Numerical study and optimization of photonic crystals

Photonic crystals (PhCs) are engineered nanostructures that enable an extraordinary control over the flow of light. These structures can be fabricated out of common semiconductors, are compatible with existing industrial fabrication technologies, and are expected to play a major role in future devices integrating photonic circuits - e.g. for telecommunications or in future quantum technologies. In this thesis, we explore a wide range of properties of the most common class of PhCs, formed by a lattice of circular holes in a semiconductor slab. To compute the electromagnetic eigenmodes of a given structure, we use fast mode-expansion methods, which are presented in detail here. The first application consists in a detailed analysis of the effects of fabrication disorder on the PhC structures. It is by now well-known that disorder is in many cases the limiting factor in device performance. Here, we shed more light on its effects, by statistically comparing various designs for PhC cavities with a high quality factor, and by analyzing the effect of irregular hole shapes on a PhC waveguide. The second application presented here stems from the fact that PhCs are in fact tremendously flexible, and their features are determined by a large number of controllable parameters. This is on one hand a great advantage, but on the other a great challenge when it comes to finding the optimal device for a given application. To face this challenge, we have developed an automated optimization procedure, using a global optimization algorithm for the exploration of an insightfully selected parameter space. This was applied to various devices of interest, and inevitably resulted in a vast improvement of their qualities. Specifically, we demonstrate various high-Q cavity designs, and a slow-light coupled-cavity waveguide with extraordinary features. We also present several experimental confirmations of the validity of our designs. Finally, we discuss two domains in which PhCs (and our optimization procedure) can be expected to play a major role. The first one is integrating quantum dots with the goal of long-range, photon-assisted dot-dot coupling, with implications for quantum information processing. We develop a semi-classical formalism, and analyze the magnitude and attenuation length of this coupling in large PhC cavities, as well as in a waveguide. The second outlook is in the field of topological photonics. We describe an array of resonators, in which an effective gauge field for photons can be induced through an appropriate time-periodic modulation of the resonant frequencies. This results in a Quantum Hall effect for light, and, in a finite system, one-directional edge states immune to fabrication disorder are predicted. We discuss the possibilities for a practical implementation, for which a PhC slab is among the most promising platforms.