NONLINEAR OPTICS
WILEY SERIES IN PURE AND APPLIED OPTICS
Founded by Stanley S. Ballard, University of Florida
EDITOR: Glenn Boreman, University of Central Florida, CREOL & FPCE
A complete list of the titles in this series appears at the end of this volume.
P&Aoptics_P&Aoptics.qxd 6/7/2012 9:07 AM Page 1
NONLINEAR OPTICSPHENOMENA, MATERIALS, ANDDEVICES
George I. StegemanRobert A. Stegeman
Copyright � 2012 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
ISBN: 978-1-118-07272-1
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
CONTENTS
PREFACE xi
1. Introduction 1
1.1 What is Nonlinear Optics and What is it Good for? 1
1.2 Notation 2
1.3 Classical Nonlinear Optics Expansion 4
1.4 Simple Model: Electron on a Spring and its Application to
Linear Optics 6
1.5 Local Field Correction 10
Suggested Further Reading 13
PART A: SECOND-ORDER PHENOMENA 15
2. Second-Order Susceptibility and Nonlinear Coupled
Wave Equations 17
2.1 Anharmonic Oscillator Derivation of Second-Order
Susceptibilities 18
2.2 Input Eigenmodes, Permutation Symmetry, and
Properties of w(2) 23
2.3 Slowly Varying Envelope Approximation 25
2.4 Coupled Wave Equations 26
2.5 Manley–Rowe Relations and Energy Conservation 31
Suggested Further Reading 38
3. Optimization and Limitations of Second-Order Parametric
Processes 39
3.1 Wave-Vector Matching 39
3.2 Optimizing dð2Þeff 53
3.3 Numerical Examples 59
References 67
Suggested Further Reading 67
v
4. Solutions for Plane-Wave Parametric Conversion Processes 69
4.1 Solutions of the Type 1 SHG Coupled Wave Equations 69
4.2 Solutions of the Three-Wave Coupled Equations 77
4.3 Characteristic Lengths 80
4.4 Nonlinear Modes 81
References 84
Suggested Further Reading 85
5. Second Harmonic Generation with Finite Beams
and Applications 86
5.1 SHG with Gaussian Beams 86
5.2 Unique and Performance-Enhanced Applications
of Periodically Poled LiNbO3 (PPLN) 98
References 107
Suggested Further Reading 107
6. Three-Wave Mixing, Optical Amplifiers, and Generators 108
6.1 Three-Wave Mixing Processes 108
6.2 Manley–Rowe Relations 110
6.3 Sum Frequency Generation 111
6.4 Optical Parametric Amplifiers 113
6.5 Optical Parametric Oscillator 119
6.6 Mid-Infrared Quasi-Phase Matching Parametric Devices 128
References 139
Selected Further Reading 140
7. w(2) Materials and Their Characterization 141
7.1 Survey of Materials 141
7.2 Oxide-Based Dielectric Crystals 143
7.3 Organic Materials 144
7.4 Measurement Techniques 149
Appendix 7.1: Quantum Mechanical Model for
Charge Transfer Molecular Nonlinearities 153
References 157
Suggested Further Reading 158
PART B: NONLINEAR SUSCEPTIBILITIES 159
8. Second- and Third-Order Susceptibilities: Quantum
Mechanical Formulation 161
8.1 Perturbation Theory of Field Interaction with Molecules 162
8.2 Optical Susceptibilities 169
vi CONTENTS
Appendix 8.1: wð3Þijk‘ Symmetry Properties for Different
Crystal Classes 192
Reference 196
Suggested Further Reading 196
9. Molecular Nonlinear Optics 197
9.1 Two-Level Model 198
9.2 Symmetric Molecules 210
9.3 Density Matrix Formalism 215
Appendix 9.1: Two-Level Model for Asymmetric
Molecules—Exact Solution 216
Appendix 9.2: Three-Level Model for Symmetric
Molecules—Exact Solution 218
References 222
Suggested Further Reading 223
PART C: THIRD-ORDER PHENOMENA 225
10. Kerr Nonlinear Absorption and Refraction 227
10.1 Nonlinear Absorption 228
10.2 Nonlinear Refraction 238
10.3 Useful NLR Formulas and Examples
(Isotropic Media) 243
Suggested Further Reading 250
11. Condensed Matter Third-Order Nonlinearities due to
Electronic Transitions 251
11.1 Device-Based Nonlinear Material Figures of Merit 252
11.2 Local Versus Nonlocal Nonlinearities in Space and Time 253
11.3 Survey of Nonlinear Refraction and Absorption
Measurements 255
11.4 Electronic Nonlinearities Involving Discrete States 256
11.5 Overview of Semiconductor Nonlinearities 266
11.6 Glass Nonlinearities 281
Appendix 11.1: Expressions for the Kerr, Raman, and
Quadratic Stark Effects 284
References 286
Suggested Further Reading 289
12. Miscellaneous Third-Order Nonlinearities 290
12.1 Molecular Reorientation Effects in Liquids and
Liquid Crystals 291
CONTENTS vii
12.2 Photorefractive Nonlinearities 300
12.3 Nuclear (Vibrational) Contributions to n2||(�o; o) 306
12.4 Electrostriction 310
12.5 Thermo-Optic Effect 312
12.6 w(3) via Cascaded w(2) Nonlinear Processes: Nonlocal 314
Appendix 12.1: Spontaneous Raman Scattering 317
References 328
Suggested Further Reading 329
13. Techniques for Measuring Third-Order Nonlinearities 330
13.1 Z-Scan 332
13.2 Third Harmonic Generation 339
13.3 Optical Kerr Effect Measurements 343
13.4 Nonlinear Optical Interferometry 344
13.5 Degenerate Four-Wave Mixing 345
References 346
Suggested Further Reading 346
14. Ramifications and Applications of Nonlinear Refraction 347
14.1 Self-Focusing and Defocusing of Beams 348
14.2 Self-Phase Modulation and Spectral
Broadening in Time 352
14.3 Instabilities 354
14.4 Solitons (Nonlinear Modes) 363
14.5 Optical Bistability 372
