Plasma Physics Research Advances [1 ed.] 9781617285622, 9781604561364

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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PLASMA PHYSICS RESEARCH ADVANCES

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PLASMA PHYSICS RESEARCH ADVANCES

SERGEI P. GROMOV

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EDITOR

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All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Plasma physics research advances / Sergei P. Gromov, editor. p. cm. ISBN H%RRN 1. Plasma (Ionized gases)--Research. I. Gromov, Sergei P. QC718.P528 2008 530.4'4--dc22 2008010248

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CONTENTS Preface

vii

Short Communication Statistical Analysis of the Electrical Breakdown Time Delay Distributions in Gas Tubes at Low Pressures Čedomir A. Maluckov, Jugoslav P. Karamarković and Miodrag K. Radović

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Research and Review Studies

1 3

21

Chapter 1

Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas: Beyond the Range of One-Dimensional Theory Alexander Shalashov

23

Chapter 2

Discharges in Gas Flow Philip Rutberg

75

Chapter 3

Generation Mechanisms and Physical Properties of Electrical Discharges in and above Water P. Bruggeman and C. Leys

121

Chapter 4

Recent Advances in the Theory and Application of Kinetic Alfvén Waves L. Yang and D.J. Wu

153

Chapter 5

Effect of External Fields on Properties of Macroscopic Plasmas from the Quantum Mechanical Viewpoint D.F. Miranda, A.F. Guimarães, O.A.C. Nunes, D.A. Agrello and A.L.A. Fonseca

185

Chapter 6

Advances in Research on Low-Pressure Capacitively Coupled Plasmas H.C. Kim, S.J. You, G.Y. Park, and J.K. Lee

235

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Contents

Chapter 7

Mass-Spectrometry Studies in RF Capacitively Coupled Discharges Magdalena Aflori

259

Chapter 8

Theory and Application of the Generalized Balescu-Lenard Transport Formalism Renato Gatto and Harry E. Mynick

289

Chapter 9

Chaotic Motion of Relativistic Electrons Driven by Whistler Waves G.V. Khazanov, A.A. Tel’nikhin and T.K. Kronberg

333

Chapter 10

Thermodynamic and Transport Properties of Non-ideal Complex Plasma T.S. Ramazanov, K.N. Dzhumagulova, M.T. Gabdullin and Y.A. Omarbakiyeva

357

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Index

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PREFACE In physics and chemistry, a plasma is typically an ionized gas. Plasma is considered to be a distinct state of matter, apart from gases, because of its unique properties. Ionized refers to presence of one or more free electrons, which are not bound to an atom or molecule. The free electric charges make the plasma electrically conductive so that it responds strongly to electromagnetic fields. Plasma typically takes the form of neutral gas-like clouds (e.g. stars) or charged ion beams, but may also include dust and grains (called dusty plasmas). They are typically formed by heating and ionizing a gas, stripping electrons away from atoms, thereby enabling the positive and negative charges to move more freely. This book presents new and important research from around the globe. Results of the statistical analysis of the electrical breakdown time delay in neon and krypton at small pressures are presented in this Short Communication. The Townsend electrical breakdown mechanism is proposed as the governing breakdown mechanism. Description of the breakdown time delay density distributions is based on the convolution time delay distribution model. This model is established on the convolution of two independent random variables: the breakdown statistical time delay with exponential distribution and breakdown formative time with Gaussian distribution. Parameters of these distributions are obtained numerically by statistical Monte Carlo method. The convolution model is tested including the experimentally obtained breakdown time delay distributions for neon and krypton for different overvoltages, pressures, interelectrode distances, relaxation times, auxiliary glows, and under UV and gamma radiation. The experimental breakdown time delay distributions are obtained on the base of 200 or 1000 successive and independent measurements. The transition of distribution shape, from Gaussian-like to the exponentiallike, is investigated by calculating the corresponding skewness and excess kurtosis parameters. It is shown that the convolution model fits experimentally obtained time delay distributions and describes the separation of the total breakdown time delay to the statistical and formative time delay. The theoretical breakdown time delay distributions are in good correspondence with the distributions of experimental data. Recent achievements are reviewed in the theory of linear mode coupling in electron cyclotron frequency range that has been stimulated by the progress of optimized stellarators and spherical tokamaks in which methods based on the linear mode conversion are especially important for electron cyclotron resonance heating and diagnostics of overdense magnetized plasma. The developed theory is generalized for two-dimensionally inhomogeneous plasma

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Sergei P. Gromov

configurations and describes the conversion of ordinary and extraordinary plasma modes in the vicinity of the plasma cutoff while taking into account variation of a magnetic field on flux surfaces. This problem can not be treated within the standard one-dimensional approach of wave tunneling though the evanescent region since the mode coupling region assumes essentially two-dimensional topology when two cutoff surfaces intersect. Asymptotic analytical solutions of wave equations in the vicinity of the two-dimensional transformation region have been obtained that allow investigation of the structure of the transformed and reflected waves and corresponding transformation efficiencies for an arbitrary field distribution in the incident beam. New effects are found that are absent in the onedimensional theory. In Chapter 1, these theoretical results are considered as a solution of a new standard problem in plasma physics that can be used outside the scope of purely thermonuclear applications. In the current range from 10 to 10000 A, the conditions were determined for the existence and transition between contracted and diffuse regimes of burning electric arcs in application to the multi-phase electric arc systems in dependence on the gas dynamic parameters in the electric discharge chambers. The main parameters of the arcs (conductivity, concentration of the current carriers, and so on) and the parameters of the generated plasma jets have been defined. Physics processes in high-current electric discharges in gas flows of working gas consumption from several grams to several kilograms per second are considered. Different gases were used as the working ones. The working gases were hydrogen, argon, helium, nitrogen, air, CO2, and others. Conditions of heat and energy exchange resulting in the generation of plasma jets are studied, as well as phenomena at electrodes under such high current. Phenomena in the near-electrode plasma and the character of emission from the electrodes’ surface were investigated in Chapter 2. The elaborated and applied complex methods for diagnostics were described. On the basis of the implemented investigations, a set of stationary plasma generators was constructed with a power range from 5 KW up to 6 MW (in inert environment and hydrogen) and to 600 KW (in oxidizing atmospheres). The description of developed powerful plasmatrons and their parameters are presented. Novel plasma technologies for waste treatment and neutralization, coal gasification for renewable energy and liquid fuel production are described. During the last two decades, electrical discharges in or near water have been drawing a lot of attention in view of environmental and medical applications. Indeed, the simultaneous generation of intense UV radiation, shock waves and active radicals makes these discharges particularly suitable for decontamination, sterilization and purification purposes. The physics of these complex discharges is not fully understood, although in the last few years considerable progress has been made. Chapter 3 reviews the current status (end 2007) of research on the generation mechanisms and physical properties of electrical discharges in water. Four discharge types covering the main basic physics of electrical discharges in water will be discussed thoroughly. (1) The pulsed direct liquid phase discharge consisting of the streamer and spark/arc discharge, generated by high voltage pulses on sharp electrodes in bulk water. (2) Electrical discharges in gas/vapor bubbles surrounded with water or in contact with a metal electrode. (3) Discharges in the gas phase (typically at atmospheric pressure) where either one or both metallic electrodes are replaced by a ’water electrode’. (4) Diaphragm or capillary discharges,

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Preface

ix

where a conductive current is forced to flow through a submerged diaphragm or capillary, leading to evaporation with subsequent electrical breakdown and plasma formation in the vapor phase. Next, different experimental configurations and generation methods will be discussed, with an assessment of their technical and physical features in relation to the application potential. Kinetic Alfvén waves (KAWs) are dispersive Alfvén waves with a small perpendicular wavelength, and have been experimentally verified in space plasmas. Due to their potential importance in various energization phenomena of plasma particles, KAWs have been extensively applied to the acceleration and heating of particles in both terrestrial auroral and solar coronal plasmas. In Chapter 4, we review recent advances in the theory and application of KAWs and focus mainly on nonlinear solitary structures of KAWs. The linear theory of KAWs has been well reviewed in the standard two-fluid model, so we only give a concise introduction to the KAW, including the wave dispersion relation and its main characteristics. Then, linear and nonlinear interactions of KAWs with heavy ions are systematically investigated in a multicomponent plasma, where heavy ions can obviously affect the wave phase velocity and the solitary structure of KAWs, and maybe have a cyclotron resonant interaction with a linear KAW. Finally, the interaction of the nonlinear solitary KAWs with heavy ions is applied to the anisotropic and mass-dependent energization of minor ions in the solar corona, which can reasonably interpret the observations obtained by Ultraviolet Coronagraph Spectrometer (UVCS) on board Solar and Heliospheric Observatory (SOHO). The anisotropic and mass-dependent energization by KAWs is of potential importance for better understanding the microphysical processes occurring in the extended solar corona. The effects of external fields on the properties of a macroscopic plasma within quantum mechanics formalism are studied. The quantum approach seems to be more suitable than the classical one, as it enables us to perform unitary transformations in order to solve timedependent Hamiltonians as, for instance, electron motion in radiation and strong magnetic fields. This kind of approach has been employed successfully to study a number of classical plasma systems. The radiation field effects enter the problem straightforwardly from the outset through the electron time-dependent Schrödinger equation. The obtained time-dependent electron wave function is then used to calculate the electronic state in a local potential which allows us to derive the dielectric response (dynamical screening) of the system in the classical limit. After the determination of the dielectric function, we derive the average kinetic energy absorption rate by the plasma electrons in the simultaneous presence of a strong magnetic field and two weak laser fields taking into account the effects of the dynamical screening. The results show that the average kinetic energy absorption rate is quadratic on the laser field intensities. The cases in which resonances and cutoffs of the energy absorption rate occur have been considered. As a final task, in Chapter 5 we study the modulation of the dielectric properties and plasmon frequencies for non-magnetized and magnetized plasmas. In the former case, it was shown that the Landau damping term undergoes modulations which lead to instabilities in the regime of evanescent waves. For magnetized plasmas, on the other hand, we find strong modulations in the regime of circularly polarized waves. At the cyclotron resonance condition, the harmonics of electron cyclotron frequency were found.

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Chapter 6 discusses capacitively coupled plasmas (CCPs), which are among the most widely used plasma sources for low-pressure material processing due to their simple hardware design and robust uniformity over large areas. Increasingly smaller device sizes with increased aspect ratios have demanded more severe requirements such as the independent controllability of ion flux and ion bombardment energy onto the electrodes. Various attempts have been made to circumvent the drawbacks of conventional capacitive reactors and improve their performance. One of their variants is known as the magnetically enhanced CCP, which is composed of a dc magnetic field parallel to the surface of rf electrodes, achieving higher ion flux and lower ion bombardment energy. A dual or triple frequency CCP operated with additional power supplies driven by different frequencies has attracted much attention as a mainstream plasma source for dielectric etchers, achieving the independent control of ion flux and ion bombardment energy. Widespread practical applications of CCPs have accelerated laboratory research in this area. Changes in the electron energy distribution function have been reported under varying discharge conditions. It has been explained that these phenomena can be attributed to the transition of the electron heating mode (collisionless/collisional heating, low-voltage/high-voltage mode, and electron resonance effect) and to the nonlocality of electron kinetics. Radio-frequency glow discharges are widely used in thin film and surface technology. They allow the homogeneous treatment of surface areas at rather low temperatures. Sputtering, anisotropic etching of semiconductor surfaces and the deposition of thin films are typical applications. The fluxes of particles created in plasma or at its edge drive the interactions between plasma and surfaces. The mass-resolved ion energy distribution (IED) has been investigated at the grounded electrode of a capacitively coupled rf (13.56 MHz) discharges in pure argon, oxygen, hydrogen, and argon-oxygen, argon-hydrogen mixtures. The investigated rf plasma (13.56 MHz) was confined in the plasma chamber of an asymmetrical industrial OTP Plasmalab 100 capacitively coupled system. The mass-resolved IED was measured at the grounded electrode (included the chamber walls), that was much larger than the driven electrode. Ion and neutrals kinetic energy distributions were measured with a Hiden EQP Plasma probe which uses an electrostatic ion-energy analyzer followed by a triple section quadrupole mass spectrometer for mass analysis. Mass spectrometry yields first, the nature of ions created in the plasma and second, the corresponding ion energy distribution function. The IEDs for investigated ions exhibit a double hump shape. These features result from both the creation of thermal ions in the sheath, by charge exchange processes and from rf modulation of the sheath potential. Two types of ions can be considered: those created in the sheath, respectively the ions that enter into the sheath from plasma and reach the anode surface without collisions. Both groups are related to the periodic extinction of the electric field, leading to a periodic penetration of the electrons into the sheath. Only low-energy ions came to the surface of the grounded electrode because of the low sheath voltage between the plasma and the ground and because of the asymmetry of the device. The processes and reactions which take place in plasma were discussed in Chapter 7. In the first part of Chapter 8, a comprehensive overview is given of the ”generalized Balescu-Lenard” (gBL) transport theory, a self-consistent action-angle transport formalism arising from the gBL collision operator. By adopting the adiabatic invariants of the unperturbed particle motion and their conjugate angles as phase-space coordinates, the theory

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describes the degradation of plasma confinement as a result of the breaking of the invariants by wave-particle resonance. A key feature of the formalism is that, by including the dynamic friction term, which describes the back-reaction of the particles on the fluctuations, it possesses the correct conservation laws, and, in particular, the conservation of toroidal angular momentum in axisymmetric systems, which implies intrinsic ambipolarity. The theory is philosophically and structurally similar to quasilinear (QL) theory in that it does not determine the fluctuation spectrum, but rather accepts whatever spectrum one believes to correctly describe a particular transport mechanism. This provides a common framework to view the effects on transport of perturbations of very short wavelength, which give rise to collisional symmetric transport, or of longer wavelengths in the turbulent range. For a two species plasma, turbulent transport is accordingly characterized by properties that are well known for symmetric transport. Several conclusions can be obtained from the theory before specialization to a particular spectrum is made. The inclusion of the friction term leads to anomalous pinch terms, as well as to an energy exchange between the two species that can yield predictions very different in magnitude from those obtainable from QL theories. Also, the magnitude of the ratio between heat and particle fluxes shows an increasing trend when going from parameter regimes typical of high-confinement tokamak discharges to those characterizing low-confinement modes, in accord with experimental findings. In the second part of this article, applications of the gBL transport theory obtained by specifying the details of the underlying perturbing spectra are presented. The resulting explicit transport models can be immediately related to experimental conditions. For trapped electron mode turbulence, the anomalous energy exchange between species is evaluated and compared with the analogous expression obtained from QL theory, clarifying the parameter range in which the two expressions differ. For magnetic turbulence, a comprehensive electron transport model which includes the generalized Ohm’s law is formulated and discussed. The article ends with a formal, fully turbulent generalization of the gBL operator. Canonical equations governing an electron motion in electromagnetic field of the whistler mode waves propagating along the direction of an ambient magnetic field are derived. The physical processes on which the equations of motion are based are identified. It is shown that relativistic electrons interacting with these fields demonstrate chaotic motion, which is accompanied by the particle stochastic heating and significant pitch angle diffusion. Evolution of distribution functions is described by the Fokker-Planck-Kolmogorov equations. A coefficient of diffusion is calculated from the equations of motion. The results in Chapter 9 indicate that phase flow of our dynamical system is structural stable strange attractor. Chaotic motion on the attractor gives rise to an irreversible process (the so-called deterministic diffusion), which leads actually to establishing a steady-state energy spectra and results in such important and easily observable effects as the stochastic heating and pitch angle scattering of plasma particles. Under conditions typical of this mechanism, the heating region is determined by the boundaries of the attractor, and the heating rate is governed by the nature of the kinetics, which in turn depends on the canonical variables on the attractor. The same conditions impose limitations on the timescales of macroscopic effects and feasible extent of heating, so that understanding of this condition is of great practical interest. It is shown that the whistler mode waves could provide a viable mechanism for stochastic energization of electrons with energies up to 50 MeV in the Jovian magnetosphere. In Chapter 10, the thermodynamic and kinetic properties of dense semi-classical plasma are investigated on the basis of effective potentials of interparticle interaction. These

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potentials take into account correlation effects and quantum-mechanical diffraction. Plasma compositions, thermodynamic functions of hydrogen and helium plasmas are obtained for a wide region of coupling parameters. Collision processes in partially ionized plasma are considered; some kinetic characteristics such as phase shifts, scattering cross section, bremsstrahlung cross section and absorption coefficient are investigated. The last part of the chapter is devoted to computer simulation of dynamic and transport properties of dusty plasma; it also presents a description of the method of Langevin dynamics. Based on this method, velocity autocorrelation functions were obtained for dusty particles, and transport properties were also considered. All results are new and are presented here in comparison with the results of other works.

