Hermann Haken. Laser Theory. Corrected Printing. With 72 Figures

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2 Hermann Haken Laser Theory Corrected Printing With 72 Figures Springer-Verlag Berlin Heidelberg New York Tokyo 1984

3 Professor Dr. Dr. h.c. HERMANN HAKEN Universitat Stuttgart, Institut fur Theoretische Physik, Pfaffenwaldring 57/IV D-7000 Stuttgart 80, Fed. Rep. of Germany This book originally appeared in hardcover as Volume XXV/2c of Encyclopedia of Physics by Springer-Verlag Berlin, Heidelberg 1970 ISBN-13: : / e-isbn-13: Library of Congress Cataloging in Publication Data. Haken, H. Laser theory. "Originally appeared in hardcover as volume XXV/2c of Encyclopedia of physics"-verso of t.p. Includes index. 1. Lasers. I. Tide. QC688.H ' This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law, where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich by Springer-Verlag Berlin Heidelberg 1970 and The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2153/

4 Dedicated to tbe memory of my parents. H.HAKEN.

5 Foreword. This book, written by one of the pioneers of laser theory, is now considered a classic by many laser physicists. Originally published in the prestigious Encyclopedia of Physics series, it is now being republished in paperback to make it available not only to professors and scientists, but also to students. It presents a thorough treatment of the theory of laser resonators, the quantum theory of coherence, and the quantization of electromagnetic fields. Especial emphasis is placed on the quantum-mechanical treatment of laser light by means of quantum-mechanical Langevin equations, the density matrix equation, and the Fokker-Planck equation. The semiclassical approach and the rate equation approach are also presented. The principles underlying these approaches are used to derive the relevant equations, from which, in turn, the various properties of laser light are derived.

6 Preface. The concept of the laser came into existence more than a decade ago when SCHAWLOW and TOWNES showed that the maser principle could be extended to the optical region. Since then this field has developed at an incredible pace which hardly anybody could have foreseen. The laser turned out to be a meeting place for such different disciplines as optics (e.g. spectroscopy). optical pumping, radio engineering, solid state physics, gas discharge physics and many other fields. The underlying structure of the laser theory is rather simple. The main questions are: what are the light intensities (a), what are the frequencies (b), what fluctuations occur (c), or, in other words, what are the coherence properties. Roughly speaking these questions are treated by means of the rate equations (a), the semiclassical equations (b). and the fully quantum mechanical equations (c), respectively. The corresponding chapters are written in such a way that they can be read independently from each other. For more details about how to proceed, the reader is advised to consult Chap When a theoretical physicist tries to answer the above questions in detail and in a satisfactory way he will find that the laser is a fascinating subject from whatever viewpoint it is treated. Indeed, mathematical methods from such dlfferent fields as resonator theory, nonlinear circuit theory or nonlinear wave theory, quantum theory including quantum electrodynamics, spin resonance theory and quantum statistics had to be applied or were even newly developed for the laser, e.g. several methods in quantum statistics applicable to systems far from thermal equilibrium. A number of these concepts and methods can certainly be used in other branches of physics, such as nonlinear optics, nonlinear spin wave theory, tunnel diodes, Josephson junctions, phase transitions etc. Thus it is hoped that physicists working in those fields, too, will find the present article useful. I became acquainted with the theoretical problems of the laser during a stay at the Bell Telephone Laboratories in spring and summer 1960, shortly before the first laser was made to work. I am grateful to Prof. WOLFGANG KAISER who drew my attention to this problem and with whom I had the first discussions on this subject. The main part of the present article had been completed in 1966, when I became ill. I have used the delay to include a number of topics which have developed in the meantime, e.g. the Fokker-Planck equation referring to quantum systems and the theory of ultrashort pulses. I am indebted to my colleagues, co-workers and students for many stimulating discussions, in particular to my friend and colleague, W. WEIDLICH. The manuscript has been read critically and checked by several of them, and lowe thanks

7 VIII Preface. besides to H. GEFFERS, U. GNUTZMANN, R. GRAHAM, F. HAAKE, Mrs. HUBNER PELIKAN, K. KAUFMANN, P. REINEKER, H. RISKEN, H. SAUERMANN, C. SCHMID, H. D. VOLLMER and K. ZElLE. In addition, several of them made a series of valuable suggestions for improving the manuscript, in particular H. RISKEN and H. D. VOLLMER. The manuscript would never have been completed, however, without the tireless assistance of my secretary, Mrs. U. FUNKE, who not only typed several versions of it with great patience, but also prepared the final form in a perfect way. Stuttgart, February, H. HAKEN.

