patience, advise and collaboration introduced me in the world of coding and numerical ..... turn, was the basis of Einstein's general description of the electrodynamics of .... Maxwell's equations in a chargeâ and current free space are given by: Dt ...... Reconstruction, using fifth-order WENO compute the piecewise elements.
Exploring Heterogeneous Time-Varying Materials for Photonic Applications, Towards Solutions for the Manipulation and Confinement of Light
Thesis by Damián Pablo San-Román-Alerigi
In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy in Electrical Engineering
Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi Arabia
November 2014
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iii The thesis of Damián Pablo San-Román-Alerigi is approved by the examination committee
Committee Committee Committee Committee
Chairperson: Professor Boon Siew Ooi Member: Professor David Isaac Ketcheson Member: Professor Ian Foulds Member: Professor Seng-Tiong Ho
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Copyright ©2014 Damián Pablo San-Román-Alerigi All Rights Reserved
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ABSTRACT Exploring Heterogeneous Time-Varying Materials for Photonic Applications Damián Pablo San-Román-Alerigi Over the past several decades our understanding and meticulous characterization of the transient and spatial properties of materials evolved rapidly. The results present an exciting field for discovery, and craft materials to control and reshape light that we are just beginning to fathom. State-of-the-art nanodeposition processes, for example, can be utilized to build stratified waveguides made of thin dielectric layers which put together result in a material with effective abnormal dispersion. Moreover, materials once deemed well-known are revealing astonishing properties, verbi gratia chalcogenide glasses undergo an atomic reconfiguration, when illuminated with electrons or photons, that results in a modification of its permittivity and permeability, up to ⇡ 40%. Moreover, these changes can be modulated in time. This work revolves around the characterization and model of spacetimevarying materials and their applications, revisits Maxwell’s equations in the context of nonlinear spacetime-varying media, and based on them introduces a numerical scheme that can be used to model waves in this kind of media, finally some interesting applications for light confinement and beam transformations are shown.
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ACKNOWLEDGEMENTS I am indebted to my parents, friends, colleagues and professors who have taught me so much during my academic life. Their encouragement and words enkindled the light of discovery and the joy of doing science. It would be impossible to name them all, and thus I will circumspect to those who had a direct impact in this work. My advisor and committee chair, Dr. Boon S. Ooi, for his invaluable support and guidance during the course of my research, his exhaustive revisions and discussions were always a source of new questions and an adventure into applied physics. Special thanks are also due to Dr. David Ketcheson whose patience, advise and collaboration introduced me in the world of coding and numerical methods, the result has been a numerical scheme presented for the first time in this manuscript. I want to also to thank Dr. Ian Foulds for his counseling, support and teachings. Along with some others, they revised this manuscript, and the articles in which it is based. Thanks to them readers can be spared from squandering time translating inconsistent ideas and phrases. Special mention also to Dr. Seng-Tiong Ho, for agreeing to review this thesis and enrich it through his suggestions. Special mention deserve Tien Khee Ng, Yaping Zhang, Ahmed Ben Slimane, Benson Muite, Lisandro Dalcin, Dalaver Anjum, Mohammad Alsunaidi, Daniel Acevedo and Shuaib U. Arshad for their continuous collaboration, support and suggestions. I want also to thank Carlos Argaez, Jose Antonio Alcantara and Tatiana Niembro for their inspiring conversations and help through out this career. My appreciation also goes to Hakan Bagci, Yiannis Hadjimichael, Victor Velazquez, Jorge G. Hirsch, Edna Hernandez, Enrique Lopez, Marcela Grether, Arturo Olvera, Rami Afandy, Ahmed A. Aljabr, Mohammed Zahed Khan, Andrea Di Falco, Susanne C. Kehr, Ulf Leonhardt, Antonia Forshaw, Naseer Aijaz, Khathijah Syed Osman, as well as my friends and colleagues and the department faculty and staff for their insightful discussions and support. My heartfelt gratitude to my parents Gerardo San Roman and Hilda Alerigi, and to my brother Fabian San Roman Alerigi, for encouraging me to look at the world with curiosity and for their vital support through my academic life.
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TABLE OF CONTENTS Examination Committee Approval
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Copyright
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Abstract
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Acknowledgements
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List of Abbreviations
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List of Figures
xvii
List of Tables
xxi
Introduction
I
xxiii
Prolegomena
1 Revisiting Maxwell’s electromagnetics 1.1 Frames of reference, notation and definitions . 1.2 Maxwell’s equations . . . . . . . . . . . . . . 1.2.1 Constitutive relations . . . . . . . . . 1.2.2 Twofold electromagnetic problem . . . 1.3 Analytic solution . . . . . . . . . . . . . . . . 1.3.1 Characteristic curves . . . . . . . . . . 1.3.2 Lorentz transformations . . . . . . . . 1.4 Analytic results: Media in motion . . . . . . 1.4.1 Moving jump . . . . . . . . . . . . . . 1.4.2 Gaussian moving perturbation . . . .
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2 Numerical modeling 2.1 A finite volume method for non-linear hyperbolic PDEs with spacetime-varying coefficients . . . . . . . . . . . . . . . . . . . . 2.1.1 Finite Volume schemes . . . . . . . . . . . . . . . . . . . . 2.1.2 Notation, Riemann problem and high order reconstruction 2.1.3 Flux: f-wave Riemann solvers . . . . . . . . . . . . . . . . 2.1.4 Spacetime-varying sources . . . . . . . . . . . . . . . . . . 2.1.5 Spacetime-varying capacity functions . . . . . . . . . . . . 2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 1D f -wave Riemann solver for Maxwell’s equations . . . .
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2.3
2.2.3 Averages of capacity function and source Convergence and benchmark test . . . . . . . . 2.3.1 Measure of error . . . . . . . . . . . . . 2.3.2 1D Convergence . . . . . . . . . . . . . 2.3.3 2D Convergence . . . . . . . . . . . . .
3 Paraxial approximation and light paths 3.1 Light paths and Lagrange-Euler equation . . 3.1.1 Photon’s path equations . . . . . . . . 3.2 Inverse paraxial problem: confinement of light 3.3 Confinement of light: dynamic equillibrium . 3.3.1 Orbit stability . . . . . . . . . . . . .
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4 Beam transformation and the inverse problem 39 4.1 Helmholtz equation for q . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 An inverse problem for q . . . . . . . . . . . . . . . . . . . . . . . 40
II
Results & Analysis
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5 Fullwave solution: spacetime-varying media 5.1 Derived quantities . . . . . . . . . . . . . . . 5.2 EMClaw . . . . . . . . . . . . . . . . . . . . . 5.3 One-dimensional electromagnetics . . . . . . . 5.3.1 Time-dependent material . . . . . . . 5.3.2 Flowing media . . . . . . . . . . . . . 5.4 Nonlinearities and spacetime-varying media . 5.5 Vibrating media - 2D . . . . . . . . . . . . . . 6 Paraxial solution, light 6.1 Single attractor . . . 6.2 Mexican hat . . . . . 6.3 Binary systems . . .
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attractors 59 . . . . . . . . . . . . . . . . . . . . . . . . . 60 . . . . . . . . . . . . . . . . . . . . . . . . . 61 . . . . . . . . . . . . . . . . . . . . . . . . . 62
7 Beam Transformation: Gauss to Bessel-Gauss beams 65 7.1 Methodology approach . . . . . . . . . . . . . . . . . . . . . . . . 66 7.2 Beam transformation . . . . . . . . . . . . . . . . . . . . . . . . . 67
III
Material Proposal
8 Material proposal 8.1 Understanding chalcogenide glasses . . . . . 8.2 Electron Energy Loss Spectroscopy . . . . . 8.3 Experimental arrangement . . . . . . . . . . 8.4 Electron-induced permittivity variation . . 8.4.1 Making sense of the physical process 8.4.2 Prospective deduction, the refractive
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IV
Epilogue
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9 Closing Remarks
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References and Appendices
References Appendices A.1 2D Riemann solver . B.1 Clawpack Simulation C.1 Journal publications C.2 Conference articles .
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LIST OF ABBREVIATIONS EELS
Electron Energy Loss Spectroscopy
FDTD FEM FVM FWHM
Finite Difference Time Domain Finite Element Method Finite Volume Method full-width-half-maximum
ODE
Ordinary Differential Equation
PDE PICs PSM
Partial Differential Equation Photonic Integrated Circuits [Pseudo–]Spectral Method
WENO
Weighted Essentially Non-Oscillatory
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LIST OF FIGURES 1
Diagram – Fizeau’s experiment, 1860. . . . . . . . . . . . . . . . xxiv
1.1
Analytical solution: characteristic curves for a moving jump in ⌘¯ measured in the laboratory frame . . . . . . . . . . . . . . . . . . Analytical solution: characteristics of a moving jump in ⌘¯ measured in the co-moving frame . . . . . . . . . . . . . . . . . . . . Analytical solution: characteristics of a moving Gaussian in ⌘¯ measured in the laboratory frame . . . . . . . . . . . . . . . . . . Analytical solution: characteristics for moving media in the comoving frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 1.3 1.4 2.1 2.2
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Schematic representation of 1D uniform cell grid. . . . . . . . . . Analytic convergence of the Finite Volume Method (FVM) (1D) in time-varying material . . . . . . . . . . . . . . . . . . . . . . . Self convergence of the FVM (1D) in time-varying media . . . . . Analytic convergence of the FVM (1D) in spacetime-varying material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self convergence of the FVM (1D) . . . . . . . . . . . . . . . . . Self convergence of the FVM (2D) . . . . . . . . . . . . . . . . .
