Electromagnetic wave chaos in photonic crystals

Similarities in the form of the Schrodinger equation that governs the behaviour of electronic wavefunctions, and Maxwell’s equations which govern the behaviour of electromagnetic waves, allow ideas that originated in solid state physics to be easily applied to electromagnetic waves in photonic structures. While electrons moving through a semiconductor experience a periodic variation in charge, in a photonic crystal electromagnetic waves experience a periodic variation in refractive index. This leads to ideas such as bandstructure being applicable to the one and two dimensional photonic crystals used in this work. The following work will contain theoretical and experimental studies of the transmission through, and electric fields within, one dimensional photonic crystals. A slow variation in the structure of these crystals will lead to the bandstructure shifting, with an photonic analogy of electronic Bloch oscillations and Wannier-Stark ladders being seen in these structures. The two dimensional photonic crystals will be shown, through Hamiltonian ray tracing, to support both stable and chaotic ray paths. Examination of the phase space reveals the existence of ‘Dynamical Barriers’, regions in phase space supporting stable ray trajectories that divide separate regions in which the ray trajectories are chaotic. Various manners in which the bandstructure may be varied will be presented, along with a proposed switch that may be made using these structures. While the ray tracing will be carried out in photonic crystals in the limit of infinitesimally thin dielectric sheets, the model will then be developed to show the bandstructure of a photonic crystal made from finite width dielectric sheets, with examples of dispersion surfaces for these structures being presented