Handbook of Nonlinear Optics, Second Edition, (Optical Engineering)_Richard L.Sutherland

Handbook of
Nonlinear
Optics
Second Edition, Revised and Expanded

Richard L Sutherland
Science Applications International Corporation
Dayton, Ohio, U.S.A.

with contributions by

Daniel G. McLean
Science Applications International Corporation
Dayton, Ohio, U.S.A.

Sean Kirkpatrick
Air Force Research Laboratory
Wright Patterson Air Force Base, Ohio, U.S.A.

MARCEL

ffi

MARCEL DEKKER, INC.

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OPTICAL ENGINEERING
Founding Editor
Brian J. Thompson
University of Rochester
Rochester, New York

Editorial Board
Toshimitsu Asakura
Hokkai-Gakuen University
Sapporo, Hokkaido, Japan

Nicholas F. Borrelli
Corning, Inc.
Corning, New York

Chris Dainty
Imperial College of Science,
Technology, and Medicine
London, England

Bahrain Javidi
University of Connecticut
Storrs, Connecticut

Mark Kuzyk
Washington State University
Pullman, Washington

Hiroshi Murata

The Furukawa Electric Co , Ltd.
Yokohama, Japan

Edmond J. Murphy
JDS/Umphase
Bloomfield, Connecticut

Joseph Shamir
Technion-Israel Institute
of Technology
Hafai, Israel

Dennis R. Pape
Photomc Systems Inc.
Melbourne, Florida

David S. Weiss
Heidelberg Digital L.L.C.
Rochester, New York


1. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E. Murr
2. Acousto-Optic Signal Processing: Theory and Implementation, edited
by Norman J. Berg and John N. Lee
3. Electro-Optic and Acousto-Optic Scanning and Deflection, Milton
Gottlieb, Clive L. M. Ireland, and John Martin Ley
4. Single-Mode Fiber Optics: Principles and Applications, Luc B. Jeunhomme
5. Pulse Code Formats for Fiber Optical Data Communication: Basic
Principles and Applications, David J. Morris
6. Optical Materials: An Introduction to Selection and Application, Solomon Musikant

7. Infrared Methods for Gaseous Measurements: Theory and Practice,
edited by Joda Wormhoudt
8. Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data
Storage Optics, edited by Gerald F. Marshall
9. Opto-Mechanical Systems Design, Paul R. Yoder, Jr.
10. Optical Fiber Splices and Connectors: Theory and Methods, Calvin M.
Miller with Stephen C. Mettler and lan A. White
11. Laser Spectroscopy and Its Applications, edited by Leon J. Radziemski, Richard W. Solarz, and Jeffrey A. Paisner
12. Infrared Optoelectronics: Devices and Applications, William Nunley
and J. Scott Bechtel
13. Integrated Optical Circuits and Components: Design and Applications,
edited by Lynn D. Hutcheson
14. Handbook of Molecular Lasers, edited by Peter K. Cheo
15. Handbook of Optical Fibers and Cables, Hiroshi Murata
16. Acousto-Optics, Adrian Korpel
17. Procedures in Applied Optics, John Strong
18. Handbook of Solid-State Lasers, edited by Peter K. Cheo
19. Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon
20. Laser Applications in Physical Chemistry, edited by D. K. Evans
21. Laser-Induced Plasmas and Applications, edited by Leon J. Radziemski and David A. Cremers
22. Infrared Technology Fundamentals, Irving J. Spiro and Monroe
Schlessinger
23. Single-Mode Fiber Optics: Principles and Applications, Second Edition,
Revised and Expanded, Luc B. Jeunhomme
24. Image Analysis Applications, edited by Rangachar Kasturi and Mohan
M. Trivedi
25. Photoconductivity: Art, Science, and Technology, N. V. Joshi
26. Principles of Optical Circuit Engineering, Mark A. Mentzer
27. Lens Design, Milton Laikin
28. Optical Components, Systems, and Measurement Techniques, Rajpal

S. Sirohi and M. P. Kothiyal
29. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Second Edition, Revised and Expanded, Lawrence E. Murr


