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Laser Physics
Laser Physics
Milonni P.W., Eberly J.H.
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Although the basic principles of lasers have remained unchanged in the past 20 years, there has been a shift in the kinds of lasers generating interest. Providing a comprehensive introduction to the operating principles and applications of lasers, this second edition of the classic book on the subject reveals the latest developments and applications of lasers. Placing more emphasis on applications of lasers and on optical physics, the book's selfcontained discussions will appeal to physicists, chemists, optical scientists, engineers, and advanced undergraduate students.
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Год:
2010
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2
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Wiley
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english
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850
ISBN 10:
0470387718
ISBN 13:
9780470387719
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radiation^{771}
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optical^{569}
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propagation^{384}
transition^{376}
cavity^{371}
quantum^{370}
gaussian^{331}
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photons^{319}
atomic^{295}
pulses^{294}
cos^{293}
wavelength^{284}
electrons^{278}
velocity^{269}
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polarization^{264}
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density^{244}
magnetic^{240}
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dispersion^{212}
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molecules^{201}
probability^{195}
spatial^{194}
oscillator^{194}
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formula^{186}
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nonlinear^{183}
mirrors^{175}
a21^{175}
harmonic^{168}
pumping^{168}
diffraction^{165}
transitions^{163}
bandwidth^{163}
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LASER PHYSICS PETER W. MILONNI JOSEPH H. EBERLY 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 Jersey Published simultaneously in Canada 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, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate percopy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 7508400, fax (978) 7504470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 7486011, fax (201) 7486008, or online at http://www.wiley. com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 7622974, outside the U; nited States at (317) 5723993 or fax (317) 5724002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress CataloginginPublication Data: Milonni, Peter W. Laser physics / Peter W. Milonni, Joseph H. Eberly p. cm. Includes bibliographical references and index. ISBN 9780470387719 (cloth) 1. Lasers. 2. Nonlinear optics. 3. Physical optics. I. Eberly, J. H., 1935 II. Title. QC688.M55 2008 621.360 6—dc22 2008026771 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To our wives, MeiLi and Shirley CONTENTS Preface xiii 1 Introduction to Laser Operation 1.1 1.2 1.3 1.4 1.5 1.6 1 Introduction, 1 Lasers and Laser Light, 3 Light in Cavities, 8 Light Emission and Absorption in Quantum Theory, 10 Einstein Theory of Light–Matter Interactions, 11 Summary, 14 2 Atoms, Molecules, and Solids 17 2.1 Introduction, 17 2.2 Electron Energy Levels in Atoms, 17 2.3 Molecular Vibrations, 26 2.4 Molecular Rotations, 31 2.5 Example: Carbon Dioxide, 33 2.6 Conductors and Insulators, 35 2.7 Semiconductors, 39 2.8 Semiconductor Junctions, 45 2.9 LightEmitting Diodes, 49 2.10 Summary, 55 Appendix: Energy Bands in Solids, 56 Problems, 64 3 Absorption, Emission, and Dispersion of Light 67 3.1 Introduction, 67 3.2 Electron Oscillator Model, 69 vii viii CONTENTS 3.3 Spontaneous Emission, 74 3.4 Absorption, 78 3.5 Absorption of Broadband Light, 84 3.6 Thermal Radiation, 85 3.7 Emission and Absorption of Narrowband Light, 93 3.8 Collision Broadening, 99 3.9 Doppler Broadening, 105 3.10 The Voigt Proﬁle, 108 3.11 Radiative Broadening, 112 3.12 Absorption and Gain Coefﬁcients, 114 3.13 Example: Sodium Vapor, 118 3.14 Refractive Index, 123 3.15 Anomalous Dispersion, 129 3.16 Summary, 132 Appendix: The Oscillator Model and Quantum Theory, 132 Problems, 137 4 Laser Oscillation: Gain and Threshold 141 4.1 Introduction, 141 4.2 Gain and Feedback, 141 4.3 Threshold, 143 4.4 Photon Rate Equations, 148 4.5 Population Rate Equations, 150 4.6 Comparison with Chapter 1, 152 4.7 ThreeLevel Laser Scheme, 153 4.8 FourLevel Laser Scheme, 156 4.9 Pumping Three and FourLevel Lasers, 157 4.10 Examples of Three and FourLevel Lasers, 159 4.11 Saturation, 161 4.12 SmallSignal Gain and Saturation, 164 4.13 Spatial Hole Burning, 167 4.14 Spectral Hole Burning, 169 4.15 Summary, 172 Problems, 173 5 Laser Oscillation: Power and Frequency 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 Introduction, 175 UniformField Approximation, 175 Optimal Output Coupling, 178 Effect of Spatial Hole Burning, 180 Large Output Coupling, 183 Measuring Gain and Optimal Output Coupling, 187 Inhomogeneously Broadened Media, 191 Spectral Hole Burning and the Lamb Dip, 192 Frequency Pulling, 194 Obtaining SingleMode Oscillation, 198 The Laser Linewidth, 203 Polarization and Modulation, 207 175 CONTENTS ix 5.13 Frequency Stabilization, 215 5.14 Laser at Threshold, 220 Appendix: The FabryPérot Etalon, 223 Problems, 226 6 Multimode and Pulsed Lasing 229 6.1 Introduction, 229 6.2 Rate Equations for Intensities and Populations, 229 6.3 Relaxation Oscillations, 230 6.4 Q Switching, 233 6.5 Methods of Q Switching, 236 6.6 Multimode Laser Oscillation, 237 6.7 PhaseLocked Oscillators, 239 6.8 Mode Locking, 242 6.9 AmplitudeModulated Mode Locking, 246 6.10 FrequencyModulated Mode Locking, 248 6.11 Methods of Mode Locking, 251 6.12 Ampliﬁcation of Short Pulses, 255 6.13 Ampliﬁed Spontaneous Emission, 258 6.14 Ultrashort Light Pulses, 264 Appendix: Diffraction of Light by Sound, 265 Problems, 266 7 Laser Resonators and Gaussian Beams 269 7.1 Introduction, 269 7.2 The Ray Matrix, 270 7.3 Resonator Stability, 274 7.4 The Paraxial Wave Equation, 279 7.5 Gaussian Beams, 282 7.6 The ABCD Law for Gaussian Beams, 288 7.7 Gaussian Beam Modes, 292 7.8 Hermite –Gaussian and Laguerre –Gaussian Beams, 298 7.9 Resonators for He–Ne Lasers, 306 7.10 Diffraction, 309 7.11 Diffraction by an Aperture, 312 7.12 Diffraction Theory of Resonators, 317 7.13 Beam Quality, 320 7.14 Unstable Resonators for HighPower Lasers, 321 7.15 Bessel Beams, 322 Problems, 327 8 Propagation of Laser Radiation 8.1 8.2 8.3 8.4 8.5 8.6 Introduction, 331 The Wave Equation for the Electric Field, 332 Group Velocity, 336 Group Velocity Dispersion, 340 Chirping, 351 Propagation Modes in Fibers, 355 331 x CONTENTS 8.7 SingleMode Fibers, 361 8.8 Birefringence, 365 8.9 Rayleigh Scattering, 372 8.10 Atmospheric Turbulence, 377 8.11 The Coherence Diameter, 379 8.12 Beam Wander and Spread, 388 8.13 Intensity Scintillations, 392 8.14 Remarks, 395 Problems, 397 9 Coherence in AtomField Interactions 401 9.1 Introduction, 401 9.2 TimeDependent Schrödinger Equation, 402 9.3 TwoState Atoms in Sinusoidal Fields, 403 9.4 Density Matrix and Collisional Relaxation, 408 9.5 Optical Bloch Equations, 414 9.6 Maxwell –Bloch Equations, 420 9.7 Semiclassical Laser Theory, 428 9.8 Resonant Pulse Propagation, 432 9.9 SelfInduced Transparency, 438 9.10 Electromagnetically Induced Transparency, 441 9.11 TransitTime Broadening and the Ramsey Effect, 446 9.12 Summary, 451 Problems, 452 10 Introduction to Nonlinear Optics 457 10.1 Model for Nonlinear Polarization, 457 10.2 Nonlinear Susceptibilities, 459 10.3 SelfFocusing, 464 10.4 SelfPhase Modulation, 469 10.5 SecondHarmonic Generation, 471 10.6 Phase Matching, 475 10.7 ThreeWave Mixing, 480 10.8 Parametric Ampliﬁcation and Oscillation, 482 10.9 TwoPhoton Downconversion, 486 10.10 Discussion, 492 Problems, 494 11 Some Speciﬁc Lasers and Ampliﬁers 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Introduction, 497 ElectronImpact Excitation, 498 Excitation Transfer, 499 He–Ne Lasers, 502 Rate Equation Model of Population Inversion in He–Ne Lasers, 505 Radial Gain Variation in He–Ne Laser Tubes, 509 CO2 ElectricDischarge Lasers, 513 GasDynamic Lasers, 515 497 CONTENTS xi 11.9 Chemical Lasers, 516 11.10 Excimer Lasers, 518 11.11 Dye Lasers, 521 11.12 Optically Pumped SolidState Lasers, 525 11.13 Ultrashort, Superintense Pulses, 532 11.14 Fiber Ampliﬁers and Lasers, 537 11.15 Remarks, 553 Appendix: Gain or Absorption Coefﬁcient for VibrationalRotational Transitions, 554 Problems, 558 12 Photons 561 12.1 What is a Photon, 561 12.2 Photon Polarization: All or Nothing, 562 12.3 Failures of Classical Theory, 563 12.4 Wave Interference and Photons, 567 12.5 Photon Counting, 569 12.6 The Poisson Distribution, 573 12.7 Photon Detectors, 575 12.8 Remarks, 585 Problems, 586 13 Coherence 589 13.1 Introduction, 589 13.2 Brightness, 589 13.3 The Coherence of Light, 592 13.4 The Mutual Coherence Function, 595 13.5 Complex Degree Of Coherence, 598 13.6 QuasiMonochromatic Fields and Visibility, 601 13.7 Spatial Coherence of Light From Ordinary Sources, 603 13.8 Spatial Coherence of Laser Radiation, 608 13.9 Diffraction of Laser Radiation, 610 13.10 Coherence and the Michelson Interferometer, 611 13.11 Temporal Coherence, 613 13.12 The Photon Degeneracy Factor, 616 13.13 Orders of Coherence, 619 13.14 Photon Statistics of Lasers and Thermal Sources, 620 13.15 Brown–Twiss Correlations, 627 Problems, 634 14 Some Applications of Lasers 14.1 14.2 14.3 14.4 14.5 14.6 Lidar, 637 Adaptive Optics for Astronomy, 648 Optical Pumping and SpinPolarized Atoms, 658 Laser Cooling, 671 Trapping Atoms with Lasers and Magnetic Fields, 685 Bose–Einstein Condensation, 690 637 xii CONTENTS 14.7 Applications of Ultrashort Pulses, 697 14.8 Lasers in Medicine, 718 14.9 Remarks, 728 Problems, 729 15 Diode Lasers and Optical Communications 735 15.1 Introduction, 735 15.2 Diode Lasers, 736 15.3 Modulation of Diode Lasers, 754 15.4 Noise Characteristics of Diode Lasers, 760 15.5 Information and Noise, 774 15.6 Optical Communications, 782 Problems, 790 16 Numerical Methods for Differential Equations 793 16.A Fortran Program for Ordinary Differential Equations, 793 16.B Fortran Program for PlaneWave Propagation, 796 16.C Fortran Program for Paraxial Propagation, 799 Index 809 PREFACE Judged by their economic impact and their role in everyday life, and also by the number of Nobel Prizes awarded, advances in laser science and engineering in the past quartercentury have been remarkable. Using lasers, scientists have produced what are believed to be the coldest temperatures in the universe, and energy densities greater than in the center of stars; have tested the foundations of quantum theory itself; and have controlled atomic, molecular, and photonic states with unprecedented precision. Questions that previous generations of scientists could only contemplate in terms of thought experiments have been routinely addressed using lasers. Atomic clock frequencies 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 processes in action, and the recent availability of attosecond pulses is allowing the study of phenomena on the time scale of electron motion in atoms. Frequency stabilization and the frequencycomb spectra of modelocked lasers have now made practical the measurement of absolute optical frequencies and promise ever greater precision in spectroscopy and other areas. Lasers are being used in adaptive optical systems to obtain image resolution with groundbased telescopes that is comparable to that of telescopes in space, and they have become indispensable in lidar and environmental studies. Together with optical ﬁbers, diode lasers have fueled the explosive growth of optical networks and the Internet. In medicine, lasers are ﬁnding more and more uses in