Principles of lasers 5th by orzio svelto

Principles of Lasers
FIFTH EDITION


Principles of Lasers
FIFTH EDITION

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

Translated from Italian and edited by

David C. Hanna
Southampton University
Southampton, England

123


Orazio Svelto
Politecnico di Milano
Dipto. Fisica
Piazza Leonardo da Vinci, 32
20133 Milano
Italy

ISBN 978-1-4419-1301-2
e-ISBN 978-1-4419-1302-9

DOI 10.1007/978-1-4419-1302-9
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009940423
1st edition: c Plenum Press, 1976
2nd edition: c Plenum Publishing Corporation, 1982
3rd edition: c Plenum Publishing Corporation, 1989
4th edition: c Plenum Publishing Corporation, 1998
c Springer Science+Business Media, LLC 2010
All rights reserved. This work may not be translated or copied in whole or in part without the written permission
of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except
for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information
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Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)


To my wife Rosanna
and to my sons Cesare and Giuseppe


Preface
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 Well lasers, diode-pumped solid-state lasers, new concepts for both stable
and unstable resonators, femtosecond lasers, ultra-high-brightness lasers etc. The basic aim

of the book 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 text-book for a senior-level or first-year graduate course and/or
as a reference book.
This edition corrects several errors introduced in the previous edition. The most relevant
additions or changes to since the third edition can be summarized as follows:

1. A much-more detailed description of Amplified Spontaneous Emission has been
given [Chapt. 2] and a novel simplified treatment of this phenomenon both for
homogeneous or inhomogeneous lines has been introduced [Appendix C].
2. A major fraction of a chapter [Chapt. 3] is dedicated to the interaction of radiation
with semiconductor media, either in a bulk form or in a quantum-confined structure
(quantum-well, quantum-wire and quantum dot).
3. A modern theory of stable and unstable resonators is introduced, where a more extensive use is made of the ABCD matrix formalism 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 [Chapt. 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 [Chapt. 6].
5. Spatially-dependent rate equations are introduced for both four-level and quasi-threelevel lasers and their implications, for longitudinal and transverse pumping, are also
discussed [Chapt. 7].

vii


viii

Preface
6. Laser mode-locking is considered at much 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 are also discussed at some
length [Chapt. 8].
7. New tunable solid-state lasers, such as Ti: sapphire and Cr: LISAF, as well as
new rare-earth lasers such as Yb3C , Er3C , and Ho3C are also considered in detail
[Chapt. 9].
8. Semiconductor lasers and their performance are discussed at much greater length
[Chapt. 9].
9. The divergence properties of a multimode laser beam as well as its propagation
through an optical system are considered in terms of the M 2 -factor and in terms of
the embedded Gaussian beam [Chapt. 11 and 12].
10. The production of ultra-high peak intensity laser beams by the technique of
chirped-pulse-amplification and the related techniques of pulse expansion and pulse
compression are also considered in detail [Chapt. 12].
The book also contains numerous, thoroughly developed, examples, as well as many
tables and appendixes. The examples either refer to real situations, as found in the literature
or encountered through my own laboratory experience, or describe a significative 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 solving 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 more accessible.
The basic philosophy of the book 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 consists 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 most relevant features of these elements, separately. With the combined knowledge about
these constituent elements, chapters 7 and 8 then allow a discussion of continuos-wave and

transient laser behavior, respectively. Chapters 9 and 10 then describe the most 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 view-point examining the properties of the output
beam as well as some relevant laser beam transformations, such as amplification, frequency
conversion, pulse expansion or compression.
With so many topics, examples, tables and appendixes, it is clear that the entire content
of the book could not be covered in only 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 mostly
concentrated on the first section of the book [Chapt. 1–5 and Chapt. 7–8]. 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 Chap. 6 and 9–12. The level of understanding


ix

Preface
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 edition
can serve the pressing need for a general introductory course to the laser field.
ACKNOWLEDGMENTS. 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, G¨unther
Huber, Gerard Mourou, 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.

Milano

Orazio Svelto


Contents

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

xix

1. Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.
1.2.

Spontaneous and Stimulated Emission, Absorption
The Laser Idea . . . . . . . . . . . . . .

1.3.

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1.4.2. Coherence . . . .
1.4.3. Directionality . . .
1.4.4. Brightness . . . .

1.4.

Pumping Schemes

Properties of Laser Beams
1.4.1. Monochromaticity

1.5.

1.4.5. Short Time Duration
Types of Lasers . . . . . .

1.6.

Organization of the Book

Problems

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2. Interaction of Radiation with Atoms and Ions . . . . . . . . . . . . . . . . . .
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2.2.2. The Rayleigh-Jeans and Planck Radiation Formula . . .
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 . . . . . . . . . .
2.1.

