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 SeriesWelcome 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 SPIE Field Guides 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. 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 SeriesKeep information at your fingertips with all of the titles in the Field Guide Series:
Field Guide to Nonlinear OpticsThis 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 ContentsGlossary 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 AcronymsConstantsc 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) kB Boltzmann constant (1.3807 × 10−23 J/K) me Electron mass (9.1094 × 10−31 kg)
Vacuum permittivity (8.8542 × 10−2 F/m)
Vacuum permeability (4π × 10−7 N/A2) General3PA 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 (2zR) B Magnetic induction scalar
Magnetic induction vector BBO b-BaB2O4, b-barium borate BIBO BiB3O6, bismuth triborate BPM Birefringent phase matching CARS Coherent anti-Stokes Raman spectroscopy c.c. Complex conjugate cgs Centimeter-gram-second D Electric displacement vector deff Effective second-order nonlinearity DFG Difference-frequency generation DFWM Degenerate four-wave mixing dijk d tensor DKDP KD2PO4, 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 ESC 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 gB Brillouin intensity gain factor gR Raman intensity gain factor GVD Group velocity dispersion
Magnetic field HHG High harmonic generation i
I Light intensity Io On-axis intensity j f Free current density k Wavevector magnitude k Wavevector K Acoustic wavevector KDP KH2PO4, potassium dihydrogen phosphate KTP KTiOPO4, potassium titanyl phosphate L Interaction length LBO LiB3O5, lithium triborate Lc Coherence length LiNbO3 Lithium niobate LiTaO3 Lithium tantalite LN Lithium niobate
Magnetization MPA Multi-photon absorption n Refractive index/index of refraction ne(θ) Extraordinary refractive index
Nonlinear index intensity coefficient NL Nonlinear NLA Nonlinear absorption NLSE Nonlinear Schrödinger equation no Ordinary refractive index nsec Nanosecond (10−9 sec)
Principal indices (or eigenindices) o Ordinary wave (o-wave)
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)
Threshold power q Gaussian beam parameter QPM Quasi-phase-matching QWP Quarter-wave plate r Electro-optic coefficient R Power reflectivity
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 Up Ponderomotive energy vg Group velocity wo Radius at the beam waist w(z) Beam radius Z Optic axis ZGP ZnGeP2, zinc germanium phosphide ZnTe Zinc telluride zR Rayleigh range α Linear absorption coefficient β Waveguide propagation coefficient β (TPA) Two-photon absorption coefficient Δ Miller’s delta Δk k-vector mismatch
Permittivity tensor θPM Phase matching angle (polar angle) Λ Quasi-phase-matching periodicity λ Vacuum wavelength λI (λ3) Idler wavelength λP (λ1) Pump wavelength λS (λ2) Signal wavelength ξ Focusing parameter ρ Poynting vector walk-off angle
Free charge density σR Raman scattering cross-section
Pulse duration φ Azimuthal angle
Linear susceptibility tensor
Effective nonlinear susceptibility
Second-order nonlinear susceptibility tensor
Third-order nonlinear susceptibility tensor
Brillouin susceptibility
Raman susceptibility ψ Noncollinear angle ω Angular frequency ωAS Anti-Stokes angular frequency ωS Signal (parametric process) or Stokes frequency (Raman process) ΩB Brillouin frequency |
CITATIONS
Nonlinear optics
Nonlinear crystals
Crystals
Laser crystals
Phase matching
Adaptive optics
Atmospheric optics