surgery and clinical procedures. Simply put, laser physics is an integral part of contemporary science 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 at more than a command of tables and formulas. Thus all of the chapters incorporate explanations of the central elements of optical engineering and physics that are required for a basic and detailed understanding of laser operation. Applications are important and we discuss how laser radiation interacts with matter, and how coherent and often very intense laser radiation is used in research and in the ﬁeld. We presume that the reader xiii xiv PREFACE has been exposed to classical electromagnetic theory and quantum mechanics at an undergraduate or beginning graduate level, but we take opportunities throughout to review parts of these subjects that are particularly important for laser physics. The perceptive reader will notice that there is substantial overlap with a book we wrote 20 years ago called simply Lasers, also published by Wiley and still in print without revision or addition. Many readers and users of that book have told us that they particularly appreciated the frequent concentration on background optical physics as well as explanations of the physical basis for all aspects of laser operation. Naturally a book about lasers that is two decades old needs many new topics to be added to be even approximately current. However, while recognizing that additions are necessary, we also wanted to resist what is close to a law of nature, that a second book must weigh signiﬁcantly more than its predecessor. We believe we have accomplished these goals by describing some of the most signiﬁcant recent developments in laser physics together with an illustrative set of applications based on them. The basic principles of lasers have not changed in the past twenty years, but there has been a shift in the kinds of lasers of greatest general interest. Considerable attention is devoted to semiconductor lasers and ﬁber lasers and ampliﬁers, and to considerations of noise and dispersion in ﬁberoptic communications. We also treat various aspects of chirping 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 other applications mentioned above. We introduce the most important concepts needed to understand the propagation of laser radiation in the turbulent atmosphere; this is an important topic for freespace communication, for example, but it has usually been addressed only in more advanced and specialized books. We have attempted to present it in a way that might be helpful for students as well as laser scientists and engineers with no prior exposure to turbulence theory. The book is designed as a textbook, but there is probably too much material here to be covered in a onesemester course. Chapters 1–7 could be used as a selfcontained, elementary introduction to lasers and laser—matter interactions. In most respects the remaining chapters are selfcontained, while using consistent notation and making reference to the same fundamentals. Chapters 9 and 10, for example, can serve as introductions to coherent propagation effects and nonlinear optics, respectively, and Chapters 12 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 will likely be of interest for many years to come. We are grateful to A. AlQasimi, 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 for comments on some of the chapters or for contributing in other ways to this effort. USEFUL TABLES TABLE 1 Physical Constants Velocity of light in vacuum Electron charge Coulomb force constant Electron rest mass Proton rest mass Bohr radius Planck’s constant Avogadro’s number Boltzmann constant Universal gas constant Stefan–Boltzmann constant c ¼ 2.998 108 m/s e ¼ 1.602 10219 C 1/4pe 0 ¼ 8.988 109 Nm2/C2 e 2/4pe 0 ¼ 1.440 eVnm me ¼ 9.108 10231 kg mp ¼ 1.672 10227 kg a0 ¼ 0.528 Å ¼ 0.0528 nm h ¼ 6.626 10234 Js h ¼ h/2p ¼ 1.054 10234 Js hc ¼ 1240 eVnm NA ¼ 6.023 1023 k ¼ 1.380 10223 J/K R ¼ NAk ¼ 8.314 J/K s ¼ 5.670 1028 Watt/m2K4 TABLE 2 Conversion Factors 1 electron volt (eV) ¼ 1:602 1019 joule (J) ¼ 1:16 104 K ¼ 2:42 1014 Hz ¼ 8:07 103 cm1 1 300 K ¼ 2:59 102 eV 40 eV 760 Torr ¼ 1.013 105 N/m2 TABLE 3 The Electromagnetic Spectrum Typical Wavelength (cm) Longwave radio AM radio FM radio Radar Microwave Infrared Light (orange) Ultraviolet Xrays Gamma rays Cosmicray photons 3 105 3 104 300 3 0.3 3 1024 6 1025 3 1026 3 1028 3 10211 3 10213 Frequency (Hz) 105 106 108 1010 1011 1014 5 1014 1016 1018 1021 1023 Photon Energy (eV) 4 10210 4 1029 4 1027 4 1025 4 1024 0.4 2 40 4000 4 106 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.” 1 1.1 INTRODUCTION TO LASER OPERATION INTRODUCTION The word laser is an acronym for the most signiﬁcant feature of laser action: light ampliﬁcation by stimulated emission of radiation. There are many different kinds of laser, but they all share a crucial element: Each contains material capable of amplifying radiation. This material is called the gain medium because radiation gains energy passing through it. The physical principle responsible for this ampliﬁcation is called stimulated emission and was discovered by Albert Einstein in 1916. It was widely recognized that the laser would represent a scientiﬁc and technological step of the greatest magnitude, even before the ﬁrst one was constructed in 1960 by T. H. Maiman. The award of the 1964 Nobel Prize in physics to C. H. Townes, N. G. Basov, and A. M. Prokhorov carried the citation “for fundamental work in the ﬁeld of quantum electronics, which has led to the construction of oscillators and ampliﬁers based on the maserlaser principle.” These oscillators and ampliﬁers have since motivated and aided the work of thousands of scientists and engineers. In this chapter we will undertake a superﬁcial introduction to lasers, cutting corners at every opportunity. We will present an overview of the properties of laser light, with the goal of understanding what a laser is, in the simplest terms. We will introduce the theory of light in cavities and of cavity modes, and we will describe an elementary theory of laser action. We can begin our introduction with Fig. 1.1, which illustrates the four key elements of a laser. First, a collection of atoms or other material ampliﬁes a light signal directed through it. This is shown in Fig. 1.1a. The amplifying material is usually enclosed by a highly reﬂecting cavity that will hold the ampliﬁed light, in effect redirecting it through the medium for repeated ampliﬁcations. This reﬁnement is indicated in Fig. 1.1b. Some provision, as sketched in Fig. 1.1c, must be made for replenishing the energy of the ampliﬁer that is being converted to light energy. And some means must be arranged for extracting in the form of a beam at least part of the light stored in the cavity, perhaps as shown in Fig. 1.1d. A schematic diagram of an operating laser embodying all these elements is shown in Fig. 1.2. It is clear that a welldesigned laser must carefully balance gains and losses. It can be anticipated with conﬁdence that every potential laser system will present its designer with 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 amplifying medium to the extraction of light from the cavity is an opportunity for energy loss Laser Physics. By Peter W. Milonni and Joseph H. Eberly Copyright # 2010 John Wiley & Sons, Inc. 1 2 INTRODUCTION TO LASER OPERATION in out 1 2 Atoms Atoms 4 (a) 3 (b) Atoms (d ) (c) Figure 1.1 Basic elements of a laser. and entropy gain. One can say that the success of masers and lasers came only after physicists learned how atoms could be operated efﬁciently as thermodynamic engines. One of the challenges in understanding the behavior of atoms in cavities arises from the strong feedback deliberately imposed by the cavity designer. This feedback means that a small input can be ampliﬁed in a straightforward way by the atoms, but not indeﬁnitely. Simple ampliﬁcation occurs only until the light ﬁeld in the cavity is strong enough to affect the behavior of the atoms. Then the strength of the light as it acts on the amplifying atoms must be taken into account in determining the strength of the light itself. This sounds like circular reasoning and in a sense it is. The responses of the light and the atoms to each other can become so strongly interconnected that they cannot be determined independently but only selfconsistently. Strong feedback also means that small perturbations can be rapidly magniﬁed. Thus, it is accurate to anticipate that lasers are potentially highly erratic and unstable devices. In fact, lasers can provide dramatic exhibitions of truly chaotic behavior and have been the objects of fundamental study for this reason. For our purposes lasers are principally interesting, however, when they operate stably, with welldetermined output intensity and frequency as well as spatial mode structure. 100% Mirror 90% Mirror Transparent medium or cell with atoms, and light being amplified Output of laser High power flash lamp Figure 1.2 Complete laser system, showing elements responsible for energy input, ampliﬁcation, and output. 1.2 LASERS AND LASER LIGHT 3 The selfconsistent interaction of light and atoms is important for these properties, and we 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 within its own domain and has so many familiar features that it may be said to be “obvious.” It is also signiﬁcant to observe what is not explained by this theory and to observe the ways in which it is not fundamental but only empirical. These gaps and missing elements are an 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. In this section we provide a brief overview of these properties, contrasting them with the properties 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 conventional limits of the optical spectrum.1 Over a wide portion of the available range laser light is “tunable.” This means that some lasers (e.g., dye lasers) have the property of emitting light at any wavelength chosen within a range of wavelengths. The longest laser wavelength can be taken to be in the far infrared, in the neighborhood of 100– 500 mm. Devices producing coherent light at much longer wavelengths by the “maser–laser principle” are usually thought of as masers. The search for lasers with ever shorter wavelengths is probably endless. Coherent stimulated emission in the XUV (extreme ultraviolet) or soft Xray 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” light photon of the same wavelength. A green–yellow photon, roughly in the middle of the optical spectrum, has an energy of about 2.5 eV (electron volts). This is the same as about 410219 J ( joules) ¼ 410212 erg. The large exponents in the last two numbers make it clear that electron volts are a much more convenient unit for laser photon energy than joules or ergs. From the infrared to the Xray region photon energies vary from about 0.01 eV to about 100 eV. For contrast, at room temperature the thermal unit of 1 eV ¼ 0:025 eV. This is two orders of magnitude smaller than the energy is kT 40 typical optical photon energy just mentioned, and as a consequence thermal excitation plays only a very small role in the physics of nearly all lasers. 1 A list of laser wavelengths may be found in M. J. Weber, Handbook of Laser Wavelengths, CRC, Boca Raton, FL, 1999. 