Introduction

2.2.

Summary of Blackbody Radiation Theory

2.2.1. Modes of a Rectangular Cavity .

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1
4
6
8
9
9
10
11

13
14
14
15

17
17
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19
22
24
26
26
30
31

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xii

Contents
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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 . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . .
2.7.1. 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. Decay of an Optically Dense Medium . . . . . . . . . . . . . .
2.9.1. Radiation Trapping . . . . . . . . . . . . . . . . . .
2.9.2. Amplified Spontaneous Emission . . . . . . . . . . . .
2.10. Concluding Remarks . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.

Absorption and Stimulated Emission

2.4.1.

Rates of Absorption and Stimulated Emission

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3. Energy Levels, Radiative and Nonradiative Transitions in Molecules
and Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . .
3.2.5. Absorption and Gain Coefficients . . . .
3.1.

Molecules

3.1.1.
3.1.2.

3.2.6.

Energy Levels . . . . . . . . . . .
Level Occupation at Thermal Equilibrium

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Spontaneous Emission and Nonradiative Decay

3.2.7. Concluding Remarks .
Semiconductor Quantum Wells

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3.3.
3.3.1. Electronic States . . . .
3.3.2. Density of States . . . .

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32
32
36
37
41
43
43
47
49
50
50

56
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64
64
67
69
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71
71
76
77
78

81
81
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91
93
93
97
98
101
104
110
112
113

113
116


xiii

Contents
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118

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

131

3.3.3.

Level Occupation at Thermal Equilibrium

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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 . . . . . . . . . . . . . . . .
3.3.4.

Stimulated Transitions

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4.3. Wave Reflection and Transmission at a Dielectric Interface .
4.4. Multilayer Dielectric Coatings . . . . . . . . . . . .
4.5. The Fabry-Perot Interferometer . . . . . . . . . . .
4.5.1. Properties of a Fabry-Perot Interferometer . . . .
4.5.2. The 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 the ABCD Law . . . . . .
4.7.4. Higher-Order Modes . . . . . . . . . . . .
4.8. Conclusions . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . .
4.1.
4.2.

Introduction . . . . . . . . . . . .
Matrix Formulation of Geometrical Optics

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5. Passive Optical Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.3. Photon Lifetime and Cavity Q .
5.4. Stability Condition . . . . .
5.5. Stable Resonators . . . . . .
5.1.
5.2.

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5.5.1. Resonators with Infinite Aperture .
5.5.1.1. Eigenmodes . . . . .
5.5.1.2. Eigenvalues . . . . . .
Introduction

Eigenmodes and Eigenvalues

5.5.1.3.

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Standing- and Traveling-Waves in a Two-Mirror Resonator

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5.6.
5.6.1. Geometrical-Optics Description . . . . . . . .
5.6.2. Wave-Optics Description . . . . . . . . . . .
5.5.2.

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Effects of a Finite Aperture

5.5.3. Dynamically and Mechanically Stable Resonators
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xiv

Contents
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196

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

205

5.6.3.

Advantages and Disadvantages of Hard-Edge Unstable Resonators

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5.7. Concluding Remarks . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . .
5.6.4.

Variable-Reflectivity Unstable Resonators

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6.2.2. Absorption of Pump Light . . . . . .
6.2.3. Pump Efficiency and Pump Rate . . . .
6.3. Laser Pumping . . . . . . . . . . . . . .
6.3.1. Laser Diode Pumps . . . . . . . . .
6.3.2. Pump Transfer Systems . . . . . . .
6.3.2.1. Longitudinal Pumping . . .

6.3.2.2. Transverse Pumping . . . .
6.3.3. Pump Rate and Pump Efficiency . . . .
6.1.

Introduction

6.2.

Optical Pumping by an Incoherent Light Source
6.2.1. Pumping Systems . . . . . . . .

6.3.4.
6.4.

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6.4.3. Electron Energy Distribution . . . . . .
6.4.4. The Ionization Balance Equation . . . . .
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Threshold Pump Power for Four-Level and Quasi-Three-Level Lasers

6.3.5. Comparison Between Diode-pumping and Lamp-pumping
Electrical Pumping . . . . . . . . . . . . . . . . . . .
6.4.1.

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Electron Impact Excitation . . . . . .
6.4.1.1. Electron Impact Cross Section

Scaling Laws for Electrical Discharge Lasers

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Problems . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . .
6.4.6. Pump Rate and Pump Efficiency
Conclusions . . . . . . . . . . .

