Theoretical and Computational Analysis of Wave Confinement in Periodic Media

In this PhD thesis, we study the propagation and behavior of waves in crystalline systems, with the emphasis on electromagnetic waves in photonic crystals and their confinement. In Chapter 1, we introduce the topic of wave manipulation by crystals and the basics of the mathematical description of electromagnetic waves in photonic crystals. Chapter 2 introduces a general scaling theory of identification and classification of wave confinement in crystal superlattices. Starting from the confinement energy and the mode volume, we use finite-size scaling to find that ratios of these quantities raised to certain powers yield the confinement dimensionality of each band. In Chapter 3, we investigate the possibility of improving the accuracy of our scaling method for classification of wave confinement in very small supercells by means of machine learning. We therefore slightly reformulate the scaling theory of Chapter 2 and use it to implement a physics- based clustering algorithm to classify wave confinement in photonic crystals. In Chapter 4, we employ the results of the previous chapters to analyze the light-confining properties of inverse woodpile photonic superlattices with respect to their structural parameters, namely the regular and defect pore radii. We create so-called confinement maps, depicting the dependence of the number and energy concentration of 3D confined bands on the crystal structural parameters. In Chapter 5 we investigate the problem of light propagation through realistically large photonic crystals - a notoriously computationally difficult task. To this end, we develop a size-robust discontinuous Galerkin finite element method, where we approximate the solution of a large but finite crystal by a set of Bloch modes, representing the solutions of an infinite crystal. We conclude in Chapter 6 by summarizing the results of this PhD thesis and providing suggestions for future research.