4 INTRODUCTION TO LASER OPERATION Directionality The output of a laser can consist of nearly ideal plane wavefronts. Only diffraction imposes a lower limit on the angular spread of a laser beam. The wavelength l and the area A of the laser output aperture determine the order of magnitude of the beam’s solid angle (DV) and vertex angle (Du) of divergence (Fig. 1.3) through the relation DV l2 (Du)2 : A (1:2:1) This represents a very small angular spread indeed if l is in the optical range, say 500 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 one wavelength, laser light would be fully monochromatic. This is not possible, in principle as well as for practical reasons. We will designate by Dl the range of wavelengths included in a laser beam of main wavelength l. Similarly, the associated range of frequencies will be designated by Dn, the bandwidth. In the optical region of the spectrum we can take n 51014 Hz (hertz, i.e., cycles per second). The bandwidth of sunlight 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 in lost intensity would usually be prohibitive. (See the discussion on spectral brightness below.) For lasers, a very low value of Dn is 1 Hz, while a bandwidth around 100 Hz is spectroscopically practical in some cases (Fig. 1.4). For Dn ¼ 100 Hz the relative spectral purity of a laser beam is quite impressive: Dn/n 100/(51014) ¼ 210213. Dq A Figure 1.3 Sketch of a laser cavity showing angular beam divergence Du at the output mirror (area A). Laser Sun Dn ~ 100 Hz Dn ~ 1014 Hz n Figure 1.4 Spectral emission bands of the sun and of a representative laser, to indicate the much closer approach to monochromatic light achieved by the laser. 1.2 LASERS AND LASER LIGHT 5 This exceeds the spectral purity (Q factor) achievable in conventional mechanical and electrical resonators by many orders of magnitude. Coherence Time The existence of a ﬁnite bandwidth Dn means that the different frequencies present in a laser beam can eventually get out of phase with each other. The time required for two oscillations differing in frequency by Dn to get out of phase by a full cycle is obviously 1/Dn. After this amount of time the different frequency components in the beam 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, not restricted to laser light, but the extremely small values possible for Dn in laser light make the coherence times of laser light extraordinarily long. For example, even a “broadband” laser with Dn 1 MHz has the coherence time Dt 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 the limit of spectral purity are nevertheless effectively 100% pure on the relevant spectroscopic time scale. By way of contrast, sunlight has a bandwidth Dn almost as great as its central frequency (yellow light, n ¼ 51014 Hz). Thus, for sunlight the coherence time is Dt 210215 s, so short that unﬁltered sunlight cannot be considered temporally coherent at all. Coherence Length The speed of light is so great that a light beam can travel a very great distance within even a short coherence time. For example, within Dt 1 ms light travels Dz (3108 m/s) (1 ms) ¼ 300 m. The distance Dz ¼ c Dt is called the beam’s coherence length. Only portions of the same beam that are separated by less than Dz are capable of interfering constructively with each other. No fringes will be recorded by the ﬁlm in Fig. 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 spectral brightness bn of the source, which is deﬁned (Fig. 1.6) to be the power ﬂow per unit Beam splitter L L Film Figure 1.5 Twobeam interferometer showing interference fringes obtained at the recording plane if the coherence length of the light is great enough. 6 INTRODUCTION TO LASER OPERATION DW A Figure 1.6 Geometrical construction showing source area and emission solid angle appropriate to discussion of spectral brightness. area, unit bandwidth, and steradian, namely bn ¼ Pn/A DV Dn. Notice that Pn/A Dn is the 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 the blackbody formula for r(n), the spectral energy density (J/m3Hz): r(n) ¼ 8pn2 hn : c3 ehn=kB T 1 (1:2:2) The spectral intensity (W/m2Hz) is thus cr, and cr/DV is the desired spectral intensity per steradian. Taking DV ¼ 4p for a blackbody, we have bn ¼ 2n2 hn : 2 hn=k c e B T 1 (1:2:3) The temperature of the sun is about T ¼ 5800K 20(300K). Since the main solar emission is in the yellow portion of the spectrum, we can take hn 2.5 eV. We recall 1 eV for T ¼ 300K, so hn/kBT 5, giving ehn=kB T 150 and ﬁnally that kB T 40 bn 1:5 108 W=m2 srHz (sun): (1:2:4) Several different estimates can be made for laser radiation, depending on the type of laser considered. Consider ﬁrst a lowpower 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 beam crosssectional area and solid angle is just l2, which for He–Ne light is l2 (6328 10210 m)2 410213 m2. Combining these, we ﬁnd bn 2:5 105 W=m2 srHz (He – Ne laser): (1:2:5) Another common laser is the modelocked neodymium–glass laser, which can easily reach power levels around 104 MW. The bandwidth of such a laser is limited by the pulse duration, say tp 30 ps (3010212 s), as follows. Since the laser’s coherence time 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 dividing by 2p and get Dn 5109 Hz. The wavelength of a Nd : glass laser is 1.06 mm, so l2 10212 m2. The result of combining these, again using A DV ¼ l2, is bn 2 1012 W=m2 srHz (Nd : glass laser): (1:2:6) 1.2 LASERS AND LASER LIGHT 7 Recent developments have led to lasers with powers of terawatts (1012 W) and even petawatts (1015 W), so bn can be even orders of magnitude larger. It is clear that in terms of brightness there is practically no comparison possible between lasers and thermal light. Our sun is 20 orders of magnitude less bright than a modelocked laser. This raises an interesting question of principle. Let us imagine a thermal light source ﬁltered and collimated to the bandwidth and directionality of a He–Ne laser, and the He –Ne laser attenuated to the brightness level of the thermal light. The question is: Could the two light beams with equal brightness, beam divergence, polarization, and bandwidth be distinguished in any way? The answer is that 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. These ﬂuctuations can reﬂect the quantum nature of the light source and are detected by photon 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 listing is hardly possible. Gases, liquids, and solids of every sort have been made to lase (a verb contributed to science by the laser). The origin of laser photons, as shown in Fig. 1.7, is most 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 semiconductor diode laser depends on transitions between electron bands rather than discrete states, and understanding the freeelectron laser does not require quantum states at all. Type of Laser Cavity All laser cavities share two characteristics that complement each other: (1) They are basically linear devices with one relatively long optical axis, and (2) the sides parallel to 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 lasers the long axis actually bends and closes on itself (Fig. 1.8). Despite what may seem E2 hn E2 – E1 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 standingwave and travelingwave (ring) conﬁgurations on left and right, respectively. 8 INTRODUCTION TO LASER OPERATION obvious, it is not always best to design a cavity with the lowest loss. In the case of Q switching an extra loss is temporarily introduced into the cavity for the laser to overcome, and very highpower lasers sometimes use mirrors that are deliberately designed to 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 lasers are well known by now to most people, such as for various surgical procedures, for holography, in ultrasensitive gyroscopes, to provide straight lines for surveying, in supermarket checkout scanners and compact disc players, for welding, drilling, and scribing, in compact deathray pistols, and so on. (The sophisticated student knows, even before reading this book, that one of these “wellknown” applications has never been realized outside the movie theater.) 1.3 LIGHT IN CAVITIES In laser technology the terms cavity and resonator are used interchangeably. The theory and 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 that is certain to be at least partly familiar to most readers. This simpliﬁcation allows us to introduce the concept of cavity modes and to infer certain features of cavity modes that remain valid in more general circumstances. We also describe the great advantage of open, rather than closed, cavities for optical radiation. We will consider only the case of a rectangular “empty cavity” containing radiation but no matter, as sketched in Fig. 1.9. The assumption that there is radiation but no matter inside the cavity is obviously an approximation if the cavity is part of a working laser. This approximation is used frequently in laser theory, and it is accurate enough for many purposes because laser media are usually only sparsely ﬁlled with active atoms or molecules. x Lx z 0 Ly Lz y Figure 1.9 Rectangular cavity with side lengths Lx, Ly, Lz. 1.3 LIGHT IN CAVITIES 9 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 kz z, (1:3:1) where E0 is a constant. However, here we are interested only in the simplest features of the 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 exactly an 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 ¼ 1, 2, . . . , is a positive integer and L is the cavity length. If we use the relation between wave vector and wavelength, k ¼ 2p/l, this is the same as kz ¼ p n, L (1:3:2) for the z component of the wave vector. By substitution into the solution (1.3.1) we see 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 y components of the wave vector. As we will show later, if the three dimensions are equivalent in this sense, the number of available modes grows extremely rapidly as a function of frequency. For example, a cubical threedimensionally reﬂecting cavity 1 cm on a side has about 400 million resonant frequencies within the useful gain band of a He–Ne laser. Then lasing could occur across the whole band, eliminating any possibility of achieving the important narrowband, nearly monochromatic character of laser light that we emphasized in the preceding sections. The solution to this multimode dilemma was suggested independently in 1958 by Townes and A. L. Schawlow, R. H. Dicke, and Prokhorov. They recognized that a onedimensional rather than a threedimensional cavity was desirable, and that this could be achieved with an open resonator consisting of two parallel mirrors, as in Fig. 1.10. The difference in wave vector between two modes of a linear cavity, according to Eq. (1.3.2), is just p/L, so the mode spacing is given by Dk ¼ (2p/c) Dn, or Dn ¼ c/2L. For L ¼ 10 cm we ﬁnd Dn ¼ 3 108 m=s ¼ 1500 MHz, (2)(0:10 m) (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 1500 MHz ¼ 1: 1500 MHz (a) (1:3:4) (b) Figure 1.10 Sketch illustrating the advantage of a onedimensional cavity. Stable modes are associated only with beams that are retroreﬂected many times. 