30. Handbook of Infrared Optical Materials, edited by Paul Klocek
31. Optical Scanning, edited by Gerald F. Marshall
32. Polymers for Lightwave and Integrated Optics: Technology and Applications, edited by Lawrence A. Homak
33. Electro-Optical Displays, edited by Mohammad A. Karim
34. Mathematical Morphology in Image Processing, edited by Edward R.
Dougherty
35 Opto-Mechamcal Systems Design: Second Edition, Revised and Expanded, Paul R. Yoder, Jr.
36. Polarized Light: Fundamentals and Applications, Edward Colleti
37. Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J. F.
Digonnet
38. Speckle Metrology, edited by Rajpal S. Sirohi
39. Organic Photoreceptors for Imaging Systems, Paul M. Borsenberger
and David S. Weiss
40. Photonic Switching and Interconnects, edited by Abdellatif Marrakchi
41. Design and Fabrication of Acousto-Optic Devices, edited by Akis P.
Goutzoulis and Dennis R. Pape
42. Digital Image Processing Methods, edited by Edward R Dougherty
43. Visual Science and Engineering: Models and Applications, edited by D,
H. Kelly
44. Handbook of Lens Design, Daniel Malacara and Zacarias Malacara
45. Photonic Devices and Systems, edited by Robert G. Hunsperger
46. Infrared Technology Fundamentals: Second Edition, Revised and Expanded, edited by Monroe Schlessinger
47. Spatial Light Modulator Technology: Materials, Devices, and Applications, edited by Uzi Efron
48. Lens Design: Second Edition, Revised and Expanded, Milton Laikin
49. Thin Films for Optical Systems, edited by Frangois R. Flory
50. Tunable Laser Applications, edited by F. J. Duarte

51. Acousto-Optic Signal Processing: Theory and Implementation, Second
Edition, edited by Norman J. Berg and John M. Pellegrino
52. Handbook of Nonlinear Optics, Richard L. Sutherland
53. Handbook of Optical Fibers and Cables: Second Edition, Hiroshi
Murata
54. Optical Storage and Retrieval. Memory, Neural Networks, and Fractals,
edited by Francis T. S. Yu and Suganda Jutamutia
55. Devices for Optoelectronics, Wallace B. Leigh
56. Practical Design and Production of Optical Thin Films, Ronald R,
Willey
57. Acousto-Optics: Second Edition, Adrian Korpel
58. Diffraction Gratings and Applications, Erwin G. Loewen and Evgeny
Popov
59. Organic Photoreceptors for Xerography, Paul M. Borsenberger and
David S. Weiss
60 Characterization Techniques and Tabulations for Organic Nonlinear
Optical Materials, edited by Mark Kuzyk and Carl Dirk


61. Interferogram Analysis for Optical Testing, Daniel Malacara, Manuel
Servin, and Zacarias Malacara
62. Computational Modeling of Vision: The Role of Combination, William
R. Uttal, Ramakrishna Kakarala, Sriram Dayanand, Thomas Shepherd,
Jagadeesh Kalki, Charles F. Lunskis, Jr., and Ning Liu
63. Microoptics Technology: Fabrication and Applications of Lens Arrays
and Devices, Nicholas F. Borrelli
64. Visual Information Representation, Communication, and Image Processing, Chang Wen Chen and Ya-Qin Zhang
65. Optical Methods of Measurement: Wholefield Techniques, Rajpal S.
Sirohi and Fook Siong Chau
66. Integrated Optical Circuits and Components: Design and Applications,

edited by Edmond J. Murphy
67. Adaptive Optics Engineering Handbook, edited by Robert K. Tyson
68. Entropy and Information Optics, Francis T. S. Yu
69. Computational Methods for Electromagnetic and Optical Systems,
John M. Jarem and Partha P. Banerjee
70. Laser Beam Shaping: Theory and Techniques, edited by Fred M. Dickey and Scott C. Holswade
71. Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Revised and Expanded, edited by Michel J. F. Digonnet
72. Lens Design: Third Edition, Revised and Expanded, Milton Laikin
73. Handbook of Optical Engineering, edited by Daniel Malacara and Brian
J. Thompson
74. Handbook of Imaging Materials, edited by Arthur S. Diamond and David S. Weiss
75. Handbook of Image Quality: Characterization and Prediction, Brian W.
Keelan
76. Fiber Optic Sensors, edited by Francis T. S. Yu and Shizhuo Yin
77. Optical Switching/Networking and Computing for Multimedia Systems,
edited by Mohsen Guizani and Abdella Battou
78. Image Recognition and Classification: Algorithms, Systems, and Applications, edited by Bahram Javidi
79. Practical Design and Production of Optical Thin Films: Second Edition,
Revised and Expanded, Ronald R. Willey
80. Ultrafast Lasers: Technology and Applications, edited by Martin E.
Fermann, Almantas Galvanauskas, and Gregg Sucha
81. Light Propagation in Periodic Media: Differential Theory and Design,
Michel Neviere and Evgeny Popov
82. Handbook of Nonlinear Optics, Second Edition, Revised and Expanded, Richard L Sutherland
Additional Volumes in Preparation
Optical Remote Sensing: Science and Technology, Walter Egan