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SHORT COMMUNICATION

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In: Plasma Physics Research Advances Editor: Sergei P. Gromov, pp. 3-19

ISBN 978-1-60456-136-4 c 2009 Nova Science Publishers, Inc.

S TATISTICAL A NALYSIS OF THE E LECTRICAL B REAKDOWN T IME D ELAY D ISTRIBUTIONS IN G AS T UBES AT L OW P RESSURES

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ˇ Cedomir A. Maluckov∗ Technical Faculty in Bor, University of Belgrade Vojske Jugoslavije 24, 19210 Bor, Serbia Jugoslav P. Karamarkovi´c Faculty of Civil Engineering and Architecture, University of Niˇs, Beogradska 14, 18000 Niˇs, Serbia Miodrag K. Radovi´c Faculty of Sciences and Mathematics, University of Niˇs, PO Box 224, 18001 Niˇs, Serbia

Abstract Results of the statistical analysis of the electrical breakdown time delay in neon and krypton at small pressures are presented in this paper. The Townsend electrical breakdown mechanism is proposed as the governing breakdown mechanism. Description of the breakdown time delay density distributions is based on the convolution time delay distribution model. This model is established on the convolution of two independent random variables: the breakdown statistical time delay with exponential distribution and breakdown formative time with Gaussian distribution. Parameters of these distributions are obtained numerically by statistical Monte Carlo method. The convolution model is tested including the experimentally obtained breakdown time delay distributions for neon and krypton for different overvoltages, pressures, interelectrode distances, relaxation times, auxiliary glows, and under U V and gamma radiation. The experimental breakdown time delay distributions are obtained on the base of 200 or 1000 successive and independent measurements. The transition of distribution shape, from Gaussian-like to the exponential-like, is investigated by calculating the corresponding skewness and excess kurtosis parameters. It is shown that the convolution model fits experimentally obtained time delay distributions and describes the separation of the total breakdown time delay to the statistical and formative time delay. The ∗

E-mail address: [email protected] ([email protected])

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c theoretical breakdown time delay distributions are in good correspondence with the distributions of experimental data.

1. Introduction The electrical breakdown in gases represents the transition of the discharge in gas from nonself-sustaining to self-sustaining form [1]. The voltage for initialization of the electrical breakdown in gases is defined as the breakdown voltage or sparking potential. Investigation of electrical breakdown in gases is significant for theoretical analysis of included physical processes and applications [2], [3]. The electrical breakdown in gases can be interpreted as a macroscopic event with a stochastic nature. Investigation of the stochastic nature of the electrical breakdown dates from von Laue [4], and develops in papers of many researches [5]-[7]. The statistical theory of the electrical breakdown is established on the presumption of the Townsend breakdown mechanism [1]. This presumption corresponds to the small pressure and small overvoltage regimes in gases when the influence of the space charge is neglected. The breakdown criterion in Townsend theory is: γ

"Z

d

#

exp(αdx) − 1 = 1

(1)

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0

where α is the primary ionization coefficient and γ is the effective secondary ionization coefficient which includes secondary processes induced by positive ionsγi , metastable atoms (molecules) γm , and photons γph , i.e., γ = γi + γm + γph . The electrical breakdown mechanism in gases can be considered as a combination of two distinct type of processes. The first type corresponds to the occurrence of one, or several physical events leading to the creation of an initial free electron. This process is the Poisson random process and time for the electron appearance is the breakdown statistical time delay tS . Therefore, the breakdown statistical time delay tS is characterized with the exponential distribution [1], [4]. The second type of processes are the process of ionization and carrier multiplication in gas, which result in the development of a low impedance conducting plasma. Corresponding characteristic time is the breakdown formative time delay tF . In most cases, the ionization events, which lead to breakdown, are the predictable sequences of action. This implies that the multiplication process has a high probability and the breakdown formative time delay is well described by a normal distribution [8]. Therefore, the electrical breakdown time delay tD consists of the breakdown statistical time delay tS and the breakdown formative time delay tF (tD = tS + tF ) [1]. Shape of the breakdown time delay distribution depends on experimental conditions. In many experiments tF  tS [1], [9], [10], so that, the breakdown time delay distributions are determined only with the breakdown statistical time delay and have exponential shape. However, in some cases breakdown formative time delay cannot be neglected. More exact analysis assumes that tF is a single value and that the distribution of the breakdown time delay is a shifted exponential distribution [1], [11], [12]. This is a good approximation for tF < tS , even for comparable tS and tF if tF is characterized with a sufficiently small variance. In the opposite cases, when the breakdown formative time delay is a dominant

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part of the total time delay, the breakdown time delay distributions can be approximated by Gaussian distributions. This corresponds to the conditions with well defined discharge initialization process, e.g. the high pressure [8] or the large electron yield conditions [13]. In this paper, we present the results of the statistical analysis of the electrical breakdown time delay obtained by applying the convolution-based model which is originally proposed by the authors and published in a few papers [14]-[25]. The main advantage of our model is that it ensures the statistical treatment of the electrical breakdown time delay over the full range of the breakdown statistical time delay and breakdown formative time delay values. The convolution-based model takes into account the Gaussian distribution of the breakdown formative time delay and the exponential distribution of the breakdown statistical time delay. In other words, we model the breakdown time delay as a sum of two random variables, one with the exponential distribution and the other with the Gaussian distribution. We analyzed the model in details using statistical methods and tested it on the breakdown time delay distributions recorded in the neon and krypton filled tubes, for various experimental conditions, relaxation times, overvoltages, the intensity ofγ and U V radiation and auxiliary glow current.

2. Convolution Based Statistical Model

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The convolution approach to the description of the electrical breakdown time delay distribution is based on the following presumptions. The total breakdown time delay is a random variable tD , with values tD (measured values of total time delay), and the random variable tD is the sum of two independent random variables tS and tF (tD = tS + tF ). The first random variable tS (with values tS ) is the breakdown statistical time delay characterized with the exponential distribution fS : fS =





1 t . exp − E(tS ) E(tS )

(2)

The distribution parameter is given as E(tS ) = 1/(Y P ), where Y is the electron yield, and P is the probability that breakdown is initialized by an initial electron [1]. The productY P can be defined as the rate of the triggering events which initiate the discharge. The second variable tF (with values tF ) is breakdown formative time delay characterized with Gaussian distribution fF : 1 (t − E(tF ))2 exp − fF = p 2σ(tF ) 2πσ(tF )

!

.

(3)

The distribution parameters are the mathematical expectationE(tF ) and standard deviation σ(tF ). The density distribution of the random variabletD can be obtained as the convolution of the density distributions of the random variables tS and tF : f (tD ) =

Z tD

fS (t)fF (tD − t)dt.

(4)

0

The density distributions are obtained by the numerical integration of equation (4) adopting the ”mechanical quadrature” method [26]. In order to obtain the appropriate values Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c . Uw R3

relay

DCM IG

ARTS

mA

R1

DC

UW

R2

VSS

(a)

(b)

I/O

PC

Figure 1. Schematic diagram of the experimental layout. of the exponential and Gaussian distribution parameters, the Monte Carlo simulation algorithm is applied. The numerical subroutines for the pseudo random numbers, the Gaussian distribution (the Box-Muller method), and the exponential distribution (the method of the inverse function) are taken from reference [27]. The parameters of the statistical time delay and formative time delay distribution are considered as fitting parameters associated with the experimental breakdown time delay mean value and experimental standard deviation tD = E(tS ) + E(tF ), σ(tD )2 = E(tS )2 + σ(tF )2 .

(5)

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The consistency of the numerically generated and experimental distributions is checked by the χ2 test [14]. The fitting procedure is reduced to the variation of only one parameter until the χ2 sum is minimized. For more details about the numerical procedure and the calculation of the distributions parameters E(tS ), E(tF ) and σ(tF ) see references [14][25].

3. Experiment The measurements of the breakdown time delay are performed on the neon and krypton filled diodes at the pressure from 1 mbar to 13.3 mbar and room temperature. The tube bulb is made of molybdenum glass, with molybdenum electrode carrier. Before the gas has been admitted (neon and krypton), the tubes were baked out at 350◦ C and pumped down to the pressure of 10−7 mbar mbar in a process similar to that used for the production of X-ray and other electron tubes. The tube was filled with Matheson research grade neon and krypton. The measurements of the breakdown time delay have been performed using the electronic systems schematically illustrated in Fig. 1 (a) and Fig. 1 (a). The system (a) consists of three subsystems: (1) the voltage supply and sense (VSS) subsystem, (2) the analog relaxation time setting (ARTS) subsystem, and (3) the digital control and measurement (DCM) subsystem. The VSS subsystem is composed of (i) the regulated dc power supply 1001000 V, 50 mA, (ii) the steady state current regulation resistor, and (iii) the sensor resistor for the selection of the appropriate start-stop measuring level. The voltage risetime (measured by Textronix TDS 2012 digital storage oscilloscope) is 53 µs. The DCM subsystem is composed at a personal computer with the ED 2000 data

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acquisition card under the control of the computer program written C. The ARTS subsystem is based on timer 555 and the demultiplexer CD4051, with the primary purpose being the establishment of manual and semiautomatic modes (see [20]). The voltage in the experimental system (b) was applied to the diode via the electromechanical relay controlled by computer and start of the measurement is indicated by appearance of voltage on potentiometer The voltage rise time on potentiometerR1 (measured by Textronix TDS 2012 digital storage oscilloscope) is about 80 µs. This time is obtained as the time interval from moment when voltage pulse attained 5%, to the moment when the voltage pulse attained 95% of working voltage. Once the current through the diode was established, it has given the voltage signal on the potentiometer indicating the end of measurement. The current in the tube is limited by potentiometer to the steady state value of 0.2 mA in order to have the most adequate current in relay to reach its highest sensitivity. Since the current’s response of the current in the tube is oscillatory, it is common to set the current which defines the end of measurement lower than its steady state value. In this experiment, we considered the breakdown time delay as the time elapsed from the moment when the high voltage is applied to the diode until the moment when the current through the diode reached 90% of its steady state value. Electrical details of the circuit and the experiment procedure are the same as in [17]. In both experimental systems the personal computer with proper interfaces is used to control the basic parameters of the experiment (the glow discharge timetG and the relaxation time τ of the diode between two successive measurements), the acquisition of the data and their analysis. The discharge glow time, the relaxation time and the glow current IG were kept constant in order to set identical initial conditions for each breakdown and, consequently, the independence of all measurements in each measurement’s series.

4. Application of the Statistical Model on the Breakdown Time Delay Distributions The convolution model is tested on the experimentally obtained time delay distributions in gas diodes filled with neon and krypton at the pressure from 1 mbar to 13.3 mbar, for various values of experimental parameters as the relaxation time, overvoltage, the intensity of γ and U V radiation, distance between electrodes (minimum and right part of Paschen curve) and auxiliary glow current (see references [14]-[25]). Before the analysis of the electrical breakdown time delay distributions is performed, the randomness of the experimental data is investigated. The randomness of the experimental data samples is checked by Wilcoxon or Mann-Whitneys test [17], [28], [29]. The analysis shows that the systematic trends are not observed on both confidence levels of 95% and 99% which confirms the randomness of data samples.

4.1. Time Delay Distributions for Various Relaxation Times The result of preliminary measurements of the electrical breakdown time delay versus relaxation time τ , for gas diodes filled with neon and krypton (UW = 500 V for neon tube and UW = 500 V for krypton tube) are shown in Fig. 2. The neon filed diode is at pressure

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c

of 6.5 mbar (see reference [20]), and the krypton filled diode is at 2.6 mbar (see reference [24]). Each point on this curve (so called the ”memory curve” [30]) is the mean value of 100 successive and independent measurements. tD (ms) 10

tD (ms)

100 measur. 1000 measur.

3

10

100

3

100

10

10

1

1

0.1 1

10

100

10 3

10 4

0.1 10 5 τ (ms) 1

10

neon

100

10 3 10 4

10 5 τ (ms)

krypton

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Figure 2. The breakdown time delay mean values tD vs relaxation time. The number of measurements per point is indicated in figures. Complete memory curve can be divided in three major parts. The first part of the memory curve, which corresponds to the relaxation time from 3 ms to 80 ms for neon and 3 ms to 40 ms for krypton, possesses the characteristic plateau. This plateau is the consequence of the presence of positive ions in the gas. Actually, the breakdown probability is almost independent of the ion concentration and tD values are small and almost constant because ions have high drift velocities and reach the cathode almost instantly after the application of the voltage releasing the secondary electrons (Auger electrons). Then the production of electrons is very high and the time delay is approximately equal to the breakdown formative time tD ' tF . For these τ values, the considerable concentrations of the neutral active state atoms are also present in the gas. But, due to the electrical neutrality of these states and the value of their diffusion time, which is higher than the drift time of the positive ions, the role of neutral states in the secondary emission of electrons from the cathode is not significant [31]. The second part of the memory curve corresponds to the relaxation times0.15 ≤ τ ≤ 30 m s for neon and 0.04 ≤ τ ≤ 150 s for krypton. As it was shown in Ref. [31], for these relaxation times, the recombination of ions has already finished. Therefore the emission of secondary electrons from the cathode is mostly induced by neutral active states. However, in literature the character of neutral active states which are the most involved in memory effect in rear gases is doubtful. For example in [20], [24], [31] – [35] the neutral active states are 3 P2 and 3 P0 metastable atoms of rare gases which de-excite at the cathode surface and release the secondary electrons [36]. The secondary electrons can initiate the breakdown when the voltage is applied on the tube. On the contrary, in [37] and [38] the neutral active states which initiate secondary electron emission are remanent nitrogen atom states. They exist in gas diodes after manufacturing. However, this dilemma does not influence the statistical approach applied in this paper, since it detects the secondary electrons without considering the mechanisms of their creation. The third part of the memory curve is the characteristic plateau, for relaxation times τ ≥ 30 s for neon and 150 s for krypton, as a consequence of the significant decrease of

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Statistical Analysis of the Electrical Breakdown Time Delay Distributions ...