10 INTRODUCTION TO LASER OPERATION Gain curve 1500 MHz Cavity mode frequencies n Figure 1.11 Mode frequencies separated by 1500 MHz, corresponding to a 10cm onedimensional cavity. A 1500MHz gain curve overlaps only 1 mode. The maximum number, including two choices of polarization, is therefore 2, considerably smaller than the estimate of 400 million obtained for threedimensional cavities. These results do not include the effects of diffraction of radiation at the mirror edges. Diffraction determines the x, y dependence of the ﬁeld, which we have ignored completely. Accurate calculations of resonator modes, including diffraction, are often done with computers. Such calculations were ﬁrst made in 1961 for the planeparallel resonator of Fig. 1.10 with either rectangular or circular mirrors. Actually lasers are seldom designed with ﬂat mirrors. Laser resonator mirrors are usually spherical surfaces, for reasons to be discussed in Chapter 7. A great deal about laser cavities can nevertheless be understood without worrying about diffraction or mirror shape. In particular, for most practical purposes, the modefrequency spacing is given accurately enough 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 by Einstein in 1905 in his theory of the photoelectric effect. Einstein assumed the difference in energy of the electron before and after its photoejection to be equal to the energy hn of the photon absorbed in the process. This picture of light absorption was extended in two ways by Bohr: to apply to atomic electrons that are not ejected during photon absorption but instead take on a higher energy within their atom, and to apply to the reverse process of photon emission, in which case the energy of the electron should decrease. These extensions of Einstein’s idea 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 electrons are restricted to a certain ﬁxed set of orbits around the atomic nucleus. This set of orbits was shown to correspond to a ﬁxed set of allowed electron energies. The idea of a “quantum jump” was introduced to describe an electron’s transition between two allowed 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 long as an outershell electron is doing the jumping. This is the ordinary case, so atoms usually absorb and emit photons in or near the optical region of the spectrum. Jumps by innershell atomic electrons usually require much more energy and are associated with Xray photons. On the other hand, quantum jumps among the socalled Rydberg energy levels, those outerelectron levels lying far from the ground level and near to 1.5 EINSTEIN THEORY OF LIGHT–MATTER INTERACTIONS 11 the ionization limit, involve only a small amount of energy, corresponding to farinfrared or even microwave photons. Molecules have vibrational and rotational degrees of freedom whose quantum jumps are smaller (perhaps much smaller) than the quantum jumps in free atoms, and the same is often true of jumps between conduction and valence bands in semiconductors. Many crystals are transparent in the optical region, which is a sign that they do not absorb or emit optical photons, because they do not have quantum energy levels that permit jumps in the optical range. However, colored crystals such as ruby have impurities that 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 optical quantum 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 microscopic transducers. That is, each atom accepts energy and jumps to a higher orbit as a result of some input or “pumping” process and converts it into other forms of energy—for example, into light energy (photons)—when it jumps to a lower orbit. At the same time, each atom must deal with the photons that have been emitted earlier and reﬂected back by the mirrors. These prior photons, already channeled along the cavity axis, are the 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 simplicity we focus our attention on quantum jumps between two energy levels, 1 and 2, of an atom. The ﬁve distinct energy conversion diagrams of Fig. 1.12 are interpreted as follows: (a) Absorption of an increment DE ¼ E2 2 E1 of energy from the pump: The atom is raised from level 1 to level 2. In other words, an electron in the atom jumps from an inner orbit to an outer orbit. (b) Spontaneous emission of a photon of energy hn ¼ E2 2 E1: The atom jumps down from level 2 to the lower level 1. The process occurs “spontaneously” without any external inﬂuence. (c) Stimulated emission: The atom jumps down from energy level 2 to the lower level 1, and the emitted photon of energy hn ¼ E2 2 E1 is an exact replica of a photon already present. The process is induced, or stimulated, by the incident photon. (d) Absorption of a photon of energy hn ¼ E2 2 E1: The atom jumps up from level 1 to 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 E2 2 E1 must appear in some other form [e.g., increased vibrational or rotational energy in the case of a molecule, or rearrangement (“shakeup”) of other electrons in the atom]. All these processes occur in the gain medium of a laser. Lasers are often classiﬁed according to the nature of the pumping process (a) which is the source of energy for the output laser beam. In electricdischarge lasers, for instance, the pumping occurs 12 INTRODUCTION TO LASER OPERATION E2 E2 E1 (a) hn E1 (b) E2 hn E1 hn hn (c) E2 E2 E1 E1 hn (d ) (e) Figure 1.12 Energy conversion processes in a lasing atom or molecule: (a) absorption of energy DE ¼ E2 2 E1 from the pump; (b) spontaneous emission of a photon of energy DE; (c) stimulated emission of a photon of energy DE; (d) absorption of a photon of energy DE; (e) nonradiative deexcitation. 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 as the absorption process (d), except that the pumping photons are supplied by a lamp or perhaps another laser. In a diode laser an electric current at the junction of two different semiconductors produces electrons in excited energy states from which they can jump into 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 the most favorable situation the lower level (level 1) of the laser transition is empty. To maintain this situation a mechanism must exist to remove downward jumping electrons from level 1 to another level, say level 0. In this situation there can be no detrimental absorption of laser photons due to transitions upward from level 1 to level 2. In practice the number of electrons in level 1 cannot be exactly zero, but we will assume for simplicity that the rate of deexcitation of the lower level 1 is so large that the number of atoms remaining in that level is negligible compared to the number in level 2; this is a reasonably good approximation for many lasers. Under this approximation laser action can be described in terms of two “populations”: the number n of atoms in the upper 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. 1.5 EINSTEIN THEORY OF LIGHT–MATTER INTERACTIONS 13 Thus, we can write a (provisional) equation for the rate of change of the number of photons, incorporating the gain and loss described in (i) and (ii) as follows: dq ¼ anq bq: dt (1:5:1) That is, the rate at which the number of laser photons changes is the sum of two separate rates: the rate of increase (ampliﬁcation or gain) due to stimulated emission, and the rate of decrease (loss) due to imperfect mirror reﬂectivity. As Eq. (1.5.1) indicates, the gain of laser photons due to stimulated emission is not only proportional to the number n of atoms in level 2, but also to the number q of photons already in the cavity. The efﬁciency of the stimulated emission process depends on the type of atom used and other factors. These factors are reﬂected in the size of the ampliﬁcation or gain coefﬁcient a. The rate of loss of laser photons is simply proportional to the number of laser photons present. We can also write a provisional equation for n. Both stimulated and spontaneous emission cause n to decrease (in the former case in proportion to q, in the latter case not), and the pump causes n to increase at some rate we denote by p. Thus, we write dn ¼ anq fn þ p: dt (1:5:2) Note that the ﬁrst term appears in both equations, but with opposite signs. This reﬂects the central role of stimulated emission and shows that the decrease of n (excited atoms) due to stimulated emission corresponds precisely to the increase of q (photons). Equations (1.5.1) and (1.5.2) describe laser action. They show how the numbers of lasing atoms and laser photons in the cavity are related to each other. They do not indicate what happens to the photons that leave the cavity, or what happens to the atoms when their electrons jump to some other level. Above all, they do not tell how to evaluate the coefﬁcients a, b, f, p. They must be taken only as provisional equations, not well justiﬁed although intuitively reasonable. It is important to note that neither Eq. (1.5.1) nor (1.5.2) can be solved independently of the other. That is, (1.5.1) and (1.5.2) are coupled equations. The coupling is due physically to stimulated emission: The lasing atoms of the gain medium can increase the number of photons via stimulated emission, but by the same process the presence of photons will also decrease the number of atoms in the upper laser level. This coupling between the atoms and the cavity photons is indicated schematically in Fig. 1.13. We also note that Eqs. (1.5.1) and (1.5.2) are nonlinear. The nonlinearity (the product of the two variables nq) occurs in both equations and is another manifestation of Atoms Equation (1.5.1) Equation (1.5.2) Cavity photons Figure 1.13 Selfconsistent pair of laser equations. 14 INTRODUCTION TO LASER OPERATION stimulated emission. No established systematic methods exist for solving nonlinear differential equations, and there is no known general solution to these laser equations. However, they have a number of welldeﬁned limiting cases of some practical importance, and some of these do have known solutions. The most important case is steady state. In steady state we can put both dq/dt and dn/dt equal to zero. Then (1.5.1) reduces to n¼ b ; nt , a (1:5:3) which can be recognized as a threshold requirement on the number of upperlevel atoms. That is, if n , b/a, then dq/dt , 0, and the number of photons in the cavity decreases, terminating laser action. The steady state of (1.5.2) also has a direct interpretation. From dn/dt ¼ 0 and n ¼ nt ¼ b/a we ﬁnd q¼ p f : b a (1:5:4) This equation establishes a threshold for the pumping rate, since the number of photons q cannot be negative. Thus, the minimum or threshold value of p compatible with steadystate operation is found by putting q ¼ 0: pt ¼ fb ¼ f nt : a (1:5:5) In words, the threshold pumping rate just equals the loss rate per atom times the number of atoms present at threshold. In Chapters 4–6 we will return to a discussion of laser equations. We will deal there with steady state as well as many other aspects of laser oscillation in twolevel, threelevel, and fourlevel quantum systems. 1.6 SUMMARY The theory of laser action and the description of cavity modes presented in this chapter can be regarded only as caricatures. In common with all caricatures, they display outstanding features of their subject boldly and simply. All theories of laser action must address the questions of gain, loss, steady state, and threshold. The virtues of our caricatures in addressing these questions are limited. They do not even suggest matters such as linewidth, saturation, output power, mode locking, tunability, and stability. Obviously, one must not accept a caricature as the truth. Concerning the many aspects of the truth that are distorted or omitted by these ﬁrst discussions, it will take much of this book to get the facts straight. This is not only a matter of dealing with details within the caricatures, but also with concepts that are larger than the caricatures altogether. One should ask whether lasers are better described by photons or electric ﬁelds. Also, is Einstein’s theory always satisfactory, or does Schrödinger’s wave equation play a role? Are Maxwell’s equations for electromagnetic waves signiﬁcant? The answer to these 1.6 SUMMARY 15 questions is no, yes, yes. Laser theory is usually based on Schrödinger’s and Maxwell’s equations, neither of which was needed in this chapter. From a different point of view another kind of question is equally important in trying to understand what a laser is. For example, why were lasers not built before 1960? Are there any rules of thumb that can predict, approximately and without detailed calculation, how much one can increase the output power or change the operating frequency? What are the most sensitive design features of a gas laser? a chemical laser? a semiconductor laser? Is a laser essentially quantum mechanical, or can classical physics explain all the important features of laser operation? It will not be possible to give detailed answers to all of these questions. However, these questions guide the organization of the book, and many of them are addressed individually. In the following chapters the reader should encounter the concepts of physics and engineering that are most important for understanding laser action in general and that provide the background for pursuing further questions of particular theoretical or practical interest. 