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H. HAKEN - - LASER TIGHTDYNAMICS NOR]H HOttAND LIGHT Volume 2 LASER LIGHT DYNAMICS H. HAKEN It! jrirut fur Theoretische Physik, Stuttgart \ORTH-HOLLAND PHYSICS PUBLISHING if1STERDAM. NEW YORK . OXFORD. TOKYO Preface to the Preface Dear Reader, Before you read this book, and even its preface, the following remarks might be useful to you. Since this book is " Volume 2" you may be inclined to believe that you must know all the contents of " Volume 1" before you can start reading (and, of course, understanding) " Volume 2". But this is not the case. The present " Volume 2" again starts at a rather elementary level, and then proceeds step by step to more difficult matters. Only at these later stages some more advanced theoretical background is required which then can be taken from " Volume I". I have chosen this way of presentation to make the theory of laser light accessible to a broad audience - ranging from students at the beginning of their graduate studies to professors and scientists interested in recent developments. For details on the relations between the chapters of these books consult the list at the end of the introduction. H. Haken Preface This book is a text which applies to students and professors of physics. Because it offers a broad view on laser physics and presents most recent results on the dynamics of laser light, such as self - pulsing and chaos, it will be of interest also to scientists and engineers engaged in laser research or development. This text starts at a rather elementary level and will smoothly lead the reader into the more difficult problems of laser physics, including the basic features of the coherence and noise properties of laser light. In the introductory chapters, typical experimental set - ups and laser materials will be discussed, but the main part of this book will be devoted to a theoretical treatment of a great variety of laser processes. The laser, or the optical maser, as it was originally called, is one of the most important inventions of this century and has found a great number of important applications in physics, chemistry, medicine, engineering, telecommunica - tions, and other fields. It bears great promises for further applications, e.g. in computers. But also from the point of view of basic research, a study of the physical processes which produce the unique properties of laser light are equally fascinating. The laser is a beautiful example of a system far from thermal equilibrium which can achieve a macroscopically ordered state through " self - organization " . It was the first example for a nonequili- brium phase transition, and its study eventually gave birth to synergetics, a new interdisciplinary field of research. I got involved in laser physics at a rather early stage and under most fortunate circumstances. In 1960 I was working as visiting scientist at the Bell Telephone Laboratories, Murray Hill. There I soon learned that these laboratories were searching for a revolutionary new light source. Two years earlier, in 1958, this source had been proposed by Schawlow and Townes, who derived in particular the laser condition and thus demonstrated the feasibility of this new device. At Bell Telephone Laboratories I soon got involved in a theoretical study of the laser processes and continued it at Stuttgart University. I developed a laser theory whose basic features I published in 1962 and which I then applied to various concrete problems, viii Preface jointly with my coworkers. At about the same time, in 1964, Willis Lamb published his theory, which he and his coworkers applied to numerous problems. It is by now well known that these two theories, which are called semiclassical and which were developed independently, are equivalent. The next step consisted in the development of the laser quantum theory which allows one to predict the coherence and noise properties of laser light (and that of light from lamps). This theory which I published in 1964 showed for the first time that the statistical properties of laser light change dramati - cally at laser threshold. In the following years my group in Stuttgart carried this work further, e.g. to predict the photon statistics close to laser threshold. From 1965 on, Scully and Lamb started publishing their results on the quantum theory of the laser, using a different approach, and Lax and Louise11 presented their theory. Again, all of these theories eventually turned out to be more or less equivalent. In those years experimental laser physics developed (and is still developing) at an enormous pace, but because I shall mainly deal with laser theory in this book, I have to cut out a representation of the history of that field. From my above personal reminiscences it may transpire that laser theory and, perhaps still more, laser physics in general have been highly competitive fields of research. But, what counts much more, laser physics has been for us all a fascinating field of research. When one looks around nowadays, one can safely say that is has lost nothing of its original fascination. Again and again new laser materials are found, new experimental set - ups invented and new effects predicted and discovered. Undoubtedly, for many years to come, laser physics will remain a highly attractive and important field of research, in which fundamental problems are intimately interwoven