Abstract Wave Chaos in Dielectric Resonators: Asymptotic and Numerical Approaches Hakan E. TureciÄ 2004 Dielectric optical micro-resonators and micro-lasers represent a realization of a wave- chaotic system, where the lack of symmetry in the resonator shape leads to non- integrable ray dynamics in the short-wavelength limit. Understanding and controlling the emission properties of such resonators requires the investigation of the correspon- dence between classical phase space structures of the ray motion inside the resonator and wave-functions. Semi-classical approaches to the resonances of deformed cylin- drical resonators are analyzed ¯rst within the closed limit, which corresponds to the quantum billiard problem from the ¯eld of quantum chaos. The results are then generalized to the dielectric case. We develop an e±cient numerical algorithm to calculate the quasi-bound modes of dielectric resonators, which play a crucial role in determining the emission properties of micro-lasers based on dielectric resonators. Resonances based on stable periodic ray orbits of dielectric cavities are constructed in the short-wavelength limit using the parabolic equation method, and an asso- ciated wavevector quantization rule for the complex wavenumbers is derived. The e®ect of discrete symmetries of the resonator is analyzed and shown to give rise to quasi-degenerate multiplets. A recent experiment on lasing emission from deformed GaN micro-cavities is analyzed, leading to the appearance of scarred modes and non-specular e®ects in the far¯eld emission pattern. A framework is presented for treating the non-linear laser equations in a form suitable for treating the dielectric micro-lasers. Wave Chaos in Dielectric Resonators: Asymptotic and Numerical Approaches A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Hakan E. TureciÄ Dissertation Director: A. Douglas Stone May 2003 Copyright c 2004 by Hakan E. TureciÄ All rights reserved. ii Hayat³ma yÄon veren ve te»sekkurÄ etmeye f³rsat bulamad³¸g³m iki insana: Yurdanur TureciÄ ve Tahir Zozan Balka»s'a iii Contents Acknowledgments xiv 1 Introduction 1 1.1 Overview . 1 1.2 Brief Overview of this Thesis . 12 2 Asymptotic Physics: Where rays and waves meet 18 2.1 Introduction . 18 2.2 Semi-classical limit of the Helmholtz equation . 20 2.3 Eikonal Equation as an initial value problem . 22 2.4 A paradigm for integrable systems: The circle billiard . 29 2.5 Generic Billiards and appearance of Chaos . 38 2.6 Hamiltonian description of Ray Motion . 40 2.7 Poincar¶e Surface of Section and Discrete Maps . 44 2.8 The monodromy matrix and periodic orbits . 48 2.9 KAM Transition { from integrability to Chaos . 52 2.10 Semiclassics in a mixed phase space . 60 2.11 Quantum Poincar¶e-SOS . 63 2.12 Breakdown of local expansions . 66 2.13 PUSC and Quantum Scars . 67 2.14 Refractive Billiards . 69 3 Scattering quantization method for optical billiards 77 3.1 Introduction . 77 3.2 Reduction of Maxwell's equations . 78 3.3 Scattering Quantization-Philosophy and Methodology . 81 3.4 Numerical Implementation . 86 3.5 Root-search strategy . 88 3.6 Scattering eigenstates and classical phase space structures . 98 3.7 Role of discrete symmetries . 100 3.8 Conclusion and Outlook . 102 i CONTENTS ii 4 Gaussian Optics for Microcavities 104 4.1 Introduction . 104 4.2 Gaussian Optical approach to the Dirichlet cavity . 108 4.3 Boundary Conditions . 111 4.4 Ray dynamics in phase space . 113 4.5 Single-valuedness and quantization . 114 4.6 Transverse excited modes . 117 4.7 Perturbation Theory . 119 4.8 Opening the cavity - The dielectric resonator . 121 4.9 Symmetry Analysis and Quasi-Degeneracy . 125 4.10 Symmetrized modes for the quadrupole . 126 4.11 Simple Rule for Quasi-Degeneracy . 128 4.12 Conclusions and Relation to Experiment . 132 5 Non-classical phenomena in Dielectric Resonators 136 5.1 Introduction . 136 5.2 Blue Lasers: Experimental Setup and Results . 137 5.3 Con¯guration Space theory of Scars . 143 5.4 Non-specular e®ects in emission in refractive billiards . 151 6 Towards a Non-Linear Theory of Dielectric Resonators 159 6.1 Introduction . 159 6.2 Single-mode Lasing . 161 6.3 Adiabatic Elimination . 164 6.4 Multi-mode Lasing . 165 6.4.1 Diagonal Inversion: Non-degenerate lasing . 167 6.4.2 Non-Diagonal Inversion: Degenerate lasing . 