Laser Systems and Applications [1 ed.] 9781783320134, 9781842657850

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Laser Systems and Applications

Laser Systems and Applications

V.K. Jain

α Alpha Science International Ltd. Oxford, U.K.

Laser Systems and Applications 264 pgs. | 106 figs. | 08 tbls.

V.K. Jain Department of Applied Science IIMT College of Engineering Meerut Copyright © 2013 ALPHA SCIENCE INTERNATIONAL LTD. 7200 The Quorum, Oxford Business Park North Garsington Road, Oxford OX4 2JZ, U.K.

www.alphasci.com All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. Printed from the camera-ready copy provided by the Author. ISBN 978-1-84265-785-0 E-ISBN 978-1-78332-013-4 Printed in India

To

Seeyal, Ira and Piyush

   

Preface 

‘Laser Systems and Applications’ have been prepared to serve as a text for graduate and postgraduate students of physics as well as for B. Tech. students. The topics covered in this book will also be of interest to beginners as all the topics are covered right from the introductory level. This book begins with the review of experimental basis of quantum mechanics that indicated the failure of classical physics at the microscopic scale and establish the new approach. Schrödinger equation is developed and is applied to particle in a box, harmonic oscillator and hydrogen atom. Chapter 2 describes the concept of atomic and molecular spectra. In chapter 3, fundamentals of lasers are discussed. The basic processes such as stimulated emission, the Einstein A and B coefficients and rate equation governing lasing action are described. Apart from this, key laser mechanism such as pumping, requirement of feedback, gain and losses in laser are also given. The lasing transitions are examined in detail and selective pumping mechanism and laser energy level schemes (three levels and four level lasers), classification of lasers, continuous wave and pulsed lasers are described. It provides the reader an introduction to the basics of lasers. Chapter 4 describes the various gas lasers which include neutral atom, ionic and molecular lasers. It provides case studies allowing the reader to see the practical application of laser. The laser chosen represents the vast majority of commercially available lasers (CO2, Ar+, He-Ne, N2, and excimer). Chapter 5 describes the solid state lasers including semiconductor laser. This deals with ruby, Nd:YAG, Nd:glass, Ti: sapphire lasers. The method of Q switching is also included. Next chapter deals with liquid laser (dye laser) and method of mode locking is described to obtain ultra short pulses. The last chapter describes the applications of laser in various fields including material processing, medicine and surgery, holography, meterology, optical communications, military, isotope separation, Lidar. A large number of short questions with answers, solved examples and problems are given at the end of the each chapter to supplement the text. The vectors are denoted with bold faces in the text. In developing the book I have

viii

Preface

  consulted a large number of books by various authors to whom I am grateful. A list of some of these is given at the end of the book. The persons whom I must acknowledge is Manu, Manish and my dear wife Minakshi who patiently supported and encouraged when at times I felt like giving up. Finally I must thank Mr. N.K. Mehra, Managing Director, Narosa Publishing House for readily agreeing to undertake this project. V.K. Jain

Contents 

Preface

vii

1. Foundation of Quantum Mechanics

1.1

1.1 Particle Properties of Electromagnetic Radiation Black Body Radiation The Old Quantum Theory Photoelectric Effect Compton Effect 1.2 Line Spectra and Atomic Structure Inadequacy of Old Quantum theory 1.3 Franck-Hertz Experiment 1.4 Stern Gerlach Experiment 1.5 Wave Nature of Matter de-Broglie Hypothesis Davisson and Germer Experiment Double Slit Experiment 1.6 Uncertainty Relation 1.7 The Time Dependent Schrödinger Equation in One Dimension 1.8 The Time Dependent Schrödinger Equation in Three Dimensions 1.9 The Wave Function Statistical Interpretation Normalization of the Wave Function 1.10 Time Independent Schrödinger Equation 1.11 Stationary States 1.12 Boundary Conditions 1.13 Particle in One Dimension Box 1.14 Harmonic Oscillator 1.15 Hydrogen Atom Examples Short Questions Problems

2. Concepts of Atomic and Molecular Spectra 2.1 The Quantization of Energy 2.2 Regions of the Spectrum

1.1 1.1 1.3 1.4 1.6 1.10 1.13 1.14 1.15 1.17 1.17 1.18 1.20 1.21 1.24 1.27 1.28 1.29 1.29 1.30 1.31 1.32 1.33 1.39 1.41 1.43 1.54 1.58

2.1 2.2 2.3

x

Contents

2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Quantum Numbers Pauli Exclusion Principle and Electron Configuration Coupling Schemes Spectroscopic Notation of Atomic States Spectroscopic Notations of Molecular States Jablonski Diagram Selection Rules Frank Condon Principle Examples Short Questions Problems

3. Introduction to Lasers 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

3.14

3.15 3.16

3.17

Spontaneous Emission Stimulated Emission Absorption Fluorescence Einstein Coefficients Population of States Population Inversion Metastable State Active Material Principle of Laser Gain Threshold Gain Condition Pumping Schemes Two Level System Three Level System Four Level system Rate Equations 3.14.1 A two level system 3.14.2 Three level laser 3.14.3 Four level laser Efficiency Methods of Obtaining Population Inversion Electrical Pumping Optical Pumping Pump Geometry Chemical Thermal Nuclear Pumping Laser Resonators

2.3 2.4 2.6 2.7 2.9 2.11 2.11 2.12 2.12 2.13 2.14

3.1 3.1 3.2 3.3 3.4 3.5 3.7 3.8 3.9 3.10 3.10 3.11 3.14 3.15 3.15 3.16 3.16 3.17 3.17 3.18 3.21 3.24 3.24 3.25 3.25 3.27 3.29 3.30 3.30 3.30

Contents

3.17.1 3.17.2 3.17.3 3.17.4

3.18 3.19 3.20 3.21

Plane - parallel (or Fabry – Perot) resonator Concentric (or spherical) resonator Confocal resonator Resonator using a combination of plane and spherical mirrors 3.17.5 Generalized spherical resonator Elements of Lasers Continuous Wave (CW) and Pulsed Lasers Classification of Lasers Properties of Laser Beams 3.21.1 Monochromaticity 3.21.2 Coherence 3.21.3 Directionality 3.21.4 Brightness Examples Short Questions Problems

4. Gas Lasers 4.1 4.2 4.3 4.4 4.5 4.6

He-Ne laser CO2 Laser Nitrogen laser Excimer Laser Ar+ Ion Laser Krypton Laser Examples Short Questions Problems

5. Solid State Lasers 5.1 5.2 5.3 5.4 5.5 5.6

Ruby Laser Nd: YAG Lasers Nd:Glass Lasers Ti:Sapphire Lasers Semiconductor Lasers Q-Switching Methods Examples Short Questions Problems

6. Liquid Lasers 6.1 Dye Laser 6.2 Mode Locking

xi

3.31 3.32 3.32 3.32 3.33 3.35 3.36 3.38 3.41 3.41 3.41 3.43 3.44 3.44 3.54 3.65

4.1 4.1 4.4 4.9 4.10 4.12 4.17 4.17 4.20 4.26

5.1 5.2 5.4 5.6 5.9 5.10 5.15 5.19 5.22 5.25

6.1 6.1 6.5

xii

Contents

Examples Short Questions Problems

6.9 6.10 6.14

7. Laser Applications

7.1

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12

Material Processing Lasers in Medicine and Surgery Meterology LIDAR Environmental Studies Optical Communication Holography Laser Isotope Separation Laser Induced Fusion Military Applications Photolithography Stereolithography Examples Short Questions Problems

Selected Bibliography Subject Index

7.1 7.5 7.10 7.13 7.13 7.15 7.21 7.22 7.25 7.27 7.28 7.29 7.29 7.31 7.35 B.1 SI.1

 

CHAPTER 1 

Foundation of Quantum Mechanics 

By the 19th century the laws of physics were based on mechanics and law of gravitation from Newton, Maxwell equation describing electricity and magnetism and statistical mechanics described the state of large collection of matter. These laws of physics described nature very well under most conditions. However, during the later part of nineteenth century and in the early years of twentieth century, experimental evidence accumulated which required new concepts radically different from those of classical physics represented by Newton’s law of motion and Maxwell’s electromagnetic equation. There were difficulty associated with the understanding spectral distribution of thermal radiation from a black body, the low temperature specific heat of solids, appearance of only five degrees of freedom in the motion of free diatomic molecules at ordinary temperature, the photoelectric effect, Compton scattering, atomic stability and atomic spectroscopy etc.

1.1

PARTICLE PROPERTIES OF ELECTROMAGNETIC RADIATION

Black Body Radiation

An ideal black body is an object that absorbs all wavelength of electromagnetic radiation incident upon it. Reflecting no radiation, the object appears to be black. The black body can be approximated by a hollow cavity whose internal walls perfectly reflect electromagnetic radiation and which has a very small hole on its surface. The area of the hole is very small in comparison with the area of the inner surface of the cavity. Radiation incident upon the hole from the outside enters the cavity and is reflected back and forth by the inner walls of the cavity and gets absorbed on these walls. If the walls of the cavity are uniformly heated to a temperature T, walls will emit radiation (due to thermal agitation or vibration of the electrons in the metallic walls) which will fill the cavity. The small fraction of this radiation which is incident on the hole will pass through the hole. The hole will act as an emitter of thermal radiation. The radiation leaving the hole of a

1.2

Laser Systems and Applications

heated hollow cavity is termed as black body distribution. The spectral energy density , in the frequency range ν to ν + dν at different temperatures as a function of frequency is shown in Fig.1.1

ρ (ν, t)

5000 K

4000 K 3000 K 1500 K ν Fig. 1.1 Spectral distribution density of black body radiation at different temperature as a function of frequency

The spectral distribution shows a pronounced maximum at a given frequency with temperature, that is, the peak of the radiation spectrum occurs at a frequency that is proportional to the temperature. The peak moves towards higher frequency with increasing temperature. A Number of attempts aimed at explaining the origin of continuous radiation spectrum were carried out. In 1900, Rayleigh and Jeans by applying classical thermodynamics and electromagnetic give the formula for electromagnetic field energy density , emitted from black body at absolute temperature T as ,

(1.1)

where kB = 1.3807 × 10-23 JK-1 is Boltzmann constant. The Eq. (1.1) predicts a physically impossible infinite radiative energy density as ν → ∞ whereas experimentally it must be finite. Moreover, if we integrate the Eq. (1.1) over all frequencies, the integral diverges. This implies that cavity contains an infinite amount of energy. This divergence in radiative energy density with increasing frequency is called the classical ultraviolet catastrophe. The experimental behaviour of the black body radiation is explained by Planck in 1900. He assumed that exchange of energy between radiation and matter must be

Foundation of Quantum Mechanics

1.3

discrete. He postulated that the energy of radiation of frequency ν emitted by the oscillating charge (from the wall of the cavity) in integral multiple of hν, that is 0,1,2, …

(1.2)

where h is a Planck constant having the value 6.6262 ×10-34 Js. hν is energy of quantum of radiation called photon. This gives a radiative energy density ,

(1.3)

This is known as Planck’s distribution. It gives an exact fit to the various experimental distribution as shown in Fig.1.1. The Eq. (1.3) solves the problem posed by the ultraviolet catastrophe and agrees with Rayleigh Jean result in the limit of low electromagnetic radiation frequencies. For low ν, we have u (ν ,t ) =

8π ν 2 k BT c3

(1.4)

which is Rayleigh-Jeans formula (1.1). In the limit of high frequency hν >> kBT and exp (hν k BT ) → ∞ which means u(ν) → 0 is observed. Thus the spectrum of the black body radiation revels the quantization of radiation notably the particle behaviour of the electromagnetic waves.

The old quantum theory The old quantum theory was initiated when Max Planck gives his theoretical derivation of distribution law for black body radiation. He explained that the experimental results on the distribution of energy with frequency of radiation in equilibrium with matter at a given temperature can be explained by postulating that the vibrating particles of matter do not emit or absorb radiation continuously but instead only in discrete quantities of magnitude hν proportional to frequency of radiation emitted, h is Planck constant having dimension of energy × time. In 1905 Einstein suggested that the quantity of radiation energy hν emitted is not in all direction but is unidirectional like a particle. With the concept of energy and statistical mechanics, Einstein and Debye explained the specific heat of solids. Bohr and Sommerfeld were able to explain the atomic spectra of hydrogen atom. The basic idea of old quantum theory is that the motion in an atomic system is quantized or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motion which obey the old quantum condition pi = momenta of the system qi = corresponding coordinate

1.4

Laser Systems and Applications

The quantum number ni are integers and the integral is taken over one period of the motion at constant energy, the integral is an area in phase space which is a quantity called action and quantized in the units of Planck’s constant.

Photoelectric Effect Hertz in 1887 observed that electrons are emitted from the metal when light of appropriate frequency fall on it. This effect is known as photoelectric effect. Figure 1.2 shows a diagram of an apparatus for studying the photoelectric effect. It consists of an evacuated quartz tube containing two electrodes connected to a source of variable voltage with metal plate whose surface is irradiated as the anode. Monochromatic light incident on metal plate A and liberate electrons. The electrons can be detected as a current if they are attracted to the metal plate B by means of a potential difference V applied between A and B. The sensitive ammeter serves to measure the photoelectric current. It is observed that 1. There exists a critical cutoff frequency νc for the incident light (and corresponding cutoff wavelength λc = c/νc). If the frequency of the incident light is less than νc, no electrons are liberated from the metal irrespective of the intensity of light. The critical frequency depends on the properties of the metal. Light A

B 

A  Fig. 1.2 Experimental observation of the photoelectric effect

2. At any ν >> νc, the number of electrons ejected per second is directly proportional to the light intensity, that is, current is directly proportional to intensity of light 3. For a given frequency of light illuminating the metal, if we increase the source potential difference V, no increase in the current occurs. The electric field established in the tube by the battery effectively pulls all the ejected electrons to the positive terminal to be collected measured and by the ammeter. If the polarity of the potential difference is reversed, the direction of electric field is reversed. As the value of the reversed potential difference increases, the current in the circuit decreases to zero. The electric field now prevents some of the electrons from reaching the new negative terminal. When reversed electric field is large enough, no electrons are collected and the current is zero. This voltage denoted by VS at which this happens is known as the stopping potential, since the electric field it creates in the tube stops all the ejected electrons from reaching the

Foundation of Quantum Mechanics

1.5

opposite terminal, even the most energetic one. The value of stopping potential is directly proportional to the frequency of the incident light for ν > νc (Fig. 1.3). When ν > νc, the ejected electrons appear instantly, that is, the electrons ejected from the metal appear instantaneously even when the incident light is of very low intensity. According to classical physics any (continuous) amount of energy can be exchanged with matter. That is, since the intensity of an electromagnetic wave is proportional to the square of the amplitude, any frequency with sufficient intensity can supply the energy to eject the electron from the metal. There are three major features of photoelectric effect that cannot be explained in terms of classical wave theory of light; notably, the dependence of the effect on the threshold frequency. VS 

νc 

ν 

Fig. 1.3 Stopping potential as a function of frequency

1. Wave theory requires that oscillating electric vector E of light wave increases in amplitude as the intensity of light beam increases. Since the force applied to the electron is eE, this suggests that kinetic energy of ejected electron should increase as the light beam is made more intense. However, it is observed that maximum kinetic energy of the ejected electron is independent of light intensity. 2. The energy of classical wave is proportional to the square of the amplitude and frequency. Thus, electrons should be able to absorb energy from incident electromagnetic radiation of any frequency. Therefore, the photoelectric effect should be independent of frequency. However, the effect is strongly frequency dependent. 3. For weak light source (that is, light waves of very small amplitude or energy), an electron would keep on absorbing energy at continuous rate until it accumulates enough to become liberated from the metallic surface. Thus for very low level of light wave illumination, the photoelectric effect would not take place for a long time possibly hours until an electron gradually accumulated the necessary amount of energy. However, no such time lag is observed with low illumination. The electrons always appear promptly. Einstein in 1905 gave an explanation of photoelectric effect. He considered that light of frequency ν can be considered a stream of photon. Each photon has an energy E given by (1.5)

1.6

Laser Systems and Applications

According to this model, a photon could be absorbed as a unit by an electron. When the photon energy is transferred to an electron in a metal, the energy acquired by the electron is hν. If hν is larger than the metal’s work function W (the energy required to liberate the electron from the metal), the electrons are ejected. No electron can be ejected from the metal surface unless hν > W. Thus kinetic energy of the ejected electrons

(1.6)

For ν = νc, kinetic energy = 0 and (1.7) Depending on the particular circumstances of an individual electron, it may have to give up part or all of its newly acquired additional energy to get out of metal surface. The emerging electrons thus appear with a variety of kinetic energies. From conservation of energy, the electrons that escape from metal with minimum amount of energy loss must be the ones with maximum amount of kinetic energy after escape. Thus for these electrons maximum kinetic energy

(1.8)

On the basis of Einstein theory 1. The photoelectric effect is not observed below νc follows from the fact that the energy of the photon must be greater than or equal to work function. If the energy of the incident photon is not equal to or greater than the work function, the electrons will never be ejected from the surface irrespective of the intensity of the incident light. 2. The independence of kinetic energy with light intensity is explained as follows. If the light intensity is doubled, the number of photons doubled, which doubles the number of photoelectrons emitted. However, their kinetic energy which is equal to depends only on the light frequency and work function, not on the light intensity. 3. The observation of increase in maximum kinetic energy with increasing frequency can be understood from Eq. (1.8) 4. The observation that electrons are emitted almost instantaneously is consistent with the particle theory of light in which the incident energy appears as photons and there is one to one interaction between photons and electrons.

Compton Effect When a beam of X-rays strike matter some of the radiation is scattered through an angle θ. The scattered radiation contains the same wavelength as well as wavelength slightly longer than the wavelength of the incident X-rays. The change in wavelength depends on the angle through which the radiation is scattered. This effect is known as Compton effect. The change in wavelength is called the Compton shift. It varies with the scattering angle. The frequency of the scattered radiation is independent of the material. The experimental setup used for Compton scattering is shown in Fig. 1.3. Compton in his experiment irradiated a

Foundation of Quantum Mechanics

1.7

graphite target with a nearly monochromatic beam of X-rays of wavelength λ. The intensity of scattered radiation was measured as a function of wavelength. It is observed that part of the scattered radiation has the same wavelength λ. Beside this there was also a second component of wavelength λ′ such that λ′ > λ. The shift in the wavelength ∆λ = λ′ – λ was found to vary with the angle of scattering θ. From further investigations it was found that ∆λ was found to be independent of both λ and of the material used. The expression for Compton shift is obtained by assuming the elastic scattering is between two particles. It is assumed that 1. electron participating in the scattering are free 2. electrons are stationary Consider X-rays of wavelength λ incident along x (Fig. 1.5). The momentum of the X-ray photon is (h/λ)i. Since the electron is assumed to be initially at rest it has zero momentum. Total momentum of the system of two particles before collision is Lead  Collimating  slits

Crystal  Detector 

X‐ray  source  θ Incident  beam 

Scatterer 

Fig. 1.4 Scattered photon of  wavelength λ’ 

θ Incident photon 

φ v

Recoiling electron 

Fig. 1.5 Compton scattering of a photon off a free stationary electron

(1.9) The photon of wavelength λ′ emerging at an angle θ to the x-axis has momentum

1.8

Laser Systems and Applications

(1.10) The electron, recoiling from the collision at speed v and making an angle φ with x axis has relativistic momentum 1

1

(1.11) (1.12) Total momentum after the collision by using Eqs. (1.10) and (1.11) is (1.13) Momentum conservation from Eqs. (1.9) and (1.13)

(1.14) Since two vectors are equal if and only if their respective components are equal, that is, (1.15) 0

(1.16)

Eq. (1.15) is written as (1.17) From Eq. (1.16) (1.18) The energy of the incident photon is (1.19) The electron assumed to be initially at rest, has a total (relativistic) energy equal to mc2. Thus the total initial energy is (1.20) After the collision, emerging photon has the energy

Foundation of Quantum Mechanics

1.9

(1.21) emerging electron has relativistic energy (1.22) Final total energy from Eqs. (1.21) and (1.22) (1.23) From energy conservation using Eqs. (1.20) and (1.23)

1

(1.24)

Squaring Eqs. (1.17) and (1.18) and adding (1.25) On squaring Eq. (1.24) 1

2

(1.26)

Subtracting Eqs. (1.25) and (1.26) 2

2

1

2 1

1

1

2

1

1

2

1

1 Using Eq. (1.24) 2

1

2

2 1

(1.27)

1

2

(1.28)

The change in wavelength depends only on the scattering angle θ. 2.426

10

(1.29)

h/mc has dimension of length and is known as Compton wavelength of the electron. It is not a wavelength of light. It has dimension of wavelength. In summary, the Compton effect confirms that photons behave like particles, they

1.10

Laser Systems and Applications

collide with electrons like material particles. The existence of unmodified component of scattering radiation which has the same wavelength as the incident radiation can be explained by assuming that it results from the scattering by electrons so tightly bound that the entire atom recoils. In this case the mass to be used is M, the mass of the entire atom, and since M >> m, the Compton shift ∆ λ is negligible. There is no Compton effect for light in visible region because the photon energy in this case is not large enough compared with the binding energy of even the loosely bound electrons. The mass of the electron m appears in the denominator of h/mc. This is the reason that change in wavelength is more apparent for scattering from electron rather than proton or nucleus. The proton and nuclear masses are much larger than the electron mass, so shift in the frequency of photon in the scattering from proton or nuclei is much less than from electrons. From Eq. (1.28) it is clear that 1. Change (increase) in wavelength depends only on the scattering angle θ and universal constant h/mc and is independent of wavelength of incident radiation and nature of scattering substance. 2. When θ = 0°, cos θ = 1 and λ′ – λ = 0. 3. When θ = 90°, cos θ = 0 and λ′ – λ = h/mc. 4. When θ = 180°, cos θ = –1 and λ′ – λ = 2h/mc.

1.2 LINE SPECTRA AND ATOMIC STRUCTURE Besides failing to explain blackbody radiation, the Compton, photoelectric and pair production effects, classical physics also fails to account for the discrete number of energy levels in atoms. Rutherford proposed a model for the atom. Rutherford considered the atom to consist of electrons orbiting around a positively charged nucleus. According to the classical laws of electromagnetism, the accelerated electrons in their motion around the nucleus would continue to radiate until they had exhausted all possible energy and would fall into the nucleus. Further, electrons would radiate spectral lines whose frequencies would continuously change as they get closer and closer to the nucleus in contrast to the observed sharp spectral lines of definite frequencies. The model therefore suffer from two deficiencies: atoms are unstable and they radiate energy over a continuous range of frequencies. Neils Bohr developed a quantitative atomic model for hydrogen atom which satisfactorily explained the observed spectrum. The model incorporated the nuclear model of the atom proposed by Rutherford, as well as the concept of light photon developed by Einstein to explain the photoelectric effect. The Bohr theory is based on the following postulates: 1. An electron in an atom moves in circular orbits about the nucleus under the action of a coulomb field of force.

Foundation of Quantum Mechanics

1.11

2. Of the infinite number of orbits of an electron about an atomic nucleus which would be possible in classical mechanics, only those for which the orbital angular momentum L is equal to an integer times ħ (Planck’s constant h divided by 2π) are allowed that is 1,2,3, . .

(1.30)

3. Electron in such an allowed circular orbits is in a stable state and would not radiate despite constantly accelerating. 4. Electromagnetic radiations is emitted or absorbed by a transition of the electron from one orbit of total energy Ei to another object of total energy Ef by a quantum jump. The frequency ν of radiation is equal to Ei – Ef divided by Planck’s constant i.e. (1.31) Consider a hydrogen atom consisting of a nucleus of charge +e and mass M and a single electron of charge –e and mass m. It is assumed that electron revolves around the nucleus in circular orbit of radius r. Since mass of the electron is negligible in comparison with the mass of the nucleus; therefore, in the first approximation it is assumed that nucleus is at rest. The acceleration of the electron in circular orbit is centripetal acceleration ac given by (1.32) where v is the speed of the electron and r is the radius of its circular orbit. The force F producing the acceleration of the electron is the Coulomb force exerted by the nuclear charge e on the charge –e on the electron. The magnitude of the force is (1.33) According to Newton’s second law: F = ma

(1.34)

Since force F and acceleration a are in the same direction, we take their magnitudes to obtain (1.35) Solving Eqs. (1.30) and (1.35) for v and r, it is found that electron orbiting the nucleus of hydrogen atom can have only certain discrete values for the orbital radius and velocity and are given by 1,2, …

(1.36)

1.12

Laser Systems and Applications

5.29

10

(1.37)

which is radius of orbit of hydrogen in the ground state and is known as first Bohr radius of hydrogen. The velocity is 1,2,3, …

(1.38)

The total energy E of the electron with orbital radius r and velocity v is just the sum of kinetic energy and potential energy (1.39) From Eqs. (1.36), (1.38) and (1.39) .

(1.40)

The energies are negative because the electron is in a bound state. E goes to zero, as r approaches infinity i.e. total energy is zero when atom is ionized. Since n is an integer, therefore only certain values of energy are permitted. According to Eqs. (1.31) and (1.40) 13.6

(1.41)

and .

(1.42)

where Rydberg constant R is .

