Nonlinear Optics E-Book

George I. Stegeman

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Pure and Applied Optics
• Sprache: Englisch

Seitenzahl: 541

Veröffentlichungsjahr: 2012

Beschreibung

Clear, integrated coverage of all aspects of nonlinear optics--phenomena, materials, and devices Coauthored by George Stegeman, one of the most highly respected pioneers of nonlinear optics--with contributions on applications from Robert Stegeman--this book covers nonlinear optics from a combined physics, optics, materials science, and devices perspective. It offers a thoroughly balanced treatment of concepts, nonlinear materials, practical aspects of nonlinear devices, and current application areas. Beginning with the presentation of a simple electron on a spring model--to help readers make the leap from concepts to applications--Nonlinear Optics gives comprehensive explanations of second-order phenomena, derivation of nonlinear susceptibilities, third-order nonlinear effects, multi-wave mixing, scattering, and more. Coverage includes: * Nonlinear response of materials at the molecular level * Second-order nonlinear devices, their optimization and limitations * The physical origins of second- and third-order nonlinearities * Typical frequency dispersion of nonlinearities, explained in terms of simple two- and three-level models * Ultrafast and ultrahigh intensity processes * Practice problems demonstrating the design of such nonlinear devices as frequency doublers and optical oscillators Based on more than twenty years of lectures at the College of Optics and Photonics (CREOL) at the University of Central Florida, Nonlinear Optics introduces all topics from the ground up, making the material easily accessible not only for physicists, but also for chemists and materials scientists, as well as professionals in diverse areas of optics, from laser physics to electrical engineering.

Leseprobe

Contents

Cover

Wiley Series in Pure and Applied Optics

Title Page

Copyright

Preface

Reference

Chapter 1: Introduction

1.1 What is Nonlinear Optics and What is It Good For?

1.2 Notation

1.3 Classical Nonlinear Optics Expansion

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

1.5 Local Field Correction

Problems

Suggested Further Reading

Part A: Second-Order Phenomena

Chapter 2: Second-Order Susceptibility and Nonlinear Coupled Wave Equations

2.1 Anharmonic Oscillator Derivation of Second-Order Susceptibilities

2.2 Input Eigenmodes, Permutation Symmetry, and Properties of χ(2)

2.3 Slowly Varying Envelope Approximation

2.4 Coupled Wave Equations

2.5 Manley–Rowe Relations and Energy Conservation

Problems

Suggested Further Reading

Chapter 3: Optimization and Limitations of Second-Order Parametric Processes

3.1 Wave-Vector Matching

3.2 Optimizing

3.3 Numerical Examples

Problems

References

Suggested Further Reading

Chapter 4: Solutions for Plane-Wave Parametric Conversion Processes

4.1 Solutions of the Type 1 SHG Coupled Wave Equations

4.2 Solutions of the Three-Wave Coupled Equations

4.3 Characteristic Lengths

4.4 Nonlinear Modes

Problems

References

Suggested Further Reading

Chapter 5: Second Harmonic Generation with Finite Beams and Applications

5.1 SHG with Gaussian Beams

5.2 Unique and Performance-Enhanced Applications of Periodically Poled LiNbO3(PPLN)

Problems

References

Suggested Further Reading

Chapter 6: Three-Wave Mixing, Optical Amplifiers, and Generators

6.1 Three-Wave Mixing Processes

6.2 Manley–Rowe Relations

6.3 Sum Frequency Generation

6.4 Optical Parametric Amplifiers

6.5 Optical Parametric Oscillator

6.6 Mid-Infrared Quasi-Phase Matching Parametric Devices

Problems

References

Selected further reading

Chapter 7: χ Materials and Their Characterization

7.1 Survey of Materials

7.2 Oxide-Based Dielectric Crystals

7.3 Organic Materials

7.4 Measurement Techniques

Appendix 7.1: Quantum Mechanical Model for Charge Transfer Molecular Nonlinearities

References

Suggested Further Reading

Part B: Nonlinear Susceptibilities

Chapter 8: Second- and Third-Order Susceptibilities: Quantum Mechanical Formulation

8.1 Perturbation Theory of Field Interaction with Molecules

8.2 Optical Susceptibilities

Appendix 8.1:

Reference

Suggested Further Reading

Chapter 9: Molecular Nonlinear Optics

9.1 Two-Level Model

9.2 Symmetric Molecules

9.3 Density Matrix Formalism

Appendix 9.1: Two-Level Model for Asymmetric Molecules—Exact Solution

Appendix 9.2: Three-Level Model for Symmetric Molecules—Exact Solution

Problems

References

Suggested Further Reading

Part C: Third-Order Phenomena

Chapter 10: Kerr Nonlinear Absorption and Refraction

10.1 Nonlinear Absorption

10.2 Nonlinear Refraction

10.3 Useful NLR Formulas and Examples (Isotropic Media)

Problems

Suggested Further Reading

Chapter 11: Condensed Matter Third-Order Nonlinearities due to Electronic Transitions

11.1 Device-Based Nonlinear Material Figures of Merit

11.2 Local Versus Nonlocal Nonlinearities in Space and Time

11.3 Survey of Nonlinear Refraction and Absorption Measurements

11.4 Electronic Nonlinearities Involving Discrete States

11.5 Overview of Semiconductor Nonlinearities

11.6 Glass Nonlinearities

Appendix 11.1: Expressions for the Kerr, Raman, and Quadratic Stark Effects

Problems

References

Suggested Further Reading

Chapter 12: Miscellaneous Third-Order Nonlinearities

12.1 Molecular Reorientation Effects in Liquids and Liquid Crystals

12.2 Photorefractive Nonlinearities

12.3 Nuclear (Vibrational) Contributions to n2||(−ω; ω)

12.4 Electrostriction

12.5 Thermo-Optic Effect

12.6 χ(3) Via Cascaded χ(2) Nonlinear Processes: Nonlocal

Appendix 12.1: Spontaneous Raman Scattering

Problems

References

Suggested Further Reading

Chapter 13: Techniques for Measuring Third-Order Nonlinearities

13.1 Z-Scan

13.2 Third Harmonic Generation

13.3 Optical Kerr Effect Measurements

13.4 Nonlinear Optical Interferometry

13.5 Degenerate Four-Wave Mixing

Problems

References

Suggested Further Reading

Chapter 14: Ramifications and Applications of Nonlinear Refraction

14.1 Self-Focusing and Defocusing of Beams

14.2 Self-Phase Modulation and Spectral Broadening in Time

14.3 Instabilities

14.4 Solitons (Nonlinear Modes)

14.5 Optical Bistability

14.6 All-Optical Signal Processing and Switching

Problems

References

Suggested Further Reading

Chapter 15: Multiwave Mixing

15.1 Degenerate Four-Wave Mixing

15.2 Degenerate Three-Wave Mixing

15.3 Nondegenerate Wave Mixing

Problems

Reference

Suggested Further Reading

Chapter 16: Stimulated Scattering

16.1 Stimulated Raman Scattering

16.2 Stimulated Brillouin Scattering

Problems

References

Suggested Further Reading

Chapter 17: Ultrafast and Ultrahigh Intensity Processes

17.1 Extended Nonlinear Wave Equation

17.2 Formalism for Ultrafast Fiber Nonlinear Optics

17.3 Examples of Nonlinear Optics in Fibers

17.4 High Harmonic Generation

References

Suggested Further Reading

Appendix: Units, Notation, and Physical Constants

A.1. Units of Third-Order Nonlinearity

A.2. Values of Useful Constants

Index

Wiley Series in Pure and Applied Optics

Founded by Stanley S. Ballard, University of Florida

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

BARRETT AND MYERS · Foundations of Image Science

BEISER · Holographic Scanning

BERGER-SCHUNN · Practical Color Measurement

BOYD · Radiometry and The Detection of Optical Radiation

BUCK -Fundamentals of Optical Fibers, Second Edition

CATHEY · Optical Information Processing and Holography

CHUANG · Physics of Optoelectronic Devices

DELONE AND KRAINOV · Fundamentals of Nonlinear Optics of Atomic Gases

DERENIAK AND BOREMAN · Infrared Detectors and Systems

DERENIAK AND CROWE · Optical Radiation Detectors

DE VANY · Master Optical Techniques

ERSOY · Diffraction, Fourier Optics and Imaging

GASKILL · Linear Systems, Fourier Transform, and Optics

GOODMAN · Statistical Optics

HOBBS · Building Electro-Optical Systems: Making It All Work, Second Edition

HUDSON · Infrared System Engineering

IIZUKA · Elements of Photonics, Volume I: In Free Space and Special Media

IIZUKA · Elements of Photonics, Volume II: For Fiber and Integrated Optics

JUDD AND WYSZECKI · Color in Business, Science, and Industry, Third Edition

KAFRI AND GLATT · The Physics of Moire Metrology

KAROW · Fabrication Methods for Precision Optics

KLEIN AND FURTAK · Optics, Second Edition

MA AND ARCE · Computational Lithography

MALACARA · Optical Shop Testing, Third Edition

MILONNI AND EBERLY · Lasers

NASSAU · The Physics and Chemistry of Color: The Fifteen Causes of Color, Second Edition

NIETO-VESPERINAS · Scattering and Diffraction in Physical Optics

OSCHE · Optical Detection Theory for Laser Applications

O'SHEA · Elements of Modern Optical Design

OZAKTAS · The Fractional Fourier Transform

PRATHER, SHI, SHARKAWY, MURAKOWSKI, AND SCHNEIDER · Photonic Crystals: Theory, Applications, and Fabrication

SALEH AND TEICH · Fundamentals of Photonics, Second Edition

SCHUBERT AND WILHELMI · Nonlinear Optics and Quantum Electronics

SHEN · The Principles of Nonlinear Optics

STEGEMAN AND STEGEMAN · Nonlinear Optics: Phenomena, Materials, and Devices

UDD · Fiber Optic Sensors: An Introduction for Engineers and Scientists

UDD · Fiber Optic Smart Structures

VANDERLUGT · Optical Signal Processing

VEST · Holographic Interferometry

VINCENT · Fundamentals of Infrared Detector Operation and Testing

WEINER · Ultrafast Optics

WILLIAMS AND BECKLUND · Introduction to the Optical Transfer Function

WU AND REN · Introduction to Adaptive Lenses

WYSZECKI AND STILES · Color Science: Concepts and Methods, Quantitative Data and Formulae, Second Edition

XU AND STROUD · Acousto-Optic Devices

YAMAMOTO · Coherence, Amplification, and Quantum Effects in Semiconductor Lasers

YARIV AND YEH · Optical Waves in Crystals

YEH · Optical Waves in Layered Media

YEH · Introduction to Photorefractive Nonlinear Optics

YEH AND GU · Optics of Liquid Crystal Displays, Second Edition

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

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

ISBN: 978-1-118-07272-1

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 the Nobel 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 the American Institute of Physics, the Optical Society of America, and other scientific organizations.

The fundamental principles of nonlinear optics are now well 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 applications and experiments. Of the two branches of nonlinear optics—nonlinear phenomena 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 helpful, it is not a requirement. When quantum mechanics 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 to Maxwell's equations lead to “orthogonal” eigenmodes, i.e., summing the fields due to two or more 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 a waveguide 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 susceptibilities 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 sum of the Maxwell fields and all the induced dipole fields at the site of an atom or a molecule. A rigorous and satisfactory approach to accurately estimating the local fields has been a subject of continuing discussion for many years. Hence this approach does not necessarily yield reliable numbers for measured nonlinear susceptibilities, but does give fairly accurately many key features, including the frequency spectrum of 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 () is , the mass of an electron is , the reduced mass for the βth vibration in a molecule is , 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.

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).

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 in my 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). A very 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.

Figure 1.1 (a) Second and third harmonic generation. (b) Nonlinear transmission. (c) Nonlinear fringe shift between low and high intensity inputs.

Figure 1.2 (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.

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 amplifiers (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 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).

Figure 1.3 (a) An optical parametric oscillator (OPO) for producing two tunable frequencies. (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.

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 been made to be consistent and to introduce more descriptive notations. The assumptions and notation used here are as follows:

1.Plane waves will 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

(1.1)

3. The unit vector is written as and has components , where i = x, y, z.

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

5. Quantities with a “bar” above, e.g., , refer to individual molecular properties in the absence of interaction with other molecules as well as parameters in a single molecule's frame of reference.

6. Quantities with a “tilde” above, e.g., , identify parameters and coefficients in the “zero (nonresonant) frequency” limit (Kleinman limit) , i.e., at frequencies much smaller than any resonant frequency of the material.

7. SI units are used throughout. Here intensity is used to mean power per unit area, usually in units of watts 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 is not the Fourier transform of and its use is restricted to Eq. 1.1. For a monochromatic wave of frequency , is the notation used for the Fourier transform of the field and . The relations between the two can be derived easily from the unitary Fourier transform equations:

(1.2)

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

(1.3)

If fields have a distribution of frequencies, then the δ functions are replaced by g(ω − ωa) and g(ω + ωa), 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 and most general expansion of the nonlinear polarization induced by the mixing of optical fields is

(1.4)

To understand the physical implications of this formula, consider the first nonlinear term due to the second-order susceptibility, i.e., . The polarization is created at time t and position by two separate interactions of the total electromagnetic field at time and position and at time and position in a material in which χ(2) ≠ 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

(1.5)

and

(1.6)

Only near a “resonance” does a noninstantaneous response typically occur for Kerr-type nonlinearities. A noninstantaneous time response translates into a frequency dependence for all the susceptibilities. An example of how a noninstantaneous response occurs is shown in Fig. 1.4 for a simple two-level model.

Figure 1.4 (a) Two-level model with all electrons initially in the ground state N0. (b) Incidence of a short pulse () causes many electron transitions to the excited state. (c) Excited-state population after the pulse passes.

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.

Figure 1.5 (a) Spectral distribution of refractive index and absorption before the incidence of the pulse (t = 0). (b)–(d) Time evolution of refractive index and absorption: (b); (c), and (d).

Equation (1.4) is not the one normally used because of its complexity. Assuming plane waves of the form

(1.7)

and expanding again in terms of the total field gives

(1.8)

In each case, 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

(1.9)

in which is the charge on the electron (−1.6 × 10−19 C), ε0 is the permittivity of free space (8.85 × 10−12 F/m), (=9.11 × 10−31 kg) is the electron mass, and is Planck's constant. Equation (1.9) gives the order of magnitude of . It is reasonable to adopt this field as an approximate field at which nonlinear optics becomes important. Since (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):

(1.10)

This is a reasonable estimate for the lower limit value of the second-order susceptibility, especially since the field was based on hydrogen, which has only a single electron and proton. Following the same approximations but now assuming that

(1.11)

As will become clear later, these approximate values are close to the minimum values found for these susceptibilities.

1.4 Simple Model: Electron on a Spring and 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 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

(1.12)

is applied along the molecular axis (+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 , where and are the electron mass and its displacement and and are the nuclear mass and its displacement, respectively. Since , only the displacements of the electrons are important for inducing dipoles.

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.

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 above the ground state and an excited-state lifetime . They move in “orbits” around nuclei described by probability density functions , with giving the probability that the electron “exists” at time t in the volume element dx dy dz at position . 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 displacement. With these approximations, the dipole moment induced by an electromagnetic field is , as shown in Fig. 1.6. For the most general case, , where is 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 of this representative electron are defined in terms of these axes and so . (Note, however, that for very anisotropic crystal classes the 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 the motion of 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.

Figure 1.7 (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.

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

(1.13)

From classical mechanics, the restoring force is given by

(1.14)

in which the spring constant is defined in terms of the excited state's energy by and the restoring force is given by The inertial force is . Therefore, the force balance equation describing the electron motion in a simple harmonic oscillator model is

(1.15)

Assuming

(1.16)

where is the resonance denominator. When , the amplitude of the displacement is enhanced. Note that and that in the zero-frequency limit, and is just a net steady-state displacement of the electron.

For a dilute medium with N noninteracting atoms (molecules) per unit volume, the induced linear polarization and the first-order susceptibility are given as follows:

(1.17)

The fact that is a diagonal tensor is a direct consequence of choosing a coordinate system in which the polarizability tensor is diagonal.

The first-order susceptibility can easily be defined in terms of . From Eq. 1.4,

(1.18)

Substituting for the field gives

(1.19)

i.e., is the Fourier transform of .

Decomposing into its real and imaginary components yields

(1.20a)

This equation is always valid. It can be simplified near and on resonance to give

(1.20b)

and off resonance to give

(1.20c)

Figure 1.8 shows the frequency dispersion in the imaginary and real parts of for a single excited state.

Figure 1.8 Spectral dispersion of the (a) imaginary and (b) real parts of .

The refractive index and the absorption coefficient (for the field) are defined in the usual way by and , respectively. Note that the absorption spectrum, i.e., , has contributions only from transitions from the ground state that are electric dipole allowed. For symmetric molecules in which the states are described by wave 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 , decreases faster with increasing frequency difference from the resonance than does . This will also be the case for 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.

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 . 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, which must be added to the “averaged” field to obtain the total (“local”) field 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 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.

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 . (c) Artificial spherical cavity assumed around the molecule of interest, embedded in a uniform medium with polarization .

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 produce a Maxwell polarization 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 the molecule in the cavity (see standard texts on electrostatics):

(1.21)

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

(1.22)

From the Clausius–Mossotti relation that connects the macroscopic relative dielectric constant to the molecular polarizability,

(1.23)

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

(1.24)

respectively, where . Since the field driving the electron is now

(1.25)

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

(1.26)

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 and a phenomenological decay constant .

a. Assuming that and , show that χ(1) can be written as

b. Find the maximum change in the real part of the susceptibility and show that the ratio R of change at frequency ω to maximum change occurs at a frequency shift given by for .

c. Find the maximum change in the imaginary part of the susceptibility and show that the ratio P of the change at frequency ω to maximum change is given by

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 and Magnetism, 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).

Part A

Second-Order Phenomena

Chapter 2

Second-Order Susceptibility and Nonlinear Coupled Wave Equations

It will be shown in Chapter 8 that the electron on a spring model does not accurately give the frequency dispersion of nonlinear susceptibilities. Nevertheless, for nonlinear optics in second-order materials, it does provide an approximate spectral distribution of susceptibilities but not their actual magnitudes. This is adequate at this stage since the magnitudes of the coefficients are normally obtained experimentally and they are used near the nonresonant limit, i.e., suitably far from the resonances associated with the electronic molecular states.

The simple harmonic oscillator model for the linear susceptibility obtained in Chapter 1 is now extended by adding an additional (cubic) term to the potential well. An expression for the second-order susceptibility is then derived by using an anharmonic oscillator model. The exact quantum mechanical derivation of second-order susceptibilities in terms of measurable molecular parameters will be given in Chapter 8. The anharmonic oscillator model is sufficiently accurate in the spectral regions far away from the resonances associated with transitions between energy levels. For applications, the measured susceptibilities are used since there is no reliable way to calculate the nonlinear force constant required in the anharmonic oscillator model.