2 2.1 ATOMS, MOLECULES, AND SOLIDS INTRODUCTION It is frequently said that quantum physics began with Max Planck’s discovery of the correct blackbody radiation formula in 1900. But it was more than a quarter of a century before Planck’s formula could be fully derived from a satisfactory theory of quantum mechanics. Nevertheless, once formulated, quantum mechanics answered so many questions that it was adopted and reﬁned with remarkable speed between 1925 and 1930. By 1930 there were new and successful quantum theories of atomic and molecular structure, electromagnetic radiation, electron scattering, and thermal, optical, and magnetic properties of solids. Lasers can be understood without a detailed knowledge of the quantum theory of matter. However, several consequences of the quantum theory are essential. This chapter provides a review of some results of quantum theory applied to simple models of atoms, molecules, and semiconductors. 2.2 ELECTRON ENERGY LEVELS IN ATOMS In 1913 Niels Bohr discovered a way to use Planck’s radiation constant h in a radically new, but still mostly classical, theory of the hydrogen atom. Bohr’s theory was the ﬁrst quantum theory of atoms. Its importance was recognized immediately, even though it raised as many questions as it answered. One of the most important questions it answered had to do with the Balmer formula: l¼ bn2 , n2 4 (2:2:1) where n denotes an integer. This relation had been found in 1885 by Johann Jacob Balmer, a Swiss school teacher. Balmer pointed out that if b were given the value 3645.6, then l equaled the wavelength (measured in Ångstrom units, 1 Å ¼ 10210 m) of a line in the hydrogen spectrum1 for n ¼ 3, 4, 5, and 6 (and possibly for higher integers as well, but no measurements existed to conﬁrm or deny the possibility). 1 Historically, the term spectral “line” arose because lines appeared as images of slits in spectrometers. Laser Physics. By Peter W. Milonni and Joseph H. Eberly Copyright # 2010 John Wiley & Sons, Inc. 17 18 ATOMS, MOLECULES, AND SOLIDS For almost 30 years the Balmer formula was a small oasis of regularity in the ﬁeld of spectroscopy—the science of measuring and cataloging the wavelengths of radiation emitted and absorbed by different elements and compounds. Unfortunately, the Balmer formula could not be explained, or applied to any other element, or even applied to other known wavelengths emitted by hydrogen atoms. It might well have been a mere coincidence, without any signiﬁcance. Bohr’s model of the hydrogen atom not only explained the Balmer formula, but also gave scientists their ﬁrst glimpse of atomic structure. It still serves as the basis for most scientists’ working picture of an atom. Bohr adopted Rutherford’s nuclear model that had been successful in explaining scattering experiments with alpha particles between 1910 and 1912. In other words, Bohr assumed that almost all the mass of a hydrogen atom is concentrated in a positively charged nucleus, allowing most of the atomic volume free for the motion of the much lighter electron. The electron was assumed attracted to the nucleus by the Coulomb force law governing opposite charges (Fig. 2.1). In magnitude this force is F¼ 1 e2 : 4pe 0 r 2 (2:2:2) Bohr also assumed that the electron travels in a circular orbit about the massive nucleus. Moreover, he assumed the validity of Newton’s laws of motion for the orbit. Thus, in common with every planetary body in a circular orbit, the electron was assumed to experience an inward (centripetal) acceleration of magnitude a¼ v2 : r (2:2:3) Newton’s second law of motion, F ¼ ma, then gives mv2 1 e2 ¼ , r 4pe 0 r 2 (2:2:4) which is the same as saying that the electron’s kinetic energy, T ¼ 12 mv2 , is half as great as the magnitude of its potential energy, V¼2e 2/4pe 0r. In the Coulomb ﬁeld of the e r p Figure 2.1 The electron in the Bohr model is attracted to the nucleus with a force of magnitude F ¼ e 2/4pe 0r 2. 2.2 ELECTRON ENERGY LEVELS IN ATOMS 19 nucleus the electron’s total energy is therefore E ¼T þV ¼ 1 e2 : 4pe 0 2r (2:2:5) These results are familiar consequences of Newton’s laws. Bohr then introduced a single, radical, unexplained restriction on the electron’s motion. He asserted that only certain circles are actually used by electrons as orbits. These orbits are the ones that permit the electron’s angular momentum L to have one of the values L¼n h , 2p (2:2:6) where n is an integer (n ¼ 1, 2, 3, . . .) and h is the constant of Planck’s radiation formula: h 6:625 1034 Js: (2:2:7) With the deﬁnition of angular momentum for a circular orbit, L ¼ mvr, (2:2:8) it is easy to eliminate r between (2.2.4) and (2.2.8) and ﬁnd v¼ 1 e2 1 2pe2 ¼ : 4pe 0 L 4pe 0 nh (2:2:9) Then the combination of (2.2.4) and (2.2.5), namely E¼ mv2 , 2 (2:2:10) together with (2.2.9), gives the famous Bohr formula for the allowed energies of a hydrogen electron: 1 2 me4 : (2:2:11) En ¼ 4pe 0 2n2 h2 This can be seen, by comparison with (2.2.5), to be the same as a formula for the allowed values of electron orbital radius: rn ¼ 4pe 0 n2 h2 : me2 (2:2:12) In both (2.2.11) and (2.2.12) we have adopted the modern notation for Planck’s constant: h ¼ h 1:054 1034 Js: 2p (2:2:13) The ﬁrst thing to be said about Bohr’s model and his unsupported assertion (2.2.6) is that they were not contradicted by known facts about atoms, and, for small values of n, 20 ATOMS, MOLECULES, AND SOLIDS En En ¢ hn + Figure 2.2 A radiative transition of an atomic electron in the Bohr model. the allowed radii deﬁned by (2.2.12) are numerically about right.2 For example, the smallest of these radii (conventionally called “the Bohr radius” and denoted a0) is a0 ¼ r1 ¼ 4pe 0 h2 : 0:53 A me2 (2:2:14) This might have been an accident without further consequences. Since no way existed to measure such small distances with any precision, Bohr needed a connection between (2.2.12) and a possible laboratory experiment. A second unsupported assertion supplied the connection. Bohr’s second assertion was that the atom was stable when the electron was in one of the permitted orbits, but that jumps from one orbit to another were possible if accompanied by light emission or absorption. To be speciﬁc, Bohr combined earlier ideas of Planck and Einstein and stated that a jump from a higher to a lower orbit would ﬁnd the decrease of the electron’s energy transformed into a quantum of radiation that would be emitted in the process. In other words, Bohr postulated that (DE)n,n0 ¼ hn ¼ energy of emitted photon: (2:2:15) Here (DE)n,n0 denotes the energy lost by the electron in switching orbits from rn to rn0 and n is the frequency of the photon emitted in the process (Fig. 2.2). The relation (2.2.15) led immediately to a connection between Bohr’s theory and all the spectroscopic data known for atomic hydrogen. By using (2.2.11) for two different orbits, that is, for two different values of the integer n, we easily ﬁnd for the energy decrement (DE)n,n0 the expression En En0 ¼ 1 2 me4 1 1 2 n02 n2 : 4pe 0 2h (2:2:16) Furthermore, the connection between the frequency and wavelength of a light wave is c l¼ : n 2 (2:2:17) Lord Rayleigh (1890) was able to estimate molecular dimensions by dropping olive oil onto a water surface. Assuming that an oil drop spreads until it forms a layer one molecule thick (a molecular monolayer), he could give a reasonable estimate of a molecular diameter from the area of the layer and the volume of the original drop. A century earlier, Benjamin Franklin tried the same oilonwater experiment on a pond in London while on diplomatic assignment there. 2.2 ELECTRON ENERGY LEVELS IN ATOMS 21 Thus, Bohr’s statement (2.2.15) and his energy formula (2.2.11) are actually equivalent to the postulate that all spectroscopic wavelengths of light associated with atomic hydrogen ﬁt the formula hc 4ph c n2 n02 ¼ (4pe 0 )2 , DEn,n0 me4 n2 n02 3 l¼ (2:2:18) where n and n0 are integers to be chosen, but where all the other parameters are ﬁxed. It is obvious that if n0 ¼ 2 and n . 2, then (2.2.18) becomes 16ph c n2 , l ¼ (4pe0 ) me4 n2 4 3 2 (2:2:19) which is exactly the Balmer formula (2.2.1). The numerical value of the product of the coefﬁcients in (2.2.19) is 3645.6 Å, just what Balmer had said the constant b was, 28 years earlier. Bohr’s expression (2.2.18) was quickly found, for values of n0 not equal to 2, to agree with other wavelengths associated with hydrogen but which had not ﬁtted the Balmer formula (Problem 2.1). Bohr’s theory opened a new viewpoint on atomic spectroscopy. All observed spectroscopic wavelengths could be interpreted as evidence for the existence of certain allowed electron orbits in all atoms, even if formulas corresponding to Bohr’s (2.2.18) were not known for any atom but hydrogen. In Chapter 3 we will see that, for many aspects of the interaction of light and matter, atoms can be regarded as a set of electrons acting as harmonic oscillators. It might be supposed that the oscillation frequencies of the electrons in this classical electron oscillator model of an atom, which was used with considerable success before quantum theory, are associated somehow with the “transition frequencies” n in Bohr’s formula (2.2.15). In this way the most useful features of the classical oscillator model of an atom survived the quantum revolution unchanged. How this is possible, in view of the obvious fact that the assumed Coulomb force (2.2.2) between electron and nucleus is not a harmonic oscillator force, will be explained in Chapter 3. It is easy to have second thoughts about Bohr’s model, no matter how successful it is. For example, one can ask why (2.2.6) does not include the possibility n ¼ 0. There is no apparent reason why zero angular momentum must be excluded, except that the energy formula (2.2.11) is not deﬁned for n ¼ 0. This point, whether physical signiﬁcance can be assigned to the orbit with zero angular momentum, cannot be clariﬁed within the Bohr theory, and it proved puzzling to physicists for more than a decade until quantum mechanics was developed. In a similar fashion, one can ask how Bohr’s results are modiﬁed by relativity. The kinetic energy formula used above, T ¼ 12 mv2 , is certainly nonrelativistic. Again, this point was not fully answered until the development of quantum mechanics. † Relativistic corrections to Newtonian physics become important when particle velocities approach the velocity of light. If v is the velocity of a particle, then typically the ﬁrst correction terms are found to be proportional to (v/c)2, where c is the velocity of light. The 22 ATOMS, MOLECULES, AND SOLIDS value of (v/c)2 can easily be estimated within the Bohr theory. It follows from (2.2.4) that v2 1 e2 ¼ : (2:2:20) c 4pe 0 rmc2 By inserting r from (2.2.12) and taking the square root, the ratio v/c can be found for any of the allowed orbits: v 1 e2 : ¼ c 4pe 0 nh c (2:2:21) Equation (2.2.21) shows that the largest velocity to be expected in the Bohr atom is associated with the lowest orbit, n ¼ 1. The ratio of this maximum velocity to the velocity of light is given by the remarkable (dimensionless) combination of electromagnetic and quantum mechanical constants, e2 =4pe 0 h c. The numerical value of this parameter is easily found: 1 e2 (1:602 1019 C)2 ¼ 4pe 0 h c (4p)(8:854 1012 C=Vm)(1:054 1034 Js)(2:998 108 m=s) ¼ 0:007297 (¼ 1=137:04): (2:2:22) The value found in (2.2.22) is small enough that corrections to the Bohr model from relativistic effects are of the relative order of magnitude 1024 or smaller, and thus negligible in most circumstances. Spectroscopic measurements, however, are commonly accurate to ﬁve signiﬁcant ﬁgures. Arnold Sommerfeld, in the period 1915 –1920, studied the relativistic corrections to Bohr’s formulas and showed that they accounted accurately for some of the ﬁne details or ﬁne structure in observed spectra. For this reason the parameter e2 =4pe 0 h c is called Sommerfeld’s ﬁnestructure constant. The ﬁnestructure constant appears so frequently in expressions of atomic radiation physics that it is very useful to remember its numerical value. Because the value given in (2.2.22) is very nearly equal to 1/137, it is in this form that its value is memorized † by physicists. Quantum States and Degeneracy In the Bohr model a state of the electron is characterized by the quantum number n. Everything the model can say about the allowed states of the electron is given in terms of n. The full quantum theory of the hydrogen atom also yields the allowed energies (2.2.11). However, in the quantum theory a state of the electron is characterized by other quantum numbers in addition to the principal quantum number n appearing in (2.2.11). The results of the quantum theory for the hydrogen atom, in addition to (2.2.11), are mainly the following: (i) For each principal quantum number n (¼1, 2, 3, . . .) there are n possible values of the orbital angular momentum quantum number ‘. The allowed values of ‘ are 0, 1, 2, . . . , n21. Thus, for n ¼ 1 we can have only ‘ ¼ 0, whereas for n ¼ 2 we can have ‘ ¼ 0 or 1, and so on. (ii) For each ‘ there are 2‘ þ 1 possible values of the magnetic quantum number m. The possible values of magnetic quantum number m are ‘, ‘ þ 1, . . . , 1, 0, 1, . . . , ‘ 1, ‘. 