14.6 All-Optical Signal Processing and Switching 375
References 382
Suggested Further Reading 383
15. Multiwave Mixing 384
15.1 Degenerate Four-Wave Mixing 385
15.2 Degenerate Three-Wave Mixing 397
15.3 Nondegenerate Wave Mixing 399
Reference 413
Suggested Further Reading 413
16. Stimulated Scattering 414
16.1 Stimulated Raman Scattering 415
16.2 Stimulated Brillouin Scattering 431
References 441
Suggested Further Reading 442
viii CONTENTS
17. Ultrafast and Ultrahigh Intensity Processes 443
17.1 Extended Nonlinear Wave Equation 444
17.2 Formalism for Ultrafast Fiber Nonlinear Optics 448
17.3 Examples of Nonlinear Optics in Fibers 452
17.4 High Harmonic Generation 460
References 462
Suggested Further Reading 463
Appendix: Units, Notation, and Physical Constants 465
A.1 Units of Third-Order Nonlinearity 465
A.2 Values of Useful Constants 467
Reference 467
INDEX 469
CONTENTS ix
PREFACE
The field of nonlinear optics came into being in the 1960s, stimulated essentially by
the invention of the laser. Its impact has been acknowledged by the award of theNobel
Prize in physics to one of its pioneers Nicholas Bloembergen in 1981 (1) and by other
Nobel prizes for work enabled by nonlinear optics in chemistry and physics and
multiple awards by theAmerican Institute of Physics, the Optical Society of America,
and other scientific organizations.
The fundamental principles of nonlinear optics are nowwell known and have been
elucidated in excellent nonlinear optics textbooks by Ron Shen, Doug Mills, Robert
Boyd, Govind Agrawal, and others over the last 20 years. These books served two
purposes: to discuss basic principles and to give an overview of interesting applica-
tions and experiments. Of the two branches of nonlinear optics—nonlinear phenom-
ena and nonlinear materials—that have evolved over the years, the latter has
accounted for most of the progress in the field over the last few decades whereas
the former has been the subject of most textbooks. In fact, the nonlinear materials
evolution has been spectacular. Since the early days of nonlinear optics, the
requirements for some nonlinear phenomena have been reduced from kilowatt lasers
to milliwatt lasers. Our particular goal for this textbook, in addition to elucidating the
fundamentals of nonlinear optics from our own perspective, was to discuss nonlinear
materials, new nonlinear phenomena developed in the last few decades such as
solitons, and the practical aspects of the most common nonlinear devices.
This textbook is based on a nonlinear optics course initially developed at CREOL
(Center for Research in Electro-Optics and Lasers, now a part of the College of Optics
and Photonics), University of Central Florida, in the 1980s by Eric Van Stryland and
David Hagan. After George I. Stegeman joined CREOL in 1990, he took over this
course, expanded and continuously revised it, put it into the PowerPoint format, and
added new problems every year. Robert A. Stegeman, the coauthor, has used this
course in his professional career, which involves current applications of nonlinear
optics, primarily in the mid-infrared region of the spectrum, and he has contributed
most of the application discussions to this text.
This course deals with the physics and applications of optical phenomena that
occur at intensity levels at which optical processes become dependent on optical
intensity or integrated flux. It is designed for graduate students, postdoctoral fellows,
and others with prior knowledge of electromagnetic wave propagation in materials as
well as for professionals in the field who are looking for newly developed fields and
concepts. Although a rudimentary knowledge of quantum mechanics would be
xi
helpful, it is not a requirement.When quantummechanics is used, it is reviewed at the
level needed for the course.
Nonlinear optics is not just a simple extension of linear optics. A keystone concept
in linear optics is that electromagnetic waves do not interact with one another.
Solutions toMaxwell’s equations lead to “orthogonal” eigenmodes, i.e., summing the
fields due to two ormore overlapping field solutions and calculating the intensity leads
to a net intensity that is the sum of the intensities of the individual waves. Nonlinear
optics is all about interactions that occur between light and matter at high intensities.
Hence, the solutions to the nonlinear wave equations do not lead to eigenmodes, just
nonlinear modes. As will be discussed in the later part of the textbook, the modes of
nonlinear optics are solitons, spatial solitons for continuous waves that do not diffract
in space, temporal solitons for noncontinuous waves that do not disperse (spread) in
time and are confined in some kind of awaveguide that inhibits spatial diffraction, and
spatiotemporal solitons that spread neither in time nor in space. Such modes, in
general, exchange energy when they interact so that the reader should be prepared to
give up notions such as the superposition principle, which may be satisfied only
approximately.
