Nonlinear Optics: Basic Concepts

188 Pages·1991·3.428 MB·English

by  Professor D. L. Mills Ph.D. (auth.)

Nonlinear Optics
D.L.Mills
Nonlinear
Optics
Basic Concepts
With 32 Figures
Springer-Verlag
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London Paris Tokyo
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Professor D. L. Mills, Ph.D.
Department ofPhysics
University ofCalifomia
Irvine, CA 92717, USA
ISBN 978-3-662-00215-5 ISBN 978-3-662-00213-1 (eBook)
DOI 10.1007/978-3-662-00213-1
Library of Congress Cataloging-in-Publication Data.
MilIs, D.L.
Nonlinear optics: basic concepts/D.L.MilIs. p. cm.
Inc1udes bibliographical references and index.
ISBN 3-540-54192-6 (alk. paper) - ISBN 0-387-54192-6 (U-S.: alk. paper)
1. Nonlinear optics. 1. Title
QC446.2.M55 1991
535.2-dc20 9l-31420 CIP
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Preface
One intriguing aspect of physics is its dynamic and rapidly evolving
nature; exciting new fields can become moribund within relatively few
years, only to revive and grow again in a dramatic and expolisve manner
in response to new developments. This has been the case for the fields of
optics and atomic physics. In the 1950s, and perhaps into the early 1960s,
both fields appeared mature, fully developed, and perhaps even a bit dull
as a consequence. The appearance of the laser has tumed both of these
fields into dynamic areas of research, within which fundamental and
profound questions are being explored. The research of the past two or
three decades has led also to very important applications and to new
devices. The dye laser, which enables a very narrow line to be tuned over
an appreciable spectral range, has led to a virtual revolution in the spec
troscopy of atoms, molecules, and the condensed phases of matter.
A parallel development, readily detectable in the recent literature of
theoretical physics, has been the substantial advance in OUf understand
ing of highly nonlinear phenomena. Numerous texts are devoted to
exposition of the theoretical methods which may be used to extract
useful information from the important equations encountered in the
various fields of physics.
A survey of the contemporary literature of nonlinear optics shows
that in this area one encounters a large fraction of the basic equations
and principles of nonlinear physics. For example, analysis of self
induced transparency leads to the sine-Gordon equation, and that of
soliton propagation in optical fibers to the nonlinear Schrödinger equa
tion. Other examples can be found as welt As a consequence, one has in
hand real data which illustrate basic properties ofthose solutions ofthese
important equations with no counterpart in linear response theory or in
perturbation theoretic analyses of nonlinear terms. Other concepts
central to nonlinear optics, such as the role of phase matching in various
wave mixing experiments, are of very considerable importance in other
subfields ofp hysics. The field of nonlinear optics is thus a superb labora
tory within which the student may encounter and explore key notions of
nonlinear physics ofg eneral importance, while at the same time leaming
the foundations of a most important and fundamental area of con
temporary physics.
Furthermore, the above issues are not addressed in many introduc
tory graduate courses in electromagnetic theory; the emphasis is usually
VI Preface
placed on more classical topics developed in an earlier era. This volume
has its origin in a course given at Irvine by the author, directed toward
the student who has completed the first year electromagnetic theory
sequence. It is intended as an extension of interest not only to students
who wish to pursue thesis research in optics or laser spectroscopy copy
but also to the general student whose ultimate research specialty may lie
within a very different subfield. The focus is then on general principles,
with many technical points that are important to the specialist played
down or set aside. It is the author's experience that existing texts on
nonlinear optics use specialized terms that are not defm~d fully and
present introductions to important basic issues that are perhaps too
concise to be grasped easily by the general reader. The purpose of this
volume is to bridge the gap between the classic texts on electromagnetic
theory, which omit systematic exploration of modern optics, and the
(often excellent!) specialized texts full of discussion essential to those
who are pursuing research in the area, but which are too detailed and too
terse for the general student of physics.
The specialist may thus fmd important topics omitted from this
volume, or discussions of a number of technical points a bit incomplete.
The intention here is to provide an overview; the literature on nonlinear
optics is sufficiently complete and accessible that those who wish to
pursue particular aspects in more detail can proceed further without
difficulty.
It is important also for the reader to acquire an understanding ofthe
optical properties of various solid materials, to appreciate the reasons for
the choice of sampies for a given experiment, and the constraints that
limit one's ability to explore various phenomena. Also, one must acquire
a certain vocabulary in this arena to understand the literature. Chapter 2,
which is quite lengthy, presents an overview ofthe optical properties of
materials, since it is the experience of the author that many students
have a very limited grasp of this area, though they may know the mathe
matical details associated with various models of matter.
It is hoped that this volume will broaden the horizons of graduate
students in the physical sciences, by introducing them to the fascinating
field of nonlinear optics, and at the same time provide them with an
introduction to general aspects of the physics of nonlinear systems.
Irvine, CA D.L.Mills
March 1991
Contents
1. Introductory Remarks ....................................... 1
Problems.................................................... 6
2. Linear Dieleetrie Response of Matter ....................... 7
2.1 Frequeney Dependenee of the Dieleetric Tensor. . . . . . . . . . 11
2.2 Wave Veetor Dependenee of the Dielectrie Tensor. . . . . .. 27
2.3 Electromagnetie Waves in Anisotropie Dieleetrics. . . . . . . . . 30
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
3. Nonlinear Dielectrie Response of Matter.... ..... .. . . . . ... ... 37
3.1 Frequeney Variation of the Nonlinear Suseeptibilities . . . .. 39
3.2 Wave Veetor Dependenee
of the Nonlinear Suseeptibilities. . .. .. .. .. .. . . . . . . . . .. . . .. 45
3.3 Remarks on the Order of Magnitude
of the Nonlinear Susceptibilities. . . . . .... . . . . . . . . . . . . . . . .. 47
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
4. Basie Principles of Nonlinear Wave Interaetions:
Seeond "Harmonie Generation and Four Wave Mixing . . . . . . .. 51
4.1 Perturbation Theoretie Analysis
of Seeond Harmonie Generation.. . .. .. .. . .. .. .. .. . . . . .. . 51
4.2 Methods of Aehieving the Phase Matching Condition..... 57
4.3 Evolution of the Seeond Harmonie Wave
under Phase Matehed Conditions ........................ 59
4.4 Other Examples of Nonlinear Wave Interaetions. . . . . . . . .. 63
4.4.1 Four Wave Mixing Speetroseopy.................... 63
4.4.2 Optical Phase Conjugation. . . . . . . . . . . . . . . . . . . . . . . . .. 69
Problems.................................................... 71
5. Inelastie Scattering of Light from Matter:
Stimulated Raman and Brillouin Scattering . . . . . . . . . . . . . . . . . . 73
5.1 Quantum Theory of Raman Scattering .. . . . . . . . . . . . . . . . . . 77
5.2 Stimulated Raman Effeet................................. 82
5.3 Contribution to Four Wave Mixing
from the Raman Nonlinearity ............................ 84
VIII Contents
5.4 Brillouin Scattering of Light. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Problems.................................................... 91
6.Interaction of Atoms with Nearly Resonant Fields:
Self-Induced Transparency .................................. 93
6.1 Description of the Wave Function
under Near Resonant Conditions. . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Bloch Equations: Power Broadening
and Saturation Effects in Absorption Spectra. ............. 101
6.3 Self-Induced Transparency ............................... 107
6.4 Area Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112
6.5 Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 117
7. Self-Interaction Effects in One-Dimensional
Wave Propagation: Solitons in Optical Fibers
and in Periodic Stmctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119
7.1 Normal Modes of Optica1 Fibers ......................... 121
7.2 Nonlinear Schrödinger Equation .......................... 128
7.3 Linear Theory of Pulse Propagation
in a Dispersive Medium: Application to Optica1 Fibers . . .. 132
7.4 Solitons and the Nonlinear Schrödinger Equation ......... 135
7.5 Gap Solitons in Nonlinear Periodic Structures ............ 139
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
8. Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 155
8.1 Duffmg Oscillator: Transition to Chaos ................... 156
8.2 Routes to Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 161
8.3 Experimental Observations of Chaos in Optica1 Systems .. 162
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165
Appendix A: Skucture of the Wave Vector
and Frequency Dependent Dielectric Tensor. ...... 167
Appendix B: Aspects of the Sine-Gordon Equation ............. 175
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183
1. Introductory Remarks
Throughout most of this volume, we shall be describing the interaction of elec
tromagnetic radiation with matter within a macroscopic framework; we shall
always be concerned with electric fields whose spatial variation involves length
scales very large compared to the size of the atoms and molecules that are the
basic constituents of the material of interest. In this regard, we note that in the
visible portion of the spectrum, the wavelength of light is roughly 5 X 10-5
cm, while a typical molecular bond length or crystal lattice constant is 3 x
10-8 cm. We may then proceed by studying the solutions of Maxwell's equa
tions, which have the form
v . D = 0, V . B = 0 , (1.1 a)
1 iJB 1 iJD
V x E = ---, VxH=--. (1.1 b)
c iJt c iJt
We assume no external charge is present, and no external current is present as
weIl, so we set p = J = O.
To proceed, we require relations between the various fields which enter
Maxwell's equations. In the limit of interest, we have [1.1]
D = E + 47TP (1.2 a)
and
B = H + 47TM, (1.2b)
where P and M are the electric dipole moment per unit volume, and magnetic
moment per unit volume, respectively. We shall confine our attention here
entirely to nonmagnetic media, and set M == O. Nonlinearities very similar to
those explored here have their origin in the magnetic degrees of freedom of
appropriate materials. Their study constitutes the very important field of mag
netooptics, a topic treated elsewhere [1.2].