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Light geodesics - trapping analysis: ratio of external energy supply 36
5.1
Staggered time plot of I as measured from the laboratory frame (vibrating media) . . . . . . . . . . . . . . . . . . . . . . . . . . . Position and velocity of Imax as measured from the laboratory frame (vibrating media) . . . . . . . . . . . . . . . . . . . . . . . Staggered time plot of I and n as measured from the laboratory frame (flowing media) . . . . . . . . . . . . . . . . . . . . . . . . Position and velocity of Imax as measured from the laboratory frame (flowing media) . . . . . . . . . . . . . . . . . . . . . . . . Position and velocity of Imax as measured from the co-moving frame (flowing media) . . . . . . . . . . . . . . . . . . . . . . . . Position and velocity of Imax as measured in the co-moving frame at different velocities (flowing media) . . . . . . . . . . . . . . . . Staggered time plot of I in nonlinear media as measured from the laboratory frame (nonlinear media) . . . . . . . . . . . . . . . . . Staggered time plot of I and n as measured from the laboratory frame (flowing and nonlinear media) . . . . . . . . . . . . . . . . Position and velocity of Imax as measured from the laboratory frame (flowing and nonlinear media) . . . . . . . . . . . . . . . .
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5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
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xviii 5.10 Position and velocity of Imax as measured from the co-moving frame (flowing and nonlinear media) . . . . . . . . . . . . . . . . 5.11 Time rate of change of Imax as measured from the laboratory frame (flowing and nonlinear media) . . . . . . . . . . . . . . . . 5.12 Grid size comprison for time rate of change of Imax as measured from the laboratory (flowing and nonlinear media) . . . . . . . . 5.13 Energy flux magnitude in heterogeneous stationary media (vibrating media, n! = 0) . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Energy flux magnitude in heterogeneous stationary media (vibrating media, n! = 16) . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8.1 8.2 8.3 8.4 8.5 8.6
Paraxial approximation: light paths for trapped light in stationary media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light trapping (paraxial approximation): single attractor - radius versus background refractive index nc . . . . . . . . . . . . . . . Light trapping (paraxial approximation): single attractor - light confinement and polar path schematic . . . . . . . . . . . . . . . Light trapping (paraxial approximation): single attractor - phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light trapping (paraxial approximation): mexican hat attractor - geodesics and polar schematic . . . . . . . . . . . . . . . . . . . Paraxial approximation: mexican hat attractor - phase space . . Light trapping (paraxial approximation): binary system - light geodescis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam transformation: rendering of ideal transformation scheme . Beam transformation - analysis: transformation characteristic functions schematic . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of proposed refractive index map for beam transformation, Bessel to Gauss Bessel . . . . . . . . . . . . . . . . . . . . . . . . High resolution refractive index map for beam transformation, Bessel to Gauss Bessel . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of transverse electric field for the transformed beam . Magnitude of transverse electric field for the transformed beam with varying grid size . . . . . . . . . . . . . . . . . . . . . . . . Magnitude of transverse electric field for scattered field . . . . . . As2 S3 characterization - " and n via Electron Energy Loss Spectroscopy (EELS): electron energy loss raw measurements . . . . . As2 S3 characterization - " and n via EELS: electron energy loss analysis of peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . As2 S3 characterization - " and n via EELS: "1 and "2 calculation via Kramer-Kronig . . . . . . . . . . . . . . . . . . . . . . . . . . As2 S3 characterization - " and n via EELS: "1 and "2 dispersion functions in UV and visible range . . . . . . . . . . . . . . . . . . As2 S3 characterization - " and n via EELS: permittivity versus electron dosage (rate of change) . . . . . . . . . . . . . . . . . . . As2 S3 characterization - " and n via EELS: schematic description of bond breaking process . . . . . . . . . . . . . . . . . . . . . . .
54 55 55 56 57 59 60 61 61 62 62 63 66 67 69 69 70 71 72 79 79 80 81 85 85
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As2 S3 characterization - " and n via EELS: deduction of refractive index and absorption functions . . . . . . . . . . . . . . . . . . . 86 As2 S3 characterization - " and n via EELS: n and k peak analysis 86
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LIST OF TABLES 2.1 2.2 2.3 2.4 8.1
Model parameters for convergence analysis in 1D and 2D . . . . Summary of errors for convergence in 1D time-varying material . Summary of errors for convergence in 1D spacetime-varying material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of errors for convergence numerical study in 2D . . . .
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As2 S3 characterization - " and n via EELS: experiment irradiation sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction I must had been five or four years old the first time I heard that light moved so fast, that it would go around the planet eight times before I could count one second. I was marveled, light could traverse all lands, go anywhere, transform and break in colorful errant streaks that met here and there in flaring beans on the kitchen floor. I began to play with mirrors and concocted an intricate plan to catch it, as if light was a firefly. The system would barter the bright light of Summer days into dozen glaring drops that I could use later to lit the night. Time passed, college came, and with it new knowledge and ideas. Light is ubiquitous and restless, our modern world relies on it and on its interactions with matter. Tailoring these interactions might be the key to stop light, to harvest it and transfigure it into the tiny beads I once imagined. Materials can change our perception of light, take for example a straw halfsubmerged in a glass with water. The straw appears to be broken, and yet we know that is not the case; rather, it is the refraction of light rays traversing water what creates the illusion. Refraction is the underlying principle to understand the path followed by light rays as they traverse any material. One may wonder, then, if a skillful craftsmen could not join different liquids together to render an unimaginable illusion. What if this could be done in ever so small scales? And what if this imagined artifact could change also in time as light traverses it? What would happen then? This manuscript is an attempt to solve some of these questions. As such It encompasses a set of discussions, journal publications, conferences, and whitepapers concerned with spacetime-varying materials and their photonic applications. My intention has been to build a platform to study and model light propagation in these media, one that can be expanded and used to design photonic devices and explore new physical phenomena.
Fresnel, Fizeau, and time-varying media Refraction and reflection were the starting knot in a thread of wave phenomena originating in the interaction of light with matter: interference, refraction, polarization, birefringence, dispersion. . . In the XIX century Maxwell’s work, summarized in his famed equations, together with Hertz’ experiments, showed that light can be described as an electromagnetic wave [1–4]. Their discovery laid the ground for many of our modern technologies. It changed forever our understanding of light-matter interactions, and in turn originated theories and methods that revolutionize our comprehension about the electromagnetic properties of materials and the fundamental building blocks of light itself. Yet, in the context of spacetime-varying media light-matter interactions remain a mater of debate and discovery. To illustrate, let us go back to refraction.
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Figure 1: Fizeau’s experimental setup to observe the change in the speed of light propagating in moving water.
In a letter to Arago, Fresnel posited that light rays traversing a moving fluid would experience a change in their velocity that could not be accounted for solely by refraction [5–8]. He calculated that light rays would be dragged by the moving medium by a factor that depended on the medium’s velocity, V =
v 1 1+
1 n2 v cn
(1)
where v is the velocity of the medium and V the speed of light in it. Moved by Fresnel’s work, Fizeau conceived an experiment using a moving fluid and mirrors to measure the speed of light in flowing water (see Figure 1). He found that the velocity of light in the liquid was lessened by a factor that matched Fresnel’s predictions. To explain the result, Fizeau and Fresnel posited a theory that conjured the existence of a stationary æther which dragged along moving substances but not air. However, the existence of æther was refuted by Michelson’s and Morley’s experiment [9], contradicting Fizeau’s and Fresnel’s hypothesis and experimental results. The conundrum was brought to terms by Lorentz, he proposed a mechanism of time-space contraction co-linear to the direction of motion that correctly explained Fresnel’s results [10]. This work, in turn, was the basis of Einstein’s general description of the electrodynamics of moving bodies [11], followed by the demonstration of the invariance of Maxwell’s equations under Lorentz transformations, ultimately leading to the derivation of the Minkowsky–Maxwell equations [12–14]. In special relativity, the paraxial approximation of the Minkowsky–Maxwell equations predicts Fizeau’s result. In particular, Laue showed that in Fresnel’s experiment the wave velocity would be given by [15] : v 1 n12 V = 1+ v | {zcn }
Einstein Landau
✓
◆ 1 ⇡ v 1+ 2 v⌧c n | {z }
(2)
Fresnel0 sdrag
These results demonstrated a fascinating side of spacetime-varying media and wave interactions. Yet, until recently they were seldom revisited in applied science and engineering, perhaps owing to the many complications of creating [meta]materials with time-varying properties. The panorama, however, is rapidly changing. As technology has evolved, so did our comprehension of the inter- and intra- atomic laws governing material properties; leading the design of sophisticated characterization and fabrication
xxv methods, that give us the ability to craft materials that can be used to control light in ways that were unthinkable before. State-of-the-art nano-deposition processes, for example, can be employed in building stratified waveguides; made of dozen thin dielectric layers, each 10nm, these waveguides show an effective abnormal dispersion with huge potential applications for telecommunications. That being said, meticulous characterizations of transient and spatial properties of materials, are revealing that these can be modulated in time at rates akin to electromagnetic wave oscillations, see e.g. [16,17]. Known as spacetimevarying or time-varying media, they present fascinating challenges and opportunities in all areas of research. In particular, wave-matter interactions, numerical models and applications remain a matter of debate and are largely unexplored. Notwithstanding, some novel effects have been identified: trapping, confinement, ultra-short pulse coupling, beam transformation, negative refraction, eigen–polarization, and the optical Bohm–Aharonov effect [18–26]. In terms of modeling, until not long ago only one general numerical scheme had been proposed for linear time-varying materials [27]. This work presents, to the best of my knowledge, the first non-linear, full wave, numerical solver for Maxwell’s equations in spacetime-varying media. Furthermore,the numerical scheme described in this thesis can be readily extended to other wave phenomena [28]. Within the realm of photonic applications, spacetime-varying materials could open the door to manipulate light in new and exciting ways. Hence the need to introduce a platform that can be used to elucidate and model novel technologies to control light for Photonic Integrated Circuits (PICs) applications. The results of such model could be used to test diverse [meta]materials to enable myriad photonic devices, either feasible or imagined. In turn, these technologies could become the heart of the next generation PICss, [all-optical] super-computer processors, optical memories, ultra-high bandwidth telecommunications, noninvasive therapies and scanners for medicine, and even Alcubierre’s like warp drives [29].