9

concentration of neutral active states in gas. The breakdown is initiated by the secondary electrons created by cosmic rays and/or natural radioactivity. Since the flux of those rays during the experiment does not vary significantly, the breakdown time delay valuestD are approximately constant (see Ref. 18). We are specially interested in the statistical investigation of the breakdown time delay distributions in ionic (first) part, and transition between ionic (first) and metastable (second) part of the memory curve. In order to obtain more reliable distributions the series of 1000 measurements were performed for τ from 3 ms to 600 m. Note also the higher density of points on the time axis in this case as can be seen in Fig. 2 (curve with triangles). The slight difference between two curves in Fig. 2 is a consequence of different number of measurements. This difference is on the statistical fluctuation level. The experimentally obtained breakdown time delay density distributions of the electrical breakdown in neon and krypton are shown in Fig. 3. Experimentally obtained the breakdown time delay distributions are fitted by the theoretical density distributions (see Sec. 2.) which are presented in Fig. 3 by dashed curves. The theoretical density distributions are obtained for the parameters shown in Fig. 4. Here is worth to mention that the numerical procedures for the determination of distribution parameters and density distributions are described in Sec. 2.. In Fig. 5 the distribution coefficients, skewness γ3 and kurtosis γ4 , with respect to the Gaussian distributions, are plotted as functions of the relaxation time. The agreement of theoretically modelled and experimentally obtained the time delay distributions is obvious in Fig. 3 – Fig. 5. Investigations of the convolution method applicability for various values of the relaxation time are presented in references [14], [16], [18], [21], [22], [24]. The model validity is confirmed. In addition, the dominant influence to the shaping of the distribution is attached to the neutral active states of neon and krypton atoms independently on their origin. In other words, the active states which lasted from the previous discharges and those which are created after the voltage was applied in order to initialize discharge, equally influence the distribution shape (see references [20] and [24]).

4.2. Time Delay Distribution for Various Overvoltages and Distances between Electrodes The Paschen curve (dependence US = f (pd)) for neon tube at 13.3 mbar [17] is presented in Fig. 6. The values of the static breakdown voltage are determined according to the definition where the static breakdown voltage is the highest voltage at which breakdown did not take a place [11]. The estimations of the static breakdown voltage are represented in Fig. 7. Each point in this figure represents mean value of 200 consecutive and independent measurements. Thus, the estimated static breakdown voltage values are US = 177 V (d = 2 mm) and US = 218 V mm (d = 8 mm), as indicated in Fig. 7. The Paschen curve has similar shape as curves in other gases, for similar electrode’s geometry [1], [39], [40]. All those curves show qualitative difference in comparison with curves recorded in plane electrodes geometry [41], in area of small values of pd. The application of the convolution model to the various values of overvoltage is pub-

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c

f(tD )

f(tD ) 0.4

τ = 3 ms

0.3 0.2 0.1 0.0

0.8

τ = 3 ms

0.4

122 124 126 128 tD (µs)

f(tD )

τ = 20 ms

0.075

0 102 f(tD )

104

tD (µs)

106

τ = 20 ms

0.16

0.050 0.08

0.025 0.000 120 f(tD )

140

160 tD (µs)

τ = 80 ms

0.004

0 100 f(tD )

110

120

tD (µs)

τ = 100 ms

0.015 0.01

0.002

0.005

0.000 0

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f(tD )

400

800 tD (µs)

τ = 600 ms

0

100 200 300

f(tD )

400 tD (µs)

τ = 600 ms

0.00012

0.0002

0.00004 0.00000

experiment model

0.0004

0.00008

0 10 20 30 40 tD (ms)

0

0

neon

2

4

6

tD (ms)

krypton

Figure 3. The breakdown time delay density distributions for indicated relaxation times. lished in references [14], [15], [17], [19]-[22], [24]. Analysis of the distribution parameters, skewness and kurtosis, shown that smaller values of overvoltage correspond to more symmetric distributions. It is related to the breakdown formative time delay dominance at low overvoltages. In addition, it is shown that distribution shape depends on the relation between the breakdown statistical time delay and breakdown formative time delay [14], [15], [17], [19]-[22], [24]. The validity of the convolution method is considered with respect to the various distances between electrodes and overvolatages from 1% to 50%, in neon gas diode at

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Statistical Analysis of the Electrical Breakdown Time Delay Distributions ...

time (µs)

1000

tD ; E(tS) E(tF) ;

σE

1000

σ(tF)

1000 time (µs)

10000

100 10

tD σE

11

E(tS) E(tF) σ(tF)

100 10 1

1 10

τ (ms)

100

0.1 10

neon

100

τ (ms)

krypton

Figure 4. Distribution parameters, the mean values and the experimental standard deviations vs τ . 13.3 mbar [14], [15], [17] (see figure Fig. 8). All distributions are drawn on the basis of 1000 consecutive and independent measurements. The agreement between theoretical and experimental results is confirmed. In the area of the minimum of the Paschen curve the time delay distribution is exponential, and in the right from the minimum the time delay distribution is Gaussian. This is associated with the ratio of the statistical and formative time in the total time delay.

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4.3. Time Delay Distribution for Various Values of Auxiliary Glow Current The dependence of the breakdown time delay mean valuestD on the auxiliary glow current IA is presented in Fig. 9 [21]. Each point is plotted on the basis of 50 successive and independent time delay measurements. The measurements were performed with overvoltages of 3% and relaxation time τ = 1 s. The decrease of the breakdown time delay values with increase of the auxiliary glow current is clearly observed. A reason for such dependence (tD = f (IA )) can be the influence of metastable neon atoms and reflected photons from auxiliary discharge. During the glow in auxiliary discharge variety of charged and neutral active particles are created. Some of them, as metastable neon atoms moving diffusively

γ3; γ4

γ4

γ3

Ne Kr

Ne Kr

γ4 - exponential

6 4

γ3 - exponencijal

2

γ3 i γ4

0 10

100

- Gaussian

τ (ms)

Figure 5. Skewness γ3 and kurtosis γ4 of the experimental distributions vs. τ . Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

12

C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c US (V) 260 240 220 200 180 160

0

5

10 15 20 pd (mbar.cm)

25

Figure 6. Experimental Paschen curve in neon at 13.3 mbar.

tD (ms) d = 2 mm d = 8 mm

100 10 1 US 150 200 250 300

350 UW (V)

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Figure 7. Time delay versus applied voltage.

can reach the main diode gap. This event increases the electron yieldY in the main diode gap, leading to the decrease of the breakdown time delay mean values tD . For auxiliary glow current values grater than 10 µA, tD becomes approximately constant. This is different from the results obtained in nitrogen where characteristic minimum oftD appears [42], [43]. The similar effects can be assigned to resonant light originated in the auxiliary glow. Some of these photons can reach the main diode gap by reflection on the glass tube walls. In order to investigate the influence of auxiliary discharge on breakdown time delay distribution, distributions for auxiliary glow current IA = 10 µs and IA = 100 µs are compared with distribution without auxiliary glow (see reference [21]). The corresponding density distributions are illustrated in Fig. 10, where each experimental distribution is drawn on the basis of 200 consecutive and independent measurements. It is confirmed that by increasing values of the auxiliary glow current more symmetric distributions are obtained. This is due to increasing influence of the active neutral states from the auxiliary discharge. In addition, the time delay distributions from the convolution method nicely agree with the experimentally obtained distributions.

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Statistical Analysis of the Electrical Breakdown Time Delay Distributions ... f (tD)

f (tD)

0.03

0.3

∆U/US = 1 %

0.02

0.2

0.01

0.1

0

13

0

50

0

150 tD (ms)

100

f (tD)

∆U/US = 1 %

60

70

90 tD (ms)

80

f (tD)

0.2

∆U/US = 10 %

∆U/US = 5 %

0.75 0.50

0.1

0.25 0

0

10

20

30 tD (ms)

f (tD) ∆U/US = 50 %

1.2

5

7.5

10 tD (ms)

∆U/US = 20 %

1.2 0.8

0.8 experiment model

0.4 0

0 2.5 f (tD)

0.4 0

0

1

2 3 tD (ms) d = 2 mm

0

1

2 3 tD (ms) d = 8 mm

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Figure 8. The breakdown time delay density distributions for indicated overvoltages and gaps.

4.4. Time Delay Distribution in the Presence of the U V and γ Radiation In order to investigate the influence of the additional electron yield on the total breakdown time delay and its constituents, the gas diode is irradiated withU V and γ radiation (see references [14], [20], [23], [24]). The breakdown time delay density distributions, obtained in neon filled diode at 6.5 mbar under γ radiation from 60 27 Co radio-active source (see [23]), are shown in Fig. 11. The experimental density distributions are drawn on base of 1000 measurements. The analysis of distribution parameters [23] shows that the statistical time delay decreases with the growth of RDe , while the formative time is not significantly changed. In order to investigate the influence of the additional electron yield on the total breakdown time delay and its constituents, the gas diode is irradiated withU V lamp. The intensity of U V radiation I is changed varying the distances between theU V lamp and gas diode (see ref [24]). The experimentally obtained breakdown time delay density distributions, on bases of 1000 measurements, for different intensities of U V radiation and without irradiation, are shown in Fig. 12. The relative intensity I/I0 , indicated in figure, is calculated with respect to the intensity observed at the shortest distance betweenU V lamp and diode, I0 . The distributions are obtained for relaxation time τ = 300 ms and voltage 450 V. The

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c tD (ms) IA = 0 µA

30 20 10 0 0.1

1

100 IA (µA)

10

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Figure 9. The time delay vs. auxiliary current. theoretical density distributions of tD are shown in Fig. 12 with dashed lines. Good correspondence between theoretical and experimental density distributions can be seen in Fig. 12. Main influence of the U V radiation on the breakdown time delay distributions is the yield’s increase in the gas diode. The mechanisms of the electron creation is photoeffect (see [24]). The increase of the intensity of the U V radiation (reported as relative intensity I/I0 ) causes the increase of the electron yield which changes the shape of distribution and values of the distribution parameters. The contribution of the U V radiation to the total electron yield can be estimated calculating yield according to the expression E(tS ) = 1/(Y P ) [1]. Since the applied voltage is constant in all measurements, the breakdown probability P is kept constant and estimated to be P = 0.54. Then the U V radiation’s influence on the process can be numerically quantified as the relative yield Y /Y0 = E(tS, 0 )/E(tS ). In the last relation the Y0 and E(tS, 0 ) denote yield and breakdown statistical time delay expected value without irradiation, respectively. The relative yield vs. relative intensity of the U V radiation I/I0 is shown in Fig. 13. From there the yield increase is only one order of magnitude when theU V source is on the shortest distance from the gas diode. This yield increase is not sufficient to suppress f(tD )

IA = 100 µA

3 2 1 0 2.25

2.5

f(tD ) 2 1.5 1 0.5 0 2.2

f(tD ) 0.1 2.75 3.0 tD (ms) 0.075 0.05 IA = 10 µA 0.025

0 20 2.8

3.4

IA = 0 µA

30

4.0 tD (ms)

40

50 tD (ms)

experiment model

Figure 10. The breakdown time delay density distributions for indicated auxiliary current. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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E(tS ) and the corresponding breakdown time delay distribution still holds characteristics of both the exponential and Gaussian distributions.

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5. Conclusion In this paper the convolution-based statistical model of the breakdown time delay distribution is presented. The convolution model is based on the numerical Monte Carlo calculations of the breakdown statistical time delay and breakdown formative time delay distributions which are constituents of the total electrical breakdown time delay distribution. The model validity is confirmed by interpreting the experimental results on the breakdown time delay in gas diode filled by neon and krypton at the low pressure. More precisely, the breakdown time delay distributions generated by the convolution-based model are compared with the corresponding experimental distributions obtained for various values of the relaxation time, the overvoltage, the distance between electrodes, the auxiliary current and exposition doses of γ and U V radiation. The comparison shows that the numerically calculated time delay distributions fit well to the corresponding experimental distributions. This confirms the basic assumption of the convolution model, the exponential distribution of the breakdown statistical time delay and Gaussian distribution of the breakdown formative discharge time delay. In addition, the significant result is the possibility of clear separation of the total breakdown time delay to the breakdown statistical time delay and the breakdown formative time delay in the framework of the convolution-based model. Physically the transformation of the distribution shape is associated with the initialization of the secondary electron emission and consecutive breakdown by particles of the different type in gases. For example, as long as the breakdown time delay distribution is Gaussian-like, positive ions induce secondary electron emission. On the other hand, as the relaxation time increases, positive ions are recombined, and neutral active states take the

f(tD )

f(tD )

R De = 0

0.0012

experiment model

0.0008

0.001

0

f(tD )

1

2

3 tD (ms) kg. s

0.001

0.002 0.5

1.0

400 800 1200 tD (µs) -9 C

RDe = 2.65 . 10 kg. s

0.006 0.004

0

0

f(tD )

-12 C

R De = 7.33 . 10

0

0.002

0

kg. s

0.002

0.0004 0

-10 C

R De = 4.23 . 10

0.003

1.5 tD (ms)

0

0

200

400

tD (µs)

Figure 11. The breakdown time delay density distributions for indicated exposition dose rate RDe. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

16

C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c f(tD )

I/I0 = 1

0.02 0.01

f(tD ) 0

280 tD (µs) 0.002

200

120

without UV radiation

f(tD )

I/I0 = 0.05 0.006

0.001 0

0.003 0 0

300

600

0

0.6

1.2

1.8

tD (ms)

experiment model

tD (µs)

Figure 12. The breakdown time delay density distributions for the intensity ofU V radiation I/I0 indicated on the figures.

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role in the initialization of the secondary electron emission. The result is the exponentiallike breakdown time delay distribution. In other words, the convolution approach gives decomposition of the breakdown total time delay with respect to two distinct processes involved in the electrical breakdown mechanism. As a consequence, range of the relaxation time where the Poisson processes of the secondary electron emission dominate the breakdown time delay can be estimated. It is worth to note, that the convolution approach is not affected by the doubt about the type of neutral active states involved in the memory effect in rare gases (metastables or impurity atoms). The further application of the convolution model to other rare gases and nitrogen, and the model extension to include the streamer breakdown mechanism are the next challenges for the authors.

Y/Y0 8 4 0

0

0.01

0.1

1 I/I0

Figure 13. The relative yield Y /Y0 vs. relative intensity of U V radiation I/I0 . Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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17

Acknowledgments This work was supported by the Ministry of Science and Environmental Protection of the Republic of Serbia under Contract 141008.