Preface to Second Edition

The science of optics, the branch of physics that deals with the properties and
phenomena of visible and invisible light, has generated a wealth of knowledge
that makes its use pervasive in other physical sciences, biology, medicine,
forensics, agriculture, art, industry, and the military. This has spawned a
technology called photonics, a name based on the quantum of energy in the
electromagnetic field, the photon. The domain of photonics extends from energy

generation to detection to communications and information processing, and
includes all means of generating and harnessing light for useful purposes.
Both the science and technology aspects of optics have and continue to be
vastly influenced by the field of nonlinear optics. It is a discipline that has enhanced
our understanding of fundamental light-matter interactions as well as provided
the means for accomplishing a variety of engineering tasks. The purpose of this
book is to provide a balanced treatment of second- and third-order nonlinear optics,
covering areas useful to the practicing scientist and engineer. The intent is to serve
as a ready source of information useful to those researchers performing
characterization of nonlinear materials, using the methods of nonlinear optics in
scientific studies, and exploiting nonlinear optical phenomena in photonics
This edition of the Handbook of Nonlinear Optics has been updated and
new material has been added It is evident from a perusal of the scientific
literature that advances in nonlinear optics continue at a rapid pace. For example,
frequency conversion in new bulk and quasi-phase-matched materials as well as
the development of new optical parametric oscillators are areas in which progress
is continuing. Ultrafast optics and the sub-picosecond domain of optical
characterization offer interesting and challenging avenues for probing the
properties of materials and developing new applications. Furthermore, new
techniques are continually being developed to measure and modify the properties
of materials for diverse applications such as optical limiting, nonlinear
fluorescent imaging, and two-photon photopolymenzation

n

Preface to the Second Edition

As in the previous edition, selection of topics for inclusion was based on a
certain bias for what has been important to me as a general practitioner of
nonlinear optics In this regard, I have chosen to add work done in my group that
is relevant primarily to the characterization and application of nonlinear
materials However, in the interest of properly setting the stage for the bulk of the
book, and because it so often seems to be a point of confusion for beginners, I

have expanded the first chapter, which deals with elements of nonlinear optical
theory Chapter 2, "Frequency Doubling and Mixing," and Chapter 3, "Optical
Parametric Generation, Amplification, and Oscillation," so important in the

generation of light for other nonlinear optics applications, have been expanded
and updated primarily to include new results reported in the literature Chapters 6
("Nonlinear Index of Refraction") 7 ("Characterization of Nonlinear Refractive
Index Materials"), 9 ("Nonlinear Absorption"), and 10 ("Experimental
Techniques in Nonlinear Absorption') all incorporate new material Several of
the chapters tabulating materials data (Chapters 5, 8, and 13) have also been
updated Chapter 13 replaces Chapter 11 in the previous edition Two new
chapters (Chapter 11, "Ultrafast Characterization Techniques," and Chapter 12,
' Laser Flash Photolysis") have been added, covering important topics in the
expanding characterization requirements of nonlinear materials Finally Chapter
17, "Electro-Optic Effects," has been added because the effect plays such a
central role in several devices used in optics, as well as in the photorefractive
effect, and because it is arguably a nonlinear effect, depending as it does on the
interaction of two or more electric fields

This second edition also afforded the opportunity to correct errors and
misprints that occurred in the first edition My gratitude goes to those who have
graciously pointed these out to me
As always, I am indebted to several people who have been of great help in
preparing this work Not the least of these is my family, which has stood beside
me with patience and support I would also like to thank my employer, SAIC, and
the U S Air Force Research Lab (AFRL/MLPJ) for their encouragement of this
project Finally, I acknowledge my colleagues for their helpful advice and
criticism, especially Scan Kirkpatrick and Daniel G McLean, who authored
Chapters 11 and 12, respectively, and Suresh Chandra, who contributed to
Chapter 3
Richard L Sutherland


Preface to the First Edition

Shortly after the demonstration of the first laser in 1960, Peter Frankin and
coworkers ushered in nonlinear optics (NLO) with the observation of second
harmonic generation in a quartz crystal. Since then, NLO has burgeoned into a
mature field of science and engineering. The scope of this discipline includes all
phenomena in which the optical parameters of materials are changed with
irradiation by light. Generally, this requires high optical intensities, which is the
main reason that NLO matured in parallel with laser technology. Judging by the
growth and continued good health of publications and international conferences
on the subject, NLO appears to have a strong future in areas of photonics devices
and scientific investigations.
The impact of NLO on science and technology has been twofold. First, it
has enhanced our understanding of fundamental light-matter interactions.
Second, it has been a driving force in the rejuvenation of optical technology for
several areas of science and engineering. NLO has matured in the sense of being a

