LASER PHYSICS

PETER W. MILONNI

JOSEPH H. EBERLY

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LASER PHYSICS

LASER PHYSICS

PETER W. MILONNI

JOSEPH H. EBERLY

Copyright # 2010 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Milonni, Peter W.Laser physics / Peter W. Milonni, Joseph H. Eberly

p. cm.Includes bibliographical references and index.ISBN 978-0-470-38771-9 (cloth)1. Lasers. 2. Nonlinear optics. 3. Physical optics. I. Eberly, J. H., 1935- II. Title.QC688.M55 2008621.3606—dc22

2008026771

Printed in the United States of America10 9 8 7 6 5 4 3 2 1

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To our wives, Mei-Li and Shirley

CONTENTS

Preface xiii

1 Introduction to Laser Operation 1

1.1 Introduction, 11.2 Lasers and Laser Light, 31.3 Light in Cavities, 81.4 Light Emission and Absorption in Quantum Theory, 101.5 Einstein Theory of Light–Matter Interactions, 111.6 Summary, 14

2 Atoms, Molecules, and Solids 17

2.1 Introduction, 172.2 Electron Energy Levels in Atoms, 172.3 Molecular Vibrations, 262.4 Molecular Rotations, 312.5 Example: Carbon Dioxide, 332.6 Conductors and Insulators, 352.7 Semiconductors, 392.8 Semiconductor Junctions, 452.9 Light-Emitting Diodes, 492.10 Summary, 55Appendix: Energy Bands in Solids, 56Problems, 64

3 Absorption, Emission, and Dispersion of Light 67

3.1 Introduction, 673.2 Electron Oscillator Model, 69

vii

3.3 Spontaneous Emission, 743.4 Absorption, 783.5 Absorption of Broadband Light, 843.6 Thermal Radiation, 853.7 Emission and Absorption of Narrowband Light, 933.8 Collision Broadening, 993.9 Doppler Broadening, 1053.10 The Voigt Profile, 1083.11 Radiative Broadening, 1123.12 Absorption and Gain Coefficients, 1143.13 Example: Sodium Vapor, 1183.14 Refractive Index, 1233.15 Anomalous Dispersion, 1293.16 Summary, 132Appendix: The Oscillator Model and Quantum Theory, 132Problems, 137

4 Laser Oscillation: Gain and Threshold 141

4.1 Introduction, 1414.2 Gain and Feedback, 1414.3 Threshold, 1434.4 Photon Rate Equations, 1484.5 Population Rate Equations, 1504.6 Comparison with Chapter 1, 1524.7 Three-Level Laser Scheme, 1534.8 Four-Level Laser Scheme, 1564.9 Pumping Three- and Four-Level Lasers, 1574.10 Examples of Three- and Four-Level Lasers, 1594.11 Saturation, 1614.12 Small-Signal Gain and Saturation, 1644.13 Spatial Hole Burning, 1674.14 Spectral Hole Burning, 1694.15 Summary, 172Problems, 173

5 Laser Oscillation: Power and Frequency 175

5.1 Introduction, 1755.2 Uniform-Field Approximation, 1755.3 Optimal Output Coupling, 1785.4 Effect of Spatial Hole Burning, 1805.5 Large Output Coupling, 1835.6 Measuring Gain and Optimal Output Coupling, 1875.7 Inhomogeneously Broadened Media, 1915.8 Spectral Hole Burning and the Lamb Dip, 1925.9 Frequency Pulling, 1945.10 Obtaining Single-Mode Oscillation, 1985.11 The Laser Linewidth, 2035.12 Polarization and Modulation, 207

viii CONTENTS

5.13 Frequency Stabilization, 2155.14 Laser at Threshold, 220Appendix: The Fabry-Pérot Etalon, 223Problems, 226

6 Multimode and Pulsed Lasing 229

6.1 Introduction, 2296.2 Rate Equations for Intensities and Populations, 2296.3 Relaxation Oscillations, 2306.4 Q Switching, 2336.5 Methods of Q Switching, 2366.6 Multimode Laser Oscillation, 2376.7 Phase-Locked Oscillators, 2396.8 Mode Locking, 2426.9 Amplitude-Modulated Mode Locking, 2466.10 Frequency-Modulated Mode Locking, 2486.11 Methods of Mode Locking, 2516.12 Amplification of Short Pulses, 2556.13 Amplified Spontaneous Emission, 2586.14 Ultrashort Light Pulses, 264Appendix: Diffraction of Light by Sound, 265Problems, 266

7 Laser Resonators and Gaussian Beams 269

7.1 Introduction, 2697.2 The Ray Matrix, 2707.3 Resonator Stability, 2747.4 The Paraxial Wave Equation, 2797.5 Gaussian Beams, 2827.6 The ABCD Law for Gaussian Beams, 2887.7 Gaussian Beam Modes, 2927.8 Hermite–Gaussian and Laguerre–Gaussian Beams, 2987.9 Resonators for He–Ne Lasers, 3067.10 Diffraction, 3097.11 Diffraction by an Aperture, 3127.12 Diffraction Theory of Resonators, 3177.13 Beam Quality, 3207.14 Unstable Resonators for High-Power Lasers, 3217.15 Bessel Beams, 322Problems, 327

8 Propagation of Laser Radiation 331

8.1 Introduction, 3318.2 The Wave Equation for the Electric Field, 3328.3 Group Velocity, 3368.4 Group Velocity Dispersion, 3408.5 Chirping, 3518.6 Propagation Modes in Fibers, 355

