Laser Theory

336 Pages·1983·15.58 MB·English

by  Professor Dr. Dr. h.c. Hermann Haken (auth.)

Hermann Haken
Laser Theory
Corrected Printing
With 72 Figures
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
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: 978-3-540-12188-6 e-ISBN-13: 978-3-642-45556-8
001: 10.1007/978-3-642-45556-8
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.H34 1983 535.5'8 83-458
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
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© by Springer-Verlag Berlin Heidelberg 1970 and 1984.
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.
2153/3130-543210
of
Dedicated to tbe memory my parents.
H.HAKEN.
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 equa
tion 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.
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 fluctua
tions 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. 1.4.
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
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 manu
script has been read critically and checked by several of them, and lowe thanks
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, 1969. H. HAKEN.
Contents.
1. Introduction ...... .
1.1. The maser principle 1
1.2. The laser condition . 2
1.3. Properties of laser light 5
a) Spatial coherence . 5
b) Temporal coherence 6
c) Photon statistics 7
d) High intensity 7
e) Ultrashort pulses 7
1.4. Plan of the article 7
II. Optical resonators 9
11.1. Introduction 9
II.2. The Fabry-Perot resonator with plane parallel reflectors 11
a) Spatial distribution of modes 11
b) Diffraction losses . . . . . 17
c) Three-dimensional resonator 18
II.3. Confocal resonator . . . . . 19
a) Field outside the resonator . 20
b) Field inside the resonator 21
c) Far field pattern of the confocal resonator 21
d) Phase shifts and losses . . . . . . . . . 21
IIA. More general configurations . . . . . . . . 22
a) Confocal resonators with unequal square and rectangular
apertures ................ . 22
b) Resonators with reflectors of unequal curvature 23
01:) Large circular apertures 23
fJ) Large square aperture. . . . . . . . . . 23
II.5. Stability . . . . . . . . . . . . . . . . . . 23
III. Quantum mechanical equations of the light field and the atoms without losses 24
IlL 1. Quantization of the light field . . . . . . . . . . . . . 24
IIL2. Second quantization of the electron wave field. . . . . . . . . . 27
IIL3. Interaction between radiation field and electron wave field . . . . 28
IlIA. The interaction representation and the rotating wave approximation 29
IlLS. The equations of motion in the Heisenberg picture . . . . . . . . 30
IIL6. The formal equivalence of the system of atoms each having 2 levels
with a system of t spins . . . . . . . . . . . . . . . . .. 31
IV. Dissipation and fluctuation of quantum systems. The realistic laser equations 33
IV.1. Some remarks on homogeneous and inhomogeneous broadening 33
a) Naturallinewidth . . . . . 33
b) Inhomogeneous broadening. 33
01:) Impurity atoms in solids 33
fJ) Gases . . . . . . . . . 34
y) Semiconductors 34
c) Homogeneous broadening 34
01:) Impurity atoms in solids 34
fJ) Gases . . . . . 34
y) Semiconductors 34
x
Contents.
IV.2. A survey of IV.2.-IV.11 • . . 35
a) Definition of heatbaths (reservoirs) 35
b) The role of heatbaths . . . . . . 35
c) Classical Langevin and Fokker-Planck equations. 36
ex) Langevin equations . . . . . . . . . . . . 36
p) The Fokker-Planck equation . . . . . . . . 36
d) Quantum mechanical formulation: the total Hamiltonian. 37
e) Quantum mechanical Langevin equations, Fokker-Planck
equation and density matrix equation . 38
ex) Langevin equations . . . . . . . . 38
P) Density matrix equation. . . . . . 38
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) . . . . . . . . . . . . . 40
b) Electrons ("atoms") . . . . . . . . . . . . . 42
IV.4. The requirement of quantum mechanical consistency 44
a) The field . . . . . . . . . . . . . . . . . . 44
b) Dissipation and fluctuations of the atoms. . . . 45
IV.5. The explicit form of the correlation functions of Langevin forces 46
a) The field . . . . . . . . 46
b) The N-Ievel atom . . . . . . . . . . . . . . . . . . .. 46
