Laser Theory

Hermann Haken is Professor of the Institute for Theoretical Physics at the University of Stuttgart. He is known as the founder of synergetics. His research has been in nonlinear optics (in particular laser physics), solid state physics, statistical physics, and group theory. After the implementation of the first laser in 1960, Professor Haken developed his institute to an international center for laser theory. The interpretation of the laser principles as self organization of non equilibrium systems paved the way to the development of synergetics, of which Haken is recognized as the founder. Hermann Haken has been visiting professor or guest scientist in England, France, Japan, USA, Russia, and China. He is the author of some 23 textbooks and monographs that cover an impressive number of topics from laser physics to synergetics, and editor of a book series in synergetics. For his pathbreaking work and his influence on academic research, he has been awarded many-times. Among others, he is member of the Order "Pour le merite" and received the Max Planck Medal in 1990.

I. Introduction.- 1.1. The maser principle.- 1.2. The laser condition.- 1.3. 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.- II.1. 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.- II.4. More general configurations.- a) Confocal resonators with unequal square and rectangular apertures.- b) Resonators with reflectors of unequal curvature.- ?) Large circular apertures.- ?) Large square aperture.- II.5. Stability.- III. Quantum mechanical equations of the light field and the atoms without losses.- III.1. Quantization of the light field.- III.2. Second quantization of the electron wave field.- III.3. Interaction between radiation field and electron wave field.- III.4. The interaction representation and the rotating wave approximation.- III.5. The equations of motion in the Heisenberg picture.- III.6. The formal equivalence of the system of atoms each having 2 levels with a system of ½ spins.- IV. Dissipation and fluctuation of quantum systems. The realistic laser equations.- IV.1. Some remarks on homogeneous and inhomogeneous broadening.- a) Natural linewidth.- b) Inhomogeneous broadening.- ?) Impurity atoms in solids.- ?) Gases.- ?) Semiconductors.- c) Homogeneous broadening.- ?) Impurity atoms in solids.- ?) Gases.- ?) Semiconductors.- IV.2. A survey of IV.2.-IV.11.- a) Definition of heatbaths (reservoirs).- b) The role ofheatbaths.- c) Classical Langevin and Fokker-Planck equations.- ?) Langevin equations.- ?) The Fokker-Planck equation.- d) Quantum mechanical formulation: the total Hamiltonian.- e) Quantum mechanical Langevin equations, Fokker-Planck equation and density matrix equation.- ?) Langevin equations.- ?) Density matrix equation.- ?) Generalized Fokker-Planck equation.- IV.3. Quantum mechanical Langevin equations: origin of quantum mechanical Langevin forces (the effect of heatbaths).- a) The field (one mode).- b) Electrons ("atoms").- IV.4. The requirement of quantum mechanical consistency.- a) The field.- b) Dissipation and fluctuations of the atoms.- IV. 5. The explicit form of the correlation functions of Langevin forces.- a) The field.- b) The N-level atom.- IV. 6. The complete laser equations.- a) Quantum mechanically consistent equations for the operators b?+ and (ai+ak)?.- ?) The field equations.- ?) The matter equations.- b) Semiclassical equations.- ?) The field equations.- ?) The matter equations.- IV.7. The density matrix equation.- a) General derivation.- b) Specialization of Eq. (IV.7.31).- ?) Light mode.- ?) Atom.- ?) 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.- ?) Wigner distribution function and related representations.- ?) Transforms of the distribution functions: characteristic functions.- ?) Calculation of expectation values by means of the distribution functions.- b) Electrons.- ?) Distribution functions for a single electron.- ?) Characteristic functions.- ?) Electrons and fields.- IV. 10. Equation for thelaser distribution function (IV.9.22).- a) Comparison of the advantages of the Heisenberg and the Schrödinger representations.- ?) The Heisenberg representation.- ?) The Schrödinger representation.- b) Final form of the generalized Fokker-Planck equation.- IV.11. The calculation of multi-time correlation functions by means of the distribution function.- V. Properties of quantized electromagnetic fields.- V.1. Coherence properties of the classical and the quantized electromagnetic field.- a) Classical description: definitions.- ?) The complex analytical signal.- ?) The average.- ?) The mutual coherence function.- b) Quantum theoretical coherence functions.- ?) Elementary introductions.- ?) Coherence functions.- ?) 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.- ?) Heuristic considerations.- ?) Exact treatment.- c) Field strength.- V.3. Spontaneous and stimulated emission and absorption.- a) Spontaneous emission.- b) Stimulated emission.- c) Comparison between spontaneous and stimulated emission rates.- d) Absorption.- V.4. Photon counting.- a) Quantum mechanical treatment, correlation functions.- 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.- VI.1. Disposition.- VI.2. Summary of theoretical results and comparison with the experiments.- a) Qualitative discussion of the characteristic features of the laser output: homogeneously broadened line.- b) Quantitative results: single mode action.- ?) The spectroscopic linewidth wellabove threshold.- ?) The spectroscopic linewidth somewhat below threshold.- ?) The intensity (or amplitude) fluctuations.- ?) Photon statistics.- VI.3. The quantum mechanical Langevin equations for the solid state laser.- a) Field equations.- b) Matter equations.- ?) The motion of the atomic dipole moment.- 1. Dipole moment between levels j and k.- 2. Dipole moment between levels l and l?k, j and between levels k and l=j, k.- 3. Dipole moment between levels i? k, j and l ? k, j.- ?) The occupation numbers change.- 1. For the laser levels j and k.- 2. For the non-laser levels.- VI.4. Qualitative discussion of single mode operation.- a) The linear range (subthreshold region).- b) The nonlinear range (at threshold and somewhat above).- ?) Phase diffusion.