CONTENTS ix

8.7 Single-Mode Fibers, 3618.8 Birefringence, 3658.9 Rayleigh Scattering, 3728.10 Atmospheric Turbulence, 3778.11 The Coherence Diameter, 3798.12 Beam Wander and Spread, 3888.13 Intensity Scintillations, 3928.14 Remarks, 395Problems, 397

9 Coherence in Atom-Field Interactions 401

9.1 Introduction, 4019.2 Time-Dependent Schrödinger Equation, 4029.3 Two-State Atoms in Sinusoidal Fields, 4039.4 Density Matrix and Collisional Relaxation, 4089.5 Optical Bloch Equations, 4149.6 Maxwell–Bloch Equations, 4209.7 Semiclassical Laser Theory, 4289.8 Resonant Pulse Propagation, 4329.9 Self-Induced Transparency, 4389.10 Electromagnetically Induced Transparency, 4419.11 Transit-Time Broadening and the Ramsey Effect, 4469.12 Summary, 451Problems, 452

10 Introduction to Nonlinear Optics 457

10.1 Model for Nonlinear Polarization, 45710.2 Nonlinear Susceptibilities, 45910.3 Self-Focusing, 46410.4 Self-Phase Modulation, 46910.5 Second-Harmonic Generation, 47110.6 Phase Matching, 47510.7 Three-Wave Mixing, 48010.8 Parametric Amplification and Oscillation, 48210.9 Two-Photon Downconversion, 48610.10 Discussion, 492Problems, 494

11 Some Specific Lasers and Amplifiers 497

11.1 Introduction, 49711.2 Electron-Impact Excitation, 49811.3 Excitation Transfer, 49911.4 He–Ne Lasers, 50211.5 Rate Equation Model of Population Inversion in He–Ne Lasers, 50511.6 Radial Gain Variation in He–Ne Laser Tubes, 50911.7 CO2 Electric-Discharge Lasers, 51311.8 Gas-Dynamic Lasers, 515

x CONTENTS

11.9 Chemical Lasers, 51611.10 Excimer Lasers, 51811.11 Dye Lasers, 52111.12 Optically Pumped Solid-State Lasers, 52511.13 Ultrashort, Superintense Pulses, 53211.14 Fiber Amplifiers and Lasers, 53711.15 Remarks, 553Appendix: Gain or Absorption Coefficient for Vibrational-RotationalTransitions, 554Problems, 558

12 Photons 561

12.1 What is a Photon, 56112.2 Photon Polarization: All or Nothing, 56212.3 Failures of Classical Theory, 56312.4 Wave Interference and Photons, 56712.5 Photon Counting, 56912.6 The Poisson Distribution, 57312.7 Photon Detectors, 57512.8 Remarks, 585Problems, 586

13 Coherence 589

13.1 Introduction, 58913.2 Brightness, 58913.3 The Coherence of Light, 59213.4 The Mutual Coherence Function, 59513.5 Complex Degree Of Coherence, 59813.6 Quasi-Monochromatic Fields and Visibility, 60113.7 Spatial Coherence of Light From Ordinary Sources, 60313.8 Spatial Coherence of Laser Radiation, 60813.9 Diffraction of Laser Radiation, 61013.10 Coherence and the Michelson Interferometer, 61113.11 Temporal Coherence, 61313.12 The Photon Degeneracy Factor, 61613.13 Orders of Coherence, 61913.14 Photon Statistics of Lasers and Thermal Sources, 62013.15 Brown–Twiss Correlations, 627Problems, 634

14 Some Applications of Lasers 637

14.1 Lidar, 63714.2 Adaptive Optics for Astronomy, 64814.3 Optical Pumping and Spin-Polarized Atoms, 65814.4 Laser Cooling, 67114.5 Trapping Atoms with Lasers and Magnetic Fields, 68514.6 Bose–Einstein Condensation, 690

CONTENTS xi

14.7 Applications of Ultrashort Pulses, 69714.8 Lasers in Medicine, 71814.9 Remarks, 728Problems, 729

15 Diode Lasers and Optical Communications 735

15.1 Introduction, 73515.2 Diode Lasers, 73615.3 Modulation of Diode Lasers, 75415.4 Noise Characteristics of Diode Lasers, 76015.5 Information and Noise, 77415.6 Optical Communications, 782Problems, 790

16 Numerical Methods for Differential Equations 793

16.A Fortran Program for Ordinary Differential Equations, 79316.B Fortran Program for Plane-Wave Propagation, 79616.C Fortran Program for Paraxial Propagation, 799

Index 809

xii CONTENTS

PREFACE

Judged by their economic impact and their role in everyday life, and also by thenumber of Nobel Prizes awarded, advances in laser science and engineering inthe past quarter-century have been remarkable. Using lasers, scientists have producedwhat are believed to be the coldest temperatures in the universe, and energy densitiesgreater than in the center of stars; have tested the foundations of quantum theoryitself; and have controlled atomic, molecular, and photonic states with unprecedentedprecision.

