 2 p S F M ·φ1 + τ )·(c·σ1 ·φ1 + τ ) l c2 f + c · σ1 · φ1∗ + σ2 · 2 (η ·φ ∗ + 1 )·c·σ SF M

1

τc2

2

1 τf

dφ1 φ∗ lcr · c · σ1 · N ∗ − 1 − η S F M · φ1∗ · φ2∗ |Case B = 0 =0⇒ dt l1 τc1 lcr φ∗ · c · σ1 · N ∗ − 1 − η S F M · φ1∗ l1 τc1    ⎧ l cr ⎨ l · c · σ2 · R p − η S F M · φ1∗ + τ1 · c · σ1 · φ1∗ + 2 c2   · ⎩ 1 η S F M · φ1∗ + τc2 · c · σ2

1 τf

⎫ ⎬ ⎭

=0

We get two fixed points coordinate equations: (φ1∗ , N ∗ ), it is recommended to solve it numerically to find the third and fourth fixed coordinates φ1∗ , N ∗ . Then find φ2∗ |Case B . Stability analysis: The standard local stability analysis about any one of the equilibrium point of the Diode-pumped Q-switched Nd:YVO4 yellow laser system consists in adding to coordinate [N , φ1 , φ2 ] arbitrarily small increments of exponential form [n, φ1 , φ2 ] · eλ·t and retaining the first order terms in N , φ1 , φ2 . The system of three homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into Diode-pumped Q-switched Nd:YVO4 yellow laser system equations. The Diode-pumped Q-switched Nd:YVO4 yellow laser system fixed values with arbitrarily small increments of exponential form [n, φ1 , φ2 ] · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. [2, 3]. ( j)

N (t) = N ( j) + n · eλ·t ; φ1 (t) = φ1 + φ1 · eλ·t ( j)

( j)

φ2 (t) = φ2 + φ2 · eλ·t ; φ1 (t − τ1 ) = φ1 + φ1 · eλ·(t−τ1 )

806

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

φ2 (t − τ2 ) = φ2 + φ2 · eλ·(t−τ2 ) ;

d N (t) = n · λ · eλ·t dt

dφ1 (t) dφ2 (t) = φ1 · λ · eλ·t ; = φ2 · λ · eλ·t dt dt

We choose these expressions for ourselves N (t), φ1 (t), φ2 (t) as a small displacement [n, φ1 , φ2 ] from the Diode-pumped Q-switched Nd:YVO4 yellow laser system ( j) fixed points in time t = 0. N (t = 0) = N ( j) + n; φ1 (t = 0) = φ1 + φ1 ; φ2 (t = ( j) 0) = φ2 + φ2 . We have three possible cases: (1) τ1 = τ ; τ2 = 0 (2) τ1 = 0; τ2 = τ (3) τ1 = τ ; τ2 = τ , and we choose to analyze the first case, τ1 = τ ; τ2 = 0. N dN = R p − c · N · [σ1 · φ1 (t − τ1 ) + σ2 · φ2 (t − τ2 )] − dt τf ( j)

n · λ · eλ·t = R p − c · (N ( j) + n · eλ·t ) · [σ1 · (φ1 + φ1 · eλ·(t−τ ) ) ( j)

+ σ2 · (φ2 + φ2 · eλ·t )] −

N ( j) + n · eλ·t τf ( j)

n · λ · eλ·t = R p − c · [σ1 · (N ( j) + n · eλ·t ) · (φ1 + φ1 · eλ·(t−τ ) ) ( j)

+ σ2 · (N ( j) + n · eλ·t ) · (φ2 + φ2 · eλ·t )] −

N ( j) n · eλ·t − τf τf

( j)

( j)

n · λ · eλ·t = R p − c · [σ1 · (N ( j) · φ1 + N ( j) · φ1 · eλ·(t−τ ) + φ1 · n · eλ·t + n · φ1 · eλ·t · eλ·(t−τ ) ) ( j)

( j)

+ σ2 · (N ( j) · φ2 + N ( j) · φ2 · eλ·t + φ2 · n · eλ·t + n · φ2 · eλ·t · eλ·t )] −

N ( j) 1 −n· · eλ·t τf τf

Assumption: n · φ1 ≈ 0; n · φ2 ≈ 0 ( j)

( j)

n · λ · eλ·t = R p − c · [σ1 · N ( j) · φ1 + σ1 · (N ( j) · φ1 · eλ·(t−τ ) + φ1 · n · eλ·t ) ( j)

+ σ2 · N ( j) · φ2 + σ2 · (N ( j) · φ2 · eλ·t ( j)

+ φ2 · n · eλ·t )] −

N ( j) 1 −n· · eλ·t τf τf ( j)

( j)

n · λ · eλ·t = R p − c · N ( j) · [σ1 · φ1 + σ2 · φ2 ] −

N ( j) τf

( j)

− [c · σ1 · (N ( j) · φ1 · eλ·(t−τ ) + φ1 · n · eλ·t ) ( j)

+ c · σ2 · (N ( j) · φ2 · eλ·t + φ2 · n · eλ·t )] − n ·

1 · eλ·t τf

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser … dN dt

At fixed point:

807

( j)

( j)

= 0 ⇒ R p − c · N ( j) · [σ1 · φ1 + σ2 · φ2 ] −

N ( j) τf

=0

( j)

n · λ · eλ·t = −[c · σ1 · N ( j) · φ1 · eλ·(t−τ ) + c · σ1 · φ1 · n · eλ·t ( j)

+ c · σ2 · N ( j) · φ2 · eλ·t + c · σ2 · φ2 · n · eλ·t ] − n ·

1 · eλ·t τf

( j)

− c · σ1 · N ( j) · eλ·(t−τ ) · φ1 − c · σ1 · φ1 · eλ·t · n − c · σ2 · N ( j) · eλ·t · φ2 1 ( j) − c · σ2 · φ2 · eλ·t · n − · eλ·t · n − n · λ · eλ·t = 0 τf   1 ( j) ( j) − c · [σ1 · φ1 + σ2 · φ2 ] + · n − n · λ − c · σ1 · N ( j) · e−λ·τ · φ1 τf − c · σ2 · N ( j) · φ2 = 0 lcr φ1 (t − τ1 ) dφ1 = · c · σ1 · N − − η S F M · φ1 (t − τ1 ) · φ2 (t − τ2 ) dt l1 τc1 ( j)

lcr (φ + φ1 · eλ·(t−τ ) ) · c · σ1 · (N ( j) + n · eλ·t ) − 1 l1 τc1

φ1 · λ · eλ·t =

( j)

( j)

− η S F M · (φ1 + φ1 · eλ·(t−τ ) ) · (φ2 + φ2 · eλ·t ) φ1 · λ · eλ·t =

( j)

lcr lcr φ φ1 · eλ·(t−τ ) · c · σ1 · N ( j) + n · · c · σ1 · eλ·t − 1 − l1 l1 τc1 τc1 ( j)

( j)

( j)

( j)

− η S F M · (φ1 · φ2 + φ1 · φ2 · eλ·t + φ2 · φ1 · eλ·(t−τ ) + φ1 · φ2 · eλ·t · eλ·(t−τ ) ) Assumption: φ1 · φ2 ≈ 0 ( j)

lcr lcr φ · c · σ1 · N ( j) + n · · c · σ1 · eλ·t − 1 l1 l1 τc1 φ1 · eλ·(t−τ ) ( j) ( j) ( j) − − η S F M · (φ1 · φ2 + φ1 · φ2 · eλ·t τc1

φ1 · λ · eλ·t =

( j)

+ φ2 · φ1 · eλ·(t−τ ) ) ( j)

lcr φ ( j) ( j) · c · σ1 · N ( j) − 1 − η S F M · φ1 · φ2 l1 τc1 lcr φ1 · eλ·(t−τ ) +n· · c · σ1 · eλ·t − l1 τc1

φ1 · λ · eλ·t =

( j)

( j)

− η S F M · (φ1 · φ2 · eλ·t + φ2 · φ1 · eλ·(t−τ ) )

808

7 Dual-Wavelength Laser Systems Stability Analysis …

At fixed points:

lcr l1

· c · σ1 · N ( j) −

φ1 · λ · eλ·t = n ·

( j)

φ1 τc1

( j)

( j)

− η S F M · φ1 · φ2 = 0.

lcr φ1 · eλ·t · e−λ·τ · c · σ1 · eλ·t − l1 τc1 ( j)

( j)

− η S F M · (φ1 · φ2 + φ2 · φ1 · e−λ·τ ) · eλ·t lcr φ1 · e−λ·τ ( j) ( j) · c · σ1 − − η S F M · φ1 · φ2 − η S F M · φ2 · φ1 · e−λ·τ l1 τc1 − φ1 · λ = 0

n·

lcr · c · σ1 · n − l1



 1 ( j) · e−λ·τ · φ1 − λ · φ1 + η S F M · φ2 τc1

( j)

− η S F M · φ1 · φ2 = 0 lcr dφ2 1 = · c · σ2 · φ2 (t − τ2 ) · N − · φ2 (t − τ2 ) dt l2 τc2 − η S F M · φ1 (t − τ1 ) · φ2 (t − τ2 ) lcr ( j) · c · σ2 · (φ2 + φ2 · eλ·t ) · (N ( j) + n · eλ·t ) l2 1 ( j) − · (φ2 + φ2 · eλ·t ) τc2

φ2 · λ · eλ·t =

( j)

( j)

− η S F M · (φ1 + φ1 · eλ·(t−τ ) ) · (φ2 + φ2 · eλ·t ) lcr ( j) ( j) · c · σ2 · (φ2 · N ( j) + φ2 · n · eλ·t l2 + N ( j) · φ2 · eλ·t + φ2 · n · eλ·t · eλ·t ) 1 1 ( j) ( j) ( j) − · φ2 − · φ2 · eλ·t − η S F M · (φ1 · φ2 τc2 τc2

φ2 · λ · eλ·t =

( j)

( j)

+ φ1 · φ2 · eλ·t + φ2 · φ1 · eλ·(t−τ ) + φ1 · φ2 · eλ·t · eλ·(t−τ ) ) At fixed points: φ2 · n ≈ 0; φ1 · φ2 ≈ 0. lcr ( j) ( j) · c · σ2 · (φ2 · N ( j) + φ2 · n · eλ·t + N ( j) · φ2 · eλ·t ) l2 1 1 ( j) ·φ − · φ2 · eλ·t − τc2 2 τc2

φ2 · λ · eλ·t =

( j)

( j)

( j)

( j)

− η S F M · (φ1 · φ2 + φ1 · φ2 · eλ·t + φ2 · φ1 · eλ·(t−τ ) )

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

809

lcr ( j) · c · σ2 · φ2 · N ( j) l2 lcr lcr ( j) + · c · σ2 · φ2 · n · eλ·t + · c · σ2 · N ( j) · φ2 · eλ·t l2 l2 1 1 ( j) ( j) ( j) − ·φ − · φ2 · eλ·t − η S F M · φ1 · φ2 τc2 2 τc2

φ2 · λ · eλ·t =

( j)

( j)

− η S F M · (φ1 · φ2 · eλ·t + φ2 · φ1 · eλ·(t−τ ) ) lcr 1 ( j) ( j) ( j) ( j) · c · σ2 · φ2 · N ( j) − · φ − η S F M · φ1 · φ2 l2 τc2 2 lcr ( j) + · c · σ2 · φ2 · n · eλ·t l2 lcr 1 + · c · σ2 · N ( j) · φ2 · eλ·t − · φ2 · eλ·t l2 τc2

φ2 · λ · eλ·t =

( j)

( j)

− η S F M · φ1 · φ2 · eλ·t − η S F M · φ2 · φ1 · eλ·(t−τ ) At fixed points:

lcr l2

( j)

· c · σ2 · φ2 · N ( j) −

1 τc2

( j)

( j)

( j)

· φ2 − η S F M · φ1 · φ2 = 0.

lcr lcr ( j) · c · σ2 · φ2 · n · eλ·t + · c · σ2 · N ( j) · φ2 · eλ·t l2 l2 1 ( j) − · φ2 · eλ·t − η S F M · φ1 · φ2 · eλ·t τc2 ( j)

− η S F M · φ2 · φ1 · eλ·t · e−λ·τ − φ2 · λ · eλ·t = 0 lcr ( j) ( j) · c · σ2 · φ2 · n − η S F M · φ2 · e−λ·τ · φ1 l2   lcr 1 ( j) · φ2 − φ2 · λ = 0 + · c · σ2 · N ( j) − − η S F M · φ1 l2 τc2 We can summary our Diode-pumped Q-switched Nd:YVO4 yellow laser system arbitrarily small increments equations:   1 ( j) ( j) · n − n · λ − c · σ1 · N ( j) · e−λ·τ · φ1 − c · [σ1 · φ1 + σ2 · φ2 ] + τf − c · σ2 · N ( j) · φ2 = 0 lcr · c · σ1 · n − l1



 1 ( j) · e−λ·τ · φ1 + η S F M · φ2 τc1 ( j)

− λ · φ1 − η S F M · φ 1 · φ 2 = 0

810

7 Dual-Wavelength Laser Systems Stability Analysis …

lcr ( j) ( j) · c · σ2 · φ2 · n − η S F M · φ2 · e−λ·τ · φ1 l2   lcr 1 ( j) · φ2 − φ2 · λ = 0 + · c · σ2 · N ( j) − − η S F M · φ1 l2 τc2  We define for simplicity ( j) ( j) − c · [σ1 · φ1 + σ2 · φ2 ] + τ1f

some

global

parameters:

2 = c · σ1 · N ( j) ; 3 = c · σ2 · N ( j) ; 4 = 5 =

1

lcr · c · σ1 l1

1 ( j) ( j) + η S F M · φ2 ; 6 = η S F M · φ1 τc1 lcr ( j) ( j) · c · σ2 · φ2 ; 8 = η S F M · φ2 l2 lcr 1 ( j) 9 = · c · σ2 · N ( j) − − η S F M · φ1 l2 τc2 7 =

1 · n − n · λ − 2 · e−λ·τ · φ1 − 3 · φ2 = 0 4 · n − 5 · e−λ·τ · φ1 − λ · φ1 − 6 · φ2 = 0 7 · n − 8 · e−λ·τ · φ1 + 9 · φ2 − φ2 · λ = 0 ⎞ ⎛ n⎞ − 2 · e−λ·τ − 3 ( 1 − λ) ⎜ ⎟ ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ · ⎝ φ1 ⎠ = 0 7 − 8 · e−λ·τ ( 9 − λ) φ2 ⎛

⎛

⎞ ( 1 − λ) − 2 · e−λ·τ − 3 A − λ · I = ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠; det(A − λ · I ) = 0 7 − 8 · e−λ·τ ( 9 − λ) ⎞ ⎛ − 2 · e−λ·τ − 3 ( 1 − λ) det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ)   −( 5 · e−λ·τ + λ) − 6 = ( 1 − λ) · det − 8 · e−λ·τ ( 9 − λ)   4 − 6 + 2 · e−λ·τ · det 7 ( 9 − λ)   4 −( 5 · e−λ·τ + λ) − 3 · det − 8 · e−λ·τ 7

=

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

811

⎛

⎞ − 2 · e−λ·τ − 3 ( 1 − λ) det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = ( 1 − λ) · [−( 5 · e−λ·τ + λ) · ( 9 − λ) − 8 · e−λ·τ · 6 ] + 2 · e−λ·τ · [ 4 · ( 9 − λ) + 7 · 6 ] − 3 · [− 8 · 4 · e−λ·τ + 7 · ( 5 · e−λ·τ + λ)] ⎛

⎞ ( 1 − λ) − 2 · e−λ·τ − 3 det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = ( 1 − λ) · [(− 5 · 9 − 8 · 6 + 5 · λ) · e−λ·τ − λ · 9 + λ2 ] + 2 · e−λ·τ · [( 4 · 9 + 7 · 6 ) − 4 · λ] − 3 · [( 7 · 5 − 8 · 4 ) · e−λ·τ + 7 · λ] ⎛

⎞ ( 1 − λ) − 2 · e−λ·τ − 3 det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = 1 · (− 5 · 9 − 8 · 6 + 5 · λ) · e−λ·τ − λ · 1 · 9 + 1 · λ2 − (− 5 · 9 − 8 · 6 + 5 · λ) · λ · e−λ·τ + λ2 · 9 − λ3 + ( 4 · 9 + 7 · 6 ) · 2 · e−λ·τ − 4 · 2 · λ · e−λ·τ − 3 · ( 7 · 5 − 8 · 4 ) · e−λ·τ − 3 · 7 · λ ⎛

⎞ ( 1 − λ) − 2 · e−λ·τ − 3 det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = (− 1 · 5 · 9 − 1 · 8 · 6 + 1 · 5 · λ) · e−λ·τ − λ · 1 · 9 + 1 · λ2 + 5 · 9 · λ · e−λ·τ + 8 · 6 · λ · e−λ·τ − 5 · λ2 · e−λ·τ + λ2 · 9 − λ3 + ( 4 · 9 + 7 · 6 ) · 2 · e−λ·τ − 4 · 2 · λ · e−λ·τ − 3 · ( 7 · 5 − 8 · 4 ) · e−λ·τ − 3 · 7 · λ ⎛

⎞ ( 1 − λ) − 2 · e−λ·τ − 3 det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = [− 1 · 5 · 9 − 1 · 8 · 6 + ( 4 · 9 + 7 · 6 ) · 2

812

7 Dual-Wavelength Laser Systems Stability Analysis …

− 3 · ( 7 · 5 − 8 · 4 ) + ( 1 · 5 + 5 · 9 + 8 · 6 − 4 · 2 ) · λ − 5 · λ2 ] · e−λ·τ − ( 1 · 9 + 3 · 7 ) · λ + ( 1 + 9 ) · λ2 − λ3 We define for simplicity some global parameters: ϒ1 = − 1 · 5 · 9 − 1 · 8 · 6 + ( 4 · 9 + 7 · 6 ) · 2 − 3 · ( 7 · 5 − 8 · 4 ) ϒ2 = 1 · 5 + 5 · 9 + 8 · 6 − 4 · 2 ; ϒ3 = − 5 ϒ4 = −( 1 · 9 + 3 · 7 ); ϒ5 = 1 + 9 ⎛

⎞ − 2 · e−λ·τ − 3 ( 1 − λ) det(A − λ · I ) = det ⎝ 4 −( 5 · e−λ·τ + λ) − 6 ⎠ 7 − 8 · e−λ·τ ( 9 − λ) = ϒ4 · λ + ϒ5 · λ2 − λ3 + (ϒ1 + ϒ2 · λ + ϒ3 · λ2 ) · e−λ·τ det(A − λ · I ) = ϒ4 · λ + ϒ5 · λ2 − λ3 + (ϒ1 + ϒ2 · λ + ϒ3 · λ2 ) · e−λ·τ D(λ, τ ) = ϒ4 · λ + ϒ5 · λ2 − λ3 + (ϒ1 + ϒ2 · λ + ϒ3 · λ2 ) · e−λ·τ D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n = 3; m = 2; n > m Pn (λ, τ ) = ϒ4 · λ + ϒ5 · λ2 − λ3 ; n = 3 Q m (λ, τ ) = ϒ1 + ϒ2 · λ + ϒ3 · λ2 ; m = 2 Pn (λ, τ ) =

n=3 

pk (τ ) · λk = p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3

k=0

p0 (τ ) = 0; p1 (τ ) = ϒ4 ; p2 (τ ) = ϒ5 ; p3 (τ ) = −1 Q m (λ, τ ) =

m=2 

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 ; q0 (τ ) = ϒ1

k=0

q1 (τ ) = ϒ2 ; q2 (τ ) = ϒ3 The homogeneous system for N , φ1 , φ2 leads to a characteristic equation for the eigenvalue λ having the form D(λ, τ ) = P(λ, τ ) + Q(λ, τ ) · e−λ·τ = 0; and P(λ) =

