handbook of nonlinear optics

Handbook ofNonlinearOpticsSecond Edition, Revised and Expanded

Richard L SutherlandScience Applications International Corporation

Dayton, Ohio, U.S.A.

with contributions by

Daniel G. McLeanScience Applications International Corporation

Dayton, Ohio, U.S.A.

Sean KirkpatrickAir Force Research Laboratory

Wright Patterson Air Force Base, Ohio, U.S.A.

M A R C E L

MARCEL DEKKER, INC. NEW YORK BASELffi

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

OPTICAL ENGINEERING

Founding EditorBrian J. Thompson

University of RochesterRochester, New York

Editorial Board

Toshimitsu Asakura Nicholas F. BorrelliHokkai-Gakuen University Corning, Inc.Sapporo, Hokkaido, Japan Corning, New York

Chris Dainty Bahrain JavidiImperial College of Science, University of Connecticut

Technology, and Medicine Storrs, ConnecticutLondon, England

Mark Kuzyk Hiroshi MurataWashington State University The Furukawa Electric Co , Ltd.

Pullman, Washington Yokohama, Japan

Edmond J. Murphy Dennis R. PapeJDS/Umphase Photomc Systems Inc.

Bloomfield, Connecticut Melbourne, Florida

Joseph Shamir David S. WeissTechnion-Israel Institute Heidelberg Digital L.L.C.

of Technology Rochester, New YorkHafai, Israel


1. Electron and Ion Microscopy and Microanalysis: Principles and Ap-plications, Lawrence E. Murr

2. Acousto-Optic Signal Processing: Theory and Implementation, editedby Norman J. Berg and John N. Lee

3. Electro-Optic and Acousto-Optic Scanning and Deflection, MiltonGottlieb, Clive L. M. Ireland, and John Martin Ley

4. Single-Mode Fiber Optics: Principles and Applications, Luc B. Jeun-homme

5. Pulse Code Formats for Fiber Optical Data Communication: BasicPrinciples and Applications, David J. Morris

6. Optical Materials: An Introduction to Selection and Application, Sol-omon Musikant

7. Infrared Methods for Gaseous Measurements: Theory and Practice,edited by Joda Wormhoudt

8. Laser Beam Scanning: Opto-Mechanical Devices, Systems, and DataStorage 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. White11. Laser Spectroscopy and Its Applications, edited by Leon J. Rad-

ziemski, Richard W. Solarz, and Jeffrey A. Paisner12. Infrared Optoelectronics: Devices and Applications, William Nunley

and J. Scott Bechtel13. Integrated Optical Circuits and Components: Design and Applications,

edited by Lynn D. Hutcheson14. Handbook of Molecular Lasers, edited by Peter K. Cheo15. Handbook of Optical Fibers and Cables, Hiroshi Murata16. Acousto-Optics, Adrian Korpel17. Procedures in Applied Optics, John Strong18. Handbook of Solid-State Lasers, edited by Peter K. Cheo19. Optical Computing: Digital and Symbolic, edited by Raymond Arra-

thoon20. Laser Applications in Physical Chemistry, edited by D. K. Evans21. Laser-Induced Plasmas and Applications, edited by Leon J. Rad-

ziemski and David A. Cremers22. Infrared Technology Fundamentals, Irving J. Spiro and Monroe

Schlessinger23. Single-Mode Fiber Optics: Principles and Applications, Second Edition,

Revised and Expanded, Luc B. Jeunhomme24. Image Analysis Applications, edited by Rangachar Kasturi and Mohan

M. Trivedi25. Photoconductivity: Art, Science, and Technology, N. V. Joshi26. Principles of Optical Circuit Engineering, Mark A. Mentzer27. Lens Design, Milton Laikin28. Optical Components, Systems, and Measurement Techniques, Rajpal

S. Sirohi and M. P. Kothiyal29. Electron and Ion Microscopy and Microanalysis: Principles and Ap-

plications, Second Edition, Revised and Expanded, Lawrence E. Murr


30. Handbook of Infrared Optical Materials, edited by Paul Klocek31. Optical Scanning, edited by Gerald F. Marshall32. Polymers for Lightwave and Integrated Optics: Technology and Ap-

plications, edited by Lawrence A. Homak33. Electro-Optical Displays, edited by Mohammad A. Karim34. Mathematical Morphology in Image Processing, edited by Edward R.

Dougherty35 Opto-Mechamcal Systems Design: Second Edition, Revised and Ex-

panded, Paul R. Yoder, Jr.36. Polarized Light: Fundamentals and Applications, Edward Colleti37. Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J. F.

Digonnet38. Speckle Metrology, edited by Rajpal S. Sirohi39. Organic Photoreceptors for Imaging Systems, Paul M. Borsenberger

and David S. Weiss40. Photonic Switching and Interconnects, edited by Abdellatif Marrakchi41. Design and Fabrication of Acousto-Optic Devices, edited by Akis P.

Goutzoulis and Dennis R. Pape42. Digital Image Processing Methods, edited by Edward R Dougherty43. Visual Science and Engineering: Models and Applications, edited by D,

H. Kelly44. Handbook of Lens Design, Daniel Malacara and Zacarias Malacara45. Photonic Devices and Systems, edited by Robert G. Hunsperger46. Infrared Technology Fundamentals: Second Edition, Revised and Ex-

panded, edited by Monroe Schlessinger47. Spatial Light Modulator Technology: Materials, Devices, and Appli-

cations, edited by Uzi Efron48. Lens Design: Second Edition, Revised and Expanded, Milton Laikin49. Thin Films for Optical Systems, edited by Frangois R. Flory50. Tunable Laser Applications, edited by F. J. Duarte51. Acousto-Optic Signal Processing: Theory and Implementation, Second

Edition, edited by Norman J. Berg and John M. Pellegrino52. Handbook of Nonlinear Optics, Richard L. Sutherland53. Handbook of Optical Fibers and Cables: Second Edition, Hiroshi

Murata54. Optical Storage and Retrieval. Memory, Neural Networks, and Fractals,

edited by Francis T. S. Yu and Suganda Jutamutia55. Devices for Optoelectronics, Wallace B. Leigh56. Practical Design and Production of Optical Thin Films, Ronald R,

Willey57. Acousto-Optics: Second Edition, Adrian Korpel58. Diffraction Gratings and Applications, Erwin G. Loewen and Evgeny

Popov59. Organic Photoreceptors for Xerography, Paul M. Borsenberger and

David S. Weiss60 Characterization Techniques and Tabulations for Organic Nonlinear

Optical Materials, edited by Mark Kuzyk and Carl Dirk


61. Interferogram Analysis for Optical Testing, Daniel Malacara, ManuelServin, and Zacarias Malacara

62. Computational Modeling of Vision: The Role of Combination, WilliamR. Uttal, Ramakrishna Kakarala, Sriram Dayanand, Thomas Shepherd,Jagadeesh Kalki, Charles F. Lunskis, Jr., and Ning Liu

63. Microoptics Technology: Fabrication and Applications of Lens Arraysand Devices, Nicholas F. Borrelli

64. Visual Information Representation, Communication, and Image Pro-cessing, 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. Tyson68. Entropy and Information Optics, Francis T. S. Yu69. Computational Methods for Electromagnetic and Optical Systems,

John M. Jarem and Partha P. Banerjee70. Laser Beam Shaping: Theory and Techniques, edited by Fred M. Dick-

ey and Scott C. Holswade71. Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Re-

vised and Expanded, edited by Michel J. F. Digonnet72. Lens Design: Third Edition, Revised and Expanded, Milton Laikin73. Handbook of Optical Engineering, edited by Daniel Malacara and Brian

J. Thompson74. Handbook of Imaging Materials, edited by Arthur S. Diamond and Da-

vid S. Weiss75. Handbook of Image Quality: Characterization and Prediction, Brian W.

Keelan76. Fiber Optic Sensors, edited by Francis T. S. Yu and Shizhuo Yin77. Optical Switching/Networking and Computing for Multimedia Systems,

edited by Mohsen Guizani and Abdella Battou78. Image Recognition and Classification: Algorithms, Systems, and Appli-

cations, edited by Bahram Javidi79. Practical Design and Production of Optical Thin Films: Second Edition,

Revised and Expanded, Ronald R. Willey80. Ultrafast Lasers: Technology and Applications, edited by Martin E.

Fermann, Almantas Galvanauskas, and Gregg Sucha81. Light Propagation in Periodic Media: Differential Theory and Design,

Michel Neviere and Evgeny Popov82. Handbook of Nonlinear Optics, Second Edition, Revised and Ex-

panded, 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 andphenomena of visible and invisible light, has generated a wealth of knowledgethat makes its use pervasive in other physical sciences, biology, medicine,forensics, agriculture, art, industry, and the military. This has spawned atechnology called photonics, a name based on the quantum of energy in theelectromagnetic field, the photon. The domain of photonics extends from energygeneration to detection to communications and information processing, andincludes all means of generating and harnessing light for useful purposes.

