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Nonlinear Wave Propagation and Imaging in Deterministic and Random Media Nonlinear Wave Propagation and Imaging in Deterministic and Random Media by Wei Li A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Applied and Interdisciplinary Mathematics) in the University of Michigan 2016 Doctoral Committee: Professor Liliana Borcea, Co-Chair Professor John C. Schotland, Co-Chair Professor Peter D. Miller Professor Theodore B. Norris Emeritus Professor Jeffrey B. Rauch c Wei Li 2016 All Rights Reserved Acknowledgements I would like to express my sincere gratitude to my advisors Professor Liliana Borcea and Professor John C. Schotland, for all the time that they invested, all the support that they offered and all the guidance and insight that they provided. I would also like to thank the rest of my committee members, Professor Peter D. Miller, Professor Theodore B. Norris and Professor Jeffrey B. Rauch, for their valuable discussions and suggestions. ii Table of Contents Acknowledgements ii List of Figures viii Abstract x Chapter 1 Introduction 1 Chapter 2 Point Scatterers in Nonlinear Optics 5 2.1 Introduction . .5 2.2 Preliminaries . .7 2.3 Second-order nonlinearities . 10 2.3.1 Second-harmonic generation . 11 2.3.2 Sum- and difference-frequency generation . 20 2.3.3 Three-wave mixing . 21 2.4 Third-order nonlinearities . 21 2.4.1 Kerr effect . 22 2.4.2 Bistability . 27 2.4.3 Third-harmonic generation . 29 2.4.4 Four-wave mixing . 30 2.5 Numerical results . 30 2.5.1 Linear dielectric . 31 2.5.2 Second-harmonic generation . 33 2.5.3 Sum-difference frequency generation . 35 2.5.4 Three-wave mixing . 37 2.5.5 Kerr effect . 40 2.5.6 Third-harmonic generation . 40 2.5.7 Four-wave mixing . 41 2.6 Discussion . 44 Appendices 45 Appendix 2.A Bistability . 45 Appendix 2.B Sum-difference frequency generation . 48 Appendix 2.C Three-wave mixing . 51 Appendix 2.D Third-harmonic generation . 53 Appendix 2.E Four-wave mixing . 55 Chapter 3 Optical Theorem for Nonlinear Media 59 3.1 Introduction . 59 iii 3.2 Preliminaries . 61 3.3 Optical theorem . 65 3.4 Second- and third-order nonlinearities . 68 3.4.1 Second-order nonlinearity . 69 3.4.2 Third-order nonlinearity . 71 3.5 Numerical results . 72 3.5.1 Linear response . 72 3.5.2 Second-harmonic generation . 74 3.5.3 Kerr effect . 74 3.6 Application to near-field microscopy . 75 3.7 Discussion . 78 Appendices 82 Appendix 3.A Conservation of energy . 82 3.A.1 Quadratic nonlinearity . 82 3.A.2 Cubic nonlinearity . 83 Appendix 3.B Calculation of local fields . 83 3.B.1 Second-harmonic generation . 83 3.B.2 Kerr effect . 87 3.B.3 Evaluation of the Integral (3.73) . 89 Appendix 3.C Near-field scanning optical microscopy . 90 3.C.1 Second-harmonic generation . 90 3.C.2 Third-harmonic generation . 93 Chapter 4 Second-harmonic Imaging in Random Media 98 4.1 Introduction . 98 4.2 Formulation of the problem . 102 4.2.1 Random model of the array data . 102 4.2.2 Migration imaging . 103 4.2.3 Coherent interferometric imaging . 107 4.3 Analysis of the migration and CINT point spread functions . 110 4.3.1 Scaling . 111 4.3.2 The random geometrical optics model . 114 4.3.3 The linearized data model in the random medium . 119 4.3.4 Analysis of migration imaging . 120 4.3.5 Analysis of CINT imaging . 124 4.4 Numerical results . 130 Appendices 134 Appendix 4.A Statistical moments . 134 4.A.1 Moments of the random phases . 134 4.A.2 Second moments of the wave fields . 137 Appendix 4.B Statistics of the migration image . 141 4.B.1 Imaging of the linear susceptibility . 142 4.B.2 Imaging of the quadratic susceptibility . 143 iv Appendix 4.C Statistics of the CINT image . 144 4.C.1 CINT image of the quadratic susceptibility . 144 4.C.2 CINT image of the linear susceptibility . 149 Appendix 4.D Numerical solution of the forward problem . 