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NONLINEAR OPTICS

WILEY SERIES IN PURE AND APPLIED OPTICS

Founded by Stanley S. Ballard, University of Florida

EDITOR: Glenn Boreman, University of Central Florida, CREOL & FPCE

A complete list of the titles in this series appears at the end of this volume.

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NONLINEAR OPTICSPHENOMENA, MATERIALS, ANDDEVICES

George I. StegemanRobert A. Stegeman

Copyright � 2012 by John Wiley & Sons, Inc. All rights reserved

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CONTENTS

PREFACE xi

1. Introduction 1

1.1 What is Nonlinear Optics and What is it Good for? 1

1.2 Notation 2

1.3 Classical Nonlinear Optics Expansion 4

1.4 Simple Model: Electron on a Spring and its Application to

Linear Optics 6

1.5 Local Field Correction 10

Suggested Further Reading 13

PART A: SECOND-ORDER PHENOMENA 15

2. Second-Order Susceptibility and Nonlinear Coupled

Wave Equations 17

2.1 Anharmonic Oscillator Derivation of Second-Order

Susceptibilities 18

2.2 Input Eigenmodes, Permutation Symmetry, and

Properties of w(2) 23

2.3 Slowly Varying Envelope Approximation 25

2.4 Coupled Wave Equations 26

2.5 Manley–Rowe Relations and Energy Conservation 31


3. Optimization and Limitations of Second-Order Parametric

Processes 39

3.1 Wave-Vector Matching 39

3.2 Optimizing dð2Þeff 53

3.3 Numerical Examples 59

References 67


v

4. Solutions for Plane-Wave Parametric Conversion Processes 69

4.1 Solutions of the Type 1 SHG Coupled Wave Equations 69

4.2 Solutions of the Three-Wave Coupled Equations 77

4.3 Characteristic Lengths 80

4.4 Nonlinear Modes 81

References 84


5. Second Harmonic Generation with Finite Beams

and Applications 86

5.1 SHG with Gaussian Beams 86

5.2 Unique and Performance-Enhanced Applications

of Periodically Poled LiNbO3 (PPLN) 98

References 107


6. Three-Wave Mixing, Optical Amplifiers, and Generators 108

6.1 Three-Wave Mixing Processes 108

6.2 Manley–Rowe Relations 110

6.3 Sum Frequency Generation 111

6.4 Optical Parametric Amplifiers 113

6.5 Optical Parametric Oscillator 119

6.6 Mid-Infrared Quasi-Phase Matching Parametric Devices 128

References 139

Selected Further Reading 140

7. w(2) Materials and Their Characterization 141

7.1 Survey of Materials 141

7.2 Oxide-Based Dielectric Crystals 143

7.3 Organic Materials 144

7.4 Measurement Techniques 149

Appendix 7.1: Quantum Mechanical Model for

Charge Transfer Molecular Nonlinearities 153

References 157


PART B: NONLINEAR SUSCEPTIBILITIES 159

8. Second- and Third-Order Susceptibilities: Quantum

Mechanical Formulation 161

8.1 Perturbation Theory of Field Interaction with Molecules 162

8.2 Optical Susceptibilities 169

vi CONTENTS

Appendix 8.1: wð3Þijk‘ Symmetry Properties for Different

Crystal Classes 192

Reference 196


9. Molecular Nonlinear Optics 197

9.1 Two-Level Model 198

9.2 Symmetric Molecules 210

9.3 Density Matrix Formalism 215

Appendix 9.1: Two-Level Model for Asymmetric

Molecules—Exact Solution 216

Appendix 9.2: Three-Level Model for Symmetric

Molecules—Exact Solution 218

References 222


PART C: THIRD-ORDER PHENOMENA 225

10. Kerr Nonlinear Absorption and Refraction 227

10.1 Nonlinear Absorption 228

10.2 Nonlinear Refraction 238

10.3 Useful NLR Formulas and Examples

(Isotropic Media) 243


11. Condensed Matter Third-Order Nonlinearities due to

Electronic Transitions 251

11.1 Device-Based Nonlinear Material Figures of Merit 252

11.2 Local Versus Nonlocal Nonlinearities in Space and Time 253

11.3 Survey of Nonlinear Refraction and Absorption

Measurements 255

11.4 Electronic Nonlinearities Involving Discrete States 256

11.5 Overview of Semiconductor Nonlinearities 266

11.6 Glass Nonlinearities 281

Appendix 11.1: Expressions for the Kerr, Raman, and

Quadratic Stark Effects 284

References 286


12. Miscellaneous Third-Order Nonlinearities 290

12.1 Molecular Reorientation Effects in Liquids and

Liquid Crystals 291

CONTENTS vii

12.2 Photorefractive Nonlinearities 300

12.3 Nuclear (Vibrational) Contributions to n2||(�o; o) 306

12.4 Electrostriction 310

12.5 Thermo-Optic Effect 312

12.6 w(3) via Cascaded w(2) Nonlinear Processes: Nonlocal 314

Appendix 12.1: Spontaneous Raman Scattering 317

References 328


13. Techniques for Measuring Third-Order Nonlinearities 330

13.1 Z-Scan 332

13.2 Third Harmonic Generation 339

13.3 Optical Kerr Effect Measurements 343

13.4 Nonlinear Optical Interferometry 344

13.5 Degenerate Four-Wave Mixing 345

References 346


14. Ramifications and Applications of Nonlinear Refraction 347

14.1 Self-Focusing and Defocusing of Beams 348

14.2 Self-Phase Modulation and Spectral

Broadening in Time 352

14.3 Instabilities 354

14.4 Solitons (Nonlinear Modes) 363

14.5 Optical Bistability 372

14.6 All-Optical Signal Processing and Switching 375

References 382


15. Multiwave Mixing 384

15.1 Degenerate Four-Wave Mixing 385

15.2 Degenerate Three-Wave Mixing 397

15.3 Nondegenerate Wave Mixing 399

Reference 413


16. Stimulated Scattering 414

16.1 Stimulated Raman Scattering 415

16.2 Stimulated Brillouin Scattering 431

References 441


viii CONTENTS

17. Ultrafast and Ultrahigh Intensity Processes 443

17.1 Extended Nonlinear Wave Equation 444

17.2 Formalism for Ultrafast Fiber Nonlinear Optics 448

17.3 Examples of Nonlinear Optics in Fibers 452

17.4 High Harmonic Generation 460

References 462


Appendix: Units, Notation, and Physical Constants 465

A.1 Units of Third-Order Nonlinearity 465

A.2 Values of Useful Constants 467

Reference 467

INDEX 469

CONTENTS ix