Questions that previous generations of scientists could only contemplate in terms ofthought experiments have been routinely addressed using lasers. Atomic clock frequen-cies can be measured to an accuracy exceeding that of any other physical quantity.The generation of femtosecond pulses has made it possible to follow chemical processesin action, and the recent availability of attosecond pulses is allowing the study ofphenomena on the time scale of electron motion in atoms. Frequency stabilizationand the frequency-comb spectra of mode-locked lasers have now made practical themeasurement of absolute optical frequencies and promise ever greater precision in spec-troscopy and other areas. Lasers are being used in adaptive optical systems to obtainimage resolution with ground-based telescopes that is comparable to that of telescopesin space, and they have become indispensable in lidar and environmental studies.Together with optical fibers, diode lasers have fueled the explosive growth of opticalnetworks and the Internet. In medicine, lasers are finding more and more uses in surgeryand clinical procedures. Simply put, laser physics is an integral part of contemporaryscience and technology, and there is no foreseeable end to its progress and application.

The guiding theme of this book is lasers, and our intent is for the reader to arrive atmore than a command of tables and formulas. Thus all of the chapters incorporateexplanations of the central elements of optical engineering and physics that are requiredfor a basic and detailed understanding of laser operation. Applications are important andwe discuss how laser radiation interacts with matter, and how coherent and often veryintense laser radiation is used in research and in the field. We presume that the reader

xiii

has been exposed to classical electromagnetic theory and quantum mechanics at anundergraduate or beginning graduate level, but we take opportunities throughout toreview parts of these subjects that are particularly important for laser physics.

The perceptive reader will notice that there is substantial overlap with a book wewrote 20 years ago called simply Lasers, also published by Wiley and still in printwithout revision or addition. Many readers and users of that book have told us thatthey particularly appreciated the frequent concentration on background optical physicsas well as explanations of the physical basis for all aspects of laser operation. Naturally abook about lasers that is two decades old needs many new topics to be added to be evenapproximately current. However, while recognizing that additions are necessary, wealso wanted to resist what is close to a law of nature, that a second book must weighsignificantly more than its predecessor. We believe we have accomplished these goalsby describing some of the most significant recent developments in laser physics togetherwith an illustrative set of applications based on them.

The basic principles of lasers have not changed in the past twenty years, but therehas been a shift in the kinds of lasers of greatest general interest. Considerable attentionis devoted to semiconductor lasers and fiber lasers and amplifiers, and to considerationsof noise and dispersion in fiber-optic communications. We also treat various aspects ofchirping and its role in the generation of extremely short and intense pulses of radiation.Laser trapping and cooling are explained in some detail, as are most of the otherapplications mentioned above. We introduce the most important concepts needed tounderstand the propagation of laser radiation in the turbulent atmosphere; this is animportant topic for free-space communication, for example, but it has usually beenaddressed only in more advanced and specialized books. We have attempted to presentit in a way that might be helpful for students as well as laser scientists and engineers withno prior exposure to turbulence theory.

The book is designed as a textbook, but there is probably too muchmaterial here to becovered in a one-semester course. Chapters 1–7 could be used as a self-contained,elementary introduction to lasers and laser—matter interactions. In most respects theremaining chapters are self-contained, while using consistent notation and makingreference to the same fundamentals. Chapters 9 and 10, for example, can serve as intro-ductions to coherent propagation effects and nonlinear optics, respectively, and Chapters12 and 13 can be read separately as introductions to photon detection, photon counting,and optical coherence. Chapters 14 and 15 describe some applications of lasers that willlikely be of interest for many years to come.

We are grateful to A. Al-Qasimi, S. M. Barnett, P. R. Berman, R. W. Boyd, L. W.Casperson, C. A. Denman, R. Q. Fugate, J. W. Goodman, D. F. V. James, C. F.Maes, G. H. C. New, C. R. Stroud, Jr., J. M. Telle, I. A. Walmsley, and E. Wolf forcomments on some of the chapters or for contributing in other ways to this effort.

xiv PREFACE

USEFUL TABLES

TABLE 3 The Electromagnetic Spectrum

Typical Wavelength(cm)

Frequency(Hz)

Photon Energy(eV)

Longwave radio 3 �105 105 4 �10210AM radio 3 �104 106 4 �1029FM radio 300 108 4 �1027Radar 3 1010 4 �1025Microwave 0.3 1011 4 �1024Infrared 3 �1024 1014 0.4Light (orange) 6 �1025 5 �1014 2Ultraviolet 3 �1026 1016 40X-rays 3 �1028 1018 4000Gamma rays 3 �10211 1021 4 �106Cosmic-ray photons 3 �10213 1023 4 �108

Human eyes are sensitive to only a rather narrow band of wavelengths ranging from about 430 to 690 nm.Figure 9.11 shows the wavelength sensitivity of the human eye for a “standard observer.”

TABLE 1 Physical Constants

Velocity of light in vacuum c ¼ 2.998 �108 m/sElectron charge e ¼ 1.602 �10219 CCoulomb force constant 1/4pe0 ¼ 8.988 �109 N-m2/C2

e2/4pe0 ¼ 1.440 eV-nmElectron rest mass me ¼ 9.108 �10231 kgProton rest mass mp ¼ 1.672 �10227 kgBohr radius a0 ¼ 0.528 Å ¼ 0.0528 nmPlanck’s constant h ¼ 6.626 �10234 J-s

h� ¼ h/2p ¼ 1.054 �10234 J-shc ¼ 1240 eV-nm

Avogadro’s number NA ¼ 6.023 �1023Boltzmann constant k ¼ 1.380 �10223 J/KUniversal gas constant R ¼ NAk ¼ 8.314 J/KStefan–Boltzmann constant s ¼ 5.670 �1028 Watt/m2-K4