Abridged chronology This text is organized into self-contained parts that discuss the fundamentals and photonic applications of spacetime-varying media. The first introduces the fundamental physics and numerical methods. The second presents the applications. Finally, the third advances a material platform to build them. This order is intended to introduce the reader progressively to the subject –time-varying materials. However, historically this text began to forge in the search for timevarying materials. The choice, ultimately fell on Chalcogenide glasses, because it is easy to incorporate in traditional nano-fabrication methods. Moreover, spectroscopic analysis revealed an unknown feature: its refractive index decreased by as much as ⇡ 40% when illuminated with electrons. This reversible change can be modulated at a pace akin to the frequency of electromagnetic radiation [17]. Moved by these results, and given the raising interest in PICs, I designed designed a beam transformation system based in a linear refractive index mapping. The device could be one day fabricated using chalcogenides and used as the light source in hand held tomography scanners [30]. Further, given the ris-
xxvi ing interest in all-optical computers, I investigated the possibility of crafting a time delay systems based on similar premiss . The result was a set of centrosymmetrical refractive index mappings that can trap light in perennial orbits redolent of celestial mechanics [31]. Concomitantly, to these findings I began working with David Ketcheson on a fullwave numerical scheme to model light propagation in non-linear, heterogeneous and time-varying media. The results premier in this work [28]. The full list of publications and conferences related to this project can be found in Appendix C.
1
Part I
Prolegomena
3
Chapter 1
Revisiting Maxwell’s electromagnetics Maxwell’s equations are a set of non-linear hyperbolic Partial Differential Equations (PDEs) that describe the propagation of electromagnetic radiation and its interaction with materials or vacuum. Usually, materials are thought to vary in space, but time-varying materials have seldom been considered in the literature. In the last decade interest in time-varying materials surged as a result of advances in characterization techniques and material sciences. Recent publications have demonstrated that there exists a wide miscellany of materials where electromagnetic properties can be modulated in time, with rates akin to electromagnetic wave oscillations [16,17]. The physical consequences and the practical applications ensuing from these discoveries are just beginning to be explored. Effects stemming from wave propagation in spacetime-varying media include: trapping, confinement, ultra-short pulse coupling, beam transformation, negative refraction, eigen–polarization, and optical Bohm–Aharonov [14, 18–26]. The advent of these materials paves the way to control and manipulate light in ways yet to be imagined. PICs where light confinement and trapping is crucial, could become a reality through these media; enabling the next generation of computers and optical data systems, and even Alcubierre’s like warp drives [29]. Whence derives the need to create numerical schemes that can incorporate transient material properties and allow us to further the realm of photonics. We begin this chapter by deriving Maxwell’s equations in spacetime-varying, heterogeneous, anisotropic and non-linear media, followed by a discussion on [semi–]analytical solutions for elementary configurations. It is our objective, however, to go beyond the scope of the rather modest set of analytical solutions. Thus, in the next chapter we describe a numerical scheme to model wave propagation in these settings. Later, we elaborate on the paraxial approximation to this problem using the formalism provided by Lagrange-Euler equations for spacetime geometries; introduce the notion of the inverse problem for paraxial equations, and discuss how it can be used to develop trapping refractive index conditions. To conclude these prolegomena we study a solution the primordial description of the inverse scattering problem in electromagnetics, and use it to craft a Gauss to Bessel beam converter.
4
1.1
Frames of reference, notation and definitions
Since we are concerned with spacetime-varying materials it is useful to define some frames of reference: 1. laboratory frame, it is the one where the laboratory is at rest, usually it is the frame of reference used to make measurements, 2. co-moving frame, it is the one attached to the object being measured. Consequently, in this frame the object is stationary⇤ , Unless otherwise stated, we will assume that all derived quantities are measured from the laboratory frame. Noting that in the co-moving frame these quantities can be obtained using either Minkowsky–Maxwell constitutive equations or through Lorentz transformation [32]. For simplicity, throughout this manuscript we adopt the following convention when referring to spacetime coordinates: time will be denoted by either t or x0 , while space will be represented by either x, y, z, or xi , i 1. ¯ The electric and magnetic fields Second-rank tensors will be indicated by ⇤. will be denoted by qe and qh , respectively. And vacuum properties are comprised in the diagonal tensor ⌘¯l0 2 Rm⇥m , with diagonal entries "o 2 R for l = e, and µo 2 R for l = h. Finally, let us introduce the vector q to represent the conserved quantities; particular to electromagnetics it takes the form: ✓ ◆ qe q⌘ , qh where qe,h : Rm ⇥ R3+1 ! Rm and q : Rn ⇥ R3+1 ! Rn , with n = 2m.
1.2
Maxwell’s equations
Maxwell’s equations in a charge– and current free space are given by: Dt
r ⇥ qh = 0,
Bt + r ⇥ qe = 0,
(1.1a) (1.1b)
where the relation between D, B and qe,h is governed by the constitutive equations [32]: D = ⇣e (¯ ⌘ e , qe ) ,
(1.2a)
B = ⇣h (¯ ⌘ h , qh ) .
(1.2b)
where ⌘¯l : Rm⇥m ⇥ R3+1 ! Rm⇥m , with l = e, h, is a symmetrical second-rank tensors with non-zero off-diagonal entries. Whereas, ⇣l : Rm⇥m ⇥ Rm ⇥ R3+1 ! Rm is a nonlinear, real-valued, function. Note that ⌘¯e,h describes the electric, l = e, or magnetic, l = h properties of the material; customarily referred to as permittivity and permeability, respectively. ⇤ Proper frames can be inertial or non-inertial. An intertial frame is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. All intertial frames do not accelerate with respect to one another. Therefore, measurements in one inertial frame can be transformed to another by means of Lorentz’ transformations.
5 Substituting relations (1.2) into (1.1), using the chain rule and grouping similar terms reduces the latter to the homogeneous system: (1.3)
⇣(¯ , q)t + fi (q, r)xi = 0,
where ⇣ : Rn⇥n ⇥Rn ⇥R3+1 ! Rn is the column vector with entries (⇣e , ⇣h ), and ¯ : Rn⇥n ⇥ R3+1 ! Rn⇥n is a second-rank tensor whose entries are the electric and magnetic properties of the material. In some cases one might be interested in dealing with the field q direction, i.e. the non-homogeneous version of (1.3). Expanding the time derivative and simplifying we obtain: (1.4)
¯ (q, r, t) · qt + fi (q; r)xi = (q, r, t),
where we have used Einstein’s summation rules; as described earlier, q refers to the the fields qe,h , i.e. the conserved quantities; and f : Rn ⇥R3 ! Rn described the spatial derivatives of the field, i.e. the fluxes. The factor ¯ : Rn⇥n ⇥R3+1 ! n⇥n R is commonly known as the capacity function, it comprises the material properties ⌘¯i † . Finally, : Rn ⇥ R3+1 ! Rn is known as the source term, contains the time derivatives of the material properties ⌘¯l , i.e. the non-hyperbolic terms. Note that ¯ and may be nonlinear in q, but do not depend on derivatives of q. Equation (1.4) is a non-linear hyperbolic PDE whose solution describes wave propagation in heterogeneous, anisotropic, non-linear, and time-varying media: given some initial condition q(r, t = 0). Let {l, m} = {e, h} ⌘ { 1, 1}. Then the terms in (1.4) can expanded explicitly using (1.2), so that ¯ encompasses: ¯ = @qli ⇣li . The i-th element of the flux function fl is defined by: n o j k fli = ✏ijk @xj qm6 @xk qm6 =l =l sgn (l) = And the source term,
(1.5)
@t
i l.
(1.6)
, is defined by: (q, r, t) =
@⌘li ⇣li @t ⌘li... .
(1.7)
Where again, for compactness, we have used Einstein’s summation convention.
