principles of lasers fourth edition orazio svelto springe

Principles of Lasers
FOURTH EDITION

Orazio Svelto
Polytechnic Institute of Milan
and National Research Council
Milan, Italy

Translated from Italian and edited by

David C. Hanna
Southampton University
Southamptont England

~ Springer


Library of Congress Cataloging in Publication Data:
Svelte, Drazle.
[Prlnclpl del laser. Engllsh]
Prlnclples of lasers I Drazlo Svelte
translated frem Itallan and
edlted by Davld C. Hanna. -- 4th ed.
p.
cm.
Includes bibliegraphlcal references and lndex.
ISBN 0-306-45748-2
1. Lasers.
I. Hanna, D. C. ( Da v 1 de. ), 194 1. II. T1 t 1e .
QC688.S913 1998

621.36'6--dc21
98-5077
CIP

Front cover photograph: The propagation of an ultraintense pulse in air results in
self-trapping of the laser beam. The rich spectrum of colors produced is the result of the high
14
2
intensity (;~ 10 W/cm ) within the self-focused filament, producing nonlinear phenomena
such as self-phase modulation, parametric interactions, ionization, and conical emission due
to the beam collapse. The rainbowlike display with its sequenced color is due to diffraction
of the different colors (copyright 1998 William Pelletier, Photo Services, Inc.),
Back cover photograph: Interaction of an ultraintense (~1 0 20 W/cm 2) laser pulse with a
target consisting of plastic and aluminum layers. The 4S0-fs pulse, with peak power of 1200
TW, is produced by the petawatt laser at the Lawrence Livermore National Laboratory.
Numerous nonlinear and relativistic-phenomena are observable including copious second
harmonic generation (green light in photo) (courtesy of M. D. Perry, Lawrence Livermore
National Laboratory),

© 1998, 1989, 1982, 1976 Springer Science+Business Media, Inc.
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To my wife Rosanna
and to my sons Cesare and Giuseppe


Preface to the Fourth Edition

This book is motivated by the very favorable reception given to the previous editions as well
as by the considerable range of new developments in the laser field since the publication of
the third edition in 1989. These new developments include, among others, quantum-well and
multiple-quantum-welllasers, diode-pumped solid-state lasers, new concepts for both stable
and unstable resonators, femtosecond lasers, ultra-high-brightness lasers, etc. This edition
thus represents a radically revised version of the preceding edition, amounting essentially to a
new book in its own right. However, the basic aim has remained the same, namely to provide
a broad and unified description of laser behavior at the simplest level which is compatible
with a correct physical understanding. The book is therefore intended as a textbook for a
senior-level or first-year graduate course and/or as a reference book.
The most relevant additions or changes to this edition can be summarized as follows:
1. A much-more detailed description of Amplified Spontaneous Emission has been
given (Chapter 2) and a novel simplified treatment of this phenomenon, both for
homogeneous and inhomogeneous lines, has been introduced (Appendix C).
2. A major fraction of a new chapter (Chapter 3) is dedicated to the interaction of
radiation with semiconductor media, either in a bulk fonn or in a quantum-confined
structure (quantum-well, quantum-wire and quantum dot).
3. A modem theory of stable and unstable resonators is introduced, where a more
extensive use is made of the ABCD matrix fonnalism and where the most recent

topics of dynamically stable resonators as well as unstable resonators, with mirrors
having Gaussian or super-Gaussian transverse reflectivity profiles, are considered
(Chapter 5).
4. Diode-pumping of solid-state lasers, both in longitudinal and transverse pumping
configurations, are introduced in a unified way and a comparison is made with
corresponding lamp-pumping configurations (Chapter 6).
5. Spatially dependent rate equations are introduced for both four-level and quasithree-level lasers and their implications, for longitudinal and transverse pumping,
are also discussed (Chapter 7).
vii


VIII

Preface to the Fourth Edition

6. Laser mode-locking is considered at tlluch greater length to account for, e.g., new
mode-locking methods, such as Kerr-lens mode locking. The effects produced by
second-order and third-order dispersion of the laser cavity and the problem of
dispersion compensation, to achieve the shortest pulse-durations, arc also discussed
at some length (Chapter 8).
7. New tunable solid-state lasers, such as Ti: sapphire and Cr: LiSAF, as well as new
rare-earth lasers such as Yb 3+, Er3+, and H0 3 + are also considered in detail
(Chapter 9).
8. Semiconductor lasers and their perfonnance are discussed at much greater length
(Chapter 9).
9. The divergence properties of a multimode laser beam as well as its propagation
through an optical system are considered in tenns of the M2 factor and in terms of
the embedded Gaussian beam (Chapters 11 and 12).
10. The production of ultra-high peak intensity laser beams by the technique of chirpedpulse-amplification and the related techniques of pulse expansion and pulse
compression are also considered in detail (Chapter 12).