TABLE 2 Conversion Factors

1 electron volt (eV) ¼ 1:602� 10�19 joule (J)¼ 1:16� 104 K¼ 2:42� 1014 Hz¼ 8:07� 103 cm�1

300K ¼ 2:59� 10�2 eV � 140 eV760 Torr ¼ 1.013 �105 N/m2

1 INTRODUCTION TO LASER OPERATION

1.1 INTRODUCTION

The word laser is an acronym for the most significant feature of laser action: lightamplification by stimulated emission of radiation. There are many different kinds oflaser, but they all share a crucial element: Each contains material capable of amplifyingradiation. This material is called the gain medium because radiation gains energy pas-sing through it. The physical principle responsible for this amplification is called stimu-lated emission and was discovered by Albert Einstein in 1916. It was widely recognizedthat the laser would represent a scientific and technological step of the greatest magni-tude, even before the first one was constructed in 1960 by T. H. Maiman. The award ofthe 1964 Nobel Prize in physics to C. H. Townes, N. G. Basov, and A. M. Prokhorovcarried the citation “for fundamental work in the field of quantum electronics, whichhas led to the construction of oscillators and amplifiers based on the maser-laserprinciple.” These oscillators and amplifiers have since motivated and aided the workof thousands of scientists and engineers.

In this chapter wewill undertake a superficial introduction to lasers, cutting corners atevery opportunity. We will present an overview of the properties of laser light, with thegoal of understanding what a laser is, in the simplest terms. Wewill introduce the theoryof light in cavities and of cavity modes, and we will describe an elementary theory oflaser action.

We can begin our introduction with Fig. 1.1, which illustrates the four key elements ofa laser. First, a collection of atoms or other material amplifies a light signal directedthrough it. This is shown in Fig. 1.1a. The amplifying material is usually enclosed bya highly reflecting cavity that will hold the amplified light, in effect redirecting it throughthe medium for repeated amplifications. This refinement is indicated in Fig. 1.1b. Someprovision, as sketched in Fig. 1.1c, must be made for replenishing the energy of theamplifier that is being converted to light energy. And some means must be arrangedfor extracting in the form of a beam at least part of the light stored in the cavity, perhapsas shown in Fig. 1.1d. A schematic diagram of an operating laser embodying all theseelements is shown in Fig. 1.2.

It is clear that a well-designed laser must carefully balance gains and losses. It can beanticipated with confidence that every potential laser system will present its designerwith more sources of loss than gain. Lasers are subject to the basic laws of physics,and every stage of laser operation from the injection of energy into the amplifyingmedium to the extraction of light from the cavity is an opportunity for energy loss

Laser Physics. By Peter W. Milonni and Joseph H. EberlyCopyright # 2010 John Wiley & Sons, Inc.

1

and entropy gain. One can say that the success of masers and lasers came only afterphysicists learned how atoms could be operated efficiently as thermodynamic engines.

One of the challenges in understanding the behavior of atoms in cavities arises fromthe strong feedback deliberately imposed by the cavity designer. This feedback meansthat a small input can be amplified in a straightforward way by the atoms, but not inde-finitely. Simple amplification occurs only until the light field in the cavity is strongenough to affect the behavior of the atoms. Then the strength of the light as it acts onthe amplifying atoms must be taken into account in determining the strength of thelight itself. This sounds like circular reasoning and in a sense it is. The responses ofthe light and the atoms to each other can become so strongly interconnected that theycannot be determined independently but only self-consistently. Strong feedback alsomeans that small perturbations can be rapidly magnified. Thus, it is accurate to anticipatethat lasers are potentially highly erratic and unstable devices. In fact, lasers can providedramatic exhibitions of truly chaotic behavior and have been the objects of fundamentalstudy for this reason.

For our purposes lasers are principally interesting, however, when they operate stably,with well-determined output intensity and frequency as well as spatial mode structure.

Atoms

(a) (b)

(d )

(c)

Atoms

Atoms

1 2

4 3

in out

Figure 1.1 Basic elements of a laser.

High power flash lamp

Transparent medium or cellwith atoms, and light beingamplified

100%Mirror

90%Mirror

Outputof laser

Figure 1.2 Complete laser system, showing elements responsible for energy input, amplification,and output.

2 INTRODUCTION TO LASER OPERATION

The self-consistent interaction of light and atoms is important for these properties, andwe will have to be concerned with concepts such as gain, loss, threshold, steady state,saturation, mode structure, frequency pulling, and linewidth.

In the next few sections we sketch properties of laser light, discuss modes in cavities,and give a theory of laser action. This theory is not really correct, but it is realistic withinits own domain and has so many familiar features that it may be said to be “obvious.” Itis also significant to observe what is not explained by this theory and to observe the waysin which it is not fundamental but only empirical. These gaps and missing elements arean indication that the remaining chapters of the book may also be necessary.

1.2 LASERS AND LASER LIGHT

Many of the properties of laser light are special or extreme in one way or another. Inthis section we provide a brief overview of these properties, contrasting them with theproperties of light from more ordinary sources when possible.

Wavelength

Laser light is available in all colors from red to violet and also far outside these conven-tional limits of the optical spectrum.1 Over a wide portion of the available range laserlight is “tunable.” This means that some lasers (e.g., dye lasers) have the property ofemitting light at any wavelength chosen within a range of wavelengths. The longestlaser wavelength can be taken to be in the far infrared, in the neighborhood of 100–500mm. Devices producing coherent light at much longer wavelengths by the“maser–laser principle” are usually thought of as masers. The search for lasers withever shorter wavelengths is probably endless. Coherent stimulated emission in theXUV (extreme ultraviolet) or soft X-ray region (10–15 nm) has been reported.Appreciably shorter wavelengths, those characteristic of gamma rays, for example,may be quite difficult to reach.