well-developed and systematic theory as well as providing applications for a
vanety of engineering tasks. Second and third order phenomena and devices are
now at a stage of understanding and development such that a coherent description
and summary of these areas forming the core of the subject are now possible and
desirable.
The rapid development of the subject has created the need for a
handbook that summarizes technical details concerning core areas impacting
several engineering and scientific endeavors. The general practitioner of NLO
requires information in at least four critical areas: (1) mathematical formulas
applicable to a variety of experimental and design situations, (2) examples of
ways NLO is applied to specific technical problems, (3) a survey of device and
matenals data for comparison purposes and numerical evaluation of formulas,
and (4) in-depth descriptions of methods required for characterizing new
matenals. When seeking this information, novice and expert alike are often


w

Pnface to the First Edition

bewildered by a lack of continuity in style notation, content, and physical
units contained m the literature Textbooks tend to develop the subject in
depth, with an emphasis on pedagogical style and with considerable
mathematical detail This inherently limits the scope of the material covered
Useful results are scattered throughout the text, usually without any helpful
summary of important and useful formulas Moreover, discussions of
applications and experimental methods as well as materials and device data,
are often sparse When seeking information, what a practicing scientist or
engineer (or student) needs is often more than a cursory treatment of a subject,

but not one lost in mathematical detail
While a few handbooks and treatises on NLO exist, some of these are
dated and some lack continuity m style, nomenclature, and use of physical
units, primarily because of multiple authorship Some are rich in materials
data but are lacking in the other four areas I describe above Finally, some
treat a limited scope of phenomena, such as only second order effects or a
single application area What is needed is a balanced treatment of both second
and third order NLO covering areas useful to the practicing scientist and
engineer
The purpose of this book is to fulfill this need by providing a ready source
of information to applied scientists, engineers, students, and others interested in
the applications of NLO Important formulas, experimental methods, and
materials data are summarized in the form of a handy reference to several aspects
of the field The scope of the book includes experimental applications and a
discussion of devices
This book is an outgrowth of my years as a general practitioner of
NLO As the leader of an optical characterization group for optical limiting
applications, I have been involved in the tasks of conceptualizing
devices based on NLO, searching for materials to transform the concepts
to practice characterizing the nonlinear properties of materials, and testing
prototypes Also, on occasion I have had the opportunity to teach a graduatelevel course in NLO The material for this book was gathered from my
lecture notes, on-the-job experience, and specific research directed toward
this work It is my intention that the contents should largely fulfil] the needs
of those researchers performing characterization of nonhneai materials, using
NLO methods in scientific studies and exploiting NLO phenomena in
photomcs devices
The richness and vitality of NLO dictate against a fully comprehensive
treatment at any given point in time, simply not everything can be covered I
admit to a certain bias, and the material given here is largely what has been of
importance to me as a general practitioner of NLO Therefore, this work

concentrates on what I consider the core of the subject, including second order
phenomena involving frequency conversion, and the third order phenomena of


Preface to the First Edition

vn

nonlinear phase modulation, nonlinear absorption, and nonlinear scattering.
The book treats technologically significant phenomena and presents a
summary of important formulas useful in the understanding and application
of NLO. Succinct physical interpretations of the mathematics are also given,
with an emphasis on conceptual understanding. Experimental methods for
charactenzing nonlinear parameters are described for both second and third
order materials. A discussion of well-accepted as well as novel, less wellknown methods is included. Finally, technical data on selected materials are
also summarized.
Differences in notation in the literature can often lead to confusion.
Therefore, I feel it imperative to clanfy some of the mathematical notation that I
strive to use consistently in the book. First, an optical wave propagating in space
and time is a real quantity and is represented by a real mathematical expression.
The simplest sinusoidal wave has the form

E H (r,0=A'cos(kT-cor)
In NLO, the product of two or more waves appears in many formulas, and it is
thus convenient to give this expression in complex exponential form:
E (r, f) = - A' exp[((k-r — wf)] + complex conjugate

(1)

Note that the addition of the complex conjugate (c.c.) keeps the quantity real. It is

evident from this expression that the product of several waves will involve 1/2
raised to some power. It is common to avoid this by suppressing the factor of 1/2
and rewriting the equation as
EM(r, t) = A exp[f(k-r - w?)] + c.c.