References [1] Meek, J. M.; Craggs, J. D.; Electrical Breakdown of Gases; Wiley: New York, 1978. [2] Kristansen, M.; Guenter, A. H. Plasma Applications, in: Electrical Breakdown and Discharges in Gases; Kunhart, E. E.; Luessen, L. H.; Plenum: New York, 1983. [3] Lieberman, M. A.; Lichtenberg, A. J. Principles of Plasma Discharges and Materials Processing; Wiley: New York, 1994. [4] von Laue, Annu. Phys. (Leipzig) 1925, vol. 76, 261-265. [5] Loeb, L. B. Rev. Mod. Phys. 1948, vol. 20, 151. [6] Wijsman, R. A. Phys. Rev. 1949, vol. 75, 833-838. [7] Farquhar, R. L. ; Ray, B.; Swift, J. D. J. Phys. D: Appl. Phys. 1980, vol. 13, 20672075. [8] Moreno, J.; Zambra, M.; Favre, M. IEEE Trans. Plasma Sci. 2002, vol. 30, 417-442.

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- . A.; Nikoli´c, Z. J. Phys. D: Appl. Phys. 1982, vol. 15, [9] Pejovi´c, M. M.; Boˇsan, D 867-872. - . A. J. Phys. D: Appl. Phys. 1983, vol. 16, [10] Pejovi´c, M. M.; Mijovi´c, B. J.; Boˇsan, D 1953-1957. ˇ A. J. Phys. D: Appl. Phys. 1998, [11] Radovi´c, M. K.; Stepanovi´c, O. M.; Maluckov, C. vol. 31, 1206-1211. ˇ A.; Stepanovi´c O. M. Contrib. [12] Radovi´c, M. K.; Jovanovi´c, T. V.; Maluckov, C. Plasma Physics, 2005, vol. 43, 78-87. ˇ A. J. Phys. D: Appl. [13] Spasi´c, I. V.; Radovi´c, M. K.; Pejovi´c, M. M.; Maluckov, C. Phys. 2003, vol. 36, 2515-2520. ˇ A. Investigation of the statistical nature and structure of the electri[14] Maluckov, C. cal breakdown time delay in gas diodes filled with neon, PhD Thesis, Faculty of Electrical energineering, University of Niˇs (2004). ˇ A.; Karamarkovi´c, J. P.; Radovi´c,M. K. Statistical analysis of elec[15] Maluckov, C. trical breakdown time delay in neon at 13.3 mbar pressure; Contributed papers of 21st International Symposium on the Physics of Ionized Gases; Soko Banja, 2002; Radovi´c, M. K.; Jovanovi´c, M. S. Ed.; pp. 414-417.

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C.A. Maluckov, Jugoslav P. Karamarkovi´c and Miodrag K. Radovi´c

ˇ A., Karamarkovi´c, J. P.; Radovi´c,M. K. Statistical analysis of elec[16] Maluckov, C. trical breakdown time delay in neon filled diode at 13.3 mbar; Proceedings of 5th General Conference of the Balkan Physical Union; Vrnjaˇ cka Banja, 2003; Joki´c, S.; Miloˇsevic, I.; Balaˇz, A.; Nikoli´c, Z. Ed.; pp. 1093-1096. ˇ A.; Karamarkovi´c, J. P.; Radovi´c, M. K. IEEE Trans. Plasma Sci. [17] Maluckov, C. 2003, vol. 31, 1344-1348. ˇ A.; et all, Application of Convolution Model on the Electrical Break[18] Maluckov, C. down Time Delay Distribution in Neon Filled-Diode at 6.5 mbar; Contributed papers of 22nd International Symposium on the Physics of Ionized Gases; National Park Tara, 2004; Hadˇzijevski, Lj. Ed.; pp. 381-384. ˇ Radovi´c, M.; Karamarkovi´c, J. Convolution model of breakdown [19] Maluckov, C.; time delay distribution in neon at 13.3 mbar; Contributed papers of 11th Symposium of Physicians in Serbian and Montenegro; Petrovac na Moru, 2004; Konjevi´ c, N.; Vujiˇci´c; Miranovi´c, P. Ed.; , vol. 3, pp. 79-83. (In Serbian) ˇ A.; et all Phys. Plasmas 2004, vol. 11, 5328-5334. [20] Maluckov, C. ˇ A.; Karamarkovi´c, J. P.; Radovi´c, M. K. Contrib. Plasma Physics [21] Maluckov, C. 2005, vol 45, 118-129.

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ˇ A.; et all The application of convolution-based statistical model on [22] Maluckov, C. the breakdown time delay distributions in krypton, Conference Record-Abstracts of 2005 IEEE International Conference on Plasma Science; Monterey, California, 2005; p. 201. ˇ A.; et all IEEE Trans. Plasma Sci. 2006, vol. 34, 2-6. [23] Maluckov, C. ˇ A. et all Physics of Plasmas 2006, vol. 13, 083502 (1-9). [24] Maluckov, C. ˇ A.; Karamarkovi´c, J. P.; Radovi´c,M. K. Investigation of the statisti[25] Maluckov, C. cal nature and structure of the electrical breakdown time delay in gas diodes filled with neon; The Physics of Ionized Gases, Invited Lectures, Topical Invited Lectures and Progress Reports; National Park Kopaonik, Serbia, 2006;, Hadˇ zijevski, Lj.; Marinkovi´c, B. P.; Simonovi´c, N. S. Ed.; pp. 317-324. [26] Knuth, D. E. Seminumerical Algorithms, 2nd ed, The Art of Computer Programming; Addison-Wesley: Reading, 1981; Vol. 2. [27] Press, W. H.; et all, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., Fortran Numerical Recipes; Cambridge University Press: Cambridge, 1997, Vol. 1. [28] Wackerly, D. D.; Mendenhall W. III; Scheaffer, R. L. Mathematical Statistics with Applications; Duxbury: Belmont, 1996. ˇ A. IEEE Trans. Plasma Sci. 2001, vol. 29, 832-836. [29] Radovi´c, M. K.; Maluckov, C. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Statistical Analysis of the Electrical Breakdown Time Delay Distributions ...

19

- . A.; Pejovi´c, M. M. J. Phys. D: Appl. Phys. 1979, vol 12, 1699-1702. [30] Boˇan, D [31] Pejovi´c, M. M.; Risti´c, G. S.; Karamarkovi´c, J. P. J. Phys. D: Appl. Phys. 2002, vol. 35, 91-103. - .; Radovi´c, M. K.; Krmpoti´c, D - . J. Phys. D: Appl. Phys. 1986, 2343-2349. [32] Boˇsan, D [33] Pejovi´c, M. M.; Risti´c, G. S. Phys. Plasmas 2002, vol. 9, 364-370. [34] Pejovi´c, M. M.; Risti´c, G. S. J. Phys. D: Appl. Phys. 2000, vol. 33, 2786-2790. [35] Pejovi´c, M. M.; Pejovi´c, M. M. Plasma Sources Sci. Technol 2005, vol. 14, 492-500. [36] von Engel, A. Ionized Gases; Claredon: Oxford, 1965. [37] Kudrle, V.; LeDuc, E.; Faitare, M. J. Phys. D: Appl. Phys. 1999, vol. 32, 2049-2055. [38] Markovi´c, V. Lj.; Goci´c, S. R.; Stamenkovi´c, S. N.; Petrovi´c, Z. Lj., Phys. Plasmas 2005, vol. 12, 073502 (1-9). [39] Osmokrovi´c, P. IEEE Trans. Plasma Sci. 1993, vol. 21, 645-653. [40] Osmokrovi´c, V.; Krivokapi´c, I.; Krsti´c,S.; IEEE Trans. Dielect. Elect. Insulation 1994, vol. 1, 77-81. [41] Miller, H. C. J. Appl. Phys. 1963, vol. 34, 3418.

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[42] Pejovi´c, M. M., Dimitrijevi´c, B. J. Phys. D: Appl. Phys. 1982, vol. 15, 87-90. - . A.; Golubovi´c, S. M. Acta Phys. Hungarica 1986, vol. [43] Pejovi´c, M. M.: Boˇsan, D 59, 273-278.

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RESEARCH AND REVIEW STUDIES

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In: Plasma Physics Research Advances Editor: Sergei P. Gromov, pp. 23-74

ISBN: 978-1-60456-136-4 © 2009 Nova Science Publishers, Inc.

Chapter 1

LINEAR COUPLING OF ELECTRON CYCLOTRON WAVES IN MAGNETIZED PLASMAS: BEYOND THE RANGE OF ONE-DIMENSIONAL THEORY* Alexander Shalashov Institute of Applied Physics of Russian Academy of Sciences, Ulyanova str. 46, 603950 Nizhny Novgorod, Russia

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Abstract Recent achievements are reviewed in the theory of linear mode coupling in electron cyclotron frequency range that has been stimulated by the progress of optimized stellarators and spherical tokamaks in which methods based on the linear mode conversion are especially important for electron cyclotron resonance heating and diagnostics of overdense magnetized plasma. The developed theory is generalized for two-dimensionally inhomogeneous plasma configurations and describes the conversion of ordinary and extraordinary plasma modes in the vicinity of the plasma cutoff while taking into account variation of a magnetic field on flux surfaces. This problem can not be treated within the standard one-dimensional approach of wave tunneling though the evanescent region since the mode coupling region assumes essentially two-dimensional topology when two cutoff surfaces intersect. Asymptotic analytical solutions of wave equations in the vicinity of the two-dimensional transformation region have been obtained that allow investigation of the structure of the transformed and reflected waves and corresponding transformation efficiencies for an arbitrary field distribution in the incident beam. New effects are found that are absent in the one-dimensional theory. In the present review, these theoretical results are considered as a solution of a new standard problem in plasma physics that can be used outside the scope of purely thermonuclear applications.

PACS 52.35.Hr, 41.20.Jb, 52.50.Sw, 52.70.Gw

*

Reviewed by A.V. Timofeev, Institute of Nuclear Fusion, Russian Research Centre 'Kurchatov Institute', Kurchatova sq. 1, 123182 Moscow, Russia, E-mail address: [email protected]

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Alexander Shalashov

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1. Introduction The problem of linear coupling (also known as linear interaction or mode conversion) between high-frequency normal waves propagating in smoothly inhomogeneous magnetized plasmas has a long history. The studies were initiated by Ginzburg in his works on the “tripling” of radio signals in the ionosphere [1–3]. Under astrophysical conditions, the linear interaction between waves is of considerable importance in describing the mechanism of radiation escape from dense plasmas, for example, from the solar corona [4]. In recent years, interest in linear mode conversion in the electron cyclotron resonance frequency range has increased appreciably in connection with the problems of high-frequency plasma heating and diagnostics in toroidal magnetic traps used in studies of controlled thermonuclear fusion. This is related to achievement of improved plasma confinement in modern tokamaks and to the development of alternative systems like spherical tokamaks and optimized stellarators, which are characterized by comparatively low magnetic fields and high plasma densities. The confinement regimes in which the density in the central regions of a plasma column exceeds its critical value for the propagation of electromagnetic waves used in conventional electron cyclotron (EC) plasma heating and diagnostic schemes, such as the ordinary (O) wave at the fundamental EC harmonic or the extraordinary (X) wave at the second harmonic, are realized increasingly often. The resonance absorption of the electromagnetic waves in spherical tokamaks has also proven to be inefficient, since the EC absorption can be realized only at fairly high harmonics of the electron gyrofrequency due to a weak magnetic field. One of the most promising ways to overcome these difficulties is based on conversion of the electromagnetic waves into the short wavelength electrostatic waves, the so-called electron Bernstein (B) waves [5], which have no density cutoffs and may be effectively absorbed at high cyclotron harmonics. In large-scale magnetic fusion devices, electron Bernstein waves may be effectively excited via a two-step mode conversion process (O–X–B conversion) in which an ordinary wave launched from the low field side is converted in the vicinity of the cutoff surface to a slow extraordinary wave, which, in turn, is converted near the upper hybrid resonance to an electron Bernstein wave [6–8]. The efficiency of the entire process is determined mainly by the O to X mode (O–X) conversion which is sensitive to the value of the parallel refractive index of the incident O wave with respect to an external magnetic field in the coupling region. The O–X–B mode conversion has been demonstrated experimentally via plasma heating, or current drive, or measurement of plasma emission related to the reversal B–X–O process, in a number of toroidal installations, such as the FT-1 tokamak [9], the Heliotron-DR device [10], the W7-AS stellarator [11–15], the MAST spherical tokamak [16, 17], the TCV tokamak [18–20], the NSTX spherical tokamak [21], and the LHD stellarator [22]. A recent review of experiments with electron Bernstein waves in fusion devices may be found in [23]. Most of the theoretical results related to the linear transformation of waves in the electron cyclotron frequency range have been obtained within the one-dimensional approximation, in which plasma density and magnetic field were assumed to vary along one direction [7, 8, 24– 32]. However, with the off-equatorial beam launch in a toroidal magnetic trap, the density and magnetic field gradients are not parallel which requires at least two-dimensional consideration of the problem for tokamaks and, in general, three-dimensional consideration for stellarators. The importance of non-planar inhomogeneity in converting electromagnetic

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Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

25

waves to electron Bernstein waves near the upper hybrid resonance (X–B coupling) was pointed out long ago [33]. Some attempts to consider this problem have been done in recent research [34]. However, the role of non-one-dimensionality in the O–X mode conversion has remained unstudied. In recent years it has been understood that the one-dimensional approximation may be insufficient to describe the O–X mode conversion in actual devices [35–45]. Such analysis has been initiated in the pioneer work by Weitzner [35], where it was found that in a twodimensional case, O–X conversion occurs in the essentially wider range of incident beam parameters as compared with the one-dimensional model in which effective transformation is possible only in a narrow range of the parallel refractive indexes. Moreover, all solutions of the wave equations obtained by Weitzner correspond to wave structures that undergo complete transformation, i.e. with no reflection of the ordinary mode from the transformation region. However, these solutions could be hardly applicable to describe a realistic case since they correspond to the infinitely increasing wave fields inside the evanescent region. Gospodchikov, Shalashov and Suvorov [36–41] developed an alternative theory of the O–X coupling in a two-dimensionally inhomogeneous tokamak-like geometry allowing for analytical solutions which include reflected waves, thus being free of the above-mentioned limitations of work by Weitzner. The theory is based on a new set of reduced reference wave equations in the transformation region that differs from that studied in [35] by making several physically clear simplifications. First, a class of solutions that correspond to the complete transformation was analyzed in [36]. Later, a complete solution of the reference wave equations that includes both the transmitted and reflected waves was obtained in [37–39], that allowed one to derive explicit expressions for the transformation and reflection coefficients, and the wave amplitude distributions were studied while taking into account phase relations in [40]. As a result, compact expressions for the wave fields that are transmitted through and reflected from the transformation region were formulated, and, in particular, a procedure was determined for coupling the quasi-optical solutions for the incident, transmitted, and reflected beams far from the transformation region. All of the mentioned results were obtained assuming that an external magnetic field has constant direction in the transformation region and plasma is homogeneous along this direction. Being applied to a toroidal magnetic configuration this means that a poloidal component of the magnetic field and a curvature of a magnetic flux surface were neglected, thus the two-dimensional inhomogeneity was fully attributed to the variation of a toroidal magnetic field intensity on a flux surface within the transformation region. However, the poloidal component of a magnetic field and its variation may be of importance [30], especially in spherical tokamaks. In a two-dimensional geometry, the poloidal component of a magnetic field was first accounted by Popov and Piliya [42, 43]. Under assumptions of the small poloidal component as compared to the toroidal field and the constant direction of a magnetic field, the reduced wave equations obtained in [42, 43] had the same form as were studied independently in [36–40]. Therefore, general solutions for the wave fields and the transformation coefficients presented in [42, 43] were essentially the same as derived previously in the zero poloidal field limit. In [41] the generalized wave equations were derived in which an arbitrary poloidal magnetic field is taken into account including, in particular, a magnetic shear. Conditions when the generalized wave equations can be converted to the “canonical” form (as for zero poloidal field) were analyzed in detail in [41]. This allowed reconsidering all previous results while taking into account the small but finite