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

813

3

 a j · λ j ; Q(λ) = 2j=0 c j · λ j . The coefficients {a j (qi , qk , τ ), c j (qi , qk , τ )} ∈ R depend on qi , qk and delay τ. qi , qk are any Diode-pumped Q-switched Nd:YVO4 yellow laser system’s parameters, other parameters kept as a constant [2, 3]. j=0

a0 = 0; a1 = ϒ4 ; a2 = ϒ5 ; a3 = −1; c0 = ϒ1 ; c1 = ϒ2 ; c2 = ϒ3 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ = 0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang and Cong 2005) [2] can also be verified in the present case: 1. If λ " = i" · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. " " is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. 2. If " Q(λ) P(λ) " 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω:Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and Pn (λ = i · ω, τ ) = p0 (τ ) + p1 (τ ) · i · ω − p2 (τ ) · ω2 − p3 (τ ) · i · ω3 = p0 (τ ) − p2 (τ ) · ω2 + ( p1 (τ ) · ω − p3 (τ ) · ω3 ) · i Pn (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + ( p1 (τ ) · ω − p3 (τ ) · ω3 ) · i Pn (λ = i · ω, τ ) = −ϒ5 · ω2 + (ϒ4 · ω + ω3 ) · i Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω − q2 (τ ) · ω2 Q m (λ = i · ω, τ ) = q0 (τ ) − q2 (τ ) · ω2 + q1 (τ ) · i · ω Q m (λ = i · ω, τ ) = ϒ1 − ϒ3 · ω2 + ϒ2 · i · ω Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + ( p1 (τ ) · ω − p3 (τ ) · ω3 ) · i + q0 (τ ) − q2 (τ ) · ω2 + q1 (τ ) · i · ω = 0 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = ϒ1 − (ϒ5 + ϒ3 ) · ω2 + [(ϒ4 + ϒ2 ) · ω + ω3 ] · i = 0

814

7 Dual-Wavelength Laser Systems Stability Analysis …

|P(i · ω, τ )|2 = ϒ52 · ω4 + (ϒ4 · ω + ω3 )2 = ϒ42 · ω2 + (ϒ52 + 2 · ϒ4 ) · ω4 + ω6 |Q(i · ω, τ )|2 = (ϒ1 − ϒ3 · ω2 )2 + ϒ22 · ω2 = ϒ12 + (ϒ22 − 2 · ϒ1 · ϒ3 ) · ω2 + ϒ32 · ω4 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2 = −ϒ12 + (ϒ42 − ϒ22 + 2 · ϒ1 · ϒ3 ) · ω2 + (ϒ52 + 2 · ϒ4 − ϒ32 ) · ω4 + ω6 We define the following parameters for simplicity: 0 , 2 , 4 , 6 0 = −ϒ12 ; 2 = ϒ42 − ϒ22 + 2 · ϒ1 · ϒ3 ; 4 = ϒ52 + 2 · ϒ4 − ϒ32 ; 6 = 1  Hence F(ω, τ ) = 0 implies 3k=0 2·k ·ω2·k = 0 and its roots are given by solving the above polynomial. PR (iω, τ ) = −ϒ5 ·ω2 ; PI (iω, τ ) = ϒ4 ·ω+ω3 ; Q R (iω, τ ) = ϒ1 − ϒ3 · ω2 ; Q I (iω, τ ) = ϒ2 · ω. sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

We use different terminology from our last characteristics parameters definition: k → j; pk (τ ) → a j ; qk (τ ) → c j ; n, m ∈ N0 ; n = 3; m = 2; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ); P(λ) =

3 

aj · λj

j=0

Q(λ) =

2 

cj · λj

j=0

P(λ) = a0 +a1 · λ+a2 · λ2 +a3 · λ3 ; Q(λ) = c0 +c1 · λ+c2 · λ2 P(λ) = ϒ4 · λ + ϒ5 · λ2 − λ3 ; Q(λ) = ϒ1 + ϒ2 · λ + ϒ3 · λ2 n, m ∈ N0 ; n > m and a j , c j :R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “—” denoted complex and conjugate. P(λ), Q(λ) are analytic functions in λ and differentiable in τ. The coefficients a j (R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . .) ∈ R and c j (R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . .) ∈ R depend on Diode-pumped Q-switched

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

815

Nd:YVO4 yellow laser system’s parameters, R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 [2, 3]. a0 + c0 = ϒ1 = − 1 · 5 · 9 − 1 · 8 · 6 + ( 4 · 9 + 7 · 6 ) · 2 − 3 · ( 7 · 5 − 8 · 4 ) = 0 ∀ R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see Kuang [2], Sect. 3.4) can also be verified in the present case. 1. If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire "∀ R p ," σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . domain of interest. " P(λ) " 2. " Q(λ) " is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. Indeed, " " " " " " " c0 +c1 ·λ+c2 ·λ2 " in the limit: " Q(λ) = " a0 +a1 ·λ+a2 ·λ2 +a3 ·λ3 ". P(λ) " 2 3. The following expressions exist: F(ω) = |P(i · ω)|2 . 3· ω)| − |Q(i 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω6 ). 4. Each positive root ω(R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . .) of F(ω) = 0 is continuous and differentiable with respect to R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . and the condition can only be assessed numerically.

In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i ·ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . ., Reλ may, at the crossing, change its ( j) ( j) sign from (−) to (+). i.e. from a stable focus E ( j) = (N ( j) , φ1 , φ2 ); j = 0, 1, 2 to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . and any system parameters.  ∂Reλ ; σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . . = const  (R p ) = ∂ R p λ=i·ω   ∂Reλ −1 (σ1 ) = ; R p , τ f , lcr , l1 , η S F M , τc1 , τ, . . . = const ∂σ1 λ=i·ω −1



816

7 Dual-Wavelength Laser Systems Stability Analysis …

−1 (τ f ) =

 

∂Reλ ∂τ f

 ; σ1 , R p , lcr , l1 , η S F M , τc1 , τ, . . . = const λ=i·ω

 ∂Reλ  (lcr ) = ; σ1 , τ f , R p , l1 , η S F M , τc1 , τ, . . . = const ∂lcr λ=i·ω   ∂Reλ −1 (η S F M ) = ; σ1 , τ f , lcr , l1 , R p , τc1 , τ, . . . = const ∂η S F M λ=i·ω   ∂Reλ −1 (τ ) = ; R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . . = const ∂τ λ=i·ω −1

P(λ) = PR (λ) + i · PI (λ); Q(λ) = Q R (λ) + i · Q I (λ), When writing and inserting λ = i · ω into Diode-pumped Q-switched Nd:YVO4 yellow laser system’s characteristic equation ω must satisfy the following equations: sin(ω · τ ) = g(ω) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos(ω · τ ) = h(ω) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

where |Q(iω, τ )|2 = 0 in view of requirement (1) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ. Now it is important to notice that if τ ∈ / I (assume that / I , ω(τ ) is I ⊆ R+0 is the set where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I, ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ · ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) + 2 · n · π ∀ n ∈ N0 Hence we can define the maps: )+2·n·π ; n ∈ N0 ; τ ∈ I . τn : I → R+0 , is given by τn (τ ) = θ(τ ω(τ ) Let us introduce the function I → R; Sn (τ ) = τ − τn (τ ); τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ. In the following, the subscripts λ, ω, R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . . indicate the corresponding partial derivatives. Let us first concentrate on (x), remember in λ(R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . .) and ω(R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . .), and keeping all parameters except one (x) and τ. The derivation closely follows that in reference [BK]. Differentiating Diode-pumped Q-switched Nd:YVO4 yellow laser system’s characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , τ, . . .

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …



∂λ ∂x

−1

=

817

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: ( ∂∂λx )−1 |λ=i·ω =   i·Pω (i·ω,x)·P(i·ω,x)+i·Q λ (i·ω,x)·Q(λ,x)−τ ·|P(i·ω,x)|2 . Px (i·ω,x)·P(i·ω,x)−Q x (i·ω,x)·Q(i·ω,x) Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω +i · PI ω ; Q ω = Q Rω +i · Q I ω ; Px = PRx +i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any Diode-pumped Q-switched Nd:YVO4 yellow laser system’s parameters R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delay x = τ : PR = −ϒ5 · ω2 ; PI = ϒ4 · ω + ω3 ; Q R = ϒ1 − ϒ3 · ω2 Q I = ϒ2 · ω; PRω = −2 · ϒ5 · ω; PI ω = ϒ4 + 3 · ω2 ; Q I ω = ϒ2 ; Q Rω = −2 · ϒ3 · ω; PRτ = 0; PI τ = 0; Q Rτ = 0. Q I τ = 0; Fτ = 0; PR · PI τ − PI · PRτ = 0 Q R · Q I τ − Q I · Q Rτ = 0 V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0 Fω = . . . Elements: PRω · PR = 2 · ϒ52 · ω3 PI ω · PI = (ϒ4 + 3 · ω2 ) · (ϒ4 · ω + ω3 ) = ϒ42 · ω + 4 · ϒ4 · ω3 + 3 · ω5 Q Rω · Q R = −2 · ϒ3 · ω · (ϒ1 − ϒ3 · ω2 ) = −2 · ϒ1 · ϒ3 · ω + 2 · ϒ32 · ω3 Q I ω · Q I = ϒ22 · ω U = . . . Elements: PR · PI ω = −(ϒ5 · ϒ4 · ω2 + ϒ5 · 3 · ω4 ) PI · PRω = (ϒ4 · ω + ω3 ) · (−2 · ϒ5 · ω) = −2 · ϒ5 · ϒ4 · ω2 − 2 · ϒ5 · ω4 Q R · Q I ω = ϒ2 · ϒ1 − ϒ2 · ϒ3 · ω2 ; Q I · Q Rω = −2 · ϒ2 · ϒ3 · ω2 We can summary our last results in the next table (Table 7.11).

818

7 Dual-Wavelength Laser Systems Stability Analysis …

Table 7.11 Diode-pumped Q-switched Nd:YVO4 yellow laser system’s stability analyses Fω , U elements

Fω , U elements

Expression

Fω , U elements

Expression

PRω · PR

2 · ϒ52 · ω3

PR · PI ω

−(ϒ5 · ϒ4 · ω2 + ϒ5 · 3 · ω4 )

PI ω · PI

ϒ42 · ω + 4 · ϒ4 · ω3 + 3 · ω5

PI · PRω

−2 · ϒ5 · ϒ4 · ω2 − 2 · ϒ5 · ω4

Q Rω · Q R −2 · ϒ · ϒ · ω + Q R · Q I ω ϒ · ϒ − ϒ · 1 3 2 1 2 2 · ϒ32 · ω3 ϒ3 · ω2 QIω · QI

ϒ22 · ω

Q I · Q Rω −2 · ϒ2 · ϒ3 · ω2

F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · = − FFωτ I ⇒ ∂ω ∂τ

∂ω ∂τ

+ Fτ = 0; τ ∈

 ∂Reλ ∂ω Fτ = ωτ = − ; ∂τ λ=iω ∂τ Fω  ! 2 −2 · [U + τ · |P| + i · Fω −1  (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ]   ! ∂Reλ −1 sign{ (τ )} = sign ∂τ λ=iω , + ∂ω V + · U ∂ω ∂τ ·τ sign{−1 (τ )} = sign{Fω } · sign +ω+ ∂τ |P|2 −1 (τ ) =



We shall presently examine the possibility of stability transitions (bifurcations) Diode-pumped Q-switched Nd:YVO4 yellow laser system, about the equilibrium ( j) ( j) points, E ( j) ; j = 0, 1, 2, . . ., E ( j) = (N ( j) , φ1 , φ2 ); j = 0, 1, 2. The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameter τ. Reλ, may at the crossing, changes its sign from – to +, i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [2, 3]. −1 (τ ) =



∂Reλ ∂τ

 λ=i·ω

; R p , σ1 , τ f , lcr , l1 , η S F M , τc1 , . . . = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = [−ϒ5 · ϒ4 · ω2 − ϒ5 · 3 · ω4 + 2 · ϒ5 · ϒ4 · ω2 + 2 · ϒ5 · ω4 ] − [ϒ2 · ϒ1 − ϒ2 · ϒ3 · ω2 + 2 · ϒ2 · ϒ3 · ω2 ]

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser …

U = −ϒ2 · ϒ1 + (ϒ5 · ϒ4 − ϒ2 · ϒ3 ) · ω2 − ϒ5 · ω4 =

819 2 

A2k · ω2·k

k=0

A4 = −ϒ5 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) =

2 

A2k · ω2·k

k=0

A0 = −ϒ2 · ϒ1 ; A2 = ϒ5 · ϒ4 − ϒ2 · ϒ3 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [(2 · ϒ52 · ω3 + ϒ42 · ω + 4 · ϒ4 · ω3 + 3 · ω5 ) − (−2 · ϒ1 · ϒ3 · ω + 2 · ϒ32 · ω3 + ϒ22 · ω)] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · (ϒ42 + 2 · ϒ1 · ϒ3 − ϒ22 ) · ω + 4 · (ϒ52 + 2 · ϒ4 − ϒ32 ) · ω3 + 6 · ω5 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

2 

B2·k+1 · ω2·k+1

k=0

B1 = 2 · (ϒ42 + 2 · ϒ1 · ϒ3 − ϒ22 ) B3 = 4 · (ϒ52 + 2 · ϒ4 − ϒ32 ); B5 = 6  Then we get the expression for Fω = 2k=0 B2·k+1 · ω2·k+1 , Diode-pumped Qswitched Nd:YVO4 yellow laser system parameter values. We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Diodepumped Q-switched Nd:YVO4 yellow laser system.   ! ∂Reλ −2 · [U + τ · |P|2 ] + i · Fω  (τ ) = = Re ∂τ λ=iω Fτ + 2i · [V + ω · |P|2 ]   ∂Reλ 2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = −1 (τ ) = ∂τ λ=iω Fτ2 + 4 · (V + ω · P 2 )2 −1



The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when

820

7 Dual-Wavelength Laser Systems Stability Analysis …

ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all Diode-pumped Q-switched Nd:YVO4 yellow laser system’s parameters are known and the stability switch due to various time delay values τ is described in the following expression: sign{−1 (τ )} = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ )  U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + |P(ω(τ ))|2 Remark: we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ (τ ) = . . . . and cos θ (τ ) = . . . .; sin(ω · τ ) = −PR ·Q I +PI ·Q R R +PI ·Q I ) |Q|2 = Q 2R + Q 2I . Finally we plot the ; cos(ω · τ ) = − (PR ·Q|Q| 2 |Q|2 stability switch diagram g(τ ) = −1 (τ ) = ( ∂Reλ ) ∂τ λ=iω 



2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} Fτ2 + 4 · (V + ω · P 2 )2 λ=iω   ∂Reλ sign[g(τ )] = sign[−1 (τ )] = sign[ ] ∂τ λ=iω   2 · {Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )} = sign Fτ2 + 4 · (V + ω · P 2 )2 −1

g(τ ) =  (τ ) =

∂Reλ ∂τ

=

Fτ2 + 4 · (V + ω · P 2 )2 > 0 ⇒ sign[−1 (τ )] = sign[Fω · (V + ω · P 2 ) − Fτ · (U + τ · P 2 )] !   Fτ Fτ sign[−1 (τ )] = sign [Fω ] · (V + ω · P 2 ) − · (U + τ · P 2 ) ; ωτ = − Fω Fω  −1 ∂ω ∂ F/∂ω ωτ = =− ∂τ ∂ F/∂τ !   V + ωτ · U + ω + ω · τ ; sign[P 2 ] > 0 sign[−1 (τ )] = sign [Fω ] · [P 2 ] · τ P2 !   V + ωτ · U sign[−1 (τ )] = sign [Fω ] · + ω + ω · τ τ P2   V + ωτ · U −1 + ω + ωτ · τ sign[ (τ )] = sign[Fω ] · sign P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]

7.4 A Diode-Pumped Q-Switched Nd:YVO4 Yellow Laser … Table 7.12 Diode-pumped Q-switched Nd:YVO4 yellow laser system sign of sign[−1 (τ )]

=

2 

821

sign[Fω ]

& ' τ ·U sign V +ω + ω + ωτ · τ P2

sign[−1 (τ )]

±

±

+

±

∓

–

B2·k+1 · ω2·k+1

k=0

We check the sign of −1 (τ ) according to the following rule (Table 7.12). If, sign[−1 (τ )] > 0, then the crossing proceeds from (−) to (+) respectively (stable to unstable). If, sign[−1 (τ )] < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

7.5 Questions 1.

In the four-level system of a Ti:sapphire laser, n 2 ≈ n 4 ≈ 0, the population inversion density n ≈ n 3 − gg23 · n 2 ≈ n 3 = N . We consider that A32  S32 (the transition from level E3 to E2 level is dominating and non-radiative transitions are neglected), S43  S41 then S43  S41 (the population in level 4 ≈ 0 (the population E4 decays immediately to level E3 , non-radiative), and dn dt of the pump level de-excites rapidly to the upper laser level E3 . Additionally, n 4 · S43 = n 1 · W14 = W p , W p is the pumping rate. Furthermore, A32 = A B A B + A32 ; S32 = S32 + S32 and τ0 = A132 (self-emission lifetime). σ A = A32 A A32 ·g(v,v A ) ; σB n v A ·v

=

B A32 ·g(v,v B ) , we can summarize our system differential equations: n v B ·v

dN N (t − τ N ) = W p − (σ A ·  A + σ B ·  B ) · v · N (t − τ N ) − dt τ0 d A l  A d B = σ A · v · N (t − τ N ) ·  A · − ; dt LA τ R A dt l B − = σ B · v · N (t − τ N ) ·  B · LB τRB N is the population inversion density (PID) and due to unideal population inversion process, there is shifting in time, N (t) → N (t − τ N ) but it is not affect the derivative ddtN ; N = N (t);  A =  A (t);  B =  B (t). 1.1 Find system fixed points for τ N = 0, and discuss stability and stability switching for different system parameter values.

822

7 Dual-Wavelength Laser Systems Stability Analysis …

1.2 Discuss stability and stability switching for different values of τ N parameter (τ N = 0; τ N > 0). 1.3 N (t) → N (t −τ Nk ); k > 1; k ∈ N, Discuss stability and stability switching for different values of k parameter. 1.4 The four-level system of Ti:sapphire laser self-emission life time τ0 =  · τ N ;  ∈ N, Discuss stability and stability switching for different values of τ N and  parameters.0 √ 1.5 Repeat (1.4) for τ0 =  · τ N ;  ∈ N and discus system dynamical behavior. 2.

We have a four-level system of a Ti:sapphire laser, same as in (1), but we do the following scaling in the variables: N → n = NN0 ;  A → ϕ A = N0A ;  B →

ϕ B = NB0 , and W p → ω p = N0p , where N0 is the concentration of Ti3+ . The normalized rate equations of a pulsed dual-wavelength laser are n = n(t); ϕ A = ϕ A (t); ϕ B = ϕ B (t). W

dn n(t − τn ) = ω p − (σ A · ϕ A + σ B · ϕ B ) · v · N0 − dt τ0 dϕ A l ϕ A dϕ B = σ A · v · n(t − τn ) · N0 · ϕ A · − ; dt LA τ R A dt l ϕB − = σ B · v · n(t − τn ) · N0 · ϕ B · LB τRB where, n is the normalized population inversion. Due to uncomplete population inversion there is a shift in time (delay) in n, n(t) → n(t − τn ), but it is not . affect the derivative of n with time, dn dt 2.1 Find system fixed points and how they changed for different values of ω p parameter. 2.2 Discuss stability and stability switching for different values of τn parameter. 2.3 We assume that the pumping rate is a Gaussian ω p = ω p (t), ω p (t) = 2

− 2·t2

· (1 − e−α p ·l ) · e T0 , where E p is the pumping energy, T0 is the pulse width of the pump laser, and α p is the absorption efficiency of the Ti:sapphire crystal at the wavelength of the pump laser. Analyze the system dynamic for t → ∞, find system fixed points and discuss stability and stability switching for different values of τn and σ A , σ B parameters. 2.4 The absorption efficiency α p is dependent on the normalized population   inversion, n, α p = ln[ n(t−τ ];  ∈ R;  > 0, Hint: e−α p ·l = e−l·log[ n(t−τn ) ] , n) 2·E p √ 2·π ·T0 ·h·v·N0





−1

n(t−τn ) l

n) l e−α p ·l = e−l·ln[ n(t−τn ) ] = el·ln[ n(t−τn ) ] = eln[  ] = [ n(t−τ ] . And  we consider that The pumping rate is not attenuated for time t, ∀ t ≥

7.5 Questions

823 2

− 2·t2

−

2·t02 2

t0 , e T0 = e T0 , Find system fixed points and how they changed for different values of t0 parameter. 2.5 Discuss stability and stability switching (2.4) for different values of τn parameter. 3.