Both the science and technology aspects of optics have and continue to bevastly influenced by the field of nonlinear optics. It is a discipline that has enhancedour understanding of fundamental light-matter interactions as well as providedthe means for accomplishing a variety of engineering tasks. The purpose of thisbook 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 serveas a ready source of information useful to those researchers performingcharacterization of nonlinear materials, using the methods of nonlinear optics inscientific studies, and exploiting nonlinear optical phenomena in photonics

This edition of the Handbook of Nonlinear Optics has been updated andnew material has been added It is evident from a perusal of the scientificliterature that advances in nonlinear optics continue at a rapid pace. For example,frequency conversion in new bulk and quasi-phase-matched materials as well asthe development of new optical parametric oscillators are areas in which progressis continuing. Ultrafast optics and the sub-picosecond domain of opticalcharacterization offer interesting and challenging avenues for probing theproperties of materials and developing new applications. Furthermore, newtechniques are continually being developed to measure and modify the propertiesof materials for diverse applications such as optical limiting, nonlinearfluorescent imaging, and two-photon photopolymenzation


As in the previous edition, selection of topics for inclusion was based on acertain bias for what has been important to me as a general practitioner ofnonlinear optics In this regard, I have chosen to add work done in my group thatis relevant primarily to the characterization and application of nonlinearmaterials However, in the interest of properly setting the stage for the bulk of thebook, and because it so often seems to be a point of confusion for beginners, Ihave expanded the first chapter, which deals with elements of nonlinear opticaltheory Chapter 2, "Frequency Doubling and Mixing," and Chapter 3, "OpticalParametric Generation, Amplification, and Oscillation," so important in thegeneration of light for other nonlinear optics applications, have been expandedand updated primarily to include new results reported in the literature Chapters 6("Nonlinear Index of Refraction") 7 ("Characterization of Nonlinear RefractiveIndex Materials"), 9 ("Nonlinear Absorption"), and 10 ("ExperimentalTechniques in Nonlinear Absorption') all incorporate new material Several ofthe chapters tabulating materials data (Chapters 5, 8, and 13) have also beenupdated Chapter 13 replaces Chapter 11 in the previous edition Two newchapters (Chapter 11, "Ultrafast Characterization Techniques," and Chapter 12,' Laser Flash Photolysis") have been added, covering important topics in theexpanding characterization requirements of nonlinear materials Finally Chapter17, "Electro-Optic Effects," has been added because the effect plays such acentral role in several devices used in optics, as well as in the photorefractiveeffect, and because it is arguably a nonlinear effect, depending as it does on theinteraction of two or more electric fields

This second edition also afforded the opportunity to correct errors andmisprints that occurred in the first edition My gratitude goes to those who havegraciously pointed these out to me

As always, I am indebted to several people who have been of great help inpreparing this work Not the least of these is my family, which has stood besideme with patience and support I would also like to thank my employer, SAIC, andthe U S Air Force Research Lab (AFRL/MLPJ) for their encouragement of thisproject Finally, I acknowledge my colleagues for their helpful advice andcriticism, especially Scan Kirkpatrick and Daniel G McLean, who authoredChapters 11 and 12, respectively, and Suresh Chandra, who contributed toChapter 3

Richard L Sutherland


Preface to the First Edition

Shortly after the demonstration of the first laser in 1960, Peter Frankin andcoworkers ushered in nonlinear optics (NLO) with the observation of secondharmonic generation in a quartz crystal. Since then, NLO has burgeoned into amature field of science and engineering. The scope of this discipline includes allphenomena in which the optical parameters of materials are changed withirradiation by light. Generally, this requires high optical intensities, which is themain reason that NLO matured in parallel with laser technology. Judging by thegrowth and continued good health of publications and international conferenceson the subject, NLO appears to have a strong future in areas of photonics devicesand scientific investigations.

The impact of NLO on science and technology has been twofold. First, ithas enhanced our understanding of fundamental light-matter interactions.Second, it has been a driving force in the rejuvenation of optical technology forseveral areas of science and engineering. NLO has matured in the sense of being awell-developed and systematic theory as well as providing applications for avanety of engineering tasks. Second and third order phenomena and devices arenow at a stage of understanding and development such that a coherent descriptionand summary of these areas forming the core of the subject are now possible anddesirable.

The rapid development of the subject has created the need for ahandbook that summarizes technical details concerning core areas impactingseveral engineering and scientific endeavors. The general practitioner of NLOrequires information in at least four critical areas: (1) mathematical formulasapplicable to a variety of experimental and design situations, (2) examples ofways NLO is applied to specific technical problems, (3) a survey of device andmatenals data for comparison purposes and numerical evaluation of formulas,and (4) in-depth descriptions of methods required for characterizing newmatenals. When seeking this information, novice and expert alike are often


bewildered by a lack of continuity in style notation, content, and physicalunits contained m the literature Textbooks tend to develop the subject indepth, with an emphasis on pedagogical style and with considerablemathematical detail This inherently limits the scope of the material coveredUseful results are scattered throughout the text, usually without any helpfulsummary of important and useful formulas Moreover, discussions ofapplications and experimental methods as well as materials and device data,are often sparse When seeking information, what a practicing scientist orengineer (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 aredated and some lack continuity m style, nomenclature, and use of physicalunits, primarily because of multiple authorship Some are rich in materialsdata but are lacking in the other four areas I describe above Finally, sometreat a limited scope of phenomena, such as only second order effects or asingle application area What is needed is a balanced treatment of both secondand third order NLO covering areas useful to the practicing scientist andengineer

The purpose of this book is to fulfill this need by providing a ready sourceof information to applied scientists, engineers, students, and others interested inthe applications of NLO Important formulas, experimental methods, andmaterials data are summarized in the form of a handy reference to several aspectsof the field The scope of the book includes experimental applications and adiscussion of devices

This book is an outgrowth of my years as a general practitioner ofNLO As the leader of an optical characterization group for optical limitingapplications, I have been involved in the tasks of conceptualizingdevices based on NLO, searching for materials to transform the conceptsto practice characterizing the nonlinear properties of materials, and testingprototypes Also, on occasion I have had the opportunity to teach a graduate-level course in NLO The material for this book was gathered from mylecture notes, on-the-job experience, and specific research directed towardthis work It is my intention that the contents should largely fulfil] the needsof those researchers performing characterization of nonhneai materials, usingNLO methods in scientific studies and exploiting NLO phenomena inphotomcs devices

The richness and vitality of NLO dictate against a fully comprehensivetreatment at any given point in time, simply not everything can be covered Iadmit to a certain bias, and the material given here is largely what has been ofimportance to me as a general practitioner of NLO Therefore, this workconcentrates on what I consider the core of the subject, including second orderphenomena involving frequency conversion, and the third order phenomena of


nonlinear phase modulation, nonlinear absorption, and nonlinear scattering.The book treats technologically significant phenomena and presents asummary of important formulas useful in the understanding and applicationof NLO. Succinct physical interpretations of the mathematics are also given,with an emphasis on conceptual understanding. Experimental methods forcharactenzing nonlinear parameters are described for both second and thirdorder materials. A discussion of well-accepted as well as novel, less well-known methods is included. Finally, technical data on selected materials arealso 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 Istrive to use consistently in the book. First, an optical wave propagating in spaceand time is a real quantity and is represented by a real mathematical expression.The simplest sinusoidal wave has the form

EH(r,0=A'cos(kT-cor)

In NLO, the product of two or more waves appears in many formulas, and it isthus 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 isevident from this expression that the product of several waves will involve 1/2raised to some power. It is common to avoid this by suppressing the factor of 1/2and 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 opticalwaves throughout this book. This is important to note because the use of the othercomplex form of the field would lead to a different numerical prefactor in thedefinition of optical intensity, the key parameter connecting theory tomeasurements

Another important note is the definition of scalars, vectors, and tensorsBoth vectors and tensors are presented as bold symbols, such as E. Whether thesymbol represents a vector or a tensor should be obvious from the context. Scalarquantities are represented by nonbold symbols.