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7. Continuous Wave Laser Behavior . . . . . . . . . . . . . . . . . . . . . . . .
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7.3. Threshold Conditions and Output Power: Four-Level Laser .
7.3.1. Space-Independent Model . . . . . . . . . . .
7.3.2. Space-Dependent Model . . . . . . . . . . . .
7.1.
7.2.

7.4.

Introduction .
Rate Equations

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7.2.2. Quasi-Three-Level Laser .

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Threshold Condition and Output Power: Quasi-Three-Level Laser

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7.5. Optimum Output Coupling . . . . . . . . . . . . . . .
7.4.1.
7.4.2.

Space-Independent Model

Space-Dependent Model .

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255
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xv

Contents
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7.8. Single-Mode Selection . . . . . . . . .
7.8.1. Single-Transverse-Mode Selection .

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

313

7.6.

7.7.

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7.8.2. Single-Longitudinal-Mode Selection .

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7.8.2.1. Fabry-Perot Etalons as Mode-Selective Elements .

Laser Tuning

Reasons for Multimode Oscillation

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References . . . . . . . . . . . . . . . . . . .

7.11. Intensity Noise and Intensity Noise Reduction
7.12. Conclusions
Problems . . . .

8.2.
8.3.
8.4.

8.5.
8.6.

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Single Mode Selection via Unidirectional Ring Resonators

7.9. Frequency-Pulling and Limit to Monochromaticity . . .
7.10. Laser Frequency Fluctuations and Frequency Stabilization

8.1.

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Dynamical Instabilities and Pulsations in Lasers . .
Q-Switching . . . . . . . . . . . . . . . .
8.4.1. Dynamics of the Q-Switching Process . .
8.4.2. Methods of Q-Switching . . . . . . . .
8.4.2.1. Electro-Optical Q-Switching . .
8.4.2.2. Rotating Prisms . . . . . . .
8.4.2.3. Acousto-Optic Q-Switches . . .
8.4.2.4. Saturable-Absorber Q-Switch . .
8.4.3. Operating Regimes . . . . . . . . . .
8.4.4. Theory of Active Q-Switching . . . . .
Gain Switching . . . . . . . . . . . . . . .
Mode-Locking . . . . . . . . . . . . . . .
8.6.1. Frequency-Domain Description . . . . .
8.6.2. Time-Domain Picture . . . . . . . . .
8.6.3. Methods of Mode-Locking . . . . . . .
8.6.3.1. Active Mode-Locking . . . . .
8.6.3.2. Passive Mode Locking . . . .

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8.6.4. The Role of Cavity Dispersion in Femtosecond Mode-Locked Lasers .
Introduction

Relaxation Oscillations . .

8.2.1. Linearized Analysis

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Phase-Velocity, Group-Velocity and Group-Delay-Dispersion

8.6.4.2.
8.6.4.3.

Limitation on Pulse Duration due to Group-Delay Dispersion
Dispersion Compensation . . . . . . . . . . . . . .

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8.7.
8.8. Concluding Remarks . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
8.6.4.4.

Soliton-type of Mode-Locking

8.6.5. Mode-Locking Regimes and Mode-Locking Systems
Cavity Dumping . . . . . . . . . . . . . . . . .

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285
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328
329
337
339
340
344
346
346
350
356
356
358
360
361
364
368
369
370
372


xvi

Contents
9. Solid-State, Dye, and Semiconductor Lasers . . . . . . . . . . . . . . . . . . .
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9.2.2.2. Nd:Glass
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9.2.2.3. Other Crystalline Hosts
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9.2.3. Yb:YAG . . . . . . . . . .
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9.2.4. Er:YAG and Yb:Er:glass . . . .
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9.2.5. Tm:Ho:YAG . . . . . . . .
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9.2.6. Fiber Lasers . . . . . . . . .
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9.2.7. Alexandrite Laser . . . . . .
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9.2.8. Titanium Sapphire Laser . . . .
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9.2.9. Cr:LISAF and Cr:LICAF . . . .
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9.3. Dye Lasers . . . . . . . . . . . .
9.3.1. Photophysical Properties of Organic Dyes .
9.3.2. Characteristics of Dye Lasers . . . . . .
9.4. Semiconductor Lasers . . . . . . . . . . . .

9.1.
9.2.

Introduction . .
Solid-State Lasers

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9.2.1. The Ruby Laser . .
9.2.2. Neodymium Lasers .
9.2.2.1. Nd:YAG .

9.4.1.

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Principle of Semiconductor Laser Operation

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9.4.4. Quantum Well Lasers . . . . . .
9.4.5. Laser Devices and Performances . .

9.4.2.
9.4.3.