Photon Energy

The energy of a laser photon is not different from the energy of an “ordinary” lightphoton of the same wavelength. A green–yellow photon, roughly in the middle of theoptical spectrum, has an energy of about 2.5 eV (electron volts). This is the same asabout 4�10219 J ( joules) ¼ 4�10212 erg. The large exponents in the last two numbersmake it clear that electron volts are a much more convenient unit for laser photon energythan joules or ergs. From the infrared to the X-ray region photon energies vary fromabout 0.01 eV to about 100 eV. For contrast, at room temperature the thermal unit ofenergy is kT � 140 eV ¼ 0:025 eV. This is two orders of magnitude smaller than thetypical optical photon energy just mentioned, and as a consequence thermal excitationplays only a very small role in the physics of nearly all lasers.

1A list of laser wavelengths may be found in M. J. Weber, Handbook of Laser Wavelengths, CRC, BocaRaton, FL, 1999.

1.2 LASERS AND LASER LIGHT 3

Directionality

The output of a laser can consist of nearly ideal plane wavefronts. Only diffractionimposes a lower limit on the angular spread of a laser beam. The wavelength land the area A of the laser output aperture determine the order of magnitude of thebeam’s solid angle (DV) and vertex angle (Du) of divergence (Fig. 1.3) through therelation

DV � l2

A� (Du)2: (1:2:1)

This represents a very small angular spread indeed if l is in the optical range, say500 nm, and A is macroscopic, say (5 mm)2. In this example we compute DV �(500)2�10218 m2/(52�1026 m2) ¼ 1028 sr, or Du ¼ 1/10 mrad.

Monochromaticity

It is well known that lasers produce very pure colors. If they could produce exactly onewavelength, laser light would be fully monochromatic. This is not possible, in principleas well as for practical reasons. We will designate by Dl the range of wavelengthsincluded in a laser beam of main wavelength l. Similarly, the associated range offrequencies will be designated by Dn, the bandwidth. In the optical region of the spec-trum we can take n � 5�1014 Hz (hertz, i.e., cycles per second). The bandwidth of sun-light is very broad, more than 1014 Hz. Of course, filtered sunlight is a different matter,and with sufficiently good filters Dn could be reduced a great deal. However, the cost inlost intensity would usually be prohibitive. (See the discussion on spectral brightnessbelow.) For lasers, a very low value of Dn is 1 Hz, while a bandwidth around 100 Hzis spectroscopically practical in some cases (Fig. 1.4). ForDn ¼ 100Hz the relative spec-tral purity of a laser beam is quite impressive: Dn/n � 100/(5�1014) ¼ 2�10213.

A

Dq

Figure 1.3 Sketch of a laser cavity showing angular beam divergence Du at the output mirror(area A).

SunDn ~ 1014 Hz

Dn ~ 100 Hz

n

Laser

Figure 1.4 Spectral emission bands of the sun and of a representative laser, to indicate the muchcloser approach to monochromatic light achieved by the laser.

4 INTRODUCTION TO LASER OPERATION

This exceeds the spectral purity (Q factor) achievable in conventional mechanical andelectrical resonators by many orders of magnitude.

Coherence Time

The existence of a finite bandwidth Dn means that the different frequencies present ina laser beam can eventually get out of phase with each other. The time required fortwo oscillations differing in frequency by Dn to get out of phase by a full cycle isobviously 1/Dn. After this amount of time the different frequency components in thebeam can begin to interfere destructively, and the beam loses “coherence.” Thus,Dt ¼ 1/Dn is called the beam’s coherence time. This is a general definition, notrestricted to laser light, but the extremely small values possible for Dn in laser lightmake the coherence times of laser light extraordinarily long.

For example, even a “broadband” laser with Dn � 1 MHz has the coherence timeDt� 1 ms. This is enormously longer than most “typical” atomic fluorescence lifetimes,which are measured in nanoseconds (1029 s). Thus even lasers that are not close to thelimit of spectral purity are nevertheless effectively 100% pure on the relevantspectroscopic time scale. By way of contrast, sunlight has a bandwidth Dn almost asgreat as its central frequency (yellow light, n ¼ 5�1014 Hz). Thus, for sunlight thecoherence time isDt � 2�10215 s, so short that unfiltered sunlight cannot be consideredtemporally coherent at all.

Coherence Length

The speed of light is so great that a light beam can travel a very great distance withineven a short coherence time. For example, within Dt � 1ms light travels Dz �(3�108 m/s)� (1ms) ¼ 300 m. The distance Dz ¼ c Dt is called the beam’s coherencelength. Only portions of the same beam that are separated by less than Dz are capable ofinterfering constructively with each other. No fringes will be recorded by the film inFig. 1.5, for example, unless 2L, c Dt ¼ Dz.

Spectral Brightness

A light beam from a finite source can be characterized by its beam divergence DV,source size (usually surface area A), bandwidth Dn, and spectral power density Pn(watts per hertz of bandwidth). From these parameters it is useful to determine the spec-tral brightness bn of the source, which is defined (Fig. 1.6) to be the power flow per unit

Film

Beam splitter

L

L

Figure 1.5 Two-beam interferometer showing interference fringes obtained at the recording plane ifthe coherence length of the light is great enough.