109737

(1.43)

Following Wolfgang Grotrian, Bohr’s permitted values of energy (Eq. 1.40) for hydrogen atom are represented graphically on an energy level diagram (Fig. 1.6). The vertical axis of the graph is an energy (eV or wave number) scale, there is no horizontal scale, but horizontal lines are drawn to show the position of energy levels. The ground level (n = 1 for hydrogen) lies near the bottom of the diagram, and the remaining (excited) levels are ranked by energy above this level. The distance between energy level rapidly decreases. Each spectral line is represented by a vertical line joining two energy levels. The length of any arrow is proportional to the frequency (or the reciprocal wavelength) for the corresponding spectral lines. The ionization limit is defined as the zero point of the energy scale. With this energies of the discrete excited states are negative numbers, whereas the positive energies occur only when the atom is ionized. An electron in an orbit with ni > 1 and energy Ei , in making a transition to an orbit of lower energy Ef produces a photon with specific energy, equivalently with a definite frequency or wavelength. The discrete emission spectrum of hydrogen therefore corresponds to electrons cascading down to lower energy

Foundation of Quantum Mechanics

1.13

levels in hydrogen atoms. All their spectral lines can be grouped in to five series depending on the values of nf and ni. These series are named after their discoverer. In a series spacing and intensity of lines decrease regularly. The series limit is derived from Eq. (1.42) for every series by putting ni = ∞. The spectral lines lie in various spectral regions.

 E(ev)   

 0 

 n = ∞  Pfund series 

 ‐ 0.54 

Brackett  i Paschen series

 ‐ 0.85  

 n = 5 

 n = 4 

n = 3 

 ‐ 1.51   Balmer series  ‐ 3.40  

≈ 

n = 2 

Lyman series

 ‐ 13.6  

n = 1  

Fig. 1.6 Energy levels of the hydrogen atom according to the Bohr theory

Normally, only lines corresponding to Lyman Series appear in the absorption spectrum as atom is always initially in the ground state n = 1, so that only absorption process from n = 1 to n > 1 can occur. However at high temperature (T ~ 1015 K) owing to collisions some of the atoms will initially be in the first excited state (n = 2) and absorption lines corresponding to the Balmer Series will be observed. It is observed that for every line in the absorption spectrum there is a corresponding (same wavelength) line in its emission spectrum, however the reverse is not true. If hydrogen atom is initially in excited state n > 1 then in going to its ground state it can follow different paths and as a result of this it will emit [n ( n – 1 )]/2 number of different wavelengths.

Inadequacy of Old Quantum Theory The old quantum theory encountered practical difficulties in several different aspects, some of these are: 1. It fails to explain normal and excited states of helium atom, normal state of hydrogen molecule ion.

1.14

Laser Systems and Applications

2. It fails to explain dispersion of light. 3. It fails to provide a method for calculating transition probabilities and intensity for spectral lines. 4. It cannot explain the influence of magnetic field on the dielectric constant of a gas. 5. It fails to explain half integral values for quantum numbers in place of integral values for certain system, in order to observe agreement with experiment. For example pure rotation spectra of hydrogen halide molecules require half integral values for quantum numbers in place of integer values of quantum numbers.

1.3 FRANCK-HERTZ EXPERIMENT The quantization of internal energy state of atom is confirmed by the Frank and Hertz experiment performed in 1914. The apparatus used by Franck and Hertz is shown in Fig. 1.7, a glass envelope is filled with mercury vapours. Electrons produced by the filament were accelerated by a voltage towards grid which is kept at a positive potential V. Relative to grid the collecting plate is at a slightly negative voltage. This prevents electrons having energy less than a certain minimum amount from reaching the collecting plate and contributing to the current I. The electrons are moving under the influence of positive potential. During its motion they collide with the heavy mercury atoms in the vapour. Since the mass of the electron is much less than the atoms of mercury, the electrons merely bounces off during elastic collision with almost no loss of kinetic energy. Heated cathode  produce electrons 

Grid

Plate

Mercury vapours

V0 

A 

I 

V Fig. 1.7 Schematic diagram of the Franck-Hertz experiment

The number of electrons reaching the collecting plate increases with increase in accelerating voltage and thus current I in the ammeter also increases. In the experiment it is observed that when accelerating voltage reaches 4.9 V, the current sharply drops. This suggests that an electron colliding with one of the atom near the grid gives up some or all of its kinetic energy to excite the atom to an energy level above its ground state. Such a collision is called inelastic and is between accelerating electron and atomic electron in the mercury atom. The critical electron energy equals the energy needed to raise the atom to its lower

Foundation of Quantum Mechanics

1.15

excited state. Then, as the accelerating potential V is further raised, the current I again increases. This is because the electrons now have enough energy left to overcome the negative potential of the collecting plate after undergoing an inelastic collision on the way. With increase in accelerating voltage another sharp drops in the current I occurs which arise from the excitation of the same energy level in another atom or due to excitation of higher discrete levels by the electrons. Figure 1.8 shows a series of critical potential of a given atomic vapour. Thus the higher potentials results from two or more inelastic collisions and are multiple of the lowest one. It is found that for mercury 4.9 eV are required to excite the atom from ground state to first excited state.

Plate Current 

Acceleration Voltage V  Fig. 1.8 Variation of the plate current as a function of accelerating voltage V in the Franck-Hertz experiment showing regular maxima and minima

1.4 STERN GERLACH EXPERIMENT Stern-Gerlach experiment measures (the z component) of magnetic moment and the experiment gives the evidence that the angular momentum is quantized. The experimental set up is shown in Fig. 1.9. A beam of neutral silver atoms is produced by evaporating silver from an oven. The beam is collimated with a system of slits and passed between the poles of magnet. One of the pole pieces is flat; the other has a sharp tip. Such a magnet produces an inhomogeneous magnetic field. Screen

Spin up

S

Beam of silver atoms Spin down Magnet

Fig. 1.9 The Stern-Gerlach experiment

N

1.16

Laser Systems and Applications

When a neutral silver atom with a magnetic moment µ enters the magnetic field B, the energy of interaction (potential energy) is W = −µ ⋅ B

(1.44)

The system experiences a torque N where N = µ× B

(1.45)

F = − ∇W

(1.46)

F = ∇(µ ⋅ B)

(1.47)

and a net force F where

Substituting Eq. (1.44) in (1.45)

The component of F are Fx = µ ⋅

∂B ∂B ∂B , Fy = µ ⋅ , Fz = µ ⋅ ∂x ∂y ∂z

(1.48)

Here Bz >> Bx, By. Hence x and y component of magnetic field can be neglected. For an inhomogeneous magnetic field in the z direction Fz = µ z

∂ Bz ∂z

(1.49)

If ∂ B z ∂ z is negative and also µz is negative, then Fz will be positive and atoms are deflected in the positive z direction. Classically

µ z = µ cos θ

(1.50)

where θ is the angle that all magnetic moment µ makes with the z axis. Due to high temperature, the silver atoms have their magnetic moments distributed uniformly in all directions which is to say that distribution of µz would take a continuum of values ranging from +µ to –µ. The beam of silver atoms is directed right along the magnet, therefore atoms feel a vertical force in the inhomogeneous magnetic field. A silver atom with its magnetic moment directed horizontally have no force on it and would go straight pass the magnet without deflection. For a silver atom, whose magnetic moment is exactly vertical would have a force pulling it up toward the sharp edge of the magnet. On the other hand silver atom whose magnetic moment is pointed downward would feel a downward push. After coming out of magnet, the atom would be spread out according to their vertical component of the magnetic moment. Since all angles are possible in classical physics, therefore when the silver atoms are collected by deposition on the plate, one would expect a continuum of deflections. However, it is observed that the beam divided into two discrete components. One component being bent in the positive z direction and the other bent in the negative z direction. Silver atom has 47 electrons; 46 of them form a spherically symmetric charge distribution and 47th electron occupies a 5s orbital. The nucleus make a

Foundation of Quantum Mechanics

1.17

very small contribution to the magnetic moment of the atom because the mass of the nucleus is so much large than the mass of the electron. Therefore, the magnetic moment of the silver atom is effectively due to the magnetic moment of the single electron. The magnetic moment of the electron due to orbital motion is

− β e g l ( l / h ) where β e = ( eh 2me ) is Bohr Magneton and orbital g factor g l is 1 the z component of magnetic moment is accordingly proportional to ml and can take (2l + 1) values. Since l is an integer, (2l + 1) is odd. If the magnetic moment

of an atom is associated only with the orbital motion, there would be an odd number of discrete traces including the undeflected one of ml = 0. For silver atom in the ground state l = 0, one expects an undeflected beam. The observation of splitting of beam into two discrete components without undeflected beam is explained by Goudsmit and Uhlenbeck in 1925. They postulated that in addition to its orbital angular momentum, the electron possesses an intrinsic angular momentum. By analogs with the orbital angular momentum of a particle, spin angular momentum is characterized by quantum number s and its projection ms on the z axis where ms = − s,….,+ s. Since only two components were observed in the Stern Gerlach experiment, we must have 2s + 1 = 2 or s = 1/2 and ms = 1/2 and −1/2. Stern Gerlach experiment confirms 1. discrete character of microphysical world 2. determination of total angular momentum 3. confirms the existence of spin Stern Gerlach experiment can be used to prepare a quantum state.

1.5 WAVE NATURE OF MATTER de-Broglie Hypothesis In 1923-24 L. de-Broglie proposed that just as radiation has particle like properties, electrons and other material particle posses wave like properties. He assumed that wavelength associated with any particle is related to the magnitude of its momentum by the relation

λ=

h p particle

(1.51)

For a particle of mass m moving with nonrelativistic speed p particle = m v

(1.52)

The wavelength associated with a particle is called de Broglie wavelength (it is not a wavelength of light nor it is associated with Compton wavelength of a particle). The matter waves are not associated with electric and magnetic fields and they require medium for propagation and hence cannot move in vacuum. The

1.18

Laser Systems and Applications

speed of the matter waves is not the same as that of light. Moreover it is not constant for all matter waves. They do not leave the source. He also assumed that the energy E of the particle is proportional to the frequency ν of the associated wave given by E = hν

(1.53)

just as for photon. For a photon p photon =

h

λ

or λ =

h p photon

(1.54)

The particle speed v (which determines its momentum) is not the same as the speed v′ of the de-Broglie wave associated with the particle. The de Broglie hypothesis implies that wave particle duality has a universal and symmetrical character. Waves have particle property, particles have wave property. The very small value of h explains why the wave like nature of the matter is very difficult to demonstrate on a macroscopic scale. Thus the particle mass must be sufficiently small to obtain a measurable wavelength.

Davisson and Germer Experiment Wave like behaviour in the motion of particles will only be apparent when de Broglie wavelength λ = h / p is of the order or larger than the characteristic dimensions of the system used to investigate the motion. Since h is very small, thus unless p is also very small, λ will be so small as to preclude any hope of observing the postulated effect. The experimental verification of the wavelike aspects of electrons was made by Davisson and Germer in studies of reflection of electrons from a nickel target. In Davisson and Germer experiment a parallel beam of monoenergetic low energy electron was produced by accelerating electrons thermally emitted from a heated filament were accelerated through a voltage drop V and emerged from the electron gun. The electrons were incident normal to the surface of nickle crystal. Some of the electrons were scattered back from the surface of the crystal. The number N(θ) of electrons scattered at an angle θ to the incident direction was measured with a detector. A retarding potential Vr slightly less than V, allowed only electrons scattered from the crystal with little energy loss to reach detector and produce a current. Low energy electrons lose energy rapidly in traversing a solid, so the detected electrons must have been scattered essentially from the surface of the crystal. The periodic distribution of the atoms of that surface is shown in the Fig.1.10. For a Ni crystal, the repetition length d of periodic structure is 0.215 nm which is comparable to de-Broglie wavelength of the incident electrons.

Foundation of Quantum Mechanics

1.19

Electron Gun P D

M

θ

O 

O

O

O

O

O

Ni

Fig. 1.10 Schematic diagram of the Davisson-Germer experiment

Since the essential feature of a diffraction grating is its periodicity, the pattern of electrons scattered from the crystal surface should show diffraction effect if the de Broglie postulate is correct. Figure 1.11 shows the number of scattered electrons N(θ) as a function of θ, measured for an accelerated voltage of 54.0 Volt. The angular distribution becomes very large for small θ, that is, most of the scattered electrons are specularly reflected. However, this information is of no use because such a behaviour is characteristic of both particle and wave motion. However, at an angle of 50° a peak is observed in angular distribution. The existence of this peak proves the qualitative validity of the de Broglie postulate because such a peak can only be explained as a constructive interference of waves scattered by the periodically placed atoms of the crystal surface. Energy of the electron = 54 eV

N(θ)

O 

50o

90o

θ Fig. 1.11 Number of scattered electrons N(θ) as a function of θ for accelerating voltage of 54 V

This is not an interference between waves associated with one electron and wave associated with another. It is an interference between waves, associated with a single electron that have been scattered from various parts of the crystal. According to data of Davisson and Germer

1.20

Laser Systems and Applications

λ=

h h = = 0.167 nm p 2mE

(1.55)

The wavelength of the wave which is interfering constructively can also be evaluated from the equation d sin θ = n λ (n = 1,2,...)

(1.56)

Assuming that the peak corresponding to n = 1 (that is first order diffraction) we have

λ = 0.215nm × sin 50° = 0.165 nm

(1.57)

The two values of λ obtained agree to within accuracy of the experiment. de Broglie idea was also confirmed by the Thomson experiment. In this experiment, the transmission of electrons through a polycrystalline thin film was studied. A beam of monoenergetic lectrons passing through the thin film is scattered and struck a photographic plate. The thin film consists of many small randomly oriented microcrystal. Classically, electron behaves only as particles yield a blurred image. However, a series of concentric rings were observed indicating a wave like behaviour of the electrons.

Double Slit experiment The evidence of the wave nature of light was deduced from Young’s double slit experiment. Experiments with electrons using double slit arrangement (Fig. 1.12) indicate that they possess wave like characteristic. Consider electrons directed towards double slits A and B in an impenetrable screen. After passing through the screen the electrons fall on to a photographic plate. Let the slit B is closed and slit A is open. The electrons after passing through A fall on the photographic plate and blacken it at the point of contact. However, it is observed that there are regions on the photographic plate in which the electrons never arrive. These regions have the characteristic of concentric rings of finite width. For the regions in which electrons do arrive form a system of concentric rings with forbidden rings. By carrying out the experiment for a sufficient long time, one obtained a pattern identical with the case when the light is diffracted from a slit. Similar behaviour is observed when A is closed and B is open. A Electron B Fluorescent  Fig. 1.12 Double slit experiment

Foundation of Quantum Mechanics

1.21

Passing through A or B, the electrons cause blackening at a definite point on the photographic plate. When A and B are both open the final diffraction pattern should be a simple superposition of the intensities of the blackening arising when electrons are let through A or B. However, the distribution of intensities of blackening is of complete different character and show interference pattern. Thus electron like a wave possesses interference properties. If we try to know through which slit the particle is passed, we will destroy the interference pattern as we are investigating the particle aspect of its behaviour and wave nature of the particle cannot be observed. Conversely, while studying the wave nature, the particle nature cannot be observed simultaneously. The electron will behave as a particle or a wave but both aspects of its behaviour can not be observed simultaneously. This is the basis for the principle of complementarity which asserts that the complete description of a particle entity such as electron, neutron etc. cannot be made in terms of only particle properties and wave properties but that both aspects of its behaviour must be considered.

1.6 UNCERTAINTY RELATION In classical mechanics the equation of motion of a system with a given force can be solved to give the position and momentum of a particle at all values of time t. The precise position and momentum of the particle at any time t, for example at t = 0, is known then future motion is determined exactly. During the process of making observations, the observer interact with the system and thus the motion of the particle is disturbed, however, for macroscopic objects the disturbance can be ignored and one can take the measurement very precisely. However, same is not true for microscopic system. The wave particle duality limits the simultaneous measurement of position and momentum of particle exactly. The Heisenberg uncertainty principle states that experiment cannot simultaneous determine the exact value of a component of momentum say px say, of a particle and also the exact value of the corresponding coordinate x. Instead, our precision of measurement is inherently limited by the measurement process itself such that ħ

(1.58)

where the momentum px is known with an uncertainty ∆ p x and position x at the same time with an uncertainty ∆ x . The principle indicates that, although it is possible to measure the momentum or position of a particle accurately, it is not possible to measure these two observable simultaneously to an arbitrary accuracy. That is, we cannot localize a microscopic particle without giving to it a rather large momentum. We cannot measure the position without disturbing it; there is no way to carry out such measurement passively as it is bound to change the momentum. Similarly, ħ

(1.59)

ħ

(1.60)

1.22

Laser Systems and Applications

there is likewise a minimum for the product of uncertainties of the energy and time and between angular momentum and angle ħ ħ

However, if the coordinates are not canonical conjugate, then uncertainty relation does not exist between them or in other words, those coordinates can be measured very precisely and simultaneously, that is 0

The energy E is related with the momentum p of the particle of mass m

2

On differentiation 2 2

Using Eq. (1.58) we have

∆ E ∆t ≥ h

(1.61)

This relation states that if we make two measurements of energy of a system and if these measurements are separated by a time interval ∆t, the measured energies will differ by an amount ∆E which can in no way smaller than ħ/∆t. If the time interval between the two measurements is large, the energy difference will be small. This can be attributed to the fact that when the first measurement is made, the system becomes perturbed and it takes it a long time to return to its initial, unperturbed state. The Heisenberg uncertainty principle is not a statement about the inaccuracy of measurement instruments, nor a reflection on the quality of experimental methods, it arises from the wave properties inherent in quantum mechanical description of nature. Even with perfect instruments and techniques, the uncertainty is inherent in the nature of things. Suppose we try to measure the position of the particle by illuminating it with a radiation of wavelength λ and using a microscope of aperture θ (Fig. 1.13). Since radiation has wave properties, therefore the size of the image observed in the microscope will be governed by the resolving power of the microscope. The position of the particle, for example electron, is therefore uncertain by an amount given by the resolving power of the microscope, that is,

Foundation of Quantum Mechanics

∆x=

λ

1.23

(1.62)

sin θ

Since the radiation is composed of photons, it means that each time the particle is struck by a photon it recoils as in Compton scattering. For simplicity let us consider the case when only one photon is scattered into the microscope. Since the aperture of lens is finite, therefore precise direction in which the photon is scattered into the microscope is not known. In the scattering process the momentum p transferred from photon to particle. The momentum of quantum of wavelength is p=

h

(1.63)

λ

This is the order of recoil momentum imparted to the particle. The x component of the momentum cannot be known exactly as we do not know through which point on the lens the photon entered the microscope. The x component of momentum can vary from p sin θ to – p sin θ and is uncertain after the scattering by an amount

∆ p x = 2 p sin θ

(1.64)

uncertainty in x component of the particle’s (and photon) recoil momentum is ∆ px ≈

2h

λ

sin θ

(1.65)

Microscope y x Objective lens

θ

Particle Light source Fig. 1.13 The Heisenberg microscope

From Eqs. (1.62) and (1.65)

∆ x ∆ px ≈

λ 2h sin θ ≈ 2 h sin θ λ

is in reasonable agreement with the limit ħ set by uncertainty principle.

(1.66)

1.24

Laser Systems and Applications

1.7

THE TIME DEPENDENT SCHRÖDINGER EQUATION IN ONE DIMENSION

The various experimental finding reveal that electron and other subatomic particles exhibit properties similar to those commonly associated with waves. Let this wave be Ψ (r, t) of free electrons of momentum p and energy E = (p2/2m), one can consider these to be free plane waves, that is, Ψ (r, t) takes the form

Ψ ( r , t ) = A exp [i (k ⋅ r − ω t )]

(1.67)

E = hω

(1.68)

p = hk

(1.69)

where k = (2 π /λ). In one dimension Eqs. (1.67) and (1.69) takes the form

Ψ ( x , t ) = A exp [i (k x − ω t )] p =hk

(1.70) (1.71)

The Eq. (1.70) is a solution to the wave equation and form of the wave equation applicable to many classical waves, for example, transverse waves on a string or plane sound waves as in a gas, is ∂2Ψ ∂t

2

= c2

∂2Ψ ∂ x2

(1.72)

where c is real constant equal to the wave velocity. Differentiating Eq. (1.70) with respect to x and t, ∂Ψ = i kΨ ∂x

∂2Ψ ∂x

2

= − k 2Ψ

∂Ψ = − i ωΨ ∂t

∂2Ψ ∂t2

= −ω 2Ψ

(1.73) (1.74) (1.75)

Substituting Eqs. (1.73) and (1.75) in Eq. (1.72)

− ω 2 Ψ = − c2 k 2 Ψ c2 =

ω2 k2

ω=c k

(1.76)

Foundation of Quantum Mechanics

1.25

From Eqs. (1.68) and (1.76)

E = hc k

(1.77)

for non-relativistic free particles of mass m we have E=

p2 2m

(1.78)

Therefore, Eq. (1.72) cannot be wave equation governing the matter waves. We therefore, discard the differential Eq. (1.72). We note that differentiation with respect to x of wave function like those of Eq. (1.70) has the general effect of multiplication of the function by k whereas the differentiation with respect to t has the general effect of multiplication by ω. Then from Eqs. (1.68) and (1.78), it is observed that the differential equation should have first derivative with respect to t and second derivative with respect to x, that is ∂Ψ ∂2Ψ =α ∂t ∂ x2

(1.79)

where α is a constant. Substituting Eqs. (1.73) and (1.74) in Eq. (1.79)

− i ω Ψ = −α k 2 Ψ α=

iω

(1.80)

k2

From Eqs. (1.68), (1.71) and (1.80) we have

α=

ihE p

2

=

i h p2

2m p

2

=

ih 2m

(1.81)

Substituting Eq. (1.81) in (1.79) i h ∂2Ψ ∂Ψ = ∂ t 2 m ∂ x2

or ih

h2 ∂2Ψ ∂Ψ =− 2 m ∂ x2 ∂t

(1.82)

This is one dimensional form of time dependent Schrödinger wave equation. The Schrödinger equation is linear in Ψ and the coefficients of the equation involves constants such as h and mass m and is independent of parameters of a particular kind of motion of the particle, for example, momentum, energy, k and frequency. The Schrödinger equation is fundamental equation of quantum

1.26

Laser Systems and Applications

mechanics in the sense that the second law of motion is the fundamental equation of Newtonian mechanics. Suppose that particle is not free, but moves in a potential so that instead of E = (p2/2m) we have E=

p2 + V ( x, t ) 2m

The potential energy V(x, t) does not depend on p and E. The equation EΨ =

p2 Ψ 2m

suggest that the Eq. (1.82) is generalized to ih

∂Ψ h2 2 =− ∇ Ψ + V ( x , t )Ψ 2m ∂t

(1.83)

which is one dimensional time dependent Schrödinger wave equation for a particle in a field characterized by the potential energy V(x, t). If the particle cannot move to infinity then bound state occur. That is, the particle is bound at all energies to move within a finite and limited region of space. The Schrödinger equation in this region admits only solutions that are discrete. Unbound state occurs in those cases where the motion of the system is not confined; for example a free particle. Potentials that confine the particle for only some energies give rise to mixed spectra. For example the mixed spectra are observed for finite square well potential. In obtaining the Schrödinger wave equation we started with the experimental knowledge concerning the properties of free particle and their associated plane waves and ended with equation for wave function associated with a particle moving under the influence of a potential. The Eq. (1.83) is obtain by method of induction, where we start from a particular example to a more general law in contrast with deduction where a particular result is obtained from a general law. The Eq. (1.83) is linear in Ψ. As a result, it satisfies the superposition principle. According to superposition principle; linear combination of possible wave functions is also a possible wave function. If Ψ1(x, t) and Ψ2(x, t) separately satisfy Eq. (1.83), then the wave function

Ψ ( x , t ) = c1 Ψ 1 ( x , t ) + c 2 Ψ 2 ( x , t )

(1.84)

also satisfy the Schrödinger wave Eq. (1.83), where c1 and c2 are complex numbers. In general

Ψ ( x , t ) = ∑ ci Ψ i

(1.85)

i

also satisfy Eq. (1.83). The fact that the superposition principle applies is directly related to the wave nature of the matter, and in particular to the existence of

Foundation of Quantum Mechanics

1.27

interference effects for de-Broglie waves. Since Eq. (1.83) is of first order in time derivative therefore, once the initial value of the wave function Ψ is given at some time, its value at all times can be found.