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The Homojunction Laser . . . .
The Double-Heterostructure Laser

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Distributed Feedback and Distributed Bragg Reflector Lasers

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9.5. Conclusions . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . .

9.4.7.
9.4.8.

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Vertical Cavity Surface Emitting Lasers
Applications of Semiconductor Lasers

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10. Gas, Chemical, Free Electron, and X-Ray Lasers . . . . . . . . . . . . . . . .
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10.2.1. Neutral Atom Lasers . . . . .
10.2.1.1. Helium-Neon Lasers .
10.2.1.2. Copper Vapor Lasers .
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. The CO2 Laser . . .
10.2.3.2. The CO Laser . . . .
10.2.3.3. The N2 Laser . . . .
10.2.3.4. Excimer Lasers . . .

10.1. Introduction
10.2. Gas Lasers

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375
375
375
377
380
380
383
384
384
386
387
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396
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397
401
405
405
407
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416
419

423
425
427
427
429

431
431
431
432
432
437
439
439
442
444
444
454
456
457


xvii

Contents
. . . . .
. . .
10.4. The Free-Electron Laser . .
10.5. X-ray Lasers . . . . . . .
10.6. Concluding Remarks . . .

Problems . . . . . . . . . . .
References . . . . . . . . . .
10.3. Chemical Lasers

10.3.1. The HF Laser

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11. Properties of Laser Beams . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11.3.1. Degree of Spatial and Temporal Coherence .

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11.3.2. Measurement of Spatial and Temporal Coherence .

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11.3.3. Relation Between Temporal Coherence and Monochromaticity .
11.3.4. Nonstationary Beams . . . . . . . . . . . . . . . . .

11.1. Introduction

11.2. Monochromaticity .
11.3. First-Order Coherence

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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

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11.4.2. Beams with Partial Spatial Coherence . . . . . . . . . . . . . .

11.4.3. The M 2 Factor and the Spot-Size Parameter of a Multimode Laser Beam
11.5. Laser Speckle . . . . . . . . . . . . . . . . . . . . . . . . . .

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Problems . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . .
11.6. Brightness

11.7. Statistical Properties of Laser Light and Thermal Light
11.8. Comparison Between Laser Light and Thermal Light .

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12. Laser Beam Transformation: Propagation, Amplification, Frequency
Conversion, Pulse Compression and Pulse Expansion . . . . . . . . . . . . . .
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12.3.1. Examples of Laser Amplifiers: Chirped-Pulse-Amplification . . .

12.1. Introduction

12.2. Spatial Transformation: Propagation of a Multimode Laser Beam
12.3. Amplitude Transformation: Laser Amplification . . . . . . .

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12.4. Frequency Conversion: Second-Harmonic Generation and Parametric Oscillation
12.4.1. Physical Picture . . . . . . . . . . . . . . . . . . . . . .

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12.4.2.1. Parametric Oscillation . . . .
12.4.2.2. Second-Harmonic Generation .
12.4.1.1. Second-Harmonic Generation

12.4.1.2. Parametric Oscillation
12.4.2. Analytical Treatment . . . .

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461
461
465
469
471
471
473

475
475
475
476
477
480
483
485
485
488
489
489
491
492
495

498
499
501
503
504

505
505
506
507
512
516
516
517
524
526
528
532


xviii

Contents
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535

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547

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547

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553

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557
560

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561

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565

12.5. Transformation in Time: Pulse Compression and Pulse Expansion

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12.5.2. Pulse Expansion . .
Problems . . . . . . . . . . .
References . . . . . . . . . .
12.5.1. Pulse Compression

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A.

Semiclassical Treatment of the Interaction of Radiation with Matter

B.

Lineshape Calculation for Collision Broadening

C.

Simplified Treatment of Amplified Spontaneous Emission
References . . . . . . . . . . . . . . . . . .

D.

Calculation of the Radiative Transition Rates of Molecular Transitions

E.

Space Dependent Rate Equations
E.1.
E.2.

Four-Level Laser . . .
Quasi-Three-Level Laser

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536
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543
544

565
571

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

607

Physical Constants and Useful Conversion Factors

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595

I.

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

Higher-Order Coherence

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593

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Propagation of a Laser Pulse Through a Dispersive Medium or a Gain Medium
References . . . . . . . . . . . . . . . . . . . . . . . . . .

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589

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References

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583
587

Active Mode-Locking
Passive Mode-Locking

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F.1.
F.2.

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575

Theory of Mode-Locking: Homogeneous Line

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575
580
581


List of Examples

Chapter 2
2.1. Estimate of sp and A for electric-dipole allowed and forbidden transitions
2.2. Collision broadening of a He-Ne laser . . . . . . . . . . . . . .