1.2 LASERS AND LASER LIGHT 5

area, unit bandwidth, and steradian, namely bn ¼ Pn/A DVDn. Notice that Pn/A Dn isthe spectral intensity, so bn can also be thought of as the spectral intensity per steradian.

For an ordinary nonlaser optical source, brightness can be estimated directly from theblackbody formula for r(n), the spectral energy density (J/m3-Hz):

r(n) ¼ 8pn2

c3hn

ehn=kBT�1 : (1:2:2)

The spectral intensity (W/m2-Hz) is thus cr, and cr/DV is the desired spectral intensityper steradian. Taking DV ¼ 4p for a blackbody, we have

bn ¼2n2

c2hn

ehn=kBT�1 : (1:2:3)

The temperature of the sun is about T ¼ 5800K � 20�(300K). Since the main solaremission is in the yellow portion of the spectrum, we can take hn � 2.5 eV. We recallthat kBT � 140 eV for T ¼ 300K, so hn/kBT � 5, giving ehn=kBT � 150 and finally

bn � 1:5� 10�8 W=m2-sr-Hz (sun): (1:2:4)

Several different estimates can bemade for laser radiation, depending on the type of laserconsidered. Consider first a low-power He–Ne laser. A power level of 1 mW is normal,with a bandwidth of around 104 Hz. From (1.2.1) we see that the product of beamcross-sectional area and solid angle is just l2, which for He–Ne light is l2 � (6328�10210 m)2 � 4�10213 m2. Combining these, we find

bn � 2:5� 105 W=m2-sr-Hz (He–Ne laser): (1:2:5)

Another common laser is the mode-locked neodymium–glass laser, which can easilyreach power levels around 104 MW. The bandwidth of such a laser is limited by thepulse duration, say tp � 30 ps (30�10212 s), as follows. Since the laser’s coherencetime Dt is equal to tp at most, its bandwidth is certainly greater than 1/tp �3.3�1010 s21. We convert from radians per second to cycles per second by dividingby 2p and get Dn � 5�109 Hz. The wavelength of a Nd : glass laser is 1.06mm, sol2 � 10212 m2. The result of combining these, again using ADV ¼ l2, is

bn � 2� 1012 W=m2-sr-Hz (Nd : glass laser): (1:2:6)

A

DW

Figure 1.6 Geometrical construction showing source area and emission solid angle appropriate todiscussion of spectral brightness.

6 INTRODUCTION TO LASER OPERATION

Recent developments have led to lasers with powers of terawatts (1012 W) and evenpetawatts (1015 W), so bn can be even orders of magnitude larger.

It is clear that in terms of brightness there is practically no comparison possiblebetween lasers and thermal light. Our sun is 20 orders of magnitude less bright thana mode-locked laser. This raises an interesting question of principle. Let us imagine athermal light source filtered and collimated to the bandwidth and directionality ofa He–Ne laser, and the He–Ne laser attenuated to the brightness level of the thermallight. The question is: Could the two light beams with equal brightness, beamdivergence, polarization, and bandwidth be distinguished in any way? The answer isthat they could be distinguished, but not by any ordinary measurement of optics.Differences would show up only in the statistical fluctuations in the light beam. Thesefluctuations can reflect the quantum nature of the light source and are detected byphoton counting, as discussed in Chapter 12.

Active Medium

The materials that can be used as the active medium of a laser are so varied that a listingis hardly possible. Gases, liquids, and solids of every sort have been made to lase (a verbcontributed to science by the laser). The origin of laser photons, as shown in Fig. 1.7, ismost often in a transition between discrete upper and lower energy states in the medium,regardless of its state of matter. He–Ne, ruby, CO2, and dye lasers are familiar examples,but exceptions are easily found: The excimer laser has an unbound lower state, the semi-conductor diode laser depends on transitions between electron bands rather than discretestates, and understanding the free-electron laser does not require quantum states at all.

Type of Laser Cavity

All laser cavities share two characteristics that complement each other: (1) They arebasically linear devices with one relatively long optical axis, and (2) the sides parallelto this axis can be open, not enclosed by reflecting material as in a microwave cavity.There is no single best shape implied by these criteria, and in the case of ring lasersthe long axis actually bends and closes on itself (Fig. 1.8). Despite what may seem

hn E2 – E1

E2

E1

Figure 1.7 Photon emission accompanying a quantum jump from level 2 to level 1.

Figure 1.8 Two collections of mirrors making laser cavities, showing standing-wave and traveling-wave (ring) configurations on left and right, respectively.

1.2 LASERS AND LASER LIGHT 7

obvious, it is not always best to design a cavity with the lowest loss. In the case of Qswitching an extra loss is temporarily introduced into the cavity for the laser to over-come, and very high-power lasers sometimes use mirrors that are deliberately designedto deflect light out of the cavity rather than contain it.

Applications of Lasers

There is apparently no end of possible applications of lasers. Many of the uses of lasersare well known by now to most people, such as for various surgical procedures, forholography, in ultrasensitive gyroscopes, to provide straight lines for surveying, insupermarket checkout scanners and compact disc players, for welding, drilling, andscribing, in compact death-ray pistols, and so on. (The sophisticated student knows,even before reading this book, that one of these “well-known” applications has neverbeen realized outside the movie theater.)