We have a system of dual-wavelength VECSELs. Inhomogeneous pumping of the active region contains non identical Quantum Wells (QWs). The carrier ( j) ( j+1) ( j) density across the QW layer to be continuous, (C1 − C1 ) · tanh(X QW ) = ( j+1) ( j) ( j) − C2 , where C1,2 are constants which are related to the j-th barrier C2 ( j) region, X QW is the position of the j-th QW layer. We have two rate equations which determined the carrier balance in the continuous state above the QW and ( j) in the QW below lasing threshold reads, respectively, n ( j) = n ( j) (t); n QW = ( j) n QW (t). ( j)

( j)

n QW dn QW dn ( j) 1 n ( j) ( j) ( j) = ( j) · (J− − J+ ) − + ; dt τc τe dt t QW =

n ( j) 1 1 ( j) − n QW · ( + ) τc τe τr ( j)

( j)

J−,+ = −

t QW Da dn ( j) τe ( j) ( j) | X ( j) ∓0 ; J− − J+ = ξ0 · n ( j) ; ξ0 = · ·( ) QW La d X τc τe + τr

( j)

( j)

t

( j)

e Then J− − J+ = ξ0 · n ( j) = QW · ( τe τ+τ ) · n ( j) . τc r Remark: j index is for the Quantum Well (QW) layer number. ( j) J−,+ is the density of carrier flux into the continuous state above the quantum ( j) confined states of the j-th QW from the left and right hand sides (− and +), t QW ( j) ( j) is the QW thickness, and n QW is the carrier density in the QW; n ( j) = n(X QW ) is the carrier density above the QW, τc is the capture time for a charge carrier from the continuous state into the QW, and τe is the escape time describing the ( j) ( j) reverse process, τ1r = A + B · n QW + C · (n QW )2 where A, B, and C are the monomolecular, bimolecular, and Auger recombination coefficients.

( j)

3.1 Draw the graph for X QW versus the ration between the differences ( j) related to the first barrier region and the second barrier region X QW = ( j+1)

−C

( j)

( j)

2 ]. Is there any intersection that X QW = 0? tanh−1 [ 2( j) ( j+1) C1 −C1 3.2 Find system fixed points and discuss stability for different values of τc , τe , and τr . 3.3 When the carrier balance is not perfect in our system in the continuous state above the QW and in the QW below lasing threshold reads, respectively ( j) ( j) then n ( j) (t) → n ( j) (t − τ1 ); n QW (t) → n QW (t − τ2 ), but it doesn’t affect

C

824

7 Dual-Wavelength Laser Systems Stability Analysis … ( j)

dn

( j)

the derivatives dndt , dtQW . We have four cases: (1) τ1 = τ ; τ2 = 0, (2) τ1 = 0; τ2 = τ , (3) τ1 = τ ; τ2 = τ 2 . Discuss stability and stability switching for different values of τ parameter in the three cases. 3.4 Return (3.3) for τc  τe , explain how the system dynamic is changed. 3.5 Return (3.3) for τe  τr , explain how the system dynamic is changed. 4.

The rate equations in (3),

dn ( j) dt

=

1

( j) t QW

( j)

( j)

· (J− − J+ ) −

n ( j) τc

+

( j)

n QW τe

;

( j)

dn QW dt

=

( j) ( j) ( j+1) ( j) n ( j) − n QW · ( τ1e + τ1r ) and the recursive equation, (C1 − C1 ) · tanh(X QW ) = τc ( j) t ( j+1) ( j) ( j) ( j) e C2 − C2 , and J− − J+ = ξ0 · n ( j) = QW · ( τe τ+τ ) · n ( j) establish τc r ( j) the relationship between C1,2 related to the barrier layers adjacent to the j-th

quantum well (QW). )

( j+1)

C1

( j+1)

*

) =M·

( j)

C1

)

*

( j)

+ ξ · nG · e

( j)

−a·X QW

·

( j)

cosh(X QW )

*

( j)

sinh(X QW ) ) * ( j) ( j) 1 2 1 + 2 · ξ · sinh(2 · X QW ) ξ · cosh (X QW ) M= ( j) ( j) −ξ · sinh2 (X QW ) 1 − 21 · ξ · sinh(2 · X QW ) C2

C2

( j)

t QW ξ0 · L a ξ0 · L a = ; ξ0 = · ξ= Da Da τc ( j)



τe τe + τr



( j)

4.1 Draw a graph values of C1 and C2 , 1 ≤ j < K ; K > 0; K ∈ N; j ∈ ( j=1) ( j=1) ( j=1) N; j = 1, 2, 3, . . . start from some initial values, X QW , C1 , C2 ( j) ( j) and parameters, ξ, n G , a. Are the values of C1 and C2 converge to specific values for j → K ∀ K  1 ? 4.2 Discuss the convergence in (4.1) for different values of a parameter, ( j) (e−a·X QW ). How the system dynamic is changed for a = 0? Draw the relative graph. 4.3 Parameter ξ = 0, How the results (4.1 and 4.2) are changed? Draw the relative graph. ( j) 4.4 Discuss (4.1 and 4.2) if X QW is a geometric series. ( j) 4.5 Discuss (4.1 and 4.2) if X QW is an algebraic series. 5.

We have an EDFL in dual wavelength system which is described as a two-mode laser system. The existence of a frequency is lower than that of the relaxation oscillations is characteristic of a two-mode laser. The laser is modeled with four equations: two for the population’s inversion and two for the corresponding laser fields. The two sub-systems are coupled with a cross-saturation parameter. The dynamical behavior of an ion is described with two equations. The nonlinear system for the dual wavelength EDFL with ion pairs differential equations:

7.5 Questions

825

dd1 =  − a2 · (1 + d1 ) − 2 · d1 · (I1 + β · I2 ) dt dd2 = γ ·  − a2 · (1 + d2 ) − 2 · d2 · (I2 + β · I1 ) dt dd+ a22 = a12 · (1 − d+ ) − · (d+ + d− ) + y · (I1 + I2 ) · (2 − 3 · d+ ) dt 2 dd− a22 =  − a12 · (1 − d+ ) − · (d+ + d− ) + y · (I1 + I2 ) · d− dt 2 d I1 = I1 · (−1 + A · (1 − 2 · x) · (d1 + β · d2 ) + A · x · y · d− ) dt d I2 = I2 · (−1 + A · (1 − 2 · x) · (d2 + β · d1 ) + A · x · y · d− ) dt The system parameters are described in the next Table. Variable/Parameter

Description

I1 , I2

Two normalized laser intensities

d1 , d2

Normalized population inversions

β

Cross saturation parameter

γ (γ = 1 for perfect symmetry between the two modes – antiphase effect vanish)

A dichroism in the pumping process in order to take into account the small anisotropies of the laser



Pumping rate

x

Ion pair concentration

y

Ratio between the absorption cross-sections of an ion pair and isolated ion

A = σl · N0 · τl

N0 —Erbium concentration l—fiber length, τl [s]—photon lifetime σl [cm3 s−1 ]—Absorption cross section for the isolated ion

d+ = n 22 + n 11

Summation of populations in levels |11 and |22. n 22 is the population of level |22 and n 11 is the population in level |11

d+ = n 22 − n 11

Difference between the population of level |22 and the population of level |11

ai j = a2 =

τl τi j τl τ2

The ratio between the photon lifetime (τl ) and the lifetime of level |i j The ratio between the photon lifetime (τl ) and the life time of level |2, (τ2 )

826

7 Dual-Wavelength Laser Systems Stability Analysis …

5.1 Find system fixed points and draw there coordinates as a function of x and y parameters. 5.2 Discuss stability and stability switching for different values of  parameter. 5.3 Due to populations of levels |kl shifting in time (parasitic system effects), we get variables d+ , d− shifting in time d+ (t) → d+ (t − τ+ ); d− (t) → d− (t − τ− ) but it is not affect the derivatives dddt+ , dddt− . Discuss stability and stability switching for different values of τ+ and τ− shifting parameters. 5.4 How the system dynamic is changed if the photon lifetime is much smaller compare to the lifetime of levels |i j, τ+ , τ− > 0; ∀ τl  τi j ∃ ai j = τl → ε, (0 0. 7.

Erbium-doped fiber lasers are characterized by cw, sinusoidal and self-pulsing operation. The obtained regimes depend on three control parameters: ion-pair concentration, photon lifetime, and pumping rate. The system model described the active medium as a mixture of isolated ions and ion pairs. We describe our system by adapted laser rate equations, four first order coupled equations. The linear stability analysis demonstrates the existence of self-pulsing for a finite range of pumping rates. There are two classes of ions: (1) isolated ions and (2) ion pairs. The quenching process is associated with ion pairs and establishes the dynamical behavior of a laser. The laser operates in cw regime without pairs and their presence lead to a self-pulsing instability. The isolated ion is described as a two-level system and the ion pair as a three-level system. The system rate equations describe the fast relaxation time associated with the electronic polarizations, large homogeneous broadening due to collisions with photons. The dynamical behavior of an isolated ion is described by the rate equation: dn 1 = − + a2 · n 2 + il · (n 2 − n 1 (t − τ )) dt dn 2 =  − a2 · n 2 − il · (n 2 − n 1 (t − τ )) dt γ2 N1 N2 ; n1 = ; n2 = γl (1 − 2 · x) · N0 (1 − 2 · x) · N0 2  σl · Il il = ; Nk = (1 − 2x) · N0 γl k=1 a2 =

N0 is the erbium concentration, γ2 and γl are the relaxation rates, respectively, of the upper level of the laser transition and of the laser field, σl is the absorption cross section for the isolated ion, Il is the photon flux of the laser field, and  is the pumping rate. The time is normalized with respect to the photon lifetime τl = γ1l . Due to parasitic effect there is shifting in time for the population in 1 . level 1, n 1 (t − τ ), but it not affect the derivative dn dt 7.1 Find system fixed points and draw the relative graphs as a function of x parameter. 7.2 Discuss stability and stability switching for different values of τ parameter.

828

7 Dual-Wavelength Laser Systems Stability Analysis …

7.3 γl  γ2 ⇒ a2 → ε; 0 < ε  1, how the system dynamic is changed? Find fixed points and discuss stability switching for different values of τ parameter. 7.4 Find the expression for x parameter as a function of fixed points coordinates. 7.5 τ = 0, discuss stability and stability switching for different values of x parameter. 8.

Erbium-doped fiber lasers are characterized by cw, sinusoidal and self-pulsing operation. The obtained regimes depend on three control parameters: ion-pair concentration, photon lifetime, and pumping rate. The system model described the active medium as a mixture of isolated ions and ion pairs. We describe our system by adapted laser rate equations, four first order coupled equations. The linear stability analysis demonstrates the existence of self-pulsing for a finite range of pumping rates. There are two classes of ions: (1) isolated ions and (2) ion pairs. The quenching process is associated with ion pairs and establishes the dynamical behavior of a laser. The laser operates in cw regime without pairs and their presence lead to a self-pulsing instability. The isolated ion is described as a two-level system and the ion pair as a three-level system. The system rate equations describe the fast relaxation time associated with the electronic polarizations, large homogeneous broadening due to collisions with photons. The three-level scheme is adopted for the ion pairs and is described by three rate equations: dn 11 = − + a12 · n 12 + y · il · (n 12 − n 11 ) dt dn 22 =  − a22 · n 22 − y · il · (n 22 − n 12 ) dt dn 12 = −a12 · n 12 + a22 · n 22 − y · il · (n 12 − n 11 ) + y · il · (n 22 − n 12 ) dt y=

σl γi j Ni j ; ai j = ; ni j = ; n 11 + n 12 + n 22 = 1 σl γl x · N0

σl is the absorption cross section at the lasing wavelength for a pair, γi j is the relaxation rate of level |i j, σl is the absorption cross section for the isolated ion, γl is the relaxation rate of the laser field. 8.1 Find system fixed points and discuss stability switching for different values of N0 parameter (N0 is the erbium concentration). 8.2 a12 = γγ12l ; γl  γ12 , How our system dynamic is changed? Find fixed points and discuss stability. 8.3 a22 = γγ22l ; γl  γ22 , How our system dynamic is changed? Find fixed points and discuss stability.

7.5 Questions

829

8.4 Find the expression of parameter x as a function of system parameters and fixed points coordinates. Draw the relative graphs. 8.5 N0  N11 ; x = 1, Find system fixed points and discuss stability and stability switching for different values of  parameter. 9.

Erbium-doped fiber lasers are characterized by cw, sinusoidal and self-pulsing operation. The obtained regimes depend on three control parameters: ion-pair concentration, photon lifetime, and pumping rate. The laser field interacts with two systems: (1) the fraction 1 − 2 · x of isolated ions and (2) the fraction x of ion pairs. The dynamics of the laser intensity is described by the differential l = −il + (1 − 2 · x) · A · il · n i + x · B · il · n − ; A = σlγ·Nl 0 ; B = equation, di dt σl ·N0 ; γl

σ

y = BA = σll . The dynamical behavior of the system is described by four coupled first order differential equations: dn i = 2 ·  − a2 · (1 + n i ) − 2 · il · n i dt dn + 1 = a12 · (1 − n + ) − · a22 · (n + + n − ) + y · il · (2 − 3 · n + ) dt 2 dn − 1 = 2 ·  − a12 · (1 − n + ) − · a22 · (n + + n − ) − y · il · n − dt 2 dil = −il + (1 − 2 · x) · A · il · n i + x · B · il · n − dt σl is the absorption cross section at the lasing wavelength for a pair, γi j is the relaxation rate of level |i j, σl is the absorption cross section for the isolated ion, γl is the relaxation rate of the laser field, N0 is the erbium concentration,  is the pumping rate, a2 = γγ2l , γ2 and γl are the relaxation rates, respectively, of the upper level of the laser transition and of the laser field, n + = n 22 +n 11 ; n − = n 22 − n 11 . 9.1 Find system fixed points and draw the function of x versus fixed point’s coordinates and system parameters. 9.2 Discuss stability and stability switching for different values of  parameter. σ 9.3 y = BA = σll ; σl  σl ; y → ε; 0 ≤ ε  1, How the system dynamic is changed? Find system fixed points and discuss stability, stability switching for different values of x parameter. 9.4 il (t) → il (t − τl ), Discuss stability and stability switching for different values of τl parameter. 9.5 n + (t) → n + (t − τ+ ), Discuss stability and stability switching for different values of τ+ parameter. 10. We have a uniform multi-wavelength fiber laser which is based on hybrid Raman and Erbium-doped fiber. There is a gain competition effects in the

830

7 Dual-Wavelength Laser Systems Stability Analysis …

fiber Raman amplification (FRA) and EDF amplification. The FRA gain mechanism is suppresses the gain competition efficiency and makes the present multi-wavelength laser stable at room temperature. The hybrid gain medium increased the lasing bandwidth (compared with a pure EDF laser) and the power conversion efficiency (compared with a pure fiber Raman laser). The instability of multi wavelength lasing is mainly due to the gain competition effect. There is a gain competition (between two signal channels) for the FRA and EDFA parts of the fiber laser. The parameters of the signals and pumps are defined. Signal #A (fixed wavelength, λ A ) is injected to port (a) for both FRA and EDFA. Signal #B is at wavelength λ B (tuned from 1525 to 1580 nm) is used to study the gain competition. The gains for signal #A when signal #B is off (G A,B−o f f ) and on (G A,B−on ) are measured. The gain difference

G A ( λ) = G A,B−o f f (λ A ; λ) − G A,B−on (λ A ; λ) for signal #A in EDFA and in FRA as wavelength spacing λ = λ B − λ A varies. 10.1 Define the functions G A,B−o f f (λ A ; λ) and G A,B−on (λ A ; λ). A ( λ) A ( λ) 10.2 Write the differential equation G

λ = . . . .; ↔ d ⇒ dGd λ = . . . .. 10.3 Find fixed points and discuss stability. 10.4 G A ( λ) → G A ( λ − τλ ), discuss stability and stability switching for different values of τλ parameter. 10.5 We switch between signal #A and signal #B, what happened? How the system dynamic is changed? Discuss stability.

References 1. F. Song, J.Q. Yao, D.W. Zhou, J.Y. Qiao, G.Y. Zhang, J.G. Tian, Rate-equation theory and experimental research on dual-wavelength operation of a Ti: sapphire laser. Appl. Phys. B 72, 605–610 (2001) 2. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993) 3. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002) 4. Y.A. Morozov, L. Tomi, H. Antti, P. Markus, Simultaneous dual-wavelength emission from vertical external-cavity surface-emitting laser: a numerical modeling. IEEE J. Quantum Electron. 42(10) (2006) 5. S. Francois, L. Marc, M. Guy, L. Patrice, P.-L. Francois, Quasi-periodic route to chaos in erbiumdoped fiber laser. IEEE J. Quantum Electron. 31(3) (1995) 6. F. Sanchez, P. Le Boudec, P.-L. Francois, G. Stephan, Effects of ion pairs the dynamics of erbium-doped fiber lasers. Phys. Rev. A 48(3) (1993) 7. J. Guckenheimer, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42 8. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Text in Applied Mathematics (Hardcover) 9. Y.-F. Chen, S.W. Tsai, Diode-pumped Q-switched Nd:YVO4 yellow laser with intra-cavity sum-frequency mixing. Opt. Lett. 27(6) (2002)

Chapter 8

Dual-Wavelength Laser Systems Stability Analysis Under Parameters Variation (II)

The dynamic responses of the photon and carrier densities in an asymmetric dual quantum well laser exhibits wavelength switching. The lasers consist of two quantum wells of different energy gap, which are separated by a high and/or thick barrier layer. The barrier layer blocks carrier transport between the wells and the rate of the transport becomes comparable to the rate of the radiative recombination. The laser two different quantum wells which are isolated by a high and/or thick barrier layer results in an inhomogeneous carrier injection into the two wells. The pulse response at a voltage of the dual-wavelength lasing shows that the emission of the shorter wavelength light precedes that of the longer wavelength light by a nanosecond. The asymmetric dual quantum well (ADQW) laser diode consists of two quantum wells, well 1 and well 2, of different emission wavelengths, λ1 and λ2 , respectively (λ2 < λ1 ). The two wells are located in the core of a single optical waveguide. The well 2 of a wider band gap is located on the p-type side. We control the height and thickness of the barrier layer in a way that holes are injected into the two wells in homogeneously, and electrons homogeneously. When defining the component of the hole current which is directly injected into well 1, equal to zero, the set of rate equations for hole densities n1 , n2 in well 1 and well 2, respectively, and the photon densities Sλ1 , Sλ2 of the lasing modes at, λ1 and λ2 , then we can inspect the dynamic and stability of our system. A self-pulsing effects in Tm3+ -doped silica fiber laser is inspected and operate near 2 μm. There are various self-pulsing regimes which are observed for a range of pumping rates when the fiber is end-pumped with a high power Nd:YAG laser operating at 1.319 μm in a linear bidirectional cavity. We can define a variety of nonlinear phenomena, ranging from self-pulsing to self-Q-switching and to a modulated quasi-CW wave. The phenomenon of pair-induced quenching (PIQ) is inspected and it is compatible with excited-state absorption (ESA). The model of Tm3+ -doped silica fiber laser is based on the rate equations for a three-level system. The laser system dynamic and stability is analyzed by the cavity photon density C p (t) and through rate equations. Terahertz (THz) quantum cascade laser (QCL) is based on GaN/AlGaN quantum wells and emits at two widely separated wavelengths 33 and 52 μm simultaneously in a single active region. The large LO-phonon energy © Springer Nature Switzerland AG 2021 O. Aluf, Advance Elements of Laser Circuits and Systems, https://doi.org/10.1007/978-3-030-64103-0_8

831

832

8 Dual-Wavelength Laser Systems Stability …

(∼90 meV), the ultrafast resonant phonon depopulation of the lower radiative levels, suppression of the electrons that escape to the continuum states and selective carrier injection and extraction lead to an enhancement in the operating temperature of the structure, for temperature around 256 °K. Rate equations models describe the coupled electronic and electromagnetic system and certain structural parameters are defined for dynamic behavior and stability inspection.