Both SI and cgs (esu) systems of units are used as much as possiblethroughout the book. The system of units used in formulas is also often a sourceof much confusion in NLO Therefore, care has been taken to present formulas in


both sets of units as much as possible, the units of measure in both systems aregiven for key physical parameters, and conversion formulas between the twosystems are summarized

Chapter 1 introduces elements ot NLO theory It is useful to study thischapter to acquaint oneself with the notation used throughout this book as well asto become familiar with the underlying principles ot the subject Chapters 2 and 3deal with second order NLO phenomena These include frequency conversionand optical parametric phenomena Topics covered range from the operation ofideal devices to realistic optical beams interacting in nomdeal materials Variousaspects such as phase matching in umaxial and biaxial crystals, effectivenonlinear coefficients, temporal effects, tuning, bandwidth, and the effects ofabsorption and diffraction are discussed The materials characterizationtechniques for second order NLO coefficients is the subject of Chaptei 4 Thisis followed in Chapter 5 by a tabulation ot second order NLO parameters forseveral selected materials

The remainder of the book is devoted to third order NLO Chapters 6and 9 discuss nonlinear refraction and nonlinear absorption, respectively Asummary of different physical mechanisms contributing to these phenomenais given Detailed discussions of important applications are includedExperimental techniques for characterizing the respective nonlmeanties inmaterials form the subjects ot Chapter 7 and 10 Materials data are tabulatedfor a vanet> of gases liquids, solutions, and solids in Chapters 8 and 11Nonlinear scattering (stimulated Raman and stimulated Bnlloum scattering) istreated in Chapters 12 and 13 Brief descriptions of the utilization of thesephenomena for frequency conversion and optical beam control are givenFinally, materials data relating to these phenomena are presented inChapter 14

In compiling the information for this book, I have felt like the proverbialdiscoverer standing on the shoulders of giants I am much indebted to thecountless number of researchers who have paved the way for the rest of us andgave the time to so adequately document their results with great detail andinsight 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 interactingthrough workshops or on site \isits at Wright Patterson Air Force Base ToElsa Garmire, Tony Ganto Hyatt Gibbs, Art Smirl, M J Soileau, GeorgeStegeman, and Eric Van Stryland I give thanks for insightful discussions Inaddition, I would like to thank my colleagues at SAIC particularly DanMcLean, Bob Ephng, Paul Fleitz, and Lalgudi Natarajan, for theircontributions to this work My thanks also to the staff of Marcel Dekker,Inc , for inviting me to write this book and encouraging itscompletion, 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


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


ContentsPreface to the Second EditionPreface to the First Edition

1. Elements of the Theory of Nonlinear Optics

2. Frequency Doubling and Mixing

3 Optical Parametric Generation, Amplification, and Oscillation

4. Characterization of Second Order Nonlinear Optical Materials

5. Properties of Selected Second Order Nonlinear Optical Materials

6. Nonlinear Index of Refraction

7. Characterization of Nonlinear Refractive Index Materials

8. Optical Properties of Selected Third Order Nonlinear OpticalMaterials

9. Nonlinear Absorption

10. Experimental Techniques in Nonlinear Absorption

11. Ultrafast Characterization TechniquesScan Kirkpatrick

12. Laser Flash PhotolysisDaniel G. McLean


13 Nonlinear Absorption Properties of Selected Materials

14 Stimulated Raman Scattering

15 Stimulated Bnlloum Scattering

16 Properties of Selected Stimulated Light-Scattering Materials

17. Electro-Optic Effects


1Elements of the Theory of NonlinearOptics

Optics is an important part of everyday life. Light seems to ow or propagate

through empty space, as well as through material objects, and provides us with

visual information about our world. The familiar effects of reection, 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 frommonochromatic 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


the electronic polarization of the material induced by light. Extending the

denition 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. ELECTROMAGNETIC BASIS OF OPTICS

A. The Optical Electric Field

Light is an electromagnetic wave. It consists of electric and magnetic elds, E (,)

and H (,), respectively. The superscripted tilde (,) implies that the elds arerapidly varying in time, and the elds are real quantities. For most of optics, the

optical wave may be characterized by dening its electric eld. (The magnetic

eld is related to the electric eld through Maxwells 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 eld is a wave

propagating primarily in one direction in space. Allowance may be made for a

nite amount of beam spreading, or diffraction. This primary direction of

propagation is usually taken to be the along the z-axis. (For noncollinearpropagation 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 formof the electric eld wave is given by

E,r; t e^Ar; texpikz2 vt c:c: 1In this equation, k is the wave vector of propagation and v is the circularfrequency 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 eld vector) is given by the unit vector e. 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 specied. Finally, the notation c.c. implies

complex conjugate. It is included in the denition of Eq. (1) since the eld E (,) is

a real quantity.

Chapter 12


For a large number of problems in linear and nonlinear optics, the eld can

be assumed to be of innite extent and constant in amplitude and phase in a plane

transverse to the direction of propagation. Thus, the complex eld amplitude

becomes a function of z and t only: A(z,t ). Such a wave is called an innite planewave, or sometimes just a plane wave. Certainly this is only an approximation

since real laser beams have a nite transverse extent and vary spatially along the

transverse direction.

A common form of a nite beam is the TEM00 mode of a circular Gaussian

beam. The eld of this type of wave has the following form.

E,r; t e^Az; t w0wzexp i

kr 2

2qz kz2 tan21 z

zR

2vt

c:c: 2

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 isdened 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, dened by w0, at the plane z 0; and w(z ) is given by

wz w0 1 zzR

2" #1=23

The diameter of the beam at z 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 innite radius of curvature, and hence mimics a plane

wave. For large distances away from the waist, the radius of curvature is,z. Thequantity q(z ) is called a complex radius of curvature and is given by

qz z2 izR 4Finally, the quantity zR is called the Rayleigh range and is dened by

zR npw20

l5

where n is the index of refraction of the medium, and l is the optical wavelengthin free space. The Rayleigh range corresponds to the distance from the waist at

which the beam radius increases by a factor of2

p: The distance between the

points ^zR about the waist is called the confocal parameter b of the beamb 2zR: These parameters are also dened schematically in Fig. 1a.

B. Electric Polarization in a Dielectric Medium

When an electric eld is applied to a dielectric medium (of neutral electric

charge), a separation of bound charges is induced as illustrated in Fig. 2. This

Elements of the Theory of Nonlinear Optics 3


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 eld. The electric polarization is dened as the net average dipole

moment per unit volume and is given by

P, Nkm,l 6

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.

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

Chapter 14


By the principle of causality, P (,) must be a function of the applied eld

E (,). To an excellent approximation, at the low intensity levels of natural light

sources, the relation of the polarization to the applied eld is linear. This is the

regime of linear optics. The most general form of the electric polarization for a

homogeneous medium is given by

P,L r; t10

Z 121

Z 121x1r2r 0; t2 t 0 E,r 0; t 0dr 0dt 0 SI

Z 121

Z 121x1r2r 0; t2 t 0 E,r 0; t 0dr 0dt 0 cgs

8>>>>>:

7

where the subscript L signies a linear polarization, 10 8:8510212 farad=meter is the electric permittivity of free space, andx1r2r 0; t2 t 0 is the linear dielectric response tensor. The functional formof x1 reects the principles of space and time invariance [6]. In other words,

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



the polarization response of a medium does not depend on when (in an absolute

sense) the driving eld is applied, but only on the time since it was applied.

Consequently, x1r2r 0; t2 t 0 must be dened in such a way that it vanisheswhen t2 t 0, 0 to preserve causality. Similarly, the polarization response in ahomogeneous medium does not depend on the absolute position in space of the

applied eld, but only on the distance away from this position. A nonzero value of

x1r2r 0; t2 t 0 for r r 0 is called a nonlocal response. If there is no responseexcept within a small neighborhood where r< r 0; then the response is calledlocal. This is equivalent to saying that the linear dielectric response tensor has a

d-function spatial dependence. For the vast majority of problems in nonlinearoptics, the media of interest produce approximately a local response.

Consequently, we will ignore the spatial dependence of x1 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

eld,

PLv 10x

1v Ev SIx1v Ev cgs

(8

where x1v; the linear susceptibility tensor, is the Fourier transform of thelinear dielectric response tensor. The tensor relation in Eq. (8) can also be

written as

PL;iv 10

j

Px1ij vEjv SI

j

Px1ij vEjv cgs

8>>>>>:

9

where the subscript i signies the ith cartesian coordinate i x; y; z; and thesum is over j x; y; z: The tensor x1v thus has nine components. In anisotropic medium, there is only one independent, nonzero component, and the

susceptibility is written as a scalar quantity, x1v:

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

Chapter 16


the vector electric eld wave is given by

7 7 E, 1c22E,

t 2 2 K

c22P,

t 210

where c 3 108 m=s 3 1010 cm=s is the speed of light in a vacuum, and K isa constant depending on the system of units used, with

K 1021 SI4p cgs

8: 15

with

dij 1 i j0 i j

(16

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 eld can be considered to be a superposition of quasi-monochromatic



waves (e.g., laser beams). The total eld is then written as

E,r; t m

Xe^mAmr; texpikm r2 vmt c:c: 17

where the sum is over m waves with frequencies vm and wave vectors km. Amr; tis a slowly varying amplitude in space and time (compared to the rapidly

oscillating part of the wave). If it is sufciently slowly varying, the mthcomponent is a monochromatic wave. However, if its time duration is sufciently

short (an ultrashort pulse) such that it cannot be described as a pure

monochromatic wave, then the mth component represents a quasi-monochro-matic 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

P,NL r; t m

XPNL;mr; texp2ivmt c:c: 18

where PNL;mr; t is a slowly varying (compared to the rapidly oscillating part ofthe wave) complex polarization amplitude. Then by the linearity of the wave

equation, each frequency component (Fourier component) of the total eld also

satises 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 simplied, for an excellent approximation in

homogeneous media,

7 7 E, < 272E, 19

where 72 is the Laplacian operator. For most of the nonlinear phenomenaconsidered 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.