(2)

where A = (1/2)A'. The mathematical form given in Eq. (2) is used for optical
waves throughout this book. This is important to note because the use of the other
complex form of the field would lead to a different numerical prefactor in the
definition of optical intensity, the key parameter connecting theory to
measurements
Another important note is the definition of scalars, vectors, and tensors
Both vectors and tensors are presented as bold symbols, such as E. Whether the
symbol represents a vector or a tensor should be obvious from the context. Scalar
quantities are represented by nonbold symbols.
Both SI and cgs (esu) systems of units are used as much as possible
throughout the book. The system of units used in formulas is also often a source
of much confusion in NLO Therefore, care has been taken to present formulas in


\in

Preface to the First Edition

both sets of units as much as possible, the units of measure in both systems are
given for key physical parameters, and conversion formulas between the two
systems are summarized
Chapter 1 introduces elements ot NLO theory It is useful to study this
chapter to acquaint oneself with the notation used throughout this book as well as

to become familiar with the underlying principles ot the subject Chapters 2 and 3
deal with second order NLO phenomena These include frequency conversion
and optical parametric phenomena Topics covered range from the operation of
ideal devices to realistic optical beams interacting in nomdeal materials Various
aspects such as phase matching in umaxial and biaxial crystals, effective

nonlinear coefficients, temporal effects, tuning, bandwidth, and the effects of
absorption and diffraction are discussed The materials characterization
techniques for second order NLO coefficients is the subject of Chaptei 4 This
is followed in Chapter 5 by a tabulation ot second order NLO parameters for
several selected materials
The remainder of the book is devoted to third order NLO Chapters 6
and 9 discuss nonlinear refraction and nonlinear absorption, respectively A
summary of different physical mechanisms contributing to these phenomena
is given Detailed discussions of important applications are included
Experimental techniques for characterizing the respective nonlmeanties in
materials form the subjects ot Chapter 7 and 10 Materials data are tabulated
for a vanet> of gases liquids, solutions, and solids in Chapters 8 and 11
Nonlinear scattering (stimulated Raman and stimulated Bnlloum scattering) is
treated in Chapters 12 and 13 Brief descriptions of the utilization of these
phenomena for frequency conversion and optical beam control are given
Finally, materials data relating to these phenomena are presented in
Chapter 14
In compiling the information for this book, I have felt like the proverbial
discoverer standing on the shoulders of giants I am much indebted to the
countless number of researchers who have paved the way for the rest of us and
gave the time to so adequately document their results with great detail and
insight It would be impossible to thank them all by name I would, however,
like to acknowledge a few with whom I have had the pleasure of interacting
through workshops or on site \isits at Wright Patterson Air Force Base To

Elsa Garmire, Tony Ganto Hyatt Gibbs, Art Smirl, M J Soileau, George
Stegeman, and Eric Van Stryland I give thanks for insightful discussions In
addition, I would like to thank my colleagues at SAIC particularly Dan
McLean, Bob Ephng, Paul Fleitz, and Lalgudi Natarajan, for their
contributions to this work My thanks also to the staff of Marcel Dekker,
Inc , for inviting me to write this book and encouraging its
completion, as well as to both SAIC and U S Air Force Wright Lab
(WL/MLPJ) for their encouragement of this project Finally, my greatest debt


Preface to the First Edition

ix

of gratitude is to my wife, Marcine, and my daughters, Kari and Kendra, for
their patience, understanding, and unswerving support to me in the sometimes
seemingly endless task of completing this book.
Richard L. Sutherland



Contents
Preface to the Second Edition
Preface to the First Edition

1. Elements of the Theory of Nonlinear Optics

Hi
v

1

2.

Frequency Doubling and Mixing

3

Optical Parametric Generation, Amplification, and Oscillation

121

4. Characterization of Second Order Nonlinear Optical Materials

241

5. Properties of Selected Second Order Nonlinear Optical Materials

295

6. Nonlinear Index of Refraction

337

7. Characterization of Nonlinear Refractive Index Materials

433

8. Optical Properties of Selected Third Order Nonlinear Optical
Materials

499

9.

579

Nonlinear Absorption

33

10. Experimental Techniques in Nonlinear Absorption

627

11. Ultrafast Characterization Techniques
Scan Kirkpatrick

673

12. Laser Flash Photolysis
Daniel G. McLean

725


xn

Contents

13

Nonlinear Absorption Properties of Selected Materials

767

14

Stimulated Raman Scattering

793

15

Stimulated Bnlloum Scattering

813

16

Properties of Selected Stimulated Light-Scattering Materials

835

17.