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength Lasing Stability Analysis Under Parameters Variation and Delay Variables in Time We perform analysis of dynamic response of asymmetric dual quantum well lasers. The dynamic responses of the photon and carrier densities in an asymmetric dual quantum well laser exhibits wavelength switching. The laser consists of two different quantum wells isolated by a high and/or thick barrier layer that results in an inhomogeneous carrier injection into the two wells. The pulse response at a voltage of the dual-wavelength lasing shows that the emission of the shorter wavelength light precedes that of the longer wavelength light by a nanosecond. The monolithic laser diodes (LDs) changes the wavelength λ with a wide variable range λ and/or emit multiple-wavelength light are required for future optoelectronic applications. An asymmetric dual quantum well (ADQW) laser diode (LD) is demonstrated the discrete switching of λ = 13 nm and λ = 50 nm, quasi-continuous wavelength tuning λ = 22 nm, when λ ≈ 800 nm. The ADQW LD consists of two quantum wells, well 1 and well 2, of different emission wavelength, λ1 and λ2 , (λ2 < λ1 ), respectively. The two wells are located in the core of a single optical waveguide, where well 2 of a wider band gap are located on the p-type side. In the case that well 2 are located on the n-type side, roles of the electrons and holes are interchanged (Fig. 8.1). We control the height and thickness of the barrier layer and the holes are injected into the two wells in homogeneously, and electrons homogeneously. By using various types of ADQW structure, it is easier to inject electrons homogeneously than holes. The electron mobility is 20 times larger than that of holes. The holes are injected from the p-type cladding layer and first they trapped in well 2, and then thermally activated to be transferred over the barrier into well 1. By blocking this transfer with the high and/or thick barrier, the rate of the transfer becomes comparable with the recombination rate. It is constructed with normal multiple-quantum well (MQW) LDs, for which barrier layers are too low and thin to inject carriers uniformly across many wells. For a low and thin barrier in an ADQW LD, most injected carrier would occupy well 1 only, and the carrier density n 2 of well 2 would not reach the threshold density at a reasonable value of the injection current. For electrons, the sufficient electron densities are homogeneously distributed in both wells. In the time that sufficient electron densities are obtained in both wells, the analysis is focused on

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

833

Fig. 8.1 Schematic band diagram of an asymmetric dual quantum well (ADQW) LD

holes and photons, for which the rate equations are defined. The component of the hole current is directly injected into well 1 is zero, than the set of rate equations for hole densities n 1 , n 2 in well 1, well 2, and the photon densities s λ1 , s λ2 of the lasing modes at λ1 , λ2 is described [1]. Due to variation structure of our asymmetric dual quantum well laser there is shift in time in the hole densities  n 1 (t) → n 1 (tλ − τ1 ), λ1 λ1 t − τ ph1 ; and s 2 (t) → → n − τ and for photon densities s → s n 2 (t) (t ) (t) 2 2   λ2 s t − τ ph2 . The shifting in time does not affect the hole and photon densities λ1 λ2 1 2 , dn , dsdt , dsdt ). The component of the hole current which is derivative in time ( dn dt dt directly injected into well 1 is zero, the set of rate equations for hole densities n 1 , n 2 in well 1, well 2, respectively, and the photon densities s λ1 , s λ2 of the lasing modes at λ1 and λ2 are   J21 (n 1 (t − τ1 ), n 2 (t − τ2 )) dn 1 = − G λ1 1 (n 1 (t − τ1 )) · s λ1 t − τ ph1 dt e · d1   n 1 (t − τ1 ) − G λ1 2 (n 1 (t − τ1 )) · s λ2 t − τ ph2 − τn 1 J − J21 (n 1 (t − τ1 ), n 2 (t − τ2 )) dn 2 = dt e · d2   n 2 (t − τ2 ) − G λ2 2 (n 2 (t − τ2 )) · s λ2 t − τ ph2 − τn 2     s λ1 t − τ ph1 ds λ1 n 1 (t − τ1 ) = G λ1 1 (n 1 (t − τ1 )) · s λ1 t − τ ph1 − + β1λ1 · λ 1 dt τ τn 1     ds λ2 = G λ1 2 (n 1 (t − τ1 )) + G λ2 2 (n 2 (t − τ2 )) · s λ2 t − τ ph2 dt   s λ2 t − τ ph2 n 1 (t − τ1 ) n 2 (t − τ2 ) − + β1λ2 · + β2λ2 · λ 2 τ τn 1 τn 2

834

8 Dual-Wavelength Laser Systems Stability …

where, J21 is the hole-current density from well 2 to 1, τn 1 and τn 2 are the recombination lifetimes in well 1 and well 2, respectively, and τ λ1 and τ λ2 are the lifetimes of the λ1 and λ2 photons. The G λ1 1 , G λ1 2 , and G λ2 2 (β1λ1 , β1λ2 , and β2λ2 ) denote the optical gain coefficients (the spontaneous emission factors) of λ1 from well 1, λ2 from well 1, and λ2 from well 2, respectively. Assumption: J21  e · v · n 2 , where v is the effective velocity of holes moving over the barrier. The barrier must be high and/or thick enough and v becomes much slower than that in normal MQW LDs. β1λ2 and β2λ2 are constant, independent of the light intensity. β1λ1 of ADQW LDs is expected to be an increasing function of s λ2 , λ2 photons induce the simultaneous emission of the λ1 photon and another quantum of energy  · ω21 =  · ω2 −  · ω1 . The quantum can be either a phonon (Raman process) or a photon (electronic Raman process). The electronic Raman process is enhanced as  · ω21 approach the inter sub-band level separation in well 1. Both types of the Raman processes are modify β1λ1 into the form λ1 β1λ1 = β10 + ζ λ1 · s λ2 , where the first term in the right hand side is the usual term, and the second represents the Raman processes. It is difficult to evaluate the coefficient ζ λ1 from the first principles, since it required detailed information on level broadening, phase matching conditions. The way is to present ζ λ1 as a fitting parameter. In ADQW LDs, the λ2 light is emitted from the LD and λ2 is much different from the λ1 light. In usual LDs the spontaneous emission factor reduces the relaxation oscillation of the λ1 light. The optical gains are defined as G λ1 1 (n 1 ) = ga · (n 1 − f a ) and G λ2 2 (n 2 ) = gb ·(n 2 − f b ), where ga and gb are constants, and f a , f b are the inversion required to overcome the bulk losses. The assumption is G λ1 2 (n 1 ) = α · G λ1 1 (n 1 ), where, α is a constant. The threshold hole densities n 1th and n 2th are G λ1 1 (n 1th ) = τ1λ1   1th and G λ2 2 (n 2th ) = τ1λ2 − G λ1 2 (n 1th ), G λ1 1 (n 1 ) = n 1n−n · τ1λ1 + ga · f a + τ1λ1 ; 1th   2th G λ2 2 (n 2 ) = n 2n−n · τ1λ2 − G λ1 2 (n 1th ) + gb · f b + τ1λ2 . It is recommended to the 2th Runge–Kutta method to solve numerically the set of rate equations for hole densities n 1 , n 2 in well 1, well 2, respectively, and the photon densities s λ1 , s λ2 of the lasing modes at λ1 and λ2 . The Runge–Kutta methods are a family of implicit and explicit iterative methods, which include the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations and judiciously uses the information on the ‘slope’ at more than one point to extrapolate the solution to the future time step. The Runge–Kutta calculates four different slopes and uses them as weighted averages. The injection current J is assumed to be a rectangular pulse of height J p and width 3 · τn 1 , starting at t = 0 [1]. The typical values of parameters used are described in next table (Table 8.1). G λ1 1 (n 1 (t

G λ2 2 (n 2 (t

 1 1 n 1 (t − τ1 ) − n 1th · + ga · f a + λ − τ1 )) = λ 1 n 1th τ τ 1 J21 = J21 (n 2 (t − τ2 ))  e · v · n 2 (t − τ2 )

n 2 (t − τ2 ) − n 2th · − τ2 )) = n 2th



1 1 λ2 − G 1 (n 1th ) + gb · f b + λ λ 2 τ τ 2

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength … Table 8.1 Asymmetric dual quantum well lasers and dual-wavelength lasing system typical parameters values

835

System parameter

Typical value

ξ

0

d2 d1

1.35

λ

2

Jth2 λ

Jth1 n 2th n 1th

0.5

τn 1 , τn 2

10–9 s

τ λ1 ,

10–12 s

τ λ2

τn 1 · ga · f a , τn 2 · gb · f b

10

α

0.4

λ1 β10 , β1λ2 , β2λ2

5 × 10–3

ζ λ1

1

· n 2th

G λ1 2 (n 1 (t − τ1 )) = α · G λ1 1 (n 1 (t − τ1 )) 

 1 n 1 (t − τ1 ) − n 1th 1 + =α· · + g · f a a n 1th τ λ1 τ λ1 We can write our system full differential equations:

  n 1 (t − τ1 ) − n 1th 1 dn 1 e · v · n 2 (t − τ2 ) 1 1. − · + ga · f a + λ = dt e · d1 n 1th τ λ1 τ 1 

   1 n 1 (t − τ1 ) − n 1th 1 + · s λ1 t − τ ph1 − α · · + g · f a a n 1th τ λ1 τ λ1   n 1 (t − τ1 ) · s λ2 t − τ ph2 − τn 1 J − e · v · n 2 (t − τ2 ) dn 2 = dt e · d2 

 1 n 2 (t − τ2 ) − n 2th 1 λ2 + − · − G (n ) + g · f 1th b b 1 n 2th τ λ2 τ λ2 n 2 (t − τ2 ) · s λ2 (t − τ ph2 ) − τn 2 

 1 n 1 (t − τ1 ) − n 1th ds λ1 1 = 3. · + ga · f a + λ · s λ1 (t − τ ph1 ) dt n 1th τ λ1 τ 1 λ1 s (t − τ ph1 ) n 1 (t − τ1 ) + β1λ1 · − λ 1 τ τn 1

  n 1 (t − τ1 ) − n 1th 1 ds λ2 1 = α· + 4. · + g · f a a dt n 1th τ λ1 τ λ1 2.

836

8 Dual-Wavelength Laser Systems Stability …



 1 n 2 (t − τ2 ) − n 2th 1 λ2 + + · − G (n ) + g · f 1th b b 1 n 2th τ λ2 τ λ2 s λ2 (t − τ ph2 ) n 1 (t − τ1 ) n 2 (t − τ2 ) · s λ2 (t − τ ph2 ) − + β1λ2 · + β2λ2 · τ λ2 τn 1 τn 2 At fixed points: dn 1 dn 2 ds λ1 = 0; = 0; =0 dt dt dt ds λ2 = 0; lim n 1 (t − τ1 ) = n 1 (t) ∀ t  τ1 t→∞ dt   lim n 1 (t − τ2 ) = n 2 (t) ∀ t  τ2 ; lim s λ1 t − τ ph1 t→∞   = s λ1 (t); lim s λ2 t − τ ph2

t→∞

t→∞

λ2

= s (t) ∀ t  τ ph1 & t  τ ph2

∗   n − n 1th 1 v · n ∗2 dn 1 1 =0⇒ − 1 · + g · f + · [s λ1 ]∗ a a dt d1 n 1th τ λ1 τ λ1 

∗  1 n − n 1th 1 n∗ + · [s λ2 ]∗ − 1 = 0 −α· 1 · + g · f a a λ λ 1 1 n 1th τ τ τn 1  1   ∗ v · n ∗2 λ + ga · f a − n ∗1 · s λ1 · τ 1 d1 n 1th    λ1  ∗  λ2 ∗ τ1λ1 + ga · f a ∗ + ga · f a · s − α · n1 · s · n 1th  λ2 ∗ n ∗1 + α · ga · f a · s − =0 τn 1  

v · n ∗2 − (1 + α) · d1

1 τ λ1

+ ga · f a n 1th

 ∗  ∗ n∗ · n ∗1 · s λ2 + (α + 1) · ga · f a · s λ1 − 1 = 0 τn 1

J − e · v · n ∗2 dn 2 =0⇒ dt e · d2 

∗  1 n 2 − n 2th 1 λ2 − · − G 1 (n 1th ) + gb · f b + λ · [s λ2 ]∗ n 2th τ λ2 τ 2 ∗ n − 2 =0 τn 2 

  1 J − e · v · n ∗2 1 1 λ2 ∗ − n2 · −1 · − G 1 (n 1th ) + gb · f b + λ e · d2 n 2th τ λ2 τ 2

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

· [s λ2 ]∗ −

837

n ∗2 =0 τn 2

We define 1 =



1

τ λ2

 − G λ1 2 (n 1th ) + gb · f b , 1 is a global parameter then



  ∗ n ∗ 1 1 n ∗2 · − 1 · 1 + λ · s λ2 − 2 = 0 n 2th τ 2 τn 2   λ2 ∗ n ∗2  ∗ 1 J v 1 · s − · n ∗2 − n ∗2 · s λ2 · − −

− =0 1 e · d2 d2 n 2th τ λ2 τn 2    λ2 ∗  λ2 ∗ 1 J 1 1 v ∗ · n ∗2 = 0 − n2 · s · − − 1 · s − + (∗) e · d2 n 2th τ λ2 τn 2 d2 J − e · v · n ∗2 − e · d2



We define 2 = (1 + α) · then v · n ∗2 − (1 + α) · d1

(∗∗)



1 τ λ1

1 τ λ1

+ ga · f a n 1th

+ga · f a n 1th





, 3 = (α + 1) · ga · f a global parameters

 ∗  ∗ n∗ · n ∗1 · s λ2 + (α + 1) · ga · f a · s λ1 − 1 = 0 τn 1

 ∗  ∗ n ∗ v · n ∗2 − 2 · n ∗1 · s λ2 + 3 · s λ1 − 1 = 0 d1 τn 1

 λ ∗  

∗  λ ∗ s 1 n 1 − n 1th n∗ 1 ds λ1 1 1 · + g · f − + β1λ1 · 1 = 0 + · s =0⇒ a a dt n 1th τ λ1 τ λ1 τ λ1 τn 1

  λ1  ∗  ∗ 1 1 n∗ · s · s λ1 + β1λ1 · 1 = 0 + g · f − g · f + a a a a λ λ τ 1 τ 1 τn 1

  n ∗ − n 1th 1 ds λ2 1 =0⇒ α· 1 · + ga · f a + λ λ 1 dt n 1th τ τ 1 

∗  1 n 2 − n 2th 1 λ2 · [s λ2 ]∗ + · − G 1 (n 1th ) + gb · f b + λ n 2th τ λ2 τ 2 [s λ2 ]∗ n∗ n∗ − λ + β1λ2 · 1 + β2λ2 · 2 = 0 τ 2 τn 1 τn 2

(∗ ∗ ∗)

n ∗1 · n 1th



Since 1 =



1 τ λ2

 − G λ1 2 (n 1th ) + gb · f b then

 ∗  ∗   ∗ 1 n 2 − n 2th n 1 − n 1th 1 1 · s λ2 α· · + ga · f a + λ + · 1 + λ λ 1 1 2 n 1th τ τ n 2th τ  λ ∗ ∗ ∗ s 2 n n − λ + β1λ2 · 1 + β2λ2 · 2 = 0 τ 2 τn 1 τn 2

838



8 Dual-Wavelength Laser Systems Stability …

   ∗ 1 1 n ∗1 n ∗2 ·α· + ga · f a + · 1 + − 1 − α · ga · f a · s λ2 n 1th τ λ1 n 2th τ λ2  λ ∗ s 2 n∗ n∗ − λ + β1λ2 · 1 + β2λ2 · 2 = 0 τ 2 τn 1 τn 2

We define for simplicity new parameters: 4 = α ·

1 − α · ga · f a .



1 τ λ1

 + ga · f a ; 5 =

1 τ λ2

−

 ∗  ∗ n1 n∗ · 4 + 2 · 1 + 5 · s λ2 (∗ ∗ ∗∗) n 1th n 2th  λ ∗ ∗ 2 s n n∗ − λ + β1λ2 · 1 + β2λ2 · 2 = 0 τ 2 τn 1 τn 2 We can summary our Asymmetric dual quantum well lasers and dual-wavelength Lasing system fixed points equations: (∗)

 ∗ 1 J − n ∗2 · s λ2 · − e · d2 n 2th



  λ2 ∗ 1 1 v · s · n ∗2 = 0 −

− + 1 τ λ2 τn 2 d2

 ∗  ∗ n ∗ v · n ∗2 − 2 · n ∗1 · s λ2 + 3 · s λ1 − 1 = 0 d1 τn 1   ∗  λ1  ∗  λ1 ∗ 1 1 λ1 n 1 · s · s · + g · f − g · f + + β · =0 a a a a 1 τ λ1 τ λ1 τn 1  ∗  ∗ n1 n∗ · 4 + 2 · 1 + 5 · s λ2 (∗ ∗ ∗∗) n 1th n 2th  λ ∗ ∗ 2 s n n∗ − λ + β1λ2 · 1 + β2λ2 · 2 = 0 τ 2 τn 1 τn 2 (∗∗)

(∗ ∗ ∗)

n ∗1 n 1th

It is recommended to solve it numerically ratherthan analytic to find system fixed ( j)  ( j) ( j) ( j)  ∀ j = 0, 1, 2, . . .. points coordinate, E ( j) n 1 , n 2 , s λ1 , s λ2 Stability analysis: We have four possible cases to inspect our system stability, (1) τ1 = τ , τ2 = 0, τ ph1 = 0, τ ph2 = 0, (2) τ1 = 0, τ2 = τ , τ ph1 = 0, τ ph2 = 0, (3) τ1 = 0, τ2 = 0, τ ph1 = τ , τ ph2 = 0, (4) τ1 = 0, τ2 = 0, τ ph1 = 0, τ ph2 = τ . We choose to inspect the first case τ1 = τ , τ2 = 0, τ ph1 = 0, τ ph2 = 0. n 1 = n 1 (t); n 2 = n 2 (t); s λ1 = s λ1 (t); s λ2 = s λ2 (t). 

 1 e · v · n2 n 1 (t − τ ) − n 1th 1 dn 1 + · s λ1 = − · + g · f 1. a a dt e · d1 n 1th τ λ1 τ λ1 

 1 n 1 (t − τ ) − n 1th 1 n 1 (t − τ ) + · s λ2 − −α· · + g · f a a λ λ 1 1 n 1th τ τ τn 1

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

839

J − e · v · n2 dn 2 = dt e · d2 

 1 n 2 − n 2th 1 n2 λ2 − · − G 1 (n 1th ) + gb · f b + λ · s λ2 − λ 2 2 n 2th τ τ τn 2 

 1 n 1 (t − τ ) − n 1th 1 ds λ1 = · + ga · f a + λ · s λ1 3. dt n 1th τ λ1 τ 1 s λ1 n 1 (t − τ ) − λ + β1λ1 · τ 1 τn 1 

 1 n 1 (t − τ ) − n 1th 1 ds λ2 + = α· · + g · f 4. a a dt n 1th τ λ1 τ λ1 

 1 n 2 − n 2th 1 λ2 + · s λ2 + · − G (n ) + g · f 1th b b 1 n 2th τ λ2 τ λ2 s λ2 n 1 (t − τ ) n2 − λ + β1λ2 · + β2λ2 · τ 2 τn 1 τn 2 2.