Chapter 18


A. Isotropic Media

In an isotropic medium the approximation given in Eq. (19) is exact for a plane

wave and the susceptibility is a scalar quantity. The latter will be a complex

function of frequency, in general, and can be written as

x1 x1R ix1I 20where the subscripts R and I signify real and imaginary parts, respectively. The

dielectric constant is also complex, and1v10

r nv iav

4p21

in SI units. For cgs units, the expression is identical with the exception that

10 ! 1: In Eq. (21) n(v) is the index of refraction given by

nv

1 x1R v

qSI

1 4px1R vq

cgs

8>>>: 22

where a(v) is the intensity absorption coefcient, with

av

vx1I vnvc SI

4pvx1Invc cgs

8>>>>>>>:

23

Absorption properties of materials are often described in terms of the

absorption cross section s, which is related to the absorption coefcient bya sN; where N is the number of absorbing molecules per unit volume. Forsolutions, the molecular density is often given as a concentration in mole/liter,

with the molar extinction coefcient 1M being analogous to the cross section. Therelationship between the two is given by

s 2:3 1031M

NA24

where s is expressed in cm2, 1M is expressed in liter/mole-cm, and NA isAvagadros number.

When the medium is conducting, such as a semiconductor or an ionic

solution, the conductivity sc of the medium will contribute to the absorption



coefcient. In this case, Eq. (23) may be generalized to include conductivity by

av

vx1I v sc=10vnvc SI

4pvx1I v sc=vnvc cgs

8>>>>>>>:

25

The plane wave solution to the wave equation then has the form

E,z; t e^A0exp 2az2

expikz2 vt c:c: 26

where A0 is the amplitude of the wave at z 0; and the wave vector obeys adispersion relation given by

kv nvvc

27

The wave thus travels with a phase velocity vp c=n: In isotropic media, theelectric eld vector is always perpendicular to the wave vector, and the phase

velocity of the wave is independent of the direction of propagation.

A homogeneous, isotropic medium will also support the propagation of an

ultrashort optical pulse. An ultrashort pulse is only quasi-monochromatic and

hence is composed of several frequencies clustered about some center frequency,

which can also be considered the carrier frequency of the wave. When the

frequency spread of the pulse is such that dispersion of the refractive index

cannot be ignored, the concept of a unique phase velocity is meaningless. What

does have meaning is the velocity of the pulse itself (i.e., the group of

superimposed monochromatic waves comprising the pulse). This is characterized

by the group velocity, dened by

vg dkdv

21vc

28

where vc is the carrier frequency. When dispersion cannot be ignored, the wavewill no longer have the simple form of Eq. (26) since dispersion will cause the

pulse amplitude to decrease and spread as the pulse propagates. The group

velocity itself may also exhibit dispersion, which is characterized by the group

velocity dispersion coefcient k2, dened by

k2 d2k

dv2

vc

29

Since detectors cannot respond to the rapidly varying optical frequency,

the quantity measured experimentally is the time-averaged eld ux, where

Chapter 110


the average is over several optical cycles. The quantity of interest then is the

optical intensity (or irradiance), which is related to the eld amplitude by

Iz; t 210ncjAz; tj2 SInc

2pjAz; tj2 cgs

8>: 30

In a cw laser beam of nite cross-sectional area, the optical power is typically

measured, and is related to the intensity by

P ZA

I dA 31

where the integral is over the area of the beam, and A is not to be confused with

the complex eld amplitude. For a TEM00 Gaussian beam, the relationship at the

beam waist is

P pw20

2I0 32

where I0 is the peak, on-axis intensity of the Gaussian beam.

Another microscopic material parameter of interest is the molecular

polarizability a(1), dened by

kmvl a1v Elocalv 33where E local is the local electric eld (at the molecule), which is a superposition

of the applied eld E and the net eld due to surrounding dipoles. An analytical

expression for the local eld can be obtained for isotropic and cubic media. The

relationship between the susceptibility and the polarizability for these media is

then given by

x1 n 223

Na1=10 SI

n 223

Na1 cgs

8>>>: 34

The quantity f v n2v 2=3 is called the local eld factor.

B. The Laws of Linear Optics for Isotropic Media

Maxwells equations and the form of the electromagnetic waves may also be used

to derive the laws of optics at an interface between two dielectric media [1]. The

geometry of the interface and the incident, reected, and transmitted waves are

shown in Fig. 3. The laws of linear optics for two homogeneous, isotropic media

are given below.



Law of reection:

ur ui 35where ui is called the angle of incidence and ur the angle of reection, and bothare measured with respect to the normal of the interface. Thus, the wave reected

from the interface will leave at the same angle the incident wave impinges on the

interface.

Law of refraction (Snells Law):

ni sin ui nt sin ut 36where subscript i quantities refer to those in the medium where the wave is

incident on the interface, and t refers to the medium of the transmitted wave.

Since ni nt; the transmitted ray direction is bent on crossing the interface. Thedirection of the bend is toward the normal to the interface if ni , nt and awayfrom the normal if ni . nt:

Reection and transmission coefcients (Fresnels formulas). The amount

of electromagnetic energy reected and transmitted at an interface can be

determined by Fresnels formulas for the reection and transmission coefcients.

Thesewill depend on the polarization of the incident wave. The polarization vector

is speciedwith respect to the plane of incidence. This is the plane that contains the

incident, reected, and transmitted rays, as shown in Fig. 3. Light polarized normal

to the plane of incidence is referred to as s-polarization, while light polarized in

the plane of incidence is referred to as p-polarization. The ratio of the reected to

Figure 3 Illustration of propagating electromagnetic elds at an interface between twodifferent isotropic dielectric media.

Chapter 112


incident eld amplitudes is given by r, while t corresponds to the ratio of the

transmitted to incident eld amplitudes. Fresnels formulas are

rs ni cos ui 2 nt cos utni cos ui nt cos ut 37

ts 2ni cos uini cos ui nt cos ut 38

rp ni cos ut 2 nt cos uini cos ut nt cos ui 39

tp 2ni cos uini cos ut nt cos ui 40

Measurable quantities are the reectance and transmittance of the interface. These

are dened as the ratios of the time-averaged reected and transmitted power,

respectively, to the incident power. For both s- and p-polarization, these quantities

are related to the Fresnel coefcients by

Rs;p jrs;pj2 41

Ts;p nt cos utni cos ui

jts;pj2 42

where R and T are the reectance and transmittance, respectively.

C. Guided Waves

The form of the electromagnetic wave is somewhat different in the conned

geometry of waveguides compared to free space. The combined effects of total

internal reection and wave interference limit the number of allowed directions

and frequencies of propagating waves.

Waveguides may be considered one-dimensional (1-D) or two-dimensional

(2-D), depending on their geometry. An example of a 1-D waveguide is an

asymmetric slab waveguide, consisting of a thin lm deposited on a dielectric

slab. The opposite side of the lm is usually air. Examples of 2-D waveguides are

rectangular channel waveguides (a rectangular channel embedded in a dielectric

slab) and an optical ber. The latter consists of a cylindrical core surrounded by a

cylindrical cladding of lower refractive index. The cross section of the core can

be circular or elliptical.



It is customary to take the z-axis to coincide with the waveguide axis. Theform of the electric eld wave is given by

En;ir; t An;iz; tan;ix; yexpibz2 vt c:c: 43where i is a cartesian coordinate, and the subscript n designates the mode number.The propagation constant b takes on a discrete set of allowed values dependingon frequency and waveguide geometry. The normalized quantity an;ix; y is themodal distribution across the cross section of the waveguide. For slab

waveguides it is a function only of the coordinate normal to the lm plane. An,i is

a normalization function that is proportional to Pn1=2; where Pn is the power inthe nth mode. In Eq. (43), this quantity is written as a slowly varying function ofspace and time to generalize the result to the case in which the eld may be a

pulse, and the optical power may be changing with propagation.

The exact form of the modal distribution can be quite complicated in

situations of practical interest. Other sources should be consulted for a complete

description of these [2,3], as well as the eigenvalue equations that determine the

allowed values of the propagation constant b. For the purposes of this book,the function An;iz; twill serve as the complex amplitude of interest in computingthe interaction of waves in nonlinear optics.

D. Energy Theorem

As alluded to above, energy is stored in the electromagnetic eld and can be used

to perform work on free electric charges or bound electric charges (i.e., dipoles).

Calling uem the energy per unit volume (energy density) stored in the

electromagnetic eld in some volume V in space, the energy theorem for a

dielectric medium (no magnetization and no free currents) states

2

t

ZV

uemdV ZS

I cos u dSZV

kE, P,t

ldV 44

where cos u dS is the elemental surface area normal to the direction of energyow, and the angle brackets imply a time average over an optical cycle. The

second term on the right-hand side Eq. (44) gives the rate at which work is done

by the electric eld on the medium through the dipole moment per unit volume

(i.e., the polarization) and becomes the integral of 2ReivE P* when the timeaverage is performed. If this term is zero, the energy theorem is just a continuity

equation. It then states that the average rate of electromagnetic energy lost from a

volume V is equal to the net power owing through the surface S bounding the

volume. When the work term is not zero, some of the energy lost is transferred to

the medium via the polarization.