Besides these major additions, the contents of the book have also been greatly enriched
by numerous examples, treated in detail, as well as several new tables and several new
appendixes. The examples either refer to real situations, as found in the literature or
encountered through my own laboratory experience, or describe a significant advance in a
particular topic. The tables provide data on optical, spectroscopic, and nonlinear-optical
properties of laser materials, the data being useful for developing a more quantitative
context as well as for solvjng the problems. The appendixes are introduced to consider some
specific topics in more mathematical detail. A great deal of effort has also been devoted to
the logical organization of the book so as to make its content even more accessible. Lastly, a
large fraction of the problems has also been changed to reflect the new topics introduced and
the overall shift in emphasis within the laser field.
However, despite these profound changes, the basic philosophy and the basic
organization of the book have remained the same. The basic IJhi/osophy is to resort,
wherever appropriate, to an intuitive picture rather than to a detailed mathematical
description of the phenomena under consideration. Simple mathematical descriptions, when
useful for a better understanding of the physical picture, are included in the text while the
discussion of more elaborate analytical models is deferred to the appendixes. The basic
organization starts from the observation that a laser can be considered to consist of three
elements, namely the active medium, the resonator, and the pumping system. Accordingly,
after an introductory chapter, Chapters 2-3, 4-5, and 6 describe the 1110St relevant features of
these elements, separately_ With the combined knowledge about these constituent elements,
Chapters 7 and 8 then allow a discussion of continuous-wave and transient laser behavior,
respectively. Chapters 9 and 10 then describe the lTIOst relevant types of laser exploiting
high-density and low-density media, respectively. Lastly, Chapters 11 and 12 consider a
laser beam from the user's viewpoi nt, examining the properties of the output beam as well as
some relevant laser beam transformations, such as amplification, frequency conversion,
pulse expansion or compression.
The inevitable price paid by the addition of so many new topics, examples, tables, and
appendixes has been a considerable increase in book size. Thus, it is clear that the entire

Preface to the Fourth Edition

IX

content of the book could not be covered in just a one semester-course. However, the
organization of the book allows several different learning paths. For instance, one may be
more interested in learning the Principles of Laser Physics. The emphasis of the study should
then be concentrated on the first section of the book (Chapters 2-8 and Chapter 11). If, on
the other hand, the reader is more interested in the Principles of Laser Engineering, effort
should mostly be concentrated on the second part of the book (Chapters 5-12). The level of
understanding of a given topic may also be suitably modulated by, e.g., considering, in more
or less detail, the numerous examples, which often represent an extension of a given topic, as
well as the numerous appendixes.
Writing a book, albeit a satisfying cultural experience, represents a heavy intellectual
and physical effort. This effort has, however, been gladly sustained in the hope that this
completely new edition can now better serve the pressing need for a general introductory
course to the laser field.
ACKNOWLEDGEMENTS. I wish to acknowledge the following friends and colleagues,
whose suggestions and encouragement have certainly contributed to improving the book in a
number of ways: Christofer Barty, Vittorio De Giorgio, Emilio Gatti, Dennis Hall, GUnther
Huber, Gerard Mourou, Nice Terzi, Franck Tittel, Colin Webb, Herbert Welling. I wish also
to warmly acknowledge the critical editing of David C. Hanna, who has acted as much more
than simply a translator. Lastly I wish to thank, for their useful comments and for their
critical reading of the manuscript, my former students: G. Cerullo, S. Longhi, M.
Marangoni, M. Nisoli, R. Osellame, S. Stagira, C. Svelto, S. Taccheo, and M. Zavelani.

Milan

Orazio Svelto

Contents
List of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

1. Introductory Concepts ..................................... .
1.1.
1.2.
1.3.
1.4.

Spontaneous and Stimulated Emission, Absorption.
The Laser Idea. . . . . . . . . . . . . . . . . . . .
Pumping Schemes . . . . . . . . . . . . . . . . . .
Properties of Laser Beams. . . . . . . . . . . . . .
1.4.1. Monochromaticity . . . . . . . . . . . . . .
1.4.2. Coherence. . . . . . . . . . . . . . . . . . .
1.4.3. Directionality. . . . . . . . . . . . . . . . .
1.4.4. Brightness. . . . . . . . . . . . . . . . . . .
1.4.5. Short Pulse Duration. . . . . . . . . . . . .
1.5. Laser Types . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . .

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2
4
7
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10
11
13
14
14

2. Interaction of Radiation with Atoms and Ions. . . . . . . . . . . . . . . . . . . . . . .

17

2.1. Introduction............................................
2.2. Summary of Blackbody Radiation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Modes of a Rectangular Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Rayleigh-Jeans and Planck Radiation Fotnlula . . . . . . . . . . . . . . . . . . . . .
2.2.3. Planck's Hypothesis and Field Quantization. . . . . . . . . . . . . . . . . . . . . . .
2.3. Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1. Semiclassical Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Quantum Electrodynamics Approach. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3. Allowed and Forbidden Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Absorption and Stimulated Emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1. Absorption and Stimulated Emission Rates . . . . . . . . . . . . . . . . . . . . . . .
2.4.2. Allowed and Forbidden Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3. Transition Cross Section, Absorption, and Gain Coefficient . . . . . . . . . . . . . .
2.4.4. Einstein Thermodynamic Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5. Line-Broadening Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
17
19
22
23
25
26
29
31
32
32
36
37
42
43

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XII

Contents
2.5.1. Homogeneous Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 .2. Inhomogeneous Broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. Nonradiative Decay and Energy Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6.1. Mechanisms of Nonradiative Decay . . . , . . . . . . . . . . . . . . . . , . . . . . .
2.6.2. Combined Effects of Radiative and Nonradiative Processes . . . . . . . . . . . . . .
2.7. Degenerate or Strongly Coupled Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.l. Degenerate Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2. Strongly Coupled Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2,8, Saturation.............................................
2.8.1. Saturation of Absorption: Homogeneous Line. . . . . . . . . . . . . . . . . . . . . .
2.8.2. Gain Saturation: Homogeneous Line. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.3. Inhomogeneously Broadened Line. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9. Fluourescence Decay of an Optically Dense Medium. . . . . . . . . . . . . . . . . . . . . .
2.9.1. Radiation Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9,2. Amplified Spontaneous Emission ... , .... , ...... , . . . . . . . . . . . . .
2.10. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44
48
49
50
50
56
58
58
60
64
64
68
69
71