It proves useful to explain the philosophy adopted here. There are two approaches
to discussing nonlinear optics. One is to introduce macroscopic nonlinear suscep-
tibilities at a phenomenological level. These susceptibilities are measured in the
laboratory and used to quantify phenomena, devices, and so forth. The second
approach starts at a more physical level with the electric dipole interaction of isolated
atoms and molecules with radiation fields to find the response at the atomic level
and the molecular level. Although it is satisfying from a physical insight perspective,
difficulties occur in going to the condensed matter limit where most experiments are
done. Because there are dipolar fields induced in neighboring molecules, the “local”
fields experienced by an atom or a molecule are not just the fields given by the
macroscopic Maxwell’s equations, called the “Maxwell” fields. The “local” field is
the sumof theMaxwell fields and all the induced dipole fields at the site of an atomor a
molecule. A rigorous and satisfactory approach to accurately estimating the local
fields has been a subject of continuing discussion formany years. Hence this approach
does not necessarily yield reliable numbers for measured nonlinear susceptibilities,
but does give fairly accuratelymanykey features, including the frequency spectrumof
the nonlinear response functions, i.e., the nonlinear susceptibilities.
Here we use a combination of these approaches. Whenever possible, the nonlinear
optics response at the molecular level is treated first, approximate susceptibilities are
derived, and then measured susceptibilities are used in discussing applications. To
facilitate the separation of microscopic and macroscopic parameters, the isolated
molecule parameters are identified by a “bar,” e.g., the transition dipole moment
between energy levels i and j (�Ei and �Ej) is �mij , the mass of an electron is �me, the
reduced mass for the bth vibration in a molecule is �mb, and so on.
This book also includes appendices in which the fundamentals of a number of
concepts such as Raman scattering and two- and three-level models are presented, as
well as some tables of the relation between nonlinear susceptibilities, their conversion
between different systems of units, and crystal symmetry.
xii PREFACE
G.S. thanks his colleagues at CREOL for their helpful discussions over the years,
especially Demetrius Christodoulides, Eric Van Stryland, and David Hagan, as well
as the many graduate students who diligently corrected lecture notes and asked
probing questions.
REFERENCE
1. N. Bloembergen, Encounters in Nonlinear Optics: Selected Papers of Nicolaas
Bloembergen with Commentary (World Scientific Press, Singapore,1996).
PREFACE xiii
CHAPTER 1
Introduction
1.1 WHAT IS NONLINEAR OPTICS AND WHAT IS IT GOOD FOR?
In general, nonlinear optics takes place when optical phenomena occur in materials
that change optical properties with input power or energy and/or generate new
beams or frequencies. Examples are power-, intensity-, or flux-dependent changes in
the frequency spectrum of light, the transmission coefficient, the polarization, and/or
the phase. New beams can also be generated either by a shift in frequency from the
original frequency or by travelling in different directions relative to the incident beam.
Although one frequently refers to the intensity or power dependence of phenomena as
being signatures of nonlinear optics, there are many cases characterized by a flux
dependence, i.e., changes in beam properties that are cumulative in the illumination
time, usually accompanied by absorption.
A frequently asked question is: “How do I really know when nonlinear optics is
occurring inmy experiment?” Some examples of commonly observed phenomena are
shown in Figs 1.1 and 1.2. Figure 1.1a shows harmonic generation, a second- or third-
order nonlinear effect. Figure 1.1b shows nonlinear transmission, essentially a third-
order nonlinear effect. In an interference experiment an increase in the input intensity
can lead to a shift in fringes due to second- or third-order nonlinear optics (see
Fig. 1.1c). Avery common effect—self-focusing of light—is illustrated in Fig. 1.2, in
which a beam narrows with an increase in the input intensity due to propagation
through a sample, forming a soliton at high intensities that propagates without change
in size or shape and then breaks up into “noise” filaments, i.e., multiple nondiffracting
beams, at very high intensities.
The second most frequently asked question is: “What is nonlinear optics good
for?” A collage of applications is shown schematically in Fig. 1.3. Probably the most
frequently used nonlinear optics device is the second harmonic generator, which
doubles the frequency of light, as shown in Fig. 1.1a. Along the same lines are optical
parametric devices, also based on second-order nonlinearities, which include am-
plifiers (optical parametric amplifiers) and frequency-tunable generators (optical
parametric generators and optical parametric oscillators), and the last two are
commonly used as sources of tunable radiation (see Fig. 1.3a). Nonlinear absorption
Nonlinear Optics: Phenomena, Materials, and Devices, George I. Stegeman and Robert A. Stegeman.� 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.
1
that depends on intensity is used for the localized activation of drugs or imaging inside
media (Fig. 1.3b). A third example is an all-optical control of optical signals, e.g., for
communications (Fig. 1.3c).
1.2 NOTATION
The diversity of notations used for optical fields, nonlinear susceptibilities, and so on,
is a frequently confusing aspect of this field. A perusal of the nonlinear optics
literature shows that there is little consistency, especially when dealing with third-
order nonlinear coefficients. Here a concentrated effort has beenmade to be consistent
and to introduce more descriptive notations. The assumptions and notation used here
are as follows:
2ωω
3ω
(a) (b)
Ioutput
Iinput
Low High(c)
FIGURE 1.1 (a) Second and third harmonic generation. (b) Nonlinear transmission.
(c) Nonlinear fringe shift between low and high intensity inputs.
OutputInput
(a) (b) (c) (d)
FIGURE1.2 (a) Beam input and output geometry. (b)Diffracted output beam at low intensity
input. (c) Self-focused output beam corresponding to a spatial soliton. (d) Multiple filaments in
the output beam at very high intensity.