Electro-Optic Effects

843

Index

947


1
Elements of the Theory of Nonlinear
Optics

Optics is an important part of everyday life. Light seems to flow or propagate
through empty space, as well as through material objects, and provides us with
visual information about our world. The familiar effects of reflection, refraction,
diffraction, absorption, and scattering explain a wide variety of visual
experiences common to us, from the focusing of light by a simple lens to the
colors seen in a rainbow. Remarkably, these can be explained by assigning a
small set of optical parameters to materials. Under the ordinary experiences of
everyday life, these parameters are constant, independent of the intensity of light
that permits observation of the optical phenomena. This is the realm of what is
called linear optics.
The invention of the laser gave rise to the study of optics at high intensities,
leading to new phenomena not seen with ordinary light such as the generation of
new colors from monochromatic light in a transparent crystal, or the self-focusing
of an optical beam in a homogeneous liquid. At the intensities used to generate
these types of effects, the usual optical parameters of materials cannot be
considered constant but become functions of the light intensity. The science of
optics in this regime is called nonlinear optics.
The theory of nonlinear optics builds on the well-understood theory of
linear optics, particularly that part known as the interaction of light and matter.
Ordinary matter consists of a collection of positively charged cores (of atoms or
molecules) and surrounding negatively charged electrons. Light interacts

primarily with matter via the valence electrons in the outer shells of electron
orbitals. The fundamental parameter in this light-matter interaction theory is
1


2

Chapter 1

the electronic polarization of the material induced by light. Extending the
definition of this parameter to the nonlinear regime allows the description of a
rich variety of optical phenomena at high intensity.
This chapter presents a brief overview of the theory of nonlinear optics.
Formulas are given which generally apply to a number of phenomena discussed
in later chapters. For a more pedagogical treatment, consult the references given
at the end of this chapter.

I.
A.

ELECTROMAGNETIC BASIS OF OPTICS
The Optical Electric Field

Light is an electromagnetic wave. It consists of electric and magnetic fields, E (,)
and H (,), respectively. The superscripted tilde (,) implies that the fields are
rapidly varying in time, and the fields are real quantities. For most of optics, the
optical wave may be characterized by defining its electric field. (The magnetic
field is related to the electric field through Maxwell’s equations from
electromagnetic theory [1].)
Nonlinear optics is performed with lasers, which have a highly directional

nature. Therefore, it is common to assume that the electric field is a wave
propagating primarily in one direction in space. Allowance may be made for a
finite amount of beam spreading, or diffraction. This primary direction of
propagation is usually taken to be the along the z-axis. (For noncollinear
propagation of multiple beams, the primary change of the beams with distance is
taken to be along a single axis, again usually the z-axis.) Hence, the general form
of the electric field wave is given by
E ð,Þ ðr; tÞ ¼ e^ Aðr; tÞexp½iðkz 2 vtފ þ c:c:

ð1Þ

In this equation, k is the wave vector of propagation and v is the circular
frequency of the rapidly oscillating wave. The wave amplitude A(r,t ) may have a
space- and time-dependence, which is slowly varying compared to the rapidly
varying parts (space and time) of the oscillating wave. This amplitude is, in
general, complex and includes the possibility of phase accumulation in addition
to that contained in the exponent of Eq. (1). The polarization of the wave (i.e.,
direction of the electric field vector) is given by the unit vector eˆ. When this
vector is real, the wave is said to be plane polarized. A complex unit vector
implies that the wave is elliptically polarized. A special case of this is circular
polarization. For most of the cases in this book, plane polarized light waves will
be assumed unless otherwise specified. Finally, the notation “c.c.” implies
complex conjugate. It is included in the definition of Eq. (1) since the field E (,) is
a real quantity.


Elements of the Theory of Nonlinear Optics

3

For a large number of problems in linear and nonlinear optics, the field can
be assumed to be of infinite extent and constant in amplitude and phase in a plane
transverse to the direction of propagation. Thus, the complex field amplitude
becomes a function of z and t only: A(z,t ). Such a wave is called an infinite plane
wave, or sometimes just a plane wave. Certainly this is only an approximation
since real laser beams have a finite transverse extent and vary spatially along the
transverse direction.
A common form of a finite beam is the TEM00 mode of a circular Gaussian
beam. The field of this type of wave has the following form.
&
!'
 
w0
kr 2
ð,Þ
21 z
E ðr;tÞ ¼ e^ Aðz;tÞ
exp i
þ kz 2 tan
2 vt þ c:c: ð2Þ
zR
wðzÞ
2qðzÞ
This beam has azimuthal symmetry and its form is illustrated in Fig. 1. Note that
the beam has a Gaussian cross section with a variable radius w(z ), which is
defined as the half-width of the Gaussian curve at the point r (the radial
coordinate), where the curve is at 1=e of its maximum value, as shown in Fig. 1b.
The radius has a minimum, defined by w0, at the plane z ¼ 0; and w(z ) is given by
"
 2 #1=2

z
ð3Þ
wðzÞ ¼ w0 1 þ
zR
The diameter of the beam at z ¼ 0 is 2w0 and is called the beam waist.
The surface of constant phase for a Gaussian beam is curved. At the beam
waist the phase has an infinite radius of curvature, and hence mimics a plane
wave. For large distances away from the waist, the radius of curvature is , z. The
quantity q(z ) is called a complex radius of curvature and is given by
qðzÞ ¼ z 2 izR

ð4Þ

Finally, the quantity zR is called the Rayleigh range and is defined by
zR ¼

npw20
l

ð5Þ

where n is the index of refraction of the medium, and l is the optical wavelength
in free space. The Rayleigh range corresponds topthe
ffiffiffi distance from the waist at
which the beam radius increases by a factor of 2: The distance between the
points ^ zR about the waist is called the confocal parameter b of the beam
ðb ¼ 2zR Þ: These parameters are also defined schematically in Fig. 1a.
B.