1.8

THE TIME DEPENDENT SCHRÖDINGER EQUATION IN THREE DIMENSIONS

The one dimensional treatment can be extended to three dimensions. The wave function is then given by Eq. (1.67). Using Eqs. (1.68) and (1.69) the Eq. (1.67) is written as ⎡i (p ⋅ r − E t )⎤⎥ ⎣h ⎦

Ψ ( r , t ) = A exp ⎢

(1.86)

(

)

⎡i ⎤ Ψ (r, t ) = A exp ⎢ p x x + p y y + p z z − E t ⎥ ⎣h ⎦

(1.87)

Differentiating Eq. (1.87) with respect to t ∂Ψ i = − EΨ h ∂t

(1.88)

Differentiating Eq. (1.87) with respect to x, y and z twice, we have ∂Ψ i = px Ψ ∂x h ∂2Ψ ∂x

2

∂2Ψ ∂ y2

∂2Ψ ∂z

2

p x2

=−

h2

=−

=−

Ψ

p y2 h2

p z2 h2

(1.89)

Ψ

(1.90)

Ψ

(1.91)

On adding Eqs. (1.89) - (1.91) ∂2Ψ ∂ x2

where

∂2Ψ ∂ x2

+

+

∂2Ψ ∂y

2

∂2Ψ ∂y

2

+

+

∂2Ψ ∂ z2

∂2Ψ ∂ z2

=−

p x2 + p y2 + p z2

= ∇2

From Eqs. (1.88) and (1.92) ∇2Ψ = −

2m h

2

ih

∂Ψ ∂t

h2

(1.92)

1.28

Laser Systems and Applications

ih

∂Ψ h2 2 =− ∇ Ψ 2m ∂t

(1.93)

which is three dimensional time dependent Schrödinger equation for a free particle. A comparison of Eqs. (1.88) and (1.93) and classical energy equation E=

p2 2m

(1.94)

suggests that for a free particle the energy and momentum can be represented by differential operators that act on the wave function Ψ. The operators for E and p are E →ih p→

∂ ∂t

h ∇ = −ih ∇ i

(1.95) (1.96)

It is a postulate of wave mechanics that when the particle is not free, the dynamical variables E and p are still represented by Eqs. (1.95) and (1.96), respectively. Suppose that the particle is not free, so that instead of Eq. (1.94) we have E=

p2 + V ( r, t ) 2m

(1.97)

Since potential energy V(r, t) does not depend on p and E, then Eq. (1.97) suggests ⎛ p2 ⎞ EΨ = ⎜⎜ + V ( r , t ) ⎟⎟ Ψ ⎝ 2m ⎠

(1.98)

which in turn suggests that the wave Eq. (1.93) is generalized to ih

⎡ p2 ⎤ ∂Ψ h2 2 = ⎢ + V ( r , t )⎥ Ψ = − ∇ Ψ + V ( r , t )Ψ ∂t 2m ⎣2m ⎦

(1.99)

which is time dependent Schrödinger equation for a particle in a field characterized by potential energy V(r, t). The Eq. (1.99) is written as ih

∂Ψ = HΨ ∂t

H =−

is known as Hamiltonian operator.

h2 2 ∇ + V ( r ,t ) 2m

(1.100) (1.101)

Foundation of Quantum Mechanics

1.29

1.9 THE WAVE FUNCTION The wave function is complex consisting of real and imaginary part. Neither real nor the imaginary part of the wave function, but only the full complex expression, is a solution to the Schrödinger equation. Since the wave function is complex, therefore it cannot be measured by any actual physical instrument. A physical system is completely described by the wave function. The wave function contains all possible information that can be obtained about the system.

Statistical interpretation The wave function Ψ (r, t) is assumed to provide a quantum mechanically complete description of the behavior of a particle of mass m with the potential energy V(r, t). Since the motion of the particle is connected with the propagation of an associated wave function (the de Broglie connection), these two entities must be associated in space. That is particle must be at some location where the wave function has appreciable amplitude. Thus Ψ (r, t) is to be large where the particle is likely to be and small elsewhere. This indicates that Ψ (r, t) can be interpreted in statistical terms. M. Born in 1926 made the fundamental postulate that if the particle is described by a wave function Ψ (r, t), the probability of finding the particle within the volume element dτ = dx dy dz about the point r at the time t is P ( r, t ) d τ = Ψ ( r, t ) P ( r, t ) = Ψ ( r, t )

2

2

dτ

= Ψ * ( r , t )Ψ ( r , t )

(1.102) (1.103)

where Ψ* (r, t) is complex conjugate of Ψ (r, t) and P (r, t) is the probability density. Thus if we know the wave function associated with a physical system, we can calculate the probability of finding the particle in the vicinity of particular point.

Normalization of the wave function The probability of finding the particle somewhere in the region must be unity, therefore we can write +∞

∫ P ( r ,t ) d τ = 1

(1.104)

−∞

or +∞

∫Ψ

*

( r , t )Ψ ( r , t ) d τ = 1

(1.105)

−∞

If Ψ is a solution of the Schrödinger wave equation, then AΨ is also a solution where A is any constant. The scale of wave function can therefore always be chosen to ensure that Eq. (1.104) as well as Eq. (1.105) holds at the same time. This process is known as normalization and the wave function which obeys these

1.30

Laser Systems and Applications

are said to be normalized. Thus a wave function can always be multiplied by a phase factor of the form exp (i α) where α is arbitrary real constant, without affecting the value of any physically significant quantity. In contrast to bound states, unbound states cannot be normalized.

1.10 TIME INDEPENDENT SCHRÖDINGER EQUATION Consider a closed system in which energy is conserved and the potential energy is time independent, that is, V = V (r). The Schrödinger Eq. (1.99) is then

ih

∂Ψ h2 2 =− ∇ Ψ + V (r ) Ψ ∂t 2m

(1.106)

For convenience, we consider one dimensional case for which V (r) = V(x) and Schrödinger Eq. (1.106) ih

∂Ψ h2 ∂2Ψ =− + V ( x )Ψ ∂t 2 m ∂x2

(1.107)

Assuming that the wave function can be expressed as a product

Ψ ( x ,.t ) = ψ ( x ) f ( t )

(1.108)

On differentiating Ψ (x, t) with respect to x and t df ∂Ψ =ψ dt ∂t

(1.109)

dψ ∂Ψ = f (t ) dx ∂x

∂ 2Ψ ∂x2

= f (t )

d 2ψ d x2

(1.110)

Substituting Eqs. (1.109) and (1.110) in Eq. (1.107) ih ψ

d f h2 d 2Ψ =− f + V( x ) f ψ 2 m d x2 dt

(1.111)

Dividing both side of Eq. (1.111) by ψ (x) f (t) ih

1 d f h2 1 d 2Ψ =− + V( x ) 2 m ψ d x2 f dt

(1.112)

Each side of Eq. (1.112) is equal to certain function of x and t. The left hand side of Eq. (1.112) does not depend on x, so this function is independent of x. The right hand side does not depend on t, this function must be independent of t. Since the function is independent of both variables x and t, it must be a constant. We denote this constant by E. Equating left hand side of Eq. (1.112) to E, we have

Foundation of Quantum Mechanics

1.31

1 d f =E f dt

(1.113)

d f i =− E f h dt

(1.114)

ih

On integration ln f ( t ) = −

i Et h

(1.115)

⎛ i ⎞ f ( t ) = exp ⎜ − E t ⎟ ⎝ h ⎠

(1.116)

Equating right hand side of Eq. (1.112) to E −

h2 d 2Ψ + V ( x )ψ = Eψ 2 m d x2

(1.117)

Equation (1.117) is known as time independent Schrödinger equation for a particle of mass m moving in time independent potential V (x). ψ (x) are not necessarily complex and time independent Schrödinger equation does not contain the imaginary number. From Eq. (1.117) it is seen that E has dimension of energy. We assume that E is the energy of the system. From Eqs.(1.108) and (1.116)

⎛ i ⎞ Ψ ( x,. t ) = exp ⎜ − E t ⎟ ψ ( x) ⎝ h ⎠

(1.118)

The wave function Ψ (x, t) correspond to states of constant energy.

1.11 STATIONARY STATES The wave function in Eq. (1.118) is complex, but the quantity that is experimentally determined is the probability density Ψ ( x , t ) 2 . The probability density is *

Ψ ( x ,t )

2

⎡ ⎤ ⎡ ⎤ ⎛ i ⎞ ⎛ i ⎞ = Ψ * ( x , t )Ψ ( x , t ) = ⎢exp ⎜ − E t ⎟ ψ ( x )⎥ ⎢exp ⎜ − E t ⎟ ψ ( x )⎥ h h ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦

Ψ ( x, t )

2

= ψ * ( x )ψ ( x ) = ψ ( x )

2

(1.119)

The probability density is then independent of time. A particle in such a state will remain in that state until acted upon by some external entity that forces it out of that state. The solution (1.118) of the Schrödinger equation (1.112) for time independent potential is called stationary state. The stationary state is a state of well defined energy, E being the definite value of its energy and not only its the expectation value. For a stationary state Ψ ( x ,.t ) equal a function of time

1.32

Laser Systems and Applications

multiplied by a function of particle coordinate. For a stationary state Ψ (x,. t) is an eigenfunction of hamiltonian. Any determination of the energy of a particle which is in stationary state always yield a particular value E and only that value. Such an interpretation is in line with the Heisenberg uncertainty relation

∆ E ∆t > h which implies that a quantum state with a precise energy ( ∆ E = 0 ) is possible only if there is an infinite time available to determine that energy. Stationary states are of just such nature in view of constancy of Ψ ( x , t ) 2 in time.

1.12 BOUNDARY CONDITIONS The wave function itself has no physical interpretation, however, the square of its absolute magnitude Ψ ( x , t ) 2 evaluated at a particular place and at a particular time is proportional to the possibility of finding the particle at that time. The probability density Ψ ( x , t ) 2 is positive and real and is taken equal toΨ* (r, t) Ψ (r, t) . The wave function Ψ can take on negative values but probability density is always be positive. Besides fulfilling the normalization condition a solution of the time independent Schrödinger equation must obey the following boundary conditions. 1. The wave function must be continuous and single valued. 2.

∂ψ ∂ψ ∂ψ , and must be continuous and single valued everywhere. ∂x ∂ y ∂z

3. The integral of the square modulus of the wave function over all values x must be finite

∫ψ

*

ψ d τ = finite

that is the wave function must be square integrable. This condition means that wave function must be normalizable, that is wave function must go to zero as x (y, z) → ± ∞ in order that

∫ψ

2

dτ over all space is finite constant.

The boundary conditions ensure that the probability of finding the particle in the vicinity of any point is unambiguously defined rather than having two or more possible values. Thus the wave function is single valued and continuous. If ψ (x) and (dψ /dx) are not single valued, finite then the same is true for Ψ (x, t). since the given formula for calculating the expectation values of position and ∂ψ . We observe that in any of these cases we momentum contains ψ (x, t) and ∂t might not obtain finite and definite values when we evaluate measured quantities. The first derivative of the wave function with respect to position coordinates must be continuous everywhere except where there is an infinite discontinuity in the potential. We know any function always has an infinite derivative whenever it

Foundation of Quantum Mechanics

1.33

has a discontinuity. Let us consider the time independent Schrödinger equation (1.117) in one dimension

d 2ψ 2 m = 2 (V − E ) ψ d x2 h for finite V, E and ψ, (d2ψ /dx2) is finite. This in turn require (dψ /dx) to be continuous. A finite discontinuity in (dψ /dx) implies an infinite discontinuity in (d2ψ /dx2) and from the Schrödinger equation in V (x).

1.13 PARTICLE IN ONE DIMENSION BOX Particle in a box (also known as the infinite potential well or the infinite square well) is defined as a single point particle confined in a box where it experiences no force, that is, the potential energy is constant or zero. At the walls of the box the potential is infinite. Therefore the particle is constrained to remain in the box. Consider a particle of mass m confined in a one dimensional box of length a (Fig. 1.14). The potential is zero inside the box and infinite elsewhere, that is, V = ∞

∞

V (x)

0

a

x 

Fig. 1.14 An infinite square well potential V =0 0< x 1 we have states with different values of l corresponding to the same value of n and hence these are degenerate (when two or more eigenfunctions have the same eigen value then the eigenvalue is said to be degenerate). The degree of degeneracy of energy levels of hydrogen atom is n2

Examples 1. A 45kW broadcasting antenna emits radio waves at a frequency of 4 MHz. How many photons are emitted per second? The electromagnetic energy emitted by the antenna in one second is E = 45000J. Thus the number of photons n emitted in one second is 45000 6.63

10

4

10

1.7

10

1.44

Laser Systems and Applications

2. Radiation of wavelength 290 nm falls on a metal surface for which the work function is 4.05 eV. What potential is needed to stop the most energetic photoelectrons We have 6.63

10 2.9

3

10

/

10

6.86

The maximum kinetic energy of electron is 0.23

10

4.28

4.28 4.05

Therefore potential needed to stop the most energetic electron is 0.23eV. 3. A light of wavelength 200 nm is incident on a polished metal surface. Obtain (a) the stopping potential (b) the maximum kinetic energy of the liberated electrons and (c) the speed of the fastest electrons. The work function of the metal is 4.7 eV. Energy of the incident photon is E = hν =

hc

λ

=

6.626 × 10 −34 Js × 3 × 10 8 m / s

200 × 10 −9 m 9.94 × 10 −19 = 9.94 × 10 −19 J = eV = 6.20 eV 1.602 × 10 −19

(a) Since the energy of the incident photon is equal to work function plus the stopping potential, therefore the stopping potential is Stopping potential = E − W = 6.20 eV − 4.7 eV = 1.5 eV

(b) the maximum kinetic energy of the photoelectron correspond to stopping potential

1.5 eV = (1.602 ×10−19 ) ×1.5 J = 2.4 ×10−19 J (c) Using non relativistic form of kinetic energy T =

v2 =

1 m v 2 = 2.4 × 10 −19 J 2

2 T 2 × 2.4 × 10 −19 J = = 5.3 × 10 11 m 2 / s 2 m 9.11 × 10 −31 Kg

v = 7.3 × 105 m / s v 7.3 ×10 5 m / s = = 2..4 ×10 −3 8 c 3 ×10 m / s

Foundation of Quantum Mechanics

1.45

4. When light of wavelength λ is incident on a metallic surface, the stopping potential for the photoelectrons is 3.2 V. If a second light source whose wavelength is 2λ is used the stopping potential drops to 0.8 V. Calculate λ , work function and cutoff frequency of the metal. The stopping potential Vs causes the most energetic electrons to turn back before reaching the collector. The stopping potential is thus connected to the electrons kinetic energy by the relation 1 2

From Eq.(4.10)

The stopping potential for first wavelength λ is

The stopping potential for first wavelength 2λ is 2

and 1 2

6.6 10 2 1.6 10

1 2

2

3

10 3.2

1.6

10

/ 0.8

2.6

10

Now 2 1.6

1.6

2.56

10

The cutoff frequency is 2.56 10 6.6 10

3.9

10

5. An X-ray photon of wavelength 10-12 m in incident on a stationary electron. Calculate the wavelength of the scattered photon if it is detected at an angle of 90° to the incident radiation. From Eq. (1.27)

1.46

Laser Systems and Applications

λ′ − λ =

h (1 − cos θ ) mc

Substituting the values of h, m, c and θ

λ′ = λ +

6..626 × 10 −34 Js (1 − cos 90°) 9.11× 10 −31 Kg × 3 × 108 m / s

≈ (1 + 2.43) × 10 −12 m = 3.43 × 10 −12 m 6. Consider an X−ray beam of wavelength 0.1nm. If the radiation scattered from free electron is viewed 90° to the incident beam (a) what is Compton shift (b) what kinetic energy is given to the recoiling electron and (c) what percentage of the incident photon energy is lost in the collision. (a) From Eq. (1.27)

λ′ − λ =

h (1 − cos θ ) mc

Substituting the values of h, m, c and θ

λ ′ − λ = ∆λ =

6..626 ×10−34 Js (1 − cos 90°) ≈ 2.43 ×10−12 m 9.11×10−31 Kg × 3 ×108 m / s

(b) We have energy of incident photon equal to energy of scattered photon plus kinetic energy of electron, that is, hc

λ hc

λ K .E. =

hc

λ

−

=

=

hc + K .E . λ′

hc + K .E. λ + ∆λ

⎡ λ + ∆λ − λ ⎤ hc hc∆λ = hc ⎢ ⎥= λ + ∆λ ⎣ λ ( λ + ∆λ ) ⎦ λ ( λ + ∆λ )

Substituting the values of λ and ∆λ K .E. ≈

6.626 × 10 −34 Js × 3 × 10 8 m / s × 2.43 × 10 −12 10 −10 m (1 + 0.0243) × 10 −10 m

≈ 4.73 × 10 −17 J

(c) Incident X-ray photon energy is E = hν =

hc

λ

=

6.626 × 10 −34 Js × 3 × 10 8 m / s

10 −10 m 1.99 × 10 −15 = 1.99 × 10 −15 J = eV = 12.4keV 1.602 × 10 −19

Energy lost by photon equal the gain by the electron is equal

Foundation of Quantum Mechanics

4.73 ×10−17 J =

1.47

4.73 ×10−17 eV = 0.295keV 1.602 ×10−19

percent loss in energy is 0.295 keV × 100% = 2.4% 12.4 keV 7. An X-ray photon of wavelength 10-12 m in incident on a stationary electron. Calculate the wavelength of the scattered photon if it is detected at an angle of 900 to the incident radiation. From Eq. (1.27)

λ′ − λ =

h (1 − cos θ ) mc

Substituting the values of h, m, c and θ

λ′ = λ +

6..626 × 10 −34 Js 9.11 × 10 −31 Kg × 3 × 10 8 m / s

(1 − cos 90°)

.

≈ (1 + 2.43) × 10 −12 m = 3.43 × 10 −12 m

8. Consider an X−ray beam of wavelength 0.1nm. If the radiation scattered from free electron is viewed 90° to the incident beam (a) what is Compton shift (b) what kinetic energy is given to the recoiling electron and (c) what percentage of the incident photon energy is lost in the collision. (a) From Eq. (1.27)

λ′ − λ =

h (1 − cos θ ) mc

Substituting the values of h, m, c and θ

λ ′ − λ = ∆λ =

6..626 × 10 −34 Js (1 − cos 90°) ≈ 2.43 × 10 −12 m −31 8 9.11× 10 Kg × 3 × 10 m / s

(b) We have energy of incident photon equal to energy of scattered photon plus kinetic energy of electron, that is, hc

λ hc

λ K .E . =

hc

λ

−

=

=

hc + K .E . λ′

hc

λ + ∆λ

+ K .E.

⎡ λ + ∆λ − λ ⎤ hc hc∆λ = hc ⎢ ⎥= λ + ∆λ λ ( λ + ∆ λ ) λ ( λ + ∆λ ) ⎣ ⎦

1.48

Laser Systems and Applications

Substituting the values of λ and ∆λ K .E. ≈

6.626 × 10 −34 Js × 3 × 10 8 m / s × 2.43 × 10 −12 ≈ 4.73 × 10 −17 J 10 −10 m (1 + 0.0243) × 10 −10 m

(c) Incident X-ray photon energy is

6.626 × 10−34 Js × 3 × 108 m / s λ 10−10 m 1.99 × 10−15 = 1.99 × 10−15 J = eV = 12.4keV 1.602 × 10−19 E = hν =

hc

=

Energy lost by photon equal the gain by the electron is equal

4.73 × 10−17 J =

4.73 × 10−17 eV = 0.295keV 1.602 × 10−19

percent loss in energy is 0.295 keV × 100% = 2.4% 12.4 keV 9. Consider a photon that scatters from an electron at rest. If the Compton wavelength shift is observed to be triple the wavelength of the incident photon and if the photon scatters at 60°, calculate the wavelength of the incident photon, the energy of the recoiling electron and the angle at which the electron scatters. From Eq. (1.27) 1

The wavelength shift is 3λ and θ = 60°. Substituting the values of wavelength shift and θ in Eq. (1.27) 1

3

60

2

6

Using Eq. (1.29) 2.426

10 6

4.04

10

The energy Te of the recoiling electron can be obtained from the conservation of energy and using λ’=4λ 1

1

3 4

3 ħ 2

3

3.14 2

197.33 10 4.04 10

2.3

Foundation of Quantum Mechanics

1.49

where 197.33

ħ

10

15

4 , the angle φ at which the electron recoil

Since

cot ϕ =

7 λ ′ − λ cosθ 4λ − λ cos 60 7 = = = = 4.0415 λ sin θ λ sin 60 3 1.732 4.0415

13.89°

10. Calculate the de-Broglie wavelength for a 100g bullet moving at 900 ms-1 From de Broglie relation 6.626 10 0.1 900

7.4

10

11. Calculate the de-Broglie wavelength of a proton having a kinetic energy of 70 MeV The kinetic energy of the proton is 2

and momentum is 2

The de-Broglie wavelength is 2

We know ħ

197

938.3

where c is the velocity of light. The de-Broglie wavelength is 2

ħ

2

ħ 2

2

3.14

197 √2

938.3

70

3.4

10

12. Show that the phase velocity of a de-Broglie wave is greater than the speed of light. The de-Broglie wave speed v′ is related to the frequency and wavelength of the de-Broglie wave by

v′ = ν λ

1.50

Laser Systems and Applications

The de-Broglie wavelength is thus

λ= v′ =

h v′ = p ν

hν E = p p

The energy of the particle is

mc 2

E=

1 − (v c ) 2

Magnitude of the momentum is p=

mv 1 − (v c ) 2

Then v′ =

E c2 = p v

Thus wave speed is inversely proportional to the particle speed. Since particle speed v < c, the de Broglie waves associated with the particle apparently travel at speed > c. 13. What de-Broglie wavelength must an electron have if it is moving with a velocity of 1455 m/s. Putting pparticle = mv in de Broglie relation

λ= λ=

h mv

6.626 × 10 −34 J .s = 500 nm 9.1 × 10 −31 Kg × 1455 m / s

14. A charged pi meson has a rest energy of 140 MeV and a life time of 26 ns. Find the energy uncertainty of the pi meson. We have ħ ħ

6.58 10 26 10

2.5

10

2.5

10

15. An electron is confined to a region of space of the size of an atom (0.1 nm). What is the uncertainty in the momentum of the electron? We have

Foundation of Quantum Mechanics

1.05 10 1 10

ħ

1.05

10

1.51

/

16. Using uncertainty principle show that electron cannot exists in the nucleus. The radius of the nucleus of an atom is of the order of 10-14m. If the electron is confined within nucleus, the uncertainty in its position must not be greater than 10-14m. According to uncertainty principle ~ħ 6.625 10 2 3.14 2 10

ħ

2

5.275

10

/

If this is the uncertainty in momentum of the electron, the momentum of the electron must be at least comparable with its magnitude 5.275

10

/

The kinetic energy of the electron of mass m is given by 5.275 2

2 2

10 9

5.275 9 10

5.275 2 9

10 10 1.6

9.7

10

10 10 10

97

This means that if the electrons exists inside the nucleus, their kinetic energy must be of the order of 97 MeV. But experimental observations show that no electron in the atom possesses energy greater than 4MeV. Thus electrons do not exist in the nucleus. 17. The average life time of an atom in its excited state is 2.5×10-14s, calculate the uncertainty in the measurement of energy in this state. From uncertainty relation ħ We have ħ

1.05 10 2.5 10

4

10

4 10 1.6 10

0.025

18. For a particle in one dimension is the wave function ψ (x) = x acceptable? We know ψ (x) is the probability amplitude ψ *ψ d x is the probability of finding the particle between x and x + dx and total probability is +∞

∫ψ ψ d x = 1 *

−∞

1.52

Laser Systems and Applications

Taking ψ (x) = x we have +∞

∫x

2

dx→∞

−∞

This function is therefore not acceptable. 19. Consider a one-dimensional particle which is confined within the region 0 ≤ x ≤ a and whose wave function is

ψ ( x, t ) = sin(π x a) exp(−iω t ) Find the potential V(x). Differentiating the given wave function

dψ π = cos(π x a) exp(−iω t ) dx a d 2ψ π2 π2 = − sin( π x a ) exp( − i ω t ) = − ψ d x2 a2 a2

dψ = −i ω sin(π x a) exp(−iω t ) = − i ωψ dt

Substituting the value of

d 2ψ dψ in Eq. (1.17) and 2 dt dx

ih

h2 2 ∂ψ =− ∇ ψ + V ( x, t )ψ 2m ∂t

⎡ h2 π 2 ⎤ i h (−i ω ) ψ = ⎢ + V ⎥ψ 2 ⎣ 2m a ⎦ V =h ω −

h2 π 2 2 m a2

20. A particle of mass m moves in one dimension under the influence of V (x). Suppose it is in an energy eigenstate

ψ ( x ) = A x exp( − kx ) ( 0 ≤ x ≤ ∞, k > 0 ) Obtain A. From Eq. (1.53)

Foundation of Quantum Mechanics

1.53

+∞

∫ψ

*

( x ) ψ ( x) d x = 1

−∞

Substituting the value of ψ ∞

∫A x

2 2

exp( −2kx) dx = 1

0

Let 2 kx = y , dx =

1 dy 2k

∞

A2 y 2 exp(− y ) dy = 1 8k 3 0

∫

∞

A2 2 = 1 as ∫ y 2 exp( − y ) dy = 2 8k 3 0

A = 2k

3 2

21. An electron is confined in the ground state in a one dimensional box of width 10-10 m. Its energy is 38 eV. Calculate the energy of the electron in its first excited state. The energy levels are given by ħ

2

For first excited state, n = 2 4 2

ħ

4

4

38

152

22. For a particle in a box show that fractional difference in the energy between adjacent eigenvalues is 2

1

We have ħ 2

1 2

ħ

2 2

1

ħ

 

1.54

Laser Systems and Applications

2

1

ħ

2

1

ħ

23. Find the probability that a particle trapped in a box of length a can be found between 0.45a and 0.55a for the ground state. The ground state wave function is

ψ 1 ( x) =

πx 2 sin a a

The probability is +∞

P=

∫

ψ 1* ψ 1

−∞

0.55 a

0.55 a

2 2πx ⎛ cos 2 π x ⎞ sin dx= dx= ⎜1 − ⎟d x a a a ⎝ ⎠ 0.45 a 0.45 a

∫

∫

P = 1−

1 (sin 1.10 π − sin 0.90 π ) 2π

P = 1−

1 (sin 1.10 π − sin 0.90 π ) 2π

 

Short Questions 1. Define black body.