1.3 LIGHT IN CAVITIES

In laser technology the terms cavity and resonator are used interchangeably. The theoryand design of the cavity are important enough for us to devote all of Chapter 7 to them.In this section we will consider only a simplified theory of resonators, a theory thatis certain to be at least partly familiar to most readers. This simplification allows us tointroduce the concept of cavity modes and to infer certain features of cavity modesthat remain valid in more general circumstances. We also describe the great advantageof open, rather than closed, cavities for optical radiation.

We will consider only the case of a rectangular “empty cavity” containing radiationbut no matter, as sketched in Fig. 1.9. The assumption that there is radiation but nomatter inside the cavity is obviously an approximation if the cavity is part of a workinglaser. This approximation is used frequently in laser theory, and it is accurate enough formany purposes because laser media are usually only sparsely filled with active atoms ormolecules.

Lz

Ly

Lx

y

z

x

0

Figure 1.9 Rectangular cavity with side lengths Lx, Ly, Lz.

8 INTRODUCTION TO LASER OPERATION

In Chapter 7 full solutions for the electric field in cavities of greatest interest are given.For example, the z dependence of the x component of the field takes the form

Ex(z) ¼ E0 sin kzz, (1:3:1)where E0 is a constant. However, here we are interested only in the simplest features ofthe cavity field, and these can be obtained easily by physical reasoning.

The electric field should vanish at both ends of the cavity. It will do so if we fit exactlyan integer number of half wavelengths into the cavity along each of its axes. This means,for example, that l along the z axis is determined by the relation L ¼ n(l/2), where n ¼1, 2, . . . , is a positive integer and L is the cavity length. If we use the relation betweenwave vector and wavelength, k ¼ 2p/l, this is the same as

kz ¼ pL n, (1:3:2)

for the z component of the wave vector. By substitution into the solution (1.3.1) wesee that (1.3.2) is sufficient to guarantee that the required boundary condition is met,i.e., that Ex(z) ¼ 0 for both z ¼ 0 and z ¼ L.

If there were reflecting sides to a laser cavity, the same would apply to the x and ycomponents of the wave vector. As wewill show later, if the three dimensions are equiv-alent in this sense, the number of available modes grows extremely rapidly as a functionof frequency. For example, a cubical three-dimensionally reflecting cavity 1 cm on a sidehas about 400 million resonant frequencies within the useful gain band of a He–Nelaser. Then lasing could occur across the whole band, eliminating any possibility ofachieving the important narrow-band, nearly monochromatic character of laser lightthat we emphasized in the preceding sections.

The solution to this multimode dilemma was suggested independently in 1958by Townes and A. L. Schawlow, R. H. Dicke, and Prokhorov. They recognized that aone-dimensional rather than a three-dimensional cavity was desirable, and that thiscould be achieved with an open resonator consisting of two parallel mirrors, as inFig. 1.10. The difference in wave vector between twomodes of a linear cavity, accordingto Eq. (1.3.2), is just p/L, so the mode spacing is given by Dk ¼ (2p/c) Dn, or Dn ¼c/2L. For L ¼ 10 cm we find

Dn ¼ 3� 108 m=s

(2)(0:10 m)¼ 1500MHz, (1:3:3)

for the separation in frequency of adjacent resonator modes. As indicated in Fig. 1.11,the number of possible modes that can lase is therefore at most

1500MHz1500MHz

¼ 1: (1:3:4)

(a) (b)

Figure 1.10 Sketch illustrating the advantage of a one-dimensional cavity. Stable modes areassociated only with beams that are retroreflected many times.

1.3 LIGHT IN CAVITIES 9

The maximum number, including two choices of polarization, is therefore 2, consider-ably smaller than the estimate of 400 million obtained for three-dimensional cavities.

These results do not include the effects of diffraction of radiation at the mirroredges. Diffraction determines the x, y dependence of the field, which we have ignoredcompletely. Accurate calculations of resonator modes, including diffraction, are oftendone with computers. Such calculations were first made in 1961 for the plane-parallelresonator of Fig. 1.10 with either rectangular or circular mirrors. Actually lasers areseldom designed with flat mirrors. Laser resonator mirrors are usually sphericalsurfaces, for reasons to be discussed in Chapter 7. A great deal about laser cavitiescan nevertheless be understood without worrying about diffraction or mirror shape. Inparticular, for most practical purposes, the mode-frequency spacing is given accuratelyenough by Dn ¼ c/2L.

1.4 LIGHT EMISSION AND ABSORPTION IN QUANTUM THEORY

The modern interpretation of light emission and absorption was first proposed byEinstein in 1905 in his theory of the photoelectric effect. Einstein assumed the differencein energy of the electron before and after its photoejection to be equal to the energy hn ofthe photon absorbed in the process.

This picture of light absorption was extended in twoways by Bohr: to apply to atomicelectrons that are not ejected during photon absorption but instead take on a higherenergy within their atom, and to apply to the reverse process of photon emission, inwhich case the energy of the electron should decrease. These extensions of Einstein’sidea fitted perfectly into Bohr’s quantum mechanical model of an atom in 1913.This model, described in detail in Chapter 2, was the first to suggest that electronsare restricted to a certain fixed set of orbits around the atomic nucleus. This set oforbits was shown to correspond to a fixed set of allowed electron energies. The ideaof a “quantum jump” was introduced to describe an electron’s transition between twoallowed orbits.