71
71
76
77
78

3. Energy Levels, Radiative, and Nonradiative Transitions in Molecules
and Semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.1. Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Level Occupation at Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Stimulated Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4. Radiative and Nonradiative Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Bulk Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Electronic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3. Level Occupation at Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4. Stimulated Transitions: Selection Rules. . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5. Absorption and Gain Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6. Spontaneous Emission and Nonradiative Dccay. . . . . . . . . . . . . . . . . . . . . .
3.2.7. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Semiconductor Quantum Wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Electronic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2. Density of States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3. Level Occupation at Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4. Stimulated Transitions: Selection Rules. . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5. Absorption and Gain Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.6. Strained Quantum Wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Quantum Wires and Quantum Dots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81
81
85
87
91
92
92
96
97
101
103
109
111
112
112
115
117
118
120
123
125
126
127
128

4. Ray and Wave Propagation through Optical Media. . . . . . . . . . . . . . . . . . .

129

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129


Contents

XIII

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129
135
137
140
140
144
145
148
148
151
154
155
158
158
160

5. Passive Optical Resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1. Plane Parallel (Fabry-Perot) Resonator. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2. Concentric (Spherical) Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3. Confocal Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4. Generalized Spherical Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5. Ring Resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Eigenmodes and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Photon Lifetime and Cavity Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Stability Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5. Stable Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5.1. Resonators with Infinite Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1.1. Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1.2. Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1.3. Standing and Traveling Waves in a Two-Mirror Resonator. . . . . . . . . . .
5.5.2. Effects of a Finite Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3. Dynamically and Mechanically Stable Resonators . . . . . . . . . . . . . . . . . . . .
5.6. Unstable Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1. Geometric Optics Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2. Wave Optics Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3. Advantages and Disadvantages of Hard-Edge Unstable Resonators. . . . . . . . . . .
5.6.4. Unstable Resonators with Variable-Reflectivity Mirrors . . . . . . . . . . . . . . . . .
5.7. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161
162
163
163
163
164
165
167
169
173
173
174
178
180
181

184
187
188
190
193
194
198
198
200

6. Pumping Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

4.2.
4.3.
4.4.
4.5.

Matrix Formulation of Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wave Reflection and Transmission at a Dielectric Interface . . . . . . . . . . . . . . . . . .
Multilayer Dielectric Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fabry-Perot Interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1. Properties of a Fabry-Perot Interferometer . . . . . . . . . . . . . . . . . . . . . . .
4.5.2. Fabry-Perot Interferometer as a Spectrometer . . . . . . . . . . . . . . . . . . . . .
4.6. Diffraction Optics in the Paraxial Approximation. . . . . . . . . . . . . . . . . . . . . . . .
4.7. Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1. Lowest Order Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.2. Free-Space Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.3. Gaussian Beams and ABCD Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7.4. Higher Order Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1. Introduction . . . . . . . . . . . . . . . . . . .
6.2. Optical Pumping by an Incoherent Light Source
6.2.1. Pumping Systems. . . . . . . . . . . . .
6.2.2. Pump Light Absorption . . . . . . . . .
6.2.3. Pump Efficiency and Pump Rate. . . . .
6.3. Laser Pumping. . . . . . . . . . . . . . . . . .
6.3.1. Laser-Diode Pumps. . . . . . . . . . . .
6.3.2. Pump Transfer Systems . . . . . . . . .

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201

204
204
206
208
210
212
214


Contents

XIV

6.3 .2.1. Longitudinal Pumping. . . . . . . . . . . . . . . . . . . . . . . .
6.3.2.2. Transverse Pumping. . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3. Pump Rate and Pump Efficiency. . . . . . . . . . . . . . . . . . . . . . .
6.3.4. Threshold Pump Power for Four-Level and Quasi-Three-Level Lasers . .
6.3.5. Comparison between Diode Pumping and Lamp Pumping. . . . . . . . .
6.4. Electrical Pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1. Electron Impact Excitation. . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1.1. Electron Impact Cross Section . . . . . . . . . . . . . . . . . . .
6.4.2. Thermal and Drift Velocities. . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3. Electron Energy Distributjon. . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4. Ionization Balance Equation. . . . . . . . . . . . . . . . . . . . . . . . .
6.4.5. Scaling Laws for Electrical Discharge Lasers. . . . . . . . . . . . . . . .
6.4.6. Pump Rate and Pump Efficiency. . . . . . . . . . . . . . . . . . . . . . .
6.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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214
219
221
223
226
228
231
232
235
237
240
241
242