2.2 ELECTRON ENERGY LEVELS IN ATOMS 23 (iii) In addition to orbital angular momentum, an electron also carries an intrinsic angular momentum, which is called simply spin. The spin of an electron always has magnitude 12 (in units of h). But in any given direction the electron spin can be either “up” or “down”; that is, quantum theory says that when the component of electron spin along any direction is measured, we will always ﬁnd it to have one of two possible values.3 Because of this, an electron state must also be labeled by an additional quantum number ms, called the spin magnetic quantum number, whose only possible values are +12. Thus, for a given n, there are n possible values of ‘, and for each ‘ there are 2‘ þ 1 possible values of m, for a total of n1 X (2‘ þ 1) ¼ n2 , (2:2:23) ‘¼0 states. And each of these states is characterized further by ms, which may be þ 12 or 12. Therefore, there are 2n 2 states associated with each principal quantum number n. In contrast to the Bohr model, in which an allowed state of the electron in the hydrogen atom is characterized by n, quantum theory characterizes each allowed state by the four quantum numbers n, ‘, m, and ms; and since the electron energy depends only on n [recall (2.2.11)], there are 2n 2 states with the same energy for every value of n. These 2n 2 states are called degenerate states or are said to be degenerate in energy. Historical designations for the orbital angular momentum quantum numbers are still in use: ‘¼0 ‘¼1 ‘¼2 ‘¼3 ‘¼4 designates the socalled s orbital p orbital d orbital f orbital g orbital The ﬁrst three letters came from the words sharp, principal, and diffuse, which described the character of atomic emission spectra in a qualitative way long before quantum theory showed that they could be associated systematically with different orbital angular momentum values for an electron in the atom. The Periodic Table Although hydrogen is the only atom for which explicit expressions such as (2.2.11) can be written down, we can nevertheless understand the gross features of the periodic table of the elements. That is, we can understand the chemical regularity, or periodicity, that occurs as the atomic number Z increases. The key to this understanding is the exclusion principle of Wolfgang Pauli (1925), which forbids two electrons from occupying the qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 h ¼ 3h , s(s þ 1)h ¼ þ 1 2 2 4 qﬃﬃ i h so that its two allowed components (spin “up” and spin “down”) make angles cos1 (+ 12)= 34 ¼ 54:748 and 3 The magnitude of the spin angular momentum vector is (Section 2.4) (180 54:74) ¼ 125:268 with the axis along which the spin is measured. 24 ATOMS, MOLECULES, AND SOLIDS same quantum mechanical state. The Pauli exclusion principle may be proved only at an advanced level that is well beyond the scope of this book. We will simply accept it as a fundamental truth. But the Pauli principle alone is not sufﬁcient for an understanding of the periodic table. We must also deal with the electron–electron interactions in a multielectron atom. These interactions present us with an extremely complicated manybody problem that has never been solved. A useful approximation, however, is to assume that each electron moves independently of all the others; each electron is thought of as being in a spherically symmetric potential V(r) due to the Coulomb ﬁeld of the nucleus plus the Z21 other electrons. In this independentparticle approximation an electron state is still characterized by the four quantum numbers (n, ‘, m, ms ), as in the case of hydrogen. However, in this case the simple energy formula (2.2.11) does not apply, and in particular the energy depends on both n and ‘ (but not m or ms) as sketched in Fig. 2.3. The simplest multielectron atom, of course, is helium, in which there are Z ¼ 2 electrons. The lowest energy state for each electron is characterized in the independentparticle approximation by the quantum numbers n ¼ 1, ‘ ¼ 0, m ¼ 0, and ms ¼ + 12. Since the energy depends now on both n and ‘, we can label this particular electron conﬁguration as 1s, a shorthand notation meaning n ¼ 1 and ‘ ¼ 0. Both electrons 1 are in the shell n ¼1, one having spin up ms ¼ 2 , the other spin down ms ¼ 12 . Since 2 is the maximum number of electrons allowed by the Pauli exclusion principle for the 1s conﬁguration, we say that the 1s shell is completely ﬁlled in the helium atom. Bohr’s hydrogen Energy Generic atom Energy 3d (10) 3p (6) n=3 3s (2) 2p (6) n=2 2s (2) n=1 1s (2) Figure 2.3 The main differences between Bohr’s model for hydrogen and a generic manyelectron atom arise from the Pauli exclusion principle and the dependence of level energies on both n and ‘. In parentheses we show the number of different states, (2‘ þ 1) 2, permitted by assignment of m and ms values for each ‘. Carbon’s 6 electrons, for example, occupy the 2 states in each of the 1s and 2s levels and 2 of the 2p states. For clarity the energy separations are not properly scaled. 2.2 ELECTRON ENERGY LEVELS IN ATOMS 25 In the case Z ¼ 3 (lithium), there is one electron left over after the 1s shell is ﬁlled. The next allowed electron conﬁguration is 2s (n ¼ 2, ‘ ¼ 0), and one of the electrons in lithium is assigned to this conﬁguration. Since the 2s conﬁguration can accommodate two electrons, the 2s subshell in lithium is only partially ﬁlled. The next element is beryllium, with Z ¼ 4 electrons, and in this case the 2s subshell is completely ﬁlled, there being two electrons in this “slot.” For Z ¼ 5 (boron), the added electron goes into the 2p conﬁguration (n ¼ 2, ‘ ¼ 1). This conﬁguration can accommodate 2(2‘ þ 1) ¼ 6 electrons. Thus, there are ﬁve other elements (C, N, O, F, and Ne) in which the outer subshell of electrons corresponds to the conﬁguration 2p. The eight elements lithium through neon, for which the outermost electrons belong to the n ¼ 2 shell, constitute the ﬁrst full row of the periodic table. Inside the back cover of this book we list the ﬁrst 36 elements and their electron conﬁgurations. The conﬁgurations are assigned in a similar manner as done above for Z ¼ 1–10. Also listed is the ionization energy, deﬁned as the energy required to remove one electron from the atom. For hydrogen the ionization energy WI may be calculated from Eq. (2.2.11) with n ¼ 1, that is, WI is just the binding energy of the electron in groundstate hydrogen: WI ¼ jE1 j ¼ 1 2 me4 18 J: 2 ¼ 2:17 10 4pe 0 2h (2:2:24) We already pointed out, in connection with photon energy in Chapter 1, that such small energies are usually expressed in units of electron volts, an electron volt being the energy acquired by an electron accelerated through a potential difference of 1 volt: 1 eV ¼ (1:602 1019 C)(1 V) ¼ 1:602 1019 J: (2:2:25) The ionization energy of hydrogen is therefore WI ¼ 2:17 1018 J ¼ 13:6 eV: 1:602 1019 J=eV (2:2:26) The ionization energy of a hydrogen atom in any state (n, ‘, m, ms) is likewise (13.6 eV)/n 2. The elements He, Ne, Ar, and Kr are chemically inactive. We note that each of these atoms has a completely ﬁlled outer shell. Evidently, an atom with a ﬁlled outer shell of electrons tends to be “satisﬁed” with itself, having very little proclivity to share its electrons with other atoms (i.e., to join in chemical bonds). However, a ﬁlled outer subshell does not necessarily mean chemical inertness. Beryllium, for instance, has a ﬁlled 2s subshell, but it is not inert. Furthermore, even some of the noble gases are not entirely inert. The alkali metals Li, Na, and K have only one electron in an outer subshell, and their outer electrons are weakly bound, leading to low ionization energies of these elements. These elements are highly reactive; they will readily give up their “extra” electron. On the other hand, the halogens F, Cl, and Br are one electron short of a ﬁlled outer subshell. These atoms will readily take another electron, and so they too are quite reactive chemically and the halogens are sufﬁciently “eager” to combine with elements that can easily 26 ATOMS, MOLECULES, AND SOLIDS contribute an electron that they can form negative ions, stably but weakly binding an extra electron. This even includes H2, the negative hydrogen ion. Hydrogen is in the odd position of having some properties in common with the alkali metals and some in common with the halogens. The characterization of atomic electron states in terms of the four quantum numbers n, ‘, m, and ms, together with the Pauli exclusion principle, thus allows us to understand why Na is chemically similar to K, Mg is chemically similar to Ca, and so forth. These chemical periodicities, according to which the periodic table is arranged, are consequences of the way electrons ﬁll in the allowed “slots” when they combine with nuclei to form atoms. Of course, there is a great deal more that can be said about the periodic table. For a rigorous treatment of atomic structure, we must refer the reader to textbooks on atomic physics. As mentioned earlier, however, we can understand lasers without a more detailed understanding of atomic and molecular physics. 2.3 MOLECULAR VIBRATIONS As in the case of atoms, there are only certain allowed energy levels for the electrons of a molecule. Quantum jumps of electrons in molecules are accompanied by the emission or absorption of photons that typically belong to the ultraviolet region of the electromagnetic spectrum. For our purposes the electronic energy levels of molecules are quite similar to those of atoms. However, in contrast with atoms, molecules have vibrational and rotational as well as electronic energy. This is because the relative positions and orientations of the individual atomic nuclei in molecules are not absolutely ﬁxed. The energies associated with molecular vibrations and rotations are also quantized, that is, restricted to certain allowed values. In this section and the next we will discuss the main features of molecular vibrational and rotational energy levels. Transitions between vibrational levels lie in the infrared portion of the electromagnetic spectrum, whereas rotational spectra are in the microwave region. Some of the most powerful lasers operate on molecular vibrationalrotational transitions. Consider the simplest kind of molecule, namely a diatomic molecule such as O2, N2, or CO. There is a molecular binding force that is responsible for holding the two atoms together. To a ﬁrst (and often very good) approximation the binding force is linear, so that the potential energy function is V(x) ¼ 12(x2 x1 x0 )2 ¼ 12 k(x x0 )2 , (2:3:1) where k is the “spring constant,” x ¼ x2 2x1 is the distance between the two nuclei, and x0 is the internuclear separation for which the spring force F ¼ k(x x0 ), (2:3:2) vanishes (Fig. 2.4). In other words, if the separation x is greater than x0, the binding force is attractive and brings the nuclei closer; if x is less than x0, the force is repulsive. The separation x ¼ x0 is therefore a point of stable equilibrium. The origin of the binding 27 2.3 