Chapter 114


Obviously, if the work term is purely imaginary, then no work is done by

the eld. This is the case in linear optics when the susceptibility is purely real,

i.e., the dielectric constant is real. In other words, the eld and the polarization are

in phase (the time-derivative of the polarization and the electric eld are then p=2radians out of phase). In this case (pure refractive case), the phase of the eld will

be modied in passage through the dielectric, but the amplitude remains constant.

When the phase difference Dw Dw wpol 2 wfield between the polarization andeld is such that 0 , Dw , p; then the eld does work on the polarization, andthe eld amplitude decreases. The net power owing out of the surface of the

volume V is thus reduced. In linear optics this is due to the dielectric constant

being complex, and the eld is reduced due to absorption, which dissipates

energy in the form of heat or other radiation. For 2p , Dw , 0; the eld woulddo negative work on the polarization. This means that the medium, through the

polarization, does work on the eld, and the eld amplitude grows or is amplied,

and the power owing out through the surface increases. This can only occur if

energy is pumped into the medium from some other source, and corresponds in

linear optics to the dielectric constant having a negative imaginary part. This eld

amplication occurs quite frequently in nonlinear optics even though the

susceptibility may be purely real.

E. Anisotropic Media

Wave propagation in anisotropic dielectric media is a somewhat more

complicated affair. Generally, the electric eld vector is not perpendicular to

the wave vector (direction of propagation). However, the displacement vector

D (,), dened by

D, 10E

, P, SIE, 4pP, cgs

(45

is orthogonal to the wave vector k. In any anisotropic medium, two independent

orthogonally polarized D (,)-waves can propagate with different phase velocities.

The problem in the optics of anisotropic media is to nd the polarization of these

modes and their corresponding phase velocities.

It can be shown that the linear dielectric constant is a symmetric tensor

1ij 1ji [1]. By the laws of linear algebra, an orthogonal coordinate system canbe found in which this tensor is diagonal 1ij 1iidij; where dij is given inEq. (16)]. The axes of this system are called the principal axes, and the

corresponding diagonal elements of the dielectric tensor are called the principal

dielectric constants of the medium. These are designated 1XX, 1YY, and 1ZZ, whereupper case symbols are used to signify the principal axes. Similarly principal

refractive indices are found: nii 1ii=101=2: For the cgs system of units, 10 is



replaced by 1. Often, these principal quantities are written with a single rather

than a double subscript. The phase velocity for a wave polarized along the ith

principal axis is vpi c=ni:If the dielectric tensor elements are complex, then the principal refractive

indices are related to the real part of the square root given above. The imaginary

part of the square root is related to principal absorption coefcients ai analogousto the scalar quantity given by Eq. (25). This polarization-dependent, propagation

direction-dependent absorption is called pleochromism [1].

In general, nX nY nZ : Such a medium is called biaxial for reasonsdiscussed below. A great simplication occurs when two of the principal indices

are equal, for example, nX nY nZ : This type of medium is called uniaxial.There is a single axis of symmetry, taken to be the Z-axis, which is called the

optic axis. When light propagates along this axis, its phase velocity is

independent of polarization. The following designation is made: nX nY noand nZ ne; where the o stands for ordinary and the e for extraordinarywaves.

Uniaxial media are said to be birefringent and exhibit double refraction [1].

The birefringence of the medium is given by Dn ne 2 no: When Dn . 0Dn , 0; the medium is said to be a positive (negative) uniaxial medium. Manyuseful transparent crystals in nonlinear optics have a very small birefringence

Dn ,, 1: In these types of materials, E (,) and D (,) are nearly parallel and canbe treated as such for most practical situations.

One of the independent propagation modes in uniaxial media has its

polarization orthogonal to the optic axis and thus has phase velocity given by

vp c=no: This is called the ordinary wave. The extraordinary wave will have acomponent of its polarization along the optic axis, and its phase velocity will be

vp c=neu; where u is the direction of propagation of the extraordinary wavewith respect to the optic axis, and

1

neu2 cos2u

no2 sin2u

ne2 46

General methods for determining the orthogonal polarizations of the D (,)-waves

and the corresponding phase velocities for any uniaxial crystal by use of the so-

called optical indicatrix or index ellipsoid are found in several textbooks [1,4,5].

The general form of the index ellipsoid for anisotropic media is given by

X 2

n2X Y

2

n2Y Z

2

n2Z 1 47

It is important to note that the index ellipsoid is not a real object that exits in

space; it is a useful mathematical construct. However, its axes do correspond to

the principal axes of the medium, and thus are useful for visualizing the directions

Chapter 116

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

of propagation and polarization. When nX nY ; Eq. (46) is recovered forpropagation at some angle u with respect to the Z-axis (optic axis).

Let A X or Z and C Z or X. Then the convention for a generalanisotropic medium is to take nA , nY , nC: The index ellipsoid is thus a prolatespheroid along ZA X;C Z or an oblate spheroid along ZA Z;C X:The intersection of the AY-plane with the ellipsoid yields an ellipse with semi-

major axis nY. Conversely, the intersection of the YC-plane with the ellipsoid

yields an ellipse where nY is the semi-minor axis. It follows that there is some

angle with respect to the C-axis where the intersection of a plane containing the

Y-axis through the origin of the ellipsoid will yield a circle of radius nY. A wave

propagating normal to this plane will have a phase velocity that is independent of

polarization. Hence, this direction is called an optic axis of the medium. The

points where the plane intersects the ellipsoid in the AC-plane will satisfy

the equations A2 C 2 n 2Y and A=nA2 C=nC2 1: The angle V, that thenormal to this plane (i.e., the optic axis), makes with the C-axis is then given by

sinV nCnY

n2Y 2 n2A

n2C 2 n2A

1=248

Since the equations are quadratic, there will be two values (^ ) of A and C that

satisfy the equations. Hence, there are two optic axes symmetrically situated at an

angle V about the C-axis in the AC-plane. Such a medium is thus called biaxial.Propagation in the principal planes of a biaxial medium is similar, but not

identical to propagation in a uniaxial medium. Consider propagation in the XY- or

YZ-plane. Let J X or Z and c f or u, respectively. Then an equation likeEq. (46) applies to determine the extraordinary index, with

1

nec2 cos2c

n2Y sin

2c

n2J49

Again employing the convention nA , nY , nZ ; where A X or Z and C Z orX, propagation in the AY-plane leads to nA # n

e # nY , nC no: This is similarto propagation in a negative uniaxial medium since ne , no; however, it is notidentical since there is no angle of propagation for which ne no: Forpropagation in the YC-plane, no nA , nY # ne # nZ : Hence, this is similar topropagation in a positive uniaxial medium since ne . no: Again, however, thereis no angle corresponding to ne no: Finally, it should be noted that for lightpropagating in the AC-plane, the situation is similar to propagation in a negative

uniaxial medium if the angle of propagation with respect to the C-axis is less than

V. For angles greater than V, it is similar to a positive uniaxial medium. Thereason for this is that, in the former case, as the angle of propagation approaches

the optic axis, the extraordinary refractive index decreases toward nY. In the latter

case, the extraordinary index increases toward nY.



For light propagating in a general direction in a biaxial medium, the phase

velocities of the two allowed modes of propagation are determined by solving the

Fresnel equation [1,4,5]

sin2u cos2f

n22 2 n22X sin

2u sin2f

n22 2 n22Y cos

2u

n22 2 n22Z 0 50

where u and f are the spherical angles describing the direction of propagationwith respect to the principal axes. Generally, this equation must be solved

numerically for the two independent values of the refractive index. The use of this

equation is discussed in greater detail in Chapter 2.

In anisotropic media, the direction of wave propagation k does not

coincide with the direction of energy or Poynting vector propagation S. The

wave vector is k D,;H, while the Poynting vector is S E,;H,:The angle a between k and S, which is also the angle between E (,) and D (,),is given by

tana 1x 2 1y2E2xE

2y 1x 2 1z2E2xE2z 1y 2 1z2E2yE2z 1=21xE

2x 1yE2y 1zE2z

51

III. NONLINEAR OPTICS

In the nonlinear optics regime, the nonlinear part of the polarization can no longer

be ignored. Note by Eq. (13) that the nonlinear polarization serves as a source for

the generation of new waves, and the wave equation becomes an inhomogeneous

differential equation. Hence, an expression for P,NL is required. For most of theapplications of nonlinear optics, this quantity can be expressed as a power series

expansion in the applied elds.

A. Nonlinear Susceptibilities

It is assumed that the nonlinear polarization can be written as [68]

P,NL P,2 P,3 52

Chapter 118


where

P,2r; t 10Z 121

Z 121x2t2 t 0; t2 t 00 : E,r; t 0E,r; t 00dt 0dt 00

53

P,3r; t 10Z 121

Z 121

Z 121x3t2 t 0; t2 t 00; t2 t 000...E,r; t 0

E,r; t 00E,r; t 000dt 0 dt 00 dt00054

These expressions are given in SI units; for cgs units, 10 ! 1: It is important tonote that the eld in the equations above is the total applied eld, which can be a

superposition of many elds of different frequencies. x (n ) is called the nth orderdielectric response, and is a tensor of rank n 1: Note that, as earlier for thelinear dielectric response, it is assumed that the response is local and hence the

spatial dependence of x (n ) is suppressed.If the applied eld is a superposition of monochromatic or quasi-

monochromatic waves, then it is possible to write expressions analogous to Eq.