244
244
247

7. Continuous Wave Laser Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249

7.1. Introduction............................................
7.2. Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1. Four.. Level Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2. Quasi-Three-Level Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Threshold Conditions and Output Power: Four-Level Laser . . . . . . . . . . . . . . . . . .
7.3.1. Space-Independent Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2. Space-Dependent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4. Threshold Condition and Output Power: Quasi-Three-Level Laser. . . . . . . . . . . . . . .
7.4.1. Space-Independent Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2. Space-Dependent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5. Optimum Output Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6. Laser Tuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7. Reasons for Multimode Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8. Single-Mode Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.1. Single-Transverse-Mode Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.2. Single-Longitudinal..Mode Selection. . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.2.1. Fabry-Perot Etalons as Mode-Selective Elements . . . . . . . . . . . . . . .
7.8.2.2. Single-Mode Selection in Unidirectional Ring Resonators. . . . . . . . . . .
7.9. Frequency Pulling and Limit to Monochromaticity. . . . . . . . . . . . . . . . . . . . . . .
7.10. Laser Frequency Fluctuations and Frequency Stabilization . . . . . . . . . . . . . . . . . . .
7.11. Intensity Noise and Intensity Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
7.12. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249
249
250
255
258
258
265
273
273
274
277
279
281
284
284
285
285
288
291
293
297
300
301
303

8. Transient Laser Behavior . ............... , ....... , . . . . . . . . . . .

305

8.1.
8.2.
8.3.
8.4.

Introduction . . . . . . . . . . . . . . . . . . .
Relaxation Oscillations. . . . . . . . . . . . . .
Dynamic Instabilities and Pulsations in Lasers.
Q-Switching . . . . . . . . . . . . . . . . . . .
8.4.1. Dynamics of the Q-Switching Process. .
8.4.2. Q-Switching Methods . . . . . . . . . .
8.4.2.1. Electrooptical Q-Switching. . .
8.4.2.2. Rotating Prisms . . . . . . . . .

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305
305
310
311
311
313
313
315


Contents
8.4.2.3. Acoustooptic Q-Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4.2.4. Saturable Absorber Q-Switch. . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.3. Operating Regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4. Theory of Active Q-Switching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5. Gain Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6. Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.l. Frequency-Domain Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.2. Time-Domain Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.3. Mode-Locking Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.3.1. Active Mode Locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.3.2. Passive Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.4. Role of Cavity Dispersion in Femtosecond Mode-Locked Lasers . . . . . . . . . . .
8.6.4.1. Phase Velocity, Group Velocity, and Group-Delay Dispersion ........
8.6.4.2. Limitation on Pulse Duration Due to Group-Delay Dispersion ........
8.6.4.3. Dispersion Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.4.4. Soliton-Type Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.5. Mode-Locking Regimes and Mode-Locking System . . . . . . . . . . . . . . . . . .
8.7. Cavity Dumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

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316
317
319
321
329
330
331
336
337
337
342
347
347
350
351

353
355
359
361
361
363

9. Solid-State, Dye, and Semiconductor Lasers. . . . . . . . . . . . . . . . . . . . . . . .

365

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2. Solid-State Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1. Ruby Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2. Neodymium Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2.1. Nd: YAG Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2.2. Nd:Glass Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2.3. Other Crystalline Hosts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3. Yb:YAG Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.4. Er:YAG and Yb:Er:Glass Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.5. Tm:Ho:YAG Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.6. Fiber Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.7. Alexandrite Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.8. Titanium Sapphire Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.9. Cr:LiSAF and Cr:LiCAF Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3. Dye Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1. Photophysical Properties of Organic Dyes . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2. Characteristics of Dye Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4. Semiconductor Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1. Principle of Semiconductor Laser Operation . . . . . . . . . . . . . . . . . . . . . . .

9.4.2. Homojunction Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.3. Double-Heterostructure Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.4. Quantum Well Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.5. Laser Devices and Performances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.6. Distributed Feedback and Distributed Bragg Reflector Lasers. . . . . . . . . . . . . .
9.4.7. Vertical-Cavity Surface-Emitting Lasers . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.8. Semiconductor Laser Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365
365
367
370
370
373
373
374
376
377
378
381
383
385
386
387
391
394
394

396
398
402
405
408
411
413
415
415
417


Contents

XVI

10. Gas, Chemical, Free-Electon, and X-Ray Lasers. . . . . . . . . . . . . . . . . . . . .
10.1. Introduction. . . . . . . . . . . . . . . .
10.2. Gas Lasers . . . . . . . . . . . . . . . .
10.2.1. Neutral Atom Lasers. . . . . . .
10.2.1.1. Helium Neon Laser. .
10.2.1.2. Copper Vapor Laser. .
10.2.2. Ion Lasers . . . . . . . . . . . .
10.2.2.1. Argon Laser. . . . . .
10.2.2.2. He-Cd Laser. . . . . .
10.2.3. Molecular Gas Lasers. . . . . .
10.2.3.1. CO 2 Laser. . . . . . .
10.2.3.2. CO Laser . . . . . . .
JO.2.3.3. Nitrogen Laser. . . . .
10.2.3.4. Excimer Lasers . . . .

10.3. Chemical Lasers . . . . . . . . . . . . .
J 0.4. Free-Electron Lasers . . . . . . . . . . .
10.5. X-Ray Lasers. . . . . . . . . . . . . . .
10.6. Concluding Remarks . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . .

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419

419
420
420
425
427
427
430
432
432
442
444
445
448
452
456
458
459
460

11. Properties of Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

11. 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I 1.2. Monochromaticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3. First-Order Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1. Degree of Spatial and Temporal Coherence . . . . . . . . . . . . . . . . . . . . . .
11.3.2. Measurement of Spatial and Temporal Coherence. . . . . . . . . . . . . . . . . . .
11.3.3. Relation between Temporal Coherence and Monochromaticity. . . . . . . . . . . .
11.3.4. Nonstationary Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.3.5. Spatial and Temporal Coherence of Single-Mode and Multimode Lasers. . . . . . .
11.3.6. Spatial and Temporal Coherence of a Thermal Light Source. . . . . . . . . . . . .
11.4. Directionality. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
11.4.1. Beams with Perfect Spatial Coherence . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.2. Beams with Partial Spatial Coherence . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.3. The M2 Factor and the Spot Size Parameter of a Multimode Laser Beam. . . . . .
11.5. Laser Speckle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6. Brightness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7. Statistical Properties of Laser Light and Thennal Light. . . . . . . . . . . . . . . . . . . . .
11.8. Comparison between Laser Light and Thermal Light. . . . . . . . . . . . . . . . . . . . . .
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463
463
464
464
468
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475
476
477
479
480
483
486
487
489

491
492

12. Laser Beam Transformation: Propagation, Amplification, Frequency
Conversion, Pulse Compression, and Pulse Expansion. . . . . . . . . . . . . . . . .

493

12.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2. Spatial Transformation: Propagation of a Multimode Laser Beam . . . . . . . . .
12.3. Amplitude Transformation: Laser Amplification. . . . . . . . . . . . . . . . . . .
12.3.1. Examples of Laser Amplifiers: Chirped-Pulse-Amplification . . . . . . .
12.4. Frequency Conversion: Second-Harmonic Generation and Parametric Oscillation.

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493
494
495
500
504


Contents

XVII

12.4.1. Physical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1.1. Second Hannonic Generation. . . . . . . . . . . . . . . . . . . . . . . .
12.4.1.2. Parametric Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2. Analytical Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2.1. Parametric Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2.2. Second-Hannonic Generation . . . . . . . . . . . . . .
12.5. Transfonnation in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.1. Pulse Compression . . . . . . . . . . . . . . . .
12.5.2. Pulse Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . .
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Appendixes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

535

A. Semiclassical Treatment of the Interaction of Radiation and Matter. . . . . . . . . . . . . . . .

535

B. Lineshape Calculation for Collision Broadening . . . . . . . . . . . . . . . . . .

541

C. Simplified Treatment of Amplified Spontaneous Emission. . . . . . .
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545
548
549

E. Space-Dependent Rate Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eol. Four-Level Lasers
E.2. Quasi-Three-Level Lasers ........

553
553
559

F. Mode-Locking Theory: Homogeneous Line . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.l. Active Mode Locking . . . . . . . .
F.2. Passive Mode Locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

563
563
568
569

G. Propagation of a Laser Pulse through a Dispersive Medium or a Gain Medium. . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

571
575

H. Higher-Order Coherence . . . . . . . . . . .

577

I. Physical Constants and Useful Conversion Factors. . . . . . . . . . . . . . . . . . . . . . . . .

581

Answers to Selected Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

595

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List of Examples

Chapter 2
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.

Estimate of TIp and A for electric-dipole allowed and forbidden transitions . .
Collision broadening of a He-Ne laser . . .
Linewidth of ruby and Nd: YAG . . . . . . . .
Natural linewidth of an allowed transition. . . .
Linewidth of a Nd:glass laser. . . . . . . . . .
Doppler linewidth of a He-Ne laser. . . . . . .

Energy transfer in the Yb 3 + :Er3 + :glass laser system. . . . . . . . . . . .
Nonradiative decay from the 4F3/2 upper laser level of Nd:YAG . . . . .
Cooperative upconversion in Er ~+ lasers and amplifiers. . . . . . . . . .
Effective stimulated-emission cross section for the ;. = 1.064-j1m laser transition of Nd: YAG . . .
Effective stimulated-emission cross section and radiative lifetime in alexandrite
Directional properties of ASE . . . . . . . . . .
ASE threshold for a solid-state laser rod . . . . . . . . . . . . . . . . . . . . .

32
46
46
47
48
49
55
56
56
62
62
72
74

Chapter 3
3.1.
3.2.
3.3.
14.
3.5.
3.6.
17.

3.8.
3.9.
110.
111.
112.

Emission spectrum of the CO2 laser transition at ;. = 10.6,um
Doppler linewidth of a CO 2 laser . . . . . . . . . . . . . . .
Collision broadening of a CO 2 laser . . . . . . . . . . . .
Calculation of the quasi-Fermi energies for GaAs ..... .
Calculation of typical values of k for a thermal electron.
Calculation of the absorption coefficient for GaAs . . . . . . . .
Calculation of the transparency density for GaAs. . . . . . . . . . .
Radiative and nonradiative lifetimes in GaAs and InGaAsP .... .
Calculation of the first energy levels in a GaAs/ AIGaAs QW ... .
Calculation of the quasi-Fermi energies for a GaAs/ AIGaAs QW . . . . . . .
Calculation of the absorption coefficient in a GaAs/ AIGaAs QW . . . . .
Calculation of the transparency density in a GaAs/ AIGaAs QW.

xix

89
90
90
100
102
105
107
III
114

118
121
122


xx

List of Examples

Chapter 4
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.

Peak reflectivity calculation in multilayer dielectric coatings. . . . . .
Single-layer antireflection coating of laser materials. . . . . . . . . . .
Free-spectral range, finesse, and transmission of a Fabry-Perot etalon.
Spectral measurement of an Ar+ -laser output beam ... . . . . . . .
Gaussian beam propagation through a thin lens . . . . . . . . . . . . .
Gaussian beam focusing by a thin lens. . . . . . . . . . . . . . . . . .