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26

Alexander Shalashov

poloidal magnetic field component, but still neglecting the magnetic shear. Although the poloidal field is not presented explicitly in the “reduced wave” equations in this limit, it affects the transformation coefficients of bounded wave beams after a convolution to a “full wave” solution. In [44] it was shown that in the shearless limit the poloidal field component can be excluded from wave equations by the integral Gaussian transformation. This allows applying the analytical solutions obtained previously to describe the O–X coupling for the magnetic field with arbitrary but constant direction. Finally, analytical theory was tested with a full wave calculation by Irzak and Popov [45], and reasonable agreement was found. In this chapter, the mentioned above theoretical results are summarized and considered as a solution of a new standard problem in plasma physics that can be used outside the scope of purely thermonuclear applications. The work is structured as follows. In Section 2, peculiarities of linear interaction between the O and X waves in one- and two- dimensional cases are discussed on qualitative level. In particular, we introduce new physical effects that arise when the two-dimensional inhomogeneity of the medium is taken into account. In Section 3, we define the model geometry and formulate the reference wave equations for the two-dimensional geometry in smoothly inhomogeneous plasma. Complete analytical solution of the reference wave equations that describe the wave fields in the region of linear interaction are given in Section 4. An asymptotic behavior of the solutions obtained far from the interaction region is considered in the Wentzel–Kramers–Brillouin (WKB) limit in Section 5. O–X transformation efficiency for an arbitrary field distribution in an incident wave beam is found in Section 6. Here, the optimum wave beams that exhibit complete transformation with no reflection are analyzed as well. In Section 7, transition to the onedimensional case is performed. In Section 8, examples of the field distribution in the transformed and reflected beams are given for incident radiation in the form of a plane wave and of a Gaussian beam. For these cases the relation between the wave fields before and after the transformation region are found in a compact analytical form, that gives us an insight into the physical processes involved in O–X mode interaction. In Section 9, the transformation coefficients of Gaussian beams are analyzed in more detail, which is the important problem for practical applications. Application of the developed theory to toroidal magnetic configurations of fusion research is discussed in Sections 10 and 11. In particular, modification of the wave equations with taking into account the poloidal magnetic field is presented in Section 10. Basing on these equations, the limit of a weak poloidal field is analysed in more details in Section 11, were main analytical results of previous sections are re-examined in the weak poloidal field limit. The results are summarized in Conclusion.

2. Linear Interaction between O and X Waves in One- and TwoDimensional Cases Propagation of high frequency electron cyclotron waves in fusion plasmas is usually adequately described within the well-known geometrical-optics (Wentzel–Kramers–Brillouin, WKB) approximation [3]. Here it is assumed that the typical scale lengths of the plasma parameter variation are large as compared to the wavelength. The Maxwell’s equations are than may be reduced to the algebraic wave equations describing a polarization of a monochromatic harmonic electromagnetic field E ∝ exp(ikr − iωt ) that propagates in a locally homogeneous media:

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Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

k × k × E + k02 ε E = 0 .

27 (2.1)

Here ε is the plasma dielectric permittivity tensor, k0 = ω / c is the vacuum wave vector, and E is the complex amplitude of the electric field. We will use standard Stix representation [46] for the electric field components

E ± = ( E x ± iE y ) / 2 , E|| = Ez

(2.2)

in the Cartesian coordinate system with the z-axis directed along the external magnetic field. The sense of rotation of the components E+ and E− coincides with the sense of cyclotron rotation of ions and electrons, respectively, and E|| is the longitudinal component with respect to the magnetic field. For electromagnetic type of waves propagating outside the plasma resonances, one may adopt the permittivity tensor in “cold” plasma approximation which in Stix representation takes the diagonal form

⎛ε + ⎜ ε=⎜0 ⎜0 ⎝

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where

ε− 0

0⎞ ⎟ 0⎟, ε || ⎟⎠

(2.3)

ε || , ε + and ε − are defined as ε± = 1−

with

0

2 2 ω pe ω pe , ε || = 1 − 2 , ω (ω ± ωce ) ω

(2.4)

ωce and ω pe being correspondingly the electron cyclotron and plasma frequencies. The

solvability condition of the wave equation, det(k × k × E + k0

2

ε E) = 0 , in Stix frame gives

the following dispersion relation for the waves in cold magnetized plasma [40]:

[

]

N ⊥2 (ε + − ε || )(ε − − N 2 ) + (ε − − ε || )(ε + − N 2 ) = 2ε || (ε + − N 2 )(ε − − N 2 ) . Here N ⊥ =

(2.5)

k x2 + k y2 / k0 and N || = k z / k 0 are components of the wave refractive

index perpendicular and parallel to the external magnetic field, and N =

N ||2 + N ⊥2 .

The WKB approximation gets invalid when the wave meets the cut-off where, in particular, the O–X mode conversion takes place. Here the perpendicular refractive index goes to zero, and the wavelength is no longer negligible in comparison with the scale length of the relevant plasma parameters. Nevertheless the major features of the O–X conversion

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Alexander Shalashov

process can be still qualitatively understood within the geometrical-optics approximation [3, 6–8, 24–31]. The O–X conversion occurs in the vicinity of the O mode and the slow X mode cutoffs,

ε || = 0 and ε + = N ||2 , when both cutoffs are close to each other. Here, the following

inequalities are hold:

N ⊥ 0 and ε + > N ||2 , or to the X mode for ε || < 0 and

ε + < N ||2 . The region with ε || (ε + − N||2 ) < 0 is evanescent for the left-hand polarized

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waves. The O mode corresponds to the plasma density less than the critical density where ω = ω pe , and vice versa, the X mode corresponds to the plasma density greater than the critical density. Thus, only the O mode can be launched from vacuum. The typical dispersion curves as dependent on plasma density ( ∝ ω pe ) are shown in 2

Fig. 1. One can see, that at a certain propagation angle, N || = N || , both opt

ε || and ε + − N ||2

take zero value at the same density, see point A in Fig. 1. In this case referred further as optimal or complete transformation, the O mode launched from vacuum smoothly turns into the X mode. For not optimal N || the O and X modes are separated by the evanescent region. Once the X wave is generated, it propagate in overdense plasma up to the high-density Xmode cut-off, see point B in Fig. 1. There they are reflected back towards the upper hybrid resonance (UHR) shown in small panel. Up to this point it was sufficient to describe the wave propagation in the “cold” plasma approximation neglecting the temperature effects. At the UHR the wave length decreases, such that it reaches the size of the electron gyroradius and the ‘hot’ plasma effects (not described by used above dispersion relations) have to be taken into account. Here the X mode coincides with the electron Bernstein mode: in the linear description the X waves are completely converted into the B waves. It should be noted that the O–X–B process can only take place if the plasma density is above the O mode cutoff density.

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Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

Figure 1. Square of perpendicular refractive index

N ⊥2 (ω p / ω )

29

of a wave launched with fixed

N ||

from the vacuum in a plasma with a density gradient in the x-direction for the optimal (solid line) and not optimal (dashed lines) transformation. Transition to the electron Bernstein mode in hot plasma is

N⊥

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shown in the small panel for

x e + = N||2

(not squared) in case of optimal transformation.

x e + = N||2

X mode

y

X mode

z

e || = 0

2a

y

O mode

e || = 0

O mode (a)

Figure 2. The

ε || = 0

and

(b)

ε + = N ||2

cutoff surfaces in (a) a one-dimensional geometry, where the

plasma density and, in general, the magnetic field strength varies along the x axis, and (b) in a twodimensional geometry, where the plasma density and the magnetic field vary in the xy plane. The evanescent region for left-hand polarized waves is hatched.

A principal difference between O–X transformation processes in the one-dimensional (slab) and non-one-dimensional cases has been first pointed out in [36]. Let the system be homogeneous along a certain direction. Assuming for a while that the external magnetic field

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Alexander Shalashov

has constant direction along the direction of homogeneity, a propagating beam may be represented as a sum of waves with the constant parallel refractive indexes. The situation typical of a one-dimensional approximation is illustrated in Fig. 2a. Here the propagation regions for the O and X waves with a fixed parallel refractive index are separated by a slab evanescent region between the plane-parallel cutoff surfaces

ε || = 0 and ε + = N ||2 . The

mode conversion actually occurs as a tunneling of the electromagnetic radiation throughout the evanescent region; thus the transformation efficiency depends on the width of this region, which for a plane wave is dependent on the propagation angle with respect to the magnetic field. Analytic formulae for the tunneling efficiency window were given by Preinhaelter [24], Weitzner and Batchelor [25], Zharov [26], Mjølhus [27] and Tokman [28]. Best agreement with a full wave calculation was found in [47] for the formula of Zharov, Mjølhus and Tokman,

⎧ ⎫ ωce T ( N || , N y ) = exp ⎨− πk 0 Ln 2(1 + ωce / ω )( N || − N ||opt ) 2 + N y2 ⎬ 2ω ⎩ ⎭

[

]

(2.8)

which is the most frequently used. The key parameters here are the normalized density scale length k0 Ln calculated at the critical density1, and the optimal longitudinal refractive index,

N ||opt = ωce /(ω + ωce ) , at which the two cutoff surfaces coincide, when, additionally,

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N y = 0 . The evanescent region is then absent, which corresponds to complete (with no reflection) mode conversion. As noted in [23], the O–X conversion in one-dimensional case shows in some sense a similarity to the Brewster angle in optics, where a beam passes through an interface without reflection, with the appropriate choice of its angle of incidence and its polarization. As we see from Fig. 2b, the situation may be topologically different in the two- or threedimensional case where the cutoff surfaces intersect in space along a certain line. There is no evanescent region for a ray passing through this line, which formally corresponds to the case of complete transformation. In contrast to the one-dimensional model, such a ray exists not for one optimal value of N || , but for a certain continuous range of N || values at which the line of intersection between the cutoff surfaces exists. A change of N || in this range leads only to a spatial displacement of the transformation region following the line of intersection between the cutoff surfaces. Of course, the WKB approximation becomes invalid in the transformation region, where N ⊥ → 0 . However, a comprehensive analysis of the wave equations presented below yields essentially the same result: for each value of N || at which the cutoff surfaces intersect, there is an optimal field distribution for which the incident wave beam passes through the interaction region without reflection, i.e., undergoes complete transformation.

1

Analytical theory works for

k0 Ln ≥ 10

[47].

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31

It should be noted that the two-dimensional configuration shown in Fig. 1b has no symmetry with respect to the interchange of the

ε || = 0 and ε + = N ||2 cutoff surfaces. The

solution of the wave equation depends on whether the vectors ∇ε || , ∇ε + and the magnetic field B form a right- or left-hand triple. In particular, the type of this triple changes when the magnetic field direction is reversed, which, according to the reciprocity law for gyrotropic media, is equivalent to the reversal of the direction of wave propagation [3]. As a result, the transformation coefficients for the beams that propagate in the direction of increasing plasma density (O–X mode conversion) and in the opposite direction (X–O mode conversion) are not reversible; they turn out to be different without violation of the reciprocity law [37–45].

3. The Reference Wave Equations for the Two-Dimensional Geometry

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Let us consider first a simplified model of the two-dimensional transformation region in which the magnetic field direction is constant, while the magnitude of the magnetic field and the plasma density vary in the plane perpendicular to this direction. Later we will generalize this model to take into account variation of the magnetic field direction as well. Let us choose a coordinate system in accordance with Fig. 2b: the z axis is directed along the magnetic field, the y axis is located in the evanescence region, and the x axis is located in the propagation region of left-hand polarized waves and is directed toward the higher plasma densities along the bisector of the angle between the cutoff surfaces. Formally, the x axis is directed along the vector ∇ε || / | ∇ε || | +∇ε + / | ∇ε + | . The wave beam propagation in the positive direction along the x axis corresponds to the O–X mode conversion and propagation in the negative direction corresponds to the X–O mode conversion. We choose the (x, y) coordinate origin on the line of intersection between the cutoff surfaces for a given N || , i.e., at the “transformation point” where

ε || ( x, y ) = 0 and ε + ( x, y ) = N ||2 . The flat cutoff surfaces are assumed in the

vicinity of the transformation region as shown in Fig.2b. The wave equation for the spatial distribution of a monochromatic electromagnetic field may be obtained from Eq. (2.1) considering the wave vector as operator k = −i∇ :

rot rot E − k 02 ε E = 0 .

(3.1)

Since the model under consideration is homogeneous along the z axis, the longitudinal wavenumber N || ≡ N z is conserved, as in the one-dimensional approximation. Accordingly, the general solution of the wave equation can be found in the form

E ( X , Y , Z ) = ∫ G ( N z ) F( X − ΔX ( N z ) , Y − ΔY ( N z ) , N z )exp (ik0 N z Z ) dN z . (3.2)

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32

Alexander Shalashov Here, F( x, y , N z ) is the reduced partial solution of the wave equation corresponding to

a fixed N z ; ( X , Y ) is a laboratory coordinate system that is independent of N z , while

ΔX ( N z ) and ΔY ( N z ) define the displacement of the local ( x, y ) coordinate origin as a function of N z , and G ( N z ) is any finite weight function. Note that due to the dependence of F(…) on N z , weight function G ( N z ) can not be treated as usual Fourier image of

E (Z ) in Eq. (3.2). Following the usual procedure, the wave equation may be simplified assuming that the medium is smoothly inhomogeneous, i.e., if λ 0 ⎫ + ~ Ar ( x ′, y ′) = ∑ ⎨ ⎬ iν n Diν n+ −1 ( x ′) α < 0⎭ n =0 ⎩ − An hn +1 ( y ′),

with

(4.17)

ν n+ = (n + 1) tan | α | . The complete analytical solution of the system of reference

wave equations in the form (4.12)-(4.17) have been first found in [38, 39]. Note that the solutions obtained have a different structure depending on the sign of α . This is due to the already mentioned asymmetry with respect to the interchange of the

ε || = 0 and ε + = N ||2

surfaces which is equivalent to the change of sign of α . Solutions obtained may be represented as a mapping of the wave field from one section x ′ = x1′ onto another section x ′ = x2′ : ∞

A

t, r

(x ′2 , y ′2 ) = ∫ A t (x1′ , y1′ )G t, r (x1′ , x2′ , y1′ , y 2′ ) dy1′ , −∞

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(4.18)

38

Alexander Shalashov

~ ′ D iν ( x 2 ) , G t ( x1′ , x 2′ , y1′ , y 2′ ) = ∑ hn ( y1′ ) hn ( y 2′ ) ~ n Diν ( x1′ ) n =0 ∞

(4.19)

n

~ D + ( x 2′ ) ′ ′ ( ) ( ) + > , α 0 h y h y iν n −1 ⎧ ⎫ 1 2 n +1 n + (4.20) G r (x1′ , x ′2 , y1′ , y ′2 ) = ∑ ⎨ ⎬ iν n ~ D + ( x1′ ) n = 0 ⎩− hn ( y1′ )hn +1 ( y 2′ ) , α < 0 ⎭ iν ∞

n

t

r

Basically, G and G are the Green function for the wave field of the transmitted and t

reflected radiation that is completely determined by the distribution A in a certain section. Remind that this component corresponds to the incident wave for x ′ → −∞ in the WKB region. As in Eq. (4.16), the dependence on the longitudinal coordinate x1′ on the right-hand side of (4.18) vanishes after performing the integration over y1′ . For wave field distributions arbitrary in the z direction, one must take into account the displacement of the local ( x ′, y ′) coordinate system corresponded to different refractive indexes N z′ as in the form (3.2). Let ( X , Y ) be some laboratory coordinate system that corresponds to a certain value of N z . The point of intersection between the

ε || = 0 and

ε + = N z′ 2 surfaces in dimensionless coordinates will then be defined as

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ΔX = −

N z ΔN z N z ΔN z , , ΔY = − L∇ | ∇ε + | cos α 2 L∇ | ∇ε + | sin α

(4.21)

here only first order terms over ΔN z = N z′ − N z are retained since the shift of the coordinate origin could not be large when the flat cutoff surfaces are assumed. The local and laboratory coordinates systems are related by

x′ = X − ΔX ,

y′ = Y − ΔY .