1.2.1
Constitutive relations
The constitutive relations (1.2) in general can take any form based on the response of the material to the electromagnetic field and vice-versa. Because of this, it is customary in classical electromagnetics to expand the constitutive relations in a power series around some constant field qo [32], to obtain: ⇣ ⌘ (i) ⇣k = ⌘¯l0 · ql + ¯l · qli , i = 1, . . . (1.8) † Notice that the term ¯ may also be understood as the change in metric of the discretization space [33, 34]
6 (n)
where ⌘¯l0 here represents the vacuum electromagnetic tensor, and ¯l , the n-th order susceptibility tensor. This expansion results convenient to segregate the material response into a linear and non-linear terms, ⇣ ⌘ ⇣ ⌘ (1) (i) ⇣l = ⌘¯l0 · I¯ + ¯l · ql + ⌘¯l0 ¯l · qli i 2 = ⌘¯l0 · ⌘¯lr · ql + ⌘¯l0 · ⇠¯ln ,
(1.9)
(n)
where ⇠¯ln is the sum over ¯l , with n 2. The first term on the righthand-side of (1.9), ⌘¯lr , is known as the relative permittivity, l = e, and permeability, l = m. It represents the linear response of the medium to the propagation of electromagnetic radiation. The second term, ⇠¯l , contains the non-linear electric and magnetic susceptibility tensors, they describe the material non-linear response to electromagnetic radiation [4]. Consequently, the capacity function, ¯ , takes the form: ¯ (q, r, t) = ⌘¯l0 · ⌘¯lr + @ql ⇠¯l ⇣ (k) = ⌘¯l0 · ⌘¯lr + k ¯l · qlk
and the source term becomes:
1
⌘
⇣ (k) (q, r, t) = ⌘¯l0 @t ⌘¯lr + @t ¯l · qlk
,
k
1
⌘
· ql
2.
(1.10)
(1.11) (i)
Theoretically it is possible to imagine a material with time-varying l , however, until this moment, it remains uncertain whether time-varying susceptibilities, electric or magnetic, are possible in naturally occurring or manmade materials. Thus, in our calculations in Part II we reduce to zero the second righthand-side term in (1.11).
1.2.2
Twofold electromagnetic problem
Two family of problems ensue from equation (1.4): (i) forward problems, describe the time evolution of the field q; and (ii) the inverse [electromagnetic] problem, they depart from known fields at input and output to determine the material properties that transformed the field. In (i) the material properties, ⌘¯ and initial condition, qo , are known at all points in time and space. The evolution of q at every time step is then calculated through various methods raging from [semi] analytical solutions for the full or paraxial approximation and numerical discretization. In general, an analytical solution is rarely trivial and often numerical methods are required. Multiple methods have been developed in this area, most dealing with non-linear and space heterogeneity. However, numerical methods that encompass time-varying coefficients (capacity function and source term) to the best of our knowledge have not been derived for a generalized scenario. Our goal, henceforth, is to develop a multi-dimensional numerical solver to (1.4) in the most general case possible. As part of this development, we will also discuss the paraxial approximation to (1.4) and use it to find possible solutions to problems in light trapping and concentration. In (ii) the field q is known at different points in time and space, and the material properties are unknown. This is the starting point for the inverse
7 electromagnetic problem which we will discuss later in the text. The inverse problem in heterogeneous media is ill-defined, and hence unique solutions, if any, are not guaranteed. For the purpose of light reshaping this problem can be used to explore beam transformation systems, optical cloaks, concentrators, etc. [22, 30, 35–38].
1.3
Analytic solution
In order to provide a general panorama of the complexity and opportunities of spacetime-varying media, we turn our attention to solving the general nonlinear problem posed in (1.4). To decrease the complexity of the nonlinear equation (1.4), consider an isotropic medium in one spatial dimension, (1.12)
¯ (q, x, t) · qt + f (q; x, t)x = (q, x, t), In this context q encompasses: q(x, t) =
✓
q0 q1
◆
⌘
✓
qe qh
◆
(1.13)
where q i : R1+1 ! R. Accordingly, the flux function, f , takes the form: q1 ⌘e0 q0 0 ⌘h
f (q; x, t) =
!
(1.14)
Following from definition (5), ¯ is given by: ¯=
⌘er + k
(k) 0 k 1 e (q )
0
0 ⌘hr + k
(k) 1 k 1 , h (q )
!
,
k
2,
(1.15)
with ⌘lr : R1+1 ! R and constant (k) 2 R. Next, we can use (1.11) to determine the source term, (q; x, t) = ¯ t · q, (1.16) (i)
where we have assumed that @t l = 0 for i 2. Substituting (1.13)–(1.16) into (1.12), and simplifying yields the 1D nonlinear PDE: qt + ¯ · (A · qx ) = (ln ¯ )t · q, (1.17)
where ¯ = ¯ 1 , and A is the Jacobian of f , i.e. A = @q f . Equation (1.17) together with the initial condition q(x, t = 0) = g(x), describe wave propagation in one spatial dimension space filled with a nonlinear isotropic medium with spacetime-varying permittivity and permeability.
1.3.1
Characteristic curves
Equation (1.17) is a quasilinear PDE of hyperbolic type, therefore we can use the method of characteristics to find q [39]. In this method we consider the time
8 rate of change of q(x(t), t) as measured by a co-moving observer, x = X(t). Thus, by virtue of the chain rule we have: d dX q(X(t), t) = @t q + @X q dt dt
(1.18)
Consequently (1.17) reduces to a set of two coupled Ordinary differential equations (ODEs): dq = (ln ¯ )t · q, dt dX = ¯ · A ⌘ c(x, t). dt
(1.19a) (1.19b)
where, c(x, t) : R1+1 ! R is the wave velocity. That is, the solution to (1.17) can be found by solving (1.19a) along the trajectory defined by (1.19b), known as the characteristic curve. The propagation velocity of an initial point will be given by the righthandside of (1.19b), which is the characteristic velocity, or the local wave velocity. In general an analytic solution to either (1.19a) and (1.19b) can rarely be found. Nevertheless, we can approximate the solution by means of a semianalytic scheme. In this context we can use a Runge Kutta method to first integrate (1.19b) and obtain the characteristic curves, afterwards we can integrate (1.19a) along the characteristic curves to find the value of q. This procedure can be implemented in Matlab using ode45 and a linear interpolation to find arbitrary points within the characteristic’s path. In the next section we use an updated version of Fizeau’s and Fresnel’s experiment to study the comportment of characteristics in a medium with a flowing perturbation. Using Lagrangian mechanics, Cacciatori et al.showed that a moving Gaussian perturbation in the refractive index can trap light in co-moving singularities, noting that this could be optical analogues to black-holes [25]. A constrain of their method is that the results are only valid in the paraxial approximation to electromagnetics. A fullwave solution to this problem is of interest to PICss, where moving perturbations could be used for energy harvesting, coupling and optical delay lines.
1.3.2
Lorentz transformations
In view that we are studying time-varying media it is useful to introduce Lorentz’ transformations to move between the laboratory frame of reference and a comoving frame of reference. In one dimension they take the form: =p
1 t0 = (t 0
x = (x
1.4
1
(1.20a)
, v2 vx),
(1.20b)
vt),
(1.20c)
Analytic results: Media in motion
Now, we introduce a succinct example solutions to (1.17) for a traveling perturbation in ⌘. This compendium is intended to acquaint the reader with benchmark examples of light propagation in time-varying media; and produce a test
9 solution to study convergence in the numerical model described in the next chapter. Due to its complexity and to secure a unique solution to the PDE (1.19), for the remaining part of this chapter we assume propagation in a linear medium, i.e. ¯i = 0, i 2.
1.4.1
Moving jump
Let us begin by studying the dynamics of the characteristics in (1.19) when we introduce a perturbation of the form:
⌘(x, t) =
8 < :
x + xo < v · t + xo , vt + xo x vt + x > vt + + xo ,
⌘o
⌘
(x vt ⌘o + ⌘
xo ) + ⌘o
+ xo ,
(1.21)
300
300
250
250
200
200 x (a.u.)
x (a.u.)
where ⌘o is the background relative permittivity or permeability, ⌘ is the perturbation shift, its width, xo the offset in the x-direction, and v the perturbation’s velocity.
150
150
100
100
50
50
0 0
100
200
300
400
0 0
500
100
200
t (a.u.)
300
400
500
400
500
t (a.u.)
(a) v = 0.00
(b) v = 0.61 350
300 300 250
200
x (a.u.)
x (a.u.)
250
150
200 150
100
100
50
50
0 0
100
200
300 t (a.u.)
(c) v = 0.63
400
500
0 0
100
200
300 t (a.u.)
(d) v = 0.75
Figure 1.1: Plot of characteristic curves for a wave propagating in a medium with a linear moving jump in ⌘¯, as measured from the laboratory frame of reference. Seeding points lie in the range xo xi xo + , and the perturbation paramenters are set to varying velocity v, ⌘o = 1.5, ⌘ = 0.15, = 5 and xo = 10.
10 In this case (1.19b) has an analytical solution: 8 ⌘ x +t o o > > ⌘o > > > > ✓ > < 2 1 ⌘o v+ ⌘v t +Lw ( 1+⌘o v+ ⌘xo v )e 1+⌘o v+ x(t) = > ⌘v > > > > > > : xo (⌘o + ⌘)+t
x < vt, ⌘(xo
vt)v
◆
vt x vt + ,
x > vt + , (1.22) where Lw is the principal solution to Lambert’s W function, also called omega function. It is defined as the inverse function of [40]: ⌘o + ⌘
f (W ) = W ew .
(1.23)
The inverse can be obtained by virtue of Lagrange inversion theorem as [40,41]: W (z) =
1 X ( n)n n! n=1
1
zn.
