We study the photonic band structure of a metal–dielectric periodic structure. The metallic component is described by the Drude model; therefore, the electric permittivity is frequency dependent, i.e., dispersive. Rather than solving a nonlinear eigenvalue problem for the band structure of the material, we follow a time-dependent formulation described in Raman and Fan (Phys Rev Lett 104:087401, 2010) which leads to a linear eigenvalue problem. At issue is the question of completeness of the eigenfunctions, which is claimed but not proven in Raman and Fan (2010). We establish completeness in one dimension. We further describe the existence of accumulation points in the spectrum that lead to an infinite family of ‘zero group velocity’ waves. Numerical calculations illustrate some of the main ideas of this work.
Bibliographical noteFunding Information:
Many people have contributed to this work through discussions. These include David Dobson, Michael Weinstein, Junshan Lin, Stephen Shipman, Braxton Osting, and Eric Bonnetier. This project was initiated at Hong Kong University of Science and Technology where FS was a visiting professor in January 2019. He thanks his hosts for the hospitality during the productive stay. The research of FS is funded in part by the National Science Foundation under DMS 1440471. The research of HZ is partially funded by Hong Kong RGC Grant GRF 16306318 and GRF 16304517.
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- Bloch waves
- Eigenfunction expansions
- Metallic optical material
- Photonic band gaps
- Spectral theory