Spectral Properties of Photonic Crystals: Bloch Waves and Band Gaps

The author of this dissertation studies the spectral properties of high-contrast photonic crystals, i.e. periodic electromagnetic waveguides made of two materials (a connected phase and included phase) whose electromagnetic material properties are in large contrast. A spectral analysis of 2nd-order divergence-form partial differential operators (with a coupling constant k) is provided. A result of this analysis is a uniformly convergent power series representation of Bloch-wave eigenvalues in terms of the coupling constant k in the high-contrast limit k → ∞. An explicit radius of convergence for this power series is obtained, and can be written explicitly in terms of the Bloch-wave vector α, the Dirichlet eigenvalues of the inclusion geometry, and a lower bound on another spectrum known as the ”generalized electrostatic resonances”. This lower bound is derived from geometric properties of the inclusion geometry for the photonic crystal.