(1.24)
Diverse numerical toolboxes like Matlab or Mathematica can compute the principal solution to Lambert’s W function. Thus we can sketch the solution to the characteristics for the moving jump, and use these to calculate the values of q. Figure 1.1 shows a plot of some characteristic curves for different perturbation velocities, v. Observe that when the velocity of the moving perturbation lies in the range v < 1/(⌘o + ⌘) the dynamic of the solution is fairly simple: characteristics passing through it will be refracted, change their speed and continue to propagate otherwise unperturbed. The characteristic curves become more interesting as the speed of the perturbation becomes comparable to the speed of the wave in the medium, u. Observe Figures 1.1c and 1.1b, they depict the case when 1/(⌘o + ⌘) < v < 1/⌘o . In this scenario the characteristics beginning at the left side of the perturbation, ul = 1/⌘o ,move faster than the perturbation; hence they will eventually catch to it. As the waves and the perturbation interact, the speed of the waves decreases owing to the raise in ⌘ (see equation (1.19b)). Because of this, as the left-side waves evolve their velocity will asymptotically converge to v. Their characteristic path will converge to one parallel to the perturbation’s own characteristic. Physically, this entails that the waves are trapped into a singularity; a point that co-moves with the perturbation. An observer sitting on the perturbation would see that from his vantage point the waves slowdown until they eventually stop. In Figure 1.2 we have plotted some characteristic curves as seen by an observer in the co-moving frame. See, in particular Figures 1.2c and 1.2b, they depict the case when the characteristics converge to a line with slope zero, i.e. trapping.
1.4.2
Gaussian moving perturbation
As we will from the material characterization in Chapter 8, it is possible to create small perturbations that move in a medium by using an electron or laser beam. Because the shape of the perturbation is inherited from the energy distribution of said beam we consider in this section a small Gaussian perturbation [16, 17].
11
25 300
20 250
x’ (a.u.)
x’ (a.u.)
200 150
15
10
100
5 50
0
0 0
100
200
300
400
0
500
50
100
t’ (a.u.)
(a) v = 0.00
150 200 t’ (a.u.)
250
300
350
(b) v = 0.61 30
25
20 10
20
x’ (a.u.)
x’ (a.u.)
0
15
10
−10 −20 −30 −40
5
−50 −60
0 0
50
100
150 200 t’ (a.u.)
250
300
350
0
50
(c) v = 0.63
100
150 200 t’ (a.u.)
250
300
350
(d) v = 0.75
Figure 1.2: Plot of characteristic curves for a wave propagating in a medium with a linear moving jump in ⌘¯, as measured from the co-moving frame of reference. Seeding points lie in the range xo xi xo + , and the perturbation paramenters are set to varying velocity v, ⌘o = 1.5, ⌘ = 0.15, = 5 and xo = 10.
Let ⌘ take a Gaussian shape, ⌘(x, t) = ⌘o + ⌘e
(x vt)2 /
2
(1.25)
In this case there is no analytical solution to (1.19). However, we can integrate (1.19b) numerically. Some of these characteristic curves, as seen from the laboratory frame, for different perturbation velocities are plotted in Figure 1.3 Once again observe that when the velocity of the perturbation is v < 1/(⌘o + ⌘) characteristics passing through it will be refracted and change their speed momentarily. Because the speed of the perturbation is smaller compared to the wave’s speed the characteristics eventually pass it. As they do their velocity goes back to their original speed and continue to propagate otherwise unperturbed. As the speed of the perturbation becomes comparable to the speed of the wave in the medium, u, some conditions for trapping appear. If 1/(⌘o + ⌘) < v < 1/⌘o , characteristics initially starting to the left side of the perturbation, ul = 1/⌘o , move faster than the perturbation and catch it. As they do, their speed decreases due to the change in ⌘; this velocity continues until their speeds are equal, and the waves’ light path asymptotically converge to a singularity. Points starting to the right of the perturbation propagate at a speed 1/(⌘o + ⌘) ur < 1/⌘o . There are two possibilities: (i) ur < v, (ii) ur = v, and
350
350
300
300
250
250
200
200
x (a.u.)
x (a.u.)
12
150
150
100
100
50
50
0 0
100
200
300
400
0 0
500
100
200
t (a.u.)
300
400
500
400
500
t (a.u.)
(a) v = 0.00
(b) v = 0.61
300
300
250
250
200
200
x (a.u.)
x (a.u.)
350
150
150
100
100
50
50
0 0
100
200
300 t (a.u.)
(c) v = 0.63
400
500
0 0
100
200
300 t (a.u.)
(d) v = 0.75
Figure 1.3: Plot of characteristic curves for a wave propagating in a medium with a Gaussian-like moving perturbation in ⌘¯, as measured from the laboratory frame of reference. Seeding points lie in the range xo xi xo + , and the perturbation paramenters are set to varying velocity v, ⌘o = 1.5, ⌘ = 0.15, = 5 and xo = 10.
(iii) ur > v. Points in (i) will be caught in the perturbation and converge to a singularity located radially-opposed to the points that began began to the left. In (ii) there is only one characteristic, it will remain unperturbed and co-move with the perturbation. Physically, this means that the waves are begin trapped into singularities; a point that co-moves with the perturbation. An observer sitting on the perturbation would see that, from his vantage point, waves approaching to the perturbation slowdown until they eventually halt in two symmetrical points. As with the previous case, transforming to the co-moving frame can bring a clearer picture. See, for example, Figures 1.4b and 1.4c they illustrate the case when the curves converge to a line with slope zero.
13
60
300
50
250
40 x’ (a.u.)
x’ (a.u.)
350
200
30
150
20 100
10
50
0
0 0
100
200
300
400
500
0
50
100
t’ (a.u.)
(a) v = 0.00
250
300
350
(b) v = 0.61
45
30
40
20
35
10 0 x’ (a.u.)
30 x’ (a.u.)
150 200 t’ (a.u.)
25 20
−10 −20
15
−30
10
−40
5
−50 −60
0 0
50
100
150 200 t’ (a.u.)
(c) v = 0.63
250
300
350
0
50
100
150 200 t’ (a.u.)
250
300
350
(d) v = 0.75
Figure 1.4: Plot of characteristic curves for a wave propagating in a medium with a Gaussian-like moving perturbation in ⌘¯, as measured from the co-moving frame of reference. Seeding points lie in the range xo xi xo + , and the perturbation paramenters are set to varying velocity v, ⌘o = 1.5, ⌘ = 0.15, = 5 and xo = 10.
14
15
Chapter 2
Numerical modeling We now turn our attention to the numerical approximation to the solution of (1.4). As described in Section 1.2, this equation is a non-linear hyperbolic PDE with spacetime-varying coefficients. It describes the wave propagation problem on heterogeneous, isotropic, non-linear and time-varying media. To model time-independent media several numerical methods have been developed over the years: Finite Difference Time Domain (FDTD), Finite Element Method (FEM), [Pseudo–]Spectral Method (PSM) and FVM. FDTD was the first efficient method to numerically solve Maxwell’s equations [42] in multiple dimensions and has remained widely popular ever since. It uses two staggered grids to split Maxwell’s equations into a semi-discrete system of ODEs and integrates in time using a leap-frog scheme [43]. FEM has been widely used to calculate time-independent solutions to Maxwell’s equations with relative success, and lately explorations into PSMs and FVM for Maxwell’s are being conducted. Notwithstanding these advances, numerical solvers for Maxwell’s equations with time-varying media have been rarely attempted. In most cases schemes have been developed to fathom out solutions to specific problems in homogeneous media with time dependent conductivity, e.g. : sudden ionization, laser pulse excitation of materials or electric arc discharge [44–48]. Furthermore, Analytic solutions are possible only in a handful of cases; for example [49] found that by variable separation it was possible to find a solution in two dimensions when studying Transverse Magnetic cylindrical waves. To the best of our knowledge, only a numerical scheme to model timedependent linear media has been demonstrated [27]. Based on FDTD, Faragó et al.showed that is possible to expand the scheme in [42] to account for linear heterogeneous time-dependent media. They achieve this by applying an operator splitting method and Magnus’ series expansion [27]. Because of splitting the scheme needs to recompute iteration matrices at every time step. Moreover, the scheme can only account for linear materials. When a wave travels along a time-dependent medium every change in the material acts like a boundary between two regions that have different material properties, thus the incident wave can be split into right and left going waves. An approach is to think about the solution in terms of a flux across the interface between the two regions, which is the idea behind FVM schemes. FVM methods are ideal for heterogeneous media, where every grid cell is assigned an average of the material properties enclosed by it, and the solution is found in term of the flux across the interfaces of the cell [33].
16 In this chapter we extend the method proposed in [50] for the solution of (1.4) to incorporate time-dependent fluctuations, capacities and sources. In particular we use a high order discretization based on the method of lines, as described in [50]. This scheme allows us to use different reconstruction methods, such as Weighted Essentially Non-Oscillatory (WENO) or piecewise–polynomial reconstruction; and sundry time integrators; thus adding to its versatility, scalability and application range. Because (1.4) is a general non-linear algebraic PDE the method described in this chapter can be applied to other wave phenomena. Thus in the ensuing sections we will focus on developing a general numerical scheme to solve (1.4) and later apply it to Maxwell’s equations.
2.1
A finite volume method for non-linear hyperbolic PDEs with spacetime-varying coefficients
Let us consider (1.4) in one spatial dimension⇤ , ¯ (q, x, t) · q(x, t)t + f (q)x = (q, x, t),
(2.1)
where q : R1+1 ! Rn is the vector of conserved quantities, f : Rn ! Rn corresponds to the flux, ¯ : Rn ⇥ R1+1 ! Rn⇥n is the capacity function, and n 1+1 n : R ⇥R ! R includes any non-hyperbolic term† . If the coefficients in (2.1) are independent of time, then it can be reduced to the more familiar set of hyperbolic conservation laws: ¯ (x) · q(x, t)t + f (q)x = (q, x), for which higher order numerical methods have been developed, see e.g. [33,50– 52]. A method for high resolution wave propagation was initially introduced by LeVeque [53], and extended to higher order by Ketcheson et al. [50]. In this chapter we extend the latter to include capacity functions that depend on q and t.