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138
140
143
145
154
155

Number of modes in closed and open resonators. . . . . . . . . . . . . . . . . . . . . . .
Calculation of the cavity photon lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linewidth of a cavity resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q factor of a laser cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spot sizes for symmetric resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency spectrum of a confocal resonator . . . . . . . . . . . . . . . . . . . . . . . . .
Frequency spectrum of a near-planar and symmetric resonator. . . . . . . . . . . . . . . .
Diffraction losses of a symmetric resonator. . . . . . . . . . . . . . . . . . . . . . . . . .
Limitation on the Fresnel number and resonator aperture in stable resonators. . . . . . . .
Unstable confocal resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design of an unstable resonator with an output mirror having a Gaussian radial reflectivity
profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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168
169
169
177
179
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182
183
190

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196

Chapter 5
5. I.

5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
5.10.
5.11.

Chapter 6
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
6.7.

Pump efficiency in lamp-pumped solid-state lasers . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of an anamorphic prism-pair system to focus the light of a single-stripe diode laser.
Diode-array beam focusing onto a multi mode optical fiber. . . . . . . . . . . . . . . . . . . . .
Electron energy distribution in a CO2 laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electron energy distribution in a He-Ne laser. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thennal and drift velocities in He-Ne and CO 2 lasers . . . . . . . . . . . . . . . . . . . . . . .
Pumping efficiency in a CO 2 laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209
217
2 I8
238
239
240
243

. . . . .. . .. .
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255
261
263
271
276

279
281
287
292
294

Chapter 7
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
7.8.
7.9.
7.10.

Calculation of the number of cavity photons in typical cw lasers . . . . . . . .
CW laser behavior of a lamp-pumped high-power Nd: YAG laser. . . . . . . .
CW laser behavior of a high-power CO 2 laser . . . . . . . . . . . . . . . . . .
Threshold and output powers in a longitudinally diode-pumped Nd:YAG laser
Threshold and output powers in a longitudinally diode-pumped Yb:YAG laser.
Optimum output coupling for a lamp-pumped Nd:YAG laser . . . . . . . . . .
Free-spectral range and resolving power of a birefingent filter. . . . . . . . . .
Single-longitudinal-mode selection in Ar-ion and Nd:YAG lasers. . . . . . . .
Limit to laser linewidth in He-Ne and GaAs semiconductor lasers . . . . . . .
Long-term drift of a laser cavity. . . . . . . . . . . . . . . . . . . . . . . . . .

List of Examples

XXI

Chapter 8
8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
8.8.

Damped oscillation in aNd: YAG and a GaAs laser.
Transient behavior of a He-Ne laser. . . . . . . . .
Condition for the Bragg regime in a quartz acoustooptic modulator. . . . . . . . . . . . . . .
Output energy, pulse duration, and pulse build-up time in a typical Q-switched Nd:YAG laser
Dynamical behavior of a passively Q-switched Nd:YAG laser. . . . . . . . . . . . . . . . . .
Typical cases of gain-switched lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AM mode-locking for cw Ar and Nd:YAG lasers. . . . . . . . . . . . . . . . . . . . . . . . .
Passive mode locking of Nd:YAG and Nd:YLF lasers . . . . . . . . . . . . . . . . . . . . . .

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308
309
317
325
326
330
340
344

Chapter 9
9.1.
9.2.
9.3.
9.4.

Carrier and current densities at threshold for a DH GaAs laser. . . . . .
Carrier and current densities at threshold for a GaAs/ AIGaAs QW laser
Output power and external quantum efficiency of a semiconductor laser.
Threshold carrier density and threshold current for a VCSEL. . . . . . .

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401
403
407
413

11.1. Calculation of the fringe visibility in Young's interferometer. . . . . . . . . . .
11.2. Coherence time and bandwidth for a sinusoidal wave with random phase jumps.
11.3. Spatial coherence for a laser oscillating in many transverse modes. . . . . . . .
11.4. M2 factor and spot-size parameters of a broad-area semiconductor laser. . . . .
11.5. Grain size of the speckle pattern as seen by a human observer. . . . . . . . . .

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469
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494
500
510

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519

Chapter 11

Chapter 12
12.1.
12.2.
12.3.
12.4.

Focusing of a multimode Nd:YAG laser beam by a thin lens. .
.. . . .. . .. ..
Maximum energy that can be extracted from an amplifier. . . . . . . . . . . . . . . . . . . .
Calculation of the phase-matching angle for a negative uniaxial crystal . . . . . . . . . . . .
Calculation of threshold intensity for the pump beam in a doubly resonant optical parametric
oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



1
Introductory Concepts

In this introductory chapter, the fundamental processes and the main ideas behind laser
operation are introduced in a very simple way. The properties of laser beams are also briefly
discussed. The main purpose of this chapter is thus to introduce the reader to many of the
concepts that will be discussed in the following chapters, and therefore help the reader to
appreciate the logical organization of the book.
Following the discussion presented in this chapter, in fact, the organization of the book is
based on the observation that a laser can be considered to consist of three elements: an active
material, a pumping scheme, and a resonator. Accordingly, after this introductory chapter,
Chaps. 2 and 3 deal with the interaction of radiation with matter, starting from the simplest
cases, i.e., atoms or ions in an essentially isolated situation (Chap. 2), then going on to the