PREFACE

The field of nonlinear optics came into being in the 1960s, stimulated essentially by

the invention of the laser. Its impact has been acknowledged by the award of theNobel

Prize in physics to one of its pioneers Nicholas Bloembergen in 1981 (1) and by other

Nobel prizes for work enabled by nonlinear optics in chemistry and physics and

multiple awards by theAmerican Institute of Physics, the Optical Society of America,

and other scientific organizations.

The fundamental principles of nonlinear optics are nowwell known and have been

elucidated in excellent nonlinear optics textbooks by Ron Shen, Doug Mills, Robert

Boyd, Govind Agrawal, and others over the last 20 years. These books served two

purposes: to discuss basic principles and to give an overview of interesting applica-

tions and experiments. Of the two branches of nonlinear optics—nonlinear phenom-

ena and nonlinear materials—that have evolved over the years, the latter has

accounted for most of the progress in the field over the last few decades whereas

the former has been the subject of most textbooks. In fact, the nonlinear materials

evolution has been spectacular. Since the early days of nonlinear optics, the

requirements for some nonlinear phenomena have been reduced from kilowatt lasers

to milliwatt lasers. Our particular goal for this textbook, in addition to elucidating the

fundamentals of nonlinear optics from our own perspective, was to discuss nonlinear

materials, new nonlinear phenomena developed in the last few decades such as

solitons, and the practical aspects of the most common nonlinear devices.

This textbook is based on a nonlinear optics course initially developed at CREOL

(Center for Research in Electro-Optics and Lasers, now a part of the College of Optics

and Photonics), University of Central Florida, in the 1980s by Eric Van Stryland and

David Hagan. After George I. Stegeman joined CREOL in 1990, he took over this

course, expanded and continuously revised it, put it into the PowerPoint format, and

added new problems every year. Robert A. Stegeman, the coauthor, has used this

course in his professional career, which involves current applications of nonlinear

optics, primarily in the mid-infrared region of the spectrum, and he has contributed

most of the application discussions to this text.

This course deals with the physics and applications of optical phenomena that

occur at intensity levels at which optical processes become dependent on optical

intensity or integrated flux. It is designed for graduate students, postdoctoral fellows,

and others with prior knowledge of electromagnetic wave propagation in materials as

well as for professionals in the field who are looking for newly developed fields and

concepts. Although a rudimentary knowledge of quantum mechanics would be

xi

helpful, it is not a requirement.When quantummechanics is used, it is reviewed at the

level needed for the course.

Nonlinear optics is not just a simple extension of linear optics. A keystone concept

in linear optics is that electromagnetic waves do not interact with one another.

Solutions toMaxwell’s equations lead to “orthogonal” eigenmodes, i.e., summing the

fields due to two ormore overlapping field solutions and calculating the intensity leads

to a net intensity that is the sum of the intensities of the individual waves. Nonlinear

optics is all about interactions that occur between light and matter at high intensities.

Hence, the solutions to the nonlinear wave equations do not lead to eigenmodes, just

nonlinear modes. As will be discussed in the later part of the textbook, the modes of

nonlinear optics are solitons, spatial solitons for continuous waves that do not diffract

in space, temporal solitons for noncontinuous waves that do not disperse (spread) in

time and are confined in some kind of awaveguide that inhibits spatial diffraction, and

spatiotemporal solitons that spread neither in time nor in space. Such modes, in

general, exchange energy when they interact so that the reader should be prepared to

give up notions such as the superposition principle, which may be satisfied only

approximately.

It proves useful to explain the philosophy adopted here. There are two approaches

to discussing nonlinear optics. One is to introduce macroscopic nonlinear suscep-

tibilities at a phenomenological level. These susceptibilities are measured in the

laboratory and used to quantify phenomena, devices, and so forth. The second

approach starts at a more physical level with the electric dipole interaction of isolated

atoms and molecules with radiation fields to find the response at the atomic level

and the molecular level. Although it is satisfying from a physical insight perspective,

difficulties occur in going to the condensed matter limit where most experiments are

done. Because there are dipolar fields induced in neighboring molecules, the “local”

fields experienced by an atom or a molecule are not just the fields given by the

macroscopic Maxwell’s equations, called the “Maxwell” fields. The “local” field is

the sumof theMaxwell fields and all the induced dipole fields at the site of an atomor a

molecule. A rigorous and satisfactory approach to accurately estimating the local

fields has been a subject of continuing discussion formany years. Hence this approach

does not necessarily yield reliable numbers for measured nonlinear susceptibilities,

but does give fairly accuratelymanykey features, including the frequency spectrumof

the nonlinear response functions, i.e., the nonlinear susceptibilities.

