ENCYCLOPEDIA OF PHYSICS

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1 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

2 HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLOGGE BAND XXV/2c LICHT UNO MATERIE Ic BANDHERAUSGEBER L. GENZEL MIT 72 FIGUREN SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1970

3 ISBN ISBN (ebook) DOI / Das Werk ist urheberrechtlich geschlltzt. Die dadurch begrllndeten Rechte, insbesondere die der "Obersetzung, des Nachdruckes, der Entnahme von Abbildungen, der Funksendung, der Wiedergabe auf photomechanischem oder iihnlichem Wege und der Speicherung in Datenverarbeitungsanlageo b1eiben, auch bei nur auszugsweiser Verwertung, vorbehalten. Bel Vervielfaltigungen fiir gewerbliche Zwecke ist gemiui 54 UrhG eine Vergiitung an den Verlag zu zablen, deren Hohe mit dem Verlag zu vereinbaren ist. by Springer-Verlag Berlin Heidelberg Urspriinglich erschienen bei Springer-Verlag 1970 Softcover reprint of the hardcover 1st edition 1970 Library Congress Catalog Card Number A Die Wledergabe von Gebrauchsnamen, Handelsnameo, Warenbezeicbnungeo usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annabme, dab solche Namen im Sinne der Warenzeichen und Markenschutz Gesetzgebung als frei zu betrachten waren und daber von jedermann benutzt werden diirften Titei-Nr. 5800

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

5 Foreword. 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. I.4. 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 different 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 I owe thanks

6 VIII Foreword. 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.

7 Contents. Laser Theory. By Dr. rer. nat. H. HAKEN, Professor of Theoretical Physics, Institut fiir Theoretische Physik der Universitat Stuttgart (Germany). (With 72 Figures) I. Introduction The maser principle The laser condition Properties of laser light 5 a) Spatial coherence. 5 b) Temporal coherence 6 c) Photon statistics 7 d) High intensity 7 e) Ultrashort pulses Plan of the article 7 II. Optical resonators Introduction The Fabry-Perot resonator with plane parallel reflectors a) Spatial distribution of modes b) Diffraction losses..... c) Three-dimensional resonator 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 II.4. More general configurations a) Confocal resonators with unequal square and rectangular apertures.... b) Resonators with reflectors of unequal curvature tx) Large circular apertures {3) Large square aperture II.s. Stability.... Ill. Quantum mechanical equations of the light field and the atoms without losses Quantization of the light field Second quantization of the electron wave field Interaction between radiation field and electron wave field I11.4. The interaction representation and the rotating wave approximation 29 IlLS. The equations of motion in the Heisenberg picture Ill.6. 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. 33 tx) Impurity atoms in solids 33 {3) Gases y) Semiconductors 34 c) Homogeneous broadening 34 tx) Impurity atoms in solids 34 {3) Gases y) Semiconductors

8 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 at) Langevin equations Pl 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 at) Langevin equations Pl 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-level atom IV.6. The complete laser equations a) Quantum mechanically consistent equations for the operators b! and (at ak)p at) The field equations. 50 Pl The matter equations 50 b) Semiclassical equations. 51 at) The field equations. 51 Pl The matter equations 51 IV.7. The density matrix equation 51 a) General derivation 51 b) Specialization of Eq. (IV.7.31). 56 at) Light mode Pl Atom y) The density matrix equation of the complete system of M laser modes and N atoms IV.S. 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 at) 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 at) Distribution functions for a single electron 64 Pl Characteristic functions y) Electrons and fields IV.10. Equation for the laser distribution function (IV.9.22) 65 a) Comparison of the advantages of the Heisenberg and the Schriidinger representations at) The Heisenberg representation Pl The Schriidinger 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 V.t. Coherence properties of the classical magnetic field and the quantized electro