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2.9. Cooperative upconversion in Er3C lasers and amplifiers . . . .

2.3. Linewidth of Ruby and Nd:YAG

2.4. Natural linewidth of an allowed transition
2.5. Linewidth of a Nd:glass laser . . . .

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D 1.064 m laser transition of Nd:YAG

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2.11. Effective stimulated emission cross section and radiative lifetime in Alexandrite

2.12. Directional property of ASE . . . . .
2.13. ASE threshold for a solid-state laser rod

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55
56
62
62
72
74

Chapter 3
3.1. Emission spectrum of the CO2 laser transition at
3.2. Doppler linewidth of a CO2 laser. . .
3.3. Collision broadening of a CO2 laser.

D 10.6 m.

. . . . . . . . . .
. . . . . . . . . .
3.4. Calculation of the quasi-Fermi energies for GaAs. . . . . .
3.5. Calculation of typical values of k for a thermal electron. . . .
3.6. Calculation of the absorption coefficient for GaAs. . . . . .
3.7. Calculation of the transparency density for GaAs. . . . . .
3.8. Radiative and nonradiative lifetimes in GaAs and InGaAsP. .

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90

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xix


xx

List of Examples
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3.11. Calculation of the absorption coefficient in a GaAs/AlGaAs quantum well. .
3.12. Calculation of the transparency density in a GaAs quantum well. . . . . .

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5.9. Limitation on the Fresnel number and resonator aperture in stable resonators .
5.10. Unstable confocal resonators . . . . . . . . . . . . . . . . . . . .

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3.9. Calculation of the first energy levels in a GaAs / AlGaAs quantum well.

3.10. Calculation of the Quasi-Fermi energies for a GaAs/AlGaAs quantum well.

115
120
122

123

Chapter 4
. . . . .
. . . . . . . .
Free-spectral range, finesse and transmission of a Fabry-Perot etalon. . .
Spectral measurement of an ArC -laser output beam. . . . . . . . .
Gaussian beam propagation through a thin lens. . . . . . . . . . .
Gaussian beam focusing by a thin lens. . . . . . . . . . . . . . .

4.1. Peak reflectivity calculation in multilayer dielectric coatings.
4.2. Single layer antireflection coating of laser materials.
4.3.
4.4.
4.5.
4.6.

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141
142
145
147
156
157

Chapter 5
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5.7. Frequency spectrum of a near-planar and symmetric resonator .
5.8. Diffraction loss of a symmetric resonator . . . . . . . . . .

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5.11. Design of an unstable resonator with an output mirror having a Gaussian radial reflectivity profile .

5.1. Number of modes in closed and open resonators.

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5.4. Q-factor of a laser cavity . . . . . . .
5.5. Spot sizes for symmetric resonators . . .

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5.6. Frequency spectrum of a confocal resonator .

5.2. Calculation of the cavity photon lifetime.
5.3. Linewidth of a cavity resonance. . . .

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167
170
171
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181
181
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185
192
198

Chapter 6
6.1. Pump efficiency in lamp-pumped solid state lasers . . . . . . . . . . . . . . . . .
6.2. Calculation of an anamorphic prism-pair system to focus the light of a single-stripe diode laser

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7.3. CW laser behavior of a high-power . . . . . . . . . . . . . . . . . . .

7.4. Threshold and Output Powers in a Longitudinally Diode-Pumped Nd:YAG Laser .

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6.5. Electron energy distribution in a He-Ne laser . . . . . .

6.6. Thermal and drift velocities in He-Ne and CO2 lasers . . .
6.7. Pumping efficiency in a CO2 laser . . . . . . . . . .

6.3. Diode-array beam focusing into a multimode optical fiber
6.4. Electron energy distribution in a CO2 laser . . . . . .

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Chapter 7
7.1. Calculation of the number of cavity photons in typical c.w. lasers
7.2. CW laser behavior of a lamp pumped high-power Nd:YAG laser .

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268
277

xxi

List of Examples
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282

. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
8.3. Condition for Bragg regime in a quartz acousto-optic modulator . . . . . . . . . . . . . .
8.4. Output energy, pulse duration, and pulse build-up time in a typical Q-switched Nd:YAG laser . .
8.5. Dynamical behavior of a passively Q-switched Nd:YAG laser . . . . . . . . . . . . . . .
8.6. Typical cases of gain switched lasers . . . . . . . . . . . . . . . . . . . . . . . .
8.7. AM mode-locking for a cw Ar and Nd:YAG laser . . . . . . . . . . . . . . . . . . .
8.8. Passive mode-locking of a Nd:YAG and Nd:YLF laser by a fast saturable absorber . . . . . .