Here we use a combination of these approaches. Whenever possible, the nonlinear

optics response at the molecular level is treated first, approximate susceptibilities are

derived, and then measured susceptibilities are used in discussing applications. To

facilitate the separation of microscopic and macroscopic parameters, the isolated

molecule parameters are identified by a “bar,” e.g., the transition dipole moment

between energy levels i and j (�Ei and �Ej) is �mij , the mass of an electron is �me, the

reduced mass for the bth vibration in a molecule is �mb, and so on.

This book also includes appendices in which the fundamentals of a number of

concepts such as Raman scattering and two- and three-level models are presented, as

well as some tables of the relation between nonlinear susceptibilities, their conversion

between different systems of units, and crystal symmetry.

xii PREFACE

G.S. thanks his colleagues at CREOL for their helpful discussions over the years,

especially Demetrius Christodoulides, Eric Van Stryland, and David Hagan, as well

as the many graduate students who diligently corrected lecture notes and asked

probing questions.

REFERENCE

1. N. Bloembergen, Encounters in Nonlinear Optics: Selected Papers of Nicolaas

Bloembergen with Commentary (World Scientific Press, Singapore,1996).

PREFACE xiii

CHAPTER 1

Introduction

1.1 WHAT IS NONLINEAR OPTICS AND WHAT IS IT GOOD FOR?

In general, nonlinear optics takes place when optical phenomena occur in materials

that change optical properties with input power or energy and/or generate new

beams or frequencies. Examples are power-, intensity-, or flux-dependent changes in

the frequency spectrum of light, the transmission coefficient, the polarization, and/or

the phase. New beams can also be generated either by a shift in frequency from the

original frequency or by travelling in different directions relative to the incident beam.

Although one frequently refers to the intensity or power dependence of phenomena as

being signatures of nonlinear optics, there are many cases characterized by a flux

dependence, i.e., changes in beam properties that are cumulative in the illumination

time, usually accompanied by absorption.

A frequently asked question is: “How do I really know when nonlinear optics is

occurring inmy experiment?” Some examples of commonly observed phenomena are

shown in Figs 1.1 and 1.2. Figure 1.1a shows harmonic generation, a second- or third-

order nonlinear effect. Figure 1.1b shows nonlinear transmission, essentially a third-

order nonlinear effect. In an interference experiment an increase in the input intensity

can lead to a shift in fringes due to second- or third-order nonlinear optics (see

Fig. 1.1c). Avery common effect—self-focusing of light—is illustrated in Fig. 1.2, in

which a beam narrows with an increase in the input intensity due to propagation

through a sample, forming a soliton at high intensities that propagates without change

in size or shape and then breaks up into “noise” filaments, i.e., multiple nondiffracting

beams, at very high intensities.

The second most frequently asked question is: “What is nonlinear optics good

for?” A collage of applications is shown schematically in Fig. 1.3. Probably the most

frequently used nonlinear optics device is the second harmonic generator, which

doubles the frequency of light, as shown in Fig. 1.1a. Along the same lines are optical

parametric devices, also based on second-order nonlinearities, which include am-

plifiers (optical parametric amplifiers) and frequency-tunable generators (optical

parametric generators and optical parametric oscillators), and the last two are

commonly used as sources of tunable radiation (see Fig. 1.3a). Nonlinear absorption

Nonlinear Optics: Phenomena, Materials, and Devices, George I. Stegeman and Robert A. Stegeman.� 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

1

that depends on intensity is used for the localized activation of drugs or imaging inside

media (Fig. 1.3b). A third example is an all-optical control of optical signals, e.g., for

communications (Fig. 1.3c).

1.2 NOTATION

The diversity of notations used for optical fields, nonlinear susceptibilities, and so on,

is a frequently confusing aspect of this field. A perusal of the nonlinear optics

literature shows that there is little consistency, especially when dealing with third-

order nonlinear coefficients. Here a concentrated effort has beenmade to be consistent

and to introduce more descriptive notations. The assumptions and notation used here

are as follows:

2ωω

3ω

(a) (b)

Ioutput

Iinput

Low High(c)

FIGURE 1.1 (a) Second and third harmonic generation. (b) Nonlinear transmission.

(c) Nonlinear fringe shift between low and high intensity inputs.

OutputInput

(a) (b) (c) (d)

FIGURE1.2 (a) Beam input and output geometry. (b)Diffracted output beam at low intensity

input. (c) Self-focused output beam corresponding to a spatial soliton. (d) Multiple filaments in

the output beam at very high intensity.

(a) (b)

Frequency orpolarization filter

(c)

Laser OPO

FIGURE 1.3 (a) An optical parametric oscillator (OPO) for producing two tunable frequen-

cies. (b) Two-photon absorption activation of chemistry. (c) All-optical control of routing in a

nonlinear coupler. A control beam (lower input arm) is used to isolate (switch out) a single pulse

from the input pulse train.

2 INTRODUCTION

1. Planewaveswill be explicitly assumed to elucidate nonlinear phenomena in the

simplest fashion. Whenever finite beams are considered, which is normally

necessary to discuss devices and applications, this will be clearly stated.