154 Chapter 5 Conclusion 157 5.1 Closely related works . 157 5.1.1 On scattering from small nonlinear particles . 157 5.1.2 Optical theorem for nonlinear media . 158 5.1.3 Imaging of nonlinear scatterers . 158 5.2 Contributions . 159 5.3 Future research . 160 5.3.1 Rigorous asymptotics . 160 5.3.2 Time dependent nonlinear scattering . 161 5.3.3 Inverse problem of near-field scattering optical microscopy . 161 5.3.4 Radiative transfer equation in nonlinear media . 161 5.3.5 Variations of imaging nonlinear scatterers in random media . 162 Bibliography 163 v List of Figures 2.1 Optical bistability. One input intensity may give rise to multiple output in- tensities. Here ka = 0:15 and η(1) =5. ..................... 28 2.2 Illustrating the unshaded region which allows bistability. The values of ka and η(1) for Fig. 2.1 is given by the point A. The zoomed in region in (b) shows that some negative values of η(1) do not permit bistability. 29 2.3 The frequency dependence of the scattering amplitude for linear scatterers. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 31 2.4 Frequency dependence of the scattering amplitude at the incident frequency (1) (2) (1) in SHG. The parameters are given by η = 2, = η u0/η . (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 33 2.5 Frequency dependence of the scattering amplitude in SHG. The parameters are as in Fig. 2.4. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 34 2.6 Frequency dependence of the scattering amplitudes when two plane waves are incident upon linear scatterers. The linear susceptibility is η(1) = 2. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 35 2.7 Frequency dependence of the scattering amplitudes in SDFG. The parameters (1) (2) (1) are given by η = 2, = η u0/η = 0:05. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 36 2.8 Frequency dependence of the scattering amplitudes when three plane waves with wave numbers k10 + k20 = k30 are incident upon linear scatterers. The linear susceptibility is η(1) = 2. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 37 2.9 Frequency dependence of the scattering amplitudes in TWM. The parameters (1) (2) (1) are given by η = 2, = η u0/η = 0:2. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 38 vi 2.10 Frequency dependence of the scattering amplitude in the Kerr effect. The (1) (3) (1) parameters are given by η = 2, = η u0/η . (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 39 2.11 Frequency dependence of the scattering amplitude at the incident frequency in THG. The parameters are as in Fig. 2.10. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 41 2.12 Frequency dependence of the scattering amplitude at the third-harmonic fre- quency in THG. The parameters are as in Fig. 2.10. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 42 2.13 Frequency dependence of the scattering amplitudes when four plane waves with wave numbers k10 + k20 + k30 = k40 are incident upon linear scatterers. The linear susceptibility is given by η(1) = 2. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 43 2.14 Frequency dependence of the scattering amplitudes in FWM. The parameters (1) (3) (1) are given by η = 2, = η u0/η = 0:1. (a) One scatterer, (b) Two scatterers parallel to the incident wave, (c) Two scatterers perpendicular to the incident wave. 44 2.15 Illustrating the regions of multistability. 48 3.1 Frequency dependence of extinguished power for a single linear scatterer. Here 2 2 P0 = a cE0 .................................... 73 3.2 Frequency dependence of extinguished power for a single nonlinear scatterer 2 2 with SHG. Here P0 = a cE0 ........................... 75 3.3 Frequency dependence of extinguished power for a single scatterer with Kerr 2 2 nonlinearity. Here P0 = a cE0 .......................... 76 3.4 Illustrating.
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