8 Contents. 1. Introduction The maser principle 1.2. The laser condition Properties of laser light a) Spatial coherence. b) Temporal coherence c) Photon statistics d) High intensity e) Ultrashort pulses 1.4. Plan of the article II. Optical resonators Introduction II.2. The Fabry-Perot resonator with plane parallel reflectors a) Spatial distribution of modes b) Diffraction losses..... c) Three-dimensional resonator II.3. Confocal resonator..... a) Field outside the resonator. b) Field inside the resonator c) Far field pattern of the confocal resonator d) Phase shifts and losses IIA. More general configurations a) Confocal resonators with unequal square and rectangular apertures.... b) Resonators with reflectors of unequal curvature 01:) Large circular apertures fj) Large square aperture II.5. Stability III. Quantum mechanical equations of the light field and the atoms without losses 24 IlL 1. Quantization of the light field IIL2. Second quantization of the electron wave field IIL3. Interaction between radiation field and electron wave field IlIA. The interaction representation and the rotating wave approximation 29 IlLS. The equations of motion in the Heisenberg picture IIL6. The formal equivalence of the system of atoms each having 2 levels with a system of t spins IV. Dissipation and fluctuation of quantum systems. The realistic laser equations 33 IV.1. Some remarks on homogeneous and inhomogeneous broadening 33 a) Naturallinewidth b) Inhomogeneous broadening :) Impurity atoms in solids 33 fj) Gases y) Semiconductors 34 c) Homogeneous broadening 34 01:) Impurity atoms in solids 34 fj) Gases y) Semiconductors

9 x Contents. IV.2. A survey of IV.2.-IV a) Definition of heatbaths (reservoirs) 35 b) The role of heatbaths c) Classical Langevin and Fokker-Planck equations. 36 ex) Langevin equations p) The Fokker-Planck equation d) Quantum mechanical formulation: the total Hamiltonian. 37 e) Quantum mechanical Langevin equations, Fokker-Planck equation and density matrix equation. 38 ex) Langevin equations P) Density matrix equation y) Generalized Fokker-Planck equation 39 IV.3. Quantum mechanical Langevin equations: ongm of quantum mechanical Langevin forces (the effect of heatbaths). 39 a) The field (one mode) b) Electrons ("atoms") IV.4. The requirement of quantum mechanical consistency 44 a) The field b) Dissipation and fluctuations of the atoms IV.5. The explicit form of the correlation functions of Langevin forces 46 a) The field b) The N-Ievel atom IV.6. The complete laser equations a) Quantum mechanically consistent equations for the operators bt and (at ak)" ex) The field equations. 50 P) The matter equations 50 b) Semiclassical equations. 51 ex) The field equations. 51 p) The matter equations 51 IY.7. The density matrix equation 51 a) General derivation 51 b) Specialization of Eq. (IV.7.31). 56 ex) Light mode P) Atom y) The density matrix equation of the complete system of M laser modes and N atoms IV.8. The evaluation of multi-time correlation functions by the single-time density matrix IV.9. Generalized Fokker-Planck equation: definition of distribution functions a) Field ex) Wigner distribution function and related representations 61 P) Transforms of the distribution functions: characteristic functions 63 y) Calculation of expectation values by means of the distribution functions b) Electrons ex) Distribution functions for a single electron 64 p) Characteristic functions y) Electrons and fields IV.1O. Equation for the laser distribution function (IV.9.22) 65 a) Comparison of the advantages of the Heisenberg and the SchrOdinger representations ex) The Heisenberg representation P) The SchrOdinger representation b) Final form of the generalized Fokker-Planck equation. 70 IV.11. The calculation of multi-time correlation functions by means of the distribution function V. Properties of quantized electromagnetic fields 73 V.1. Coherence properties of the classical and the quantized electromagnetic field