2. Continuous-wave fields are explicitly assumed unless otherwise stated. The

electromagnetic fields are written as

~Eð~r; tÞ ¼ 1

2~EðoÞ e�iot þ c:c: ¼ 1

2~EðoÞ ei½kz�ot� þ c:c: ð1:1Þ

3. The unit vector is written as e and has components ei, where i¼ x, y, z.

4. The “Einstein” notation is used for summations over repeated indices; e.g.,

aibici ¼ axbxcx þ aybycy þ azbzcz.

5. Quantitieswith a “bar” above, e.g., �m~, refer to individualmolecular properties in

the absence of interactionwith othermolecules as well as parameters in a single

molecule’s frame of reference.

6. Quantities with a “tilde” above, e.g., ~wð2Þijk , identify parameters and coefficients

in the “zero (nonresonant) frequency” limit (Kleinman limit) o � �or, i.e., at

frequencies much smaller than any resonant frequency �or of the material.

7. SI units are used throughout. Here intensity is used to mean power per unit

area, usually in units ofwatts per square centimeter. It is equivalent to irradiance.

In the cases of pulses of light, flux per unit area is defined as the integrated

intensity of a pulse over time, typically over the duration of the pulse. Flux is

defined as the energy of a pulse integrated over both time and cross section.

It is important to realize that in this textbook~EðoÞ is not the Fourier transform of~EðtÞand its use is restricted to Eq. 1.1. For amonochromatic wave of frequencyoa,EiðoaÞis the notation used for the Fourier transform of the field and EiðoaÞ 6¼ EiðoaÞ. Therelations between the two can be derived easily from the unitary Fourier transform

equations:

EðtÞ ¼ffiffiffiffiffiffi1

2p

s ð¥�¥

EðoÞ e�iot do; EðoÞ ¼ffiffiffiffiffiffi1

2p

s ð¥�¥

Eðt0Þ eiot0 dt0

dðt�t0Þ ¼ffiffiffiffiffiffi1

2p

s ð¥�¥

e�ioðt�t0Þ do; dðo�oaÞ ¼ffiffiffiffiffiffi1

2p

s ð¥�¥

eiðo�oaÞt0Þ dt0:

ð1:2Þ

Substituting Eq. 1.1 for Eðt0Þ into the E(o) equation in Eq. 1.2 gives

EiðoÞ¼ffiffiffiffiffiffi1

2p

s1

2

ð¥�¥

EiðoaÞ eiðo�oaÞt dtþð¥�¥

E*i ðoaÞ eiðoþoaÞt dt

� �

¼ 1

2EiðoaÞ

ffiffiffiffiffiffi1

2p

s ð¥�¥

eiðo�oaÞt dtþE*i ðoaÞ

ffiffiffiffiffiffi1

2p

s ð¥�¥

eiðoþoaÞt dt

8<:

9=;

!EiðoÞ¼ 1

2EiðoaÞdðo�oaÞþEið�oaÞdðoþoaÞ½ �: ð1:3Þ

NOTATION 3

If fields have a distribution of frequencies, then the d functions are replaced by g

(o�oa) and g(o þ oa), normalized so that their integrals over frequency are unity.

Additional notation will be introduced as needed in succeeding chapters.

1.3 CLASSICAL NONLINEAR OPTICS EXPANSION

The simplest andmost general expansion of the nonlinear polarization induced by the

mixing of optical fields is

Pið~r;tÞ¼ e0

ð¥�¥

ðt�¥

wð1Þij ð~r�~r0; t�t0ÞEjð~r0;t0Þd~r0dt0�

þð¥�¥

ð¥�¥

ðt�¥

ðt�¥

wð2Þijk ð~r�~r0;~r�~r00; t�t0;t�t00ÞEjð~r0;t0ÞEkð~r00;t00Þd~r0d~r00dt0dt00

þð¥�¥

ð¥�¥

ð¥�¥

ðt�¥

ðt�¥

ðt�¥

wð3Þijk‘ð~r�~r0;~r�~r00;~r�~r000; t�t0;t�t00;t�t000Þ

�Ejð~r0;t0ÞEkð~r00;t00ÞE‘ð~r000;t000Þd~r0d~r00d~r000dt0dt00dt000 þ ��

with

ð¥�¥

d~r0 �ð¥�¥

ð¥�¥

ð¥�¥

dx0dy0dz0 ð1:4Þ

To understand the physical implications of this formula, consider the first nonlinear

term due to the second-order susceptibility, i.e., wð2Þijk ð~r�~r0;~r�~r00; t�t0; t�t00Þ. Thepolarization Pið~r; tÞ is created at time t and position~r by two separate interactions

of the total electromagnetic field at time t0 and position~r0 and at time t00 and position~r00 in a material in which w(2) 6¼ 0. This form also includes nonlocal-in-space effects,

such as thermal nonlinearities, in which the refractive index changes due to

absorption, e.g., diffuses. In most cases encountered on optics, the response is

local in space and so

wð2Þijk ð~r�~r0;~r�~r00; t�t0; t�t00Þ ! wð2Þij ðt�t0; t�t00Þdð~r�~r0Þdð~r�~r00Þ ð1:5Þ

and

Pð2Þi ð~r; tÞ ¼ e0

ð¥�¥

ð¥�¥

wð2Þijk ðt�t0; t�t00ÞEjð~r; t0ÞEkð~r; t00Þ dt0 dt00: ð1:6Þ

Only near a “resonance” does a noninstantaneous response typically occur for

Kerr-type nonlinearities. A noninstantaneous time response translates into a fre-

quency dependence for all the susceptibilities. An example of how a noninstanta-

neous response occurs is shown in Fig. 1.4 for a simple two-level model.

