Bloch Waves in Crystals and Periodic High Contrast Media

Analytic representation formulas and power series are developed describing the band structure inside periodic photonic and acoustic crystals made from high contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasi-periodic source free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. Convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation

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 -\u3e infinity. An explicit radius of convergence for this power series is obtained, and can be written explicitly in terms of the Bloch-wave vector, 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