Stability analysis: The standard local stability analysis about any one of the equilibrium point of the Asymmetric dual quantum well lasers and dual-wavelength λ1 λ2 Lasing system consists in adding  to coordinate [n 1 , n 2 , s , s ] arbitrarily small increments of exponential form n 1 , n 2 , s λ1 , s λ2 · eλ·t and retaining the first order terms in n 1 , n 2 , s λ1 , s λ2 . The system of four homogeneous equations leads to a polynomial characteristic equation in the eigenvalues. The polynomial characteristic equations accept by set the below variables and variables derivative with respect to time into Asymmetric dual quantum well lasers and dual-wavelength Lasing system equations. The Asymmetric dual quantum well lasers and dual-wavelength Lasing system fixed values with arbitrarily small increments of exponential form  n 1 , n 2 , s λ1 , s λ2 · eλ·t are: j = 0 (first fixed point), j = 1 (second fixed point), j = 2 (third fixed point), etc. [2, 3]. ( j)

n 1 (t) = n 1 + n 1 · eλ·t ; n 2 (t) ( j)

= n 2 + n 2 · eλ·t ; s λ1 (t)  ( j)  ( j) + s λ1 · eλ·t ; s λ2 (t) = s λ2 + s λ2 · eλ·t = s λ1 dn 1 (t) dn 2 (t) = n 1 · λ · eλ·t ; = n 2 · λ · eλ·t dt dt ds λ1 (t) ds λ2 (t) = s λ1 · λ · eλ·t ; = s λ2 · λ · eλ·t dt dt ( j)

n 1 (t − τ ) = n 1 + n 1 · eλ·(t−τ ) ; 6 =

1 + ga · f a τ λ1

840

8 Dual-Wavelength Laser Systems Stability …



4 = α ·

1 + g · f a a ; 4 = α · 6 τ λ1

λ1 λ2 We choosethese expressions  for ourselves n 1 (t), n 2 (t), s (t), s (t) as a small λ1 λ2 displacement n 1 , n 2 , s , s from the Asymmetric dual quantum well lasers and dual-wavelength Lasing system fixed points in time t = 0. ( j)

( j)

n 1 (t = 0) = n 1 + n 1 ; n 2 (t = 0) = n 2 + n 2  ( j) s λ1 (t = 0) = s λ1 + s λ1 ; s λ2 (t = 0)  λ2 ( j) + s λ2 = s 

e · v · n2 dn 1 n 1 (t − τ ) − n 1th 1 = − · 6 + λ · s λ1 dt e · d1 n 1th τ 1 

n 1 (t − τ ) − n 1th 1 n 1 (t − τ ) −α· · 6 + λ · s λ2 − n 1th τ 1 τn 1 ( j)

e · v · (n 2 + n 2 · eλ·t ) e · d1  ( j)  (n 1 + n 1 · eλ·(t−τ ) ) − n 1th 1 − · 6 + λ · ([s λ1 ]( j) + s λ1 · eλ·t ) n 1th τ 1  ( j)  (n 1 + n 1 · eλ·(t−τ ) ) − n 1th 1 −α· · 6 + λ · ([s λ2 ]( j) + s λ2 · eλ·t ) n 1th τ 1

n 1 · λ · eλ·t =

( j)

−

(n 1 + n 1 · eλ·(t−τ ) ) τn 1 ( j)

e · v · n2 e · v · n 2 · eλ·t + e·d e · d1  ( j)1  n 1 − n 1th n 1 · eλ·(t−τ ) 1 − · 6 + · 6 + λ n 1th n 1th τ 1

n 1 · λ · eλ·t =

· ([s λ1 ]( j) + s λ1 · eλ·t )  ( j)  n 1 − n 1th n 1 · eλ·(t−τ ) 1 · 6 + · 6 + λ −α· n 1th n 1th τ 1 · ([s λ2 ]( j) + s λ2 · eλ·t ) − n 1 · λ · eλ·t =

( j)

e · v · n2 e · v · n 2 · eλ·t + e · d1 e · d1

( j)

n1 n 1 · eλ·(t−τ ) − τn 1 τn 1

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

 −  −

( j)

n 1 − n 1th 1 · 6 + λ n 1th τ 1 ( j)

n 1 − n 1th 1 · 6 + λ n 1th τ 1

841

 · [s λ1 ]( j) 

n 1 · eλ·(t−τ ) · 6 · [s λ1 ]( j) n 1th  ( j)  n 1 − n 1th 1 −α· · 6 + λ · [s λ2 ]( j) n 1th τ 1 · s λ1 · eλ·t −

eλ·(t−τ ) · 6 · n 1 · s λ1 · eλ·t n 1th  ( j)  n 1 − n 1th 1 n 1 · eλ·(t−τ ) + · 6 + λ · s λ2 · eλ·t + · 6 · [s λ2 ]( j) n 1th τ 1 n 1th

( j) n eλ·(t−τ ) n 1 · eλ·(t−τ ) λ2 λ·t − 1 − + · 6 · n 1 · s · e n 1th τn 1 τn 1 −

Assumption: n 1 · s λ1 ≈ 0; n 1 · s λ2 ≈ 0  ( j)  ( j) λ·t n · e − n e · v · n e · v · n 1 2 1th 2 1 n 1 · λ · eλ·t = + − · 6 + λ · [s λ1 ]( j) e · d1 e · d1 n 1th τ 1  ( j)  n 1 − n 1th 1 n 1 · eλ·(t−τ ) − · 6 + λ · s λ1 · eλ·t − · 6 · [s λ1 ]( j) n 1th τ 1 n 1th  ( j)  n 1 − n 1th 1 −α· · 6 + λ · [s λ2 ]( j) n 1th τ 1  ( j)  n 1 − n 1th 1 + · 6 + λ · s λ2 · eλ·t n 1th τ 1

( j) n n 1 · eλ·(t−τ ) n 1 · eλ·(t−τ ) + · 6 · [s λ2 ]( j) − 1 − n 1th τn 1 τn 1   ( j) ( j) n 1 − n 1th e · v · n2 1 λ·t n1 · λ · e = − · 6 + λ · [s λ1 ]( j) e · d1 n 1th τ 1  ( j)  ( j) n 1 − n 1th 1 n −α· · 6 + λ · [s λ2 ]( j) − 1 n 1th τ 1 τn 1  ( j)  n 1 − n 1th e · v · n 2 · eλ·t 1 + − · 6 + λ · s λ1 · eλ·t e · d1 n 1th τ 1 n 1 · eλ·(t−τ ) · 6 · [s λ1 ]( j) n 1th  ( j)  n 1 − n 1th 1 −α· · 6 + λ · s λ2 · eλ·t n 1th τ 1 −

842

8 Dual-Wavelength Laser Systems Stability …

−α·

n 1 · eλ·(t−τ ) n 1 · eλ·(t−τ ) · 6 · [s λ2 ]( j) − n 1th τn 1

At fixed points:  ( j)  ( j) n 1 − n 1th e · v · n2 1 − · 6 + λ · [s λ1 ]( j) e · d1 n 1th τ 1  ( j)  ( j) n 1 − n 1th 1 n −α· · 6 + λ · [s λ2 ]( j) − 1 = 0 n 1th τ 1 τn 1

1 1 λ1 ( j) · e−λ·τ − n 1 · λ − n 1 · (1 + α) · · 6 · [s ] + n 1th τn 1  ( j)  e · v · n2 n 1 − n 1th 1 + − · 6 + λ · s λ1 e · d1 n 1th τ 1  ( j)  n 1 − n 1th 1 −α· · 6 + λ · s λ2 = 0 n 1th τ 1

 n 2 − n 2th 1 J − e · v · n2 dn 2 = − · − G λ1 2 (n 1th ) dt e · d2 n 2th τ λ2  n2 1 + gb · f b ) + λ · s λ2 − τ 2 τn 2 n2 · λ · e

λ·t

( j)

J − e · v · (n 2 + n 2 · eλ·t ) = e · d2  ( j)  (n 2 + n 2 · eλ·t ) − n 2th 1 − · − G λ1 2 (n 1th ) n 2th τ λ2  ( j) (n + n 2 · eλ·t ) 1 + gb · f b ) + λ · ([s λ2 ]( j) + s λ2 · eλ·t ) − 2 τ 2 τn 2 ( j)

J − e · v · n2 e · v · n 2 · eλ·t − e·d e · d2  ( j) 2   n 2 − n 2th n 2 · eλ·t 1 − + − G λ1 2 (n 1th ) · n 2th n 2th τ λ2  1 + gb · f b ) + λ · ([s λ2 ]( j) + s λ2 · eλ·t ) τ 2

n 2 · λ · eλ·t =

( j)

−

n 2 · eλ·t n2 − τn 2 τn 2

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

We already know that 1 =

1 τ λ2

843

− G λ1 2 (n 1th ) + gb · f b .

( j)

J − e · v · n2 e · v · n 2 · eλ·t − e·d e · d2  ( j) 2   n 2 − n 2th n 2 · eλ·t 1 − + · 1 + λ · ([s λ2 ]( j) + s λ2 · eλ·t ) n 2th n 2th τ 2

n 2 · λ · eλ·t =

( j)

−

n 2 · eλ·t n2 − τn 2 τn 2 ( j)

J − e · v · n2 e · v · n 2 · eλ·t − e·d e · d2  ( j) 2   n 2 − n 2th 1 n 2 · eλ·t − · 1 + λ + · 1 · ([s λ2 ]( j) + s λ2 · eλ·t ) n 2th τ 2 n 2th

n 2 · λ · eλ·t =

( j)

−

n 2 · eλ·t n2 − τn 2 τn 2 ( j)

J − e · v · n2 e · v · n 2 · eλ·t − e·d e · d2  ( j)2  n 2 − n 2th 1 − · 1 + λ · [s λ2 ]( j) n 2th τ 2  ( j)  n 2 − n 2th 1 + · 1 + λ · s λ2 · eλ·t n 2th τ 2

n 2 · λ · eλ·t =

+

n 2 · eλ·t eλ·t · 1 · [s λ2 ]( j) + · 1 · n 2 · s λ2 · eλ·t n 2th n 2th

−

n2 n 2 · eλ·t − τn 2 τn 2

( j)

Assumption: n 2 · s λ2 ≈ 0 ( j)

J − e · v · n2 e · v · n 2 · eλ·t − e·d e · d2  ( j) 2  ( j)   λ2 ( j) n 2 − n 2th n 2 − n 2th 1 − · 1 + λ · s − · 1 2 n 2th τ n 2th  ( j) 1 n 2 · eλ·t + λ · s λ2 · eλ·t − · 1 · s λ2 τ 2 n 2th

n 2 · λ · eλ·t =

( j)

−

n2 n 2 · eλ·t − τn 2 τn 2

844

8 Dual-Wavelength Laser Systems Stability …

n2 · λ · e

λ·t

 ( j) ( j) J − e · v · n2 n 2 − n 2th = − · 1 e · d2 n 2th  ( j) n (2j) 1 + λ · s λ2 − τ 2 τn  2 ( j) e · v · n 2 · eλ·t n 2 − n 2th − − · 1 e · d2 n 2th  ( j) 1 n 2 · eλ·t + λ · s λ2 · eλ·t − · 1 · s λ2 τ 2 n 2th λ·t n2 · e − τn 2

At fixed points:

s λ1



 ( j)  ( j) n (2j) n 2 − n 2th 1 · 1 + λ · s λ2 − =0 n 2th τ 2 τn 2

 λ2 ( j) e·v 1 1 + · 1 · s + − n2 · λ − n2 · e · d2 n 2th τn 2  ( j)  n 2 − n 2th 1 · 1 + λ · s λ2 = 0 − n 2th τ 2 

 ds λ1 1 n 1 (t − τ ) − n 1th 1 + · s λ1 = · + g · f a a dt n 1th τ λ1 τ λ1 s λ1 n 1 (t − τ ) − λ + β1λ1 · τ 1 τn 1   ⎡ ( j) n 1 + n 1 · eλ·(t−τ ) − n 1th  1 · λ · eλ·t = ⎣ · n 1th τ λ1     ( j) 1 +ga · f a ) + λ · s λ1 + s λ1 · eλ·t τ 1      ( j) ( j) s λ1 n 1 + n 1 · eλ·(t−τ ) + s λ1 · eλ·t − + β1λ1 · τ λ1 τn 1 ( j)

J − e · v · n2 − e · d2

We already know that 6 =  s

λ1

·λ·e

λ·t

=

( j)

1 τ λ1

+ ga · f a .

n 1 − n 1th 1 · 6 + λ n 1th τ 1



8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

 n 1 · eλ·(t−τ ) + · 6 · ([s λ1 ]( j) + s λ1 · eλ·t ) n 1th ( j)

λ·(t−τ ) s λ1 · eλ·t [s λ1 ]( j) λ1 n 1 λ1 n 1 · e − + β · + β · 1 1 τ λ1 τ λ1 τn 1 τn 1  ( j)   ( j) n 1 − n 1th 1 · λ · eλ·t = · 6 + λ · s λ1 n 1th τ 1  ( j)  n 1 − n 1th 1 + · 6 + λ · s λ1 · eλ·t n 1th τ 1

−

s λ1

+

 ( j) n 1 · eλ·(t−τ ) · 6 · s λ1 n 1th

 λ ( j) s 1 eλ·(t−τ ) λ1 λ·t + · 6 · n 1 · s · e − n 1th τ λ1 ( j)

−

λ·(t−τ ) s λ1 · eλ·t λ1 n 1 λ1 n 1 · e + β · + β · 1 1 τ λ1 τn 1 τn 1

Assumption: n 1 · s λ1 ≈ 0 

s

λ1

·λ·e

λ·t

 ( j)  ( j) n 1 − n 1th 1 = · 6 + λ · s λ1 n 1th τ 1  ( j)  n 1 − n 1th 1 + · 6 + λ · s λ1 · eλ·t n 1th τ 1  ( j) n 1 · eλ·(t−τ ) · 6 · s λ1 n 1th  λ ( j) 1 s s λ1 · eλ·t − − τ λ1 τ λ1 ( j) n n 1 · eλ·(t−τ ) + β1λ1 · 1 + β1λ1 · τn 1 τn 1  ( j)   ( j) n 1 − n 1th 1 = · 6 + λ · s λ1 n 1th τ 1  λ ( j) ( j) s 1 λ1 n 1 − + β · 1 τ λ1 τn 1  ( j)  n 1 − n 1th 1 + · 6 + λ · s λ1 · eλ·t n 1th τ 1 +

s λ1 · λ · eλ·t

+

 ( j) s λ1 · eλ·t n 1 · eλ·(t−τ ) · 6 · s λ1 − n 1th τ λ1

845

846

8 Dual-Wavelength Laser Systems Stability …

+ β1λ1 ·

n 1 · eλ·(t−τ ) τn 1

At fixed points: 

  λ ( j) ( j) ( j)  λ1 ( j) s 1 n 1 − n 1th 1 λ1 n 1 · 6 + λ · s − + β · =0 1 n 1th τ 1 τ λ1 τn 1  ( j)  n − n 1 1th 1 s λ1 · λ · eλ·t = · 6 + λ · s λ1 · eλ·t n 1th τ 1  ( j) s λ1 · eλ·t n 1 · eλ·(t−τ ) · 6 · s λ1 − n 1th τ λ1 λ·(t−τ ) n1 · e + β1λ1 · τn 1

  λ1 ( j) 1 1 λ1 · n 1 · e−λ·τ · 6 · s + β1 · n 1th τn 1 +

( j)

n 1 − n 1th · 6 · s λ1 − s λ1 · λ = 0 n 1th 

 1 n 1 (t − τ ) − n 1th ds λ2 1 + = α· · + g · f a a dt n 1th τ λ1 τ λ1 

 1 n 2 − n 2th 1 λ2 · s λ2 + · − G 1 (n 1th ) + gb · f b + λ n 2th τ λ2 τ 2 s λ2 n 1 (t − τ ) n2 − λ + β1λ2 · + β2λ2 · 2 τ τn 1 τn 2 +

We already know that 6 =

1 τ λ1

+ ga · f a ; 1 =

1 τ λ2

− G λ1 2 (n 1th ) + gb · f b .





n 2 − n 2th n 1 (t − τ ) − n 1th 1 1 ds λ2 · s λ2 = α· · 6 + λ + · 1 + λ dt n 1th τ 1 n 2th τ 2 s λ2 n 1 (t − τ ) n2 − λ + β1λ2 · + β2λ2 · τ 2 τn 1 τn 2  ⎧ ⎡  ( j) ⎤ ⎨ n 1 + n 1 · eλ·(t−τ ) − n 1th 1 λ2 λ·t s ·λ·e = α·⎣ · 6 + λ ⎦ ⎩ n 1th τ 1   ⎡ ( j) ⎤⎫ ⎬    n 2 + n 2 · eλ·t − n 2th 1 ( j) +⎣ · 1 + λ ⎦ · s λ2 + s λ2 · eλ·t n 2th τ 2 ⎭

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

−

   ( j) s λ2 + s λ2 · eλ·t

  ( j) n 1 + n 1 · eλ·(t−τ )

τ λ2

τn 1

+ β2λ2 · 

s

λ2

·λ·e

λ·t

s

·λ·e

λ·t

τn 2



  ( j) n 1 − n 1th 1 n 1 · eλ·(t−τ ) = α· · 6 + λ + · 6 n 1th τ 1 n 1th  ( j)      j) n 2 − n 2th 1 n 2 · eλ·t ( + · 1 + λ + · 1 + s λ2 · eλ·t · s λ2 n 2th τ 2 n 2th  λ ( j) ( j) s 2 s λ2 · eλ·t n 1 · eλ·(t−τ ) λ2 n 1 − + β · + β1λ2 · − 1 λ λ τ 2 τ 2 τn 1 τn 1 ( j)

+ β2λ2 ·

n2 n 2 · eλ·t + β2λ2 · τn 2 τn 2

+ β1λ2 ·

n n 1 · eλ·(t−τ ) n 2 · eλ·t + β2λ2 · 2 + β2λ2 · τn 1 τn 2 τn 2



λ2

 ( j) n 2 + n 2 · eλ·t

+ β1λ2 · 

847

  ( j)   ( j) n 1 − n 1th 1 n 1 · eλ·(t−τ ) =α· · 6 + λ + · 6 · s λ2 + s λ2 · eλ·t n 1th τ 1 n 1th  ( j)     ( j) n 2 − n 2th 1 n 2 · eλ·t + · 1 + λ + · 1 · s λ2 + s λ2 · eλ·t n 2th τ 2 n 2th  λ ( j) ( j) s 2 n s λ2 · eλ·t − + β1λ2 · 1 − λ λ τ 2 τ 2 τn 1 ( j)



s

λ2

·λ·e

λ·t

 ( j) n 1 − n 1th 1 =α· · 6 + λ · [s λ2 ]( j) n 1th τ 1  ( j)  n 1 − n 1th 1 n 1 · eλ·(t−τ ) + · 6 + λ · s λ2 · eλ·t + · 6 · [s λ2 ]( j) n 1th τ 1 n 1th 

 ( j) n 2 − n 2th eλ·(t−τ ) 1 λ2 λ·t + + · 6 · n 1 · s · e · 1 + λ · [s λ2 ]( j) n 1th n 2th τ 2  ( j)  n 2 − n 2th 1 n 2 · eλ·t + · 1 + λ · s λ2 · eλ·t + · 1 · [s λ2 ]( j) n 2th τ 2 n 2th ( j)

+

eλ·t [s λ2 ]( j) s λ2 · eλ·t n · 1 · n 2 · s λ2 · eλ·t − − + β1λ2 · 1 λ λ 2 2 n 2th τ τ τn 1