with applications of great practical importance. I hope that this book will let transpire the fascination of this field. Over the past nearly 25 years I greatly profited from the cooperation or discussion with numerous scientists and I use this oppprtunity to thank all of them. There is Wolfgang Kaiser, who was the first at BTL with whom I had discussions on the laser problem. Then there are the members of my group at Stuttgart who in the sixties, worked on laser theory and who gave important contributions. I wish to mention in particular R. Graham, H. Geffers, H. Risken, H. Sauermann, Chr. Schmid, H.D. Vollmer, and W. Weidlich. Most of them now have their own chairs at various universities. Among my coworkers who, in later years, contributed to laser theory and its applications are in particular J. Goll, A. Schenzle, H. Ohno, A. Wunderlin and J. Zorell. Over the years I enjoyed many friendly and stimulating discussions with F.T. Arecchi, W.R. Bennett, Jr., N. Bloembergen, R. Bonifacio, J.H. Eberly, C.G.B. Garret, R.J. Glauber, F. Haake, Yu. Preface ix Klimontovich, W. Lamb, M. Lax, W. Louisell, L. Lugiato, L. Mandel, L. Narducci, E.R. Pike, M. Sargent, M. Scully, S. Shimoda, S. Stenholm, Z.C. Wang, E. Wolf, J. Zhang, and many other scientists. I wish to thank my coworker, Dr. H. Ohno, for his continuous and valuable assistance in the preparation of the manuscript. In particular, he carefully checked the formulas and exercises, contributed some in addition, and drew the figures. My particular thanks go to my secretary, Mrs. U. Funke, who in spite of her heavy administrative work assisted me in many ways in writing the manuscript and typed various versions of it both rapidly and perfectly. Her indefatigable zeal constantly spurred me on to bring it to a finish. The writing of this book was greatly helped by a program of the Deutsche Forschungsgemeinschaft. This program was initiated by Prof. Dr. Maier- Leibnitz, whom I wish to thank cordially for his support for this project. H. Haken Contents Preface to the preface Preface Contents List of symbols Introduction The maser and laser principle The problems of laser theory The structure of laser theory and its representation in this book Basic properties and types of lasers The laser condition Typical properties of laser light Examples of laser systems (types of lasers and laser processes) Laser resonators Survey Modes in a confocal resonator Modes in a Fabry - Perot resonator The intensity of laser light. Rate equations Introduction The photon model of a single mode laser Relaxation oscillations Q - switching v vii xi xv xii Contents The basic rate equations of the multimode laser Hole burning. Qualitative discussion Quantitative treatment of hole burning. Single mode laser action of an inhomogeneously broadened line Spatial hole burning. Qualitative discussion The multimode laser. Mode competition and Darwin's survival of the fittest The coexistence of modes due to spatial hole burning. Quantitative treatment The basic equations of the semiclassical laser theory Introduction Derivation of the wave equation for the electric field strength The matter equations The semiclassical laser equations for the macroscopic quantities electric field strength, polarization, and inversion density The laser equations in a resonator Two important approximations: The rotating wave approximation and the slowly varying amplitude approximation The semiclassical laser equations for the macroscopic quantities electric field strength, polarization, and inversion density in the rotating wave - and slowly varying amplitude approximations Dimensionless quantities for the light field and introduction of a coupling constant The basic laser equations Applications of semiclassical theory The single mode laser. Investigation of stability Single mode laser action. Amplitude and frequency of laser light in the stationary state The single mode laser: Transients Multimode action of solid state lasers. Derivation of reduced equations for the mode amplitudes alone Simple examples of the multimode case Frequency locking of three modes The laser gyro Contents xiii The gas laser. Single mode operation 147 Derivation of the rate equations from the semiclassical laser equations 151 Ultrashort pulses 154 Some basic mechanisms. Active and passive mode locking 154 The basic equations of self - pulsing lasers 162 A general method for calculating evolving patterns close to instability points 164 Onset of ultrashort laser pulses: linear stability analysis 171 Onset of ultrashort laser pulses: nonlinear analysis 173 Solution of the order parameter equation 178 Models of lasers with saturable absorbers 183 Instability hierarchies of laser light. Chaos, and routes to chaos 187 Survey 187 The basic equations 189 The single mode laser equations and their equivalence with the Lorenz model of turbulence 189 Criteria for the presence of chaos 194 Routes to chaos 195 How to produce laser light chaos. Some theoretical models 198 Single mode laser with injected signal. Chaos, breathing, spiking 208 Optical bistability 2 15 Survey 2 15 A specific model 2 17 Steady state behavior of the model of section 9.2 219 The general case of an arbitrary susceptibility 223 Concluding remarks on chapter 9 233 Quantum theory of the laser I 234 A first approach via quantum mechanical Langevin equations. Coherence, noise and photon statistics Why quantum