169 7 Conclusion and Open Questions 171 A Uniform expressions valid on the caustic 173 B Tunneling in the dielectric cylindrical cavity 175 C Coordinate Systems on the PO 179 D Linear Ray Reection & Refraction transformations 181 E Perturbation Theory for Gaussian Modes 183 F The Lens Transform for a Cavity 188 Bibliography 191 List of Figures 1.1 A comparison of scattering and emission perspectives. Variation of the intensity scattered o® a dielectric circular cylinder with the wavenum- ber k of an incoming plane-wave is plotted on the back panel (I ¡ Re [kR] plane). The intensity is observed at 170± with respect to the incoming wave direction. The complex quasi-bound mode frequen- cies are plotted on the Re [kR] Im [kR] plane. Notice that the most prominent peaks in scattering ¡intensity are found at the values of k where a quasi-bound mode frequency is closest to the real-axis. These are the long-lived resonances of the cavity. Also visible is the contri- bution of resonances with shorter lifetimes (higher values of Im [kR]) to broader peaks and the scattering background. 3 1.2 A schematics of the Fabry-Perot type laser cavity is shown on the left. An actual Fabry-Perot mode. The right mirror is assumed to be formed by a dielectric interface. 4 1.3 (a) A scanning electron micrograph picture of a VCSEL device [1]. The horizontal scale of the device is 4¹m. (b) Schematics of the structure of such a device (courtesy of Jung Han, Yale University) . 4 1.4 (a) Fresnel reection and refraction in a planar dielectric cavity. (b) The high-reectivity mirror action is provided by total internal reection. 5 1.5 (a) A scanning electron micrograph of a microdisk laser on a pedestal [2]. The horizontal scale of the device is about 5¹m. (b) Top panel: real space false color plot of the modulus of the electric ¯eld for a cal- culated quasi-bound state of a planar microdisk cavity. Lower panel: Ray trajectories associated with the mode on the top panel. 6 1.6 The construction of the surface of section plot. Each bounce from the boundary is represented by a point in the SOS by recording the angular position of the bounce on the boundary (Á) and the angle of incidence with respect to the local outward pointing normal (sin Â). For sin  > sin Âc > 1=n, total internal reection takes place. Refractive escape results when a bounce point (bounce #4 in the ¯gure) falls below the \critical line" sin  > sin Âc. Note that sin  < 0 correspond to clockwise sense of circulation. We do not plot the sin  < 0 region-the SOS has reection symmetry. 8 iii LIST OF FIGURES iv 1.7 The Poincar¶e surface of section at an intermediate quadrupolar defor- mation of ² = 0:08. Regular motion takes place along invariant curves in the SOS, i.e. any trajectory started on such a curve will remain on the same curve throughout its motion. Note that a curve is obtained by discrete points in the large bounce limit. The regions containing scatter of points represent chaotic motion. A generic initial condition within a given connected area will come arbitrarily close to any point within that component. Note that the invariant curves act as barri- ers to the chaotic motion. This feature is limited to two-dimensional systems. 9 1.8 (a) A real space false color plot of the electric ¯eld intensity for a chaotic whispering gallery resonance at n = 2, illustrating the generic emission pattern from points of highest curvature on the boundary in the tangent direction. (b) Emission of a dynamically eclipsed state at n = 1:54 (after [3, 4]) . 10 1.9 (a) Scanning electron micrographs of the top and side-view of one of the deformed cylindrical quantum cascade micro-lasers. (b) Numeri- cally calculated resonance corresponding to the bowtie-mode observed for the deformation ² = 0:16. (c) Angular dependence of the emission intensity for deformations ² = 0 (triangles), ² = 0:14 (open circles), ² = 0:16 (¯lled circles). Dashed line represents the calculated far¯eld pattern corresponding to the resonance in (b). The right inset shows the coordinate system used and the left inset shows the the logarith- mic plot of the measured power spectrum. The free-spectral range of the peaks is found to agree with the calculated bowtie free spectral range (after [5]). 11 2.1 Schematics showing the level-curves of the Eikonal S(q), S(q) = S0 and S(q) = S1. The normals of S(q) = S0, form a normal congruence of rays and yield the next level-curve S(q) = S1. Depicted is a sit- uation where the index of refraction n(q) is monotonically increasing from the top of the ¯gure to the bottom. 21 2.2 Schematics showing the ray coordinates (s; ¿). The dashed lines are equi-coordinate curves s = constant and ¿ = constant, which are the wavefronts and rays. 23 2.3 Di®erent boundary conditions § resulting in di®erent geometry of wavefronts. 25 2.4 Schematics showing the mapping of an initial line element @D¿0 on an initial wavefront at ¿ = 0 by ray ow to its value at ¿ > 0, @D¿ .