The amount of energy involved in a quantum jump depends on the quantum system.Atoms have quantum jumps whose energies are typically in the range 1–6 eV, as longas an outer-shell electron is doing the jumping. This is the ordinary case, so atomsusually absorb and emit photons in or near the optical region of the spectrum. Jumpsby inner-shell atomic electrons usually require much more energy and are associatedwith X-ray photons. On the other hand, quantum jumps among the so-called Rydbergenergy levels, those outer-electron levels lying far from the ground level and near to

1500 MHz

Gain curve

Cavity modefrequencies

n

Figure 1.11 Mode frequencies separated by 1500 MHz, corresponding to a 10-cm one-dimensionalcavity. A 1500-MHz gain curve overlaps only 1 mode.

10 INTRODUCTION TO LASER OPERATION

the ionization limit, involve only a small amount of energy, corresponding to far-infraredor even microwave photons.

Molecules have vibrational and rotational degrees of freedom whose quantum jumpsare smaller (perhaps much smaller) than the quantum jumps in free atoms, and the sameis often true of jumps between conduction and valence bands in semiconductors. Manycrystals are transparent in the optical region, which is a sign that they do not absorbor emit optical photons, because they do not have quantum energy levels that permitjumps in the optical range. However, colored crystals such as ruby have impuritiesthat do absorb and emit optical photons. These impurities are frequently atomic ions,and they have both discrete energy levels and broad bands of levels that allow opticalquantum jumps (ruby is a good absorber of green photons and so appears red).

1.5 EINSTEIN THEORY OF LIGHT–MATTER INTERACTIONS

The atoms of a laser undergo repeated quantum jumps and so act as microscopictransducers. That is, each atom accepts energy and jumps to a higher orbit as a resultof some input or “pumping” process and converts it into other forms of energy—forexample, into light energy (photons)—when it jumps to a lower orbit. At the sametime, each atom must deal with the photons that have been emitted earlier and reflectedback by the mirrors. These prior photons, already channeled along the cavity axis, arethe origin of the stimulated component to the atom’s emission of subsequent photons.

In Fig. 1.12 we indicate some ways in which energy conversion can occur. For sim-plicity we focus our attention on quantum jumps between two energy levels, 1 and 2, ofan atom. The five distinct energy conversion diagrams of Fig. 1.12 are interpreted asfollows:

(a) Absorption of an increment DE ¼ E22E1 of energy from the pump: The atom israised from level 1 to level 2. In other words, an electron in the atom jumps froman inner orbit to an outer orbit.

(b) Spontaneous emission of a photon of energy hn ¼ E22E1: The atom jumpsdown from level 2 to the lower level 1. The process occurs “spontaneously” with-out any external influence.

(c) Stimulated emission: The atom jumps down from energy level 2 to the lowerlevel 1, and the emitted photon of energy hn ¼ E22E1 is an exact replicaof a photon already present. The process is induced, or stimulated, by theincident photon.

(d) Absorption of a photon of energy hn ¼ E22E1: The atom jumps up from level 1to the higher level 2. As in (c), the process is induced by an incident photon.

(e) Nonradiative deexcitation: The atom jumps down from level 2 to the lower level 1,but no photon is emitted so the energy E22E1 must appear in some other form[e.g., increased vibrational or rotational energy in the case of a molecule, orrearrangement (“shakeup”) of other electrons in the atom].

All these processes occur in the gain medium of a laser. Lasers are often classifiedaccording to the nature of the pumping process (a) which is the source of energy forthe output laser beam. In electric-discharge lasers, for instance, the pumping occurs

1.5 EINSTEIN THEORY OF LIGHT–MATTER INTERACTIONS 11

as a result of collisions of electrons in a gaseous discharge with the atoms (or molecules)of the gain medium. In an optically pumped laser the pumping process is the same asthe absorption process (d), except that the pumping photons are supplied by a lamp orperhaps another laser. In a diode laser an electric current at the junction of two differentsemiconductors produces electrons in excited energy states from which they can jumpinto lower energy states and emit photons.

This quantum picture is consistent with a highly simplified description of laser action.Suppose that lasing occurs on the transition defined by levels 1 and 2 of Fig. 1.12. In themost favorable situation the lower level (level 1) of the laser transition is empty. Tomaintain this situation a mechanism must exist to remove downward jumping electronsfrom level 1 to another level, say level 0. In this situation there can be no detrimentalabsorption of laser photons due to transitions upward from level 1 to level 2. In practicethe number of electrons in level 1 cannot be exactly zero, but we will assume for sim-plicity that the rate of deexcitation of the lower level 1 is so large that the number ofatoms remaining in that level is negligible compared to the number in level 2; this isa reasonably good approximation for many lasers. Under this approximation laseraction can be described in terms of two “populations”: the number n of atoms in theupper level 2 and the number q of photons in the laser cavity.

The number of laser photons in the cavity changes for two main reasons:

(i) Laser photons are continually being added because of stimulated emission.(ii) Laser photons are continually being lost because of mirror transmission, scattering

or absorption at the mirrors, etc.

E2

E1

E2

E1

hn

E2 hn

hn

hn

(a) (b)

(c)

(d ) (e)

hn E1

E2

E1

E2

E1

Figure 1.12 Energy conversion processes in a lasing atom or molecule: (a) absorption of energyDE ¼ E2 2 E1 from the pump; (b) spontaneous emission of a photon of energyDE; (c) stimulated emis-sion of a photon of energy DE; (d ) absorption of a photon of energy DE; (e) nonradiative deexcitation.

12 INTRODUCTION TO LASER OPERATION