Ans. An ideal black body is an object that absorb all wavelength of electromagnetic radiation incident upon it. Reflecting no radiation, the object appears to be black. 2. What is ultraviolet catastrophe?

Ans. The divergence in radiative energy density with increasing frequency is called ultraviolet catastrophe. 3. Which experiment indicates that internal energy states of an atom are quantized?

Ans. Franck-Hertz experiment. 4. Which experiment gives the evidence that angular momentum is conserved?

Ans. Stern Gerlach experiment. 5. Which experiment confirm the existence of spin?

Ans. Stern Gerlach experiment. 6. Which experiment verify the wavelike aspects of electrons?

Ans. Davisson and Germer experiment.

Foundation of Quantum Mechanics

1.55

7. Why wave like nature of the matter is very difficult to demonstrate on a macroscopic scale?

Ans. Because of very small value of h, it is very difficult to demonstrate the wave like behaviour on a macroscopic scale. 8. Why Compton effect is not observed for electromagnetic radiation in visible region?

Ans. There is no Compton effect for electromagnetic radiation in visible region because the photon energy in this case is not large enough compared with the binding energy of even the loosely bound electron. 9. What assumptions are made in deriving the expression for Compton shift?

Ans. The assumptions made are (i) electrons participating in the scattering are free (ii) electrons are stationary 10. What are the dimensions of Compton wavelength?

Ans. It has the dimension of length. 11. Why the wavelength shift obtained from scattering of X-rays with proton or nucleus is very small?

Ans. In Compton scattering, the Compton wavelength is given by h/mc. Since m is large for proton or nucleus, therefore h/mc is very small in comparison with scattering with electrons. Therefore, wavelength shift is also very small. 12. On what factors change of wavelength depends in the scattering of electrons with X-rays?

Ans. It depends on the scattering angle and universal constant h/mc and is independent of wavelength of incident radiation and nature of scattering substance. 13. When the change of wavelength is small in the scattering of X-rays with electrons?

Ans. When the scattering angle is 180°. 14. Can a wave function be negative?

Ans. The wavefunction can be negative, it is the probability density what we measure, which is always positive. 15. What are the two keys properties that come to distinguish classical and quantum physics?

Ans. The two key properties are (i) quantization: energy at the atomic level is not a continuous variable but comes in discrete packets called quanta (ii) wave particle duality : at the atomic level, light wave has particle like properties and atoms, electrons etc have wave like properties. 16. Which of the experimental results for the photoelectric effect suggests that light can display particle like behaviour?

1.56

Laser Systems and Applications

Ans. The observation that electrons are emitted almost instantaneously irrespective of the intensity of the light source is small or large. 17. Why does the analysis of the photoelectric effect based on classical physics predicts that the kinetic energy of electrons will increase with increasing intensity?

Ans. Wave theory requires that oscillating electric vector E of light wave increases in amplitude as the intensity of the light beam increases. Since the force applied to the electron is eE, this suggests that kinetic energy of ejected electron should increase as the light beam is made more intense. 18. Is

1 for a one particle one dimensional system?

Ans. No, as ψ is not a measurable quantity, the expression should be 1. 19. Define stationary state.

Ans. The stationary state is a state of well defined energy and | , | is constant in time. Any determination of the energy of a particle which is in stationary state always yield a particular value of energy and only that value. 20. How did Planck conclude that the discrepancy between experiments and classical theory for blackbody radiation was at high and not low frequencies?

Ans. The spectral distribution shows a pronounced maximum at a given frequency with temperature T, that is the peak of the radiation spectrum occurs at a frequency that is proportional to the temperature. The peak moves towards the higher frequency with increasing temperature. Planck’s saw that the discrepancy between experimental and classical theory occurred at high and not at low frequencies. The absence of high frequency radiation at low temperature showed that the high frequency dipole oscillations were active only at high temperature. 21. Why were investigations at the atomic and subatomic levels required to detect the wavenature of particles?

Ans. According to de-Broglie relation

. Substituting the values of h.m

and v it is known that it is difficult to obtain de Broglie wavelength much larger than 1nm even with the particles as light as electrons. Therefore, investigations about the wave nature of particles are required for atomic or subatomic particles. 22. The inability of classical theory to explain the spectral density distribution of a black body was called the ultraviolet catastrophe. Why is this name appropriate?

Ans. According to the classical expression ,

8

The radiative energy is infinite as ν→∞. If we integrate the above expression over all frequencies, the integral diverges. This implies that cavity contains an infinite

Foundation of Quantum Mechanics

1.57

amount of energy. This divergence in radiative energy density with increasing frequency is called ultraviolet catastrophe. 23. Write the uncertainty relation between ∆y and ∆pz .

Ans. ∆y∆pz = 0 24. Why boundary conditions are needed?

Ans. Schrödinger equation contains two unknown, the allowed energy E and the allowed wave function ψ. To solve for the unknowns, we need to impose additional condition called boundary conditions on ψ besides enquiring that it satisfy the Schrödinger equation. The boundary conditions determine the allowed energies since only certain values of E allow ψ to satisfy boundary conditions. 25. How the state is described in quantum mechanics?

Ans. To describe the state of the system in quantum mechanics we postulates the existence of a function of the particle’s coordinates called wave function or state function ψ. Since the state will in general, changes with time ψ is also a function of time. The wave function contains all possible information about a system. So instead of saying the state described by ψ we say that the state ψ. 26. Does in the stationary state the particle is at rest?

Ans. The term stationary should not lead into thinking that a particle in a stationary state is at rest. What is stationary is the probability density | | not the particle itself. 27 What is the dimension of |

Ans. |

,

| has dimension of

,

| in |

,

|

.

28. What is Planck’s hypothesis?

Ans. According to Planck’s hypothesis, radiation is emitted or absorbed by matter in discrete quanta, each of energy hν, where ν is the frequency of the radiation and .h is Planck’s constant. 29. What does constant h represent? What are its dimension?

Ans. It denotes an elementary quantum or quantity of action. The dimensions of h are

30. Every metal has a definite work function. Why do photoelectrons not come out all with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?

Ans. Work function is defined as the minimum energy required to remove an electron from the metal. The electrons are distributed in various energy levels in a

1.58

Laser Systems and Applications

continuous band of levels. As a result for the same incident radiation, electrons knocked off from different levels come out with different energies. 31. The work function of a metal is 4.2 eV. If two photons each of energy 2.3 eV strikes an electron of the metal, will the emission of electron be possible?

Ans. No, emission of the electron is not possible. This will be possible only when the energy of each photon is equal or more than the work function of the metal surface. 32. Out of IR, yellow, blue and Ultraviolet radiation which is more effective for causing photoelectric emission?

Ans. Since frequency or energy of the ultraviolet photon is maximum, therefore ultraviolet radiation are more effective for causing photoelectric emission. 33. Is Schrödinger equation linear in the wavefunction? What is the meaning of it? Ans. The Schrödinger equation is linear in wavefunction. This means that equation has term that contain wavefunction and its derivative but no terms independent of wavefunction or that involve higher power of wavefunction of its derivative. 34. What is the importance of normalizing a wavefunction?

Ans. As the Schrödinger equation is a linear equation, so its wave function can be multiplied by any arbitrary constants and still it remains its solution. Normalizing a wavefunction means fixing the amplitude by fixing the value of multiplying constant of the wavefunction.

Problems 1.1.

What are the shortcomings of the classical physics?

1.2.

What was Planck’s idea to explain observed spectrum of black body?

1.3.

What is Compton effect? Derive expression for Compton shift.

1.4.

What is photoelectric effect? Give Einstein’s theory to explain it.

1.5.

What are matter waves? Mention any three properties of matter waves.

1.6.

State and derive de Broglie hypothesis. How has it been verified experimentally?

1.7.

Discuss Heisenberg uncertainty principle.

1.8.

Describe how the physical implications of the Uncertainty principle can be illustrated by an experiment performed with gamma ray microscope.

1.9.

Distinguish between phase velocity and group velocity.

1.10. What was the basis of old quantum theory? Discuss the inadequacy of quantum theory. 1.11. Discuss electron diffraction from a slit.

Foundation of Quantum Mechanics

1.59

1.12. Describe Davisson and Germer experiment to demonstrate that the electrons behave as waves. 1.13. Show that ratio of kinetic energy T of the recoil electrons and energy of the incident photons in Compton scattering is given by T ( 2hν mc 2 ) sin 2 (θ 2) = E 1 + ( 2hν mc 2 ) sin 2 (θ 2)

1.14. A photon of energy h ν collides with a stationary electron of rest mass m. Show that it is not physically possible for the photon to impart all its energy to the electron. 1.15. Show that the allowed orbit of electron are those for which an integral number of electron de-Broglie wavelength fit along the circumference. 1.16. Show that the phase velocity of a de-Broglie wave is greater than the speed of light. 1.17. Show that the de-Broglie wavelength of a particle of rest mass m0 and kinetic energy T is given by

λ=

hc T [1 + (2m0 c 2 T )]1 2

1.18. Estimate the ground state radius and ground state energy of hydrogen atom using uncertainty principle. 1.19. Which among the following functions represent physically acceptable wave functions (i) A sec x (ii) 3 sin π x (iii) A exp(x 2 ). exp(−ax) (ii) exp(−bx 2 ) (iii) x exp(−bx 2 ) 1.20. Give the formulation of time dependent Schrödinger wave equation. 1.21. Explain the physical significance of wave function. 1.22. What are stationary states? 1.23. Deduce time independent Schrodinger wave equation.

CHAPTER 2   

Concepts of Atomic and  Molecular Spectra 

The spectra of substances is in general assigned to one of the three categories: (i)

The continuous spectra, which cannot be resolved into lines irrespective of the resolving power of the measuring instruments. These spectra are produced by the bodies heated to incandescent.

(ii)

The line spectra which are produced by atoms. The lines extend over the range of several hundred angstroms. These lines can be grouped into different series often overlapping. The separation of lines in each series is found to decrease as the wavelength decreases. These lines show further structure known as fine and hyperfine structure.

(iii)

Band spectra which are produced by the molecules and derive their names from the fact that in visible region under low spectroscopic resolution they appear as continuous band, bands of colour. The band consists of more or less broad wavelength regions and usually has at one end a sharp edge called a band head or band edge. At band head the intensity falls suddenly to zero, while it fades off more or less slowly towards the other end. According to the gradual fading of intensity take place toward high frequency or low frequency, the bands are said to be shaded or degraded to the violet or red. Bands are observed in some cases to have more than one head. Band spectra are observed both in emission and absorption. When band spectra are examined with spectroscopic instruments of high resolving power, they have been found to consist of a large number of discrete lines.

The spectra of a molecule can be divided into three spectral ranges from far infrared and microwave to extreme ultraviolet corresponding to different types of transitions between molecular quantum states. The molecules have spectrum in the far infrared region only when they possess a permanent electric dipole moment and this spectrum is caused by the transitions between the rotational states of the molecule. In the near infrared region the vibration rotation spectra is observed. This corresponds to radiation emitted in vibrational transition of

2.2 Laser Systems and Applications    molecules having electric dipole moment. The vibrational transitions are accompanied by change in rotational state. The bands in the visible and ultraviolet part of the spectrum are due to radiation emitted in transitions between electronic states. The electronic spectra have a fine structure determined by the rotational and vibrational state of the molecule during electronic transitions.

2.1 THE QUANTIZATION OF ENERGY Atoms, ions, and molecules can exist only in discrete energy states. A change from one energy state to another, called a transition, is associated with either the emission or the absorption of a photon. Consider two possible energy states of a atomic system, labelled E1 and E2, in Fig. 2.1. The suffixes 1 and 2 used to distinguish these levels are quantum numbers. The wavelength of the absorbed or emitted radiation is given by (2.1) ν is the frequency, and h is Planck’s constant. E2

∆E

E1 Fig. 2.1

If we take an atomic system in state 1 and a beam of electromagnetic radiation of frequency ν fall on it, the energy will be absorbed from the beam and the atom or molecule of the atomic system will jump to state 2. If the atomic system is already in state 2 and may go to state 1 with emission of electromagnetic radiation of frequency ν. The emitted or absorbed electromagnetic radiation falls in various regions of electromagnetic spectra depending on the frequency or wavelength of the emitted or absorbed radiation. The various characteristic of the emitted or absorbed radiation is measured in units of frequency, wavenumbers, electron volt etc. The relation between frequency and wavelength λ is

c ν

(2.2)

1 ν = λ c

(2.3)

λ=

2.3 

Concepts of Atomic and Molecular Spectra

ν=

c λ

(2.4)

Using Eqs. (2.1) to (2.4), the energy interval ∆E can be expressed in eV, nm (10–9 m), cm-1 (wave number) or Hz (frequency). 1cm–1 sometimes called 1 Kayser. The relation between various units are 1 Joule = 6.24146 × 1018 eV = 5.003417 × 1022cm-1 = 1.50916 × 1033 Hz

2.2 REGIONS OF THE SPECTRUM The electromagnetic radiation can be divided into various regions. The boundaries between these regions are not very precise. In decreasing frequency the regions are: 1. Gamma ray region: ~3 × 1019 Hz (wavelength < 10pm). 2. X-ray region: ~3 × 1019 –3 × 1016Hz (wavelength 10pm-10nm). 3. Vacuum ultraviolet: ~ 3 × 1019 - 1.5 × 1015Hz (wavelength 10-200 nm). 4. Ultraviolet: ~1.5 × 1015 –7.5 × 1014Hz (wavelength 200-400nm). 5. Visible: ~7.5 × 1014 –3.75 × 1014Hz (wavelength 400-800 nm). 6. Near infrared: ~ 3.75 × 1014 –1.2 × 1014Hz (wavelength 0.8-2.5 µm). 7. Infrared: ~1.2 × 1014 –6 × 1012Hz (wavelength 2.5-50 µm). 8. Far Infrared: ~6 × 1012 –3 × 1012Hz (wavelength 50-100 µm). 9. Microwave: ~3 × 1011 –3 × 109Hz (wavelength 0.1-90 cm). 10. Radio frequency: ~3 × 109 –3 × 105Hz (wavelength 0.1-1000 m).

2.3 QUANTUM NUMBERS In the process of solving the Schrödinger wave equation for hydrogen like atoms, the integer n, l and m(ml) are introduced in a logical way. These integers are called quantum numbers. In Bohr’s theory the quantum number n is introduced arbitrarily in the form of the quantized condition. The wavefunction Ψnlm(r, θ, φ) are description of states of the system. They are related to n, l and m. Thus the quantum number themselves may be said to describe the state of the system. The quantum number l and ml are related to the magnitude L of the orbital angular momentum and ml is the z component Lz. The quantum number l gives the magnitude of the angular momentum, ml the orientation of the angular momentum and n gives the quantization of energy. The quantum numbers l and m can be equal only for l = 0. However, the Schrödinger theory was inadequate to explain certain details in the spectrum of hydrogen and of other simple atoms. The quantum numbers n, l and m alone cannot explain certain experimental observations To explain these observations Uhlenbeck and Goudsmit in 1925 assigned a new angular momentum to the electron. According to them, the electron in any state spins about its own mechanical axis i.e., the electron has intrinsic angular momentum denoted by s. This spin angular momentum is in addition to the orbital angular momentum. To explain the experimental

2.4 Laser Systems and Applications    observations, they assigned to the electron a spin angular momentum ħ/2. For this value of the angular momentum, there are only two orientations of the angular momentum. The notion of electron spin s proved to be successful in explaining the various atomic effects. The magnitude of s and z component sz of the spin angular momentum are related to two quantum numbers s and magnetic quantum number ms, by quantization relation which are identical to those for orbital angular momentum.

The values of the quantum numbers n,l,ml,ms are: 1,2,3, … .. 0,1,2,3, … ,

1

0, 1, 2, … , 1 , 2

1 2

2.4 PAULI EXCLUSION PRINCIPLE AND ELECTRON CONFIGURATION In 1925, W. Pauli discovered the fundamental principle that governs the electron configurations of multielectron atoms. His exclusion principle states that “in a multielectron atom there can never be more than one electron in the same quantum state. Each electron must have a different set of quantum numbers n, l, ml and ms”. He established from the analysis of experimental data that the exclusion principle represents a property of electrons and not, particularly, of atoms. The exclusion principle operates in any system containing electrons. With the help of Pauli’s exclusion principle we can assign different quantum states to the electron in a given atom. For an atom, the electrons that have the same principal quantum number n are said to be in the same shell. For a given n, the electrons having the same value of l are said to be in the same subshell. Now we can calculate the maximum number of electrons belonging to the same shell or subshell. (i) The case of a subshell For electrons in a subshell we have same value of quantum numbers n and l. These electrons must differ either by the value of quantum number ml (which can be one of the 2l + 1 integral values between –l and +l) or by quantum number ms (which can take values +1/2 or –1/2). There exists, therefore, 2(2l + 1) distinct quantum states corresponding to the same value of n and l and therefore, there can be 2(2l + 1) electrons in a subshell of quantum number l. The maximum number of electrons in s, p, d and f subshell are therefore 2, 6, 10 and 14, respectively. A subshell containing 2(2l + 1) electrons is said to be complete.

2.5 

Concepts of Atomic and Molecular Spectra

(ii) The case of a shell For electrons in a shell we have same values of quantum number n but different quantum numbers l, ml and ms. The quantum numbers l can have all values from l to n – 1.The maximum number of electrons can be obtained by adding the maximum number of electrons in each subshell. The series may be written as

2( 2l + 1 ) = 2 + 6 + 10 + ....... + 2( 2n − 1 ) = 2n 2 + .............+

(2.5)

s electron (l = 0) +

p electron (l = 1)

Electron with l = n–1

Maximum number

Maximum number

Maximum number

2

6

2(2n–1)

The maximum number of electrons of quantum number n is therefore 2n2. The maximum numbers of electrons in various shells are given in Table 2.1 A shell containing the maximum number of electrons is called a complete shell. We may be tempted to say that electron configuration of any atom follows the general rule Table 2.1 Number of electrons in various shells Shell

N

Maximum number of electrons

K

1

2

L

2

8

M

3

18

N

4

32

1s 2 2s 2 2 p 6 3s 2 3 p 6 3d 10 4s 2 4 p 6 4d 10 4 f 14

(2.6)

But this is not true. The actual order in which the levels must be filled so that the resulting energy is minimum (corresponding to a stable atom) is as follows: 1s 2 2 s 2 2 p 6 3s 2 3 p 6 4 s 2 3d 10 4 p 6 5s 2 4d 10 5 p 6 6 s 2 4 f 14 5d 10 6 p 6 7 s 2 6d 10 (2.7)

A given electron configuration is specified by quantum numbers n and l for each electron but not by ml and ms. Therefore, to each single configuration these

2.6 Laser Systems and Applications    will correspond a certain number of descriptions, differing in the values of ml and ms of each electron. A configuration will therefore have certain degeneracy.

2.5 COUPLING SCHEMES Every electron in an atom has two possible kinds of angular momenta, one due to orbital and other due to its spin motion. The magnitude of the orbital angular momentum vector for a single electron is given by 1  ,

0,1,2, … . . ,

1

(2.8)

Similarly the magnitude of the spin-angular momentum vector for a single electron is 1  ,

1/2

(2.9)

For an electron which has both orbital and spin angular momentum there is a quantum number j associated with the total (orbital plus spin) angular momentum. This is also a vector quantity whose magnitude is given by 1

(2.10)

where j can take the values ,

1, … , |

|

(2.11)

The number of possible values of j is equal to the smaller of the two numbers 2s + 1 and 2l + 1. The angular momentum l and s interact magnetically. If there is no magnetic field, the total angular momentum j is conserved in magnitude and direction. The process of combining two angular momenta into a resultant angular momentum is generally called coupling. For one electron the total angular momentum is obtained in just one way by forming the resultant of the spin and orbital angular momentum. A configuration of two electrons with orbital angular momentum l1, l2 and spin angular momentum s1, s2 allows several way of arriving at the final resultant of the four vectors l1, s1, l2 and s2. The four vectors can be combined in pairs in six possible ways i.e. (l1, s1); (l2, s2); (l1, l2); (s1, s2); (l1, s2) and (l2, s1). Out of these last two are negligible. The spin-orbit interaction (l1, s1) and (l2, s2) are magnetic while the interaction (l1, l2) and (s1, s2) are due mainly to electrostatic force which for (s1, s2) are the so-called exchange type. The magnetic spin interaction is very small, and its influence is noticeable only in the lightest atoms. The two important coupling schemes are LS coupling and JJ coupling:

(i) LS Coupling The approximation in which the residual electrostatic interaction between electrons is assumed to be large compared with the spin orbit interaction is called LS coupling or Russell-Saunders coupling and is important for light atoms.

Concepts of Atomic and Molecular Spectra

2.7 

For two electrons in LS coupling, l1 and l2 form a vector resultant L and according to classical vector model l1 and l2 precess rapidly about their resultant L; similarly s1 and s2 precess rapidly about their resultant S. The precession of L and S about their resultant J is much slower, corresponding to the assumption that the spin-orbit energy splitting is much smaller than the splitting due to electrostatic interaction (with exchange effects). That is to say the classical motion of the orbital system L = l1 + l2 is nearly independent of that of the spin system S = s1 + s2 and L and S represent constant of the motion in this approximation. The classical vector model set up in this dynamical way, is concerned with time averages: the component of l1(l2) perpendicular to L averages to zero over the many cycles of the rapid precession of l1 (l2) about L which take place during one cycle of the slow precession of L about J. Thus only the component of l1 (l2) lying along L is taken into consideration. Similarly for s1 and s2 about S. The S and L couples together to form J = L +S with 1, … . , |

,

|

(2.12)

(ii) JJ Coupling In this coupling, spin-orbit interaction for an electron is large compared with the electrostatic interaction between electrons On the classical vector model , this is obtained by coupling together s and l for each individual electron to give possible values for their resultant j and then coupling the j vectors to give a resultant J . For two electron atom the resultant J = j1 + j2 such that 1, … . . , |

,

|

2.13)

This coupling is prominent in atoms having large atomic number Z.

2.6 SPECTROSCOPIC NOTATION OF ATOMIC STATES Different energy states of electrons are expressed by different quantum numbers. The electrons in an atom are characterized by a principal quantum number n, an orbital angular momentum l, the orientation of the angular momentum vector ml, and a spin quantum number s. The Atomic states are described by the L and S values. The symbol for the state 2S+1

L

(2.14)

where the orbital quantum numbers L = 0, 1, 2, 3, 4, 5, are expressed by the capital letters S, P, D, F, G, H, I. A superscript to the left of the letter indicates the value (2S + 1), that is, the multiplicity of the term due to possible orientation of the resultant spin S. The multiplicity of 1,2,3,4,5 … are pronounced as singlet, doublet, triplet, quartet, quintent,…. For designation of multiplet level we add J value to the term symbol as subscript and term is written as 2 S +1

LJ

(2.15)

2.8 Laser Systems and Applications    where J is the total angular momentum quantum number of atom and takes the values from L + S to L – S . For example, 4F term correspond to the state with L = 3 and 2S + 1 = 4 and therefore S=3/2. For 4F term, L+S = 9/2 and L – S = 3/2. Hence for 4F term there are four multiplet levels with J = 9/2, 7/2, 5/2, 3/2; that is, 4F9/2, 4F7/2, 4F5/2, 4F3/2. The spin multiplicity 2S + 1 in the upper left hand corner indicates the number of these multiplet levels. In order to determine all the possible terms corresponding to a given configuration, the rules for the addition of angular momenta and Pauli exclusion principle must be used. Let us consider some cases

Helium: The helium (He) has electronic configuration 1s2. Helium in its ground state have two 1s electrons. For this system, both the electrons have l = 0. Since n, l and ml are same therefore, according to the Pauli exclusion principle their spins are antiparallel. Thus for 1s2, S and L are zero and hence J = 0. The ground state is accordingly 1S0. Let us assume that one of the two electrons is excited to the 2 p orbits yielding a configuration 1s2p (n = 1 for the s electron and n = 2 for the p electron). Pauli exclusion principle does not apply, as the two electrons are non-equivalent. For the s and p electrons, l = 0 and 1, respectively. This leads to L = 1. The state is therefore, represented by the symbol P. The spins can either be parallel or antiparallel giving S = 1 and 0, respectively. In the first case multiplicity is 2S+1 = 3 (triplet) and for S = 0 the multiplicity is 1(singlet). The J values for triplet are 2,1,0 and for singlet J = 1. The possible terms are therefore, 3 P210 and 1P1. Similarly, one of the two electrons is promoted to various excited configuration and resulting states are given in Table 2.2. Table 2.2 LS terms for some configurations Normal Configuration

Excited Configuration

1s2

-

n

Terms

 

1 1

S0

2

1sns

2, 3, 4, ......

1s2

1snp

2, 3, 4, ......

1

1s2

1snd

3, 4, 5, ......

1

1s2

1snf

4, 5, 6, ......