+ β1λ2 ·

( j)

n 1 · eλ·(t−τ ) n n 2 · eλ·t + β2λ2 · 2 + β2λ2 · τn 1 τn 2 τn 2

848

8 Dual-Wavelength Laser Systems Stability …

Assumption: n 1 · s λ2 ≈ 0; n 2 · s λ2 ≈ 0 

s

λ2

·λ·e

λ·t

 ( j) n 1 − n 1th 1 =α· · 6 + λ · [s λ2 ]( j) n 1th τ 1  ( j)   n 1 − n 1th 1 n 1 · eλ·(t−τ ) λ2 λ·t λ2 ( j) + · 6 + λ · s · e + · 6 · [s ] n 1th τ 1 n 1th  ( j)  n 2 − n 2th 1 · 1 + λ · [s λ2 ]( j) + n 2th τ 2  ( j)  n 2 − n 2th 1 n 2 · eλ·t + · 1 + λ · s λ2 · eλ·t + · 1 · [s λ2 ]( j) n 2th τ 2 n 2th ( j)

−

[s λ2 ]( j) s λ2 · eλ·t n − + β1λ2 · 1 λ λ 2 2 τ τ τn 1

+ β1λ2 · 

( j)

n 1 · eλ·(t−τ ) n n 2 · eλ·t + β2λ2 · 2 + β2λ2 · τn 1 τn 2 τn 2

 ( j) n 1 − n 1th 1 · 6 + λ · [s λ2 ]( j) n 1th τ 1  ( j)  ( j) n 2 − n 2th 1 [s λ2 ]( j) λ2 n 1 + · 1 + λ · [s λ2 ]( j) − + β · 1 n 2th τ 2 τ λ2 τn 1   ( j) ( j) n n 1 − n 1th 1 + β2λ2 · 2 + α · · 6 + λ · s λ2 · eλ·t τn 2 n 1th τ 1   ( j) n 2 − n 2th n 1 · eλ·(t−τ ) 1 λ2 ( j) +α· · 6 · [s ] + · 1 + λ · s λ2 · eλ·t n 1th n 2th τ 2

s λ2 · λ · eλ·t = α ·

+

n 2 · eλ·t s λ2 · eλ·t n 1 · eλ·(t−τ ) n 2 · eλ·t · 1 · [s λ2 ]( j) − + β1λ2 · + β2λ2 · λ n 2th τ 2 τn 1 τn 2

At fixed points:  α·

( j)

n 1 − n 1th 1 · 6 + λ n 1th τ 1 ( j)

− s

· [s λ2 ]( j) +

( j)

n 2 − n 2th 1 · 1 + λ n 2th τ 2

 · [s λ2 ]( j)

( j)

[s λ2 ]( j) n n + β1λ2 · 1 + β2λ2 · 2 = 0 λ 2 τ τn 1 τn 2 

λ2





·λ·e

λ·t

=α·

( j)

n 1 − n 1th 1 · 6 + λ n 1th τ 1



· s λ2 · eλ·t

n 1 · eλ·(t−τ ) +α· · 6 · [s λ2 ]( j) + n 1th +



( j)

n 2 − n 2th 1 · 1 + λ n 2th τ 2

 · s λ2 · eλ·t

n 2 · eλ·t s λ2 · eλ·t n 1 · eλ·(t−τ ) n 2 · eλ·t · 1 · [s λ2 ]( j) − + β1λ2 · + β2λ2 · n 2th τ λ2 τn 1 τn 2

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

849

1 1 1 1 +α· · 6 · [s λ2 ]( j) · e−λ·τ · n 1 + β2λ2 · + · 1 · [s λ2 ]( j) · n 2 τn 1 n 1th τn 2 n 2th   ( j)     ( j) n 2 − n 2th n 1 − n 1th 1 1 1 + α· · 6 + λ + · 1 + λ − λ · s λ2 n 1th τ 1 n 2th τ 2 τ 2

β1λ2 ·

− s λ2 · λ = 0

We can summary our system arbitrarily small increment equations:

1 1 · e−λ·τ · 6 · [s λ1 ]( j) + 1. − n 1 · λ − n 1 · (1 + α) · n 1th τn 1  ( j)  e·v n 1 − n 1th 1 + · n2 − · 6 + λ · s λ1 e · d1 n 1th τ 1  ( j)  n 1 − n 1th 1 −α· · 6 + λ · s λ2 = 0 n 1th τ 1

e·v 1 1 + · 1 · [s λ2 ]( j) + 2. − n 2 · λ − n 2 · e · d2 n 2th τn 2  ( j)  n 2 − n 2th 1 · 1 + λ · s λ2 = 0 − n 2th τ 2 

1 1 λ1 λ1 ( j) · n 1 · e−λ·τ 3. · 6 · [s ] + β1 · n 1th τn 1 ( j)

+

4.

n 1 − n 1th · 6 · s λ1 − s λ1 · λ = 0 n 1th

1 1 1 1 β1λ2 · +α· · 6 · [s λ2 ]( j) · e−λ·τ · n 1 + β2λ2 · + · 1 · [s λ2 ]( j) · n 2 τn 1 n 1th τn 2 n 2th   ( j)   ( j)   n 1 − n 1th n 2 − n 2th 1 1 1 + α· · 6 + λ + · 1 + λ − λ · s λ2 n 1th τ 1 n 2th τ 2 τ 2 − s λ2 · λ = 0

We define in the next table (Table 8.2) some global parameters for simplicity. We can summary our system arbitrarily small increment equations: − n 1 · λ − n 1 · 1 · e−λ·τ + 2 · n 2 − 3 · s λ1 − 4 · s λ2 = 0 − n 2 · λ − n 2 ·  5 −  6 · s λ2 = 0 7 · n 1 · e−λ·τ + 8 · s λ1 − s λ1 · λ = 0 9 · e−λ·τ · n 1 + 10 · n 2 + 11 · s λ2 − s λ2 · λ = 0

850

8 Dual-Wavelength Laser Systems Stability …

Table 8.2 Asymmetric dual quantum well lasers and dual-wavelength Lasing system global parameters Parameter

Expression

1

(1 + α) ·

2

e·v e·d1

3

n 1 −n 1th n 1th

( j)



4

α·

1 n 1th

 ( j) · 6 · s λ1 +

· 6 +

( j)

n 1 −n 1th n 1th

1 τ λ1

1 τn 1



· 6 +

1

τ λ1

 ( j) · 1 · s λ2 +

5

e·v e·d2

6

n 2 −n 2th n 2th

7

1 n 1th

8

n 1 −n 1th n 1th

· 6

9

β1λ2 ·

+α·

10

 ( j) 1 β2λ2 · τn1 + n 2th · 1 · s λ2 2  ( j)  ( j) n −n n −n α · 1 n 1th 1th · 6 + τ 1λ1 + 2 n 2th 2th · 1 +

+

1 n 2th

( j)

1 τ λ2

 ( j) · 6 · s λ1 + β1λ1 ·

( j)

11

· 1 +

1 τn 1

1 τn 2

1 n 1th

1 τn 1

 ( j) · 6 · s λ2

1

τ λ2

−

1 τ λ2

⎛ ⎞ ⎞ n1 −λ − 1 · e−λ·τ 2 −3 −4 ⎜ ⎟ ⎜n ⎟ ⎜ 0 −λ − 5 0 −6 ⎟ ⎟·⎜ 2 ⎟=0 ⎜ ⎜ s λ1 ⎟ ⎠ ⎝ 7 · e−λ·τ 0 8 − λ 0 ⎝ ⎠ −λ·τ 10 0 11 − λ 9 · e s λ2 ⎛

⎛

⎞ −λ − 1 · e−λ·τ 2 −3 −4 ⎜ 0 −λ − 5 0 −6 ⎟ ⎟ A−λ· I =⎜ ⎝ 7 · e−λ·τ ⎠ 0 8 − λ 0 −λ·τ 10 0 11 − λ 9 · e det(A − λ · I ) = 0  ⎛ ⎞ −λ − 1 · e−λ·τ 2 −3 −4 ⎜ 0 0 −6 ⎟ (−λ − 5 ) ⎟ det(A − λ · I ) = det ⎜ −λ·τ ⎝ ⎠ 7 · e 0 0 (8 − λ) −λ·τ 10 0 9 · e (11 − λ)

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

851

⎛ ⎞ 0 −6 (−λ − 5 )   ⎠ = −λ − 1 · e−λ·τ · det ⎝ 0 0 (8 − λ) 0 10 (11 − λ) ⎛ ⎞ 0 0 −6 ⎠ − 2 · det ⎝ 7 · e−λ·τ (8 − λ) 0 −λ·τ 0 9 · e (11 − λ) ⎛ ⎞ 0 (−λ − 5 ) −6 ⎠ − 3 · det ⎝ 7 · e−λ·τ 0 0 −λ·τ 10 9 · e (11 − λ) ⎛ ⎞ 0 0 (−λ − 5 ) + 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ 9 · e−λ·τ

10

0

Step 1: ⎛

⎞ 0 −6 (−λ − 5 ) ⎠ det⎝ 0 0 (8 − λ) 0 10 (11 − λ) ⎛

⎞ 0 −6 (−λ − 5 ) ⎠ det ⎝ 0 0 (8 − λ) 0 10 (11 − λ)  0 (8 − λ) = (−λ − 5 ) · det 0 (11 − λ)  0 (8 − λ) − 6 · det 10 0 ⎞ ⎛ 0 −6 (−λ − 5 ) ⎠ det ⎝ 0 0 (8 − λ) 0 10 (11 − λ) = (−λ − 5 ) · (8 − λ) · (11 − λ) + 6 · 10 · (8 − λ) ⎛

⎞ 0 −6 (−λ − 5 ) ⎠ det ⎝ 0 0 (8 − λ) 0 10 (11 − λ) = [−5 · 8 · 11 + 6 · 10 · 8 ] + [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · λ

852

8 Dual-Wavelength Laser Systems Stability …

+ [11 + 8 − 5 ] · λ2 − λ3 ⎛

⎞ 0 −6 (−λ − 5 ) ⎠ (−λ − 1 · e−λ·τ ) · det ⎝ 0 (8 − λ) 0 0 (11 − λ) 10 = (−λ − 1 · e−λ·τ ) · ([−5 · 8 · 11 + 6 · 10 · 8 ] + [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · λ + [11 + 8 − 5 ] · λ2 − λ3 ) ⎛

⎞ 0 −6 (−λ − 5 ) ⎠ (−λ − 1 · e−λ·τ ) · det ⎝ 0 (8 − λ) 0 0 (11 − λ) 10 = (−λ − 1 · e−λ·τ ) · ([−5 · 8 · 11 + 6 · 10 · 8 ] + [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · λ + [11 + 8 − 5 ] · λ2 − λ3 ) = −λ · [−5 · 8 · 11 + 6 · 10 · 8 ] − [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · λ2 − [11 + 8 − 5 ] · λ3 + λ4 + {−[−5 · 8 · 11 + 6 · 10 · 8 ] · 1 − [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · 1 · λ − 1 · [11 + 8 − 5 ] · λ2 + 1 · λ3 } · e−λ·τ We define for simplicity some global parameters: ϒ1 = −5 ·8 ·11 +6 ·10 ·8 ϒ2 = −8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ; ϒ3 = 11 + 8 − 5 ; ϒ6 = 1 · [11 + 8 − 5 ] ϒ4 = −[−5 · 8 · 11 + 6 · 10 · 8 ] · 1 ; ϒ5 = [−8 · 11 + 5 · 11 + 5 · 8 − 6 · 10 ] · 1 ⎛ ⎞ 0 −6 (−λ − 5 )   ⎠ −λ − 1 · e−λ·τ · det ⎝ 0 0 (8 − λ) 0 10 (11 − λ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4   + ϒ4 − ϒ5 · λ − ϒ6 · λ2 + 1 · λ3 · e−λ·τ

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

853

Step 2: ⎛

⎞ 0 0 −6 ⎠ 2 · det ⎝ 7 · e−λ·τ (8 − λ) 0 −λ·τ 0 9 · e (11 − λ) ⎛

⎞ 0 0 −6 ⎠ 2 · det ⎝ 7 · e−λ·τ (8 − λ) 0 −λ·τ 0 9 · e (11 − λ)  −λ·τ 7 · e (8 − λ) = 2 · (−6 ) · det 9 · e−λ·τ 0 = 2 · 6 · 9 · e−λ·τ · (8 − λ) = (2 · 6 · 9 · 8 − 2 · 6 · 9 · λ) · e−λ·τ We define for simplicity some global parameters: ϒ7 = 2 · 6 · 9 · 8 ; ϒ8 = 2 · 6 · 9 ⎛

⎞ 0 0 −6 ⎠ = (ϒ7 − ϒ8 · λ) · e−λ·τ 2 · det ⎝ 7 · e−λ·τ (8 − λ) 0 −λ·τ 0 9 · e (11 − λ) Step 3: ⎞ 0 (−λ − 5 ) −6 ⎠ 3 · det ⎝ 7 · e−λ·τ 0 0 10 9 · e−λ·τ (11 − λ) ⎛ ⎞ 0 (−λ − 5 ) −6 ⎠ 3 · det ⎝ 7 · e−λ·τ 0 0 −λ·τ 10 9 · e (11 − λ)  0 7 · e−λ·τ = 3 · −(−λ − 5 ) · det 9 · e−λ·τ (11 − λ)  7 · e−λ·τ 0 −6 · det 9 · e−λ·τ 10 ⎛ ⎞ 0 (−λ − 5 ) −6 ⎠ 3 · det ⎝ 7 · e−λ·τ 0 0 −λ·τ 10 9 · e (11 − λ) ⎛

= 3 · {(λ + 5 ) · (11 − λ) · 7 − 6 · 7 · 10 } · e−λ·τ

854

8 Dual-Wavelength Laser Systems Stability …

⎛

⎞ 0 (−λ − 5 ) −6 ⎠ 3 · det ⎝ 7 · e−λ·τ 0 0 −λ·τ 10 9 · e (11 − λ) = {3 · (5 · 11 − 6 · 10 ) · 7

& +λ · (11 − 5 ) · 3 · 7 − λ2 · 3 · 7 · e−λ·τ

We define for simplicity some global parameters: ϒ9 = 3 · (5 · 11 − 6 · 10 ) · 7 ϒ10 = (11 − 5 ) · 3 · 7 ; ϒ11 = 3 · 7 ⎞ 0 (−λ − 5 ) −6   ⎠ = ϒ9 + λ · ϒ10 − λ2 · ϒ11 · e−λ·τ 3 · det ⎝ 7 · e−λ·τ 0 0 10 9 · e−λ·τ (11 − λ) ⎛

Step 4: ⎛

⎞ 0 0 (−λ − 5 ) 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ −λ·τ 9 · e 10 0 ⎞ ⎛ 0 0 (−λ − 5 ) 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ −λ·τ 9 · e 10 0  −λ·τ 7 · e (8 − λ) = −4 · (−λ − 5 ) · det 9 · e−λ·τ 0 ⎛ ⎞ 0 0 (−λ − 5 ) 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ −λ·τ 9 · e 10 0 = −4 · 9 · (λ + 5 ) · (8 − λ) · e−λ·τ ⎛

⎞ 0 0 (−λ − 5 ) 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ −λ·τ 9 · e 10 0 = {−4 · 9 · 8 · 5 − λ · 4 · 9 · [8 − 5 ] & +4 · 9 · λ2 · e−λ·τ We define for simplicity some global parameters:

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

855

ϒ12 = −4 · 9 · 8 · 5 ; ϒ13 = 4 · 9 · [8 − 5 ]

⎛

ϒ14

⎞ 0 0 (−λ − 5 ) = 4 · 9 ; 4 · det ⎝ 7 · e−λ·τ 0 (8 − λ) ⎠ −λ·τ 9 · e 10 0   −λ·τ 2 = ϒ12 − λ · ϒ13 + ϒ14 · λ · e

We can summary system determinant elements (Steps 1–4) in Table 8.3.  ⎞ ⎛ 2 −3 −4 −λ − 1 · e−λ·τ ⎜ 0 0 −6 ⎟ (−λ − 5 ) ⎟ det(A − λ · I ) = det ⎜ −λ·τ ⎠ ⎝ 7 · e 0 0 (8 − λ) −λ·τ 10 0 9 · e (11 − λ) ' 2 3 4 = −λ · ϒ1 − ϒ2 · λ − ϒ3 · λ + λ  &  + ϒ4 − ϒ5 · λ − ϒ6 · λ2 + 1 · λ3 · e−λ·τ   − (ϒ7 − ϒ8 · λ) · e−λ·τ − ϒ9 + λ · ϒ10 − λ2 · ϒ11 · e−λ·τ   + ϒ12 − λ · ϒ13 + ϒ14 · λ2 · e−λ·τ  ⎞ ⎛ −3 −4 2 −λ − 1 · e−λ·τ ⎜ 0 0 −6 ⎟ (−λ − 5 ) ⎟ det(A − λ · I ) = det ⎜ −λ·τ ⎠ ⎝ 7 · e 0 0 (8 − λ) −λ·τ 10 0 9 · e (11 − λ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4 + [(ϒ4 − ϒ7 − ϒ9 + ϒ12 ) + (ϒ8 − ϒ5 − ϒ10 − ϒ13 ) · λ + (ϒ11 − ϒ6 + ϒ14 ) · λ2 + 1 · λ3 ] · e−λ·τ We define for simplicity some global parameters:

0 = ϒ4 − ϒ7 − ϒ9 + ϒ12

1 = ϒ8 − ϒ5 − ϒ10 − ϒ13 ; 2 = ϒ11 − ϒ6 + ϒ14  ⎞ ⎛ 2 −3 −4 −λ − 1 · e−λ·τ ⎜ 0 0 −6 ⎟ (−λ − 5 ) ⎟ det(A − λ · I ) = det ⎜ −λ·τ ⎠ ⎝ 7 · e 0 0 (8 − λ) −λ·τ 10 0 9 · e (11 − λ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4

9 · e−λ·τ

10

(8 − λ)

0 ⎞

10

0

0

(−λ − 5 )

0

9 · e−λ·τ

⎜ −λ·τ 4 · det ⎜ ⎝ 7 · e

⎛

(11 − λ) ⎞ 0 (−λ − 5 ) ⎟ 0 (8 − λ) ⎟ ⎠ 10 0

0 0 −6 ⎟ ⎜ −λ·τ ( − λ) ⎟ 2 · det ⎜  · e 0 8 ⎠ ⎝ 7 9 · e−λ·τ 0 (11 − λ) ⎞ ⎛ 0 (−λ − 5 ) −6 ⎟ ⎜ −λ·τ ⎟ 3 · det ⎜ 0 0 ⎠ ⎝ 7 · e

⎛

(−λ − 1

⎜ ⎝

⎛

· e−λ·τ ) · det ⎜

Determinant element

(11 − λ)

0

−6 ⎟ ⎟ ⎠

⎞

  ϒ12 − λ · ϒ13 + ϒ14 · λ2 · e−λ·τ

  ϒ9 + λ · ϒ10 − λ2 · ϒ11 · e−λ·τ

(ϒ7 − ϒ8 · λ) · e−λ·τ

− λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4   + ϒ4 − ϒ5 · λ − ϒ6 · λ2 + 1 · λ3 · e−λ·τ

Expression

Table 8.3 Asymmetric dual quantum well lasers and dual-wavelength Lasing system determinant elements