MOLECULAR VIBRATIONS m1 m2 F F F F x0 x x (a) (b) (c) Figure 2.4 (a) When the two nuclei of a diatomic molecule are separated by the equilibrium distance x0, there is no force between them. If their separation x is larger than x0, there is an attractive force (b), whereas when x is less than x0 the force is repulsive (c). The internuclear force is approximately harmonic, that is, springlike. force is quantum mechanical; we will not attempt to explain it but will simply accept the result (2.3.1) and consider its consequences. For simplicity, let us assume that the nuclei can move only in one dimension. The total energy of a diatomic system (i.e., the sum of kinetic and potential energies) is then E ¼ 12 m1 x_ 12 þ 12 m2 x_ 22 þ 12 k(x2 x1 x0 ) 2 , (2:3:3) where the dots denote differentiation with respect to time, that is, x_ ¼ dx=dt. In terms of the reduced mass m¼ m1 m2 , M (2:3:4) where M ¼ m1 þ m2 is the total mass, and the centerofmass coordinate X¼ m1 x 1 þ m 2 x 2 , M (2:3:5) we may write (2.3.3) as (Problem 2.2) 2 E ¼ 12 M X_ þ 12 m_x 2 þ 12 k(x x0 ) 2 : (2:3:6) The ﬁrst term is just the kinetic energy associated with the centerofmass motion. We ignore it and focus our attention on the internal vibrational energy E ¼ 12 m_x2 þ 12 k(x x0 )2 : (2:3:7) The vibrational motion of a diatomic molecule must clearly be one dimensional, and so we lose nothing in the way of generality by restricting ourselves to onedimensional vibrations from the start [Eq. (2.3.3)]. The quantum mechanics of the motion associated with the energy formula (2.3.7) has much in common with that for the hydrogen atom electron. The most important result is that the allowed energies E of the oscillator are also quantized. The quantized 28 ATOMS, MOLECULES, AND SOLIDS E 9 hw E4 = — 2 7 hw E3 = — 2 5 hw E2 = — 2 3 hw E1 = — 2 1 hw E0 = — 2 Figure 2.5 The energy levels of a harmonic oscillator form a ladder with rung spacing hv. energies are given by En ¼ hv(n þ 12), with v¼ n ¼ 0, 1, 2, 3, . . . , pﬃﬃﬃﬃﬃﬃﬃﬃﬃ k=m: (2:3:8) (2:3:9) This formula is clearly quite different from Bohr’s formula for hydrogen. The quantum mechanical energy spectrum for a harmonic oscillator is simply a ladder of evenly spaced levels separated by h v (Fig. 2.5). The ground level of the oscillator corresponds to n ¼ 0. However, an oscillator in its ground level is not at rest at its stable equilibrium point x ¼ x0. Even the lowest possible energy of a quantum mechanical oscillator has ﬁnite kinetic and potential energy contributions. At zero absolute temperature, where classically all motion ceases, the quantum mechanical oscillator still has a ﬁnite energy 12 hv. For this reason the energy 12 hv is called the zeropoint energy of the harmonic oscillator. Of course, real diatomic molecules are not perfect harmonic oscillators, and their vibrational energies do not satisfy (2.3.8) precisely. Figure 2.6 shows the sort of potential Harmonic oscillator V(x) Real diatomic molecule x0 x (Internuclear separation) Figure 2.6 The potential energy function of a real diatomic molecule is approximately like that of a harmonic oscillator for values of x near x0. 2.3 MOLECULAR VIBRATIONS 29 energy function V(x) that describes the bonding of a real diatomic molecule. The Taylor series expansion of the function V(x) about the equilibrium point x0 is 2 dV 1 2 d V V(x) ¼ V(x0 ) þ (x x0 ) þ (x x0 ) dx x¼x0 2 dx2 x¼x0 3 1 d V þ (x x0 )3 þ: 6 dx3 x¼x0 (2:3:10) Here V(x0) is a constant, which we put equal to zero by shifting the origin of the energy scale. Also (dV/dx)x¼x0 ¼ 0 because, by deﬁnition, x ¼ x0 is the equilibrium separation, at which the potential energy is a minimum. Furthermore (d 2V/dx 2) at x ¼ x0 is positive if x0 is a point of stable equilibrium (Fig. 2.6). Thus, we can replace (2.3.10) by V(x) ¼ 12 k(x x0 )2 þ A(x x0 )3 þ B(x x0 )4 þ , (2:3:11) where A, B, . . . are constants and k ¼ (d2 V=dx2 )x¼x0 . From (2.3.10) we can conclude that any potential energy function describing a stable equilibrium [i.e., (dV=dx)x¼x0 ¼ 0, (d 2 V=dx2 )x¼x0 . 0] can be approximated by the harmonic oscillator potential (2.3.1) for small enough displacements from equilibrium. Of course, what is “small” is determined by the constants A, B, . . . in (2.3.11), that is, by the shape of the potential function V(x). If the terms involving third and/or higher powers of x 2x0 in (2.3.11) are not negligible, however, we have what is called an anharmonic potential. The energy levels of an anharmonic oscillator do not satisfy the simple formula (2.3.8). Real diatomic molecules have vibrational spectra that are usually only slightly anharmonic. In conventional notation the vibrational energy levels of diatomic molecules are written in the form h i 2 3 Ev ¼ hcve v þ 12 xe v þ 12 þ ye v þ 12 þ , (2:3:12) v ¼ 0, 1, 2, 3, . . . , (2:3:13) where and ve is in units of “wave numbers,” i.e., cm21; cve is the same as v/2p ¼ n in this notation. If the anharmonicity coefﬁcients xe, ye, . . . are all zero, we recover the harmonic oscillator spectrum (2.3.8). Numerical values of ve, xe, ye, . . . are tabulated in the literature.4 Values of ve, xe, ye, . . . are given for several diatomic molecules in Table 2.1. The deviations from perfect harmonicity are small until v becomes large, that is, until we climb fairly high up the vibrational ladder. The level spacing decreases as v increases, in contrast to the even spacing of the ideal harmonic oscillator (Fig. 2.7). 4 A standard source is G. Herzberg, Molecular Spectra and Molecular Structure. Volume I, Spectra of Diatomic Molecules, Robert E. Krieger, Malabar, FL, 1989. 30 ATOMS, MOLECULES, AND SOLIDS TABLE 2.1 Vibrational Constants of the Ground Electronic State for a Few Diatomic Molecules Molecule H2 O2 CO HF HCl ve (cm21) xe ye 4395.24 1580.36 2170.21 4138.52 2989.74 0.0268 0.00764 0.00620 0.0218 0.0174 6.67 1025 3.46 1025 1.42 1025 2.37 1024 1.87 1025 E Ev +1 Ev Ev –1 Figure 2.7 The vibrational energy level spacing of a real (anharmonic) diatomic molecule decreases with increasing vibrational energy. † For a simple check on our theory, consider the two molecules hydrogen ﬂuoride (HF) and deuterium ﬂuoride (DF). These molecules differ only to the extent that D has a neutron and a proton in its nucleus and H has only a proton. Since neutrons have no effect on molecular bonding, we expect HF and DF to have the same potential function V(x) and therefore the same “spring constant” k. According to (2.3.9), therefore, we should have DF 1=2 vHF m e ¼ , (2:3:14) vDF mHF e where m DF and m HF are the reduced masses of DF and HF, respectively, so that mDF mD mF ¼ mHF mD þ mF mH mF (2)(19) 1 þ 19 1:90: mH þ mF 2 þ 19 (1)(19) (2:3:15) From the value of ve for HF given in Table 2.1, therefore, we calculate 1=2 vDF ¼ 2998:64 cm1 , e (4138:52)(1:90) (2:3:16) and indeed this is very close to the tabulated value ve ¼ 2998.25 cm21 for DF.4 Regarding the zeropoint energy of molecular vibrations, consider a transition in which there is a change in both the electronic (e) and the vibrational (v) states of a diatomic molecule. 2.4 MOLECULAR ROTATIONS 31 The transition energy is approximately DEe0 v0 ,ev ¼ Ee0 Ee þ hc ve0 v0 þ 12 ve v þ 12 , (2:3:17) where the unprimed and primed labels refer to the initial and ﬁnal states, respectively, and ve and ve0 are the vibrational constants associated with the two electronic states. Now suppose that one of the nuclei of the diatomic molecule is replaced by a different isotope, for example, HF is replaced by DF. The electronic energy levels are approximately unchanged by this replacement, but the vibrational constants ve and ve0 are changed to rve and rve0 , where r is the square root of the ratio of reduced masses of the two molecules, as in our example above comparing HF and DF. For the second, isotopically different molecule, then, the transition energy (2.3.17) is replaced by (2:3:18) DEei 0 v0 ,ev ¼ Ee0 Ee þ hc rve0 v0 þ 12 rve v þ 12 : The vibrational spectra of the two isotopic molecules for the same electronic transition Ee ! Ee0 therefore differ by DEei 0 v0 ,ev DEe0 v0 ,ev ¼ hc(r 1) ve0 v0 þ 12 ve v þ 12 , (2:3:19) and in particular, for v ¼ 0 ! v0 ¼ 0, DEei 0 0,e0 DEe0 0,e0 ¼ 12 hc(r 1)(ve0 ve ): (2:3:20) This is nonzero because of zeropoint energy, that is, it would vanish if the energy levels of a harmonic oscillator were given by En ¼ nh v instead of En ¼ (n þ 12)h v. The zeropoint energy of molecular vibrations was conﬁrmed in this way by R. S. Mulliken (1924), who compared the observed vibrational spectra of B10O16 and B11O16. This was before the quantum mechanical derivation by Heisenberg (1925) of the formula (2.3.8) for the energy † levels of a harmonic oscillator. 2.4 MOLECULAR ROTATIONS The rotations of a diatomic molecule can be understood in two stages. First, we imagine the molecule to be a dumbbell consisting of two masses, m1 and m2, held together by a (massless) rigid rod of length x0 (Fig. 2.8). The dumbbell can rotate about its center of mass. The moment of inertia I is mx20 , where m is the reduced mass (2.3.4) and x0 is the distance separating the masses m1 and m2. If the angular velocity of rotation (radians per second) is vR, the angular momentum and kinetic energy are, respectively, L ¼ I vR m1 (magnitude of angular momentum vector) C.M. (2:4:1) m2 Figure 2.8 A dumbbell rotating about an axis through its center of mass serves as a classical model for the rotations of a diatomic molecule. 32 ATOMS, MOLECULES, AND SOLIDS and E ¼ 12 I v2R ¼ L2 : 2I (2:4:2) These classical formulas are the starting point of a quantummechanical treatment of the rigid dumbbell, just as similar classical formulas underlie treatments of the hydrogen atom and the vibrations of molecules. It is found that the rotational energy (2.4.2) of the molecule has the allowed values EJ ¼ h2 J(J þ 1), 2I J ¼ 0, 1, 2, . . . : (2:4:3) Actual diatomic molecules are, of course, not rigid dumbbells. In particular, the masses m1 and m2 do not stay a ﬁxed distance x0 apart. As the molecule rotates, the centrifugal force tends to increase the separation of the two masses, and therefore also the moment of inertia I. This decreases the rotational energy, the more so as the rate of rotation (i.e., J ) increases. In the notation of molecular spectroscopy this is accounted for by writing EJ ¼ hcBJ(J þ 1) hcDJ 2 (J þ 1)2 , (2:4:4) where the Jindependent quantities B and D have units of wave numbers. The fact that the molecule can vibrate also tends to increase the effective moment of inertia, the more so as the vibrational quantum number v increases. This is accounted for by writing B ¼ Be ae v þ 12 , (2:4:5) where Be and ae (in cm21) are independent of v and J. The rotational energy levels associated with the vibrational level v of a diatomic molecule are therefore written as EJ (v) ¼ hc Be ae v þ 12 J(J þ 1) hcDJ 2 (J þ 1)2 : (2:4:6) Higherorder corrections are necessary in general to explain the ﬁne details of the rotational energy spectrum of a diatomic molecule. However, (2.4.6) is often accurate enough for practical purposes, and in fact the term involving D is often negligible. The constants Be, ae, . . . for different molecules are tabulated in the spectroscopic literature. The constants Be and ae for several molecules are given in Table 2.2. For our purposes it will sufﬁce to make the rigiddumbbell approximation and write EJ (v) EJ hcBe J(J þ 1): (2:4:7) † Once again it is possible to check our theory with an example. A comparison of Eqs. (2.4.3) and (2.4.7) shows that the rotational constant Be of a diatomic molecule should be inversely proportional to its moment of inertia I ¼ mx20 . Since the equilibrium separation x0 is determined primarily by chemical (i.e., electromagnetic) forces, we expect that it should be practically the same for the two molecules HF and DF. Thus, we expect BHF mDF e ¼ HF ¼ 1:90: DF Be m (2:4:8) 2.5 EXAMPLE: CARBON DIOXIDE 33 TABLE 2.2 Rotational Constants of the Ground Electronic State for the Molecules Listed in Table 4.1 Molecule Be (cm21) ae (cm21) H2 O2 CO HF HCl 60.81 1.44567 1.9314 20.939 10.5909 2.993 0.01579 0.01749 0.770 0.3019 It follows from the data in Table 2.2, therefore, that the rotational constant for DF should 21 1 be BDF tabulated by e 11:02 cm , in excellent agreement with the value 11.007 cm Herzberg.4 Equations (2.4.2) and (2.4.3) imply that the square of the angular momentum in a state with angular momentum quantum number L is L(L þ 1)h2 rather than L2 h2 . This is a general feature of the quantum theory of angular momentum; the square of the orbital angular momentum for the electron in a hydrogen atom in a state with orbital angular momentum quantum number ‘, for example, is ‘(‘ þ 1)h2 . It can be understood as a consequence of “space quantization,” that is, the fact that the z component of angular momentum, Lz, has only the 2Lþ1 allowed values M¼ 2L,2L þ 1, . . . , L21, L. Since there is nothing special about the “z direction,” and there are three space dimensions, the average k L 2l of the square of the angular momentum must be three times the average of L2z : kL2 l ¼ 3kL2z l ¼ 3 L X 1 3 1 M2 ¼ L(L þ 1)(2L þ 1) ¼ L(L þ 1), 2L þ 1 M¼L 2L þ 1 3 (2:4:9) where we have used the general identity L X 1 M 2 ¼ L(L þ 1)(2L þ 1): 3 M¼L (2:4:10) In other words, once we accept space quantization as an experimental fact, we can understand why the square of the angular momentum must be L(L þ 1)h2 in a state with angular momentum quantum number L. † In summary, with every electronic state of a molecule there are associated vibrational constants ve, xe, ye, . . . and rotational constants Be, ae, . . . . In Tables 2.1 and 2.2 the vibrationalrotational constants are given for the ground (lowest energy) electronic state. 2.5 EXAMPLE: CARBON DIOXIDE In our treatment of molecular vibrations and rotations we have only considered the relatively simple case of diatomic molecules. Rather than now discussing general polyatomic molecules, which are more complicated but fundamentally much the same as diatomics, we will consider only the speciﬁc case of the carbon dioxide molecule. We choose this example because the CO2 laser is one of the most important molecular lasers. Carbon dioxide is a linear triatomic molecule (Fig. 2.9). Such a molecule has three socalled normal modes of vibration, shown in Fig. 2.9b. For obvious reasons these 34 ATOMS, MOLECULES, AND SOLIDS O (i) C (a) O C O O C (ii) O (iii) O O C O (b) Figure 2.9 (a) Carbon dioxide (CO2) is a linear triatomic molecule. (b) Normal modes of vibration of the CO2 molecule: (i) the asymmetric stretch mode, (ii) the bending mode, and (iii) the symmetric stretch mode. are called the asymmetric stretch, bending, and symmetric stretch modes. With each of these normal modes is associated a characteristic frequency of vibration. Each mode of vibration has a ladder of allowed energy levels associated with it, as in the case of a diatomic molecule (which has only one normal mode of vibration). The vibrational energy levels of the molecule may therefore be labeled by three integers v1, v2, and v3 (¼ 0, 1, 2, 3, . . .), and we have approximately (2) (3) 1 1 1 E(v1 , v2 , v3 ) ¼ hcv(1) (2:5:1) e v1 þ 2 þ hcve v2 þ 2 þ hcve v3 þ 2 , (2) (3) where v(1) e , ve , and ve are the normalmode frequencies in units of wave numbers. In reality each normal mode is slightly anharmonic, but the harmonicoscillator approximation will sufﬁce for our purposes. For CO2 the normalmode frequencies are 1 v(1) e ¼ v(symmetric stretch) 1388 cm , (2:5:2a) 1 v(2) e ¼ v(bending) 667 cm , (2:5:2b) 1 v(3) e ¼ v(asymmetric stretch) 2349 cm : (2:5:2c) The ﬁrst few vibrational energy levels (v1v2v3) of the CO2 molecule are indicated in Fig. 2.10. Since CO2 is a linear molecule, its rotational energy spectrum has the same character as that for diatomic molecules. The CO2 rotational energy levels are thus given to a good approximation by Eq. (2.4.7): EJ ¼ hcBe J(J þ 1), J ¼ 0, 1, 2, . . . , (2:5:3) 2.6 CONDUCTORS AND INSULATORS 35 2349 cm–1 (001) (030) 1388 cm–1 (100) (020) 667 cm–1 (010) Symmetric stretch mode Bending mode Asymmetric stretch mode (000) Figure 2.10 The ﬁrst few vibrational energy levels of the CO2 molecule. where the rotational constant Be for the CO2 molecule is Be ¼ 0:39 cm1 : 2.6 (2:5:4) CONDUCTORS AND INSULATORS In a gas the average distance between molecules (or atoms) is large compared to molecular dimensions. In liquids and solids, however, the intermolecular distance is comparable to a molecular diameter (Problem 2.3). Consequently the intermolecular forces are roughly comparable in strength to the interatomic bonding forces in the molecules. The molecules in liquids and solids are thus inﬂuenced very strongly by their neighbors. What is generally called “solidstate physics” is mostly the study of crystalline solids, that is, solids in which the molecules are arranged in a regular pattern called a crystal lattice. The central fact of the theory of crystalline solids is that the discrete energy levels of the individual atoms are split into energy bands, each containing many closely spaced levels (Fig. 2.11). Between these allowed energy bands are gaps with no allowed energies. The way this happens is easy to explain with a simple example. Imagine a sodium atom with its 11 electrons distributed according to the Pauli exclusion principle over its 1s level (2 electrons), 2s level (2 electrons), 2p level (6 electrons) and 3s level (1 electron). A second sodium atom has exactly the same energy levels occupied by 11 electrons in the same way. If the two sodium atoms are brought close together their two equalenergy 1s levels turn into two levels of “disodium,” and these two levels of disodium have slightly different energies from their Na values and slightly different energies from each other. Similarly, the two 2s levels of Na become two slightly different levels of disodium, and so on for the higher levels. 36 ATOMS, MOLECULES, AND SOLIDS Allowed energy band Forbidden energy gap Allowed energy band Forbidden energy gap Allowed energy band Figure 2.11 In a crystalline solid the allowed electron energy levels occur in bands of closely spaced levels. Between these allowed energy bands are forbidden gaps. The same process occurs for three sodium atoms, in which case the 1s label applies to three slightly separated levels, the 2s label applies to three slightly separated levels, and so on. When the number of atoms N is as large as is appropriate to a macroscopic piece of sodium metal the N slightly separated levels are so closely bunched that they constitute an effectively continuous band of energies. The relatively large gap between the 1s and 2s levels in sodium atoms becomes the “forbidden gap” between 1s and 2s bands in sodium metal, where there are no longer any distinguishable levels. This is sketched in Fig. 2.12 for 1, 2, 6, and N 1023 atoms. A onedimensional model showing how such band structure arises in quantum theory is discussed in the Appendix to this chapter. We can reach a crude understanding of the formation of energy bands by beginning with the case of two identical atoms. When the atoms are far apart and effectively noninteracting, their electron conﬁgurations and energy levels are identical. As they are brought closer together, however, the electrons of each atom begin to feel the presence 1 atom 2 atoms 6 atoms N atoms First excited state Ground state Figure 2.12 Sketch of the change in the energy values originally assigned to the ground and ﬁrst excited states of an atom as more and more atoms are combined to form a solid. Note that the band gap energy can be identiﬁed with the original atomic level spacing, but is generally different in size. 2.6 CONDUCTORS AND INSULATORS 37 En +1 En En –1 Figure 2.13 Crystalline solid energy bands formed from energy levels of isolated atoms. of the other atom, and the energy levels become those of the two atoms as a whole.5 The difference between these new energy levels depends upon the interatomic spacing (Fig. 2.12). The difference between the highest and the lowest of these N levels depends on the interatomic distances, amounting typically to several electron volts for atomic spacings of a few angstroms, typical of solids. Now if we increase N, keeping the interatomic spacing ﬁxed as in a crystalline solid, the total energy spread of the N levels stays about the same, but the levels become more densely spaced. For the large values of N typical of a solid (say, something like 1029 atoms/m3), each set of N levels thus becomes in effect a continuous energy band (as in Fig. 2.13), which in some solids can be even wider then the original atomic level spacing. The chemical and optical properties of atoms are determined primarily by their outer electrons. In solids, similarly, many important properties are determined by the electrons in the highest energy bands, the bands evolving out of the higher occupied states of the individual atoms. Consider, for instance, a solid in which the highest occupied energy band is only partially ﬁlled, as illustrated in Fig. 2.14a. In an applied electric ﬁeld the electrons in this band can readily take up energy and move up within the band. A solid whose highest occupied band is only partially ﬁlled is a good conductor of electricity. Now consider a solid whose highest occupied band is completely ﬁlled with electrons, as illustrated in Fig. 2.14b. In this case it is quite difﬁcult for an electron to move because all the energetically allowed higher states in the band already have their full measure of electrons permitted by the Pauli principle. Therefore, a solid whose highest occupied energy band is ﬁlled will be an electrical insulator; in other words, its electrons will not ﬂow freely when an electric ﬁeld is applied. Implicit in 5 These two energy levels correspond to symmetric and antisymmetric spatial wave functions for the electrons, with correspondingly antisymmetric and symmetric spin eigenfunctions. The twofold exchange degeneracy in the case of widely separated atoms is broken when their wave functions begin to overlap. 38 ATOMS, MOLECULES, AND SOLIDS Conduction band Eg Valence band (a) Conduction band Valence band Valence band (b) (c) Eg Figure 2.14 In a good conductor of electricity (a), the highest occupied band is only partially ﬁlled with electrons, whereas in a good insulator (b) it is ﬁlled. In (b) the energy gap Eg between the valence band and the conduction band is large. In the case (c) of a semiconductor, however, this gap is small, and electrons in the valence band can easily be promoted to the conduction band. this deﬁnition of an insulator is the assumption that the forbidden energy gap between the highest ﬁlled band and the next allowed energy band, denoted Eg in Fig. 2.14, is large compared to the amount of energy an electron can pick up in the applied ﬁeld. Solids in which this band gap is not so large are called semiconductors. Their band structure is indicated in Fig. 2.14c. At the absolute zero of temperature the valence band of a semiconductor is completely ﬁlled, whereas the conduction band, the next allowed energy band, is empty. At room temperature, however, electrons in the valence band may have enough thermal energy to cross the narrow energy gap and go into the conduction band. Thus, diamond, which has a band gap of about 7 eV, is an insulator, whereas silicon, with a band gap of only about 1 eV, is a semiconductor. In a metallic conductor, by contrast, there is no band gap at all; the valence and conduction bands are effectively overlapped. This characterization of solids as insulators, conductors, and semiconductors is obviously more descriptive than explanatory. To understand why a given solid is an insulator, conductor, or semiconductor, we must consider the nature of the forces binding the atoms (or molecules) together in the solid. In covalent solids the atoms are bound by the sharing of outer electrons in partially ﬁlled conﬁgurations. In a true covalent solid, there are no free el