(4.22)

From Eqs. (3.2) and (4.18), the fields in sections X 1 and X 2 , respectively, are related by

kL E (Y2 , Z 2 ) = 0 ∇ 2π t,r 2

∞ ∞ ∞

∫ ∫ ∫ E (Y , Z )G ( X t 1

t,r

1

1

1

− ΔX , X 2 − ΔX , Y1 − ΔY , Y2 − ΔY ) ×

−∞−∞−∞

(4.23)

× exp[ik0 L∇ N z′ ( Z 2 − Z1 )]dY1dZ1dN z′ ,

The dependence on X 1 on the right-hand side vanishes for the same reasons as in Eqs. (4.16) and (4.18). The X–O mode conversion, where the incident wave propagates in the negative direction of the x axis toward the lower plasma densities, can be considered in a similar way.

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Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

39

Moreover, it follows from the symmetry of Eqs. (4.3) that the change in the direction of propagation along the x axis is equivalent to the simultaneous change of sign of the angle α and the direction of the y axis; i.e., from the standpoint of our description, the O–X mode conversion for α > 0 is equivalent to the X–O mode conversion for α < 0 and vice versa, the O–X mode conversion for α < 0 is equivalent to the X–O mode conversion for α > 0 .

5. Expansion of the Green Functions in the WKB Region The most important for practical applications problem is related to reconstruction of the field distribution in the transformed beam after the passage through the region of linear interaction ( x 2′ > 0 ) for a given field distribution in the incident beam specified before the transformation region ( x1′ < 0 ). This problem may be essentially simplified when the both sections x1′ and x 2′ lie in the WKB region, i.e. x1′ → −∞ and x ′2 → +∞ ; note however, that the sections x1′ and x 2′ should still be close enough to the transformation region, so that the spatial inhomogeneities not taken into account in the reference wave equations could be t

ignored. Corresponding asymptotic representation of the Green function G allows to determine the field distribution in the transmitted wave beam excluding from analysis the region of linear interaction, | x ′ |≤ 1 , where WKB approximation breaks down. The same is r

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true as well for the reflected waves defined by the Green function G . In the WKB region of Eqs. (4.3), the parabolic cylinder functions defining the field distribution over the x ′ coordinate can be substituted by the following asymptotic expansions for | z |>> ν : 2 ~ D iν ≈ z iν e − z / 4 +

2π σ −iν −1 πν + z 2 / 4 iπ / 4 z e , z = x ′ 2 cos α ⋅ e , Γ(−iν )

(5.1)

where σ = 1 for x ′ < 0 and σ = 0 for x ′ > 0 [49, 50]. For the transmitted wave only the first term in (5.1) is significant resulting in the following asymptotic expansion when x1′ → −∞ and x2′ → +∞ :

~ Diν ( x 2′ ) ⎡ ⎤ x1′ 2 − x 2′ 2 n ′ ′ ( ) i x x i ≈ + exp ν ln / cos α ⎥ . ~ ⎢ n 2 1 2 Diν ( x1′ ) ⎣ ⎦ n

(5.2)

Using this representation and Mehler’s formula [49] n

⎡ 2 xyz − ( x 2 + y 2 ) z 2 ⎤ 1⎛z⎞ 1 ( ) ( ) = H x H y exp ⎜ ⎟ ∑ n n ⎢ ⎥ 1− z2 n = 0 n! ⎝ 2 ⎠ 1− z2 ⎣ ⎦ ∞

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40

Alexander Shalashov

with

z = exp[i ln (x ′2 / x1′ ) tan | α |] ,

(5.4)

one can perform summation in Eq. (4.19) for the Green function of the transmitted wave, and obtain the following asymptotic representation [40]:

sin α × π [1 − exp(−2π~ tan α )]

G t (x1′ , x 2′ , y1′ , y 2′ ) ≈

⎡ x ′ 2 − x 2′ 2 ( y ′ 2 + y 2′ 2 ) cosh(π~ tan | α |) − 2 y1′ y 2′ × exp ⎢i 1 cos α − 1 sin | α 2 2 sinh(π~ tan | α |) ⎣

⎤ |⎥ . ⎦

(5.5)

Here, π~ = i ln ( x1′ / x 2′ ) . In this expression, both x1′ and x ′2 should lie in the WKB region, but may be separated by the region of linear interaction where the WKB approximation is not applicable. This allows to use integral operators (4.18) and (4.23) with kernel (5.5) for calculation of the transformed beam structure from the incident beam. In this case, x1′ < 0 , x 2′ > 0 , and

π~ = π + i ln | x1′ / x 2′ | .

(5.6)

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As we will see later, the second term in (5.6) results in oscillations of the wave fields in the incident and transmitted radiation along the x ′ coordinate. However, the spatial period of these oscillations is rather large, thus they develop at distances far from the transformation region, | x ′ |≥ exp(2π cot | α |) , where the approximations used for obtaining basic equations (4.3) are violated. In most of practical cases, the second term in Eq. (5.6) can be omitted; below in most cases we will assume that π~ = π . With this approximation, dependence of the wave fields (with a fixed value of N z ) on the x ′ coordinate takes a

simple form given by Eq. (4.4). One can see that the phase shift along the x ′ axis is the same for all of the terms in sum (4.12), what means that y-structure of the wave beam is not changing in the WKB region. r

Unfortunately, the Green function for the reflected waves G can not be converted to a simple form in the WKB limit. It can be verified that the reflected wave decays

~

asymptotically as | Diν −1 |∝ 1 / x 2′ when x ′2 → +∞ ; therefore, below, we will not consider the region x 2′ > 0 for the reflected radiation. At x1′ < 0 and x2′ < 0 , the following asymptotic expansion holds for the reflected wave

~ D + ( x 2′ ) ⎡ +⎛π iν n −1 x1′ 2 + x 2′ 2 ⎤ 2π ⎞ ′ ′ + ≈ − − + ⋅ exp ln | 2 cos | cos ν α α i x x i ⎜ ⎟ ~ ⎢ n ⎥ . (5.7) 1 2 2 2 Γ(1 − iν n+ ) D + ( x1′ ) ⎝ ⎠ ⎣ ⎦ iν n

Due to the gamma function in the denominator of the right-hand side, we failed to obtain a compact expression for the sum over n in Eq. (4.20) as we did for the transmitted radiation. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Thus, the Green function of the reflected wave in the WKB region is still represented as an infinite sum:

⎧− hn +1 ( y1′ )hn ( y ′2 ) , α > 0⎫ × ⎨ ⎬× + n=0 ⎩+ hn ( y1′ )hn +1 ( y ′2 ) , α < 0⎭ n ) ⎡ x ′ 2 + x 2′ 2 y ′ 2 + y 2′ 2 ⎤ ⎛π ⎞ × exp ⎢− ν n+ ⎜ + i ln | 2 cos α ⋅ x1′ x 2′ | ⎟ + i cos α 1 − sin | α | 1 ⎥. 2 2 ⎝2 ⎠ ⎣ ⎦

G ( x1′ , x ′2 , y1′ , y ′2 ) = r

2π iν n+

∞

∑ Γ(1 − iν

(5.8)

This sum can be transformed to an integral using an appropriate integral representation of the gamma function and Mehler’s formula (5.3), but the resulting expression seems to be less convenient for both analysis and calculations. Let us consider the main properties of the asymptotic Green functions (5.5) and (5.8). t First of all, it can be verified that G → δ ( y1′ − y ′2 ) when x ′2 → x1′ , as would be expected

from the definition of the Green function (4.18). When transmitted beam can be presented as

G t ( x1′ , x ′2 , y1′ , y ′2 ) =

α → 0 , the Green function for the

⎡ x ′ 2 − x ′22 ( y1′ − y 2′ ) 2 ⎤ exp ⎢i 1 − ⎥. 2 2π 2π ⎣ ⎦

1

(5.9)

This expression corresponds to the result of the one-dimensional theory. In contrast, if tan | α | in Eq. (4.14) is not too small, then terms in sum (4.19) decrease as exp( −πν n )

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t

when n increases. In this case, we can derive an approximate expression for G by omitting all terms except n = 0 in sum (4.19):

G t ( x1′, x2′ , y1′, y2′ ) ≈

sin | α |

π

⎤ ⎡ x′2 − x2′2 y ′2 + y2′2 exp ⎢ − πν 0 + i 1 cos α − 1 sin | α |⎥ . (5.10) 2 2 ⎦ ⎣

This approximation holds if the beam structure is close to the optimal profile discussed below. Note that only the pre-exponential factor depends on the sign of α in the asymptotic Green function (5.5), which is valid for any sign of α . In particular, it follows that

G t (α ) = G t (− α )exp(π tan α ) .

(5.11)

Mathematically, the factor exp(π tan α ) appears due to the difference of

ν n for the positive and negative α in Eq. (5.2). Having in mind that the change of sign of α is equivalent to the change of the transformation direction, the latter relation can be rewritten as t t (x1′ , x 2′ , y1′ , y 2′ ) = GXO (x 2′ , x1′ , y1′ , y ′2 ) exp(π tan α ) . GOX

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(5.12)

42

Alexander Shalashov Since exp(π tan α ) does not depend on both coordinates and the wave refractive index

N z , the same relation must also hold for the field distributions in the WKB zone, provided that the distributions of the incident field for the O–X and X–O mode conversions are identical. Thus, we obtain that O–X and X–O mode conversions are not reversible, opposite to the one-dimensional case. For the reflected wave, the relationship between the O–X and X–O mode conversions appears differently. As one can see from definition (4.20), the change of sign α is equivalent to the interchange of the y variables and the change of sign of the corresponding Green function: r r (x1′, x2′ , y1′, y2′ ) = −GXO (x2′ , x1′, y2′ , y1′ ) . GOX

(5.13)

It thus follows that when passing from the O–X mode conversion to the X–O one, the absolute value of the “scalar” product of the incident and reflected wave beam distributions is conserved: ∞

∞

r t r t ∫ AOX (x1′, y′)A (x2′ , y′)dy′ = − ∫ AXO (x2′ , y ′)A (x1′, y′)dy ′ .

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−∞

(5.14)

−∞

The applicability of this relation is not limited to the WKB region, however an additional symmetry appears in the WKB region—the x variables in each integral can be interchanged. The asymptotic representations of the Green functions for the transmitted and reflected waves analyzed in this section may be useful in calculations of the wave conversion in systems with a complex nonuniform distribution of plasma parameters, such as tokamaks and stellarators. Since the solutions found allow the region of linear wave interaction to be completely excluded from analysis, standard quasi-optics beam-tracing codes may be used to simulate the wave field distribution in fusion devices [51]. Indeed, the structure of the incident radiation in some section corresponding to the WKB region of Eqs. (4.3) may be determined using these tools and, subsequently, the field of the transformed wave in section that lies in the WKB region behind the transformation region is calculated using the Green function; the subsequent evolution of the beam is again considered in terms of standard quasioptics models. For application of geometrical-optics ray-tracing models, it is sufficient to determine the O–X transformation efficiency. This separate and, in some cases, more simple then determination of a full wave problem, is considered in the next section.

6. Transformation Efficiency and the Optimum Beams Let us introduce the O–X transformation, Tn , and reflection, Rn , coefficients for the field intensity that correspond to the basis functions hn . A wave beam with a transverse structure proportional to one of the basis functions hn , conserves its shape after passing the transformation region,

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∞

⎛ ⎞ x1′ 2 − x ′22 t ⎜ ′ ′ ′ ′ ′ ′ ( ) ( ) = − + h y G x , x , y , y d y exp πν i cos α ⎟⎟ hn ( y ′2 ) . (6.1) 1 2 1 2 1 n ∫−∞ n 1 ⎜ 2 ⎝ ⎠ and is reflected in the form of a “neighboring” basis function:

2π iν ∫ h ( y ′ )G (x′ , x′ , y ′ , y ′ ) dy ′ = Γ(1 − iν

∞

r

n

−∞

1

1

2

1

2

n

1

n

)

×

⎡ x ′ 2 + x 2′ 2 ⎤ ⎧− hn −1 ( y 2′ ), α > 0 ⎞ ⎛π . × exp ⎢− ν n ⎜ + i ln | 2 cos α ⋅ x1′ x 2′ | ⎟ + i cos α 1 ⎥×⎨ 2 ⎠ ⎝2 ⎦ ⎩+ hn +1 ( y 2′ ), α < 0 ⎣

(6.2)

The coefficients in front of hn in the right-hand side are calculated in the WKB region. Since hn form orthonormal basis, see Eq. (4.15), corresponding transformation, Tn , and reflection, Rn , coefficients can be determined as squares of absolute values of the coefficients in front of hn in the right-hand part of Eqs. (6.1) and (6.2):

Tn = exp(−2πν n ) .

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Rn =

(6.3)

2πν n exp(−πν n ) = 1 − exp(−2πν n ) | Γ(1 − iν n ) | 2

(6.4)

In the latter relation, we used the identity πν n | Γ(1 − iν n ) | = sinh(πν n ) [48]. We see −2

from these formulas that Tn + Rn = 1 ; i.e., the solutions obtained satisfy the rule of energy flux conservation3. For an incident beam with the arbitrary distribution over the both transverse coordinates,

E1t (Y , Z ) in the WKB region X 1 → −∞ , the structure of the transmitted beam, E 2t (Y , Z ) , in the limit X 2 → +∞ is determined by Eqs. (4.23) and (5.5). This allows, in particular, to obtain the corresponding transformation coefficient which is defined as a ratio of power fluxes in the transmitted and incident beams, T = Π 2 / Π1 , where

3

Here, the energy conservation law has the same form as those for the waves propagating along the x axis in the one-dimensional case. Since we consider the y-limited modes the energy flux along the y axis is zero when

y → ∞;

and since the medium is homogeneous in the z direction the energy flux in this direction is

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44

Alexander Shalashov

Π1, 2 =

1 2π

∞ ∞

∫ ∫ | E (Y , Z ) | t 1, 2

2

dYdZ .