2.1.1
Finite Volume schemes
A finite volume approach is based on subdividing the spatial domain into finite volumes, often called grid cells or cells for short; and calculating an approximation to the integral of q over each of these volumes. For example, the cell average Qi (t) is given by Z x 1 i+ 1 2 Qi (t) = q(x, t)dx, (2.2) x xi 1 2
where the index i denotes the cell number and ⇤ The
x its size.
method described in this section is one dimensional and can be easily extended to higher dimensions based on the solution to the 1D problem [33, 50]. † Remember that ¯ and are in general nonlinear in q, but do not depend on derivatives of q
17
i
1
i+1
i xi
xi+ 12
1 2
Figure 2.1: Schematic representation of 1D uniform cell grid.
h Rearranging (2.1), using definition (2.2), integrating the former over xi
[t, t +
t] and dividing by x we obtain: Z t+ t Z x 1 i+ 1 2 n Qn+1 = Q ¯ 1 · (f (q)x i i x t x 1 i
where Qi (t +
(q, x, t)) dxdt.
1 2
i , xi+ 12 ⇥
(2.3)
2
t) ⇡ Qn+1 , i
and Qi (t) ⇡ Qni .
Equation (2.3) tell us how the cell average Qni gets updated at every time step, tn . To have a fully discrete scheme we need to find an adequate approximation to the integral in (2.3) in terms of Qni .
2.1.2
Notation, Riemann problem and high order reconstruction
We use the notation for the Riemann solution from [50, 53]. It originates from the consideration of the hyperbolic system: qt + A¯ · qx = 0,
(2.4)
where A¯ 2 Rn⇥n . Equation (2.4) is hyperbolic if A¯ is diagonalizable with real eigenvalues; hereafter we assume this to be the case. Let sp and rp denote the eigenvalues and corresponding eigenvectors of A for 1 p n. Hereinafter we assume the eigenvalues to be sequenced in ascending order, i.e. s1 . . . sn . The Riemann problem consists of (2.4) and the initial data: ⇢ ql , x < 0, q(x, 0) = (2.5) qr , x > 0.
Its solution for t > 0 is a piecewise-constant function with jumps proportional to rp , each moving at speed sp along the ray x = sp t. These waves, denoted by W, can be obtained in terms of the initial jump in q: X X qr ql = ↵p r p = W p. (2.6) p
p
Since we are interested in a finite volume discretization, it is convenient to specify notation for the net effect of all right- and left-going waves: X A q⌘ (sp ) W p , (2.7a) p
A
+
q⌘
X p
+
(sp ) W p ,
(2.7b)
18 where the values A± q are known as fluctuations, and the terms (sp ) the positive or negative part of sp [33, 53]:
±
denote
+
(sp ) = min (sp , 0)
(sp ) = max (sp , 0) .
To be concise we will refer to the Riemann problem with initial left state ql and right state qr as the Riemann problem with initial states (ql , qr ). Accordingly, we will sometimes use the notation W p (ql , qr ) to represent the pth wave in the solution of the Riemann problem withi initial states (ql , qr ). h Now, we integrate (2.4) over xi obtain: Qi (t +
t)
1 2
Qi (t) =
t] and divide by
, xi+ 12 ⇥ [t, t +
1 x
Z
t+ t
t
Z
xi+ 1 2
xi
x, to
(2.8)
Aqx (x, t) dx
1 2
In [50] a piecewise-polynomial approximation, accurate to order n, is used to approximate the solution q (x, t); that is at t = to ⇣ ⌘ qˆ (x, to ) = qˆi (x) , for x 2 xi 12 , xi+ 12 , (2.9) where
xn+1 .
qˆi (x) = q (x, to ) + O
(2.10)
At each interface xi 12 this approximation is discontinuous, so we must solve the Riemann problem with: ⇢ qˆi 1 , x < xi 12 , q (x, 0) = (2.11) qˆi , x > xi 12 . The solution to this problem is a set of waves obtained by decomposing the jump in Q in h i terms of the eigenvalues of A. If, then, we integrate (2.4) over xi 12 , xi+ 12 ⇥ [t, t + t] and divide by x, we obtain: Z t xi+ 12 Qi (t + t) Qi (t) = Aˆ qx (x, t + t) dx (2.12) x xi 1 2
where qˆ (x, to + t) is the exact evolution of qˆ after a time increment Let us define ⇣ ⌘ ⇣ ⌘ sL = min s1i+ 1 , 0 , sR = max sm , 1,0 i 2
and
qiR
1 2
=
lim
x!x+ i
qˆi (x) , 1 2
2
L qi+ 1 = 2
lim
x!x+
qˆi (x)
t. (2.13) (2.14)
i+ 1 2
where the superscripts L and R indicate, respectively, to the left and the right state of the interface considered. Coming back to (2.12), taking the limit as t ! 0 leads to a general semidiscrete scheme [50]: 0 1 Z x 1 +sL t i+ @Qi 1 @ + 2 = A qi 12 + A qi+ 12 + Aˆ qx dxA , @t x xi 1 +sR t 2 ⌘ 1 ⇣ + = A qi 12 + A qi+ 12 + A qi (2.15) x
19 where the fluctuations, A± q,at xi A ± qi
1 2
=
X⇣ p
are defined as
1 2
⇣ ⌘⌘± ⇣ ⌘ sp qiL 1 , qiR 1 W p qiL 1 , qiR 1 2
2
2
(2.16)
2
and A qi , the total fluctuation, takes the form: A qi =
2.1.3
X⇣ p
⇣ ⌘⌘± ⇣ ⌘ L L sp qiR 1 , qi+ W p qiR 1 , qi+ 1 1 2
2
2
(2.17)
2
Flux: f-wave Riemann solvers
Now we can return to (2.1). Ideally we seek a semidiscrete scheme akin to (2.15), so that we can use the algorithms developed to solve Riemann problems with varying fluxes that are employed in Clawpack [50, 52, 54, 55]. Let us recall the scheme developed in [50] for the homogeneous system‡ , (2.18)
qt + f (q)x = 0,
There the solution to the Riemann problem at xi 12 was obtained by computing a flux-based wave decomposition, where the flux difference is given in terms of the eigenvectors: ⇣ ⌘ ⇣ ⌘ X X p p p f qiR 1 f qiL 1 = r = Zi 1 (2.19) 2
2
p
2
p
Here the vectors Z p are called f-waves [50,52]; and rp are the eigenvectors of the flux jacobian, fq (q) ⌘ Ai (q), Once again, if qˆ(x, t + t) is the exact evolution of qˆ after a time increment t we obtain: Z t xi+ 12 Qi (t + t) Qi (t) = Ai qˆx (x, t + t)dx (2.20) x x 1 i
Dividing by
t and taking the limit as 0 @Qi 1 @ + = A qi 12 + A @t x
2
t ! 0 we obtain: qi+ 12 +
Z
xi+ 1 2
xi
1 2
1
Ai qˆx dxA
(2.21)
where the fluctuations A± qi⌥1/2 are given in term of the f -waves, Z, as: A + qi
1 2
X
=
p:sp i
1 2
>0
Zip
A
1 2
qi+ 12 =
X
p:sp
i+ 1 2
t0 a reduction in the density of trapped electrons is inexorable, i.e. ⇢(r, t, ; E) < ⇢(r, t0 , ; E). Evidence of such electronic rearrangement under energy absorption by As2 S3 chalcogenide thin film has been reported separately by Tanaka et al. and Lee et al.. Tanaka et al. [146], observed chemical and medium range re-ordering in As2 S3 film under photon energy absorption. Meanwhile, using X-rays, Lee et al. observed modification in the structure order of chalcogenide films [147]. Albeit focusing on the dynamics of photo-darkening and anisotropy, respectively, both studies show that upon energy absorption the material suffers an alteration in both the electron energy density and the bond structure; in agreement with our observations, where the alteration of the permittivity springs from the changes in the structural and electronic states of the film.
