Laser Theory

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 1/2 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 of heatbaths.- 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 the laser distribution function (IV.9.22).- a) Comparison of the advantages of the Heisenberg and the Schrodinger representations.- ?) The Heisenberg representation.- ?) The Schrodinger 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 well above 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 special case 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?1/2 (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 1/2 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 matter equations.- 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 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).- 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).