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316
317

7.5. Threshold and Output Powers in a Longitudinally Pumped Yb:YAG Laser

. . . .
7.7. Free spectral range and resolving power of a birefringent filter . . .
7.8. Single-longitudinal-mode selection in an Ar and a Nd:YAG laser . .
7.9. Limit to laser linewidth in He-Ne and GaAs semiconductor lasers . .
7.10. Long term drift of a laser cavity . . . . . . . . . . . . . . .

7.6. Optimum output coupling for a lamp-pumped Nd:YAG laser

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285
287
294
299
300

Chapter 8
8.1. Damped oscillation in a Nd:YAG and a GaAs laser
8.2. Transient behavior of a He-Ne laser . . . . . .

325
333
334
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353

Chapter 9
. . . . . . . .
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9.3. Output power and external quantum efficiency of a semiconductor laser . . . . .
9.4. Threshold current density and threshold current for a VCSEL . . . . . . . . .

9.1. Carrier and current densities at threshold for a DH GaAs laser

9.2. Carrier and current densities at threshold for a GaAs/AlGaAs quantum well laser

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412
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Chapter 11
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.

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11.3. Spatial coherence for a laser oscillating on many transverse-modes. . . . . . .
11.4. M 2 -factor and spot-size parameter of a broad area semiconductor laser. . . . . .
11.5. Grain size of the speckle pattern as seen by a human observer. . . . . . . . .

481
484
486
494
498

Chapter 12
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12.3. Calculation of the phase-matching angle for a negative uniaxial crystal. . . . . . . . . . . . .

12.1. Focusing of a multimode Nd:YAG beam by a thin lens

507

12.2. Maximum energy which can be extracted from an amplifier.

512
523

12.4. Calculation of the threshold intensity for the pump beam in a doubly resonant optical parametric
oscillator.

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531


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 later on, in the book, and therefore help the reader to appreciate
the logical organization of the book.

1.1. SPONTANEOUS AND STIMULATED EMISSION, ABSORPTION
To describe the phenomenon of spontaneous emission, let us consider two energy levels, 1 and 2, of some atom or molecule of a given material, their energies being E1 and
E2 .E1 < E2 / (Fig. 1.1a). As far as the following discussion is concerned, the two levels could
be any two out of the infinite set of levels possessed by the atom. It is convenient, however, to
take level 1 to be the ground level. Let us now assume that the atom is initially in level 2. Since
E2 > E1 , the atom will tend to decay to level 1. The corresponding energy difference, E2 E1 ,
must therefore be released by the atom. When this energy is delivered in the form of an electromagnetic (e.m. from now on) wave, the process will be called spontaneous (or radiative)
emission. The frequency 0 of the radiated wave is then given by the well known expression
0

D .E2

E1 /=h

(1.1.1)

where h is Planck’s constant. Spontaneous emission is therefore characterized by the emission of a photon of energy h 0 D E2 E1 , when the atom decays from level 2 to level 1
(Fig. 1.1a). Note that radiative emission is just one of the two possible ways for the atom
to decay. The decay can also occur in a nonradiative way. In this case the energy difference
E2 E1 is delivered in some form of energy other than e.m. radiation (e.g. it may go into
kinetic or internal energy of the surrounding atoms or molecules). This phenomenon is called
non-radiative decay.

O. Svelto, Principles of Lasers,
DOI: 10.1007/978-1-4419-1302-9 1, c Springer Science+Business Media LLC 2010

1


2

1

Introductory Concepts

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

Let us now suppose that the atom is found initially in level 2 and that an e.m. wave of
frequency D o (i.e., equal to that of the spontaneously emitted wave) is incident on the
material (Fig. 1.1b). 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 e.m. wave that adds to the
incident one. 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, the atoms emits an e.m. wave that has no definite phase relation with 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 e.m. wave, the emission of any atom adds
in phase to that of the incoming wave and along the same direction.
Let us now assume that the atom is initially lying in level 1 (Fig. 1.1c). If this is the
ground level, the atom will remain in this level unless some external stimulus is applied to
it. We shall assume, then, that an e.m. wave of frequency D o 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 E2 E1 required by the atom to undergo the transition is obtained from the energy
of the incident e.m. wave. This is the absorption process.
To introduce the probabilities for these emission and absorption phenomena, let N be the
number of atoms (or molecules) per unit volume which, at time t, are lying in a given energy

level. From now on the quantity N will be called the population of the level.
For the case of spontaneous emission, the probability for the process to occur can be
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
Ã
Â
dN2
D AN2
(1.1.2)
dt sp
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 and is called the rate of spontaneous emission
or the Einstein A coefficient (an expression for A was in fact first obtained by Einstein from
thermodynamic considerations). The quantity sp D 1= A is called the spontaneous emission
(or radiative) lifetime. Similarly, for non-radiative decay, we can often write
Â

dN2
dt

Ã
D
nr

N2
nr

(1.1.3)