more complicated cases, i.e., molecules and semiconductors (Chap. 3). As an introduction to
optical resonators, Chap. 4 considers some topics relating to ray and wave propagation in
particular optical elements, such as free space, optical lens-like media, Fabry-Perot
interferometers, and multilayer dielectric coatings. Chapter 5 treats the theory of optical
resonators, while Chap. 6 discusses pumping processes. Concepts introduced in these
chapters are then used in Chaps. 7 and 8, wnere a theory is developed for continuous wave
and transient laser behavior, respectively. The theory is based on the lowest order
approximation, i.e., using the rate equation approach. This approach is in fact applicable in
describing most laser characteristics. Since lasers based on different types of active media
have significantly different characteristics, Chaps. 9 and 10 discuss characteristic properties of
a number of laser types: Chapter 9 covers ionic crystal, dye, and semiconductor lasers, which
have a number of common features; Chap. 10 considers gas, chemical, and free-electron
lasers. By this point the reader should have acquired sufficient understanding of laser
behavior to study properties of the output beam (coherence, monochromaticity, brightness,
noise), which are considered in Chap. 11. Chapter 12 is then based on the fact
that, before being usecL a laser beam is generally transformed in some way, which includes: (1) Spatial transformation of the beam due to its propagation through, e.g., a lens
system; (2) amplitude transformation as a result of passing through an amplifier; (3) wavelength transformation, or frequency conversion, via a number of nonlinear phenomena


2

1 • Introductory Concepts

(second harmonic generation, parametric processes); (4) time transformation by, e.g., pulse
compression or pulse expansion.

1.1. SPONTANEOUS AND STIMULATED EMISSION, ABSORPTION
To describe the phenomenon of spontaneous emission (Fig. l.la), let us consider two
energy levels, 1 and 2, of some atom or molecule of a given material with energies El and E2
(E 1 < £2), respectively. In the following discussion, the two levels can be any two of an

atom's infinite set of levels. It is convenient however to take level 1 as the ground level. Let
us now assume that the atom is initially at level 2. Since E2 > E1, the atom tends to decay to
level 1. The corresponding energy difference E2 - El must therefore be released by the
atom. When this energy is delivered in the form of an electromagnetic (em) wave, the
process is called spontaneous (or radiative) emission. The frequency Vo of the radiated wave
is then given by the well known expression:
Vo =

(E2 - E1)
h

(1.1.l)

where h is Planck's constant. Spontaneous emission is therefore characterized by the
emission of a photon of energy hvo = E2 - El when the atom decays from level 2 to levell
(Fig. 1.1 a). Note that radiative emission is just one of two possible ways for the atom to
decay. Decay can also occur in a nonradiative way. In this case the energy difference
E2 - El is delivered in some form of energy other than em radiation (e.g., it may go into the
kinetic or internal energy of the surrounding atoms or molecules). This phenomenon is
called nonradiative decay.
Let us now suppose that the atom is initially found in level 2 and an em wave of
frequency v = vo (i.e., equal to that of the spontaneously emitted wave) is incident on the
material (Fig. 1.1 b). Since this wave has the same frequency as the atomic frequency, there is
a finite probability that this wave will force the atom to undergo the transition 2-+ 1. In this
case the energy difference E2 - E1 is delivered in the form of an em wave that adds to the
incident wave. This is the phenomenon of stimulated emission. There is a fundamental
difference between the spontaneous and stimulated emission processes. In the case of
spontaneous emission, atoms emit an em wave that has no definite phase relation to that
emitted by another atom. Furthermore the wave can be emitted in any direction. In the case
of stimulated emission, since the process is forced by the incident em wave, the emission of

any atom adds in phase to that of the incoming wave and in the same direction.
Let us now assume that the atom is initially lying in level 1 (Fig. 1.1 c). If this is the
ground level, the atom remains in this level unless some external stimulus is applied. We
E2

E2

2

hV=E 2-E,
hv

hI'
hv

..I\J\f\--

E2

2

-'\f\r

..JV'--

2

hv

-'W'--

hv
E1
(a)

1

1

1

E1
(b)

E1
(c)

FIG. 1.1. Schematic illustration of the three processes: (a) spontaneous emission, (b) stimulated emission, (c)

absorption.


1.1 • Spontaneous and Stimulated Emission, Absorption

3

assume that an em wave of frequency v = Vo is incident on the material. In this case there is a
finite probability that the atom will be raised to level 2. The energy difference £2 - El
required by the atom to undergo the transition is obtained from the energy of the incident em
wave. This is the absorption process.

To introduce probabilities for these emission and absorption phenomena, let Ni be the
number of atoms (or molecules) per unit volume that at time t occupy a given energy level, i.
From now on the quantity Ni is called the population of the level.
For the case of spontaneous emission, the probability that the process occurs is defined
by stating that the rate of decay of the upper state population (dN2/ dt)sp must be proportional
to the population N2 • We can therefore write
dN2)
( - dt sp = -AN2

(1.1.2)

where the minus sign accounts for the fact that the time derivative is negative. The coefficient
A, introduced in this way, is a positive constant called the rate of spontaneous emission or
the Einstein A coefficient. (An expression for A was first obtained by Einstein from
thermodynamic considerations.) The quantity ! sp = 1/A is the spontaneous emission (or
radiative) lifetime. Similarly, for nonradiative decay, we can generally write
(

dN2)
dt

(1.1.3)
nr

where ! nr is the nonradiative decay lifetime. Note that for spontaneous emission the
numerical value of A (and !sp) depends only on the particular transition considered. For
nonradiative decay, on the other hand, !nr depends not only on the transition but also on
characteristics of the surrounding medium.
We can now proceed in a similar way for stimulated processes (emission or absorption).
For stimulated emission we can write