4 INTRODUCTION

As the excited-state electrons relax back to the ground state, the induced polarization

relaxes back to the ground-state polarization, leading to time evolution in both the

refractive index and the absorption coefficient, as illustrated in Fig. 1.5. The Fourier

transform of this time evolution gives the frequency response.

Equation 1.4 is not the one normally used because of its complexity. Assuming

plane waves of the form

Pið~r; tÞ ¼ 1

2Piðo; zÞ e�iot þ c:c:; Eið~r; tÞ ¼ 1

2

Xm

Emi ðom; zÞ e�iomt þ c:c:;

Emj ð�omÞ ¼ E*m

j ðomÞ ð1:7Þ

and expanding again in terms of the total field gives

Piðo; zÞ e�iot ¼ e0

�wð1Þij ð�o; omÞ

Pm Em

j ðom; zÞ e�iomt

þ 1

2

Xm

Xp

wð2Þijk ð�o; � om; � opÞEmj ð�om; zÞEp

kð�op; zÞ e�ið�om�opÞt

þ 1

4

Xm

Xp

Xq

wð3Þijk‘ð�o; � om;�op;�oqÞEmj ð�om; zÞEp

kð�op; zÞ

�Eq‘ ð�oq; zÞ e�ið�om�op�oqÞt þ � � �

�ð1:8Þ

Input pulse ω ≈ ωi

(a)

Excited state

Ground stateN0

N1

(b)

Δt

(c)

Excited state (lifetime τ )

Ground state

hωiτhωi

FIGURE 1.4 (a) Two-level model with all electrons initially in the ground state N0.

(b) Incidence of a short pulse (Dt � �t) causes many electron transitions to the excited state.

(c) Excited-state population after the pulse passes.

Absorption

(a)

t = 0 t = τ

t = 2Δt t τ

(b)

(c)

(d)

AbsorptionRefractive index Refractive index

ω − ωi ω − ωi

FIGURE 1.5 (a) Spectral distribution of refractive index and absorption before the incidence

of the pulse (t¼ 0). (b)–(d) Time evolution of refractive index and absorption: t ¼ 2Dt (b); t ¼ �t(c), and Dt �t (d).

CLASSICAL NONLINEAR OPTICS EXPANSION 5

In each case, o ¼ �om � op � oq is the output frequency generated by the

interaction. The hat (roof) superscript is meant to emphasize that the quantity

underneath is a complex number.

A key question is the order of magnitude of the nonlinear susceptibilities. The

simplest atom is hydrogen. Its structure and spectrum of excited states is well known

and is simple to calculate since it has only one electron, the minimum needed for the

interaction of electromagnetic radiation with matter. The atomic Coulomb field

binding the electron to the proton in its orbit of Bohr radius rB is given by

Eatomic ¼ �e

4pe0r2B; rB ¼ 4pe0�h2

�me�e2; ð1:9Þ

in which �e is the charge on the electron (�1.6� 10�19 C), e0 is the permittivity of free

space (8.85� 10�12 F/m), �me (¼9.11� 10�31 kg) is the electron mass, and

h ¼ 2p�h ¼ 6:63� 10�34 J s is Planck’s constant. Equation 1.9 gives the order of

magnitude of Eatomic ¼ 1012 V=m. It is reasonable to adopt this field as an approx-

imate field at which nonlinear optics becomes important. Since wð1Þ ¼ n2�1 (where n

is the refractive index of the order of unity) for a perturbation expansion in terms of

products of electric fields to be valid, P(1) 10�P(2):

Pð1Þ

Pð2Þ¼ wð1Þ

wð2ÞE� 1

wð2ÞEatomic

� 10 ! wð2Þ � 10�13 m=V: ð1:10Þ

This is a reasonable estimate for the lower limit value of the second-order suscep-

tibility, especially since the field was based on hydrogen, which has only a single

electron and proton. Following the same approximations but now assuming that

Pð1Þ

Pð3Þ¼ wð1Þ

wð3ÞE2� wð1Þ

wð3ÞE2atomic

� 10 ! wð3Þ � 10�25 m2=V2: ð1:11Þ

Aswill become clear later, these approximate values are close to the minimum values

found for these susceptibilities.

1.4 SIMPLE MODEL: ELECTRON ON A SPRINGAND ITS APPLICATION TO LINEAR OPTICS

There are many physical mechanisms that lead to nonlinear optical phenomena.

Initially, the focus here is on transitions between the electronic states associated with

atoms and molecules in matter. Although the appropriate treatment (Chapter 8) for

completely describing the interaction of radiation with atoms and molecules involves

quantum mechanics, initially a simpler classical approach that provides a useful

description of the linear (and as it turns out exact) susceptibility is adopted.

As an example of this approach, consider the molecule O2 and its electron cloud,

as illustrated in Fig. 1.6. This molecule has inversion symmetry (i.e., a center of

6 INTRODUCTION

symmetry halfway between the oxygen atoms) and hence has no permanent dipole

moment since the centers of positive (nuclei) and negative charges are coincident.