84
8.4.2
Prospective deduction, the refractive index
Earlier we discussed the inextricable relation between the refractive index, n, and absorption, k, with the electromagnetic properties of the material, "r and µr . Often, in the deduction of the refractive index, the permeability, µr , is set to a constant value, generally 1, resulting in a set of relations widely available in the basic literature. Here we present the calculated optical parameters, n and k, based on these relations (see Figures 8.7a and 8.7b respectively). Under the former assumption, the computed results show similar behaviour for n and k to that observed for "1 and "2 , with the peak of the refractive index, n, reducing by ⇠ 23% after an irradiation time of t = 1s. The minimum change takes place at 6.5 eV where the reduction is ⇠ 8%. For energies close to the band-gap the reduction is on average ⇠ 20%. The absorption, k, also shows a striking decay, down to 40% from its peak value, and a minimum of ⇠ 10%, with an average reduction of ⇠ 35% for dispersion energies close to the band-gap. Calculation of the extrema of the refractive index and absorption coefficient are shown in Figures 8.8a and 8.8b. The observed measurements, and the calculated optical properties therein, are extraordinary, in the sense that all previously published experiments, with high energy electrons (40 keV), reported an increase in the refractive index between 3% to 8% [126–130, 148]. In contrast to the vast literature reporting photon-induced refractive index change, which has yielded as much as an 8% increase in the refractive index of As2 S3 film under illumination [112, 114], in our experiment the refractive index decreases as much as 23%. Bearing in mind that the conditions in the cited experiments are significantly different from those reported here, we believe the discrepancy could be explained by two different mechanisms. The first would state that, as described in the previous section, the permeability remains unchanged, while the number of electrons and the number of homopolar and broken heteropolar bonds increases; leading to the recombination mechanism described earlier. These structural alterations, in turn, induce nano-crystallization, ultimately leading to the generalized reduction in the number of energy traps, especially within energies in range A; all these simultaneous changes will cause reduction in the permittivity, and therefore reduce the refractive index. On the other hand, to reconcile previous published results with ours, it would be necessary to acknowledge a dynamic change of the permeability with respect to electron irradiation. This would require the electrons to alter the atoms intrinsically by means of elastic collisions, and/or induce current loops, causing the magnetic dipoles in the material to reorganize, so inducing paramagnetic states, and hence increasing the permeability. However plausible the latter explanation, an experimental confirmation is required, together with further studies on the permeability of chalcogenide glass under high energy electronand photon- irradiation.
85
(a) @t⇤ "1
(b) @t⇤ "1
(c) @t⇤ "2
(d) @t⇤ "2
Figure 8.5: Derivatives of the permittivity function with respect to the causal variable dosage, t⇤ .
Figure 8.6: Homopolar and heteropolar bond breaking process under electron irradiation in As2 S3 .
86
(a) n("1 , "2 )
(b) k("1 , "2 )
Figure 8.7: (a) Refractive index n, and (b) absorption constant k, derived from the real and imaginary permittivity.
(a) nmax ("1 , "2 )
(b) kmax ("1 , "2 )
Figure 8.8: Peak trace sampling of (a) refractive index, n, and (b) absorption constant, k, derived from the real and imaginary permittivity.
87
Part IV
Epilogue
89
Chapter 9
Closing Remarks Through this work a new characterization procedure and analysis of the permittivity of Chalcogenide glass has been presented. Based on low-loss Electron Energy Loss Spectroscopy (EELS), the results are extendable to the optical regime by means of the small angle scattering approximation. Furthermore, they allow us to calculate an approximate form of the refractive index, assuming constant permeability, and suggest the possibility of magnetic alterations induced by electron irradiation. The calculated results and observations found that high energy electrons induced a reduction in the permittivity, real and imaginary, of the material. The real permittivity underwent a maximum reduction of ⇠ 40%, while the imaginary permittivity decreased by ⇠ 50%. The results can be explained in terms of the atomic bond reconfiguration; in this model the incident electrons break the homopolar and heteropolar bonds, leading to a reduction of the former, correcting the wrong bonds. The results are significant to the development of manifold photonic applications, with applications to numerous areas of research and engineering. Namely, the observed reduction in the permittivity could enable a new range of transformation optic devices, which have been so far limited to the realm of far-IR range of the electromagnetic field. Furthermore, these results could be significant to future implementation of reconfigurable photonic circuits, infrared telecommunications, photonic crystals, and all optical conversion and computing. Based on this material properties we have demonstrated that, under the paraxial description of light in heterogeneous media, it is possible to use centrosymmetrical refractive index distributions as attractors for light, which are able to perform light trapping or confinement in open orbits. These mappings are bound within fabrication constrains, and due to the paraxial approximation they are several times larger than the mean beam width and wavelength, ⇠ 50 times larger. The proposed mappings have potential applications to transient optical memories, delays, concentrators, random cavities, and beam stirrers, to cite a few; which could enable the next generation of photonic integrated circuits, PICs, processors and computation systems. Due to their sensitivity to the change in the refractive index these devices could be employed as a sensor platform. Furthermore, since they are drawn in close analogy to celestial phenomena they could be used as laboratory setups to test a large variety of effects in celestial mechanics/ Furthermore, we have, to the best of our knowledge, theoretically demonstrated the viability of the first PIC and PLC compatible device utilizing a heterogeneous refractive index map, to achieve the transformation of a Gauss
90 beam into a Bessel-Gauss-J0 beam. The computed device has a loss of 0.457dB, and high energy focus, where 95% of the output beam energy is concentrated at the principal peak of the Bessel-Gauss-J0 beam profile. The beam has a divergence of 5% over a propagation distance of 1mm, and exhibits selfhealing when partially obstructed by an opaque object. The resulting beam has a Bessel-Gauss-J0 beam profile for both near- and far-fields. It is note that our design is based on current manufacturing techniques, such as controlled oxidation of Si/Si1y Oy /Si1xy Gex Cy or by stacking different layers of chalcogenide glasses. An integrated Gauss to Bessel-Gauss-J0 beam convertor could enable on-chip applications which take advantage of the beamâĂŹs self-healing and non-diffractive properties, such as micro optical tweezers, traps and couplers, photonic integrated circuits for telecommunications and computerized micro tomography and microscopy. Finally, we have developed a numerical scheme to model light propagation in spacetime-varying and nonlinear materials. The results suggest that our scheme is second order accurate. Some examples of light trapping and nonlinearity compensation were numerically demonstrated, together with two spatial dimensions examples of light propagation in an oscillatory medium. While our discussion revolves around Maxwell’s electromagnetic equations, the methods can be applied to other physical wave phenomena in time-varying media. This numerical scheme resulted in the development of EMClaw an extension to the Clawpack Software. An abridge summary of this thesis could be condensed in the following points: • Developed a novel fabrication and characterization techniques for time-varying materials, and structures for light confinement, • Presented a novel characterization of the electric properties of As2 S3 chalcogenide glass and • Demonstrated an electron-induced refractive index increase in As2 S3 chalcogenide glass • Developed a numerical scheme to model wave phenomena in spacetime-varying media, and • Created EMClaw, an extension to the Clawpack software, to model spacetimevarying and nonlinear media. EMClaw could be used in other problems beyond the scope of photonics, for example interest to use this package has been raised by communities in plasma and solar physics. • Devised PICss to trap and transform light that could potentially be used as time delays and power portable tomography scanners.
What is next? Through the course of this thesis ideas were drawn and later abandoned due to many reasons, some depended on external collaborators and other went beyond the scope of this thesis. However, the elements set forth by this work could be apply to the development of some ideas like • Ultra fast coupling in time-varying waveguide couplers
91 • Incorporation of dispersion and absorption in the numerical scheme for spacetimevarying materials • Analysis of chaos in light trapping orbits • Light in random-time ordered media • Moving and sudden cloaking in real-like materials, i.e. spacetime-varying cloaks. • ...
92
93
Part V
References and Appendices
95
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106
APPENDICES
107
Riemann Solver A.1
2D Riemann solver
In this case equation (1.4) takes the form: ¯ @t q(x, y, t) + @x f 1 (q; x, y, t) + @y f 2 (q; x, y, t) = , .