Electric Polarization in a Dielectric Medium

When an electric field is applied to a dielectric medium (of neutral electric
charge), a separation of bound charges is induced as illustrated in Fig. 2. This


4

Chapter 1

Figure 1 Schematic illustration of a TEM00 Gaussian beam. (a) Beam propagation
profile; (b) beam cross section.

separation of charge results in a collection of induced dipole moments m (,),
which, as designated, may be rapidly oscillating if induced by a rapidly varying
applied field. The electric polarization is defined as the net average dipole
moment per unit volume and is given by
P ð,Þ ¼ Nkm ð,Þ l

ð6Þ

where N is the number of microscopic dipoles per unit volume, and the angular
brackets indicate an ensemble average over all of the dipoles in the medium. In
what follows, any permanent dipoles within the medium will be ignored since
they will not be oscillating at optical frequencies and hence will not radiate
electromagnetic waves.


Elements of the Theory of Nonlinear Optics

5

Figure 2 Illustration of the response of a dielectric medium to an applied electric field.
(a) Without field applied; (b) field applied.

By the principle of causality, P (,) must be a function of the applied field
E . To an excellent approximation, at the low intensity levels of natural light
sources, the relation of the polarization to the applied field is linear. This is the
regime of linear optics. The most general form of the electric polarization for a
homogeneous medium is given by
8 Z 1Z 1
>
>
x ð1Þ ðr 2 r 0 ;t 2 t 0 Þ·E ð,Þ ðr 0 ;t 0 Þdr 0 dt 0
ðSIÞ
1
>
< 0 21 21
ð,Þ
ð7Þ
PL ðr;tÞ ¼ Z 1 Z 1
>
ð1Þ
0
0
ð,Þ 0 0
0 0
>
>
x
ðr

2
r
;t
2
t
Þ·E
ðr
;t
Þdr
dt
ðcgsÞ
:
(,)

21 21

where the subscript L signifies a linear polarization, 10 ¼ 8:85 £
10212 farad=meter is the electric permittivity of free space, and
x ð1Þ ðr 2 r 0 ;t 2 t 0 Þ is the linear dielectric response tensor. The functional form
of x ð1Þ reflects the principles of space and time invariance [6]. In other words,


6

Chapter 1

the polarization response of a medium does not depend on when (in an absolute
sense) the driving field is applied, but only on the time since it was applied.
Consequently, x ð1Þ ðr 2 r 0 ;t 2 t 0 Þ must be defined in such a way that it vanishes
when t 2 t 0 , 0 to preserve causality. Similarly, the polarization response in a

homogeneous medium does not depend on the absolute position in space of the
applied field, but only on the distance away from this position. A nonzero value of
x ð1Þ ðr 2 r 0 ;t 2 t 0 Þ for r – r 0 is called a nonlocal response. If there is no response
except within a small neighborhood where r < r 0 ; then the response is called
local. This is equivalent to saying that the linear dielectric response tensor has a
d-function spatial dependence. For the vast majority of problems in nonlinear
optics, the media of interest produce approximately a local response.
Consequently, we will ignore the spatial dependence of x ð1Þ in what follows.
The form of the linear dielectric response tensor allows a simpler relation to
be made between the Fourier transforms of the linear polarization and the applied
field,
(
P L ðvÞ ¼

10 x ð1Þ ðvÞ · EðvÞ

ðSIÞ

x ð1Þ ðvÞ · EðvÞ

ðcgsÞ

ð8Þ

where x ð1Þ ðvÞ; the linear susceptibility tensor, is the Fourier transform of the
linear dielectric response tensor. The tensor relation in Eq. (8) can also be
written as
8 P
>
1 xð1Þ ðvÞEj ðvÞ

>
> 0 j ij
<
PL;i ðvÞ ¼ P ð1Þ
>
xij ðvÞEj ðvÞ
>
>
: j

ðSIÞ
ðcgsÞ

ð9Þ

where the subscript i signifies the ith cartesian coordinate ði ¼ x; y; zÞ; and the
sum is over j ¼ x; y; z: The tensor x ð1Þ ðvÞ thus has nine components. In an
isotropic medium, there is only one independent, nonzero component, and the
susceptibility is written as a scalar quantity, xð1Þ ðvÞ:

C.