856 8 Dual-Wavelength Laser Systems Stability …

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

857

  + 0 + 1 · λ + 2 · λ2 + 1 · λ3 · e−λ·τ det(A − λ · I ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4   + 0 + 1 · λ + 2 · λ2 + 1 · λ3 · e−λ·τ D(λ, τ ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4   + 0 + 1 · λ + 2 · λ2 + 1 · λ3 · e−λ·τ D(λ, τ ) = Pn (λ, τ ) + Q m (λ, τ ) · e−λ·τ ; n, m ∈ N0 ; n = 4; m = 3; n > m Pn (λ, τ ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4 ; n = 4 Q m (λ, τ ) = 0 + 1 · λ + 2 · λ2 + 1 · λ3 ; m = 3 Pn (λ, τ ) =

n=4 (

pk (τ ) · λk

k=0

= p0 (τ ) + p1 (τ ) · λ + p2 (τ ) · λ2 + p3 (τ ) · λ3 + p4 (τ ) · λ4 p0 (τ ) = 0; p1 (τ ) = −ϒ1 ; p2 (τ ) = −ϒ2 ; p3 (τ ) = −ϒ3 ; p4 (τ ) = 1 Q m (λ, τ ) =

m=3 (

qk (τ ) · λk = q0 (τ ) + q1 (τ ) · λ + q2 (τ ) · λ2 + q3 (τ ) · λ3

k=0

q0 (τ ) = 0 q1 (τ ) = 1 ; q2 (τ ) = 2 ; q3 (τ ) = 1 The homogeneous system for n 1 , n 2 , s λ1 , s λ2 leads to a characteristic equation for the eigenvalue λ)having the form D(λ,) τ ) = P(λ, τ ) + Q(λ, τ ) · 4 3 j j a · λ ; Q(λ) = e−λ·τ = 0; and P(λ) = j=0 c j · λ . The coefficients ' & j=0 j a j (qi , qk , τ ), c j (qi , qk , τ ) ∈ R depend on qi , qk and delay τ . qi , qk are any Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s parameters, other parameters kept as a constant [2, 3]. a0 = 0; a1 = −ϒ1 ; a2 = −ϒ2 ; a3 = −ϒ3 ; a4 = 1 c0 = 0 ; c1 = 1 ; c2 = 2 ; c3 = 1 Unless strictly necessary, the designation of the varied arguments (qi , qk ) will subsequently be omitted from P, Q, a j and c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 for qi , qk ∈ R+ ; that is, λ = 0 is not of P(λ) + Q(λ) · e−λ·τ =

858

8 Dual-Wavelength Laser Systems Stability …

0. Furthermore, P(λ), Q(λ) are analytic functions of λ, for which the following requirements of the analysis (Kuang and Cong 2005) [2] can also be verified in the present case: 1. If λ = i · ω; ω ∈ R, then P(i · ω) + Q(i · ω) = 0. | is bounded for |λ| → ∞; Reλ ≥ 0. No roots bifurcation from ∞. 2. If | Q(λ) P(λ) 3. F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 has a finite number of zeros. Indeed, this is a polynomial in ω. 4. Each positive root ω(qi , qk ) of F(ω) = 0 is continuous and differentiable with respect to qi , qk . We assume that Pn (λ, τ ) and Q m (λ, τ ) cannot have common imaginary roots. That is for any real number ω: Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 and Pn (λ = i · ω, τ ) = p0 (τ ) + p1 (τ ) · i · ω − p2 (τ ) · ω2 − p3 (τ ) · i · ω3 + p4 (τ ) · ω4   Pn (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + p4 (τ ) · ω4 + p1 (τ ) · ω − p3 (τ ) · ω3 · i   Pn (λ = i · ω, τ ) = ϒ2 · ω2 + ω4 + −ϒ1 · ω + ϒ3 · ω3 · i Q m (λ = i · ω, τ ) = q0 (τ ) + q1 (τ ) · i · ω − q2 (τ ) · ω2 − q3 (τ ) · i · ω3   Q m (λ = i · ω, τ ) = q0 (τ ) − q2 (τ ) · ω2 + q1 (τ ) · ω − q3 (τ ) · ω3 · i   Q m (λ = i · ω, τ ) = 0 − 2 · ω2 + 1 · ω − 1 · ω3 · i Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = p0 (τ ) − p2 (τ ) · ω2 + p4 (τ ) · ω4   + p1 (τ ) · ω − p3 (τ ) · ω3 · i + q0 (τ ) − q2 (τ ) · ω2   + q1 (τ ) · ω − q3 (τ ) · ω3 · i = 0   Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = ϒ2 · ω2 + ω4 + −ϒ1 · ω + ϒ3 · ω3 · i   + 0 − 2 · ω2 + 1 · ω − 1 · ω3 · i = 0 Pn (λ = i · ω, τ ) + Q m (λ = i · ω, τ ) = 0 + (ϒ2 − 2 ) · ω2 + ω4 & ' + ( 1 − ϒ1 ) · ω + (ϒ3 − 1 ) · ω3 · i = 0  2  2 |P(i · ω, τ )|2 = ϒ2 · ω2 + ω4 + −ϒ1 · ω + ϒ3 · ω3 = ϒ22 · ω4 + 2 · ϒ2 · ω6 + ω8 + ϒ12 · ω2 − 2 · ϒ1 · ϒ3 · ω4

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

859

+ ϒ32 · ω6  2  2 |P(i · ω, τ )|2 = ϒ2 · ω2 + ω4 + −ϒ1 · ω + ϒ3 · ω3   = ϒ12 · ω2 + ϒ22 − 2 · ϒ1 · ϒ3 · ω4   + 2 · ϒ2 + ϒ32 · ω6 + ω8 2  2  |Q(i · ω, τ )|2 = 0 − 2 · ω2 + 1 · ω − 1 · ω3 = 20 − 2 · 0 · 2 · ω2 + 22 · ω4 + 21 · ω2 − 2 · 1 · 1 · ω4 + 21 · ω6 2  2  |Q(i · ω, τ )|2 = 0 − 2 · ω2 + 1 · ω − 1 · ω3   = 20 + 21 − 2 · 0 · 2 · ω2   + 22 − 2 · 1 · 1 · ω4 + 21 · ω6 F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2   = ϒ12 · ω2 + ϒ22 − 2 · ϒ1 · ϒ3 · ω4     + 2 · ϒ2 + ϒ32 · ω6 + ω8 − { 20 + 21 − 2 · 0 · 2 · ω2   + 22 − 2 · 1 · 1 · ω4 + 21 · ω6 } F(ω, τ ) = |P(i · ω, τ )|2 − |Q(i · ω, τ )|2   = − 20 + ϒ12 − 21 + 2 · 0 · 2 · ω2   + ϒ22 − 2 · ϒ1 · ϒ3 − 22 + 2 · 1 · 1 · ω4   + 2 · ϒ2 + ϒ32 − 21 · ω6 + ω8 We define the following parameters for simplicity: 0 , 2 , 4 , 6 , 8 . 0 = − 20 ; 2 = ϒ12 − 21 + 2 · 0 · 2 4 = ϒ22 − 2 · ϒ1 · ϒ3 − 22 + 2 · 1 · 1 6 = 2·ϒ2 +ϒ32 −21 ; 8 = 1. Hence F(ω, τ ) = 0 implies and its roots are given by solving the above polynomial.

)4 k=0

2·k ·ω2·k = 0

PR (iω, τ ) = ϒ2 · ω2 + ω4 ; PI (iω, τ ) = −ϒ1 · ω + ϒ3 · ω3 ; Q R (iω, τ ) = 0 − 2 · ω2 ; Q I (iω, τ ) = 1 · ω − 1 · ω3

860

8 Dual-Wavelength Laser Systems Stability …

sin θ (τ ) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos θ (τ ) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

k → j; pk (τ ) → a j ; qk (τ ) → c j ; n, m ∈ N0 ; n = 4; m = 3; n > m Pn (λ, τ ) → P(λ); Q m (λ, τ ) → Q(λ); P(λ) =

4 (

a j · λ j ; Q(λ) =

j=0

3 (

cj · λj

j=0

P(λ) = a0 +a1 · λ+a2 · λ2 +a3 · λ3 +a4 · λ4 ; Q(λ) = c0 +c1 · λ+c2 · λ2 +c3 · λ3 P(λ) = −λ · ϒ1 − ϒ2 · λ2 − ϒ3 · λ3 + λ4 ; Q(λ) = 0 + 1 · λ + 2 · λ2 + 1 · λ3 n, m ∈ N0 ; n > m and a j , c j : R0+ → R are continuous and differentiable function of τ such that a0 + c0 = 0. In the following “−” denoted complex and conjugate. in λ and differentiable in τ . The coefficients  P(λ), Q(λ) are analytic functions   λ1 λ2 λ2 a j τn 1 , τn 2 , β1 , β1 , β2 , τ, . . . ∈ R and c j τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . ∈ R depend on Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s parameters, τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . values. Unless strictly necessary, the designation of the varied arguments: τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . will subsequently be omitted from P, Q, a j , c j . The coefficients a j , c j are continuous and differentiable functions of their arguments, and direct substitution shows that a0 + c0 = 0 [2, 3]. a0 + c0 = 0 = ϒ4 − ϒ7 − ϒ9 + ϒ12 = 0 ∀ τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . ∈ R+ I.e. λ = 0 is not a root of the characteristic equation. Furthermore P(λ), Q(λ) are analytic functions of λ for which the following requirements of the analysis (see [2], Sect. 3.4) can also be verified in the present case. 1. If λ = i · ω; ω ∈ R then P(i · ω) + Q(i · ω) = 0, i.e. P and Q have no common imaginary roots. This condition was verified numerically in the entire ∀ τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . domain of interest. P(λ) | is bounded for |λ| → ∞; Re λ ≥ 0. No roots bifurcation from ∞. Indeed, 2. | Q(λ) 1 ·λ+c2 ·λ +c3 ·λ in the limit: | Q(λ) | = | a0 +ac01+c |. P(λ) ·λ+a2 ·λ2 +a3 ·λ3 +a4 ·λ4 2 · ω)|2 . 3. The following expressions exist: F(ω) = |P(i )·4 ω)| − |Q(i 2 2 2·k F(ω, τ ) = |P(i · ω, τ )| − |Q(i · ω, τ )| = k=0 2·k · ω has at most a finite number of zeros. Indeed, this is a polynomial in ω (degree in ω8 ). 2

3

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

861

  4. Each positive root ω τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . of F(ω) = 0 is continuous and differentiable with respect to τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . and the condition can only be assessed numerically. In addition, since the coefficients in P and Q are real, we have P(−i · ω) = P(i · ω) and Q(−i · ω) = Q(i · ω) thus, ω > 0 may be on eigenvalue of characteristic equations. The analysis consists in identifying the roots of the characteristic equation situated on the imaginary axis of the complex λ plane, whereby increasing the parameters: τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . ., Reλ may, at the its sign from (−) to (+). i.e. from a stable focus  crossing, change   λ ( j)  ( j) ( j) λ1 ( j) ( j) n1 , n2 , s ∀ j = 0, 1, 2, . . . to an unstable one, or vice versa. , s 2 E This feature may be further assessed by examining the sign of the partial derivatives with respect to τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . and any system parameters.    τn 1 = −1



∂Reλ ∂τn 1

λ=i·ω

; τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . = const

∂Reλ ; τn , β λ1 , β1λ2 , β2λ2 , τ, . . . = const ∂τn 2 λ=i·ω 1 1     ∂Reλ λ1 −1  β1 = ; τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . = const ∂β1λ1 λ=i·ω     ∂Reλ λ2 −1 ; τn 1 , τn 2 , β1λ1 , β2λ2 , τ, . . . = const  β1 = ∂β1λ2 λ=i·ω     ∂Reλ λ2 −1 ; τn 1 , τn 2 , β1λ1 , β1λ2 , τ, . . . = const  β2 = λ2 ∂β2 λ=i·ω  ∂Reλ −1 (τ ) = ; τn , τn , β λ1 , β1λ2 , β2λ2 , . . . = const ∂τ λ=i·ω 1 2 1   −1 τn 2 =



P(λ) = PR (λ)+i · PI (λ); Q(λ) = Q R (λ)+i · Q I (λ), When writing and inserting λ = i · ω into Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s characteristic equation ω must satisfy the following equations: sin(ω · τ ) = g(ω) =

−PR (iω, τ ) · Q I (iω, τ ) + PI (iω, τ ) · Q R (iω, τ ) |Q(iω, τ )|2

cos(ω · τ ) = h(ω) = −

PR (iω, τ ) · Q R (iω, τ ) + PI (iω, τ ) · Q I (iω, τ ) |Q(iω, τ )|2

where |Q(iω, τ )|2 = 0 in view of requirement (1) above, and (g, h) ∈ R. Furthermore, it follows above sin(ω · τ ) and cos(ω · τ ) equation that, by squaring and adding

862

8 Dual-Wavelength Laser Systems Stability …

the sides, ω must be a positive root of F(ω) = |P(i · ω)|2 − |Q(i · ω)|2 = 0. Note: F(ω) is independent on τ . Now it is important to notice that if τ ∈ / I (assume that / I , ω(τ ) is I ⊆ R+0 is the set) where ω(τ ) is a positive root of F(ω) and for, τ ∈ not defined. Then for all τ in I , ω(τ ) is satisfied that F(ω, τ ) = 0. Then there are no positive ω(τ ) solutions for F(ω, τ ) = 0, and we cannot have stability switches. For τ ∈ I where ω(τ ) is a positive solution of F(ω, τ ) = 0, we can define the angle θ (τ ) ∈ [0, 2 · π ] as the solution of sin θ (τ ) = . . . and cos θ (τ ) = . . .; the relation between the arguments θ (τ ) and τ · ω(τ ) for τ ∈ I must be describing below. τ · ω(τ ) = θ (τ ) + 2 · n · π ∀ n ∈ N0 . Hence we can define the maps: ; n ∈ N0 ; τ ∈ I . Let us introduce the τn : I → R+0 , is given by τn (τ ) = θ(τ )+2·n·π ω(τ ) function I → R ; Sn (τ ) = τ −τn (τ ) ; τ ∈ I ; n ∈ N0 that is continuous and differentiable in τ . In the following, the subscripts λ, ω, τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , . . . indicate the corresponding partial derivatives. Let   us first concentrate on(x), remember  λ1 λ2 λ2 in λ τn 1 , τn 2 , β1 , β1 , β2 , . . . and ω τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , . . . , and keeping all parameters except one (x) and τ . The derivation closely follows that in reference [BK]. Differentiating Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s characteristic equation P(λ) + Q(λ) · e−λ·τ = 0 with respect to specific parameter (x), and inverting the derivative, for convenience, one calculates: x = τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , τ, . . . 

∂λ ∂x

−1

=

−Pλ (λ, x) · Q(λ, x) + Q λ (λ, x) · P(λ, x) − τ · P(λ, x) · Q(λ, x) Px (λ, x) · Q(λ, x) − Q x (λ, x) · P(λ, x)

where Pλ = ∂∂λP , . . . etc., substituting λ = i · ω and bearing P(−i · ω) = P(i · ω); Q(−i · ω) = Q(i · ω) then i · Pλ (i · ω) = Pω (i · ω); i · Q λ (i · ω) = Q ω (i · ω) and that on the surface |P(iω)|2 = |Q(iω)|2 , one obtain: 

∂λ ∂x

−1

 |λ=i·ω =

i · Pω (i · ω, x) · P(i · ω, x) + i · Q λ (i · ω, x) · Q(λ, x) − τ · |P(i · ω, x)|2



Px (i · ω, x) · P(i · ω, x) − Q x (i · ω, x) · Q(i · ω, x)

Upon separating into real and imaginary parts, with P = PR + i · PI ; Q = Q R + i · Q I and Pω = PRω + i · PI ω ; Q ω = Q Rω + i · Q I ω ; Px = PRx + i · PI x ; Q x = Q Rx + i · Q I x ; P 2 = PR2 + PI2 , When (x) can be any Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s parameters τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , . . . and time delay τ etc. Where for convenience, we have dropped the arguments (i · ω, x), and where Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]; Fx = 2 · [(PRx · PR + PI x · PI ) − (Q Rx · Q R + Q I x · Q I )] and ωx = − FFωx . We define U and V: U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ), V = (PR · PI x − PI · PRx ) − (Q R · Q I x − Q I · Q Rx ), we choose our specific parameter as time delay x = τ : PR = ϒ2 · ω2 + ω4 ; PI = −ϒ1 · ω + ϒ3 · ω3 Q R = 0 − 2 · ω2 ; Q I = 1 · ω − 1 · ω3 ; PRω = 2 · ϒ2 · ω + 4 · ω3 ; PI ω = −ϒ1 + 3 · ϒ3 · ω2 ; Q Rω = −2 · 2 · ω Q I ω = 1 − 3 · 1 · ω2 ; PRτ = 0; PI τ = 0; Q Rτ = 0; Q I τ = 0; Fτ = 0; PR · PI τ − PI · PRτ = 0; Q R · Q I τ − Q I · Q Rτ = 0

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

863

V = (PR · PI τ − PI · PRτ ) − (Q R · Q I τ − Q I · Q Rτ ) = 0. Fω = . . . Elements PRω · PR     = 2 · ϒ2 · ω + 4 · ω3 · ϒ2 · ω2 + ω4 = 2 · ϒ2 · ϒ2 · ω3 + 6 · ϒ2 · ω5 + 4 · ω7

    PI ω · PI = −ϒ1 + 3 · ϒ3 · ω2 · −ϒ1 · ω + ϒ3 · ω3 = ϒ12 · ω − 4 · ϒ3 · ϒ1 · ω3 + 3 · ϒ3 · ϒ3 · ω5   Q Rω · Q R = −2 · 2 · ω · 0 − 2 · ω2 = −2 · 2 · 0 · ω + 2 · 22 · ω3     Q I ω · Q I = 1 − 3 · 1 · ω2 · 1 · ω − 1 · ω3 = 21 · ω − 4 · 1 · 1 · ω3 + 3 · 21 · ω5 U = . . . Elements:     PR · PI ω = ϒ2 · ω2 + ω4 · −ϒ1 + 3 · ϒ3 · ω2 = −ϒ1 · ϒ2 · ω2 + (3 · ϒ2 · ϒ3 − ϒ1 ) · ω4 + 3 · ϒ3 · ω6     PI · PRω = −ϒ1 · ω + ϒ3 · ω3 · 2 · ϒ2 · ω + 4 · ω3 = −2 · ϒ1 · ϒ2 · ω2 + 2 · (ϒ3 · ϒ2 − 2 · ϒ1 ) · ω4 + 4 · ϒ3 · ω6     Q R · Q I ω = 0 − 2 · ω2 · 1 − 3 · 1 · ω2 = 0 · 1 − (3 · 0 · 1 + 2 · 1 ) · ω2 + 3 · 2 · 1 · ω4   Q I · Q Rω = 1 · ω − 1 · ω3 · (−2 · 2 · ω) = −2 · 2 · 1 · ω2 + 2 · 1 · 2 · ω4 We can summary our last results in the next table (Table 8.4). F(ω, τ ) = 0, differentiating with respect to τ and we get Fω · = − FFωτ . τ ∈ I ⇒ ∂ω ∂τ −1

 (τ ) =



∂Reλ ∂τ

λ=iω

;

∂ω Fτ = ωτ = − ; ∂τ Fω

∂ω ∂τ

+ Fτ = 0;

864

8 Dual-Wavelength Laser Systems Stability …

Table 8.4 Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s stability analyses Fω , U elements Fω , U elements

Expression

Fω , U elements

PRω · PR

2 · ϒ2 · ϒ2 · ω3 + 6 · ϒ2 · ω5 + 4 · ω7 PR · PI ω

−ϒ1 · ϒ2 · ω2 + (3 · ϒ2 · ϒ3 − ϒ1 ) · ω4 + 3 · ϒ3 · ω6

PI ω · PI

ϒ12 ·ω−4·ϒ3 ·ϒ1 ·ω3 +3·ϒ3 ·ϒ3 ·ω5 PI · PRω

−2 · ϒ1 · ϒ2 · ω2 + 2 · (ϒ3 · ϒ2 − 2 · ϒ1 ) · ω4 + 4 · ϒ3 · ω6

Q Rω · Q R −2 · 2 · 0 · ω + 2 · 22 · ω3 QIω · QI

Expression

Q R · Q I ω · − (3 · ·  + · ) · 0 1 0 1 2 1 ω 2 + 3 · 2 · 1 · ω 4

21 · ω − 4 · 1 · 1 · ω3 + 3 · 21 · ω5 Q I · Q Rω −2 · 2 · 1 · ω2 + 2 · 1 · 2 · ω4

−2 · [U + τ · |P|2 + i · Fω  (τ ) = Re Fτ + i · 2 · [V + ω · |P|2 ]

 ' −1 & ∂Reλ sign  (τ ) = sign ∂τ λ=iω   ' −1 & V + ∂ω ·U ∂ω ∂τ ·τ sign  (τ ) = sign{Fω } · sign +ω+ |P|2 ∂τ