IV.6. The complete laser equations . . . . . . . . . . . . . . .. 49
a) Quantum mechanically consistent equations for the operators bt
and (at ak)". . . . . . 50
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 . . . . . . . 56
P) Atom . . . . . . . . . . 57
y) The density matrix equation of the complete system of M laser
modes and N atoms. . . . . . . . . . . . . . . . . .. 58
IV.8. The evaluation of multi-time correlation functions by the single-time
density matrix. . . . . . . . . . . . . . . . . . . . . . .. 59
IV.9. Generalized Fokker-Planck equation: definition of distribution
functions . . . . . . . . . . . . . . . . . . . . . . . . 60
a) Field. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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 . . . . . . . . . . . . . . 64
b) Electrons. . . . . . . . . . . . . . . . 64
ex) Distribution functions for a single electron 64
p) Characteristic functions . . . . . . . . 65
y) Electrons and fields. . . • . . • . . . 65
IV.1O. Equation for the laser distribution function (IV.9.22) 65
a) Comparison of the advantages of the Heisenberg and the
SchrOdinger representations. . . . . . . . . . . . 65
ex) The Heisenberg representation . . . . . . . . . 65
P) The SchrOdinger representation. . . . . . . . . . 67
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 . . . . . . . 71
V. Properties of quantized electromagnetic fields 73
V.1. Coherence properties of the classical and the quantized electro-
magnetic field . . . . . . . . . . . . . . . . 73
Contents. XI
a) Classical description: definitions 73
01:) The complex analytical signal 73
{J) The average . . . . . . . . 74
y) The mutual coherence function 74
b) Quantum theoretical coherence functions. 76
01:) Elementary introductions . . . . . . 76
{J) Coherence functions. . . . . . . . . 77
y) Coherent wave functions. . . . . . . 78
6) Generation of coherent fields by classical sources (the forced
harmonic oscillator) . . . . . . . . . . . 80
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 . . . 85
c) Field strength. . . . . . 87
V.3. Spontaneous and stimulated emission and absorption 88
a) Spontaneous emission . . . . . . . . . . . . 88
b) Stimulated emission . . . . . . . . . . . . . 90
c) Comparison between spontaneous and stimulated emission rates 91
d) Absorption . . . . . . . . . . . . . . . . . . . 92
V.4. Photon counting. . . . . . . . . . . . . . . . . . 93
a) Quantum mechanical treatment, correlation functions 93
b) Classical treatment of photon counting. . . . . . . 94
V.5. Coherence properties of spontaneous and stimulated emission. The
spontaneous linewidth . . . . . . . . . . . . 97
VI. Fully quantum mechanical solutions of the Jaser equations. . . . . . . . . 99
VI.1. Disposition . . . . . . . . . . . . . . . . . . . . . . . . . 99
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. . . . . . . . . 102
b) Quantitative results: single mode action . . . . . . . . 102
01:) The spectroscopic linewidth well above threshold . . . 102
{J) The spectroscopic linewidth somewhat below threshold 103
y) The intensity (or amplitude) fluctuations ..... . 104
6) Photon statistics . . . . . . . . . . . . . . . . . 107
VI.3. The quantum mechanical Langevin equations for the solid state laser 112
a) Field equations . . . . . . . . . . . . . 113
b) Matter equations . . . . . . . . . . . . 115
01:) The motion of the atomic dipole moment 115
1. Dipole moment between levels j and k 11 5
2. Dipole moment between levels j and 1= != k, j and between
levels k and 1 =j, k. . . . . . . . • . . . . • 115
3. Dipole moment between levels i =!= k, i and 1 =!= k, j . 115
(j) The occupation numbers change 115
1. For the laser levels j and k . . . . . . 115
2. For the non-laser levels . . . . . . . . 116
VIA. Qualitative discussion of single mode operation. 116
a) The linear range (subthreshold region) . . . . 118
b) The nonlinear range (at threshold and somewhat above) 119
01:) Phase diffusion . . . . . . . . . . . 120
{J) Amplitude (intensity) fluctuations. . . 120
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) . . . . . . . . . . . . . . . . . 120
a) No external signals . . . . . . . . . . 120
01:) Single-mode linewidth below threshold 123
{J) Many modes below threshold 123
b) External signals. . . . . . . . . . . . 124