(a) (b)
Frequency orpolarization filter
(c)
Laser OPO
FIGURE 1.3 (a) An optical parametric oscillator (OPO) for producing two tunable frequen-
cies. (b) Two-photon absorption activation of chemistry. (c) All-optical control of routing in a
nonlinear coupler. A control beam (lower input arm) is used to isolate (switch out) a single pulse
from the input pulse train.
2 INTRODUCTION
1. Planewaveswill be explicitly assumed to elucidate nonlinear phenomena in the
simplest fashion. Whenever finite beams are considered, which is normally
necessary to discuss devices and applications, this will be clearly stated.
2. Continuous-wave fields are explicitly assumed unless otherwise stated. The
electromagnetic fields are written as
~Eð~r; tÞ ¼ 1
2~EðoÞ e�iot þ c:c: ¼ 1
2~EðoÞ ei½kz�ot� þ c:c: ð1:1Þ
3. The unit vector is written as e and has components ei, where i¼ x, y, z.
4. The “Einstein” notation is used for summations over repeated indices; e.g.,
aibici ¼ axbxcx þ aybycy þ azbzcz.
5. Quantitieswith a “bar” above, e.g., �m~, refer to individualmolecular properties in
the absence of interactionwith othermolecules as well as parameters in a single
molecule’s frame of reference.
6. Quantities with a “tilde” above, e.g., ~wð2Þijk , identify parameters and coefficients
in the “zero (nonresonant) frequency” limit (Kleinman limit) o � �or, i.e., at
frequencies much smaller than any resonant frequency �or of the material.
7. SI units are used throughout. Here intensity is used to mean power per unit
area, usually in units ofwatts per square centimeter. It is equivalent to irradiance.
In the cases of pulses of light, flux per unit area is defined as the integrated
intensity of a pulse over time, typically over the duration of the pulse. Flux is
defined as the energy of a pulse integrated over both time and cross section.
It is important to realize that in this textbook~EðoÞ is not the Fourier transform of~EðtÞand its use is restricted to Eq. 1.1. For amonochromatic wave of frequencyoa,EiðoaÞis the notation used for the Fourier transform of the field and EiðoaÞ 6¼ EiðoaÞ. Therelations between the two can be derived easily from the unitary Fourier transform
equations:
EðtÞ ¼ffiffiffiffiffiffi1
2p
s �
EðoÞ e�iot do; EðoÞ ¼ffiffiffiffiffiffi1
2p
s �
Eðt0Þ eiot0 dt0
dðt�t0Þ ¼ffiffiffiffiffiffi1
2p
s �
e�ioðt�t0Þ do; dðo�oaÞ ¼ffiffiffiffiffiffi1
2p
s �
eiðo�oaÞt0Þ dt0:
ð1:2Þ
Substituting Eq. 1.1 for Eðt0Þ into the E(o) equation in Eq. 1.2 gives
EiðoÞ¼ffiffiffiffiffiffi1
2p
s1
2
�
EiðoaÞ eiðo�oaÞt dtþð¥�¥
E*i ðoaÞ eiðoþoaÞt dt
� �
¼ 1
2EiðoaÞ
ffiffiffiffiffiffi1
2p
s �
eiðo�oaÞt dtþE*i ðoaÞ
ffiffiffiffiffiffi1
2p
s �
eiðoþoaÞt dt
8<:
9=;
!EiðoÞ¼ 1
2EiðoaÞdðo�oaÞþEið�oaÞdðoþoaÞ½ �: ð1:3Þ
NOTATION 3
If fields have a distribution of frequencies, then the d functions are replaced by g
(o�oa) and g(o þ oa), normalized so that their integrals over frequency are unity.
Additional notation will be introduced as needed in succeeding chapters.
1.3 CLASSICAL NONLINEAR OPTICS EXPANSION
The simplest andmost general expansion of the nonlinear polarization induced by the
mixing of optical fields is
Pið~r;tÞ¼ e0
�
ðt�¥
wð1Þij ð~r�~r0; t�t0ÞEjð~r0;t0Þd~r0dt0�
þð¥�¥
�
ðt�¥
ðt�¥
wð2Þijk ð~r�~r0;~r�~r00; t�t0;t�t00ÞEjð~r0;t0ÞEkð~r00;t00Þd~r0d~r00dt0dt00
þð¥�¥
�
�
ðt�¥
ðt�¥
ðt�¥
wð3Þijk‘ð~r�~r0;~r�~r00;~r�~r000; t�t0;t�t00;t�t000Þ
�Ejð~r0;t0ÞEkð~r00;t00ÞE‘ð~r000;t000Þd~r0d~r00d~r000dt0dt00dt000 þ �� ��
with
�
d~r0 ��
�
�
dx0dy0dz0 ð1:4Þ
To understand the physical implications of this formula, consider the first nonlinear
term due to the second-order susceptibility, i.e., wð2Þijk ð~r�~r0;~r�~r00; t�t0; t�t00Þ. Thepolarization Pið~r; tÞ is created at time t and position~r by two separate interactions
of the total electromagnetic field at time t0 and position~r0 and at time t00 and position~r00 in a material in which w(2) 6¼ 0. This form also includes nonlocal-in-space effects,
such as thermal nonlinearities, in which the refractive index changes due to
absorption, e.g., diffuses. In most cases encountered on optics, the response is
local in space and so
wð2Þijk ð~r�~r0;~r�~r00; t�t0; t�t00Þ ! wð2Þij ðt�t0; t�t00Þdð~r�~r0Þdð~r�~r00Þ ð1:5Þ
and
Pð2Þi ð~r; tÞ ¼ e0
�
�
wð2Þijk ðt�t0; t�t00ÞEjð~r; t0ÞEkð~r; t00Þ dt0 dt00: ð1:6Þ
Only near a “resonance” does a noninstantaneous response typically occur for
Kerr-type nonlinearities. A noninstantaneous time response translates into a fre-
quency dependence for all the susceptibilities. An example of how a noninstanta-
neous response occurs is shown in Fig. 1.4 for a simple two-level model.