1

1s

S0 , 3S1

P1 , 3P210 D2 , 3D321 F3 , 3F432

Transition Metals Important members of the transition metal group include chromium and titanium. The chromium atom (atomic number 24) has 18 electrons in orbitals which make up the filled core. The outer configuration includes six electrons in the d and s orbitals: The ground configuration of Cr3+ is 1s22s22p63s23p63d3. It has 3 equivalent d electrons. For d electrons l = 2 and ml = 2,1,0,–1,–2 Terms arising from this configuration are 2P, 4P, 2D(2), 2F, 4F, 2G, 2H. The ground state can be determined by Hunds rule which states that

Concepts of Atomic and Molecular Spectra

2.9 

(i) of the terms arising from equivalent electrons those with the highest multiplicity lie lowest in energy. (ii) of these, the lowest is that with the highest value of L.

There are two further rules for ground term which tell us whether a multiplet arising from equivalent electron is normal or inverted. (i) normal multiplet ( the component with the smallest value of J lies lowest in energy) arise from equivalent electrons when an incomplete subshell is less than half filled, (ii) inverted multiplet (the component with largest value of J lie lowest in energy) arises from equivalent electrons when an incomplete subshell is more than half filled. According to Hund’s rule, the ground state is 4F. Of the others 2G is the lowest excited state.

2.7 SPECTROSCOPIC NOTATIONS OF MOLECULAR STATES In diatomic molecules, the electrons move in strong electrostatic field of nuclei which is cylindrically symmetric about the bond axis. The orbital angular momentum L precesses very rapidly about the direction of electrostatic field and only axial component of L is a constant of motion. The axial component is characterized by the quantum number ML where ML = L, L – 1,........, –L. Since internuclear field is of electrical nature, therefore energy is not changed by the exchange of ML ↔ –ML. The absolute value of ML is designated by the symbol Λ (Fig. 2.2) where Λ

|

|

0,1,2,3, … . . ,

(2.16)

  Fig. 2.2 The vector diagram showing the coupling of L about the electric field along the internuclear axis producing the axial component Λ

All the states are doubly degenerate except Σ state because of ML ↔ – ML symmetry. The individual electronic spins vectorially add up to produce integral or half integral spin quantum number depending on whether there is even or odd number of electrons. If S is total spin, then multiplicity is 2S + 1. The multiplicity of the term is designated by a superscript. The term symbol for a state is written as 2S+1Λ. For Λ > 0, the orbital motion of the electrons produces a magnetic field along the bond axis and S precesses about the magnetic field

Laser Systems and Applications    direction The quantized components MS of S, about the magnetic field direction, have value ħMS. The quantum number MS is designated by the symbol Σ. which can take values S, S – 1, S – 2, ...., – S. Thus, Σ can take 2S + 1 values. For the electronic state Σ (Λ=0), there is no resultant magnetic field and therefore, MS is not defined. These states have only one component, whatever be the multiplicity. Spin-orbit coupling can lift the degeneracy of 2S+1Λ state. The symbol Ω is used to designate the quantum number of z component of total (spin plus orbit) angular momentum in diatomic molecules if coupling between L and S is weak. The quantum number Ω is written as a subscript to the term symbol. The revised term symbol is 2S+1ΛΩ with Ω = Λ + S, Λ + S – 1 ,........, Λ – S. For example, a triplet ∆ state is split by spin orbit coupling into three doubly degenerate levels. 2.10

The symbols for the states are written as follows: Λ 0 1

Symbol of state Σ Π

2 3 4 -

∆ Φ Γ

In classification of molecular electronic state, symmetry properties of electronic functions must be considered besides quantum numbers Σ, Λ and Ω. The reflection operator σ acing twice in succession on electronic wavefunction must give the original wavefunction, that is,

σ 2ψe = ( + 1 )ψ e

(2.17)

The eigenvalue of σ2 is 1. Thus the two eigenvalues of the σ operator are +1 and –1. If ψe+ and ψe– represent the eigenfunction of σ with eigenvalues +1 and –1, respectively, then

σψ e+ = ( + 1 )ψ e+

(2.18)

σψ e− = ( − 1 )ψ e−

(2.19)

The function ψ+ remains unchanged, but ψ– changes sign under reflection operation. The electronic states which are doubly degenerate (Λ > 0) may be distinguished as + or –, for example, ∆+, ∆– etc. The electronic state Σ is not degenerate but can still be classified as Σ+ or Σ–. Homonuclear diatomic molecules also have inversion symmetry through the midpoint of the bond. With the midpoint taken as origin of the cartesian coordinate system and under inversion operation (xi, yi, zi) → (-xi, -yi, -zi) of electronic function twice in succession gives the same electronic function then

i 2ψ e = ( + 1)ψe

(2.20)

Concepts of Atomic and Molecular Spectra

2.11 

The eigenvalues of operator i are therefore +1 and – 1. The wavefunction either remains unchanged or changed sign under inversion. The wavefunction is called even or g (gerade) if it remains unchanged upon inversion. On the other hand if wavefunction changes its sign upon inversion it is called odd or u (ungerade). The symbols g and u are written as subscripts to the term value, for example, Σg , Σu , etc.

2.8 JABLONSKI DIAGRAM The electronic states of a molecules and transitions between them is also represented by a diagram known as Jablonski diagram. In this diagram, the energy levels are arranged vertically and grouped horizontally by spin multipliciuty. In the diagram the radiative transitions are shown by straight arrows while non radiative transitions are shown by squiggly arrows. The vibrational ground states of electronic states is shown by thick lines while the higher vibrational states are shown as thinner lines

2.9 SELECTION RULES The emission or absorption of radiation in atomic and molecular system depends on the probability of a transition between the energy levels, that is the likelihood of a system in one state to another state. The detailed calculation of absolute transition probability involves a knowledge of wavefunction of two states between which transition occurs. It is often possible to decide whether a particular transition is allowed or forbidden (whether the transition probability is non zero or zero). This process leads to selection rules, which allow to determine between which levels transition will give rise absorption or emission of radiation. (i) Atoms In multielectron system the the electric dipole transitions are: 0,

1,

0, 1

J = 0 cannot make transition to another J = 0 state (ii) Molecules In molecules rotational, vibrational and electronic spectra are observed. In pure rotational transitions may be observed in the microwave, millimeter wave or far infrared regions. They are electric dipole transitions and the allowed selection rules are 1 The selection rules for vibration motion are 1 The selection rules for rotational structure of electronic transitions are

2.12

Laser Systems and Applications 

  0, 1 And further selection rules are 0, 1,

0,

0, 1

There are restriction on symmetry character, for example, Σ+ state can undergo transition only into other Σ+ or Π+ state. Further, the transition between g and u are ,

,

2.10 FRANK CONDON PRINCIPLE The characteristic time for an electronic transition is ~10–16 sec. whereas for a nuclear vibration the time has much larger value ~ 10–13 sec. Therefore, it can be assumed that during an electronic transition, nuclei remain fixed. Franck in 1925 explained the intensity distribution in vibronic transitions. Franck Condon principle states that “An electronic transition in a molecule takes place so much rapidly than a vibrational transition that, in a vibronic transition, the internuclear distance can be regarded as fixed and velocity is nearly the same before and after the transition”.

Examples 1. Convert 2000cm–1 to µm. 1 1 2000

10 2

2000 5

10

5

10

5

2. Convert 0.15 nm into Hz. We have 3 10 0.15 10

2

10

3. Convert 9GHz into cm-1. We have

1

3 10 9 10 9 30

/

30 9

0.3

4. The term symbols for particular states of the three different atoms are quoted as 4S0, 2D7/2 and 0D1. Explain why these are erroneous.

Concepts of Atomic and Molecular Spectra

2.13 

For 4S0 state, the total spin S = 3/2 (2S + 1 = 4) hence J must be 3/2. For 2D7/2 , S = 1/2, L = 2 and J values are 3/2,5/2. Thus J cannot be greater than 5/2. For 0D1 state, 2S+1 cannot be zero. 5. The term symbol for a particular atomic state is 4D5/2. What are the values of L,S and J. What is the minimum number of electrons which could give rise to this? The value of L = 2, S = 3/2 and J = 5/2. Since S = 3/2 therefore the minimum number of electrons which could give rise to this state is 3.

Short Questions 1. If value of l is 2, what are the values of ml. Ans. The quantum number ml can take values fro –l to +l. Hence, the values of ml are: 3,2,1,0,–1,–2,–3 2. What are the maximum number of electrons in a state with principal quantum number n =3. Ans. The maximum number of electrons in a state with principal quantum number n is 2n2. Thus for n = 3, the number of electrons would be 18. 3. What kind of interaction is responsible for LS coupling? Ans. Magnetic interaction is responsible for LS coupling. 4. What kind of interaction is responsible for ll and ss coupling? Ans. Electrostatic interaction is responsible for ll and ss coupling. 5. When the wavefunction is called even or g (gerade) Ans. If the wavefunction remains unchanged upon inversion it is called even or gerade (g) 6. When the wavefunction is called odd or u (ungerade) Ans. If the wavefunction changes its sign upon inversion it is called odd or u (ungerade). 7. Is the electric dipole transitions between 11S0 and 23S1 is allowed? Ans. The electric dipole transitions are not allowed as the transitions does not satisfied the selection rules 0,

1,

8. What are the symbols of the states corresponding to L = 6, 7. Ans. The symbols are I and K respectively. 9. What are the symbols of the states corresponding to Λ = 3. Ans. The symbol is Φ.

Laser Systems and Applications    10. What kind of spectra is observed for bodies heated to incandescen. 2.14

Ans. The continuous spectra, which cannot be resolved into lines irrespective of the resolving power of the measuring instruments. 11. What is the magnitude of orbital angular momentum l? Ans. The magnitude is

ħ

1

Problems 2.1

If atom could contain electrons with principal quantum numbers up to and including n = 6, how many elements would there be? 2.2 Consider an atom with an electron configuration of 1s22s22p1. What are the values of l, s, and j, and the corresponding values of the angular momenta L, S and J respectively. 2.3 Find the possible terms for d2 configuration. 2.4 What are the configurations of the electrons of atoms of the following elements (a) Ca (i = 2 0) (b) Cl (Z = 17). 2.5 Explain on the basis of Hund’s rules why t he ground state of carbon is 3P0 and that of Oxygen is 3P2. 2.6 Consider a p electron in a one-electron system. Calculate values of l, s, j 2.7 For diatomic molecules having the following electronic configurations, determine the electronic state term, its multiplet (assuming weak Λ,S coupling) and the total degeneracy of each component of the multiplet (a) S = 1/2, Λ = 1, (b) S = 1, Λ = 2 2.8 Electron shell usually fill as 1s,2s,2p,3s and so on; however after 3p it is found that 4s level fill before 3d level since its energy is actually lower. Knowing this information, show the electronic configuration for neon (Z =10), argon (Z = 18) and titanium (Z = 22). 2.9 Explain Jablonski Diagram. 2.10 What is Frank Condon principle? 2.11 What is band spectra? 2.12 Explain LS and JJ coupling.

      CHAPTER 3   

Introduction to Lasers 

The atoms and molecules can exist only in certain energy states. The state of lowest energy is called the ground state; all other states have more energy than the ground state and are called excited states. There are many excited states. Each excited state has a fixed amount of energy above that of the ground state. Under normal conditions, almost all atoms and molecules are in their ground states. Three types of processes are possible for a two-level atomic system, which are spontaneous emission, stimulated emission and absorption. Let us consider two of these energy levels of an atom as shown in Fig. 3.1. We assume that these energy levels are non degenerate. The atom can make a transition between these two levels by emission or absorption of a photon of energy E = E 2 − E1 = hν

(3.1)

where E2 is the energy of the level 2 and E1 is the energy of level 1 such that E2 > E1 . 2 hν

1 Fig. 3.1 Energy level diagram

3.1 SPONTANEOUS EMISSION Let us assume that the atom is initially at level 2. Since E2 > E1, the atom will tend to decay to level 1 when the corresponding energy difference as given by Eq. (3.1) is emitted. If the energy is emitted in the form of photon of frequency ν, the process called spontaneous emission. The photon is emitted in a random direction with an arbitrary polarization and within a more or less broad range of

3.2

Laser Systems and Applications

frequencies. The probability of such a spontaneous emission is given by Einstein A coefficient defined as A21 = probability per second of a spontaneous jump from level 2 to level 1. It is also called radiative transition rate or radiation transition probability. A21 has a unit of 1/time. Let N2 is the number of atoms per unit volume in level 2. Then the spontaneous jump from level 2 to level 1 is N2A21 per second. The total rate at which jumps are made from level 2 to 1 is

dN 2 = − N 2 A21 dt

(3.2)

There is a negative sign because the population of level 2 is decreasing due to spontaneous emission. In case the level 2 is not the first excited level, the electron can jump to more than one lower level. On integration N 2 = constant exp( − A21t )

If at time t = 0, N2 = N20 , then

N 2 = N 20 exp(– A21t )

(3.3)

The population of level 2 decay exponentially with time. The time in which population falls to 1/e of its initial value is called natural life time of level 2, τ2, where τ2 =

1 A21

(3.4)

This lifetime is also called spontaneous lifetime of the state and is of the order of 10–8 - 10–9 sec. The shorter the spontaneous lifetime, the greater is the probability that spontaneous emission will occur. In certain materials, there are energy levels, which have the spontaneous lifetime of the order of microseconds to a few milliseconds. These levels are known as metastable levels. The probability of transitions involving metastable levels is relatively low.

3.2 STIMULATED EMISSION Consider transition between energy state 2 and 1 of an atom in the presence of an electromagnetic radiation. Let the atom be initially in the upper level 2 and electromagnetic radiation of frequency ν given by Eq. (3.1) is incident on it. Since this radiation has the same frequency as the frequency of transition between levels 2 and 1, there is a finite probability that the incident radiation will force the atom to undergo transition from 2 to 1. In this case, a photon of frequency ν is emitted preferentially in the direction of the incident electromagnetic beam, which is thereby amplified in intensity (Fig. 3.2). This behaviour contrasts markedly with the completely random direction over which spontaneous emission occurs. This is the phenomenon of stimulated emission: it is the emission, which is stimulated by other photons of the appropriate

Introduction to Lasers

3.3

frequency. Stimulated emission has three important properties (i) it is of a very precisely defined frequency (ii) the emitted radiation is in phase with the stimulating radiation (iii) stimulated radiation is coherent. 2 hν

1 Fig. 3.2 Stimulated emission process for two energy levels of an atom

Let the energy density of externally applied electromagnetic radiation at frequency ν be ρ (ν) (energy per unit volume per unit frequency interval, that is, Jm–3Hz–1). The rate of change of population of the upper level due to stimulated emission is proportional to the population of the upper level and the energy density of incident electromagnetic radiation. Therefore,

dN 2 = − N 2 B21ρ (ν ) dt where B21 is Einstein coefficient for stimulated emission and of m3J–1s–2.

(3.5) has units

3.3 ABSORPTION Consider that atom is initially lying in level 1. If this is the ground level, the atom will remain in this level. Let an electromagnetic radiation of frequency ν given by Eq. (3.1) incident on the atom. There is a finite probability that the atom will be raised to level n (Fig. 3.3). The energy difference h ν required by the atom to undergo the transition is obtained from the energy of the incident electromagnetic radiation. The rate of change of population of the upper level 2 due to absorption is proportional to the population N1 of the level 1 and to the incident radiation density ρ (ν), that is

dN 2 = N1B12 ρ (ν ) dt

(3.6)

B12 is Einstein coefficient for stimulated absorption. The coefficients B are also referred to as transition cross section. In stimulated emission and absorption processes, an applied electromagnetic radiation causes the transition between the two energy levels 2 and 1 with the emission or absorption of photon, respectively. In both processes, the atom recoils to conserve the linear momentum. Spontaneous emission process takes

3.4

Laser Systems and Applications

place without an external electromagnetic radiation. The spontaneous emission process is strictly a quantum mechanical effect. Quantum electrodynamics shows that there are fluctuations in the electromagnetic field. Because of zero point energy of the electromagnetic field, these fluctuations occur even when classically there is no field. It is these fluctuations that induce the spontaneous emission of radiation. Thus spontaneous emission corresponds to stimulated emission resulting from this zero point energy of the radiation field. 2

hν 1 Fig. 3.3 Stimulated absorption between two levels 1 and 2

3.4 FLUORESCENCE The power of various lines due to spontaneous transitions versus frequency is plotted in Fig. 3.4. The line shape is continuous and is called fluorescence line shape and its width is called fluorescence linewidth. The width is measured at half its maximum point called full width at half its maximum (FWHM). The fluorescence linewidth can be expressed in terms of frequency or wavelength of two points on the spontaneous emission curve at half the maximum height. We have ∆

Δ

|

|

, , thus can be used and where λ0 is Since Δ wavelength at the centre of emission spectrum . Thus Δ

Δ Now consider Δ where Δ line.

,

| and

| with

is central frequency of the emission

Introduction to Lasers

3.5

Intensity

Frequency

Fig. 3.4 Fluorescence line width

3.5 EINSTEIN COEFFICIENTS Consider a collection of atoms inside a cavity at temperature T. The energy density of radiation within a cavity is given by ρ (ν ) =

8π hν 3 1 3 exp (hν / k BT) − 1 c

(3.7)

Atoms in such a cavity posses many discrete energy levels. Consider two energy levels 2 and 1 with energy E2 and E1, respectively. The energy difference between two levels is hν. Let N2 and N1 be the number of atoms per unit volume in upper level 2 and lower level 1, respectively. The radiation of the cavity may interact with the atoms and a transition between the two states may occur. Spontaneous emission, stimulated emission and absorption can occur between these levels as described in Sec. 3.1. In the Fig. 3.5, the processes are expressed using the Einstein coefficients. Spontaneous emission is independent of energy density of radiation The rate of change of population from Eqs. (3.2), (3.5) and (3.6)

dN 2 dN = − 1 = N1 B12 ρ (ν ) − N 2 B21 ρ (ν ) − N 2 A21 dt dt

(3.8)

At equilibrium

dN 2 dN =− 1 =0 dt dt ⎡N ⎤ ρ (ν) ⎢ 1 B12 − B21 ⎥ = A21 ⎣ N2 ⎦

ρ (ν ) =

A21 N1 B12 − B21 N2

(3.9)

Laser Systems and Applications

3.6

2 B21N2ρ(ν)

B12N1ρ(ν)

A21N2

1 Fig. 3.5 The spontaneous and stimulated emission, absorption processes connecting the levels 1 and 2 For the system in thermal equilibrium, population densities are described by the Boltzmann distribution (assuming non degenerate levels)

⎡ (E − E1 ) ⎤ ⎡ hν ⎤ N2 = exp⎢− 2 ⎥ = exp⎢− ⎥ N1 k BT ⎦ ⎣ ⎣ k BT ⎦

(3.10)

From Eqs. (3.9) and (3.10)

ρ (ν ) =

A21 ⎛ hν ⎞ ⎟⎟ B12 − B21 exp⎜⎜ ⎝ k BT ⎠

(3.11)

comparing Eqs. (3.7) and (3.11) 8π hν 3 c3

1 ⎛ hν exp⎜⎜ ⎝ k BT

⎞ ⎟⎟ − 1 ⎠

=

A21 ⎛ hν ⎞ ⎟⎟ B12 − B21 exp⎜⎜ ⎝ k BT ⎠

we obtain

B12 = B 21

(3.12)

and A21 =

8π hν 3 B21 c3

A21 = B21

(3.13)

8π ν 2 hν c3

This can be described as

A21 = B 21 × No. of modes per unit volume per frequency interval × photon energy (3.14)

Introduction to Lasers

3.7

Equations (3.12) and (3.13) are called Einstein coefficients. Eq. (3.12) indicates that the probability for absorption and stimulated emissions are the same for a transition between states 1 and 2. Thus materials characterized by strong absorptions are also expected to exhibit also a large stimulated emission. Eq. (3.13) shows that a material, in which spontaneous emission does not take place, does not exhibit stimulated emission either. The Einstein relations determine the principal conditions, which should be fulfilled when looking for a material to be used as an active medium in lasers. From Eq. (3.13), the ratio between the number of spontaneous and stimulated emissions increases as ν3 , thus the upper level 2 will be comparatively rapidly depopulated by spontaneous emission and a very efficient pumping mechanism is necessary to achieve laser action. Therefore, it is very difficult to construct very high frequency lasers like X-ray lasers.

3.6 POPULATION OF STATES In an atomic or molecular system there are infinite set of discrete energy levels. The relative population at each energy level is governed by Boltzmann distribution function and for non degenerate energy levels is given by exp

(3.15)

where Ni is the population of atom or molecules at a given energy level i, N0 is the population of atom or molecules at the ground state, (Ei –E0) is the energy above the ground level in Joule, T is absolute temperature in Kelvin, kB is Boltzmann constant. The law predicts an exponential decrease in population of atoms or molecules at higher energies. Consider four consecutive energy levels 0, 1, 2 and 3 of an atom with 0 level as ground level. From Eq. (3.15) exp

(3.16)

exp

(3.17)

exp

(3.18)

From Eqs. (3.16)-(3.18) exp

exp

exp

exp

exp

exp

3.8

Laser Systems and Applications

Or in general the relative population of two levels is exp

(3.19)

Figure 3.6 shows the population of each energy level at thermodynamic equilibrium. From The Eq. (3.19) it is concluded that 1. The population of upper energy level can be increased by raising the temperature as heat energy lift the atoms from low energy levels to higher energy levels. 2. The two population numbers (Ni/Nj) does not depend on the values of the energy levels Ei and Ej, but only on the difference between them: Ei- Ej. 3. For a certain energy difference, the higher the temperature, the bigger the relative population. 4. The relative population can be between 0 and 1. 5. In a thermodynamic equilibrium, the population number of higher energy level is always less than the population number of a lower energy level. 6. The lower the energy difference between the energy levels, the less is the difference between the population numbers of these two levels.

3.7 POPULATION INVERSION At normal temperature, the population of atoms in various energy levels of an atom is governed by Boltzmann distribution as shown in Fig.3.6 If the temperature is increased, the population of atoms in higher energy level also increases. In thermal equilibrium the population of lower energy level is more than the upper energy level. If the population of atoms in a higher energy level i is more than the population of atoms in lower energy level j , then this non equilibrium condition is referred to as population inversion. Figure 3.7 represent a state in which N3 > N2; the state of population inversion. Consider the expression exp In the situation where there exists population inversion, the effective temperature Teff is defined by exp If N3 > N2 (Fig. 3.7) ln Therefore, theoretically, the population inversion corresponds effectively to negative temperature. Population inversion is achieved by supplying the energy. The process of supplying the energy is called pumping.

Introduction to Lasers

3.9

Fig. 3.6 Population at thermodynamic equilibrium

Fig. 3.7

3.8 METASTABLE STATE The atoms and molecules can exist only in certain energy states. The state of lowest energy is called the ground state; all other states have more energy than the ground state and are called excited states. There are many excited states. Each excited state has a fixed amount of energy above that of the ground state and have finite lifetime. The lifetime of the excited states is ≈ 10–8sec. The emission of the light take place when atom jump from higher energy state to the lower energy state and are governed by the electric dipole transitions. Only those transitions are allowed in which multiplicity of the two states involved are not changed (∆S = 0) and ∆l = ±1. If these selection rules are not followed, then probability of the radiative transition is very small resulting in relatively long

3.10

Laser Systems and Applications

lifetime of the upper state. The life time of the upper state may be ~ millisecond to microseconds (10–3sec-10–6sec) due to slow radiative and non radiative decay. The excited states having a lifetime of the order of milliseconds to microseconds are called metastable states. Consider the example of helium atom. The (1s2s) 3 S1and (1s2s) 1S0 state of helium atom are metastable as no electric dipole transition to (1s1s) 1S0 is possible. They are forbidden according to selection rules ∆S = 0, ∆l = ±1. Therefore, they have a large lifetime in comparison with usual lifetime of ≈10-8sec. The lifetime of metastable states of helium are ~104 sec. Existence of metastable state is necessary for achieving population Inversion.

3.9 ACTIVE MATERIAL The material or medium in which population inversion can be created is known as active material or active medium. This may be solid, liquid or gas. The name of the laser is derived from the name of the active medium. A population inverted medium will amplify an incoming wave of correct frequency through stimulated emission. The active material determines 1. 2. 3. 4.

Laser Wavelength. Preferred pumping method. Order of magnitude of the laser output. The efficiency of the laser system.

3.10 PRINCIPLE OF LASER Consider an active medium (gas, liquid or solid). Let E2 be the energy of metastable state and E1 is the ground state. Let both the states are nondegenerate. The active medium is pumped so that population inversion is established such that N2 > N1 (N2 is the population density of metastable state E2 and N1 is the population density of ground state E1). After sometime, when the lifetime of one of the atoms in the metastable state is over, it comes to the ground state by emitting a photon of energy hν = E2 – E1. This is spontaneous emission. This photon now triggers the other atom to come down to the ground state. The photon emitted in this transition will be similar in all respect to the photon emitted by spontaneous emission. As a result two photons will be produced. These photons also trigger the other atoms and four photons will be produced. This process continues and photons remain within the active medium. In this way the number of photon increases as 2n. As a result, the beam of non diverging, highly monochromatic, highly intense light is propagated. It is called laser light. The process of stimulated emission thus makes possible the amplification of light in lasers.