When an electric field

~Eð~r; tÞ ¼ 1

2ExðoaÞ eiðkz�oatÞ þ c:c: ð1:12Þ

is applied along the molecular axis (þ x-axis), the negative and positive charges and

their centers of charge are displaced in opposite directions by the Coulomb forces,

giving rise to the forces �me;n€xe;n, where �me and xe are the electron mass and its

displacement and �mn and xn are the nuclear mass and its displacement, respectively.

Since �mn �me, only the displacements of the electrons are important for inducing

dipoles.

The electrons are bound to the nucleus (atoms) or nuclei (molecules) by Coulomb

forces and, for isolated atoms or molecules, exist in discrete states m with an energy

�hð�om��ogÞ ¼ �h�omg above the ground state and an excited-state lifetime �tmg. They

move in “orbits” around nuclei described by probability density functions �cmð~r; tÞ,with j�cmð~r; tÞj2 dx dy dz dt giving the probability that the electron “exists” at time t in

the volume element dx dy dz at position ~r. Since “optics” usually deals with the

spectral region longer in wavelength (smaller in frequency) than the low frequency

absorption edge of the material determined by the transitions between electronic

states, the electron in the lowest lying energy level is normally the prime participant

when radiation interacts with matter; i.e., it is the electron with the largest displace-

ment. With these approximations, the dipole moment induced by an electromagnetic

field is �mx ¼ �eexe, as shown in Fig. 1.6. For themost general case, �mi ¼ �aijEj , where �aijis the polarizability tensor, and the induced dipole and the electric field are not

necessarily collinear.

In linear optics, it is possible to diagonalize the polarizability tensor. The

deflections~�q ¼ ex�qx þ ey�qy þ ez�qz of this representative electron are defined in terms

of these axes and so~�m ¼ ��e~�q. (Note, however, that for very anisotropic crystal classesthe coordinate system may be nonorthogonal and/or frequency dependent.)

The Coulomb interaction between the net positive and negative charges provides a

restoring force that oscillates at the frequency of the applied field, and so themotion of

→µ

→E

FIGURE 1.6 The oxygen molecule O2, its center of mass , and positive and negative

charges in the absence of the field. After the field is applied, is the center of the negative

charge and the electron cloud is shifted (and distorted) from its original position.

SIMPLE MODEL: ELECTRON ON A SPRING AND ITS APPLICATION TO LINEAR OPTICS 7

the electron can be described as a simple harmonic oscillator. In three dimensions, this

can be visualized as the electron attached to three orthogonal springs, as illustrated in

Fig. 1.7a, and the electron motion can be described as oscillation in a harmonic

potential well.

The equation of motion of an electron is described by a simple harmonic oscillator

with the potential

�VðmÞð�qÞ ¼ 1

2�kðmÞii �q

ðmÞi �q

ðmÞi : ð1:13Þ

From classical mechanics, the restoring force is given by

�FðmÞi ¼ � @ �V

ðmÞ

@�qðmÞi

¼ � 1

2�kðmÞii

@�qðmÞi

@�qðmÞi

�qðmÞi þ �q

ðmÞi

@�qðmÞi

@�qðmÞi

" #¼ ��k

ðmÞii �q

ðmÞi ; ð1:14Þ

in which the spring constant kðmÞii is defined in terms of the excited state’s energy by

�omg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðmÞii =�me

qand the restoring force is given by �F

ðmÞi ¼ �me �o2

mg�qðmÞi : The inertial

force is �me€�qðmÞi . Therefore, the force balance equation describing the electron motion

in a simple harmonic oscillator model is

�me

h€�qðmÞi þ 2�t�1

mg�qðmÞi þ �o2

mg�qðmÞi

i¼ ��eEi

! �o2mg�o2

a�2ioa�t�1mg

h i�qðmÞi ðtÞ ¼ � �e

�me

EiðtÞ:ð1:15Þ

Assuming

�qðmÞi ¼ 1

2�QðmÞi e�ioat þ c:c: ! �Q

ðmÞi ¼ � �eEiðoaÞ

�meDðmÞi ðoaÞ

; ð1:16Þ

(a) (b)

–

(qi)V

qiqy

qzqx

FIGURE1.7 (a) Electron connected via three springs oriented along the axes that diagonalize

the polarizability. (b) One-dimensional cut of the three-dimensional parabolic potential well

inside which the electron oscillates at the frequency of an applied field.

8 INTRODUCTION

where DðmÞi ðoaÞ ¼ �o2

a�2ioa�t�1mg þ �o2

mg is the resonance denominator. When

oa � �omg, the amplitude of the displacement is enhanced. Note that �ee ¼ ��e and

that in the zero-frequency limit, DðmÞi ðoaÞ ¼ �o2

mg and �QðmÞi ¼ ��eEiðoaÞ=�me �o2

mg is

just a net steady-state displacement of the electron.

For a dilute mediumwithN noninteracting atoms (molecules) per unit volume, the

induced linear polarization and the first-order susceptibility wð1Þii ð�oa; oaÞ are givenas follows:

PiðtÞ ¼ �N�eP

m

�1

2�QðmÞi e�ioat þ c:c:

�¼ 1

2PiðoaÞ e�ioat þ c:c:

PiðoaÞ ¼ N�e2

�me

EiðoaÞXm

1

DðmÞi ðoaÞ

¼ e0wð1Þii ð�oa;oaÞEiðoaÞ

! wð1Þii ð�oa; oaÞ ¼ N�e2

e0 �meðoaÞXm

1

DðmÞi ðoaÞ

:

ð1:17Þ

The fact that wð1Þij ð�oa; oaÞ is a diagonal tensor is a direct consequence of choosing acoordinate system in which the polarizability tensor is diagonal.