4 INTRODUCTION
As the excited-state electrons relax back to the ground state, the induced polarization
relaxes back to the ground-state polarization, leading to time evolution in both the
refractive index and the absorption coefficient, as illustrated in Fig. 1.5. The Fourier
transform of this time evolution gives the frequency response.
Equation 1.4 is not the one normally used because of its complexity. Assuming
plane waves of the form
Pið~r; tÞ ¼ 1
2Piðo; zÞ e�iot þ c:c:; Eið~r; tÞ ¼ 1
2
Xm
Emi ðom; zÞ e�iomt þ c:c:;
Emj ð�omÞ ¼ E*m
j ðomÞ ð1:7Þ
and expanding again in terms of the total field gives
Piðo; zÞ e�iot ¼ e0
�wð1Þij ð�o; omÞ
Pm Em
j ðom; zÞ e�iomt
þ 1
2
Xm
Xp
wð2Þijk ð�o; � om; � opÞEmj ð�om; zÞEp
kð�op; zÞ e�ið�om�opÞt
þ 1
4
Xm
Xp
Xq
wð3Þijk‘ð�o; � om;�op;�oqÞEmj ð�om; zÞEp
kð�op; zÞ
�Eq‘ ð�oq; zÞ e�ið�om�op�oqÞt þ � � �
�ð1:8Þ
Input pulse ω ≈ ωi
(a)
Excited state
Ground stateN0
N1
(b)
Δt
(c)
Excited state (lifetime τ )
Ground state
hωiτhωi
FIGURE 1.4 (a) Two-level model with all electrons initially in the ground state N0.
(b) Incidence of a short pulse (Dt � �t) causes many electron transitions to the excited state.
(c) Excited-state population after the pulse passes.
Absorption
(a)
t = 0 t = τ
t = 2Δt t τ
(b)
(c)
(d)
AbsorptionRefractive index Refractive index
ω − ωi ω − ωi
FIGURE 1.5 (a) Spectral distribution of refractive index and absorption before the incidence
of the pulse (t¼ 0). (b)–(d) Time evolution of refractive index and absorption: t ¼ 2Dt (b); t ¼ �t(c), and Dt �t (d).
CLASSICAL NONLINEAR OPTICS EXPANSION 5
In each case, o ¼ �om � op � oq is the output frequency generated by the
interaction. The hat (roof) superscript is meant to emphasize that the quantity
underneath is a complex number.
A key question is the order of magnitude of the nonlinear susceptibilities. The
simplest atom is hydrogen. Its structure and spectrum of excited states is well known
and is simple to calculate since it has only one electron, the minimum needed for the
interaction of electromagnetic radiation with matter. The atomic Coulomb field
binding the electron to the proton in its orbit of Bohr radius rB is given by
Eatomic ¼ �e
4pe0r2B; rB ¼ 4pe0�h2
�me�e2; ð1:9Þ
in which �e is the charge on the electron (�1.6� 10�19 C), e0 is the permittivity of free
space (8.85� 10�12 F/m), �me (¼9.11� 10�31 kg) is the electron mass, and
h ¼ 2p�h ¼ 6:63� 10�34 J s is Planck’s constant. Equation 1.9 gives the order of
magnitude of Eatomic ¼ 1012 V=m. It is reasonable to adopt this field as an approx-
imate field at which nonlinear optics becomes important. Since wð1Þ ¼ n2�1 (where n
is the refractive index of the order of unity) for a perturbation expansion in terms of
products of electric fields to be valid, P(1) 10�P(2):
Pð1Þ
Pð2Þ¼ wð1Þ
wð2ÞE� 1
wð2ÞEatomic
� 10 ! wð2Þ � 10�13 m=V: ð1:10Þ
This is a reasonable estimate for the lower limit value of the second-order suscep-
tibility, especially since the field was based on hydrogen, which has only a single
electron and proton. Following the same approximations but now assuming that
Pð1Þ
Pð3Þ¼ wð1Þ
wð3ÞE2� wð1Þ
wð3ÞE2atomic
� 10 ! wð3Þ � 10�25 m2=V2: ð1:11Þ
Aswill become clear later, these approximate values are close to the minimum values
found for these susceptibilities.
1.4 SIMPLE MODEL: ELECTRON ON A SPRINGAND ITS APPLICATION TO LINEAR OPTICS
There are many physical mechanisms that lead to nonlinear optical phenomena.
Initially, the focus here is on transitions between the electronic states associated with
atoms and molecules in matter. Although the appropriate treatment (Chapter 8) for
completely describing the interaction of radiation with atoms and molecules involves
quantum mechanics, initially a simpler classical approach that provides a useful
description of the linear (and as it turns out exact) susceptibility is adopted.
As an example of this approach, consider the molecule O2 and its electron cloud,
as illustrated in Fig. 1.6. This molecule has inversion symmetry (i.e., a center of
6 INTRODUCTION
symmetry halfway between the oxygen atoms) and hence has no permanent dipole
moment since the centers of positive (nuclei) and negative charges are coincident.