(1.1.4)
\\J'here (dN2/dt)st is the rate at which transitions 2-+ 1 occur as a result of stimulated emission
and W21 is the rate of stimulated emission. As in the case of the A coefficient defined by Eq.
(1.1.2), the coefficient W21 also has the dimension of (time)-l. Unlike A, however, W21
depends not only on the particular transition but also on the intensity of the incident em
wave. More precisely, for a plane wave, we can write
(1.1.5)
\vhere F is the photon flux of the wave and G21 is a quantity having the dimension of an area
(the stimulated emission cross section) and depending on characteristics of the given
transition.
As in Eq. (1.1.4) we can define an absorption rate W21 using the equation:
(1.1.6)


4

1 •

Introductory Concepts

where (dN} / dt)a is the rate of transitions 1~ 2 due to absorption and NI is the population of
level 1. As in Eq. (1.1.5) we can write
(1.1. 7)
where (J12 is some characteristic area (the absorption cross section), which depends only on
the particular transition.
In the preceding discussion the stimulated processes are characterized by the stimulated
emission and absorption cross-sections 0'21 and 0'12, respectively. Einstein showed at the
beginning of the twentieth century that, if the two levels are nondegenerate, one has
W21 = W12 and thus 0'21 = (J12' If levels 1 and 2 are gl-fold and g2 .. fold degenerate,
respectively, one then has:

(1.1.8)
that is
(1.1.9)

Note also that the fundamental processes of spontaneous emission, stimulated emission, and
absorption can be described in terms of absorbed or emitted photons as follows (see Fig. 1.1):
(a) In the spontaneous emission process, the atom decays from level 2 to level 1 through the
emission of a photon. (b) In the stimulated emission process, the incident photon stimulates
the transition 2--+ 1, so that there are two photons (the stimulating one and the stimulated one).
(c) In the absorption process, the incident photon is simply absorbed to produce transition
1--+ 2. Thus each stimulated emission process creates a photon, whereas each absorption
process annihilates a photon.

1.2. THE LASER IDEA
Consider two arbitrary energy levels 1 and 2 of a given material, and let Nl and N2 be
their respective populations. If a plane wave with a photon flux F is traveling in the zdirection in the material (Fig. 1.2), the elemental change dF of this flux along the elemental
length dz of the material is due to both stimulated absorption and emission processes
occurring in the shaded region of Fig. 1.2. Let S be the cross-sectional area of the beam. The
change in number between outgoing and incoming photons in the shaded volume per unit
time is thus SdF. Since each stimulated process creates a photon whereas each absorption
removes a photon, SdF must equal the difference between stimulated emission and
absorption events occurring in the shaded volume per unit time. From Eqs. (1.1.4) and
(1.1.6) we can write SdF (W21N2 - W12 N1 )(Sdz), where Sdz is the volume of the shaded
region. With the help of Eqs. (1.1.5), (1.1.7), and (1.1.9), we obtain
(1.2.1)


1.2 •

The Laser Idea

5

FIG. 1.2. Elemental change dF in the photon flux F for a plane em wave in traveling a distance dz through the
material.

Note that, in deriving Eq. (1.2.1), we did not consider radiative and nonradiative decays. In
fact nonradiative decay does not add new photons, while photons created by radiative decay
are emitted in any direction and thus give negligible contribution to the incoming photon flux

F.
Equation (1.2.1) shows that the material behaves as an amplifier (i.e., dF/dz> 0) if
N2 > g2 Nl / g l' while it behaves as an absorber if N2 < g2 Nl / g l' At thermal equilibrium
populations are described by Boltzmann statistics. Then if Nf and Ni are the thermal
equilibrium populations of the two levels:

(1.2.2)

where k is Boltzmann's constant and T is the absolute temperature of the material. In thermal
equilibrium we thus have N2 < g2Nf /gl' According to Eq. (1.2.1) the material then acts as
an absorber at frequency Vo. This is what happens under ordinary conditions. However if a
nonequilibrium condition is achieved for which N2 > g2Nl / gl' then the material acts as an
amplifier. In this case we say that there exists a population inversion in the material. This
means that the population difference N2 - (g2Nl / gr) is opposite in sign to what exists under
thermodynamic equilibrium [N2 - (g2Nl / g 1) < 0]. A material in which this population
inversion is produced is referred to as an active medium.
If the transition frequency Vo = (£2 £l)/kT falls in the microwave region, this type of
amplifier is called a maser amplifier, an acronym for microwave amplification by stimulated
emission of radiation. If the transition frequency falls in the optical region, the amplifier is
called a laser amplifier, an acronym obtained from the preceding one with light substituted

for microwave.
To make an oscillator from an amplifier, it is necessary to introduce suitable positive
feedback. In the microwave region this is done by placing the active material in a resonant
cavity having a resonance at frequency Vo. In the case of a laser, feedback is often obtained by
placing the active material between two highly reflecting mirrors~ such as the plane parallel
mirrors in Fig. 1.3. In this case a plane em wave traveling in a direction perpendicular to the
mirrors bounces back and forth between the two mirrors, and is amplified on each passage
through the active material. If one of the two mirrors (e.g. mirror 2) is partially transparent, a
useful output beam is obtained from that mirror.