−1

We shall presently examine the possibility of stability transitions (bifurcations) Asymmetric dual quantum well lasers and dual-wavelength Lasing system, about the  ( j)  ( j)  ( j) ( j)  equilibrium points, E ( j) ; j = 0, 1, 2, . . ., E ( j) n 1 , n 2 , s λ1 , s λ2 ∀j = 0, 1, 2, . . .. The analysis consists in identifying the roots of our system characteristic equation situated on the imaginary axis of the complex λ-plane, where by increasing the delay parameter τ . Reλ , may at the crossing, changes its sign from − to + , i.e. from a stable focus E(*) to an unstable one, or vice versa. This feature may be further assessed by examining the sign of the partial derivatives with respect to τ [2, 3]. −1 (τ ) =



∂Reλ ∂τ

λ=i·ω

; τn 1 , τn 2 , β1λ1 , β1λ2 , β2λ2 , . . . = const

U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = −ϒ1 · ϒ2 · ω2 + (3 · ϒ2 · ϒ3 − ϒ1 ) · ω4 + 3 · ϒ3 · ω6 & ' − −2 · ϒ1 · ϒ2 · ω2 + 2 · (ϒ3 · ϒ2 − 2 · ϒ1 ) · ω4 + 4 · ϒ3 · ω6 − ( 0 · 1 − (3 · 0 · 1 + 2 · 1 ) · ω2 & ' + 3 · 2 · 1 · ω4 − −2 · 2 · 1 · ω2 + 2 · 1 · 2 · ω4 ) U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) = − 0 · 1 + (3 · 0 · 1 + ϒ1 · ϒ2 − 2 · 1 ) · ω2

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

865

+ (ϒ2 · ϒ3 + 3 · ϒ1 − 2 · 1 ) · ω4 − ϒ3 · ω6 U=

3 (

A2k · ω2·k ; A0 = − 0 · 1 ; A2 = 3 · 0 · 1 + ϒ1 · ϒ2 − 2 · 1

k=0

A4 = ϒ2 · ϒ3 + 3 · ϒ1 − 2 · 1 A6 = −ϒ3 U = (PR · PI ω − PI · PRω ) − (Q R · Q I ω − Q I · Q Rω ) =

3 (

A2k · ω2·k

k=0

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] = 2 · [(2 · ϒ2 · ϒ2 · ω3 + 6 · ϒ2 · ω5 + 4 · ω7 + ϒ12 · ω − 4 · ϒ3 · ϒ1 · ω3 + 3 · ϒ3 · ϒ3 · ω5 ) − (−2 · 2 · 0 · ω + 2 · 22 · ω3 + 21 · ω − 4 · 1 · 1 · ω3 + 3 · 21 · ω5 )] Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )]   = 2 · ϒ12 + 2 · 2 · 0 − 21 · ω   + 2 · 2 · ϒ2 · ϒ2 − 4 · ϒ3 · ϒ1 + 4 · 1 · 1 − 2 · 22 · ω3   + 2 · 6 · ϒ2 + 3 · ϒ3 · ϒ3 − 3 · 21 · ω5 + 8 · ω7   B1 = 2 · ϒ12 + 2 · 2 · 0 − 21   B3 = 2 · 2 · ϒ2 · ϒ2 − 4 · ϒ3 · ϒ1 + 4 · 1 · 1 − 2 · 22 3 (   B2·k+1 · ω2·k+1 B5 = 2 · 6 · ϒ2 + 3 · ϒ3 · ϒ3 − 3 · 21 ; B7 = 8; Fω = k=0

Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

3 (

B2·k+1 · ω2·k+1

k=0

) Then we get the expression for Fω = 3k=0 B2·k+1 · ω2·k+1 , Asymmetric dual quantum well lasers and dual-wavelength Lasing system parameter values. We find those ω, τ values which fulfill Fω (ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω for specific τ values. τ ∈ [0.001 . . . 10], we can express by 3D function Fω (ω, τ ) = 0. We plot the stability switch diagram based on different delay values of our Asymmetric dual quantum well lasers and dual-wavelength Lasing system.

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8 Dual-Wavelength Laser Systems Stability …



−1

 (τ ) = −1 (τ ) =



∂Reλ ∂τ

∂Reλ ∂τ

λ=iω



   −2 · U + τ · |P|2 + i · Fω   = Re Fτ + 2i · V + ω · |P|2 λ=iω  &   ' 2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 = 2  Fτ2 + 4 · V + ω · P 2



The stability switch occurs only on those delay values (τ ) which fit the equation: τ = ωθ++(τ(τ)) and θ+ (τ ) is the solution of sin θ (τ ) = . . . . and cos θ (τ ) = . . . . when ω = ω+ (τ ) If only ω+ is feasible. Additionally, when all Asymmetric dual quantum well lasers and dual-wavelength Lasing system’s parameters are known and the stability switch due to various time delay values τ is described in the following expression: & ' sign −1 (τ ) = sign{Fω (ω(τ ), τ )} · sign{τ · ωτ (ω(τ )) + ω(τ )

U (ω(τ )) · ωτ (ω(τ )) + V (ω(τ )) + |P(ω(τ ))|2 Remark: we know F(ω, τ ) = 0 implies its roots ωi (τ ) and finding those delays values τ which ωi is feasible. There are τ values which give complex ωi or imaginary number, then unable to analyze stability. F(ω, τ ), function is independent on τ the parameter F(ω, τ ) = 0. The results: We find those ω, τ values which fulfill F(ω, τ ) = 0. We ignore negative, complex, and imaginary values of ω. Next is to find those ω, τ values which fulfill sin θ(τ ) = . . . . and cos θ (τ ) = . . . .; sin(ω · τ ) = −PR ·Q I +PI ·Q R R +PI ·Q I ) ; cos(ω · τ ) = − (PR ·Q|Q| |Q|2 = Q 2R + Q 2I . Finally we plot the 2 |Q|2   . stability switch diagramg(τ ) = −1 (τ ) = ∂Reλ ∂τ λ=iω    & ' 2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 g(τ ) =  (τ ) = = 2  Fτ2 + 4 · V + ω · P 2 λ=iω 

  −1  ∂Reλ sign[g(τ )] = sign  (τ ) = sign ∂τ λ=iω  '  &    2 · Fω · V + ω · P 2 − Fτ · U + τ · P 2 = sign 2  Fτ2 + 4 · V + ω · P 2 2    Fτ2 + 4 · V + ω · P 2 > 0 ⇒ sign −1 (τ )      = sign Fω · V + ω · P 2 − Fτ · U + τ · P 2 −1



∂Reλ ∂τ





 −1   Fτ    2 2 sign  (τ ) = sign [Fω ] · V + ω · P − · U +τ · P Fω  −1 Fτ ∂ω ∂ F/∂ω =− ωτ = − ; ωτ = Fω ∂τ ∂ F/∂τ

8.1 Asymmetric Dual Quantum Well Lasers and Dual-Wavelength …

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Table 8.5 Asymmetric dual quantum well lasers and dual-wavelength Lasing system sign of   sign −1 (τ )     sign[Fω ] sign −1 (τ ) τ ·U sign V +ω + ω + ωτ · τ P2 +/−

+/−

+

+/−

−/ +

−

   V + ωτ · U     sign −1 (τ ) = sign [Fω ] · P 2 · + ω + ω · τ ; sign P 2 > 0 τ 2 P 

  V + ωτ · U sign −1 (τ ) = sign [Fω ] · + ω + ω · τ τ P2 

 −1  V + ωτ · U + ω + ωτ · τ sign  (τ ) = sign[Fω ] · sign P2 Fω = 2 · [(PRω · PR + PI ω · PI ) − (Q Rω · Q R + Q I ω · Q I )] =

3 (

B2·k+1 · ω2·k+1

k=0 −1 We check the sign  of  (τ ) according to the following rule (Table 8.5).  −1 proceeds from (−) to (+) respectively If, sign  (τ ) > 0, then   the crossing (stable to unstable). If, sign −1 (τ ) < 0, then the crossing proceeds from (+) to (−) respectively (unstable to stable).

8.2 Tm3+ -Doped Silica Fibre Lasers Nonlinearity and Stability Analysis Under Parameters Variation The self-pulsing of the Tm3+ -doped silica fiber laser is operated near 2 μm. There are various self-pulsing regimes for a range of pumping rates where the fiber is endpumped with a high power Nd:YAG laser which operated at 1.319 μm in a linear bidirectional cavity. There are a rich variety of nonlinear phenomena, ranging from self-pulsing to self-Q-switching and to a modulated quasi-cw wave. The Tm3+ -doped silica glasses and fiber are prepared from mesoporous silica glass by glass phaseseparation technology. The Judd–Ofelt parameters and radiation properties of Tm3+ doped silica glasses are evaluated according to the recorded absorption spectrum. Tm3+ -doped large-core double-cladding fiber with a core NA of 0.08 and fabricated by the rod-in-tube method. The fiber has a Tm3+ concentration of 10,300 ppm by weight and excellent homogeneity in the fiber core. There is a phenomenon of pairinduced quenching (PIQ) which is compatible with excited-state absorption (ESA). There is influence of thulium concentration on ion-pair production. In highly doped

868

8 Dual-Wavelength Laser Systems Stability …

rare-earth glasses the ions can end up close to each other that, rather than emitting and absorption energy from the core glasses, they start to pass energy between themselves; this effectively removes them from externally observed energy transfer processes and is almost as if they were not in the glass-reducing the effective doping concentration. This effect can occur when two ions are co-located, and is known as pair-induced quenching, or PIQ. In fact in low-concentration erbium-doped silicate fibers (below 1000 p.p.m) a residual absorption at 980 nm cannot be saturated. The usual models for up conversion of Er/sup 3+/cannot match this behavior nor explain a fluorescence lifetime independent of pump power and erbium concentration. The Excited-state Absorption (ESA) is the absorption of light by ions or atoms in an excited electronic state, rather than in the electronic ground state. The ESA is a common problem for broadband gain media such as transition-metal-doped crystal, but less so for rare-earth-doped crystals with their relatively narrow-bandwidth transitions. ESA is more usual for laser ions with multiple electronic levels, such as erbium or thulium, whereas it is not possible for ytterbium. Rare-earth-doped optical fiber lasers are characterized by their potential applications as optical sources or amplifiers and in fields of telecommunication networks, sensing, spectroscopy, medicine and LIDAR. The rate-earth-doped fiber lasers have a large variety of dynamical behaviors, static and dynamic polarization effects, antiphase and chaotic dynamics under autonomous or pump modulation conditions. They have a highly longitudinal multimode operation due to the large inhomogeneous linewidth, fiber lasers exhibit co-operative effects between longitudinal modes; and we get a simple modeling of the dynamics by low dimensional equations [4]. The conjunction of the inherent nonlinear character of both the optical fiber and the light amplification process makes it suited for analysis of nonlinear dynamics in optical systems. Due to the opticalguiding characteristics of their amplifying medium, fiber lasers have a cavity length of the order of tens of meters, orders of magnitude higher than in most other lasers. It ensures that a large number of longitudinal modes experience gain and coexist inside the cavity, coupled through gain sharing. The fiber laser usually operates in a strongly longitudinal multimode regime. The dynamics of multimode lasers include antiphase behavior and self-pulsing organized collective oscillation. There is both antiphase dynamics in self-pulsing, and a quasi-periodic route to chaos in Er3± doped fiber laser operating simultaneously at 1.536 and 1.55 μm wave-lengths. We use a CW state for high pumping rates and the system becomes T-periodic, 2T-periodic, 3T-periodic and chaotic for decreasing pumping ratio, T is represented the fundamental periodicity of the pulses, independently of their magnitude. There is a stable self-pulsing in a linear cavity Er3+ -doped fiber laser at 1550 nm with a unidirectional pump from a 980 nm laser diode for continuous-wave pumping. When the pump laser is externally chopped at low frequencies, which is typically in 20–200 Hz. The dynamics of a high power Yb3+ -doped double-clad fiber laser is done in a various optical configurations, operating in 1.08 μm wavelength. The fiber is side-pumped with a high power laser diode using the V-groove technique. Different self-pulsing regimes exist and attributed to third-order nonlinear effects. There is also an influence of the cavity losses on the dynamical behavior of our system. The system behavior typical guidelines:

8.2 Tm3+ -Doped Silica Fibre Lasers Nonlinearity …

869

1. Self-pulsing behavior in the fiber laser is due to the presence of ion-pairs or clusters. 2. Bidirectional propagation in “high-loss cavity” and Brillouin-scattering effects in the fiber laser. 3. Ion-pairs (ion clusters) are responsible for the performances of both heavily doped erbium-fiber amplifiers or super fluorescent fiber sources and self-pulsing instability in erbium-doped fiber lasers (EDFLs) at λ = 1.55 μm. 4. For low ion-pair concentrations (x ≤ 5%) the EDFL operates continuously, where x is the proportion of ion-pairs in the doped fiber, whereas for high ion-pair concentration (x > 5%) the laser is self-pulsing. The physical mechanism of saturable absorption is a quenching effect which it is accrued between two neighboring ions in the 4 I13/2 state of Er3+ : one ion transfers its energy to the other, producing one up-converted 4 I9/2 ion and one ground-state ion. The characteristic time is associated with the process is some μsec, and the upconverted ion quickly decays to the 4 I13/2 state. We get the loss of one excited erbium ion. The quenching effect in negligible for low ion-pair concentrations which allow CW operation of the EDFL. For high ion-pair concentrations an efficient saturable absorption effect which occurs leading to self-pulsing behavior. We get output relaxation oscillations of a high power free-running diode-pumped Tm3+ -doped silica fiber laser. The laser operates on the 3 H4 → 3 H6 quasi-three-level transition and operates efficiency from 1.9 to 2.0 μm. By evaluation of the 3F4 → 3 H4 fluorescence   we get that after 1.98 μm pumping of a separate laser, the 3 H4 , 3 H4 → 3 H6 , 3 F4 up conversion process is significant in heavily Tm3+ -doped silica fibers. The process causes the saturable absorption which is indicated by the presence of the relaxation oscillations. There is a model that describes the ion-pair dynamics, relevant to the Tm3+ -doped silica system. For large emission-to-absorption cross-section ratios, which are relevant to Tm3+ -doped silica and for pump rates for which stable output is predicted, the oscillations are weakly damped before the steady state is reached. The emission behavior of CW diode pumped neodymium doped double clad fiber lasers is with and without a semiconductor saturable absorber mirror (SESAM). Without the SESAM the CW-pumped multi-longitudinal mode lasers start oscillating in a pulsed fashion. By maximizing the emission rate, the lasing system favors pulsed emission (self-pulsing) over CW emission. The simultaneous oscillation of a large number of modes in homogeneously broadened laser is not alone due to spatial hole burning but due to spontaneous mode-locking spanning a certain frequency range regardless of the number of modes contained in this range. The SESAM modulation’s self-pulsing goes over into a self-locked mode of emission. The conditions to achieve this type of emission behavior are inferred from the self-pulsing state. In the onset dynamics in erbium-doped fiber lasers, the lasing light have a build-up time when the pump is on, and it existed in the EDFL whether the laser is CW or self-pulsing. Self-pulsing is a transient phenomenon in CW lasers and it takes place at the beginning of laser action. As the pump is switched on, the gain in the active medium rises and exceeds the steady-state value. There is a mechanism that the number of photons in the cavity increases, depleting the gain below the steady-state value. After several strong peaks,

870

8 Dual-Wavelength Laser Systems Stability …

the amplitude of pulsation reduces, and the system behaves as a linear oscillator with damping. Then the pulsation decays; this is the beginning of the CW operation. The build-up time is from 1 ms to more than 10 ms. The onset behavior is characterized by delayed time, the frequency of the relaxation oscillations, and the pump power profile in the resonator. It determined the loss and gain coefficients of the laser. We target on a passively Q-switched fiber laser system with erbium-doped fiber (EDF) and samarium-doped. A linear stability analysis gives an unstable steady-state region for an erbium-doped fiber of high gain coefficient and a samarium-doped fiber of moderate absorption. The ubiquitous erbium-doped fiber amplifier (EDFA) is typically doped with rare earth elements, which are erbium and ytterbium. The obtaining of effective signal amplification with those rare-earth ions requires advanced materials. The heavy-metal and alkali/alkaline earth elements are lead, bismuth, gallium, lithium, potassium, and barium in an oxide glass doped with trivalent samarium (Sm) rate-earth ions. There are an optical properties of trivalent samarium doped silica glass fiber which this material has a narrow fluorescence of 2.2 nm f.w.h.m at a wavelength of 650 nm. The performance of the laser in continuous, Q-switched and self-mode-locked operation is very important to the operation of the laser. In the unstable steady-state region of erbium-doped fiber and a samarium-doped fiber, a self-pulsing appears and giant pulses of good Q-switched light is obtained with increasing ratio of the cross section of samarium-doped fiber to that of erbium-doped fiber. There is a bi-stable behavior between two distinct periodic orbits in a selfpulsing dual-wavelength erbium-doped fiber laser which operated simultaneously at 1.536 and 1.548 μm. The interval between the two wavelengths is around 12 nm and be varied by adjustment of a Bragg grating. When stable output intensity is required, there is a need to stabilize the laser. The laser stabilization is achieved with a Bragg grating or in a unidirectional ring cavity. The unidirectional optical isolator allows the suppression of Brillouin back scattering. A self-Q-switched Tm 3+ -doped fiber lasers is capable of generating powerful nanosecond optical pulses at wavelengths around 2 μm. Our system includes a 2 μm Tm 3+ -doped silica fiber laser, pumped by a CW-Nd:YAG laser at a wavelength of 1.319 μm. It is characterized by parameters such as the input and the output powers from which the laser threshold, slop and launch efficiency are defined for fibers of various lengths. The Tm 3+ -doped silica fiber has a Tm 3+ concentration of 1.1 wt% (7000 ppm), a core diameter of 17 μm, a numerical aperture (NA) of 0.25 and an optimized length of 3.1 m for maximum 3 power which is obtained in cw mode of operation. The T3+ m − H4 upper laser level has lifetime of ≈ 500 μs. The fiber ends are cleaved with a vibrating diamond blade to ensure flatness and incidence at right angles for minimization of losses. The source for the fiber laser is a high power continuous wave 1.319 μm Nd:YAG laser, operating in the fundamental transverse TEM00 mode at the maximum output. This laser is pumped by a (CW) high-pressure krypton arc lamp. A cooling system avoids instabilities and thermal problems (circulating water in the crystal chamber). A warm-up time establish the stable laser conditions. The pump irradiance is varied by the use of a plane polarizer, while keeping the lamp current of the Nd:YAG laser constant, then the current variations not change the pump beam characteristics but change the launch efficiency. The pump light is focused into the fiber with a NA IR microscope

8.2 Tm3+ -Doped Silica Fibre Lasers Nonlinearity …

871

objective with calculated 70% transmittance at 2 μm wavelength, and about 85% transmittance at the 1.319 μm wavelength. The cleaved ends of the fiber are located on x-y-z differential micrometer translation stages. It allowed fine and coarse alignment. The pump light is propagated only inside the core of the fiber (assumption), because only light launched into the cladding is absorbed in the outer jacket layer of the fiber. The pump beam misalignment affects the fiber laser performance and the absorption characteristics of the fiber. There are x-y-z micrometer translation stages which continuously adjusted, in order to obtain maximum fluorescence at 2 μm, and thus ensure the maximum amount of pump power launched into the core. The maximum fluorescence is achieved by observation of the blue emission from the fiber due to pump excited-state absorption (ESA). The blue up-conversion can become weaker because the stimulated emission process reduces pump exited-state absorption (PESA). For high efficiency use of pump power we choose a mirror with high reflectivity (>90%) at the fiber laser wavelength and it butted onto the output end of the fiber to provide feedback. Another mirror which characterize by a highly reflective at the fiber laser wavelength and highly transmissivity at the pump wavelength (≈80%) is placed before the input of the fiber. It formed part of the fiber cavity arrangement to guide the fiber laser radiation into a 300 lines/mm grating mono chromatic. The mono chromatic output is detected by an InAs photodiode which is cooled by liquid nitrogen and has a typical response time