Wave chaos in dielectric resonators: Asymptotic and numerical approaches

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 correspondence between classical phase space structures of the ray motion inside the resonator and wave-functions. Semi-classical approaches to the resonances of deformed cylindrical resonators are analyzed first within the closed limit, which corresponds to the quantum billiard problem from the field of quantum chaos. The results are then generalized to the dielectric case. We develop an efficient 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 associated wavevector quantization rule for the complex wavenumbers is derived. The effect 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 effects in the farfield emission pattern. A framework is presented for treating the non-linear laser equations in a form suitable for treating the dielectric micro-lasers.