(8) or Eq. (9) in terms of the Fourier transforms of the nonlinear polarization,

elds, and the dielectric response tensor, providing that the frequency

dependence of the Fourier transform of x (n ) is slowly varying in the region ofeach Fourier component of the applied eld. When the total eld is expanded in

terms of its Fourier components (e.g., its various laser frequencies), then the

nonlinear polarization will consist of several terms oscillating at various

combination frequencies. For example, if the total eld consists of two waves

oscillating at frequencies v1 and v2, the second-order nonlinear polarization willhave components oscillating at 2v1, 2v2, v1 v2; v1 2 v2; and dc terms at zerofrequency. Similarly, with three elds oscillating at frequencies v1, v2, andv3, the third-order polarization will oscillate at 3v1, 3v2, 3v3, v1 v2 v3;v1 v2 2 v3; etc.

It is common to write the Fourier components of the nonlinear polarization

in the following way. Consider a second-order polarization oscillating at v3 dueto the presence of elds oscillating at frequencies v1 and v2, with v3 v1 v2:Then the ith cartesian component of the complex polarization amplitude is

expressed as

P2i v3 10D 2jk

Xx2ijk 2v3;v1;v2Ejv1Ekv2 55



with

D 2 1 for indistinguishable fields

2 for distinguishable fields

(56

where x2ijk 2v3;v1;v2 is the second-order (complex) susceptibility and theFourier transform of x2t: The form of Eq. (55) allows for the possibility thatthe frequencies v1 and v2 are equal, or equal in magnitude and opposite in sign.In this case, there may actually be only one eld present, and the degeneracy

factor D (2) takes this into account. It should be noted, however, that the

determination of the degeneracy factor is whether the elds are physically

distinguishable or not. Two elds of the same frequency will be physically

distinguishable if they travel in different directions, for example. Also, the

negative frequency part of the real eld is considered to be distinguishable from

the positive frequency part (i.e., they have different frequencies). For negative

frequencies, it is important to note that E2v E* v since the rapidly varyingeld is a real mathematical quantity. Thus, for example, if v1 v and v2 2v;then the second-order polarization would be written as

P2i 0 210jk

Xx2ijk 0;v;2vEjvE*kv 57

This polarization drives the phenomenon known as optical rectication, wherein

an intense optical wave creates a dc polarization in a nonlinear medium. It is

important to remember that in these equations, the eld amplitude still contains

the rapidly varying spatial part, i.e., Er Arexpik r:This notation is easily extended to higher orders. When three frequencies

v1, v2, v3 are present, the third-order polarization at v4 v1 v2 v3 isgiven by

P3i v4 10D 3jkl

Xx3ijkl2v4;v1;v2;v3Ejv1Ekv2Elv3 58

where the degeneracy factor in this case becomes

D 3 1 all fields indistinguishable

3 two fields indistinguishable

6 all fields distinguishable

8>>>: 59

This form of the third-order polarization allows for various combination

frequencies even when only two elds are present, such as v1 2v2; or 2v1 2v2; etc. The degeneracy factor is just due to the number of different ways inwhich the products of the eld Fourier components appear in the expansion of

Chapter 120


the total eld to some power. For example, there is only one way that the product

for the frequency 3v1 appears: Ev1Ev1Ev1: However, there are threedifferent ways that the product for the frequency 2v1 2 v2 appears:Ev1Ev1E* v2; Ev1E* v2Ev1; and E* v2Ev1Ev1: The degen-eracy factor is thus related to the coefcients of Pascals triangle from algebra.

(The use of a degeneracy factor in these equations also relies on an intrinsic

symmetry of the susceptibility tensor, which is discussed below.)

The equations above are written in SI units. To obtain the form in cgs units

let 10 go to 1. Also, it is important to note that the nth order susceptibility isfrequency dependent and complex, in general. The reason for writing the

frequency dependence as shown in the equations above is for the purpose of

expressing symmetry relations of the susceptibility tensor. This is further

described below.

B. Symmetry Relations of the Nonlinear Susceptibility

The rst symmetry apparent from the form of Eqs. (55) and (58) is due to the fact

that it makes no difference physically in which order the product of the eld

amplitudes is given. Thus, an interchange in the order of the product

Ejv1Ekv2 [i.e., Ejv1Ekv2 $ Ekv2Ejv1 will not affect the value orsign of the ith component of the nonlinear polarization. The nonlinear

susceptibility should reect this symmetry. But, note that in the above

interchange, both frequencies and subscripts for the cartesian coordinates are

interchanged simultaneously. This is important since, for example, exchanging

the product Exv1Eyv2 with the product Exv2Eyv1 could change thenonlinear polarization, especially, for example, if the two elds are orthogonally

polarized. Thus, the symmetry property is expressed as (for third-order

susceptibilities)

x3ijkl2v4;v1;v2;v3 x3ikjl2v4;v2;v1;v3

x3ilkj2v4;v3;v2;v1 etc:60

In other words, if any of the subscripts {jkl} are permuted, then the susceptibility

will remain unchanged as long as the corresponding set of subscripts {123} are

also permuted. This holds even if any of the frequencies are negative. Note that

this does not hold for the subscript pair (i,4). The same relation holds for second

order and can be generalized to any order. This is called intrinsic permutation

symmetry and is the underlying reason why the nonlinear polarization can be

written compactly in terms of a degeneracy factor as in Eqs. (55) and (58).

At this point it is important to note another notation that is used in second-

order nonlinear optics. Often the susceptibility is represented as the so-called



d-coefcient, where d is a tensor given by

dijk 12x2ijk 61

Furthermore, the intrinsic permutation symmetry is used to contact the last two

subscripts and is written dijk ! dil: The subscripts are then written as numbersinstead of letters using the scheme

i : x 1 jk : xx 1

y 2 yy 2

z 3 zz 3

yz zy 4xz zx 5xy yx 6

62

For example, dxyz dxzy d14 and dzxx d31; etc. The utility of this notation isthat d-coefcients can be expressed as elements of a 3 6 matrix rather than a3 3 3 tensor. To use these coefcients in the nonlinear polarization ofEq. (55), just make the substitution x2ijk ! 2dil:

Another form of the permutation symmetry can be shown when the

nonlinear susceptibility is calculated quantum mechanically. This is usually done

using the density matrix method and expressions can be found in textbooks [6,7].

It can be shown generally, for example in third order, that

x3ijkl2v4;v1;v2;v3 x3*jikl 2v1;v4;2v2;2v3 etc: 63

First, it is noted that when the complex conjugate of the susceptibility is taken, it

just changes the sign of all of the frequencies. This is a consequence of the fact

that the rapidly varying nonlinear polarization is a real mathematical quantity.

Then, for any permutation of the cartesian subscripts, the new susceptibility thus

obtained is equal to the original susceptibility if the corresponding frequency

subscripts are also permuted, if all of the frequencies are changed by multiplying

by 21; and if the complex conjugate of susceptibility is taken. This importantresult states that the susceptibilities for different physical processes are simply

related. For example, Eq. (63) relates the susceptibilities for the third-order

processes of sum-frequency generation v4 v1 v2 v3 and difference-frequency generation v1 v4 2 v2 2 v3: The relations in Eq. (63) hold forany permutation and are generalized to all orders. Thus, for example, in second

Chapter 122


order

x2ijk 2v3;v1;v2 x2*kji 2v2;2v1;v3 etc: 64The above expressions hold generally when the susceptibilities are

complex. This is especially important when any single frequency or any

combination frequency is near a natural resonance frequency of the material.

However, it is often the case in many applications that all frequencies and

combination frequencies are far from any material resonance. Then the

susceptibilities can be treated as real quantities, and any susceptibility is thus

equal to its complex conjugate. In this case, Eq. (63) may be written as

x3ijkl2v4;v1;v2;v3 x3jikl2v1;v4;2v2;2v3

x3kjil2v2;2v1;v4;2v3 etc:65

Thus, under the condition that the susceptibilities are real (all frequencies far

from any resonance), the susceptibilities are unchanged for the simultaneous

permutation of subscripts from the cartesian set {ijkl} and the corresponding

subscripts from the frequency set {4123}, with the stipulation that the frequencies

carry the proper sign. Note: The rst frequency in the argument, that is, the

generated frequency, carries a negative sign. The signs on the other frequencies

must be such that the algebraic sum of all frequencies is zero. For example,

2v4 v3 v2 v1 0 implies that2v2 2 v1 v4 2 v3 0: This is calledfull permutation symmetry. This symmetry generalizes to all orders.