10 Contents. XI a) Classical description: definitions 73 01:) The complex analytical signal 73 {J) The average y) The mutual coherence function 74 b) Quantum theoretical coherence functions :) Elementary introductions {J) Coherence functions y) Coherent wave functions ) Generation of coherent fields by classical sources (the forced harmonic oscillator) V.2. Uncertainty relations and limits of measurability 83 a) Field and photon number. 83 b) Phase and photon number 85 01:) Heuristic considerations 85 {J) Exact treatment c) Field strength V.3. Spontaneous and stimulated emission and absorption 88 a) Spontaneous emission b) Stimulated emission c) Comparison between spontaneous and stimulated emission rates 91 d) Absorption V.4. Photon counting a) Quantum mechanical treatment, correlation functions 93 b) Classical treatment of photon counting V.5. Coherence properties of spontaneous and stimulated emission. The spontaneous linewidth VI. Fully quantum mechanical solutions of the Jaser equations VI.1. Disposition VI.2. Summary of theoretical results and comparison with the experiments 101 a) Qualitative discussion of the characteristic features of the laser output: homogeneously broadened line b) Quantitative results: single mode action :) The spectroscopic linewidth well above threshold {J) The spectroscopic linewidth somewhat below threshold 103 y) The intensity (or amplitude) fluctuations ) Photon statistics VI.3. The quantum mechanical Langevin equations for the solid state laser 112 a) Field equations b) Matter equations :) The motion of the atomic dipole moment Dipole moment between levels j and k Dipole moment between levels j and 1 =!= k, j and between levels k and 1 =j, k Dipole moment between levels i =!= k, i and 1 =!= k, j. 115 (j) The occupation numbers change For the laser levels j and k For the non-laser levels VIA. Qualitative discussion of single mode operation. 116 a) The linear range (subthreshold region) b) The nonlinear range (at threshold and somewhat above) :) Phase diffusion {J) Amplitude (intensity) fluctuations c) The nonlinear range at high inversion 120 d) Exact elimination of all atomic coordinates 120 VI.5. Quantitative treatment of a homogeneously broadened transition: emission below threshold (intensity, linewidth, amplification of signals) a) No external signals :) Single-mode linewidth below threshold 123 {J) Many modes below threshold 123 b) External signals

11 XII Contents. V1.6. Exact elimination of atomic variables in the case of a homogeneously broadened line. Running or standing waves 125 oc) Standing waves P) Running waves VI. 7. Single mode operation above threshold, homogeneously broadened line a) Lowest order b) First order c) Phase noise. Linewidth formula 130 d) Amplitude fluctuations oc) The special case of a moderate photon number 133 P) The special case of a big photon number V1.8. Stability of amplitude. Spiking and damped oscillations. Single-mode operation, homogeneously broadened line a) Qualitative discussion b) Quantitative treatment c) The special case W ("two level system"). 137 V1.9. Qualitative discussion of two-mode operation a) Some transformations b) Both modes well below threshold c) Modes somewhat above or somewhat below threshold 140 d) Both modes above threshold oc) IWl -w21 ">1IT P) IWl - w21 ;:SilT VI.10. Gas laser and solid-state laser with an inhomogeneously broadened line. The van der Pol equation, single-mode operation a) Solid-state laser with an inhomogeneously broadened line and an arbitrary number of levels b) Gas laser VI.11. Direct solution of the density matrix equation VI.12. Reduction of the generalized Fokker-Planck equation for single-mode action a) Expansion in powers of } r (N: ~ number of atoms) 154 b) Adiabatic elimination of the atomic variables 156 c) The Fokker-Planck equation VI. 13. Solution of the reduced Fokker-Planck equation 159 a) Steady state solution b) Transient solution VI.14. The Fokker-Planck equation for multimode action near threshold. Exact or nearly exact stationary solution a) The explicit form of the Fokker-Planck equation b) Theorem on the exact stationary solution of a Fokker-Planck equation c) Nearly exact solution of (VI.14.1) oc) Normal multimode action P) Phase locking of many modes y) A qualitative discussion of phase locking (example of three modes) VI.15. The linear and quasi-linear solution of the general Fokker-Planck equation a) Far below threshold b) Well above threshold 172 VII. The semiclassical approach and its applications. 173 VII.1. Spirit of the semiclassical approach. The equations for the solid state laser a) The field equations 174 b) The material equations c) Macroscopic treatment oc) Wave picture, inhomogeneous atomic line 178 P) Wave picture, homogeneous atomic line. 178

12 Contents. XIII y) Wave picture, homogeneous atomic line, rotating wave approximation, slowly varying amplitude approximation. 179 d) Mode picture, polarization waves d) Extension to multilevel atoms e) Systematics of the semiclassical approach 181 VII.2. Method of solution for the stationarv state 182 a) Single-mode operation, general features 183 b) Two-mode operation, general features 184 ex) Time-independent atomic response 185 (J) Time-dependent atomic response. 185 VII.3. The solid-state laser with a homogeneously broadened line. Single and multimode laser action a) Single-mode operation b) Multiple-mode operation ex) Equations for the photon densities of M modes. 187 (J) Equations for the frequency shift VilA. The solid-state laser with an inhomogeneously broadened Gaussian line. Single- and two-mode operation. 187 a) One mode ex) Equation for the frequency shift (J) Equation for the photon density b) Two modes ex) Equations for the photon densities na 189 (J) Equations for the frequency shifts 189 c) Lorentzian line shape VII.5. The solid-state laser with an inhomogeneously broadened line: multimode action a) Normal multimode action. 191 b) Combination tones 192 c) F r e q u locking. ~ n c y VII.6. Equations of motion for the gas laser 194 VII. 7. Single- and two-mode operation in gas lasers. 197 a) Single-mode operation ex) Equation for the photon density. 198 (J) Equation for the frequency shift. 199 b) Two-mode operation ex) Equations for the photon densities 200 (J) Equations for the frequency shifts 201 VII.8. Some exactly solvable problems a) Single-mode operation in solid state lasers 201 ex) Homogeneously broadened line Running waves Standing waves in axial direction 202 (J) Inhomogeneously broadened line, running waves 203 b) Single-mode in the gas laser VII.9. External fields a) The effect of a longitudinal magnetic field on the single spatial mode output b) The field equations c) The matter equations d) 30lution of the amplitude and frequency-determining Eqs. (VII.9.24), (VII.9.25) VII. 10. Ultrashort optical pulses: the principle of mode locking 213 a) Loss modulation by an externally driven modulator 215 b) Loss modulation by a saturable absorber 216 c) Gain modulation d) Frequency modulation e) Analogy to microwave circuits 217 VII.1t. Ultrashort optical pulses: detailed treatment of loss modulation 217 a) Pulse shape and pulse width b) Discussion of the results and of the range of validity. 223 c) Numerical application