When an electric field
~Eð~r; tÞ ¼ 1
2ExðoaÞ eiðkz�oatÞ þ c:c: ð1:12Þ
is applied along the molecular axis (þ x-axis), the negative and positive charges and
their centers of charge are displaced in opposite directions by the Coulomb forces,
giving rise to the forces �me;n€xe;n, where �me and xe are the electron mass and its
displacement and �mn and xn are the nuclear mass and its displacement, respectively.
Since �mn �me, only the displacements of the electrons are important for inducing
dipoles.
The electrons are bound to the nucleus (atoms) or nuclei (molecules) by Coulomb
forces and, for isolated atoms or molecules, exist in discrete states m with an energy
�hð�om��ogÞ ¼ �h�omg above the ground state and an excited-state lifetime �tmg. They
move in “orbits” around nuclei described by probability density functions �cmð~r; tÞ,with j�cmð~r; tÞj2 dx dy dz dt giving the probability that the electron “exists” at time t in
the volume element dx dy dz at position ~r. Since “optics” usually deals with the
spectral region longer in wavelength (smaller in frequency) than the low frequency
absorption edge of the material determined by the transitions between electronic
states, the electron in the lowest lying energy level is normally the prime participant
when radiation interacts with matter; i.e., it is the electron with the largest displace-
ment. With these approximations, the dipole moment induced by an electromagnetic
field is �mx ¼ �eexe, as shown in Fig. 1.6. For themost general case, �mi ¼ �aijEj , where �aijis the polarizability tensor, and the induced dipole and the electric field are not
necessarily collinear.
In linear optics, it is possible to diagonalize the polarizability tensor. The
deflections~�q ¼ ex�qx þ ey�qy þ ez�qz of this representative electron are defined in terms
of these axes and so~�m ¼ ��e~�q. (Note, however, that for very anisotropic crystal classesthe coordinate system may be nonorthogonal and/or frequency dependent.)
The Coulomb interaction between the net positive and negative charges provides a
restoring force that oscillates at the frequency of the applied field, and so themotion of
→µ
→E
FIGURE 1.6 The oxygen molecule O2, its center of mass , and positive and negative
charges in the absence of the field. After the field is applied, is the center of the negative
charge and the electron cloud is shifted (and distorted) from its original position.
SIMPLE MODEL: ELECTRON ON A SPRING AND ITS APPLICATION TO LINEAR OPTICS 7
the electron can be described as a simple harmonic oscillator. In three dimensions, this
can be visualized as the electron attached to three orthogonal springs, as illustrated in
Fig. 1.7a, and the electron motion can be described as oscillation in a harmonic
potential well.
The equation of motion of an electron is described by a simple harmonic oscillator
with the potential
�VðmÞð�qÞ ¼ 1
2�kðmÞii �q
ðmÞi �q
ðmÞi : ð1:13Þ
From classical mechanics, the restoring force is given by
�FðmÞi ¼ � @ �V
ðmÞ
@�qðmÞi
¼ � 1
2�kðmÞii
@�qðmÞi
@�qðmÞi
�qðmÞi þ �q
ðmÞi
@�qðmÞi
@�qðmÞi
" #¼ ��k
ðmÞii �q
ðmÞi ; ð1:14Þ
in which the spring constant kðmÞii is defined in terms of the excited state’s energy by
�omg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðmÞii =�me
qand the restoring force is given by �F
ðmÞi ¼ �me �o2
mg�qðmÞi : The inertial
force is �me€�qðmÞi . Therefore, the force balance equation describing the electron motion
in a simple harmonic oscillator model is
�me
h€�qðmÞi þ 2�t�1
mg�qðmÞi þ �o2
mg�qðmÞi
i¼ ��eEi
! �o2mg�o2
a�2ioa�t�1mg
h i�qðmÞi ðtÞ ¼ � �e
�me
EiðtÞ:ð1:15Þ
Assuming
�qðmÞi ¼ 1
2�QðmÞi e�ioat þ c:c: ! �Q
ðmÞi ¼ � �eEiðoaÞ
�meDðmÞi ðoaÞ
; ð1:16Þ
(a) (b)
–
(qi)V
qiqy
qzqx
FIGURE1.7 (a) Electron connected via three springs oriented along the axes that diagonalize
the polarizability. (b) One-dimensional cut of the three-dimensional parabolic potential well
inside which the electron oscillates at the frequency of an applied field.
8 INTRODUCTION
where DðmÞi ðoaÞ ¼ �o2
a�2ioa�t�1mg þ �o2
mg is the resonance denominator. When
oa � �omg, the amplitude of the displacement is enhanced. Note that �ee ¼ ��e and
that in the zero-frequency limit, DðmÞi ðoaÞ ¼ �o2
mg and �QðmÞi ¼ ��eEiðoaÞ=�me �o2
mg is
just a net steady-state displacement of the electron.
For a dilute mediumwithN noninteracting atoms (molecules) per unit volume, the
induced linear polarization and the first-order susceptibility wð1Þii ð�oa; oaÞ are givenas follows:
PiðtÞ ¼ �N�eP
m
�1
2�QðmÞi e�ioat þ c:c:
�¼ 1
2PiðoaÞ e�ioat þ c:c:
PiðoaÞ ¼ N�e2
�me
EiðoaÞXm
1
DðmÞi ðoaÞ
¼ e0wð1Þii ð�oa;oaÞEiðoaÞ
! wð1Þii ð�oa; oaÞ ¼ N�e2
e0 �meðoaÞXm
1
DðmÞi ðoaÞ
:
ð1:17Þ
The fact that wð1Þij ð�oa; oaÞ is a diagonal tensor is a direct consequence of choosing acoordinate system in which the polarizability tensor is diagonal.
The first-order susceptibility wð1Þii ð�oa; oaÞ can easily be defined in terms of
wð1Þii ð~r�~r0; t�t0Þ. From Eq. 1.4,
Pð1Þi ðtÞ ¼ e0
ðwð1Þij ðt�t0ÞEjðt0Þ dðt�t0Þ
� �¼ 1
2Pð1Þi ðoaÞ e�ioat þ c:c: ð1:18Þ
Substituting for the field Ejðt0Þ gives
Pð1Þi ðtÞ ¼ e0
2
ðwð1Þij ðt�t0Þ½EjðoaÞ e�ioat
0 þ c:c:� dðt�t0Þ� �
¼ 1
2e0EjðoaÞ e�ioat
�
wð1Þij ðt�t0Þ eioaðt�t0Þ dðt�t0Þ þ c:c:
! wð1Þij ð�oa; oaÞ ¼Ð¥�¥ w
ð1Þij ðt�t0Þ eioaðt�t0Þ dðt�t0Þ;
ð1:19Þ
i.e., wð1Þij ð�oa; oaÞ is the Fourier transform of wð1Þij ðt�t0Þ.Decomposing wð1Þii ð�oa; oaÞ into its real and imaginary components yields
wð1Þii ð�oa; oaÞ ¼ N�e2
�mee0
Xm
ð�o2mg�o2
aÞþ 2ioa�t�1mg
ð�o2mg�o2
aÞ2 þ 4o2a�t
�2mg
: ð1:20aÞ
SIMPLE MODEL: ELECTRON ON A SPRING AND ITS APPLICATION TO LINEAR OPTICS 9
This equation is always valid. It can be simplified near and on resonance ðoa � �omgÞto give
wð1Þii ð�oa; oaÞ ¼ N�e2
2�omg �mee0
Xm
�omg�oa þ i�t�1mg
ð�omg�oaÞ2 þ�t�2mg
ð1:20bÞ
and off resonance ðjomg�opj�tmg 1Þ to give
wð1Þii ð�oa; oaÞ ¼ N�e2
2�omg �mee0
Xm
�omg�oa þ i�t�1mg
ð�omg�oaÞ2: ð1:20cÞ
Figure 1.8 shows the frequency dispersion in the imaginary and real parts of
wð1Þii ð�oa; oaÞ for a single excited state.
The refractive indexand theabsorptioncoefficient (for thefield) aredefined in theusual
way by n2ðoÞ ¼ 1þRealfwð1Þii ð�o; oÞg and aðoÞ ¼ kvac=magfwð1Þii ð�o; oÞg=2nðoÞ, respectively. Note that the absorption spectrum, i.e., aðoÞ, has contributions
only from transitions from theground state that are electric dipole allowed. For symmetric
molecules inwhich the states are described bywave functions that are either symmetric or
antisymmetric in space, the linear absorption spectrum does not contain contributions
from the even-symmetry excited states because electric dipole transitions from the even-
symmetry ground state are not dipole allowed.
As stated previously, optics normally refers to electromagnetic waves in the
spectral region defined by frequencies below the lowest lying electronic
resonance due to electric dipole transitions. Assuming that j�omg�oaj�t�1mg 1,
=magfwð1Þii ð�oa; oaÞg decreases faster with increasing frequency difference from
the resonance j�omg�oaj than does Realfwð1Þii ð�oa; oaÞg. This will also be the casefor the real and imaginary parts of the nonlinear susceptibilities.
1.5 LOCAL FIELD CORRECTION
Although local field correction is discussed in most introductory textbooks on optics,
it will prove useful to repeat it here since the transition to nonlinear optics is not
straightforward. The preceding analysis for the linear susceptibility was for a single
isolated atom or molecule and, to a good approximation, for a dilute gas. The
situation is more complex in dense gases or condensed matter (liquids and solids)
where the atoms and molecules interact with one another via the dipole fields
induced by an applied optical field.
(a) (b)
Imaginary
0
ωa − ωmg Real
ωa − ωmg
FIGURE 1.8 Spectral dispersion of the (a) imaginary and (b) real parts of wð1Þii ð�o; oÞ.
10 INTRODUCTION
Experiments are usually performed with an optical field incident onto a nonlinear
material from another medium, typically air. Maxwell’s equations in the material
and the usual boundary conditions at the interface are valid for spatial averages of
the fields over volume elements small on the scale of a wavelength, but large on the
scale of a molecule. The “averaged” quantities also include the refractive index, the
Poynting vector, and the so-called Maxwell field, which has been written here as~Eð~r; tÞ. It is the Maxwell field that satisfies the wave equation for a material with the
averaged refractive index n.
However, at the site of a molecule the situation can be quite complex since the
dipoles induced by the Maxwell electric fields on all the molecules create their own
electric fields, whichmust be added to the “averaged” field to obtain the total (“local”)
field~E locð~r; tÞ acting on a molecule, as shown in Fig. 1.9. In the low density limit, the
dipolar fields decay essentially to zero with distance from their source dipole and so~E locð~r; tÞ ffi ~Eð~r; tÞ and the single molecule result is converted to a macroscopic
polarization by multiplying the molecular result by N, the number of molecules per
unit volume.