Another symmetry follows when the susceptibilities exhibit negligible

dispersion over the entire frequency range of interest. Thus, in addition to all

frequencies being far from any material resonance, this symmetry requires that

there be no resonance between any of the frequencies. This would not be the case

if, for example, v2 . v1 and for some frequency between v1 and v2 anabsorption line exists in the material. When dispersion can be ignored, the

frequencies can be freely permuted without permuting the corresponding

cartesian subscripts, and vice versa, and the susceptibility will remain unchanged.

This is known as Kleinman symmetry.

Nonlinear susceptibilities also reect the structural symmetry of the

material. This is important since in many cases this greatly reduces the number of

nonzero, independent tensor components needed to describe the medium. One

immediate consequence of this is that for all materials that possess a center of

inversion symmetry (e.g., isotropic liquids and crystals of symmetry class 432),

all elements of all even-order susceptibility tensors are identically equal to zero.

Thus, no even-order nonlinear processes are possible in these types of materials.

(This is strictly true only when the susceptibility is derived using the electric

dipole approximation in the perturbation Hamiltonian. For example, in an atomic



vapor a second order process may occur when a transition matrix element

between two equal parity states, which is forbidden in the electric dipole

approximation, is nonzero due to an electric quadrupole allowed transition. Such

a transition however, is generally very weak.)

Other simplications of the susceptibility tensors can be derived based on

specic symmetry properties of material, such as rotation axis and mirror plane

symmetries. The forms of the d-matrix for crystals of several different symmetry

classes are given in Table 1. Note that in many cases the elements are zero or

equal to ^1 times other elements. The form of the matrix when Kleinman

symmetry is valid is also given.

The specic form of third order susceptibilities may also be given for each

symmetry class. These are shown in Table 2, which gives each symmetry class,

the number of nonzero elements, and the relation between nonzero elements. For

the lowest symmetry (triclinic), all 81 (34) elements are nonzero and independent.

In all of the higher order symmetry classes several elements are equal to zero. For

classes higher in symmetry than monoclinic and orthorhombic, the number of

independent elements is less than the number of nonzero elements, with the

isotropic class having the smallest number (3) of independent elements.

C. Coupled-Wave Propagation

The optical waves are coupled through the nonlinear polarization, and the

nonlinear polarization acts as a source term in the wave equation for each

monochromatic, or quasi-monochromatic, wave. As stated earlier, because of the

linearity of the wave equation, each frequency component of the eld satises the

wave equation independently, with the source term being the Fourier component

of the nonlinear polarization corresponding to the frequency of that optical eld.

In this section, the form of the coupled-wave equations is considered when

the waves propagate in an isotropic medium or along one of the principal axes in

an anisotropic medium. When the birefringence of an uniaxial medium is small,

quite often true in applications, the forms of these equations will be the same to a

good approximation for propagation along any axis in a uniaxial medium.

Consider the interaction of m 1 waves through an mth order nonlinearpolarization. The frequency of the (m 1)th wave is given by

vm1 Xmm1

vm 66

where the set {vm} may contain both positive and negative frequencies. For anywave amplitude, A2v A* v; and for any wave vector, k2v 2kv:The wave amplitudes are assumed to vary primarily along the z-axis.

Chapter 124


Table 1 Form of the d-Matrix for Different Symmetry Classes

Symmetry

class General conditions Kleinman symmetry

Biaxial crystals

1

d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

0BB@

1CCA

d11 d12 d13 d14 d15 d16

d16 d22 d23 d24 d14 d12

d15 d24 d33 d23 d13 d14

0BB@

1CCA

2

0 0 0 d14 0 d16

d21 d22 d23 0 d25 0

0 0 0 d34 0 d36

0BB@

1CCA

0 0 0 d14 0 d16

d16 d22 d23 0 d14 0

0 0 0 d23 0 d14

0BB@

1CCA

m

d11 d12 d13 0 d15 0

0 0 0 d24 0 d26

d31 d32 d33 0 d35 0

0BB@

1CCA

d11 d12 d13 0 d15 0

0 0 0 d24 0 d12

d15 d24 d33 0 d13 0

0BB@

1CCA

222

0 0 0 d14 0 0

0 0 0 0 d25 0

0 0 0 0 0 d36

0BB@

1CCA

0 0 0 d14 0 0

0 0 0 0 d14 0

0 0 0 0 0 d14

0BB@

1CCA

mm2

0 0 0 0 d15 0

0 0 0 d24 0 0

d31 d32 d33 0 0 0

0BB@

1CCA

0 0 0 0 d15 0

0 0 0 d24 0 0

d15 d24 d33 0 0 0

0BB@

1CCA

Uniaxial crystals

3

d11 2d11 0 d14 d15 2d22

2d22 d22 0 d15 2d14 2d11

d31 d31 d33 0 0 0

0BB@

1CCA

d11 2d11 0 0 d15 2d22

2d22 d22 0 d15 0 2d11

d15 d15 d33 0 0 0

0BB@

1CCA

3m

0 0 0 0 d15 2d22

2d22 d22 0 d15 0 0

d31 d31 d33 0 0 0

0BB@

1CCA

0 0 0 0 d15 2d22

2d22 d22 0 d15 0 0

d15 d15 d33 0 0 0

0BB@

1CCA

6

d11 2d11 0 0 0 2d22

2d22 d22 0 0 0 2d11

0 0 0 0 0 0

0BB@

1CCA

d11 2d11 0 0 0 2d22

2d22 d22 0 0 0 2d11

0 0 0 0 0 0

0BB@

1CCA

(continued )



Table 1 Continued

Symmetry

class General conditions Kleinman symmetry

6m2

0 0 0 0 0 2d22

2d22 d22 0 0 0 0

0 0 0 0 0 0

0BB@

1CCA

0 0 0 0 0 2d22

2d22 d22 0 0 0 0

0 0 0 0 0 0

0BB@

1CCA

6,4

0 0 0 d14 d15 0

0 0 0 d15 2d14 0

d31 d31 d33 0 0 0

0BB@

1CCA

0 0 0 0 d15 0

0 0 0 d15 0 0

d15 d15 d33 0 0 0

0BB@

1CCA

6mm,

4mm

0 0 0 0 d15 0

0 0 0 d15 0 0

d31 d31 d33 0 0 0

0BB@

1CCA

0 0 0 0 d15 0

0 0 0 d15 0 0

d15 d15 d33 0 0 0

0BB@

1CCA

622,422

0 0 0 d14 0 0

0 0 0 0 2d14 0

0 0 0 0 0 0

0BB@

1CCA

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0BB@

1CCA

4

0 0 0 d14 d15 0

0 0 0 2d15 d14 0

d31 2d31 0 0 0 d36

0BB@

1CCA

0 0 0 d14 d15 0

0 0 0 2d15 d14 0

d15 2d15 0 0 0 d14

0BB@

1CCA

32

d11 2d11 0 d14 0 0

0 0 0 0 2d14 2d11

0 0 0 0 0 0

0BB@

1CCA

d11 2d11 0 d14 0 0

0 0 0 0 2d14 2d11

0 0 0 0 0 0

0BB@

1CCA

42m

0 0 0 d14 0 0

0 0 0 0 d14 0

0 0 0 0 0 d36

0BB@

1CCA

0 0 0 d14 0 0

0 0 0 0 d14 0

0 0 0 0 0 d14

0BB@

1CCA

Isotropic crystals

43m; 23

0 0 0 d14 0 0

0 0 0 0 d14 0

0 0 0 0 0 d14

0BB@

1CCA

0 0 0 d14 0 0

0 0 0 0 d14 0

0 0 0 0 0 d14

0BB@

1CCA

432

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0BB@

1CCA

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0BB@

1CCA

Chapter 126


Table 2 Form of the Third Order Susceptibility for Different Symmetry Classes

Symmetry

class

Nonzero

elements Form

Triclinic 1,1 81 x 3ijkli x; y; z; j x; y; z; k x; y; z; l x; y; z

Monoclinic 2,m,2/m 41 x 3iiii i x; y; zx 3iijj ;x

3ijij ; x

3ijji i x; j y; z; i y; j x; z; i z; j x; y

x 3ijyy; x3yyij;x

3iyjy;x

3yiyj; x

3yijy;x

3iyyji x; j z; i z; j x

x 3iiij ;x3iiji ; x

3ijii ;x

3jiii i x; j z; i z; j x

Orthorhombic 2mm, 21 x 3iiii i x; y; z222,mmm x 3iijj ;x

3ijij ; x

3ijji i x; j y; z; i y; j x; z; i z; j x; y

Tetragonal 4, 4; 4=m 41 x 3xxxx x 3yyyy;x 3zzzzx 3xxyy x 3yyxx;x 3xyxy x 3yxyx; x 3xyyx x 3yxxyx 3xxxy 2x 3yyyx; x 3xxyx 2x 3yyxy;x 3xyxx 2x 3yxyy;x 3yxxx 2x 3xyyy

x 3xxzz x 3yyzz;x 3zzxx x 3zzyy; x 3xzxz x 3yzyz;x 3zxzx x 3zyzy;x 3xzzx x 3yzzy; x 3zxxz x 3zyyz

x 3xyzz 2x 3yxzz;x 3zzxy 2x 3zzyx;x 3xzyz 2x 3yzxz;x 3zxzy 2x 3zyzx;x 3zxyz 2x 3zyxz; x 3xzzy 2x 3yzzx