9 Contents. XI a) Classical description: definitions 7 3 oc) The complex analytical signal 73 {J) The average y) The mutual coherence function 74 b) Quantum theoretical coherence functions. 76 oc) Elementary introductions 76 {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 a) Field and photon number. b) Phase and photon number oc) Heuristic considerations {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 laser equations Vl.t. 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 oc) 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 oc) The motion of the atomic dipole moment Dipole moment between levels j and k Dipole moment between levels j and l =1= k, j and between levels k and l = j, k Dipole moment between levels i =I= k, j and l =I= k, j {J) The occupation numbers change For the laser levels j and k For the non-laser levels VI.4. Qualitative discussion of single mode operation. 116 a) The linear range (subthreshold region) b) The nonlinear range (at threshold and somewhat above) 119 oc) Phase diffusion {J) Amplitude (intensity) fluctuations c) The nonlinear range at high inversion 120 d) Exact elimination of all atomic coordinates 120 VI. s. Quantitative treatment of a homogeneously broadened transition: emission below threshold (intensity, linewidth, amplification of signals) a) No external signals oc) Single-mode linewidth below threshold 123 {J) Many modes below threshold 123 b) External signals ~

10 XII Contents. Vl.6. Exact elimination of atomic variables in the case of a homogeneously broadened line. Running or standing waves IX) Standing waves Pl Running waves VI. 7. Single mode operation above threshold, homogeneously broadened line a) Lowest order b) First order c) Phase noise. Linewidth formula d) Amplitude fluctuations IX) The special case of a moderate photon number 133 p) The special case of a big photon number VI.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 13 ~ oo ("two level system"). 137 VI.9. Qualitative discussion of two-mode operation.. a) Some transformations b) Both modes well below threshold c) Modes somewhat above or somewhat below threshold d) Both modes above threshold IX) jco1 -coal >1/T.... Pl lco 1 -coal ;s1jt 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 1rl (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) IX) 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 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 IX) Wave picture, inhomogeneous atomic line 178 p) Wave picture, homogeneous atomic line. 178

11 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 stationary state 182 a) Single-mode operation, general features 183 b) Two-mode operation, general features 184 ot) Time-independent atomic response. 185 Pl Time-dependent atomic response VII.3. The solid-state laser with a homogeneously broadened line. Single and multimode laser action a) Single-mode operation b) Multiple-mode operation ot) Equations for the photon densities of M modes. 187 Pl Equations for the frequency shift VII.4. The solid-state laser with an inhomogeneously broadened Gaussian line. Single- and two-mode operation. 187 a) One mode ot) Equation for the frequency shift 188 Pl Equation for the photon density 189 b) Two modes ot) Equations for the photon densities na 189 Pl Equations for the frequency shifts 189 c) Lorentzian line shape Vll.S. The solid-state laser with an inhomogeneously broadened line: multimode action a) Normal multimode action. 191 b) Combination tones 192 c) Frequency locking 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 ot) Equation for the photon density. 198 Pl Equation for the frequency shift. 199 b) Two-mode operation ot) Equations for the photon densities 200 Pl Equations for the frequency shifts 201 VII.8. Some exactly solvable problems a) Single-mode operation in solid state lasers 201 ot) Homogeneously broadened line Running waves Standing waves in axial direction 202 Pl 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) Solutionoftheamplitudeandfrequency-determiningEqs.(VII.9.24), (Vll.9.25) VII. to. Ultrashort optical pulses: the principle of mode locking. a) Loss modulation by an externally driven modulator b) Loss modulation by a saturable absorber 216 c) Gain modulation d) Frequency modulation e) Analogy to microwave circuits 217 VII.! 1. 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

12 XIV Contents. VII.12. Super-radiance. Spin and photo echo 224 a) Definition of super-radiant states 224 b) Generation of super-radiant states 228 oc) 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! system and a two-level system: the fictitious spin Vll.13. Pulse propagation in laser-active media 236 a-c) Steady state and self-pulsing. 237 oc) The basic equations {J) Stationary solution y) Normalized amplitudes 238 t'5) Stability of the stationary solution 238 e) Transient build-up of the pulse 239 C) Steady state pulse 241 1)) A simplified model fj) The special case v = c. 244 d) The n-pulse e) The 2n-pulse. (Self-induced transparency). 246 Vll.14. Derivation of rate equations VIII. Rate equations and their applications VIII.t. 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 oc) The field equations {J) The matter equations b) Treatment of the steady state c) The completely homogeneous case 251 oc) General formulation 251 {J) 3-Level system, the lower transition is laser-active. 252 y) Pump power at threshold t'5) 3-Level system, the upper transition is laser-active 253 e) 4-Lev!1 system, laser action between the two middle levels. 255 VIII.2. 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) oc) Axial modes with a different frequency distance from the line center {J) Different losses b) Spatially inhomogeneous pumping, homogeneously broadened line 258 oc) Running waves {J) Standing waves c) Inhomogeneously broadened line 259 VII1.3. Laser cascades a) Matter equations 260 b) Homogeneously broadened line and standing waves (modes in axial direction) c) Inhomogeneously broadened line and standing waves 261 d) Discussion of an example VIII.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 VIII.5. The giant pulse laser a) Semiquantitative treatment 268 b) Quantitative treatment 269