(6.5)

−∞−∞

However, calculations of the transformation coefficient may be essentially simplified if to exploit the orthogonality of the transverse basis functions (4.15) and Fourier harmonics over the Z coordinate. Due to the orthogonality, there is no interference between different terms in the sum (4.12) and in the integral over N z in (4.23) when calculating the transformed wave intensity – this is essentially the Parseval’s theorem [48] for orthogonal series. Each term corresponding to fixed n and N z , is converted independently with the transformation coefficient (6.3). Therefore, the overall transformation coefficient may be obtained as a weighted sum of the partial transformation coefficients corresponding to fixed n and N z : ∞

1 T= dN z′ Π1 −∫∞

∞ ∞

∞

∑ T ∫ ∫ exp[− ik L n =0

0 ∇

n

2

N z′ Z ]hn (Y − ΔY )E (Y , Z )dYdZ , t 1

(6.6)

−∞−∞

Here the weight function is represented by the internal integrals over Y and Z that project the initial field distribution to the basis functions. We remind that displacement ΔY reflects the N z′ dependence of the local coordinate origin given by Eqs. (4.21). As for the wave fields, the sum over n in Eq. (6.6) can be performed analytically, then the result is expressed through the asymptotic Green function as ∞

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T=

∞ ∞ ∞ ∞

1 dN z′ ∫ ∫ ∫ ∫ exp[− ik0 L∇ N z′ ( Z − Z ′)] G*t (Y − ΔY , Y ′ − ΔY ) × Π1 −∫∞ −∞− ∞−∞−∞ × E1t (Y , Z )(E1t (Y ′, Z ′)) dYdZdY ′dZ ′.

(6.7)

*

Here ∞

G*t ( y1′, y 2′ ) = ∑ Tn hn ( y1′ )hn ( y 2′ ) = n =0

=

⎡ ( y ′2 + y 2′2 ) cosh(2π tan | α |) − 2 y1′ y2′ sin α sin | α exp ⎢ − 1 2 sinh( 2π tan | α |) π [1 − exp( −4π tan α )] ⎣

⎤ |⎥ (6.8) ⎦

is calculated similar as Eq. (5.5). Corresponding to Eq. (5.12), the reciprocity law for the O–X and X–O conversions manifests in the transformation coefficients as:

TOX = TXO exp(2π tan α ) .

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Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

45

Now let us consider an optimal wave beam which exhibits the ideal mode conversion with 100% efficiency. One can see that such structures are existing, e.g. from Eq. (6.3), in which ν 0 = 0 when α > 0 for O–X conversion (incident beam propagation in the direction of the plasma density increase) or

α < 0 for X–O conversion (propagation in the direction of

the plasma density decrease). This means that for every N z there is optimal field distribution (being proportional to the first basis function h0 ) that pass through the transformation region with no modification:

E

t, opt Nz

⎛ y 2 ix 2 cos α ⎞ L∇ . = C exp⎜⎜ − 2 − + ik0 N z z ⎟⎟ , a0 = 2 2 L∇ sin | α | ⎝ 2a 0 ⎠

(6.10)

Taking into account the importance of this result, we return to the dimensional coordinates here. Here a0 corresponds to the width of the optimal Gaussian beam, we will see later that this is the main scale characterizing the O-X coupling in two-dimensional geometry. Optimal wave field distribution (6.10) is not limited in the z direction. However, one can construct limited in the z direction optimal distribution by summation of the optimal beams (6.10) with different N z and taking into account displacement of the coordinate origin with N z : ∞

E t, opt =

∫ g (N ′ )E (x − Δx, y − Δy, z ) dN ′ , z

t, opt N Z′

(6.11)

z

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−∞

here, Δx = L∇ ΔX and Δy = L∇ ΔY are defined by Eqs. (4.21). By choosing a proper weight function g ( N z′ ) one can synthesize the ideally converted wave structures that are bounded in the both transverse directions. As a good practical example, one may consider the Gaussian weight function

⎛ 1 2⎞ g ( N z′ ) ∝ exp⎜ − k 02ζ 2 (N z′ − N z ) ⎟ ⎝ 2 ⎠

(6.12)

for which optimal distribution of the wave field in the incident beam takes the following form:

E

t, opt

=E

t, opt Nz

(

)

2 −1 ⎛1 ⎞ N z L−∇2 ∇ε + (ix + signα y ) − ik 0 z ⎜ ⎟. exp ⎜⎜ 2 k 2ζ 2 + N 2 L−2 ∇ε −2 (cosec | α | +i sec α ) ⎟⎟ z ∇ + ⎝ 0 ⎠

One can essentially simplify this expression noting that dimensionless quantity Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

(6.13)

ε + = N z2 in Eq. (3.8), thus the

46

Alexander Shalashov

κ≡

Nz = 2 k0 L∇ | ∇ε + |

2 | ∇ε || |

⎛ ω ⎞ ≈ 2⎜1 + ce ⎟ | ∇ε + | ω ⎠ ⎝

(6.14)

is weakly dependent of N z ; the last transition in Eq. (6.14) is valid for LB >> Ln . In the following both a0 and

κ , with the same accuracy as for L∇ and α , are assumed to be

independent of N z . Taking this into account, one can transform (6.13) to

⎛ y 2 1 (iκx + signα κy − iz ) 2 ⎞ ix 2 cos α E t, opt = C exp⎜⎜ − 2 + − + ik0 N z z ⎟⎟ . (6.15) 2 2 2 2 2 L∇ ⎝ 2a0 2 ζ + κ a0 (1 + i tan | α |) ⎠ This is a Gaussian distribution of general type with astigmatism and phase modulation. Relation between orthogonal widths of the optimal Gaussian beam in the y and z directions, a y and a z , is given by

a z2 ( a 2y − a02 ) = κ 2 a02 a 2y , and its dependence on parameter

ζ by

a z2 = ζ 2 + κ 2 a02 +

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(6.16)

κ 2 a04 tan 2 α . ζ 2 + κ 2 a02

(6.17)

Absence of a unique value for the parallel refractive index corresponding to the total conversion is an essentially two-dimensional effect found in [35] and discussed qualitatively in Section 2. Note, optimal beam definition is based on asymptotics (5.2) that are exact equalities for ν n = 0 . Therefore, in case of complete transformation, Eqs. (6.10) and (6.15) hold not only in the WKB region of the reference wave equations, but in the entire space where these equations are applicable. It should be stressed here, that solutions with strictly zero reflection are absent when ν 0 > 0 , i.e. α < 0 for O–X conversion and α > 0 for X–O conversion. More detailed analysis of the perfectly and nearly perfectly converting wave structures may be found in recent paper [52].

7. Transition to the One-Dimensional Case From geometrical considerations it follows that the considered two-dimensional model must reduce to the one-dimensional (slab) model in the limiting case of α → 0 with all other parameters being constant. From Eq. (6.9) one might suggest that the necessary condition for the slab approximation is α 0 in sum (4.12). Indeed, starting from reduced Green function (5.10) in which high harmonics are omitted, one can obtain essentially the same expression for the field behind the transformation region as given by Eqs. (8.2) and (8.3), but with tanh(π tan | α |) → 1 and sech(π tan | α |) → 0 . The functions

tanh(π tan | α |) and sech(π tan | α |) are shown in Fig. 3. We see from this figure that the harmonics with n > 0 may be omitted only at fairly large angles between the cutoff surfaces, α ≥ π / 4 . The contribution of high harmonics increases with decreasing α giving, in the limiting case α → 0 , the correct transition to the one-dimensional case. Note that an energy flux corresponding to the field structure (8.2)-(8.3) is finite. Thus, the transformation coefficient of a plane wave (possessing infinite energy flux) is always equal to zero in the two-dimensional case! However, when passing to the one-dimensional case α → 0 , the transformation coefficient takes on a nonzero value at α = 0 in a jump. This can be seen from the fact that the term y / 2a0 ∝ sin | α | in Eq. (8.3), which leads to a limitation of the 2

2

total energy flux in the beam, vanishes when α → 0 ; the remaining term closely matches the one-dimensional transformation coefficient (7.2). The field distribution in the reflected beam for a plane wave can be determined from Eqs. (4.18) and (5.8) as

⎡ y2 ⎤ Er = i tan α exp ⎢− 2 + ik0 N z z ⎥ ⎣ 2a0 ⎦

⎛ y⎞ in H n +1 (k0 N y a0 )H n ⎜⎜ ⎟⎟ ϕ n , (8.4) ∑ n n =0 2 n! ⎝ a0 ⎠ ∞

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Alexander Shalashov

where the right-hand side of Eq. (5.7) is denoted by

ϕ n . This relation is valid at α > 0 . At

α < 0 , one must interchange the indices n and n + 1 of the Hermitian polynomials and change the sign in front of the entire expression. Despite the cumbersome form, the field structure of the reflected radiation may be easily understood. Since only a y-localized 1.2 1

1.2

(à)

|Ei|

1

0.8

0.8

|Er|

0.6

0.6

0.4

0.4

|Et|

0.2 0

|Er|

0

2

4

6

8

y’

10

(c) (â)

|Et| |Ei|

0

5

10

y’ (d ) (ã)

|Er|

0.3

|Et|

0.2

|Er|

0.4

-5

|Ei|

0.4

0.8 0.6

0 -10

0.5

1

0.1

0.2 0

2

4

6

8

y’

10

0

0

Figure 4. Distributions of the amplitudes of the incident, Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

|Et|

0.2

1.2

0

(b) (á)

|Ei|

(dotted curves), and reflected,

| Er |

2

| Ei |

4

6

(solid curves), wave fields along the

ξ y = 0.2

(b) or Gaussian distribution with

y′

α = 10° ,

10

| Et |

coordinate in the WKB

Ny = 0

(a) and

N y = 0 , y0 = 0 , a z = ∞ , ξ y = 2

(c) and

(d). Since the plots in panels (a), (c), and (d) are even in

shown. The angle

y’

(dashed curves), transmitted,

region. The incident radiation is specified in the form of a plane wave with

N y = 1 / k 0 a0

8

y ′ , only the region y ′ > 0

is

the field of the incident and reflected radiations corresponds to

x1′ = −5 ; the field of the transmitted radiation corresponds to x2′ = 5 . structure with a finite energy flux penetrates through the transformation region, the infinite energy flux stored in the incident wave is converted to the reflected wave. One may expect the reflected wave to be an almost plane wave disturbed only in the region of “interaction” with the transmitted beam, | y |≤ a0 . This is illustrated in Figs. 4a and 4b, where the distributions of the amplitudes of the incident, transmitted, and reflected radiation fields were constructed for zero and nonzero values of N y . It follows from Eq. (8.3) that the transformation of the radiation is most efficient for N y = 0 . Accordingly, the greatest disturbances in the reflected beam would also be expected in this case (see Fig. 4a). For

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nonzero N y , the fraction of the radiation that “leaks” through the transformation region decreases, while the structure of the reflected radiation approaches a plane wave. Calculations show that the deviations of the reflected radiation field from a plane wave are insignificant for k0 | N y | a0 ≥ 1 over a wide range of parameters; as an example, Fig. 4b shows the boundary case that corresponds to N y = 1 / k0 a0 . Let us now assume that the incident field is specified by a Gaussian beam of general form:

Ei = exp[ −( y − y0 ) 2 / 2a 2y − z 2 / 2a z2 + ik0 N y y + ik0 N z z ] .

(8.5)

Here, the coordinate system in which the beam is shifted by the value y 0 is introduced in such a way that its origin coincides with the point of intersection between the

ε || = 0 and

ε + = N z2 cutoff surfaces. Substituting Eqs. (5.5) and (8.5) into (4.23), taking X 1, 2 = ∓b / L∇ and assuming that the size of the interaction region is large, b / L∇ >>| ΔX | , one can obtain the following expression for the wave field behind the transformation region at the point x2 = b :

[

]

E t = δ yδ z 1 + tanh(π tan α ) exp δ y2φ0 + φ y + φz + ik0 N z z ,

δ y = (1 + a02 tanh(π tan | α |) / a 2y ) Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

−1 / 2

,

(8.6) (8.7)

φ y = i (1 − δ y2 )k0 N y y0 − δ y2 ( y 2 − 2 yy0 sech(π tan | α |) + y02 )/ 2a 2y , δ z = [1 + κ 2 (1 − δ y2 )(a 2y + 2a02 tanh ( 12 π tan | α |))/ a z2 ]

−1 / 2

,

(8.8) (8.9)

φz = −δ z2 (z − Δz + iψ )2 / 2a z2 ,

(8.10)

Δz = κ ( 2b + (1 − δ y2 )k0 N y a 2y tanh ( 12 π tan α )) ,

(8.11)

ψ = κ (1 − δ y2 )( a 2y y / a02 + ( y + y0 ) tanh ( 12 π tan α )) .

(8.12)

φ 0 , φ y and φz are the terms responsible for the contribution of the plane wave and the Gaussian field structure in the y and z directions, respectively; κ is given by Eq. (6.14); the contribution of the plane wave φ 0 is defined by Eq. (8.3). For beams not bounded in the y Here,

direction

φ y = 0 and δ y = 1 ; for beams not bounded in the z direction beams φz = 0 and

δ z = 1 ; for beams not bounded in both directions the above equations are reduced to Eq. Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Alexander Shalashov

(8.2) for a plane wave. Since, in general, the parameters a y and a z take complex values, the derived relations can be applied to Gaussian beams with arbitrary focusing and propagation direction. However, below, we ignore the focusing and restrict analysis to the case of real valued a y and a z . One can see from Eq. (8.6)-(8.12), that the beam structure remains Gaussian after passing the transformation region, although the beam parameters change. In particular, for a beam not bounded in the z direction the effective width of the transformed beam and its shift in the y direction can be determined from the relation

δ y2φ0 + φ y = −( y − y0 )2 / 2a y2 + … as

a =a

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2 y

2 0

a y2 + a02 tanh(π tan | α |) a + a tanh(π tan | α |) 2 0

2 y

Figure 5. Transformed beam width

ay

, y0 = y0

(8.13)

a0 sech (π tan α )

a + a 2y tanh (π tan α ) 2 0

versus incident Gaussian beam width

The beam sizes were normalized to the scale length

L∇

(8.14)

at various angles

α.

defined by Eq. (3.8). The optimal beam size is

determined by the point of intersection of the curves with the

The function a y ( a y ) for various angles

ay

.

ay = ay

line (dashed line).

α is shown in Fig. 5. Irrespective of α , the

width of the transformed beam cannot be smaller than L∇ . The width of the transformed beam a y monotonically increases with width of the incident beam a y while changing in a

(

limited interval from a0 tanh

1/ 2

(π tan | α |)) at a y = 0 to (a0 tanh −1 / 2 (π tan | α |) ) at

a y = ∞ . The incident beam is widened if its initial width a y < a 0 and narrowed if a y > a 0 ; the optimal beam with a y = a 0 passes without any modification of its width. Thus, the spatial filter formed in a two-dimensionally inhomogeneous region of linear

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interaction imposes a certain spatial scale in the y direction that depends weakly on the initial width of the incident beam, provided that the angle α is not too small. One can also show that the width of the transformed beam cannot be smaller than a y even if the finite width of a beam in the z direction is taken into account. When passing through the transformation region, the Gaussian beam bounded along the z axis is displaced by Δz , see Eq. (8.10). The first term in the expression for Δz arises from the phase modulation proportional to ΔX in G and matches closely the well-known geometrical–optical solution for the optimal beam path [31]; the second term is related to the phase modulation of the incident beam along the y axis and arises from the dependence of ΔY on N z in (4.23). Note that the imaginary term iψ in (8.10) describes the additional t

phase modulation of the form exp(−iyz / a yz ) that arises for beams bounded in the z 2

direction. It is interesting to trace the behavior of the reflected radiation field in the case where the structure of the incident radiation approaches the optimal beam. For this purpose, let us consider the y-localized not-shifted distribution Gaussian distribution with y0 = 0 and

a z = ∞ . In this case, the reflected wave can be derived from Eqs. (4.18) and (5.8) as ⎡ y2 ⎤ ∞ 1 ⎛ y⎞ Er = i tan | α | exp ⎢ − 2 + ik0 N z z ⎥ ∑ k θ k H 2 k +1 ⎜⎜ ⎟⎟ ⎣ 2a 0 ⎦ k = 0 4 k! ⎝ a0 ⎠

(8.15)

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where

⎧ (ξ y2 − 1) k +1ξ y ϕ 2 k +1 , ⎪ 2 k +3 / 2 ⎪ (ξ y + 1) θk = ⎨ 2 k ⎪− (ξ y − 1) ξ y ϕ , ⎪ (ξ 2 + 1) k +1 / 2 2 k y ⎩ and

α >0 (8.16)

α > 1 ), the field pattern approaches the case of an incident plane wave shown in Fig. 4a.

(á)

log ( x’ )

(à) 50

50

40

40

30

30

20

20

10

10

0

0

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log ( - x’ )

-7.5

-5

-2.5

0

2.5

5

7.5

50

50

40

40

30

30

20

20

10

10

-7.5

-5

-2.5

-7.5

-5

-2.5

0

2.5

5

7.5

0

2.5

5

7.5

0

0 -7.5

-5

-2.5

0

y’

2.5

Figure 6. Amplitudes of the transmitted,

5

| Et |

7.5

y’

(upper panels), and reflected,

| Er |

(lower panels),

wave fields in the ( y ′, ln | x ′ | ) plane. The incident radiation is specified in the form of a plane wave with a nonzero

N y (a)

transformation point

and in the form of a Gaussian distribution in

y′ = 0

y′

centered relative to the

(b). The transmitted and reflected waves were constructed in the WKB

regions at x′ > 0 and x′ < 0 , respectively. The calculation was performed for the parameters in Figs. 4b and 4c.

For completeness, let us consider how the wave fields vary along the x axis, which defines the main direction of propagation. First of all, the quadratic phase dependence

± i cos α ⋅ x ′ 2 / 2 of the field in the reflected and transmitted waves, which corresponds to the linear dependence of ε + and ε || on x ′ , can be distinguished in the WKB region. This is

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an obvious effect that is also present in the one-dimensional case. In the two-dimensional geometry a new effect appears in that reflects the fact that the cutoff surfaces (shown in Fig. 2b) form an inhomogeneous waveguide channel in which the radiation can be trapped. Formally, this effect can be understood as follows. There is an additional phase shift of the form iν n ln | x ′ | that depends on the harmonic number n in the asymptotics of parabolic cylinder functions (5.2) and (5.7). As a result, after the summation over all n, the dependence on ln | x ′ | also appears in the expression for the field amplitude. For example, all of the relations derived in this section for the transmitted radiation can be easily modified to include the logarithmic dependence on x ′ — one should substitute π → π~( x ′) in all hyperbolic functions according to Eq. (5.6). The wave beam channeling is clearly seen from Fig. 6, which shows the amplitudes of the transmitted and reflected fields in the ( y ′, ln | x ′ |) plane for an incident plane wave with a nonzero N y and a Gaussian beam centered relative to

y ′ = 0 . These two situations also qualitatively reflect the general case: a sequence of foci lying on the straight line y ′ = const arises for distributions symmetric about y ′ = 0 ; the sequence of foci lies on “wriggling” trajectories for asymmetric distributions. The above oscillating structures are well reproduced by a set of geometrical–optical rays propagating along trajectories of the following parametric form

x = a x sinh (τ cos α + ϕ x ) , y = a y sin (τ sin α + ϕ y ).

(8.17)

These ray trajectories correspond to the geometrical–optical Hamiltonian

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H = N x2 + N y2 − c 2 ( x 2 cos 2 α − y 2 sin 2 α )

(8.18)

which follows from the dispersion relation (2.7) near the transformation region. We see from Eqs. (8.17) that the period of the oscillations over ln | x ′ | is 2π cot | α | , thus the scale length of the first pulsation can be estimated as δx ≈ L∇ exp(2π cot | α |) . It is unlikely that more than one oscillation can be observed in actual systems due to the exponential increase in the x coordinate. The oscillation period increases infinitely with decreasing angle α , i.e., when passing to the one-dimensional case. The field amplitude oscillations also vanish if only one harmonic hn contributes to the field. Therefore, one might expect the amplitude of the oscillations over ln | x ′ | to be small for beams close to the optimal beam.

9. Transformation Coefficients for Gaussian Beams In this Section we follow the logic of [41]. Let us start from the simplest case of the Gaussian beam not bounded in the z direction. Using general equation (6.7), one can determine the O– X transformation coefficient for the Gaussian beam (8.5) with a fixed N z and a z = ∞ as

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Alexander Shalashov

TN z =

a y a y (1 + tanh (π tan α )) a y2 + a 02 tanh (π tan α )

k 02 N y2 a 2y a02

Φ1 =

Φ2 =

exp(− Φ 1 − Φ 2 ) ,

a02 + a 2y coth(π tan α )

(9.1)

,

(9.2)

y 02 . a y2 + a 02 coth (π tan α )

1

(9.3)

Α10

TN z

0.8 0.6 Α50

0.4 0.2

Α200

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0

5

10 a y L

15

Figure 7. Transformation coefficient of the Gaussian beam as a function of

20

a y / L∇

for different

angles α ≥ 0 between the cutoff surfaces: solid lines correspond to the results of the two-dimensional model for the O–X transformation, dashed lines correspond to the same results for the X–O transformation, dot line corresponds to the result of the one-dimensional model ( α = 0 ). Transverse refractive index corresponds to its optimal value, N y

= 0,

and zero shift of the beam is assumed,

y0 = 0 . Pre-exponential term in Eq. (9.1) corresponds to the transformation coefficient of the Gaussian beam launched into the interaction region with zero shift, y0 = 0 , and with the optimal angle, N y = 0 . The dependence of this coefficient (both for O–X and X–O transformation) on the normalized incident beam width a y / L∇ for a number of

α ≥ 0 is

shown in Fig. 7. The O–X and X–O transformation coefficients are related according to Eq. (6.9). The

α = 0 corresponds to the one-dimensional limit, TN = a y / a 2y + πL2∇ , with the z

optimal value of N || . From Fig. 7 one might see that even for the relatively small angles

α

between the cutoff surfaces, the two-dimensional theory results in the transformation Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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coefficients that may be essentially greater or less than those of the one-dimensional theory depending on the “favorable” or “unfavorable” direction of beam propagation. In case shown in the figure, the favorable direction corresponds to the O–X transformation, the unfavorable direction corresponds the to X–O transformation. The maximum value of the transformation coefficient is reached at a y / L∇ = sin

−1 / 2

α (i.e. a y = a0 ). For the favorable direction of

propagation this point corresponds to the perfect conversion, TN z = 1 . For essentially narrow or essentially wide beams, the transformation coefficient degrades, TN z → 0 , when a y → 0 or a y → ∞ . Two terms Φ 1 and Φ 2 in the exponential factor in (9.1) describe the degradation of the transformation efficiency resulted from not optimal aiming, namely, not optimal beam propagation angle ( N y ≠ 0 ) and the shift in the y direction ( y 0 ≠ 0 ), correspondingly. These factors can be neglected if

Φ 1 a* .

(9.13)

For not optimal incident beams, i.e. when inequalities (9.4) are not fulfilled, the dependence of the transformation efficiency on the beam width a z is more complicated. For example, when beams are strongly shifted in the y direction, a narrow beam possessing a wide spectrum over N z′ may be transformed more effectively as compared to a wider beam Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

because of for a certain range of N z′ (being far from the central value N z ) the partial contributions TN ′z

would correspond to the not shifted beams which are effectively

transformed. However, all of such peculiarities occur in the region were the transformation coefficient is rather small, thus they are of minor importance for practical applications.

10. Application to a Toroidal Magnetic Configuration As was noted in the Introduction, the interest to the problem discussed has been stimulated by considering the O–X mode conversion in toroidal magnetic traps. In next two sections we describe some peculiarities of the two-dimensional mode coupling model in applications to actual magnetic installations, such as tokamaks. In a tokamak geometry, intersection of the cutoff surfaces naturally appears due to different spatial distributions of the magnetic field strength and plasma density in a toroidally confined plasma, so the cutoff surfaces are not parallel as show in Fig. 10. Indeed, the ε || = 0 cutoff surfaces, which are determined only by the plasma density profile, coincide with magnetic flux surfaces. The

ε + = N ||2 cutoff surfaces, which also

depend on the external magnetic field, do not coincide with the flux surfaces because of the magnetic field variation on it. The regions close to the equatorial plane, where the two Plasma Physics Research Advances, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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61

cutoff surfaces are almost parallel, constitute an exception corresponded to onedimensional case shown in Fig. 2a of Section 2. Outside the equatorial plane, effective O–X conversion is possible only in the case shown in Fig. 2b, where the cutoff surfaces intersect. This implies that the mode coupling can be essentially two-dimensional in nature even if the effects of flux surface curvature are neglected (as it was done in reference wave equations considered above).

e + = N ||2 e || = 0

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Figure 10. The cut-off surfaces in a poloidal cross-section of a tokamak. Two regions of O–X transformation are marked by arrows, the transformation efficiencies in these regions are related by Eq. (6.9).

As was shown in Section 6, the two-dimensional theory reveals the lack of reversibility in transformation efficiencies from O to X mode and backward resulted from Eq. (6.9). In a tokamak-like geometry, this asymmetry may be reformulated in an alternative form: regions where total transformation is possible for similar beams propagating into and from the dense plasma are separated spatially being above and below the equatorial plane as shown in Fig. 10. In principle, this statement may be checked experimentally by comparing B–X–O emission from these two regions. There are two additional non-one-dimensional effects that are ignored in the qualitative picture presented above. First, we neglect the curvature of the cutoff surfaces in a certain vicinity of the transformation region by assuming that all of the phenomena related to the O–X mode conversion take place in a localized region whose sizes are much smaller than the scale lengths of the plasma density and magnetic field inhomogeneities. Second, we assume above that the tokamak magnetic field is represented only by its toroidal component and, hence, effects related to a poloidal component of the external magnetic field can be disregarded. Both ignored effects do not change topologically the situation shown in Fig. 2b, although the second effect may be significant in an accurate quantitative analysis of the O–X mode conversion in fusion devices, especially in spherical tokamaks characterized by essential poloidal fields. In the following, the poloidal component of the magnetic field will be taken into account while the flux surface curvature will be ignored.

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Alexander Shalashov

x

z ­ Ñne

B q h

Figure 11. Coordinate system used for description of the two-dimensional model of mode conversion region in toroidal configurations.

Let us consider the model of the two-dimensional transformation region in which the plasma density and the magnetic field are homogeneous along the axis z; the plasma density ne (ξ ) varies along the ξ axis perpendicular to the direction of the homogeneity and the magnetic field B(ξ ,η ) varies in the plane also perpendicular to this direction. Here

(ξ ,η , z ) form the right-system of Cartesian coordinates with the ξ axis directed along the ∇ne towards the plasma density increase as shown in Fig.11. Having in mind the specific feature of toroidal magnetic configurations in which the plasma density is constant at magnetic flux surfaces, let us assume that magnetic field has no component along the direction of density gradient, B ⊥ ∇ne or, equivalently, Bξ = 0 . Then Bη and B z

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corresponds accordingly to the poloidal and toroidal components of the magnetic field. This model describes adequately the O–X transformation region in case when the curvature of flux surfaces can be neglected (corresponding to the ne (ξ ) being dependent only on one Cartesian coordinate), i.e. when the wave interaction region is small as compared to inhomogeneity scales the configuration. As before, it is natural to choose the (ξ ,η ) coordinate origin on the line of intersection between the cutoff surfaces, i.e., at the transformation point where the

ε || (ξ ,η ) = 0 and

ε + (ξ ,η ) = N ||2 lines crosses. However, in contrast to the previously studied case of purely toroidal magnetic field (magnetic field directed along the axis of homogeneity), the parallel component of the refractive index N || is not constant when the poloidal component Bη of the magnetic field is taken into account. Nevertheless, due to the tokamak symmetry the toroidal component of the refractive index N z is conserved. The uncertainty in definition of

N || at the transformation point may be resolved by using the geometrical-optics approximation, from which it follows that at this point the wave vector should be parallel to the magnetic field direction [3], i.e.

N ⊥2 = ( N z sin θ − Nη cos θ ) 2 + N ξ2 = 0 ,

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(10.2)

Linear Coupling of Electron Cyclotron Waves in Magnetized Plasmas…

63

where N ⊥ , N ξ , Nη , N z denote the components of refractive index perpendicular the magnetic field and along the coordinate orts correspondingly, θ is the angle between the magnetic field direction and the z axis (the toroidal direction) which may be defined as

tan θ (ξ ,η ) = Bη / B z .

(10.2)

In the following, the angle θ will act as a measure of the poloidal magnetic field. From Eq. (10.1) one may obtain the components of the refractive index at the transformation point related to the constant value N z :

where the angle

Nξ = 0 ,

(10.3)

Nη = N z tan θ 0 ,

(10.4)

N || = N z cos θ 0 + Nη sin θ 0 = N z sec θ 0 ,

(10.5)

θ 0 = θ (0,0) corresponds to a value of θ at the transformation point. The

(ξ ,η ) coordinate origin and the angle θ 0 are defined by the conditions followed from Eq.

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(10.5),

⎧⎪ε || = 0 . ⎨ ⎪⎩ε + = N z2 sec 2 θ

(10.6)

Now we are ready to repeat the same procedure of obtaining the reference wave equations as described in Section 3, but with taking into account the poloidal field. Again, we will use Stix representation for the electric field components, E+ , E− and E|| , related to the

constant magnetic field B 0 = B(0,0 ) at the coordinate origin. Then, transformation from the

Cartesian frame to the Stix frame takes the following form:

E± =

Eξ ± i ( Eη cos θ 0 − E z sin θ 0 )

2

, E|| = Eη sin θ 0 + E z cos θ 0 .

(10.7)

Note, that previously we use a slightly different basis to define wave polarization (2.2) [turned along the magnetic field direction on the constant angle α ]. In the vicinity of the transformation point where variation of the magnetic field direction is not too strong,

θ − θ 0