3.11 GAIN Consider a medium in which population inversion has been created. In this medium laser gain can occur. Laser gain (or optical gain) is a measure of how

Introduction to Lasers

3.11

well a medium amplifies photons by stimulated emission. Consider two energy levels 1 and 2 of a given atom. Let N1 and N2 be their respective population density. In Sec. 3.1, while describing the three radiative processes, which is spontaneous emission, stimulated emission and absorption, we have assumed that during transition between these two levels a photon of frequency ν is emitted or absorbed indicating that energy levels are perfectly sharp. From uncertainty principle it means that if atoms were excited to a perfectly well defined energy levels, it would stay there. Yet we assumed that the excited atom decay by the emission of photon. Therefore, energy level picture is to be slightly modified to take into account that atoms do radiate. Real energy levels are not infinitely sharp, they are broadened due to different broadening mechanisms. Different levels have different broadening. An atom in a given energy level can actually have energy within a finite range. The frequency spectrum of the emitted or absorbed radiation is described by a line shape function g(ν). The spectral distribution of atoms per unit frequency is then such that g(ν) dν is the probability of emission or absorption of a photon with frequency between ν and ν + dν. The function is normalised to unity ∞

∫ g (ν ) dν = 1

(3.20)

0

In Sec. 3.2 it is assumed that ρ (ν) is continuous with bandwidth of ρ (ν) much larger than the band of emission or absorption by the atoms. However, if bandwidth of ρ (ν) is much smaller than the corresponding spread expressed by the g (ν), the Eqs. (3.5) and (3.6) change accordingly to reflect this difference. The rate of change of population of level 2 as a result of a monochromatic wave at frequency ν with energy density ρν (in joules/m3) for stimulated emission is dN 2 = − N 2 B21 ρν g (ν ) dt

(3.21)

For absorption dN 2 = N1 B12 ρν g (ν ) dt

(3.22)

The energy density of a radiation field ρν can simply be related to the intensity of a plane electromagnetic wave. If the intensity of the wave is Iν (watt per unit area per frequency interval) then ρν =

Iν c

(3.23)

where c is the velocity of light in the medium. Consider an incident beam of light of intensity Iν passing through a medium of thickness ∆z and cross sectional area A as shown in Fig. 3.8. After emerging from the medium, the output consists of incident intensity plus that added by the

3.12

Laser Systems and Applications

radiative processes, that is spontaneous emission, stimulated emission and absorption. Each of these processes contribute (or subtract) a photon of energy hν. However, we neglect the spontaneous emission contribution, since it is radiated in all directions into a solid angle of 4π and this contribution is very little in the direction of the incident beam. The amount of energy per unit time added is the difference between the number of transitions per unit time from 1 to 2 and the number of transitions per unit time from 2 to 1 within the volume, multiplied by hν per transition. 2

1 dz Fig. 3.8 Elemental change in incident photon flux for electromagnetic wave while travelling a distance dz through the material I [ I ν ( z + dz ) − I ν ( z )] A = [ N 2 B21 − N1 B12 ] g (ν ) ν hν A dz c

I dI ν = ( N 2 B21 − N1 B12 ) g (ν ) ν hν c dz

using Eq. (3.12) dI ν I = ( N 2 − N1 ) B21 g (ν ) ν hν dz c

Substituting the value of B21 from Eq. (3.13) dI ν A21c 2 = g (ν ) [N 2 − N1 ] I ν dz 8π ν 2

(3.24)

dI ν ≅ γ 0 (ν ) I ν dz

(3.25)

where γ0 =

A21c 2 g (ν ) [N 2 − N1 ] 8π ν 2

(3.26)

is called the gain coefficient (m–1) with the subscript 0 indicating that the incident intensity is sufficiently small to cause negligible perturbation on the

Introduction to Lasers

3.13

population of the levels 2 and 1. Sometimes it is convenient to express γ0(ν) in terms of the stimulated emission cross section defined by σ se =

A21c 2 g (ν ) 8π ν 2

(3.27)

and Eq. (3.26) is γ0(ν ) = [N 2 − N1 ] σ se (ν ) = ∆Nσ se (ν )

(3.28)

where ∆N = N 2 − N 1

(3.29)

From Eq. (3.25) dI ν = γ0 (ν ) dz Iν

On integration I ν ( z ) = I ν (0) exp[ γ0 (ν ) z ] = G0 (ν ) I ν (0)

(3.30)

I ν ( z ) = I ν (0) exp[σ se (N 2 − N 1 ) z ]

(3.31)

where Iν (0) is the intensity of the beam as it enters the medium. Iν (z) represents the intensity at some distance z. Go(ν) is the power gain of amplifier of length z. If the value of the exponent is positive, the beam will increase in intensity or amplification will occur. If the exponent is negative, the intensity of the beam will decrease and absorption will occur. Since σse and z are always positive, the amplification will occur if N 2 > N1

(3.32)

or N2 >1 N1

The case of upper level 2 being more populated than the lower level 1 is referred to as a population inversion. Population inversion is a necessary condition for amplification. Population inversion though a necessary condition, but in itself not sufficient for producing a laser. As there are certain losses within the gain medium itself in addition to spontaneous emission. If the frequency of transitions between levels 2 and 1 falls in the microwave region then this type of amplifier is called MASER. The word MASER is an acronym for microwave amplification by stimulated emission of radiation. If the transition frequency falls in far or near infrared, in the visible, ultraviolet or even in X-ray region, the amplifier is called LASER. The word LASER is again an acronym for light amplification by stimulated emission of radiation.

3.14

Laser Systems and Applications

3.12 THRESHOLD GAIN CONDITION In any medium there will be losses due to scattering and absorption by impurities, losses caused by the cavity and tube wall itself. Given all losses, a minimum gain can be calculated that allows laser action. This is threshold gain of the laser medium. The laser threshold is defined as the input power for which the small signal gain is equal to the total cavity losses. Threshold pump power is defined as the pump power for which the laser threshold is reached. Consider a laser oscillator whose basic elements are shown in Fig. 3.9. The oscillator is formed from the laser amplifier by introducing the positive feedback. The positive feedback is provided by the resonator (mirrors). Laser oscillator is composed of two mirrors (e.g. plane mirrors) having reflectivity R1 and R2 and an active material of length L. The region bounded by the mirrors is called the laser cavity. One of the two mirrors is made partially transparent to obtain output beam. For gain R1

L

R2

Output Beam Active Material Fig. 3.9 Scheme of Laser

N 2 > N1

Let the length of the active material is L, the small signal power gain is then given by Eq. (3.30) G = exp[ γ 0 (ν ) L ]

(3.33)

In each pass through the material, the intensity increases by a factor of exp [γ0(ν)L]. At each reflection a fraction (1 – R1) or (1 – R2) of the energy is lost. Starting at one point, the electromagnetic radiation will suffer two reflections before it can pass the same point in the original direction. When the electromagnetic radiation crosses length L it is amplified by a factor of exp [γ0(ν)L]. After reflection from mirror R1, a part of it is lost because of mirror losses and R1 exp [γ0(ν)L] is reflected back. This radiation is again amplified by a factor of exp [γ0(ν)L] as it crosses length L. The radiation reaching the mirror R2 is R1 exp [2γ0(ν)L]. After reflection from mirror R2, a part of this is lost and R2 times R1 exp [2γ0(ν) L] is reflected back in the original direction that is R1 R2 exp[2γ 0 (ν ) L ]

(3.34)

Introduction to Lasers

3.15

In each pass the intensity increases by a factor of exp[2γ 0 (v) L ] . The intensity lost, due to losses within the gain medium, during the same trip will be exp[− 2α L ]

where α describes all loses per unit length in the gain medium with the exception of mirrors themselves. Thus considering the losses also, the intensity obtained after one round trip is R1 R2 exp{[ γ0 (ν0 ) − α ]2 L}

The threshold condition is established by requiring that intensity after reflection from R1 and R2 be equal to the initial intensity R1 R2 exp{[ γ0 (ν0 ) − α ]2 L} ≥ 1 γ 0 (ν ) ≥ α +

1 1 1 ln =α− ln( R1 R2 ) 2 L R1 R2 2L

(3.35)

α does not involve the particular laser level 1 and 2 but occurs at the same frequency as that of the laser beam. Thus threshold refers to a situation where gain coefficients exceed the loss over a small band of frequencies. For α = 0, ∆Nσ se (ν) = ∆N =

⎛ 1 1 ln ⎜⎜ 2 L ⎝ R1 R2

⎛ 1 1 ln ⎜⎜ 2 Lσ se ⎝ R1 R2

⎞ ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

From Eq. (3.29) N 2 − N1 =

⎛ 1 1 ln⎜⎜ 2 Lσ se ⎝ R1 R2

⎞ ⎟⎟ ⎠

(3.36)

Thus threshold is reached when population inversion is given by Eq. (3.36). N2 – N1 of Eq. (3.36) is known as critical inversion. Once the critical inversion is achieved, the oscillations will build up from the spontaneous emissions. The photons, which are spontaneously emitted in a direction along the axis of the laser cavity, initiate the amplification process.

3.13 PUMPING SCHEMES Two Level System Consider a two level system as shown in the Fig. 3.10. At thermal equilibrium the population of level 1 is more than that of level 2. An incoming radiation of appropriate frequency will cause stimulated emission and absorption. Since the population of level 1 is large in comparison with that of 2, therefore, the absorption of radiation predominates over the stimulated emission. As a result of this the population of level 2 increases and a condition is reached when the

3.16

Laser Systems and Applications

population of level 1 and 2 becomes equal. The absorption and stimulated processes will compensate each other and there will be no population inversion. 2

1 Fig. 3.10 A two level system

Three Level System Now consider a three level system as shown in Fig. 3.11. The population of various levels is governed by the Boltzmann distribution. The atoms are in some way raised from level 1 to level 3. Let the atom raised to level 3 decay very rapidly to level 2 and lifetime of level 2 is very large. Thus the atoms rise from level 1 to 3 decay rapidly to level 2. Because the lifetime of level 2 is very large, therefore the atoms start accumulating in level 2 and a population inversion can be achieved between levels 2 and 1.

Fig. 3.11 A three level laser system

Four Level System Figure 3.12 shows a four level laser system. The atoms are raised from level 0 to level 3. The lifetime of level 3 is very small; therefore the atoms, which are transferred to level 3, decay rapidly to level 2. If the lifetime of level 2 is large, a population can be built up in level 2. Since the population of various levels are governed by Boltzmann distribution, the level 1 is nearly empty. Hence a population inversion between levels 2 and 1 is achieved. For maintaining population inversion it is necessary that atoms reaching level 1 should quickly

Introduction to Lasers

3.17

decay to level 0 so that level 1 remains more or less empty. The process by which atoms are raised from level 1 to 3 in three level laser system and from level 0 to level 3 in a four level laser system is known as pumping.

Fig. 3.12 A four level laser system

3.14 RATE EQUATIONS 3.14.1 A Two Level System Consider a hypothetical atom having just two levels 1 and 2 (Fig. 3.13). Let the population of two levels be N1 and N2 per unit volume. We assume that total population of two levels is constant, that is, N1 + N 2 = N 0

(3.37)

∆N = N 1 − N 2

(3.38)

Fig. 3.13 A two level system

combining Eqs. (3.37) and (3.38) N1 = N2 =

N 0 + ∆N 2 N 0 − ∆N 2

(3.39) (3.40)

3.18

Laser Systems and Applications

Consider a monochromatic electromagnetic wave of frequency ν21. This electromagnetic radiation will be absorbed and some of the atoms will go from level 1 to level 2. Excited atoms lose their energy by radiative and non-radiative transitions. The probability for spontaneous emission is A21=1/τsp and for nonradiative transition it is 1/τnr where τsp and τnr are spontaneous emission decay time and non-radiative process lifetime. The overall time decay τ is given by 1 1 1 = + τ τ sp τ nr

(3.41)

The rate of change of population of the upper level is dN 2 N = B12 ρ (ν21 ) N1 − B21 ρ (ν21 ) N 2 − 2 dt τ

(3.42)

If the levels are non degenerate, B21 = B12 and Eq. (3.42) is dN 2 N = B12 ρ (ν21 ) ∆N − 2 dt τ

In the steady state

(3.43)

dN 2 = 0 . From Eqs. (3.40) and (3.43) dt ∆N =

N0 1 + 2 B12 τ ρ (ν 21 )

(3.44)

Thus the population difference between the two levels in a steady state depends on the decay time of the upper level and on the density of incident radiation. B12 ρ (ν21) is the probability per unit time that the atoms are excited to the upper level and is called the pumping rate denoted by Wp. Equation (3.44) becomes ∆N =

N0 1 + 2W p τ

(3.45)

whatever the value of Wp is, ∆N is always positive and hence population inversion is not possible. A two level system is not suitable for lasing action.

3.14.2 Three Level Laser Consider a three level laser system as shown in Fig. 3.14. Assume that all the levels are non-degenerate. Energy levels 1 and 2 are the levels involved in the actual laser transition. The third level is required to achieve a population inversion. The pump lifts atoms from the level 1 to level 3. The level 3 is never populated because there is nearly instantaneous, non-radiative transition from level 3 to level 2. The level 2 is required to be metastable. Thus pump effectively transfers atoms from level 1 to level 2 through level 3.

Introduction to Lasers

3.19

Fig. 3.14 A three level laser system

The rate equation for the time development of N2 is dN 2 = W p N1 + W12 N1 − W21 N 2 − A21 N 2 dt

(3.46)

where Wp is pumping rate per atom, W21 and A21 the stimulated and spontaneous rates from level 2. Because N3 ≈ 0, all the atoms exist in level 1 or 2. The first term in Eq. (3.46) represents the number of atoms arriving at level 2 from level 1 via level 3 by the pumping. It is assumed that all the atoms arriving at level 3 from level 1 are instantaneously transferred to level 2 by non-radiative transitions and level 3 remains empty. The second term represents the induced absorptions from level 1 to level 2.The third term represents the number of stimulated emissions from level 2. The last term represent the spontaneous emission. Because stimulated emission and absorption cross-section are equal, hence W12 = W21. In steady state dN 2 =0 dt

(3.47)

and from which

(W21 + A21 ) N 2 = (W p + W12 ) N1

(3.48)

Dividing both sides by N = N1 + N2 N2 N (W21 + A21 ) = 1 (W p + W12 ) N N

(3.49)

or

n2 (W21 + A21 ) = n1 (W p + W12 )

(3.50)

where n1 and n2 are normalised populations. From Eq. (3.50) n2 =

W p + W12 W21 + A21

n1

(3.51)

3.20

Laser Systems and Applications

Since n1 + n2 = 1

(3.52)

From Eqs. (3.51) and (3.52)

n2 = n1 =

W p + W12 W p + A21 + 2W12 W12 + A21 W p + A21 + 2W12

n2 − n1 = n =

W p − A21 W p + 2W12 + A21

(3.53) (3.54)

(3.55)

For amplification

W p > A21

(3.56)

Thus we must pump to level 2 faster than n2 is depleted by spontaneous emission. For Eq. (3.56) to be satisfied it is desirable that the lifetime of level 2 should be large, pumping source should be intense and material should strongly absorb the pump energy. Suppose a parallel beam of light at frequency νp of pumping light is incident on the surface of an optically thin sample of the material. The power of the pumping light is

P = ni hν p where ni is the number of incident photons. The flux density Fp Fp =

P ni hν p = A A

(3.57)

where A is the area of the sample perpendicular to the incident beam. The pumping rate is by definition Wp =

ni σp A

(3.58)

where σp is the absorption cross section of pump light. Wp is the pump power per unit area. For the threshold condition from Eqs. (3.57) and (3.58) we get

Fp =

W p hν p σp

(3.59)

The threshold condition for population inversion is obtained by putting Wp = A21

Introduction to Lasers

Fp =

A21hν p σp

=

hν p σp τ

3.21

(3.60)

Equation (3.60) is the required power density of the laser. Let us estimate the total power. Suppose the pump power absorbed at νp is equal to P. The total output power is P0 ≈ P

ν 21 νp

because in the steady state, each excitation to the pumping level results in single emission from upper laser level 2. Let us assume that laser action starts when n2 – n1 is slightly greater than zero. Then nearly half of the atoms are in the upper level and this condition will persist as long as pumping is continued. Let N1 + N2 = N0 and N1 ≈ N0/2. The total pumping power absorbed is P=

N 0V W p hν p 2

where V is the volume of the active material. For threshold condition Wp = A21, therefore P=

N 0Vhν p N 0V A21 hν p = 2 2τ

3.14.3 Four Level Laser Let us consider a laser operating on four level scheme as shown in Fig. 3.15. There is only one broad level (level 3). Level 0 is the ground level and levels 1,2 and 3 are excited levels of the system. Atoms from level 0 are excited by a pump to level 3 from which the atoms decay rapidly through some non-radiative transitions to level 2. Level 2 is a metastable level having a long lifetime (~10–3s). The level 2 forms the upper laser level and level 1 forms the lower laser level. The lifetime of the level 1 is very short. Therefore, the incoming atoms from level 2 relax down immediately from level 1 to level 0 ready for being pumped to level 3. If the rate of relaxation of atoms from level 1 to level 0 is faster than the rate of arrival of atoms into level 2, one can obtain population inversion between levels 2 and 1 even for very small pump power. Let N0, N1, N2 and N3 represent the population per unit volume of the levels 0, 1, 2 and 3, respectively. The rate of change of population of level 3 is dN 3 = B03 ρ (ν03 ) N 0 − B30 ρ (ν03 ) N 3 − N 3T32 dt dN 3 = W p ( N 0 − N 3 ) − N 3T32 dt

(3.61) (3.62)

3.22

Laser Systems and Applications

Here we have ignored the relaxation from level 3 to 1 and 0. Wp = B03 ρ (ν03) and B03 = B30. The first term represents the net rate of stimulated transitions between levels 0 and 3 caused by the pump. T32 is the relaxation rate (both radiative and non radiative) from level 3 to 2. The rate equation for N2 would be dN 2 = N 3T32 + Wl ( N1 − N 2 ) − T21 N 2 dt

(3.63)

The first term represents the rate at which atoms are transferred from level 3 to 2, the second term represents the rate of stimulated transitions from level 1 to 2 because of the presence of laser radiation. Third term represents the rate of loss of atoms from level 2 to level 1, through spontaneous emission. Transitions from level 2 to level 0 are ignored. The rate equation for N1 would be dN1 = Wl ( N 2 − N1 ) − T10 N1 + T21 N 2 dt

(3.64)

Fig. 3.15 A four level laser system

Similarly, the rate equation for N0 would be dN 0 = T10 N1 + W p ( N 3 − N 0 ) dt

(3.65)

N = N 0 + N1 + N 2 + N 3

(3.66)

Let

From

dN = 0 we have dt

Introduction to Lasers

3.23

dN 0 dN1 dN 2 dN 3 + + + =0 dt dt dt dt

(3.67)

dN 0 dN1 dN 2 dN 3 = = = =0 dt dt dt dt

(3.68)

At steady state

From

dN 3 =0 dt

N3 = From

Wp N0

(3.69)

W p + T32

dN 2 = 0 we have dt

N2 =

Similarly, from

⎡ ⎤ Wp N0 1 T32 ⎥ ⎢Wl N1 + T21 + Wl ⎣⎢ W p + T32 ⎦⎥

(3.70)

Wl + T10 N1 Wl + T21

(3.71)

dN 1 = 0 we get dt

N2 =

From Eqs. (3.70) and (3.71)

N1 = N2 =

W pT32 (W p + T32 )T10

N0

(Wl + T10 )W pT32 (Wl + T21 )(W p + T32 )T10

(3.72)

N0

N 2 Wl + T10 = N 1 Wl + T21

(3.73) (3.74)

In order to obtain a population inversion, that is, N2 > N1 we must have T10 > T21

(3.75)

that is, the rate at which atoms relax from level 1 to level 0 must be greater than the rate at which atoms relax from level 2 to level 1. Under such a condition, the creation of population inversion between level 2 and level 1 is independent of the pumping power, but the magnitude of population inversion depends on Wp. At and below threshold of oscillations Wl ≈ 0 and population difference between levels 2 and 1 is

3.24

Laser Systems and Applications

N 2 − N1 =

⎛ T10 ⎞ ⎜⎜ − 1⎟⎟ (W p + T32 )T10 ⎝ T21 ⎠ W pT32 N 0

(3.76)

From Eqs. (3.66), (3.69), (3.72) and (3.73)

N0 =

N [W p + T32 ]T10T21 2W pT21T10 + W p T32T10 + W pT32T21 + T32T21T10

(3.77)

From Eqs. (3.76) and (3.77)

N 2 − N1 =

W pT32 (T10 − T21 ) 2W pT21T10 + W pT32T10 + W pT32T21 + T32T21T10

(3.78)

In the approximation T10 > T21; T32 > T21 Wp

Wp

N 2 − N1 T21 T21 = ≅ 2W p W p W p Wp N + + +1 +1 T32 T21 T10 T21

(3.79)

Since N >> N2 – N1, we have for threshold pumping

⎛ W pt ⎞⎛ N 2 − N1 ⎞ W pt ⎟ ⎜ ⎜ T + 1⎟⎜⎝ N ⎟⎠ = T 21 ⎠ ⎝ 21 W pt ≅

N 2 − N1 T21 N

(3.80)

3.15 EFFICIENCY Consider an energy level diagram of a laser system as shown in Fig. 3.10. It is assumed that only level 2 which is upper laser level having energy E2 is only pumped and the lower laser level 1 with energy E1 decays instantaneously so that the population of level 1 tend to zero. Further stimulated emission rate from level 2 to level 1 overwhelms all other decay processes of level 2. In a pumping event, external power supply gives R2E2 watts per unit of volume where R2 is the rate. However stimulated emission can remove at the most R2 photons per second per unit volume with the energy (E2-E1). The maximum efficiency η of this laser is therefore R ( E 2 − E1 ) E 2 − E1 Pout (3.81) = η= = RE 2 E2 Pin

3.16 METHODS OF OBTAINING POPULATION INVERSION Since all laser emission involves radiation from excited states, the energy must be supplied to these atoms to produce the excited states. Methods of pumping,

Introduction to Lasers

3.25

irrespective of the system of levels (three level or four level) involved and of whether lasing is to be pulsed or continuous wave (CW) falls into following categories: (a) Electrical (b) Optical (c) Chemical reaction (d) Thermal (e) Nuclear In view of the wide variety of techniques we will discuss only electrical pumping and optical pumping.

Electrical Pumping Electrical pumping is accomplished by means of a sufficiently intense electrical discharge. In an electric discharge electrons are produced by ionization in strong electric field. The electrons achieve very high speed of 106 to 107 m/s. These electrons on collision transferred energy to atom or molecules of gas and raised them to excited states M + e→ M *+e where M and M* represent the atom in the ground and excited states, respectively. Such a process is called a collision of first kind. This method is preferred in gaseous lasers which contain one gas species. Argon ion laser uses this technique. For a gas consisting of mixture of gases say X and Y, X serve only to transfer energy from electrons to Y. X is excited to X*, a long lived metastable state, by electron impact. The X is in excited state and Y is in ground state as shown in Fig. 3.16. If the level Y to be pumped is of similar energy to that of X*, then there is an appreciable probability that after collision X* will transfer its energy to Y to raise it to excited state. The process is denoted by X * + Y → X + Y * + ∆E where the energy difference ∆E between X* and Y* will be added or subtracted. This method of transfer of energy is called resonant transfer or collision of second kind. In Helium-Neon laser, CO2 laser this technique of pumping is used.

Optical Pumping For optical pumping, the excitation involves the transfer of the energy to the system from a high intensity light source. The incident photon energy hν must be equal to the energy differences between the excited states and normal states. We can express

3.26

Laser Systems and Applications

Fig. 3.16 Energy transfer between X and Y

optical pumping as hν + X → X*, where X is the atom at normal state, X* is the corresponding atom at excited state. So if there are lasers whose light wavelengths are within the absorption bands of the active medium, we can use these laser lights for pumping. Since the bandwidth of laser light is very narrow, the pumping efficiency can be very high. Optical pumping can be accomplished by many different light sources including discharge lamps and lasers. Discharge lamps used for laser pumping can be grouped into two categories: arc lamp and flash lamps. Pumping with inert gas flash lamp usually xenon at a pressure of about 100 torr results is a pulsed laser, the repetition rate being that of pumping source. CW optical pumping may be achieved by a continuously acting krypton or high pressure mercury lamp. The second technique under optical pumping is through diode lasers .The lasers based on this type of pumping are known as Diode Pumped Solid State Lasers (DPSSL) or sometimes the all-solid state lasers. Because optical pumping is a resonant process, the wavelengths of the pumping diode lasers must be within the absorption bandwidth of the active medium to be pumped, the nearer to the absorption peak wavelength the better.

The advantages and disadvantages of discharge lamp pumping Advantages (i) The cost of operation is less in comparison with the laser pumping. (ii) High peak power of the lamp can be generated (iii) quite immune to voltage and current fluctuations Disadvantages (i) the lifetime is usually small, normally up to a few thousand hours. (ii) Flash lamp has a broad emission spectrum as a result most of the optical energy emitted by the flash lamp is wasted as heat. This heat is to be properly removed, that is, laser is to be cooled. With diode laser most of the absorbed power is used for population inversion. (iii) Electrical to optical efficiency is low. (iv) Electrical power supply requires high voltages whereas diode laser power requirement is low. (v) Lamp brightness is low (vi) With flash lamp pumping absorption efficiency is ~0.17 while diode laser absorption efficiency is

Introduction to Lasers

3.27

~0.90-0.98 (vii) Overall pumping efficiency of discharge lamp is about 1/7 of diode laser pumping. Advantages of Diode Pumping (i) The lifetime of laser diodes is long compared with that of discharge lamps. (ii) the pump laser, its power supply and cooling system is compact (iii) small power supplies are needed (iv) A high electrical to optical efficiency. (v) The optical bandwidth of diode laser is narrow thus it is possible to pump certain transition of laser active ion without losing power in other spectral region. (vi) Diode pumping makes it possible to use a very wide range of solid state gain media for different wavelength region. (vii) The end pumping of laser provide very good overlap of laser mode and pump region, leading to high beam quality and power efficiency. The main disadvantage of diode pumping, as compared to lamp pumping, is the higher cost per watt of pump power Pump Geometry The pumping geometries can be broadly classified into the following categories depending on the shape of the active material and the type of pump source used are: 1. Side-pumped 2. End-pumped 3. Face pumped 4. Edge Pumped. Side or transverse-and end-or longitudinal pumping refers to the orientation of the pump beam relative to the direction of the laser beam in the gain medium. Face- and end-pumped slabs and disks differ in the direction in which waste heat is removed; and in the resulting directions and magnitudes of the thermal gradients relative to the pump and output laser beam directions. Side pumped In this arrangement the laser rod and pump lamp are placed at each focus of the highly reflective elliptical cylinder (Fig. 3.17). This configuration is based on the geometrical theorem that rays originating from one focus of an ellipse are reflected into the other focus. For large eccentricity of the ellipse, the laser rod and lamp are separated by a fairly large distance. For small eccentricity of the elliptical cylinder, the laser rod and lamp are close together. If the elliptical Active Material Flash Lamp

Fig. 3.17

33.28

Laser Syystems and Appliications

ccylinder closelly surrounds the t lamp and rod r then it is known as a cllose-coupled eelliptical geom metry. A furtther improvem ment of this arrangement is a double eelliptical cylinnder (Fig. 3.18 8). Here two lamps l are useed in order to increase the total pumping flux to the rod d. In side pumpping with diod des, the diode array are placced along the length l of the ggain material and a the pumping is transverse to the laserr beam

Fig. 3.18

E End Pumped IIn this pumpinng, the light beam b is directted in the lonngitudinal direection of the laser gain meddium. The pum mping beam enters e the region at the endd of the gain m medium as shoown in the Fig g. 3.19. Fiibre lens Laser

Active A M Material Diode Arrays

Output Mirror

Focussin ng Lens Fig. 3.19 3

F Face Pumped Disks IIn this, the suurfaces of thee active materrial are set att Brewster anggle with the ppumping lampps and their co onnected systeem of reflector placed abovve and below ddisk surface inn such a way as not to interfere with thhe direction of the optical bbeam (Fig. 3.1120). The lam mp radiation inncident on thee disk and is transverse t to thhe beam direcction.

Introduction to Lasers

3.29

Beam

Brewster Angle Pump Fig. 3.20

Edge Pumped In this arrangement the slab is pumped from the edge as shown in Fig. 3.21. Electrical pumping is typically used in most gases and semiconductor lasers. On the other hand, optical pumping is often used in liquids (dye) lasers and crystalline solid state lasers. In solids and in liquids, electrons cannot easily be accelerated by electric field to excite the laser energy levels of impurities species. The line broadening mechanisms in solids and liquids produces an appreciable broadening. Therefore, one is usually dealing with pump band rather than levels. These bands can therefore absorb a sizeable fraction of light emitted by flash lamps whose energy occurs over a broad wavelength region. However, non laser optical pumping is not feasible in gaseous systems. The gaseous system does not have in general broad absorption bands.

Pump

Pump

Cooling

Cooling Fig. 3.21

Chemical Chemical compounds store large amount of energy. The energy may be released by exothermal chemical reaction. The exothermal chemical reaction follow with the release of energy and this energy is used for population inversion. Chemical

3.30

Laser Systems and Applications

lasers use energy of substitution reaction which produces molecules in their excited states. Consider the reaction between hydrogen and fluorine. They can be ignited by an electric spark or by chemical means. In the reaction between hydrogen molecules and fluorine atoms, the highly active fluorine reacts with the hydrogen molecule (H2) to create free hydrogen plus a molecule of HF*. The * denotes a molecule in the excited state. The free hydrogen reacts with the fluorine molecule: H2 + F → HF* + H H + F2 → HF* + H These reactions will continue as long as there are molecules of fluorine and hydrogen. Thus, gas flow into the laser cavity creates continuous laser emission. The examples of chemically pumped lasers are HF, DF, HCl.

Thermal In gas dynamic laser, cooling of hot gases by adiabatic expansion is used to produce population inversion. In this technique hot gases are expended through specially shaped nozzles from a high pressure, high temperature chamber into a low-pressure chamber, thus creating a highly non-equilibrium state in the resonator. Due to adiabatic expansion, the upper level population is frozen and the lower level population is depleted, resulting in strong population inversion. Carbon dioxide gas dynamic laser (CDGDL) is an example of a thermally excited laser.

Nuclear Pumping The nuclear pumping sources are particles (proton, alpha particles etc.) or gamma rays resulting from nuclear reactions (fission, fusion, radioisotope). This may occur in either a reactor or a nuclear explosion, thus NPLs can be grouped into two broad categories: reactor pumped lasers and nuclear device pumped lasers. Both types provide direct conversion of nuclear energy into directed optical energy. Nuclear pumped lasers (NPL) are gas lasers excited directly or indirectly by high energy. The examples of nuclear pump lasers are: He-Zn laser, He-Ar laser, Xe2 laser etc.

3.17 LASER RESONATORS A population inverted medium will amplify an incoming wave of correct frequency through stimulated emission. A laser in the visible region is very noisy because of increasing importance of spontaneous emissions. According to Eq. (3.13) the signal to background ratio will be reduced by cube of frequency. Therefore, lasers are rarely used for light amplification. Lasers are normally used as light oscillator. An amplifier becomes an oscillator if feedback is introduced. This can be achieved by placing the laser medium in a resonator consisting of

Introduction to Lasers

3.31

two mirrors (Fig. 3.9). A pair of resonator facing each other can in general form a resonant cavity. The requirements for the resonator are: (i) it should contain a sufficiently large amount of active material and (ii) it should permit amplification at only a narrow band of frequencies. Laser resonators are usually open, that is, no lateral surface is used. The dimensions of resonators are much larger than the wavelength of the laser because the resonators are open means that there would be some losses due to diffraction losses. In this section we will limit ourselves to passive resonators (which have no gain or loss). The most widely used laser resonators have either plane or spherical mirrors of rectangular or circular shape separated by some distance.

3.17.1 Plane -parallel (or Fabry-Perot) Resonator Plane parallel resonator consists of two plane mirrors set parallel to each other as shown in Fig. 3.22. To a first approximation the modes (the possible standing waves in the cavity) of this resonator can be thought of as the superposition of two plane electromagnetic waves propagating in opposite direction along the cavity axis. For electric field of electromagnetic wave to be zero on the two mirrors, the cavity length must be integral number of half wavelength, that is, L=

nλ 2

Fig. 3.22 Plane parallel resonator

ν=

nc 2L

(3.82)

The frequency difference between two consecutive modes is from Eq. (3.82) ν=

c 2L

(3.83)

This difference is referred to as the frequency difference between two consecutive longitudinal modes. The word longitudinal is used because the n indicates the number of half wavelengths of the modes along the resonator. This type of arrangement has certain advantages and disadvantages. There is optimal use of all the volume of the active material and there is no focusing of the laser

3.32

Laser Systems and Applications

beam. However, this arrangement has high diffraction losses and is highly sensitive to misalignment and therefore difficult to operate. This type of arrangement is used in pulsed lasers which needs the maximum energy.

3.17.2 Concentric (or spherical) Resonator This consists of two spherical mirrors with the same radius R separated by a distance L = 2R so that the centres are coincident (Fig. 3.23). In this case the resonance frequencies are given by Eq. (3.82). In this arrangement of mirrors, focusing of the beam is at the centre of the cavity. This has low sensitivity to misalignment of the mirrors and diffraction losses are low. However, there is a limited use of volume of active material. There is maximum focusing of the laser beam inside the optical cavity, which may cause electric breakdown or damage to the optical elements. This sort of arrangement is used in the optical pumping of dye lasers.

Fig. 3.23 Concentric (spherical) resonator

3.17.3 Confocal Resonator This consists of two spherical mirrors with the same radius of curvature R separated by a distance L as shown in Fig. 3.24. In this case foci of two mirrors coincide and centre of curvature of one mirror lie on the surface of another mirror. The resonant frequency cannot be readily obtained from geometrical optic consideration. In this arrangement of mirrors, there is little sensitivity to misalignment, has low diffraction losses, no high focusing inside the cavity and there is a medium use of volume of active material.

3.17.4 Resonator using a Combination of Plane and Spherical Mirrors Examples of these resonators are shown in Figs. (3.25) (hemiconfocal) and (3.26) (hemispherical). The hemispherical arrangement has low sensitivity to misalignment and low diffraction losses. In He-Ne laser this configuration is used.

Introduction to Lasers

3.33

3.17.5 Generalized Spherical Resonator This is formed by two spherical mirrors of the same radius of curvature separated by a distance L such that R < L < 2R. This is in between confocal and concentric resonator. The resonators can be identified as stable or unstable. A resonator is stable if the oscillatory beam is bouncing back and forth between the two mirrors without much loss due to finite size of the mirrors as shown e.g. in Fig. 3.27. On the other hand in unstable resonators, the oscillating beam spreads out of the cavity after a few traversals as shown in Fig. 3.27. Stable cavity design allows the beam to oscillate many times inside the cavity to get high gain. The focal properties and directionality are important. For high power laser, unstable cavities are often used. Laser output comes from the edge of the output mirror. Unstable cavities are suitable for high gain per round trip laser system, which do not require large number of oscillations between the mirrors. In such a cavity, no standing wave pattern is created inside the cavity. The radiation does not move in the same path between the mirrors. All the power inside the cavity is emitted out of laser, not just a small fraction of it. The laser emission is emitted out of the laser around the edge of the convex mirror. This arrangement is used in high power lasers. It is customary to define dimensionless stability parameters g1 and g2 by the equations g1 = 1 −

L R1

(3.84)

g2 = 1−

L R2

(3.85)

where R1 and R2 are the radii of the curvature of the mirrors separated by a distance L. The sign of radius of curvature is taken to be positive for concave mirrors and negative for convex mirrors.

Fig. 3.24 Confocal resonator

3.34

Laser Systems and Applications

Fig. 3.25 Hemiconfocal resonator

Fig. 3.26 Hemispherical resonator

Fig. 3.27 Unstable resonator

Thus stability criterion for lasers is

⎛ L ⎞⎛ L ⎞ ⎟⎟ ≤ 1 0 ≤ g1 g 2 = ⎜⎜1 − ⎟⎟⎜⎜1 − ⎝ R1 ⎠⎝ R2 ⎠

(3.86)

This condition can be expressed in the form of stability diagram as shown in Fig. 3.28. The clear regions are the regions for which g1 g2 > 1 and cavity is unstable. For the shaded region g1 g2 < 1 and the cavity is stable. The limiting case g1 g2 = 1 is a hyperbola. Stable resonator lies between the two branches of the hyperbola and the axes; unstable resonator lies outside the two branches. For shaded region, Eq. (3.86) is satisfied and the cavity is stable. Three particular points are of interest. These points represent cavities on the verge of instability: R1 = R2 =

L 2

(symmetric concentric)

R1 = R2 = L

(confocal)

R1 = R2 = ∞

(plane parallel )

Introduction to Lasers

3.35

All three of these are on the edge of stability in the diagram and can become extremely loosy for slight deviation from the shaded region. In short the plane parallel and concentric resonators are sensitive to mirror misalignment. The concentric resonators produce a very small spot on the resonator centre. Confocal resonators typically give a spot size that is too small for effective use of all available cross section of the laser medium. The most commonly used laser resonator uses two concave mirrors of large radius of curvature or a plane mirror and a concave mirror of large radius. The resonator gives a spot size somewhat larger than the confocal resonator and a reasonable stability against misalignment.

Fig. 3.28 Stability diagram for two mirrors with radii of curvature R1 and R2

3.18 ELEMENTS OF LASERS Four functional elements are necessary in laser to produce a coherent light by stimulated emission: Active medium, pumping, resonator or feedback mechanism and output coupler. Figure 3.29 shows these parts. (i) Active medium It is a material (gas, liquid or solid) in which population inversion can be created, that is, where most atoms are molecules are in the upper excited state than in some lower energy state. The two states chosen for the lasing transition must possess certain characteristic (a) atoms or molecules must remain in the upper laser level for a relatively long time to provide more emitted photons by stimulated emission of radiation than by spontaneous emission, that is, upper

3.36

Laser Systems and Applications

laser level must be metastable (b) there must be an effective method of pumping atoms from ground state into the upper laser level in order to increase the population of the higher energy level over the population of lower energy level. The active medium may be a gas, a liquid, a solid material or a junction between two slab semiconductors. For example ruby is solid, carbon di oxide is a gas, dye solution is a liquid, and p-n semiconductor junction of GaAs. Pumping R2

R1

Active Material

Output beam

L Fig. 3.29

(ii) Pumping It is a source of energy that excites or pumps the atoms or molecules of the active medium from lower energy levels to higher energy level in order to create the population inversion. Various kinds of pumping are used mainly electrical, optical, chemical, thermal etc. In gas lasers and semiconductor lasers the electrical pumping is used while in liquid and solid lasers optical pumping is used (iii) Resonator It consists of two mirrors one at each end of the active material, aligned in such a way that they reflect the coherent light back and forth through the active medium for further amplification by stimulated emission of radiation. (iv) Output Coupler The output coupler allows a portion of the laser light contained between the two mirrors to leave the laser in the form of a beam. One of the mirrors of the feedback mechanism allows some light to be transmitted through it at the laser wavelength.

3.19 CONTINUOUS WAVE (CW) AND PULSED LASERS Continuous Wave Lasers A laser whose output power is constant over time (Fig. 3.30) is known as CW laser. For continuous wave operation it is required for the population inversion

Introduction to Lasers

3.37

of the gain medium to be continually replenished by a steady pump source. In some active materials this is not possible. In some active materials pumping is needed at a very high continuous power level which may not be practical or destroy the laser by producing excessive heat. Such lasers cannot be run in CW mode. The examples of CW lasers are He-Ne, Argon ion, Carbon di- oxide etc.

Energy (Watt)

Time Fig. 3.30

Pulsed Lasers A laser whose power appears in pulses of some duration at some repetition rate (Fig. 3.31) is known as pulsed laser. Some lasers are simply pulsed as they cannot be run in CW mode. The examples of pulsed lasers are Ruby, Excimer, Nitrogen, Copper vapour etc. These lasers can never be operated in CW mode.

Energy (Joule)

Time Fig. 3.31

Pulsed lasers operation can be achieved, if the active material is pumped with source that is itself pulsed either through electronic charging in the case of flash lamp, or another laser which is already pulsed. In any laser the probability that a particular system will lase a CW mode is determined mainly by the lifetime of upper laser level and the lower laser level. If the upper laser level have a relatively long lifetime (metastable state) compared to lower laser level, a CW operation is probable. On the other hand if the lower laser level has a relatively long lifetime, atoms in that lower energy state remain there for larger period. This gives a good probability to atoms of lower laser levels to absorb photons and thus destroying the population

3.38

Laser Systems and Applications

inversion. In this situation a pulse laser may be possible if the upper laser level is filled quickly and preferentially over the lower energy level. However, the population of lower laser level will eventually exceeds that of the upper energy level and laser action will stop. The energy of the pulsed laser is equal to average power divided by the repetition rate. The pulse having large energy can be obtained by lowering the rate of pulses so that more energy can be built up in between the pulses. However, some time it is desired to have a large peak pulse power rather than the energy in the pulse. Pulse peak power can be increased by creating pulses of shortest possible duration utilizing the technique such as Q-switching. The optical band width of the pulse cannot be narrower than the reciprocal of the pulse width. In the case of extremely short pulses, that implies lasing over a considerable band width. The lasing medium in some dye lasers produces optical gain over a wide bandwidth making a laser possible which can thus generate pulses of light as short as few femtosecond (10–15s). Short pulses can be obtained by: (i) Q-switching In Q-switched lasers, the population inversion is allowed to build-up by introducing loss inside the resonator which exceeds the gain of the medium thus effectively reducing the quality factor Q of the cavity. Then after the pump energy stored in the laser medium has approached the maximum possible level, the introduced loss mechanism (often electro optical or accousto optical element) is rapidly removed allowing lasing to begin which rapidly obtains the stored energy in the gain medium. This results in a short pulse incorporating that energy and thus a high peak power or giant pulse. (ii) Mode locking A mode-locked laser is capable of emitting short pulses of the order of ten of picoseconds down to less than 10 femtoseconds. These pulses repeat at the round trip time (that is 2L/c if L is the separation between the mirrors of the resonator) that is time that it takes light to complete one round trip between the mirrors. (iii) Pulsed pumping Pulsed laser operation can also be achieved if the active material is pumped with a source that is itself pulsed, either through electronic charging in the case of flash lamp or another laser which is already pulsed. Pulsed pumping was used with dye lasers where the inverted population lifetime of a dye molecule was so short that a high energy fast pump was needed. Pulsed pumping is also required for three level laser in which the lower energy level rapidly becomes highly populated preventing further lasing until those atom relax to the ground state.

3.20 CLASSIFICATION OF LASERS Lasers systems can be divided on the basis of 1. Density of gain medium

Introduction to Lasers

3.39

2. 3. 4. 5.

The state of matter of the active medium: solid, liquid, gas. On the basis of output of laser beam, continuous wave (CW) or pulsed On the basis of region of wavelength: X-ray, Ultraviolet, Visible or Infrared. The number of energy levels which participates in the lasing process: three levels or four levels 6. The pumping method of the active medium: Optical pumping, Electrical pumping etc.

1. Density of Gain Medium The laser systems may be grouped on the basis of density of gain medium: low gain and high gain medium. Low Gain Medium: A gas laser comes under this category. Example of low gain medium laser are: Neutral atom lasers (for example He-Ne laser, He-Cd laser), Ionic lasers for example Argon ion, Krypton ion lasers, Molecular lasers for example carbon di-oxide laser, nitrogen laser, excimer laser (for example XeF, XeCl etc.), Far infrared lasers, chemical lasers (for example HF laser), X-ray lasers, Free electron lasers , Atom lasers (for example, rubidium atom) etc. High Gain Medium: This can be further classified into solid state lasers, semiconductor lasers and liquid lasers Solid State Lasers Solid state lasers include lasers based on paramagnetic ions (for example ruby laser, Nd:YAG, Ti:Sapphire, Cr3+ in BeAl2O4 etc.), organic dye molecules and colour centres in crystalline or amorphous hosts (for example F2+ centre in LiF). Liquid Lasers In this kind of lasers the active medium is in the form liquid. The examples are dye lasers Semiconductor Lasers

2. State of Matter The lasers may be divided on the basis of state of matter, whether they are solid, liquid or gases Solid State Lasers The examples are Ruby laser, Nd:YAG laser, Nd: Glass laser, Ti: sapphire etc.

3.40

Laser Systems and Applications

Gas Lasers They are further classified as neutral atom lasers (for example He-Ne); ionic laser (for example argon ion) and Molecular lasers (nitrogen, carbon di oxide, excimer gas lasers etc.), chemical lasers (for example HF laser) Liquid Lasers In this kind of lasers the active medium is in the form liquid. The examples are dye lasers

3. On the basis of nature of output of laser beam A laser may be classified as operating in either continuous wave (CW) or pulsed mode, depending on whether the power output is essentially continuous over time or whether its output takes the form of pulses of light. The examples of continuous wave lasers are: He-Ne laser, Nd:YAG laser, Argon ion laser. The examples of pulsed lasers are: Ruby laser, Nitrogen laser, Excimer laser, Copper vapour laser. Some of the lasers can operate both in CW as well as made to operate in pulsed mode.

4. Region of Wavelength The lasers can also be classified on the basis of spectral range of wavelength. (a) X-ray region for example Cr21+ (b) Ultraviolet: examples are Excimer laser, Nitrogen laser (c) Visible Region: for example Ruby, He-Ne, Dye, etc. (d) Infrared region: for example Nd:YAG laser, Carbon di oxide laser etc. The range of wavelength for a laser cannot be accurately fitted to a spectral region as some lasers emit wavelength in visible as well as in the infrared region for example He-Ne laser, some lasers emit radiation in the visible as well as ultraviolet region, for example He-Cd laser .

5. The number of energy levels which participates in the lasing process The lasers can also be classified depending on the number of energy levels which participates in the lasing process. This can be grouped into two categories, three level lasers and four level lasers. The example of three level lasers are ruby laser, copper vapour lasers etc. while the example of four level lasers are He-Ne, Argon ion laser, Dye laser, Nd:YAG laser, CO2 laser etc.

6. The pumping method of the active medium The lasers can also be classified on the basis of pumping used Electrical pumping: the examples are He-Ne laser, CO2 laser, Argon ion laser, Excimer lasers etc.

Introduction to Lasers

3.41

Optical Pumping: ruby laser, Nd:YAG laser , Dye laser etc. Chemical pumping: the examples are HF Thermal pumping: the examples is Carbon dioxide gas dynamic laser (CDGDL) Heavy ion pumping for example 3He-Ar, Ar-Xe, Ne-Xe-NF3 Free electron pumping

3.21 PROPERTIES OF LASER BEAMS Laser radiation is characterized by an extremely monochromaticity, coherence, directionality and brightness.

high

degree

of

3.21.1 Monochromaticity Monochromaticity of laser radiation results from the fact that all the photons are emitted because of a transition between the same two atomic or molecular energy levels, and hence have almost exactly the same frequency. Further, the laser cavity is resonant only for the frequencies of resonant cavity limits the frequency range. However, as the energy levels are not sharp, there is always a small spread to the frequency distribution, which may cover several discrete frequencies. The result is that a small number of closely spaced frequencies may appear in a laser action i.e. the light is not monochromatic. In order to achieve the optimum monochromaticity, generally an etalon is placed within the laser cavity and arranged so that only well defined wavelength can travel back and forth between end mirrors. One of the important factors which characterizes laser is the quality factor Q defined as the ratio of emission frequency ν to the linewidth, that is, Q=

ν λ = ∆ν ∆λ

(3.87)

Where ν is the central frequency of the laser beam. The degree of nonmonochromaticity ξ of laser radiation is the reciprocal of quality factor i.e. ξ=

1 ∆ν ∆λ = = Q ν λ

(3.88)

3.21.2 Coherence It is the property which most clearly distinguish laser light from other kind of light. Coherence is the property, which results from the nature of the stimulated emission process. In the laser, photons emitted by the excited laser medium are emitted in phase with those of already present in the cavity. There are two types of coherence: temporal coherence and spatial coherence. Consider two points P and Q a distance L apart in the direction of propagation of laser beam as shown in Fig. 3.32. If a definite and fixed phase relationship exists between the wave

3.42

Laser Systems and Applications

amplitudes at P and Q at time t and t + τ, then wave shows temporal coherence for time τ. The maximum separation L at which the fixed relationship is retained is called the coherence length Lc. The coherence length is related to the coherence time τc by

Fig. 3.32 Diagram for explaining temporal coherence

(3.89) From Heisenberg uncertainty relation the coherence time is related to monochromaticity of the laser by (3.90)

∆

and ∆

∆

(3.91)

Temporal coherence comes from the monochromaticity of the laser beam. The narrower the linewidth ∆λ or ∆ν of the light source the better is its temporal coherence. Spatial coherence in the simplest case tells us about the phase relationship between the field amplitudes at two points P and Q in a plane normal to the wave vector k as shown in Fig. 3.33.

Fig. 3.33 Diagram for explaining spatial coherence

Spatial coherence also referred to as transverse coherence. It described how far apart to sources, or two portion of the same source can be located in a direction transverse to the direction of propagation of light and still show coherent properties. Let the two sources are separated by a distance d in transverse direction and are at a distance L from the point where the observations are made as shown in the Fig. 3.34. Suppose the interference effect due to the

Introduction to Lasers

3.43

two sources is observed at A as well as at B. The transverse coherent length d is then from point A to B. From the Fig. 3.34

where

Spatial coherence is related to directionality. Light that is emitted from different parts of the conventional light source is not phase-related. Therefore, the light from an extended light source will not be spatially coherent. A good measure of coherence of a light source is its ability to produce stable interference fringes. Temporal coherence can be studied with a Michelson interferometer. The existence of spatial coherence between the fields at two points can be demonstrated in a Young’s slits experiment. A

d

S B R Fig. 3.34

3.21.3 Directionality Stimulated emission produces photons with almost precisely identical direction of propagation. The mirrors selectively amplify axial beam. The laser thus emits a narrow parallel beam from its output mirror. The extent of beam divergence is essentially determined by the diffraction limit of output aperture. From diffraction theory, the divergence angle θD is θD =

βλ D

(3.92)

where λ and D are wavelength and diameter of the laser beam, respectively. β is a coefficient whose value is around unity. The beam divergence increases with wavelength and decreases as beam (or output lens) diameter decreases. Thus a

3.44

Laser Systems and Applications

smaller diameter beam will suffer more divergence and greater spread with distance than a larger beam. If the partial spatial coherent beam has a given intensity distribution over the diameter D and a given coherence area Asc, its divergence is bigger than the diffraction limited divergence, then it can be shown θd =

βλ ( Asc )1/ 2

(3.93)

3.21.4 Brightness It is defined, as the power emitted from unit area of the output mirror per unit solid angle. It is extremely high. The reason for this is that although the power may be small but it is distributed over solid angle which is small. The units are watt per square meter per steradian. A steradian is the unit of solid angle Ω which is three dimensional analogue of conventional two dimensional (planar) angle expressed in radians. For small angle the relation between planar and solid angle of cone with that planar angle θ is to a good approximation is

4

Ω

θ

Fig. 3.35

Examples 1. A laser beam emits light of wavelength 632.8 nm and has a power output of 5mW. How many photons are emitted per second by this laser? Energy of the photon E = hν =

hc 6.6262 × 10 −34 Js × 2.9979 × 10 8 ms −1 = = 3.1392 × 10 −19 J −9 λ 632 .8 × 10 m

Let the number of emitted photons per second be N, the energy of these photons is equal to5 mW, i.e. 3.1392 × 10 −19 J × N = 5 × 10 −3 Js −1

Introduction to Lasers

N=

3.45

5 × 10 −3 Js −1 ≈ 1.6 × 1016 s −1 3.1392 × 10 −19 J

2. The energy band gap of a semiconductor material is 1.46 eV. Calculate the wavelength and frequency of the light produced by semiconductor. 1.46

1.60219

2.340 10 6.6262 10 3 10 / 3.53 10

2.340

10 3.53

10

10

849

3. In Nd:YAG laser , the spontaneous transition probability of the upper level of thelaser line 1.064 µm is 4.35×103s-1 . Determine the life time of the upper level. From Eq. (3.4) τn =

1 1 = = 230 × 10 -6 s = 230 µs An 4.35 × 10 3 s -1

4. A He-Ne laser is operating at 632.8 nm. Calculate the ratio of stimulated emission to spontaneous emission coefficient. From Eq. (3.13) A21 8π hν 3 8π h 8 × 3.14 × 6.6262 × 10 −34 Js = = 3 = ≈ 6.57 × 10 −14 Js/m 3 −9 3 3 B21 c λ (632.8 × 10 m) B21 1 = = 1.52 × 1013 m 3 / Js A21 6.57 × 10 −14 Js / m 3

5. Calculate the relative rate of spontaneous and stimulated process in the visible region (ν = 1015Hz) at 300 K. From Eq. (3.11) B21 ρ (ν ) 1 = A21 exp ( hν / k B T ) − 1

hν 6.6262 × 10 −34 Js × 1015 Hz = ≈ 1.6 × 10 2 − 23 −1 k BT 1.3807 × 10 JK × 300K Thus B21 ρ(ν ) 1 = ≈ 3.257 × 10 −70 2 A21 exp(1.6 × 10 ) − 1

stimulated emission is negligible.

3.46

Laser Systems and Applications

6. For a system in thermal equilibrium, calculate the wavelength at which spontaneous emission to stimulated emission rate are equal for a temperature of 4000K. From Eq. (3.11) B 21 ρ (ν ) 1 = =1 A21 exp ( hν / k B T ) − 1

1

1

ln2

ln2

6.6262 10 4000 1.3807

3

10 /

10

/ 0.693147

5.1928

7. For a system in thermal equilibrium, calculate the temperature at which spontaneous emission to stimulated emission rate are equal for a wavelength of 500 nm. From Eq. (3.11) B 21 ρ (ν ) 1 = =1 A21 exp ( hν / k B T ) − 1

exp

1

1

ln2

ln2

2

500

6.6262 10 1.3807 10

3 10

10 /

/ 0.693147

41542 8. Find the volume of the cavity if the number of modes which fall within bandwidth of 10.0 nm centred at 600.0 nm is 1.94×1012 . From Eq. (3.14), the number of modes is ∆N =

8π ν 2 ∆ν c

3

V =

8π ∆λ λ

4

V =

8 × 3.14 × 10 × 10 −9 m −9

(600 × 10 m)

4

× 10 −6 m 3 = 1.94 × 1012 modes

Introduction to Lasers

∆ 8

∆

600 10 8 3.14

∆ 8 ∆

1.94 10 10 10

3.47

10

9. A relative population of two energy levels out of which upper one corresponds to a metastable state is 1.059×10-30. Find the wavelength of light emitted at a temperature 330 K. We have exp 1.059

10

exp

6.6262 10 1.3807 10

30ln1.059

6.6262 10 30ln1.059 1.3807

3 10

10 /

3 /

10 / 330

/ 330

25.186

10. A relative population or Boltzmann ratio,1/e, is often considered representation of the ratio of the population in two energy state at 300 K. Determine the wavelength of radiation emitted at that temperature. We have

1 1 6.6262 10 1.38066

3 10

/

10 / 300

48

10

48

11. A certain transition involves an energy change of 0.0025 eV /molecule. If there are 1000 molecules in the ground state, what is the approximate equilibrium population of the excited state at 300 K. We know 1 Therefore,

1.60219

10

3.48

Laser Systems and Applications

0.0025

0.0025

1.60219

10

4.005

10

From Eq. (3.15) exp

.

1000exp

.

/

1000

0.907

1000exp

0.09674

907

12. For He-Ne laser line 632.8 nm, stimulated emission cross section is 3.0 × 10–17 m2 and inversion density is 5.0 × 1015 m–3. Determine the small signal gain coefficient. From Eqs. (3.35) and (3.36) assuming no cavity losses γ 0 (ν ) = σ se ( N 2 − N 1 ) = 3 × 10 −17 m 2 × 5 × 1015 m −3 = 0.15 m −1

13. Consider GaAs laser diode with length of the active material 500 µm and reflectivity of the mirrors equal to 0.3. Calculate the threshold gain if the cavity losses are 5.0 mm-1 From Eq. (3.35)

γ0 (ν) = α +

1 1 1 10 2 ln = 50 cm −1 + ln ≈ 74 cm −1 2L R1 R2 2 × 5 cm 0.3 × 0.3

14. A laser cavity consists of two mirrors with reflectivity R1 = 0.5 and R2 = 0.95. Calculate the threshold inversion if the length of the active material is 7.5 cm and the transition cross section is 8.8 × 10-19 cm2. From Eq. (3.36) N 2 − N1 =

N 2 − N1 =

1 1 1 1 ⎛ ⎞ ln ln⎜ = ⎟ 2 Lσ se R1 R2 2 × 7.5 cm × 8.8 × 10 −19 cm 2 ⎝ 0.5 × 0.95 ⎠

1 2 × 7.5 cm × 8.8 × 10

−19

cm

2

ln (2.1053) =

0.7444 2 × 7.5 cm × 8.8 × 10 −19 cm 2

= 5.64 × 1016 /cm3 15. For N2 laser line 337.1 nm calculate the stimulated emission cross section if small signal gain coefficient is 10.0 m-1 and inversion density is 2.5 × 1017 m–3. From Eqs. (3.35) and (3.36) assuming no cavity losses σ se =

γ 0 (ν )

(N − N ) 2 1

=

10.0 m –1 = 4 × 10 − 17 m 2 − 17 3 2.5 × 10 m

Introduction to Lasers

3.49

16. For Argon ion laser line 488 nm, small signal gain coefficient is 0.5m-1 and stimulated emission cross section is 2.5×10–16 m2 . Calculate the inversion density From Eqs. (3.35) and (3.36) assuming no cavity losses γ (ν ) 0.5 m –1 = = 2 × 1015 m − 3 N 2 − N1 = 0 – 16 2 σ 2.5 × 10 m se

17. The intensity 1Watt/cm2 is required to be amplified to 106 Watt/cm2 by making the beam travel up and down through an active material of length 5.0 cm and cross sectional area 0.1cm2. If the small signal gain coefficient is 0.1cm-1 how many number of travels are required for attaining the desired intensity. From Eqs. (3.28) and (3.31) I ν ( z ) = I ν (0) exp[γ 0 z ]

and Iν(z) = 106W/cm2, Iν(0)=1W/cm2, γ0= 0.1cm-1. Substituting these values 1

ln

1 ln10 0.1

0

6 ln10 0.1

60

2.30258

138.15

Minimum total distance travelled in the active material is 138.15 cm. Distance travelled in to and fro passage in the active material is 2 × 5 = 10 cm. Thus the number and to and fro passage in the active material would be 138.15cm/10cm = 13.81≈ 14. 18. Calculate the number of modes of the laser beam with wavelength 700 nm and spacing between the mirrors is 35 cm. We have the number of modes 2

2 0.35 700 10

10

19. One mirror with radius of curvature of 2.0 m and other with radius of curvature 3.0m are separated by a distance of 2.3m. Determine whether this mirror arrangement leads to stability or not? From Eqs. (3.84) and (3.85) we obtain g1 = -0.15 and g2 = 0.2333 and g1g2 = –0.035. Thus from Eq. (3.86) the mirror arrangement is not stable. 20. A diode laser emitting a wavelength of 800.0nm has a spectral width of 1nm. What is coherent length? and

We have ∆

∆

∆

3

10 / 1 800 10

10

4.6875

10

3.50

Laser Systems and Applications

From Eq. (3.91) 3 10 / 4.6875 10 /

∆

6.4

10

0.64

21. A LED emitting a wavelength of 600 nm has a spectral width of 60 nm. Find the coherent length. We have and ∆

∆

∆

/

3 10 / 5 10 /

∆

5

0.6

10

10

6

22. Obtain the brightness of 1µm He-Ne laser with a 1mm output diameter and divergence of 1 milliradian. The solid angle corresponding to 1 milliradian is 4

1

0.8

10

The brightness is power divided by the area of the beam and the solid angle 10 10

0.785

0.8

1.6

10

10

The brightness of a milliwatt He-Ne laser is far greater than 106 Watts/m2steradian that of the sun which emits more than 1026 Watt. 23. Calculate the emission frequency width required to have a temporal coherence length of 10 m at a source wavelength of 488 nm. We have from Eq. (3.91) ∆

3

10 10

/

3

10

24. Calculate the power density of radiation per unit area at a distance of 2 meters, from an incandescent lamp rated 100 W, compared to a HeliumNeon laser of 1 mW. The laser beam diameter at the laser output is 2 mm, and its divergence is 1 mrad. Light from incandescent lamp is radiated to all directions, so it is distributed on a surface of a sphere with a radius of 2m. The sphere surface area is: 4π R2, so the power density at a distance of 2 m is:

Introduction to Lasers

100 200

4

3.51

0.2

Compared to the incandescent lamp, the laser beam diameter at a distance of 2 m increased to 4 m 2

2 2

1

20000. tan 0.5

2.1

.

0.21

The power density of the laser radiation is: 1 0.2

8

At a distance of 2 m from the radiation source, the power density of the laser radiation is 40 times higher than from the lamp, although the power of the lamp is 5 orders of magnitude higher than the original power of the laser. 25. Determine what emission frequency width would be required to have a temporal coherence length of 10.0 m at a source wavelength of 488 nm. From Eq. (3.91)

∆ν =

c 2.9979 × 108 ms −1 = ≈ 3 ×107 Hz Lc 10m

26. Show that laser light He-Ne laser of even 1mW power is brighter than sun. The temperature of the sun is 6000K and the wavelength of He-Ne laser is 632.8 nm, divergence is 0.5 mrad, spot size is 1 mm. For Sun According to Stefens Law, the energy density is 5.6696 Brightness

10

Energy density 4

6000 7.4

7.4

10 / 4 3.14

10 5.9

/ 10

3.52

Laser Systems and Applications

For He-Ne Laser 632.8

, power

1

,

1

, divergance

0.5

Solid angle Ω is Ω

θ

mrad r

L

But

Ω

3.14 0.5 1 3.14 5

Power density

10

Areal spread Brightness

3

1.5 10

3.14 100 1273 7.85 10

10

1273 /

10

600 10 4 10 2

7.85

10 .

1.5

10

1.62

7

10

10

Brightness of He-Ne laser is ~300 times the brightness of sun. 27. A sodium lamp of 80 W power and of cylindrical surface area of 700 cm2 is emitting wavelengths between 589 nm and 589.6 nm in all direction is 4π Sr. If the power contained in 589 nm of spectral width of 0.1nm is 40 W, calculate the brightness of the lamp. What is the spectral brightness?

Introduction to Lasers

3.53

It is given that power is 40 W, surface area of emitter = 700cm2 = 700 × 10-4m2, Solid angle = 4πSr. The brightness of the lamp 700

40 10

45

4

Brightness specteal width

Spectral brightness

45 0.1

450 28. For an ordinary source the coherence time is 10–10s. Obtain the degree of non-monochromaticity for wavelength of 540 nm. We have 1

1

10

10

3 10 / 540 10

1 18

10

Degree of non-monochromaticity 18

10

1.8

10

10

29. If bandwidth is 2500 Hz, find coherence time and coherent length. From the relation 1

1 2500

4

10

Coherent length Lc is 3

10

4

10

120

10

120

30. Consider thermal light such as would be generated by incandescent matter or gas discharge with a broad spectrum. The bandwidth is of the order of 108s–1. A laser has band width ∆ν ~ 104s–1. Find the coherence time and coherence length in both the cases. The coherence time for the thermal light is 1

10

10 Coherence length

3

10

10

3

3.54

Laser Systems and Applications

The coherence time for the laser light is 1

10

10 Coherence length

3

10

10

3

10

31. Consider a sphere of plasma of diameter 100µm. This plasma radiate at a wavelength of 10 nm. Consider a point at 0.50m from the source of radiation. What will be the transverse coherence length due to this light? We have coherence length L 0.5

10 10

5

10

50

32. Consider an optical pump at 940 nm for Yb:YAG crystal placed in a cavity. The wavelength of Yb is 1030 nm. If all the photons emitted by the pump are absorbed by the crystal and used for the lasing process, calculate the maximum power output. The pump power is 1W. At best a pump photon gives a laser photon. The maximum output power is defined by

where νp and νl are pump frequency and laser frequency respectively. By taking into account the different wavelengths, the power output is 1

940 1030

912

Short Questions 1. Distinguish between spontaneous emission and stimulated emission. Ans. In the case of spontaneous emission, the emission is natural where as in the case of stimulated emission, it is induced or stimulated. In case of spontaneous emission there is no amplification as well as no phase relationship between emitted photons. However, in the case of stimulated emission there is amplification as well as phase relationship between the emitted photons. Since in the stimulated emission the emitted photons have a definite phase relationship with each other therefore coherent output is produced. Photons of spontaneous emission have no phase relationship with each other and therefore are not coherent. 2. In which direction and polarization the photon of spontaneous emission are emitted? Ans. In random directions, and with arbitrary polarization.

Introduction to Lasers

3.55

3. What is the direction of photon emitted by stimulated emission? Ans. It is emitted in the direction of the incident photon. 4. What is the natural lifetime of an energy level? Ans. The time in which population of the energy level falls to 1/e of its initial value is called natural lifetime and is related to Einstein’s A coefficient as τ = 1/A. 5. What is the unit of Einstein’s A coefficient? Ans. (Time)-1 6. What causes spontaneous emission? Ans. Spontaneous emission is a quantum mechanical effect. According to quantum electrodynamics, the electromagnetic field fluctuations (because of zero point energy) induce the spontaneous emission of radiation. 7. Is linear momentum is conserved in emission and absorption processes? Ans. The linear momentum is conserved in emission and absorption processes 8. What are the properties of stimulated emission? Ans. It has three very important qualities (i) It is of a very precisely defined frequency: the excited state does not spontaneously decay so it has a long life time, which according to Heisenberg uncertainty implies a narrow energy level (ii) the emitted radiation is in phase with the stimulating radiation: the excited state is stimulated to emit by interaction with the oscillating electromagnetic field of frequency νstim., so the maximum amplitude of the emitted wave coincides with that of νstim (iii) the stimulating and emitted radiation are coherent, that is, they travel in precisely the same direction. 9. What are the properties of spontaneous emission? Ans. The spontaneous emission has the following characteristics (i) it occur at any time, therefore each emitted photon is not necessarily in phase with any other (ii) it is emitted in any direction (iii) it is emitted within a more or less broad range of frequencies 10. Which of the two processes, spontaneous emission or stimulated emission is more likely to occur at high frequencies? Ans. For high frequencies transition (that is IR, Visible, and UV upwards) spontaneous emission is more likely to occur.

3.56

Laser Systems and Applications

11. For high frequency transition which is more prominent: spontaneous emission or stimulated emission? Ans. The spontaneous emission coefficient A to stimulated emission coefficient B, that is , A/B is proportional to cube of the frequency. Therefore, as frequency of the transition increases, the spontaneous emission become more prominent. 12. What is the interpretation of Einstein’s A and B coefficients? Ans. (1) The relation B12=B21, indicate that for stimulated emission to occur, the material should absorb strongly. (2) The relation A21=B21(8πν2/c3) hν indicates that for stimulated emission, the spontaneous emission must occur. Further, at higher frequencies the spontaneous emission is prominent. 13. What other name is given to Einstein B coefficients? Ans. The other name given to Einstein B coefficient is transition cross section. 14. What is the unit of Einstein B coefficient? Ans. The unit is m3J–1s–2. 15. Why (1s2s)3S1 and (1s2s) 1S0 state of helium are metastable? Ans. These states are metastable as no electric dipole transition to (1s1s) 1S0 is possible. They are forbidden according to selection rules ∆S = 0, ∆l = ±1. Therefore, they have a large lifetime in comparison with the usual life time ~10–8s of an excited state. 16. When lasing is possible? Ans. When population inversion exist between the levels. 17. What is active material or an active medium? Ans. A material or medium (may be gas, liquid or solid) in which population inversion can be created is called active material or active medium. 18. What is function of active material? Ans. A population inverted medium will amplify an incoming wave of correct frequency through stimulated emission. 19. Define efficiency of a laser. Ans. It is defined as Ef iciency

Output power Input power

20. What is necessary condition for amplification? Ans. Population inversion is a necessary condition for amplification.

Introduction to Lasers

3.57

21. How a laser oscillator is made from laser amplifier? Ans. To make an oscillator from an amplifier, it is necessary to introduce suitable positive feedback so that gain is increased. This increase in gain is obtained by placing an active material between the mirrors. 22. Define laser cavity. Ans. The region bounded by the mirrors is called laser cavity. 23. What is essential condition for gain? Ans. The population of upper laser level should be greater than the population of the lower laser level. 24. What is the unit of gain coefficient? Ans. The unit of gain coefficient is m-1. 25. In each passes through the active material of length L, how the intensity increases? Ans. In each passes through the material of length L, the intensity increases by a factor of exp[γ0(ν)L] where γ0 is small signal gain coefficient. 26. When the threshold condition is established? Ans. The threshold condition is established by requiring that photon density after reflection from reflector R1 and R2 be equal to the initial photon density, that is, 1

2 1 ln 2

1

27. Do distributed losses α per unit length within the gain medium involve particular laser levels? Do they occur at the same frequency as that of a laser beam? Ans. The α does not involve particular laser levels but occur at the same frequency as that of the laser beam. 28. What is critical inversion? Ans. The population inversion given by the expression .

ln

is known as critical inversion. 29. What initiate the amplification process in laser? Ans. The photons, which are spontaneously emitted in a direction along the laser cavity, initiate the amplification process

3.58

Laser Systems and Applications

30. The intensity of radiation between mirrors of a laser increases exponentially. What limit this increase in intensity? Ans. The intensity of radiation between the mirrors increases exponentially. This increase in intensity is finally limited by the ability of the pumping mechanism producing the population inversion to keep the number of atoms or molecules in the upper laser level. 31. Can two-level atomic system generate laser? Ans. No, because it cannot create population inversion 32. What is the condition for amplification in three level laser? Ans. For amplification in three level laser, the level 2 (metastable level) must be pumped faster than the population of level 2 depleted by spontaneous emission, that is, Wp > A21. 33. What is threshold condition for a three level laser? Ans. Wp = A21 34. What is necessary to obtain population inversion in four level laser? Ans. For population inversion it is necessary that the rate at which atom relax from level 1 to level 0 must be greater than the rate at which atoms relax from level 2 to level 1. 35. Is creation of population inversion between level 2 and level 1 of four level laser is independent of pumping power? Ans. Yes 36. Is the magnitude of population inversion between level 2 and level 1 of four level laser is independent of pumping power? Ans. The magnitude of population inversion depends on pumping power. 37. What is the condition of threshold pumping for four level laser? Ans. The condition is .

38. What are the advantages of four level laser over three level laser? Ans. The advantages are (i) the lasing threshold of a four level laser is lower (ii) the efficiency is high (iii) required pumping rate is lower (iv) continuous operation is possible

Introduction to Lasers

3.59

39. List at least three pumping methods. Ans. Optical pumping, electrical pumping and chemical pumping. 40. Why it is necessary to have strong pumping for lasers of high frequencies? Ans. According to Einstein’s relation spontaneous emission probability increases with frequency as ν3. Therefore, to obtain stimulated emission a strong pumping is required. 41. What kind of pumping is used for most gases and semiconductor lasers? Ans. Electrical pumping 42. Why optical pumping is often used in liquids and solid lasers? Ans. In solids and in liquids, electrons cannot be easily accelerated by electric field to excite the laser energy levels of impurity species. In solids and liquids, one usually deals with bands rather than levels. These bands can therefore, absorbs a sizeable fraction of light emitted by flash lamps whose energy occurs over a broad wavelength region. 43. Mention various geometries of optical pumping. Ans. Various geometries of optical pumping are (i) side pumping (ii) end pumping (iii) face pumping (iv) edge pumping 44. Compare diode laser pumping and lamp pumping. Why diode laser pumping is usually more efficient than other pumping methods? Ans. Diode laser pumping has much higher absorption efficiency and quantum efficiency than other pumping methods 45. What are basic components of a laser? Ans. The basic components of a laser are (i) Active material (ii) Pumping (iii) Resonators 46. Define Continuous Wave (CW) laser. Ans. A laser whose output is constant over time is known as continuous wave (CW) laser. The examples are Ar+ ion laser, He-Ne laser etc.

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Laser Systems and Applications

47. Define a pulsed laser Ans. A laser whose power appears in pulses of some duration at some repetition rate is known as pulsed laser. 48. When continuous wave (CW) operation of a laser is probable? Ans. The probability that a particular system will lase in CW mode is determined mainly by the lifetime of upper laser level and the lower laser level. If the upper laser level has a relatively long lifetime (metastable state) compared to the lower laser level and population inversion of the gain medium is continually replenished by a steady pump source. 49. When pulsed operation of a laser is probable? Ans. If the lower laser level has a relatively long lifetime, atoms in that lower energy state remain there for longer period. This gives a good probability to atoms of lower laser level to absorb photons and thus destroy the population inversion. In this situation a pulse laser may be possible if the upper laser level is filled quickly and preferentially over the lower energy level. The population of lower energy level eventually exceeds that of the upper laser level and lasing will cease. 50. What are the requirements necessary for a pair of mirrors to form resonator? Ans. The requirements are: (i) It should contain a sufficiently large amount of active material (ii) It should permit amplification at only a narrow band of frequencies. 51. Basically into what kinds the Laser resonator can be basically divided ? Ans. The laser resonators can be basically divided into stable and unstable resonator. 52. When unstable resonator is used? Ans. The unstable resonator is used when laser energy is too high inside the cavity. 53. What is meant by open resonators? Ans. It means that no lateral surfaces are used. 54. What is passive resonator? Ans. A passive resonator has no gain or loss 55. What are stable resonators? Ans. If the oscillatory laser beam is bouncing back and forth between the two mirrors without much loss due to finite size of the mirrors and allow cavity to get high gain then the mirror arrangement is called stable resonator.

Introduction to Lasers

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56. What are unstable resonators? Ans. In the unstable resonator, the arrangement of the mirrors is such that the oscillatory laser beam spread out of the cavity after a few transversal. 57. What is the disadvantage of open resonator? Ans. There would be some losses due to diffraction. 58. Where the unstable resonator arrangements suitable? Ans. For high power losses, unstable cavities are used. Laser output comes from the edge of the output mirror. They are suitable for high gain per round laser system, which do not require large number of oscillations between the mirrors. 59. What is Fabry-Perot resonator? Ans. A resonator consisting of two plane mirrors set parallel to each other at a separation L. 60. What is the frequency difference between two consecutive modes in FabryPerot or plane parallel resonator? Ans. The frequency difference ν is given by ν = c/2L where L is the separation between two plane mirrors and c is the velocity of light in the medium. 61. What is the stability criterion for mirror arrangement of resonator? 1, where g1 = 1 – (L/R1), g2 = 1– Ans. The stability criterion is 0 (L/R2), L is separation between the mirrors, R1 and R2 are radius of curvature of two mirrors used. 62. What is the separation between two mirrors corresponding to generalized spherical resonator? Ans. In this arrangement, the two spherical mirrors of same radius of curvature R are separated by a distance L such that R