13 Contents. XV IX. F u ~ methods e r. for dealing with quantum systems far from thermal equihbnum 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 Schriidinger equation. X.4. Disentangling theorem X.5. Disentangling theorem for Bose operators Sachverzeichnis (Deutsch-Englisch) Subject Index (English-German)

14 List of important notations. One of the difficulties in following up the literature about laser theory consists in the different notations used by different authors. In the present book we have tried to unify the notations as far as possible, regardless of whether we are dealing with rate equations, semiclassical equations or the fully quantum mechanical equations. We give here a list of the most important notations. A A c D E E i<±l f!±l e g Vector potential Derivative of the quantity A with respect to time Annihilation and creation operator of an electron at the atom p in the state i In classical description dimensionless time-dependent complex amplitudes of mode A; in quantum mechanical description annihilation and creation operator respectively of a photon of the mode A Velocity of light Saturated inversion of all laser atoms Unsaturated inversion of all laser atoms Saturated inversion density Unsaturated inversion density Saturated inversion of a single atom Unsaturated inversion of a single atom Threshold inversion of a single atom Vector of the electric field strength Electric field strength in the direction of the atomic polarization, if E is parallel to the polarization Positive or negative frequency part of the electric field strength in the interaction representation and the rotating wave approximation Positive or negative frequency part of the electric field strengths in the rotating wave approximation and the interaction representation after the exponential e±th has been split off from E!±l Charge on an electron Vector of polarization of mode A Coupling constant between field and electronic transition or spin

15 XVIII m N 0 p s+, s-, s. Tp T t th thr u u X z List of important notations. Coupling constant between field mode A and atom p for the electronic transition from state l to stade j. If only a single laser transition is treated the indices j and l are dropped: g;." Coupling constant between spin and external magnetic field Hamiltonian Magnetic field strength Boltzmann's constant Wave number of mode A Wave vector of mode A Mass of an electron Total number of laser-active atoms in the cavity Occupation number of a single atom p in the state j Number of photons in the mode A. In the classical treatment the mean number of photons is meant. For further definitions, such as <n> and n, consult the text Operator Vector of polarization Dipole moment of atom p Stable amplitude of light mode A above threshold, in dimensionless units. Often the index A is dropped Sum over the corresponding spin flip or atomic transition operators Time after which equilibrium of the inversion is achieved under the action of pump and incoherent decay processes if the coupling to the laser light field is switched off. Sometimes the index p is dropped Absolute temperature Time The index "th" means "thermal" The index "thr" means "threshold" Unitary operator Wave function in the cavity. Its connection with the electric and magnetic field strengths is defined in the Eqs. (11.2.1) and (11.2.2) Scalar wave function of the cavity mode A Vector wave function of the cavity mode A Volume of the cavity Incoherent transition rates from an atomic level i to an atomic level k Space coordinate in three dimensions Space coordinate in one dimension Space coordinate in one dimension

16 r YJ. Yll a (J ~ (x) List of important notations. XIX Quantum mechanical Langevin forces for atoms Halfwidth of the homogeneous part of the atomic transition i--7-k. If a special laser transition is treated, the indices i and k are dropped. Other notation for y 1 - Tp Population difference between the levels i and k of a single atom fl If LJN;k,f' is the same for all atoms, the index fl is dropped Atomic dipole moment matrix element for the transition k--7-j Dirac's function Kronecker's symbol, ~. i = 1 for i = j, ~. i = 0 for i =!= j Cavity halfwidth Index which distinguishes different spatial modes Index which distinguishes different laser-active atoms Circular frequency of the atom with index fl Circular frequency of the atomic transition from state j to state l Density matrix or density operator Electric conductivity Spin operator with components a,, a ~, Spin flip operators Spin operator for the z component with eigenvalue ± t Inversion operator for atom fl Wave function, mostly of the light field Wave function of the vacuum Wave function at the initial time t = 0 Wave function of electrons Actual circular frequency of the mode A in the loaded cavity Circular frequency of the mode A in the unloaded cavity a. Planck's constant, divided by 2n