(A.1)
Because electromagnetic waves are transversal, in two dimensions we can decompose any wave into a polarization basis, habitually called transverse electric, E, and magnetic, M modes [4], 0 11 0 11 E H qM ⌘ @ E 2 A , qE ⌘ @H 2 A . (A.2) H3 E3 If the medium is isotropic and linear, only the diagonal entries of ¯ are non-zero, with values ii = ⌘ i . Then, expanding the E mode yields: ⌘ 0 qt0 + ⌘t0 q 0
qy2 /"o = 0
(A.3a)
⌘ 1 qt1 ⌘ 2 qt2
qx2 "o = qx1 µo
0
(A.3b)
qy0 µo = 0
(A.3c)
+ +
⌘t1 q 1 ⌘t2 q 2
+ +
A similar set of equations can be calculated for M mode. In this particular case the flux functions take the form: 0 1 0 2 1 0 q /"o f 1 = @ q 2 /"o A and f 2 = @ 0 A . (A.4) q 1 /µo q 0 /µo Then we can calculate the flux Jacobian for each f i and obtain: 0 1 0 1 0 0 0 0 0 1/"o 0 1/"o A and B ⌘ fq2 = @ 0 0 0 A . (A.5) A ⌘ fq1 = @0 0 1/µo 0 1/µo 0 0
Because AB 6= BA, there is no single transformation that will simultaneously diagonalize A and B . Hence we diagonalize each matrix separately, A = Rx ⇤x (Rx )
1
,
B = Ry ⇤y (Ry )
1
(A.6)
where Ri is the matrix whose columns are the eigenvectors, ri,p ,of either A or B, and ⇤i the diagonal matrix of the corresponding eigenvalues i,p set in increasing order. The matrix A yield the right eigen-vectors: 0 1 0 1 0 1 0 1 0 rx,1 = @ z A , rx,2 = @0A , rx,3 = @z A . (A.7) 1 0 1
108 And matrix B has the right eigen-vectors: 0 1 0 1 0 1 z 0 z rx,1 = @ 0 A , rx,2 = @1A , rx,3 = @ 0 A . 1 0 1
(A.8)
Either one with eigenvalues: i,1
=
i,2
c,
i,3
= 0,
(A.9)
=c
Where the values z and c are defined as: p c = 1 "o µ o , µo z= , "o
(A.10a) (A.10b)
Note that the inverse of z is commonly known as the impedance in electromagnetic theory, while c defines the speed of the waves. From (A.7) and (??) we obtain Rx,y : 0 1 0 1 0 1 0 z 0 z Rx = @ z 0 z A and Ry = @ 0 1 0 A (A.11) 1 0 1 1 0 1 1
With inverse (Rx,y ) , 0 1 0 2z 1 x 0 (R ) = @ 1 1 0 2z
1 2
1
0 A 1 2
(Ry )
and
Now we can solve for x,y in (??) with q 2 , 0, q 0 , to obtain: x,1
=
x,2
=0
x,2
=
y,1
=
y,2
=0
y,3
=
1
0
=@
f 1 = 0,
f 1,2 + 2z
f 1,3 z
1 2z
0
1 2z
q2 ,
0 1 0
1 2
1
0 A (A.12) 1 2
q 1 and
f1 =
(A.13a) (A.13b)
f 1,2 + f 1,3 z 2z f 2,1 + f 2,3 z 2z
(A.13c)
(A.14a) (A.14b)
f
2,1
f 2z
2,3
z
(A.14c)
109
Simulation Parameters B.1
Clawpack Simulation paramters
Summary of convergence tests for maxwell variable coefficients in 1D ((emclaw:maxwell_vc_1d), calculation settings: Material shape is given by,
n + n exp
✓
(x
(v0 ⇤ t
xof f set ))2 2
◆
(B.1)
Excitation, source, shape is defined by,
q
0,1
= A0,1 exp
✓
(x
( 2.0 + v0 ⇤ t))2 2
◆
(B.2)
110
111
Publications and Conferences C.1
Journal publications
D. P. San-Roman-Alerigi, D. Ketcheson, and B. S. Ooi, A finite volume method for Maxwell’s equations in nonlinear, dispersive, and spacetime-varying media, SIAM Journal on Applied Mathematics (in preparation) D. P. San-Roman Alerigi, A. B. Slimane, T. K. Ng, M. Alsunaidi, and B. S. Ooi, A possible approach on optical analogues of gravitational attractors, Optics Express 21, 8298–8310, 2013 D. P. San-Roman-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. Alsunaidi, and B. S. Ooi, Electron irradiation induced reduction of the permittivity in Chalcogenide glass (As2 S3 ) thin film, Journal of Applied Physics 113, 044116, 2013 D. P. San-Roman-Alerigi, T. K. Ng, M. Alsunaidi, Y. Zhang, and B. S. Ooi, Generation of J0 -Bessel-Gauss beam by an heterogeneous refractive index map, Journal of the Optical Society of America A 29, 1252–1258, 2012
C.2
Conference articles
(invited talk ) D. I. Ketcheson and D. P. San-Roman-Alerigi, âĂIJHigh order parallel WENO-wave propagation algorithms for hyperbolic PDEs in three dimensions,âĂİ World Congress on Computational Mechanics (WCCM), 2014. D. P. San-Roman-Alerigi, D. Ketcheson, and B. S. Ooi, Revisiting Maxwell’s equations: a wave propagation method for electromagnetics in spacetime-varying media, SPIE Photonics Europe, Belgium, 2014. Abstract ID: 9131-38 D. P. San-Roman-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. Hedhili, M. Alsunaidi, and B. S. Ooi, Electron-induced giant refractive index reduction in chalcogenide glass thin films, 7th International Conference on Materials for Advanced Technologies (ICMAT), Singapore, 2013. Abstract ID ICMAT13-A-0568 D. P. San-Roman-Alerigi, A. B. Slimane, T. K. Ng, M. Alsunaidi, and B. S. Ooi, Photonic analogies of gravitational attractors, Saudi International Electronics, Communications and Photonics Conference (SIECPC), Saudi Arabia, 2013. Abstract ID: 189 D. P. San-Roman-Alerigi, D. H. Anjum, Y. Zhang, X. Yang, A. B. Slimane, T. K. Ng, M. Hedhili, M. Alsunaidi, and B. S. Ooi, Time depend characterization of permittivity of chalcogenide glass As2 S3 by electron energy loss spectroscopy, XXII International Materials Research Congress, Symposium on Electron Microscopy of Materials, Mexico 2013. Abstract ID: IMRC2013-SIM18-ABS001 D. P. San-Roman-Alerigi, A. B. Slimane, T. K. Ng, M. Alsunaidi, and B. S. Ooi, On a pragmatic approach to optical analogues of gravitational attractors,
112 Photonics Global Conference (PGC), Singapore, December 2012 D. P. San-Roman-Alerigi, Y. Zhang, T. K. Ng, A. B. Slimane, and B. S. Ooi, Monolithic Silicon-based Gauss to J0 -Bessel-Gauss Beam Converter, Progress in Electromagnetics Research Symposium (PIERS), Kuala Lumpur, Malaysia, 2012 Y. Zhang, D. P. San-Roman-Alerigi, T. K. Ng, B. S. Ooi, Building blocks of passive optical components fabricated in pulsed laser deposited As2 S3 chalcogenide glass film, 6th International Conference on Materials for Advanced Technologies (ICMAT), Singapore, 2011 Y. Zhang, Y. Gao, D. P. San-Roman-Alerigi, T. K. Ng, and B. S. Ooi, B. Chew and M. Hedhili, D. Zhao and H. Jain, Engineering of refractive index in sulfide chalcogenide glass by direct laser writing, Photonics Global Conference (PGC), Singapore, 2010. Abstract ID: 5705967
C.3
Book contributions
B. K. Muite, et al. Parallel Spectral Numerical Methods: Chapter Maxwell’s equations (electronic, in preparation)
C.4
Talks, seminars and presentations
D. P. San-Roman-Alerigi, The effects of time-varying permittivity in light propagation, Seminar Series on Quantum and Classical Optics, College of Sciences, National Autonomous University of Mexico (UNAM), Mexico 2013. D. P. San-Roman-Alerigi, M. Alsunaidi, and B. S. Ooi, On the electron induced modification of permittivity in chalcogenide glass, KFUPM International Physics Day: Nonlinear sciences and their applications, Saudi Arabia, 2012 D. P. San-Roman-Alerigi, Y. Zhang, T. K. Ng, A. B. Slimane, and B. S. Ooi, Optical attractors and light control on photorefractive structures, 1st KAUSTUCSB-NSF Workshop on Solid State Lighting, Saudi Arabia, 2012 D. P. San-Roman-Alerigi, Similarities between spacetime geometries and refractive index mappings, Seminar Series on Quantum and Classical Optics, College of Sciences, National Autonomous University of Mexico (UNAM), Mexico 2012. D. P. San-Roman-Alerigi, A. B. Slimane, Y. Zhang, T. K. Ng, and B. S. Ooi, Monolithic Si-based optical beam converter, KAUST-KFUPM Second International Workshop on Photonics, Saudi Arabia, 2011 D. P. San-Roman-Alerigi, T. K. Ng, and B. S. Ooi, Towards light manipulation with plasmon polaritons, KAUST-KFUPM First International Workshop on Photonics, Saudi Arabia, 2010
C.5
News features
Laser Focus World: 2D refractive-index mapping models cosmological systems, July 2013. Advances in Engineering: A possible approach on optical analogues of gravitational attractors, November 2013.
113 Variable
Value
Description Excitation (Source1D)
shape wavelength v Ey Hz
’pulse’ 2.0 1/bkg_n zo 1.0
shape transversal_shape wavelength v Ex Ey Hz
’pulse’ ’cosine’ 2.0 1/bkg_n 0.0 zo 1.0
Gaussian shaped pulse. Wavelength, . It defines the pulse width, i.e. Relative velocity of wave in the medium Amplitude of electric field, E ⌘ q 0 Amplitude of magnetic field, H ⌘ q 1 Excitation (Source2D) Gaussian shaped pulse in x direction. Transversal shape, y direction. Wavelength, . It defines the pulse width, i.e. Relative velocity of wave in the medium Amplitude of electric field, E ⌘ q 0 Amplitude of electric field, E ⌘ q 1 Amplitude of magnetic field, H ⌘ q 2 Vacuum
eo mo co zo
Permittivity of vacuum, "0 Permeability of vacuum, µ0 Speed of the wave in vacuum c Impedance of vaccum
1.0
Material (background) shape bgk_er bgk_mr bkg_n chi2_e/m chi3_e/m
’moving_gauss’ 1.5 sqrt(bkg_er*bkg_mr)
Shape of material, corresponds to a moving perturbation Relative permittivity "r ⌘ ⌘ 11 /"0 Relative permeability µr ⌘ ⌘ 22 /µ0 p Refractive index, n = "r µr Second order non-linearity, Third order non-linearity,
0.0
(2) (3)
Material (perturbation), .a.k.a RIP prip delta_n offset_e/m sigma_e/m
0.1 prip*bkg_n 25.0 5.0
Relative amplitude of perturbation (RIP) from the background Absolute amplitude of perturbation, n Perturbation offset at t0 Width of perturbation Simulation space 1D
x_lower x_upper mx
0.0 300.0 2p , p = 7, 8, 9, . . . , 17
Lower bound for x-dimension Upper bound for x-dimension Number of cells, nc Simulation space 2D
x_lower x_upper y_lower y_upper mx my
0.0 180.0 0.0 180.0 2p , p = 7, 8, 9, . . . , 17 2p , p = 7, 8, 9, . . . , 17
Lower bound for x-dimension Upper bound for x-dimension Lower bound for y-dimension Upper bound for y-dimension Number of cells, nc Number of cells, nc