Tetragonal 422,4mm, 21 x 3xxxx x 3yyyy;x 3zzzz42m; 4=mm x 3xxyy x 3yyxx;x 3xyxy x 3yxyx; x 3xyyx x 3yxxy

x 3xxzz x 3yyzz;x 3zzxx x 3zzyy; x 3zxxz x 3zyyzx 3xzzx x 3yzzy;x 3xzxz x 3yzyz; x 3zxzx x 3zyzy

Trigonal 3; 3 73 x 3xxxx x 3yyyy;x 3zzzzx 3xxxx x 3xxyy x 3xyyx x 3xyxyx 3xxyy x 3yyxx;x 3xyyx x 3yxxy;x 3xyxy x 3yxyxx 3xxzz x 3yyzz;x 3zzxx x 3zzyy;x 3xzzx x 3yzzy;x 3zxxz x 3zyyz;x 3xzxz x 3yzyz;x 3zxzx x 3zyzy

x 3xyzz 2x 3yxzz;x 3zzxy 2x 3zzyx;x 3xzzy 2x 3yzzx; x 3xzyz 2x 3yzxz;x 3zxzy 2x 3zyzx;x 3zxyz 2x 3zyxz

x 3xxxy 2x 3yyyx x 3yyxy x 3yxyy x 3xyyy;x 3xxyx 2x 3yyxy;x 3xyxx 2x 3yxyy;x 3xyyy 2x 3yxxx;

x 3xxxz 2x 3xyyz 2x 3yxyz 2x 3yyxz;x 3xxzx 2x 3xyzy 2x 3yxzy 2x 3yyzx

x 3xzxx 2x 3yzxy 2x 3yzyx 2x 3xzyy;x 3zxxx 2x 3zxyy 2x 3zyxy 2x 3zyyx

(continued)



Table 2 Continued

Symmetry

class

Nonzero

elements Form

x 3yyyz 2x 3yxxz 2x 3xyxz 2x 3xxyz;x 3yyzy 2x 3yxzx 2x 3xyzx 2x 3xxzy

x 3yzyy 2x 3yzxx 2x 3xzyx 2x 3xzxy;x 3zyyy 2x 3zyxx 2x 3zxyx 2x 3zxxy

Trigonal 32; 3m; 3m 37 x 3xxxx x 3yyyy;x 3zzzzx 3xxxx x 3xxyy x 3xyyx x 3xyxyx 3xxyy x 3yyxx;x 3xyyx x 3yxxy;x 3xyxy x 3yxyxx 3xxzz x 3yyzz;x 3zzxx x 3zzyy; x 3xzzx x 3yzzy;x 3zxxz x 3zyyzx 3xxxz 2x 3xyyz 2x 3yxyz 2x 3yyxz;x 3xxzx 2x 3xyzy 2x 3yxzy 2x 3yyzx

x 3xzxx 2x 3xzyy 2x 3yzxy 2x 3yzyx;x 3zxxx 2x 3zxyy 2x 3zyxy 2x 3zyyx

x 3xzxz x 3yzyz;x 3zxzx x 3zyzyHexagonal 6; 6; 6=m 41 x 3xxxx x 3yyyy;x 3zzzz

x 3xxxx x 3xxyy x 3xyyx x 3xyxyx 3xxyy x 3yyxx;x 3xyyx x 3yxxy;x 3xyxy x 3yxyxx 3xxzz x 3yyzz;x 3zzxx x 3zzyy; x 3xzzx x 3yzzy;x 3zxxz x 3zyyz;x 3xzxz x 3yzyz; x 3zxzx x 3zyzy

x 3xyzz 2x 3yxzz;x 3zzxy 2x 3zzyx;x 3xzyz 2x 3yzxz;x 3zxzy 2x 3zyzx;x 3xzzy 2x 3yzzx;x 3zxyz 2x 3zyxz

x 3xxxy 2x 3yyyx x 3yyxy x 3yxyy xxyyyx 3xxyx 2x 3yyxy;x 3xyxx 2x 3yxyy;x 3xyyy 2x 3yxxx

Hexagonal 622; 6mm; 6m2; 21 x 3xxxx x 3yyyy;x 3zzzz6=mmm x 3xxxx x 3xxyy x 3xyyx x 3xyxy

x 3xxyy x 3yyxx;x 3xyyx x 3yxxy;x 3xyxy x 3yxyxx 3xxzz x 3yyzz;x 3zzxx x 3zzyy; x 3xzzx x 3yzzy;x 3zxxz x 3zyyz;x 3xzxz x 3yzyz; x 3zxzx x 3zyzy

Cubic 23,m3 21 x 3xxxx x 3yyyy x 3zzzzx 3xxyy x 3yyzz x 3zzxxx 3yyxx x 3zzyy x 3xxzzx 3xyxy x 3yzyz x 3zxzxx 3yxyx x 3zyzy x 3xzxz

(continued)

Chapter 128


For practically every problem in nonlinear optics, the slowly varying

amplitude approximation may be used [68]. This assumes that the magnitude

and phase of the wave amplitude vary slowly in space and time over an optical

wavelength and period, respectively. For any wave amplitude this implies that

2A

z2

,, k Az

67

2A

t 2

,, v At

68

and for the complex amplitude of the Fourier component of the nonlinear

polarization,

2Pm

t 2

,, v Pmt

,, jv2Pmj 69

In all of the wave equations given below, the following denitions apply

xmeff e^m1 xm2vm1;v1;v2; . . .;vm...e^1e^2 e^m 70

Table 2 Continued

Symmetry

class

Nonzero

elements Form

x 3xyyx x 3yzzy x 3zxxzx 3yxxy x 3zyyz x 3xzzx

Cubic 432;m3m; 43m 21 x 3xxxx x 3yyyy x 3zzzzx 3xxyy x 3yyxx x 3xxzz x 3zzxx x 3yyzz x 3zzyyx 3xyxy x 3yxyx x 3xzxz x 3zxzx x 3yzyz x 3zyzyx 3xyyx x 3yxxy x 3xzzx x 3zxxz x 3yzzy x 3zyyz

Isotropic 21 x 3xxxx x 3yyyy x 3zzzzx 3xxxx x 3xxyy x 3xyyx x 3xyxyx 3xxyy x 3yyxx x 3xxzz x 3zzxx x 3yyzz x 3zzyyx 3xyxy x 3yxyx x 3xzxz x 3zxzx x 3yzyz x 3zyzyx 3xyyx x 3yxxy x 3xzzx x 3zxxz x 3yzzy x 3zyyz



where em is a unit vector pointing in the direction in space of the polarization of

the mth eld,

Dk Xmm1

km 2 km1

" # z^ 71

K 0 1 SI4p cgs

(72

and the frequency-dependent index of refraction and absorption coefcient,

respectively, are

nm1 nvm1 73

am1 avm1 74The following equations are the form of the wave equation for Am1

encountered under various situations. The terms quasi-cw, short pulse, and ultra-

short pulse have the following meanings. Quasi-cw may refer to a true cw wave

or to a pulse with a full-width at half-maximum (tFWHM) such that the physical

length of the medium is small compared to the distance ctFWHM/n. A short pulse

will have a width such that the physical length of the medium is comparable to or

larger than ctFWHM/n. An ultrashort pulse is only quasi-monochromatic, and

hence is composed of several frequencies clustered about some carrier frequency.

When the frequency spread of the pulse is such that dispersion in the refractive

index cannot be ignored, the concept of a unique phase velocity is meaningless.

The pulse is then treated as a superposition of monochromatic waves clustered

about the carrier frequency, which move as a group with group velocity vg dk=dv21: This term is encountered when the physical length of the medium iscomparable to or larger then the distance vgtFWHM.

The wave equation is thus give by

1. Innite plane waves, no absorption, quasi-cw, propagation primarily

along z.dAm1dz

i K0vm1

2nm1cD mxmeff

Ymm1

AmexpiDkz 75

2. Innite plane waves, absorption, quasi-cw, propagation primarily

along z.dAm1dz

am12

Am1 i K0vm1

2nm1cD mxmeff

Ymm1

AmexpiDkz 76

Chapter 130


3. Innite plane waves, absorption, quasi-cw, propagation primarily

along 2z.

dAm1dz

2am12

Am1 i K0vm1

2nm1cD mxmeff

Ymm1

AmexpiDkz 77

Dk Xmm1

km km1 !

z^ 78

4. Finite beams, no absorption, quasi-cw, propagation primarily alongz.

72t 2ikm1Am1z

2K0v2m1c2

D mxmeffYmm1

AmexpiDkz 79

72t

2

x2

2

y2rectangular coordinates

1

r

rr

r

1r 22

f2cynlindrical coordinates

8>>>>>

handbook of nonlinear optics

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