The first-order susceptibility wð1Þii ð�oa; oaÞ can easily be defined in terms of

wð1Þii ð~r�~r0; t�t0Þ. From Eq. 1.4,

Pð1Þi ðtÞ ¼ e0

ðwð1Þij ðt�t0ÞEjðt0Þ dðt�t0Þ

� �¼ 1

2Pð1Þi ðoaÞ e�ioat þ c:c: ð1:18Þ

Substituting for the field Ejðt0Þ gives

Pð1Þi ðtÞ ¼ e0

2

ðwð1Þij ðt�t0Þ½EjðoaÞ e�ioat

0 þ c:c:� dðt�t0Þ� �

¼ 1

2e0EjðoaÞ e�ioat

ð¥�¥

wð1Þij ðt�t0Þ eioaðt�t0Þ dðt�t0Þ þ c:c:

! wð1Þij ð�oa; oaÞ ¼Ð¥�¥ w

ð1Þij ðt�t0Þ eioaðt�t0Þ dðt�t0Þ;

ð1:19Þ

i.e., wð1Þij ð�oa; oaÞ is the Fourier transform of wð1Þij ðt�t0Þ.Decomposing wð1Þii ð�oa; oaÞ into its real and imaginary components yields

wð1Þii ð�oa; oaÞ ¼ N�e2

�mee0

Xm

ð�o2mg�o2

aÞþ 2ioa�t�1mg

ð�o2mg�o2

aÞ2 þ 4o2a�t

�2mg

: ð1:20aÞ

SIMPLE MODEL: ELECTRON ON A SPRING AND ITS APPLICATION TO LINEAR OPTICS 9

This equation is always valid. It can be simplified near and on resonance ðoa � �omgÞto give


2�omg �mee0

Xm

�omg�oa þ i�t�1mg

ð�omg�oaÞ2 þ�t�2mg

ð1:20bÞ

and off resonance ðjomg�opj�tmg 1Þ to give


2�omg �mee0

Xm

�omg�oa þ i�t�1mg

ð�omg�oaÞ2: ð1:20cÞ

Figure 1.8 shows the frequency dispersion in the imaginary and real parts of

wð1Þii ð�oa; oaÞ for a single excited state.

The refractive indexand theabsorptioncoefficient (for thefield) aredefined in theusual

way by n2ðoÞ ¼ 1þRealfwð1Þii ð�o; oÞg and aðoÞ ¼ kvac=magfwð1Þii ð�o; oÞg=2nðoÞ, respectively. Note that the absorption spectrum, i.e., aðoÞ, has contributions

only from transitions from theground state that are electric dipole allowed. For symmetric

molecules inwhich the states are described bywave functions that are either symmetric or

antisymmetric in space, the linear absorption spectrum does not contain contributions

from the even-symmetry excited states because electric dipole transitions from the even-

symmetry ground state are not dipole allowed.

As stated previously, optics normally refers to electromagnetic waves in the

spectral region defined by frequencies below the lowest lying electronic

resonance due to electric dipole transitions. Assuming that j�omg�oaj�t�1mg 1,

=magfwð1Þii ð�oa; oaÞg decreases faster with increasing frequency difference from

the resonance j�omg�oaj than does Realfwð1Þii ð�oa; oaÞg. This will also be the casefor the real and imaginary parts of the nonlinear susceptibilities.

1.5 LOCAL FIELD CORRECTION

Although local field correction is discussed in most introductory textbooks on optics,

it will prove useful to repeat it here since the transition to nonlinear optics is not

straightforward. The preceding analysis for the linear susceptibility was for a single

isolated atom or molecule and, to a good approximation, for a dilute gas. The

situation is more complex in dense gases or condensed matter (liquids and solids)

where the atoms and molecules interact with one another via the dipole fields

induced by an applied optical field.

(a) (b)

Imaginary

0

ωa − ωmg Real

ωa − ωmg

FIGURE 1.8 Spectral dispersion of the (a) imaginary and (b) real parts of wð1Þii ð�o; oÞ.

10 INTRODUCTION

Experiments are usually performed with an optical field incident onto a nonlinear

material from another medium, typically air. Maxwell’s equations in the material

and the usual boundary conditions at the interface are valid for spatial averages of

the fields over volume elements small on the scale of a wavelength, but large on the

scale of a molecule. The “averaged” quantities also include the refractive index, the

Poynting vector, and the so-called Maxwell field, which has been written here as~Eð~r; tÞ. It is the Maxwell field that satisfies the wave equation for a material with the

averaged refractive index n.

However, at the site of a molecule the situation can be quite complex since the

dipoles induced by the Maxwell electric fields on all the molecules create their own

electric fields, whichmust be added to the “averaged” field to obtain the total (“local”)

field~E locð~r; tÞ acting on a molecule, as shown in Fig. 1.9. In the low density limit, the

dipolar fields decay essentially to zero with distance from their source dipole and so~E locð~r; tÞ ffi ~Eð~r; tÞ and the single molecule result is converted to a macroscopic

polarization by multiplying the molecular result by N, the number of molecules per

unit volume.