The situation is more complex in condensed matter. It is very difficult to
calculate the “local” field accurately because it depends on crystal symmetry,
intermolecular interactions, and so on. Standard treatments such as Lorenz–Lorenz
are only approximately valid even for isotropic and cubic crystal media. Nevertheless,
they are universally used. Here the usual formulation found in standard
electromagnetic textbooks will be followed. The dipole moments of the molecules
induced by the Maxwell field ~Eð~r; tÞ produce a Maxwell polarization~Pð~r; tÞ ¼ e0~w
ð1Þ : ~Eð~r; tÞ in the material. Consider a spherical cavity around the
molecule of interest to find the local field acting on the molecule (see Fig. 1.9c).
Assuming that the effects of the induced dipoles inside the cavity average to zero, the
polarization field outside the cavity induces charges on the walls of the cavity, which
produce an additional electric field on themolecule in the cavity (see standard texts on
electrostatics):
X~Edipolesð~r; tÞ
D E¼ 1
3e0~Pð~r; tÞ ! ~E locð~r; tÞ ¼ ~Eð~r; tÞþ 1
3e0~P
ð1Þð~r; tÞ:ð1:21Þ
(a) (b) (c)
Moleculeof interest
P(r,t)Cavity
→ →
FIGURE 1.9 (a) The local fields created by the induced dipoles in a medium. (b) Dipoles
induced everywhere in the material. The average gives the Maxwell polarization ~Pð~r; tÞ.(c) Artificial spherical cavity assumed around the molecule of interest, embedded in a uniform
medium with polarization ~Pð~r; tÞ.
LOCAL FIELD CORRECTION 11
The induced dipole on a molecule at the center of the cavity is now given by
~�pð1Þ ð~r; tÞ ¼ ~~�a ��~Eð~r; tÞþ 1
3e0~P
ð1Þð~r; tÞ�
!~Pð1Þð~r; tÞ ¼ N~~�a �
�~Eð~r; tÞþ 1
3e0~P
ð1Þð~r; tÞ�
!�1� 1
3N~~�a
��~Pð1Þð~r; tÞ ¼ N~~�a �~Eð~r; tÞ: ð1:22Þ
From the Clausius–Mossotti relation that connects themacroscopic relative dielectric
constant to the molecular polarizability,
er�1
er þ 2¼ 1
3N�ah i ! ~P
ð1Þð~r; tÞ ¼ er þ 2
3N~~�a �~Eð~r; tÞ; ð1:23Þ
and so the local field and the local field correction f (1) is defined as
~E locð~r; tÞ ¼ erðoaÞþ 2
3~Eð~r; tÞ; f ð1Þ ¼ er þ 2
3; ð1:24Þ
respectively, where erðoaÞ ¼ eðoaÞ=e0. Since the field driving the electron is now
�QðmÞi ¼ �f ð1Þ
�eEiðoaÞ�meD
ðmÞi ðoaÞ
; ð1:25Þ
the linear susceptibility from Eq. 1.20a, including the local field correction, becomes
wð1Þii ð�oa; oaÞ ¼ N�e2f ð1ÞðoaÞ�mee0
Xm
ð�o2mg�o2
aÞþ 2ioa�t�1mg
ð�o2mg�o2
aÞ2 þ 4o2a�t
�2mg
: ð1:26Þ
PROBLEMS
1. The purpose of this problem is to show that absorption decreases much faster than
refractive index with frequency difference from a resonance. Consider an isolated
molecule with a single excited state with a transition frequency �oi and a
phenomenological decay constant �t�1i .
(a) Assuming that �t�1i �oi 1 and o � �oi, show that w(1) can be written as
wð1Þii ð�o; oÞ ¼ Realfwð1Þii gþ i =magfwð1Þii g
¼ Ne2ð�oi�oÞ2�oi �me0½ð�oi�oÞ2 þ�t�2
i �
þ iNe2o�ti�1
2�o2i �me0½ð�oi�oÞ2 þ�t�2
i � :
12 INTRODUCTION
(b) Find the maximum change in the real part of the susceptibility and show that
the ratioR of change at frequencyo tomaximum change occurs at a frequency
shift given by �oi�o ffi 2�t�1i =R for j�oi�oj 2�t�1
i .
ABR =
BA ω − ωi
ωi
(c) Find themaximum change in the imaginary part of the susceptibility and show
that the ratio P of the change at frequency o to maximum change is given by
P ¼ �t�2i
½ð�oi�oÞ2 þð�t�2i Þ� :
A
B
P =BA
ω − ωi
ωi
(d) How small is P for values of R equal to 10%: 1%?
Although you have calculated this difference for the linear susceptibility, the
results are also typical of what is obtained for higher order susceptibilities. A
10% or less “remnant” in the susceptibility is the upper limit for calling the
value “nonresonant.”
SUGGESTED FURTHER READING
B. I. Bleaney and B. Bleany, Electricity andMagnetism, 2nd Edition (Oxford University Press,
London, 1968).
M. Born and E. Wolf, Principles of Optics, 7th Edition (Cambridge University Press,
Cambridge, UK, 1999).
R. W. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, Burlington, MA, 2008).
F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Volume 1: Linear Optics
(John Wiley & Sons, New York, 1985).
D. L. Mills, Nonlinear Optics: Basic Concepts, 2nd Edition (Springer, New York, 1998).
Y. Ron Shen, Principles of Nonlinear Optics (John Wiley & Sons, New York, 1984).
SUGGESTED FURTHER READING 13