1.1

3

Spontaneous and Stimulated Emission, Absorption

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

dN2
dt

Ã
D

W21 N2

(1.1.4)

st

where .dN2 =dt/st is the rate at which transitions 2 ! 1 occur as a result of stimulated emission
and W21 is called the rate of stimulated emission. Just as in the case of the A coefficient defined
by Eq. (1.1.2) the coefficient W21 also has the dimension of .time/ 1 . Unlike A, however, W21
depends not only on the particular transition but also on the intensity of the incident e.m.
wave. More precisely, for a plane wave, it will be shown that we can write

W21 D

21 F

(1.1.5)

where F is the photon flux of the wave and 21 is a quantity having the dimension of an
area (the stimulated emission cross section) and depending on the characteristics of the given
transition.
In a similar fashion to Eq. (1.1.4), we can define an absorption rate W21 by means of the
equation
Â

dN1
dt

Ã
D

W12 N1

(1.1.6)

a

where .dN1 =dt/a is the rate of the 1 ! 2 transitions due to absorption and N1 is the population
of level 1. Furthermore, just as in Eq. (1.1.5), we can write
W12 D

12 F

(1.1.7)

where 12 is some characteristic area (the absorption cross section), which depends only on
the particular transition.
In what has just been said, the stimulated processes have been characterized by the stimulated emission and absorption cross-sections, 21 and 12 , respectively. Now, it was shown
by Einstein at the beginning of the twentieth century that, if the two levels are non-degenerate,
one always has W21 D W12 and 21 D 12 . If levels 1 and 2 are g1 -fold and g2 -fold degenerate,
respectively one has instead
g2 W21 D g1 W12

(1.1.8)

i.e.
g2

21

D g1

12

(1.1.9)

Note also that the fundamental processes of spontaneous emission, stimulated emission
and absorption can readily be described in terms of absorbed or emitted photons as follows


4

1

Introductory Concepts

(see Fig. 1.1). (1) In the spontaneous emission process, the atom decays from level 2 to level 1
through the emission of a photon. (2) In the stimulated emission process, the incident photon
stimulates the 2 ! 1 transition and we then have two photons (the stimulating plus the stimulated one). (3) In the absorption process, the incident photon is simply absorbed to produce
the 1 ! 2 transition. Thus we can say that each stimulated emission process creates while
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 N1 and N2 be their
respective populations. If a plane wave with a photon flux F is traveling along the z direction in
the material (Fig. 1.2), the elemental change, dF, of this flux along the elemental length, dz, of
the material will be due to both the stimulated 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, will thus be SdF. Since
each stimulated process creates while 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 (1.1.4) and (1.1.6) we can thus write SdF D .W21 N2 W12 N1 /.Sdz/ where
Sdz is, obviously, the volume of the shaded region. With the help of Eqs. (1.1.5), (1.1.7) and
(1.1.9) we obtain
dF D

21 F ŒN2

.g2 N1 =g1 / dz

(1.2.1)

Note that, in deriving Eq. (1.2.1), we have not taken into account the radiative and nonradiative decays. In fact, non-radiative decay does not add any new photons while the photons
created by the radiative decay are emitted in any direction and do not contribute 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 N1 =g1 , while it behaves as an absorber if N2 < g2 N1 =g1 . Now, at thermal equilibrium, the
populations are described by Boltzmann statistics. So, if N1e and N2e are the thermal equilibrium

FIG. 1.2. Elemental change dF in the photon flux F fro a plane e.m. wave in traveling a distance dz through the
material.


1.2

5

The Laser Idea

populations of the two levels, we have
N2e
g2
D
exp
N1e
g1

Ä

E1

E2

kT

(1.2.2)

