We identify explicit conditions on geometry and material contrast for creating band gaps in two-dimensional photonic and three-dimensional acoustic crystals. This approach is novel and makes use of the electrostatic and quasi-periodic source free resonances of the crystal. The source free modes deliver a spectral representation for solution operators associated with propagation of electromagnetic and acoustic waves inside periodic high contrast media. An accurate characterization of the quasi-periodic and electrostatic resonance spectrum in terms of the shape and geometry of the scatters is possible. This information together with the Dirichlet and a Neumann-like spectra associated with the inclusions deliver conditions sufficient for opening band gaps at finite contrast. The theory provides a systematic means for the identification of photonic and phononic band gaps within a specified frequency range.
Publication Source (Journal or Book title)
Multiscale Modeling and Simulation
Lipton, R., & Viator, R. (2017). Creating band gaps in periodic media. Multiscale Modeling and Simulation, 15 (4), 1612-1650. https://doi.org/10.1137/16M1083396