Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Thursday, July 15, 16:00 ~ 16:30 UTC-3

Quantitative Estimates of Frequency Band-gaps in Photonic Crystals

Robert Lipton

Louisiana State University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A frequency band-gap inside a photonic crystal is an interval of frequencies where there is no wave propagation through the crystal. The size and location of a frequency band-gap depends on the geometry of the periodic array of scatterers. We obtain explicit and rigorously quantitive estimates on band-gap location and size in terms of the geometry of the scatters. Examples are provided for different scatterer configurations and shapes.

These estimates are given in terms of formulas that explicitly depend on the eigenvalues of the Neumann-Poincare operator defined on the boundary of the scatters, the Dirichlet spectrum of the scatter, the scatter configuration, and the contrast in dielectric properties between scatterer and connected phase. The methodology is operator theoretic and uses a new analytic representation of the time harmonic wave equation together with analytic perturbation theory about high contrast. These methods deliver rigorously convergent power series in the contrast from which estimates can be made.

Joint work with Robert Viator (Swarthmore College, Pennsylvania).

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