13 XIV Contents. VIL12. Super-radiance. Spin and photo echo 224 a) Definition of super-radiant states 224 b) Generation of super-radiant states 228 (l() Classical treatment of the spin motion. 228 (J) Quantum theoretical treatment 229 c) Classical description of super-radiant emission. 231 d) The spin-echo experiment e) The photo-echo experiment f) A further analogy between a spin t system and a two-level system: the fictitious spin VII. 13. Pulse propagation in laser-active media 236 a-c) Steady state and self-pulsing. 237 (l() The basic equations (J) Stationary solution y) Normalized amplitudes 238 6) Stability of the stationary solution 238 e) Transient build-up of the pulse 239 ~ ) Steady state pulse 241 1'}) A simplified model ) The special case v = c d) The n-pulse e) The 2n-pulse. (Self-induced transparency). 246 VII.14. Derivation of rate equations VIII. Rate equations and their applications VIlLi. Formulation of rate equations and solution for the steady state (especially: threshold condition, pump power requirement, single versus multimode laser action) 249 a) The rate equations 249 (l() The field equations (J) The matter equations b) Treatment of the steady state c) The completely homogeneous case 251 (l() General formulation 251 (J) 3-Level system, the lower transition is laser-active. 252 y) Pump power at threshold ) 3-Level system, the upper transition is laser-active 253 8) 4-Level system, laser action between the two middle levels 255 VIlL2. The coexistence of modes on account of spatial inhomogeneities or an inhomogeneously broadened line a) Homogeneous line, but space-dependent modes (represented by standing waves) (l() Axial modes with a different frequency distance from the line center (J) Different losses b) Spatially inhomogeneous pumping, homogeneously broadened line 258 (l() Running waves (J) Standing waves c) lnhomogeneously broadened line 259 VIlL3. Laser cascades a) Matter equations 260 b) Homogeneously broadened line and standing waves (modes in axial direction) c) lnhomogeneously broadened line and standing waves 261 d) Discussion of an example VIlI.4. Solution of the time-dependent rate equations. Relaxation oscillations a) The 3-level system with laser action between the two lower levels 264 b) 3-Level system, laser action between the two upper levels 265 c) 4-Level system d) Approximate solution for small oscillations 266 VIIL5. The giant pulse laser a) Semiquantitative treatment 268 b) Quantitative treatment 269

14 Contents. xv IX. Further methods for dealing with quantum systems far from thermal equilibrium IX.1. The general form of the density matrix equation IX.2. Exact generalized Fokker-Planck equation: definition of the distribution function IX.3. The exact generalized Fokker-Planck equation IX.4. Derivation of the exact generalized Fokker-Planck equation. 276 IX.5. Projection onto macroscopic variables IX.6. Exact elimination of the atomic operators within quantum mechanical Langevin equations IX.7. Rate equations in quantized form IX.8. Exact elimination of the atomic operators from the density matrix equation IX.9. Solution of the generalized field master Eq. (IX.8.12) 290 X. Appendix. Useful operator techniques.... X.1. The harmonic oscillator X.2. Operator relations for Bose operators X.3. Formal solution of the Schrodinger equation. X.4. Disentangling theorem X.5. Disentangling theorem for Bose operators Sachverzeichnis (Deutsch-Englisch) Subject Index (English-German)

ENCYCLOPEDIA OF PHYSICS

ENCYCLOPEDIA OF PHYSICS CHIEF EDITOR S. FLOGGE VOLUME XXV/2c LIGHT AND MATTER Ic EDITOR L. GENZEL WITH 72 FIGURES SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1970 HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLOGGE