Library of Congress Cataloging-in-Publication Data

Powers, Peter E.

Field guide to nonlinear optics / Peter E. Powers.

pages cm. – (The field guide series; FG29)

Includes bibliographical references and index.

ISBN 978-0-8194-9635-5—ISBN 978-0-8194-9636-2—ISBN 978-0-8194-9637-9

1. Nonlinear optics. I. Title.

QC446.2.P688 2013

621.36'94−dc23

2013014471

Published by

SPIE

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: +1.360.676.3290

Fax: +1.360.647.1445

Email: books@spie.org

The content of this book reflects the work and thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

First printing

## Introduction to the Series

Welcome to the * SPIE Field Guides*—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at

**fieldguides@SPIE.org**.

John E. Greivenkamp, **Series Editor**

College of Optical Sciences

The University of Arizona

## The Field Guide Series

, Second Edition, Robert Tyson & Benjamin Frazier**Adaptive Optics**, Larry Andrews**Atmospheric Optics**, Paul Yoder, Jr. & Daniel Vukobratovich**Binoculars and Scopes**, Yakov Soskind**Diffractive Optics**, John Greivenkamp**Geometrical Optics**, Angelo Arecchi, Tahar Messadi, & John Koshel**Illumination**, Khan M. Iftekharuddin & Abdul Awwal**Image Processing**, 2nd Edition, Arnold Daniels**Infrared Systems, Detectors, and FPAs**, Eric Goodwin & Jim Wyant**Interferometric Optical Testing**, Rüdiger Paschotta**Laser Pulse Generation**, Rüdiger Paschotta**Lasers**, Julie Bentley & Craig Olson**Lens Design**, Tomasz Tkaczyk**Microscopy**, Peter Powers**Nonlinear Optics**, Ray Williamson**Optical Fabrication**, Rüdiger Paschotta**Optical Fiber Technology**, Chris Mack**Optical Lithography**, Ronald Willey**Optical Thin Films**, Katie Schwertz & James Burge**Optomechanical Design and Analysis**, Daniel Smith**Physical Optics**, Edward Collett**Polarization**, Larry Andrews**Probability, Random Processes, and Random Data Analysis**, Barbara Grant**Radiometry**, Larry Andrews**Special Functions for Engineers**, David Ball**Spectroscopy**, Créidhe O’Sullivan & J. Anthony Murphy**Terahertz Sources, Detectors, and Optics**, Jim Schwiegerling**Visual and Ophthalmic Optics**

## Field Guide to Nonlinear Optics

This Field Guide is designed for those looking for a condensed and concise source of key concepts, equations, and techniques for nonlinear optics. Topics covered include technologically important effects, recent developments in nonlinear optics, and linear optical properties central to nonlinear phenomena. The focus of each section is based on my research, my interactions with colleagues in the field, and my experiences teaching nonlinear optics.

Examples throughout this Field Guide illustrate fundamental concepts while demonstrating the application of key equations. Equations are presented without proof or derivation; however, the interested reader may refer to the bibliography for a list of resources that go into greater detail. In addition to the overview of nonlinear phenomena, this Field Guide includes an appendix of material properties for some commonly used nonlinear crystals.

This Field Guide features notations commonly encountered in nonlinear optics literature. All equations are written in SI units for convenience when comparing calculations to laboratory measurements. The formalism of writing equations using complex variable notation introduces ambiguity in defining the electric field’s complex amplitude. Though one convention is used throughout this text, conventions and conversions are presented as part of the first topic. Equations in terms of experimentally measured quantities such as power, intensity, and energy, have no ambiguity.

Peter E. Powers

University of Dayton

March 2013

## Table of Contents

**Glossary of Terms and Acronyms x**

**Electromagnetic Waves and Crystal Optics 1**

Conventions and Conversions 1

Maxwell’s Equations and the Wave Equation 2

Uniaxial Crystals 3

Biaxial Crystals 4

**Nonlinear Susceptibility 5**

Nonlinear Polarization for Parametric Interactions 5

Classical Expressions for Nonlinear Susceptibility 6

Nonlinear Susceptibilities 7

* d* Matrices 8

Working with * d* Matrices and SHG 9

Effective Nonlinearities 10

Tabulation of * d* Matrices 11

**Electro-optic Effect 13**

Electro-optic Effect 13

* r* Matrices 14

Electro-optic Waveplates 16

Q Switches 17

Amplitude and Phase Modulators 18

Electro-optic Sampling for Terahertz Detection 19

Photorefraction 20

**χ ^{(2)} Parametric Processes 21**

χ^{(2)} Coupled Amplitude Equations 21

χ^{(2)} Processes with Focused Gaussian Beams 22

DFG and OPA 23

Sum-Frequency Generation 24

Second-Harmonic Generation 25

Three-Wave Mixing Processes with Depletion 26

Optical Parametric Generation 27

Optical Parametric Oscillator 28

Singly Resonant Optical Parametric Oscillator 29

**Phase Matching 30**

Birefringent Phase Matching 30

e- and o-Wave Phase Matching 31

DFG and SFG Phase Matching for Uniaxial Crystals 32

SHG Phase Matching for Uniaxial Crystals 33

Biaxial Crystals in the * XY* Plane 34

Biaxial Crystals in the * YZ* Plane 35

Biaxial Crystals in the * XZ* Plane 36

Quasi-phase-matching 37

Birefringent versus Quasi-phase-matching 38

Noncollinear Phase Matching 39

Tuning Curves 40

**Phase Matching Bandwidth 41**

Bandwidths for DFG and SFG 41

Bandwidth Calculation Aids 42

SFG and DFG Bandwidth Formulae 43

SHG Bandwidth Formulae 46

Graphical Approach for Bandwidths 48

Noncollinear Bandwidth 49

**χ ^{(2)} Waveguides 50**

χ^{(2)} Waveguide Interactions 50

χ^{(2)} Waveguide Devices 51

Waveguide Phase Matching 52

**χ ^{(3)} Parametric Processes 53**

Four-Wave Mixing 53

Degenerate Four-Wave Mixing 54

Third-Harmonic Generation 55

χ^{(3)} Parametric Amplifier 56

Noncollinear Phase Matching for χ^{(3)} Processes 57

**Nonlinear Refractive Index 58**

Nonlinear Refractive Index 58

Nonlinear Absorption 59

Calculations of Nonlinear Index 60

Self-Phase Modulation 61

* z* Scan 62

Optical Bistability 63

**Raman and Brillouin Processes 64**

Spontaneous Raman Scattering 64

Stimulated Raman Scattering 65

Anti-Stokes Raman Scattering 66

Raman Microscopy 67

Photo-acoustic Interactions 68

Stimulated Brillouin Scattering 69

**Ultrafast Nonlinear Effects 70**

Saturable Absorption 70

Temporal Solitons 71

Spatial Solitons 72

High Harmonic Generation 73

Ultrashort-Pulse Measurement 74

**Appendices 75**

Gaussian Beams 75

Sellmeier Equations for Selected χ^{(2)} Crystals 76

Properties of Selected χ^{(2)} Crystals 78

References for Selected χ^{(2)} Crystals Tables 79

**Equation Summary 81**

**Bibliography 90**

**Index 93**

#### Glossary of Terms and Acronyms

#### Constants

**c**

Speed of light in vacuum (299,792,458 m/sec)

**e**

Elementary charge (1.6022 × 10^{−19} C)

**h**

Planck’s constant (6.6261 × 10^{−34} J · sec)

ℏ

* h*/2π (1.0546 × 10

^{−34}J · sec)

**k _{B}**

Boltzmann constant (1.3807 × 10^{−23} J/K)

**m _{e}**

Electron mass (9.1094 × 10^{−31} kg)

${\text{\epsilon}}_{0}$

Vacuum permittivity (8.8542 × 10^{−2} F/m)

${\text{\mu}}_{0}$

Vacuum permeability (4π × 10^{−7} N/A^{2})

#### General

3PA

Three-photon absorption

**A**

Electric field’s complex amplitude scalar

**A**

Electric field’s complex amplitude vector

AC

Autocorrelation

AO

Acousto-optic

**b**

Confocal distance (2* z_{R}*)

**B**

Magnetic induction scalar

$B$

Magnetic induction vector

BBO

b-BaB_{2}O_{4}, b-barium borate

BIBO

BiB_{3}O_{6}, bismuth triborate

BPM

Birefringent phase matching

CARS

Coherent anti-Stokes Raman spectroscopy

**c.c.**

Complex conjugate

cgs

Centimeter-gram-second

**D**

Electric displacement vector

**d _{eff}**

Effective second-order nonlinearity

DFG

Difference-frequency generation

DFWM

Degenerate four-wave mixing

**d _{ijk}**

* d* tensor

DKDP

KD_{2}PO_{4}, deuterated potassium dihydrogen phosphate

DR-OPO

Doubly-resonant optical parametric oscillator

**e**

Extraordinary wave (e-wave)

**E**

Electric field scalar

**E**

Electric field vector

EO

Electro-optic

**E _{SC}**

Space-charge electric field

**f**

Focal length of a lens

FROG

Frequency-resolved optical gating

fsec

Femtosecond (10^{–15} sec)

FWHM

Full-width at half-maximum

FWM

Four-wave mixing

GaAs

Gallium arsenide

GaP

Gallium phosphide

**g _{B}**

Brillouin intensity gain factor

**g _{R}**

Raman intensity gain factor

GVD

Group velocity dispersion

$H$

Magnetic field

HHG

High harmonic generation

**i**

$\sqrt{-1}$

**I**

Light intensity

**I _{o}**

On-axis intensity

**j**
_{f}

Free current density

**k**

Wavevector magnitude

**k**

Wavevector

**K**

Acoustic wavevector

KDP

KH_{2}PO_{4}, potassium dihydrogen phosphate

KTP

KTiOPO_{4}, potassium titanyl phosphate

**L**

Interaction length

LBO

LiB_{3}O_{5}, lithium triborate

**L _{c}**

Coherence length

LiNbO_{3}

Lithium niobate

LiTaO_{3}

Lithium tantalite

LN

Lithium niobate

$M$

Magnetization

MPA

Multi-photon absorption

**n**

Refractive index/index of refraction

* n_{e}*(θ)

Extraordinary refractive index

${n}_{2}^{I}$

Nonlinear index intensity coefficient

NL

Nonlinear

NLA

Nonlinear absorption

NLSE

Nonlinear Schrödinger equation

**n _{o}**

Ordinary refractive index

nsec

Nanosecond (10^{−9} sec)

${n}_{X},\text{\hspace{0.17em}}{n}_{Y},\text{\hspace{0.17em}}{n}_{Z}$

Principal indices (or eigenindices)

**o**

Ordinary wave (o-wave)

$\widehat{o}$

Ordinary wave unit vector

OPA

Optical parametric amplification

OPG

Optical parametric generation

OPO

Optical parametric oscillator

**P**

Optical power

**P**^{(1)}

Linear material polarization

**P**^{(2)}

Second-order nonlinear polarization

**P**^{(3)}

Third-order nonlinear polarization

PCM

Phase-conjugate mirror

pm

Picometer

**P**^{(NL)}

Nonlinear polarization

PPLN

Periodically poled lithium niobate

psec

Picosecond (10^{−12} sec)

${P}_{TH}$

Threshold power

**q**

Gaussian beam parameter

QPM

Quasi-phase-matching

QWP

Quarter-wave plate

**r**

Electro-optic coefficient

**R**

Power reflectivity

$S$

Poynting vector

SBS

Spontaneous Brillouin scattering

SESAM

Semiconductor saturable absorber mirror

SFG

Sum-frequency generation

SHG

Second-harmonic generation

sinc(* x*)

sin(* x*)/

**x**SPM

Self-phase modulation

SPS

Spontaneous parametric scattering

SR-OPO

Singly-resonant optical parametric oscillator

SVEA

Slowly varying envelope approximation

**T**

Temperature

THG

Third-harmonic generation

TPA

Two-photon absorption

**U _{p}**

Ponderomotive energy

**v _{g}**

Group velocity

**w _{o}**

Radius at the beam waist

* w*(

*)*

**z**Beam radius

**Z**

Optic axis

ZGP

ZnGeP_{2}, zinc germanium phosphide

ZnTe

Zinc telluride

**z _{R}**

Rayleigh range

α

Linear absorption coefficient

β

Waveguide propagation coefficient

β (TPA)

Two-photon absorption coefficient

Δ

Miller’s delta

Δk

* k*-vector mismatch

${\text{\epsilon}}_{ij}$

Permittivity tensor

θ_{PM}

Phase matching angle (polar angle)

Λ

Quasi-phase-matching periodicity

λ

Vacuum wavelength

λ*_{I}* (λ

_{3})

Idler wavelength

λ*_{P}* (λ

_{1})

Pump wavelength

λ*_{S}* (λ

_{2})

Signal wavelength

ξ

Focusing parameter

ρ

Poynting vector walk-off angle

${\text{\rho}}_{f}$

Free charge density

σ_{R}

Raman scattering cross-section

$\tau $

Pulse duration

φ

Azimuthal angle

${\chi}_{ij}^{\left(1\right)}$

Linear susceptibility tensor

${\chi}_{\mathit{eff}}^{\left(2\right)}$

Effective nonlinear susceptibility

${\chi}_{ijk}^{\left(2\right)}$

Second-order nonlinear susceptibility tensor

${\chi}_{ijkl}^{\left(3\right)}$

Third-order nonlinear susceptibility tensor

${\chi}_{B}^{\left(3\right)}$

Brillouin susceptibility

${\chi}_{R}^{\left(3\right)}$

Raman susceptibility

ψ

Noncollinear angle

ω

Angular frequency

ω_{AS}

Anti-Stokes angular frequency

ω_{S}

Signal (parametric process) or Stokes frequency (Raman process)

Ω_{B}

Brillouin frequency