Wave Equation

For the majority of situations considered in nonlinear optics, and for every case
treated in this book, it can be assumed that there is no macroscopic magnetization
in the dielectric medium (no microscopic magnetic dipoles). The medium is also
electrically neutral and nonconducting so that no free charge or current density
exists. Under these conditions, the wave equation describing the propagation of

Elements of the Theory of Nonlinear Optics

7

the vector electric field wave is given by
7 £ 7 £ E ð,Þ þ

1 ›2 E ð,Þ
K ›2 P ð,Þ
¼2 2
2
2
c ›t
c ›t 2

ð10Þ

where c ¼ 3 £ 108 m=s ð3 £ 1010 cm=sÞ is the speed of light in a vacuum, and K is
a constant depending on the system of units used, with
8
< ð10 Þ21
ðSIÞ
K¼
ð11Þ
: 4p
ðcgsÞ
When the intensity of the light is sufficiently high (e.g., from a laser), a
small additional polarization will appear, so that the total polarization can be
written as

P ð,Þ ¼ PLð,Þ þ Pð,Þ
NL

ð12Þ

where Pð,Þ
NL is a nonlinear function of the applied field. Substituting this
expression into Eq. (10), the wave equation becomes
7 £ 7 £ E ð,Þ þ

1 ›2 E ð,Þ K ›2 Pð,Þ
K ›2 Pð,Þ
L
NL
þ
¼
2
c 2 ›t 2
c 2 ›t 2
c 2 ›t 2

ð13Þ

When the nonlinear polarization is negligible, the left-hand side of Eq. (13) will
be recognized as the homogeneous wave equation for linear optics. This is
generally given in terms of the Fourier transform of the electric field by
7 £ 7 £ Eðr; vÞ þ

v2
kðvÞ · Eðr; vÞ ¼ 0

c2

where k is the linear dielectric tensor 1 (cgs) or 1=10 (SI),with
8 

ð1Þ
>
ðSIÞ
< 10 dij þ xij ðvÞ
1ij ðvÞ ¼
>
: dij þ 4pxijð1Þ ðvÞ
ðcgsÞ

ð14Þ

ð15Þ

with
(

dij ¼

1 ði ¼ jÞ
0 ði – jÞ

ð16Þ

We see then that the nonlinear polarization acts as a source term for an
inhomogeneous wave equation. For most situations in nonlinear optics, the total

electric field can be considered to be a superposition of quasi-monochromatic


8

Chapter 1

waves (e.g., laser beams). The total field is then written as
E ð,Þ ðr; tÞ ¼

X
m

e^ m Am ðr; tÞexp½iðk m · r 2 vm tފ þ c:c:

ð17Þ

where the sum is over m waves with frequencies vm and wave vectors km. Am ðr; tÞ
is a slowly varying amplitude in space and time (compared to the rapidly
oscillating part of the wave). If it is sufficiently slowly varying, the mth
component is a monochromatic wave. However, if its time duration is sufficiently
short (an ultrashort pulse) such that it cannot be described as a pure
monochromatic wave, then the mth component represents a quasi-monochromatic wave with carrier frequency vm.
For the typical case when the nonlinear polarization represents a small
perturbation to the total polarization, it can also be written as
Pð,Þ
NL ðr; tÞ ¼

X
m

P NL;m ðr; tÞexpð2ivm tÞ þ c:c:

ð18Þ

where P NL;m ðr; tÞ is a slowly varying (compared to the rapidly oscillating part of
the wave) complex polarization amplitude. Then by the linearity of the wave
equation, each frequency component (Fourier component) of the total field also
satisfies Eq. (13), with the corresponding frequency component of the nonlinear
polarization appearing on the right-hand side of the equation. Thus, there will be
m inhomogeneous wave equations describing the interaction. Also, the dielectric
constant in each wave equation is evaluated at the frequency vm.
This is the most general form of the wave equation. Under usual conditions,
the left-hand side can be simplified, for an excellent approximation in
homogeneous media,
7 £ 7 £ E ð,Þ < 272 E ð,Þ

ð19Þ

where 72 is the Laplacian operator. For most of the nonlinear phenomena
considered in this book, this approximation will be used.

II.

LINEAR OPTICS

In the linear optics regime, the nonlinear part of the polarization may be
neglected (i.e., set equal to zero). The wave equation, Eq. (13), then becomes a
homogeneous differential equation. Its solutions are given in the form of Eq. (1).
The simplest waves to consider are plane waves in an isotropic medium.