where k is Boltzmann’s constant and T the absolute temperature of the material. In thermal
equilibrium we thus have N2e < g2 N1e =g1 . According to Eq. (1.2.1), the material then acts as
an absorber at frequency . This is what happens under ordinary conditions. If, however, a
non-equilibrium condition is achieved for which N2 > g2 N1 =g1 then the material will act as
an amplifier. In this case we will say that there exists a population inversion in the material,
by which we mean that the population difference N2 .g2 N1 =g1 / is opposite in sign to that
which exists under thermodynamic equilibrium ŒN2 .g2 N1 =g1 / < 0. A material in which
this population inversion is produced will be called an active material.
If the transition frequency 0 D .E2 E1 /= kT falls in the microwave region, this type
of amplifier is called a maser amplifier. The word maser is 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. The word laser is again an acronym,
with the letter l (light) substituted for the letter m (microwave).
To make an oscillator from an amplifier, it is necessary to introduce a suitable positive feedback. In the microwave region this is done by placing the active material in a
resonant cavity having a resonance at frequency 0 . In the case of a laser, the feedback is
often obtained by placing the active material between two highly reflecting mirrors (e.g.
plane parallel mirrors, see Fig. 1.3). In this case, a plane e.m. wave traveling in the direction perpendicular to the mirrors will bounce back and forth between the two mirrors and
be amplified on each passage through the active material. If one of the two mirrors is made
partially transparent, a useful output beam is obtained from this mirror. It is important to
realize that, for both masers and lasers, a certain threshold condition must be reached. In
the laser case, for instance, the oscillation will start when the gain of the active material
compensates the losses in the laser (e.g. the losses due to the output coupling). According to Eq. (1.2.1), the gain per pass in the active material (i.e. the ratio between the output
and input photon flux) is exp f ŒN2 .g2 N1 =g1 /lg where we have denoted, for simplicity, D 21 , and where l is the length of the active material. Let R1 and R2 be the power
reflectivity of the two mirrors (Fig. 1.3) and let Li be the internal loss per pass in the laser
cavity. If, at a given time, F is the photon flux in the cavity, leaving mirror 1 and traveling
toward mirror 2, then the photon flux, F 0 , again leaving mirror 1 after one round trip will be

F 0 D F exp f ŒN2 .g2 N1 =g1 /lg .1 Li /R2 exp f ŒN2 .g2 N=g1 /lg .1 Li /R1 . At threshold we must have F 0 D F, and therefore R1 R2 .1 Li /2 exp f2 ŒN2 .g2 N1 =g1 /lg D 1. This
equation shows that threshold is reached when the population inversion, N D N2 .g2 N1 =g1 /,
reaches a critical value, known as the critical inversion, given by
Nc D

Πln R1 R2 C 2 ln .1

Li /= 2 l

FIG. 1.3. Scheme of a laser.

(1.2.3)


6

1

Introductory Concepts

The previous expression can be put in a somewhat simpler form if we define
1

D

ln R1 D

ln .1

T1 /

(1.2.4a)

2

D

ln R2 D

ln .1

T2 /

(1.2.4b)

i

D

ln .1

Li /

(1.2.4c)

where T1 and T2 are the two mirror transmissions (for simplicity mirror absorption has been
neglected). The substitution of Eq. (1.2.4) in Eq. (1.2.3) gives
Nc D = l

(1.2.5)

where we have defined
D

i

C.

1

C

2 /= 2

(1.2.6)

Note that the quantities i , defined by Eq. (1.2.4c), may be called the logarithmic internal loss
of the cavity. In fact, when Li
1 as usually occurs, one has i Š Li . Similarly, since both T1
and T2 represent a loss for the cavity, 1 and 2 , defined by Eq. (1.2.4a and b), may be called
the logarithmic losses of the two cavity mirrors. Thus, the quantity defined by Eq. (1.2.6)
will be called the single pass loss of the cavity.
Once the critical inversion is reached, oscillation will build up from spontaneous emission. The photons that are spontaneously emitted along the cavity axis will, in fact, initiate
the amplification process. This is the basis of a laser oscillator, or laser, as it is more simply
called. Note that, according to the meaning of the acronym laser as discussed above, the word
should be reserved for lasers emitting visible radiation. The same word is, however, now commonly applied to any device emitting stimulated radiation, whether in the far or near infrared,
ultraviolet, or even in the X-ray region. To be specific about the kind of radiation emitted one
then usually talks about infrared, visible, ultraviolet or X-ray lasers, respectively.

1.3. PUMPING SCHEMES

We will now consider the problem of how a population inversion can be produced in a
given material. At first sight, it might seem that it would be possible to achieve this through
the interaction of the material with a sufficiently strong e.m. wave, perhaps coming from a
sufficiently intense lamp, at the frequency D o . Since, at thermal equilibrium, one has
g1 N1 > g2 N2 g1 , absorption will in fact predominate over stimulated emission. The incoming
wave would produce more transitions 1 ! 2 than transitions 2 ! 1 and we would hope
in this way to end up with a population inversion. We see immediately, however, that such a
system would not work (at least in the steady state). When in fact the condition is reached such
that g2 N2 D g1 N1 , then the absorption and stimulated emission processes will compensate one
another and, according to Eq. (1.2.1), the material will then become transparent. This situation
is often referred to as two-level saturation.