The situation is more complex in condensed matter. It is very difficult to

calculate the “local” field accurately because it depends on crystal symmetry,

intermolecular interactions, and so on. Standard treatments such as Lorenz–Lorenz

are only approximately valid even for isotropic and cubic crystal media. Nevertheless,

they are universally used. Here the usual formulation found in standard

electromagnetic textbooks will be followed. The dipole moments of the molecules

induced by the Maxwell field ~Eð~r; tÞ produce a Maxwell polarization~Pð~r; tÞ ¼ e0~w

ð1Þ : ~Eð~r; tÞ in the material. Consider a spherical cavity around the

molecule of interest to find the local field acting on the molecule (see Fig. 1.9c).

Assuming that the effects of the induced dipoles inside the cavity average to zero, the

polarization field outside the cavity induces charges on the walls of the cavity, which

produce an additional electric field on themolecule in the cavity (see standard texts on

electrostatics):

X~Edipolesð~r; tÞ

D E¼ 1

3e0~Pð~r; tÞ ! ~E locð~r; tÞ ¼ ~Eð~r; tÞþ 1

3e0~P

ð1Þð~r; tÞ:ð1:21Þ

(a) (b) (c)

Moleculeof interest

P(r,t)Cavity

→ →

FIGURE 1.9 (a) The local fields created by the induced dipoles in a medium. (b) Dipoles

induced everywhere in the material. The average gives the Maxwell polarization ~Pð~r; tÞ.(c) Artificial spherical cavity assumed around the molecule of interest, embedded in a uniform

medium with polarization ~Pð~r; tÞ.

LOCAL FIELD CORRECTION 11

The induced dipole on a molecule at the center of the cavity is now given by

~�pð1Þ ð~r; tÞ ¼ ~~�a ��~Eð~r; tÞþ 1

3e0~P

ð1Þð~r; tÞ�

!~Pð1Þð~r; tÞ ¼ N~~�a �

�~Eð~r; tÞþ 1

3e0~P

ð1Þð~r; tÞ�

!�1� 1

3N~~�a

��~Pð1Þð~r; tÞ ¼ N~~�a �~Eð~r; tÞ: ð1:22Þ

From the Clausius–Mossotti relation that connects themacroscopic relative dielectric

constant to the molecular polarizability,

er�1

er þ 2¼ 1

3N�ah i ! ~P

ð1Þð~r; tÞ ¼ er þ 2

3N~~�a �~Eð~r; tÞ; ð1:23Þ

and so the local field and the local field correction f (1) is defined as

~E locð~r; tÞ ¼ erðoaÞþ 2

3~Eð~r; tÞ; f ð1Þ ¼ er þ 2

3; ð1:24Þ

respectively, where erðoaÞ ¼ eðoaÞ=e0. Since the field driving the electron is now

�QðmÞi ¼ �f ð1Þ

�eEiðoaÞ�meD

ðmÞi ðoaÞ

; ð1:25Þ

the linear susceptibility from Eq. 1.20a, including the local field correction, becomes

wð1Þii ð�oa; oaÞ ¼ N�e2f ð1ÞðoaÞ�mee0

Xm

ð�o2mg�o2

aÞþ 2ioa�t�1mg

ð�o2mg�o2

aÞ2 þ 4o2a�t

�2mg

: ð1:26Þ

PROBLEMS

1. The purpose of this problem is to show that absorption decreases much faster than

refractive index with frequency difference from a resonance. Consider an isolated

molecule with a single excited state with a transition frequency �oi and a

phenomenological decay constant �t�1i .

(a) Assuming that �t�1i �oi 1 and o � �oi, show that w(1) can be written as

wð1Þii ð�o; oÞ ¼ Realfwð1Þii gþ i =magfwð1Þii g

¼ Ne2ð�oi�oÞ2�oi �me0½ð�oi�oÞ2 þ�t�2

i �

þ iNe2o�ti�1

2�o2i �me0½ð�oi�oÞ2 þ�t�2

i � :

12 INTRODUCTION

(b) Find the maximum change in the real part of the susceptibility and show that

the ratioR of change at frequencyo tomaximum change occurs at a frequency

shift given by �oi�o ffi 2�t�1i =R for j�oi�oj 2�t�1

i .

ABR =

BA ω − ωi

ωi

(c) Find themaximum change in the imaginary part of the susceptibility and show

that the ratio P of the change at frequency o to maximum change is given by

P ¼ �t�2i

½ð�oi�oÞ2 þð�t�2i Þ� :

A

B

P =BA

ω − ωi

ωi

(d) How small is P for values of R equal to 10%: 1%?

Although you have calculated this difference for the linear susceptibility, the

results are also typical of what is obtained for higher order susceptibilities. A

10% or less “remnant” in the susceptibility is the upper limit for calling the

value “nonresonant.”

SUGGESTED FURTHER READING

B. I. Bleaney and B. Bleany, Electricity andMagnetism, 2nd Edition (Oxford University Press,

London, 1968).

M. Born and E. Wolf, Principles of Optics, 7th Edition (Cambridge University Press,

Cambridge, UK, 1999).

R. W. Boyd, Nonlinear Optics, 3rd Edition (Academic Press, Burlington, MA, 2008).

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Volume 1: Linear Optics

(John Wiley & Sons, New York, 1985).

D. L. Mills, Nonlinear Optics: Basic Concepts, 2nd Edition (Springer, New York, 1998).

Y. Ron Shen, Principles of Nonlinear Optics (John Wiley & Sons, New York, 1984).

SUGGESTED FURTHER READING 13

nonlinear optics - startseite€¦ · molecular nonlinear optics 197 9.1 two-level model 198 9.2...

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