theory of the laser? The laser Hamiltonian Quantum mechanical Langevin equations xiv Contents Coherence and noise The behavior of the laser at its threshold. Photon statistics Quantum theory of the laser I1 A second approach via the density matrix equation and quantum classical correspondence The density matrix equation of the iaser A short course in quantum classical correspondence. The example of a damped field mode (harmonic oscillator) Generalized Fokker - Planck equation of the laser Reduction of the generalized Fokker - Planck equation Concluding remarks A theoretical approach to the two - photon laser Introduction Effective Hamiltonian, quantum mechanical Langevin equations and semiclassical equations Elimination of atomic variables Single mode operation, homogeneously broadened line and running wave The laser - trailblazer of synergetics What is synergetics about? Self - organization and the slaving principle Nonequilibrium phase transitions References and further reading Subject Index [...]... possible Therefore we have to where according to Vol 1, eq (2. 56) the number p is form W = l / ( ~ p ) , given by $2. 1 The laser condition 17 In it V is the volume of the laser, v the laser light frequency, Av the atomic line-width and c the velocity of light in the laser medium By means of the formulas derived above we may immediately present the laser condition Laser action sets in if n increases exponentially... chapters In conclusion of this chapter I present a table showing which knowledge of Volume 1 is required for an understanding of the chapters of the present book Chapters of Volume 1 needed (if not known otherwise): Present Vol 2 number of chapter Vol 1 needed chapters 1 What is light? 2 The nature of light 2 The nature of light 2 + 3 The nature of matter 6,7.1-7.6,8.1,9.1-9.4 Quantization of field and electron-wave... established In this new development the laser played the role of a trailblazer Within the frame of synergetics it became possible to make further predictions on the behavior of laser light For instance, on account of analogies between fluid dynamics and laser light the phenomenon of laser light chaos was predicted (Haken, 1975) Various routes to chaotic laser light could be discovered experimentally... not only in laser physics but also in nonlinear optics, we shall present it in section 11 .2 1 .2. 5 The laser - trailblazer of synergetics New vistas on laser theory were opened in 1968 when it was recognized that the transition from light from thermal sources to laser light within an individual laser bears a striking resemblance to phase transitions of systems in thermal equilibrium Thus the laser became... others): line-widths of laser light, phase, amplitude and intensity fluctuations (noise), coherence, photon statistics, and all problems quoted under 1 and 2 List of sections for a jirst reading 2. 1 -2. 3 3.1 4 .2 4.4 5.1-5.6,5.8-5.9 6.1-6.3 6.8 , Basic properties and types of lasers Laser resonators Photon model of single mode laser Q-switching Semiclassical equations Single mode laser action including... coupling to heatbaths 5, 6, 7.1-7.6, 8.1, 9.1-9.5 Chapters 5 and 6 of Vol 2 Chapter 2 Basic Properties and Types of Lasers 2. 1 The laser condition Let us consider the laser depicted in fig 2. 1 more closely, and let us discuss the tasks of its individual parts The two mirrors mounted at the endfaces fulfil the following functions When we treat light as a wave, between the two mirrors only standing waves can... suggested: 12 1 Introduction Table 1 .2 The structure of laser theory 1 Rate equations for photon numbers and atomic occupation numbers These equations allow the treatment of the following problems: laser condition, intensity distribution over the modes, single mode laser action, multi-mode laser action (coexistence and competition of modes), laser cascades, Q-switching, relaxation oscillations 2 Semiclassical... which I published in 1964, showed that laser light differs basically from light from conventional lamps Whereas light from conventionnal lamps consists of individual incoherent wave tracks, laser light essentially consists of a single wave whose phase and amplitude are subject to small fluctuations Subsequent measurements of the intensity fluctuations of laser light below and above threshold by Armstrong... photons which run in other directions leave the laser quickly Thus the mirrors serve for a selection of photons with respect to their lifetimes in the laser FI shtube \ Fig 2. 1 The first experimental set-up of the ruby laser according to Maiman The ruby rod in the middle is surrounded by a flashlamp in form of a spiral 42. 1 The laser condition 15 Fig 2. 2 The energy W of a two-level atom with the energy... further "mac~oscopic properties", frequency locking, ultrashort pulses, chaos, etc.: 6.4-6.5 6.6 Multimode laser Frequency locking I 0 1.3 The structure of laser theory 13 6.7 7.1 8.1 8 .2 8.3 9.1 9 .2 Laser gyro (perhaps) Ultrashort pulses Some basic mechanisms Laser light chaos (now needed 7 .2) Optical bistability Readers can also proceed by reading the chapters individually if they want to get to . H. HAKEN - - LASER TIGHTDYNAMICS NOR]H HOttAND LIGHT Volume 2 LASER LIGHT DYNAMICS H. HAKEN It! jrirut fur Theoretische. types of lasers The laser condition Typical properties of laser light Examples of laser systems (types of lasers and laser processes) Laser resonators