- ?) Amplitude (intensity) fluctuations.- c) The nonlinear range at high inversion.- d) Exact elimination of all atomic coordinates.- 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.- ?) Many modes below threshold.- b) External signals.- VI.6. Exact elimination of atomic variables in the case of a homogeneously broadened line. Running or standing waves.- ?) Standing waves.- ?) 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.- ?) The special case of a moderate photon number.- ?) 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 specialcase w13?? ("two level system").- 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.- ?) |?1 ? ?2| ? 1/T.- ?) |?1 ? ?2| ? 1/T.- 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 N?½ (N: number of atoms).- b) Adiabatic elimination of the atomic variables.- c) The Fokker-Planck equation.- VI. 13. Solution of the reduced Fokker-Planck equation.- 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).- ?) Normal multimode action.- ?) Phase locking of many modes.- ?) 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.- VII.1. Spirit of the semiclassical approach. The equations for the solid state laser.- a) The field equations.- b) The material equations.- c) Macroscopic treatment.- ?) Wave picture, inhomogeneous atomic line.- ?) Wave picture, homogeneous atomic line.-?) Wave picture, homogeneous atomic line, rotating wave approximation, slowly varying amplitude approximation.- ?) Mode picture, polarization waves.- d) Extension to multilevel atoms.- e) Systematics of the semiclassical approach.- VII.2. Method of solution for the stationary state.- a) Single-mode operation, general features.- b) Two-mode operation, general features.- ?) Time-independent atomic response.- ?) 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.- ?) Equations for the photon densities of M modes.- ?) Equations for the frequency shift.- VII.4. The solid-state laser with an inhomogeneously broadened Gaussian line. Single-and two-mode operation.- a) One mode.- ?) Equation for the frequency shift.- ?) Equation for the photon density.- b) Two modes.- ?) Equations for the photon densities n??.- ?) Equations for the frequency shifts.- c) Lorentzian line shape.- VII.5. The solid-state laser with an inhomogeneously broadened line: multimode action.- a) Normal multimode action.- b) Combination tones.- c) Frequency locking.- VII.6. Equations of motion for the gas laser.- VII.7. Single-and two-mode operation in gas lasers.- a) Single-mode operation.- ?) Equation for the photon density.- ?) Equation for the frequency shift.- b) Two-mode operation.- ?) Equations for the photon densities.- ?) Equations for the frequency shifts.- VII.8. Some exactly solvable problems.- a) Single-mode operation in solid state lasers.- ?) Homogeneously broadened line.- 1. Running waves.- 2. Standing waves in axial direction.- ?) Inhomogeneously broadened line, running waves.- 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) Solution of the amplitude and frequency-determining Eqs. (VII.9.24), (VII.9.25).- VII. 10. Ultrashort optical pulses: the principle of mode locking.- a) Loss modulation by an externally driven modulator.- b) Loss modulation by a saturable absorber.- c) Gain modulation.- d) Frequency modulation.- e) Analogy to microwave circuits.- VII.11. Ultrashort optical pulses: detailed treatment of loss modulation.- a) Pulse shape and pulse width.- b) Discussion of the results and of the range of validity.- c) Numerical application.- VII. 12. Super-radiance. Spin and photo echo.- a) Definition of super-radiant states.- b) Generation of super-radiant states.- ?) Classical treatment of the spin motion.- ?) Quantum theoretical treatment.- c) Classical description of super-radiant emission.- d) The spin-echo experiment.- e) The photo-echo experiment.- f) A further analogy between a spin ½ system and a two-level system: the fictitious spin.- VII. 13. Pulse propagation in laser-active media.- a-c) Steady state and self-pulsing.- ?) The basic equations.- ?) Stationary solution.- ?) Normalized amplitudes.- ?) Stability of the stationary solution.- ?) Transient build-up of the pulse.- ?) Steady state pulse.- ?) A simplified model.- ?) The special case v =c.- d) The ?-pulse.- e) The 2?-pulse. (Self-induced transparency).- VII. 14. Derivation of rate equations.- VIII. Rate equations and their applications.- VIII. 1. Formulation of rate equations and solution for the steady state (especially: threshold condition, pump power requirement, single versus multimode laser action).- a) The rate equations.- ?) The field equations.- ?) The matterequations.- b) Treatment of the steady state.- c) The completely homogeneous case.- ?) General formulation.- ?) 3-Level system, the lower transition is laser-active.- ?) Pump power at threshold.- ?) 3-Level system, the upper transition is laser-active.- ?) 4-Level system, laser action between the two middle levels.- 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).- ?) Axial modes with a different frequency distance from the line center.- ?) Different losses.- b) Spatially inhomogeneous pumping, homogeneously broadened line.- ?) Running waves.- ?) Standing waves.- c) Inhomogeneously broadened line.- VIII.3. Laser cascades.- a) Matter equations.- b) Homogeneously broadened line and standing waves (modes in axial direction).- c) Inhomogeneously broadened line and standing waves.- 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.- b) 3-Level system, laser action between the two upper levels.- c) 4-Level system.- d) Approximate solution for small oscillations.- VIII.5. The giant pulse laser.- a) Semiquantitative treatment.- b) Quantitative treatment.- 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.- IX.5. Projection onto macroscopic variables.- IX.6. Exact elimination of the atomic operators withinquantum 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).- X. Appendix. Useful operator techniques.- X.1. The harmonic oscillator.- X.2. Operator relations for Bose operators.- X.3. Formal solution of the Schrödinger equation.- X.4. Disentangling theorem.- X.5. Disentangling theorem for Bose operators.- Sachverzeichnis (Deutsch-Englisch).- Subject Index (English-German).