We then have
V . E + 47TV . P = 0 (1.3 a)
and with B = H, the two remaining Maxwell equations are easily combined
to give
V 2E - V(V . E) - -1 -iJ2E - -47TiJ-2p = 0 . (1.3 b)
c2 iJt2 c2iJt2
2 1. Introductory Remarks
To proceed, we require information on the relationship between P and E.
In principle, one requires a full microscopic theory of the response of a par
ticular material to relate E, the macroscopic electric field [1.1], to the dipole
moment per unit volume.
It would appear that by our neglect of the term involving the current density
J on the right-hand side of the V x H Maxwell equation, our attention will be
confined exclusively to insulating materials, rather than conductors such as
metals or doped semiconductors. We shall see shortly that this is not the case;
the conduction currents stimulated by time varying electric fields can in fact
be viewed as contributing to the dipole moment per unit volume, as we shall
come to appreciate. It is the case, however, that many of the nonlinear optical
phenomena to be explored here require that the various beams involved have
a long path length in the medium of interest. Thus, our attention will be di
rected largely toward applications to insulating materials that are nominally
transparent at the relevant wavelengths. The reader should keep in mind that
one encounters very interesting nonlinear optical phenomena in metals and in
doped semiconductors, as discussed in the text by Shen [1.3].
One may proceed by noting the following: The largest electric fields en
countered in practice fall into the range of 106 V I cm; most forms of matter
exhibit electrical breakdown for fields in excess of this value. An electron bound
to an atom or molecule, or moving through asolid or dense liquid, experiences
electric fields in the range of 109 V Icm. This follows by noting that, over
distances the order of an angstrom, the change in electrostatic potential can be
several electron volts. The laboratory fields of interest are then small compared
to the electric fields experienced by the electrons in the atoms and molecules
from which dense matter is constructed. In this circumstance, we can expand
the dipole moment per unit volume in a Taylor series in powers of the mac
roscopic field E. For the moment, we suppose the dipole moment per unit
volume P(r, t), depends on the electric field E at the same point r, and at the
same time. We shall see shortly that in real materials, this assumption is overly
restrictive. There are important qualitative implications of a more realistic re
lation between P and E. But this simple assertion will allow us to begin the
discussion. Then the ath Cartesian component of the dipole moment per unit
volume, P a(r, t), is a function of the three Cartesian components of the electric
field, Eß(r, t). Here a and ß range over x, y, and z. The Taylor series then
takes the form
(1.4)
In most of the common materials we encounter, the first term, which is the
electric dipole moment per unit volume in zero electric field, vanishes iden-
I. Introductory Remarks 3
tically. The usual situation is a dielectric material, within which the dipole
moment in zero field vanishes; any dipole moment present is then induced by
the external field.
There are an important dass of materials, known as ferroelectrics, which
possess a spontaneous electric dipole moment in zero field. These are electrical
I
analogues of the better known ferromagnets, which possess a spontaneous mag
netization per unit volume M. In a ferromagnet, as the material is heated, the
spontaneous magnetization M decreases in magnitude, to vanish at a certain
temperature Tc known as the Curie temperature. Ferroelectrics behave in a sim
ilar manner; the polarization P will decrease with increasing temperature, again
to vanish above a critical temperature.
In the common ferroelectrics, the origin of the spontaneous dipole moment
is in the shift of an ion from a high symmetry site to a low symmetry site as
the temperature is lowered. For example, in the ferroelectric material BaTi03,
in the high temperature phase for which p~) = 0, the Ti2+ ion sits at the center
of an oxygen octahedron. As the temperature is lowered below the critical tem
perature, the Ti2+ shifts off this high symmetry site, so the Ti-O complex
acquires a net electric dipole moment. This is a collective phenomenon; all the
Ti2+ ions shift in a coherent manner, so the crystal as a whole acquires an
electric dipole moment. Similarly, in KH2P04, it is the H that shifts.
Such materials are influenced strongly by dc or low frequency electric fields,
particularly near their transition temperature, because the unstable ionic species
respond strongly to an applied electric field. Ferroelectrics are thus useful in
a variety of electro-optic devices.1
In a ferroelectric, the dipole moment per unit volume, in the absence
p(O),
of an electric field, is independent of time, but may vary with position in the
sampIe. In general, the presence of a nonzero polarization leads to the presence
of a static, macroscopic electric field, This static field obeys the two
E(O)(r).
equations
V X E(O) = 0 (1.5 a)
and
V . E(O) = -47TV. p(O) • (1.5b)
Such fields may be analyzed by the methods of electrostatics [1.4]. We may
write, from (1.5 a), = -Vc/J(O), and (1.5 b) becomes Poisson's equation
EO)
with effective charge density Pp = - V . p(O).
Our interest will be in the study of various time-dependent phenomena, in
response to externally applied electric fields, usually with their origin in in
cident laser radiation. Thus the first term in (1.4) and the static fields it may
generate in a ferroelectric are of little interest. The second and subsequent terms
I In the ferroelectric phase, the materials necessarily lack an inversion center. They then exhibit
the phenomenon of piezoelectricity, a property that is most useful in a variety of device appli
cations.