Abstract
We identify explicit conditions on geometry and material contrast for creating band gaps in 2D photonic and 3D acoustic crystals made from composites. This approach is new and makes use of the electrostatic and quasiperiodic 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. This approach is motivated by the analytic representation method introduced in the seminal papers [3], [25], developed for bounds on effective properties of composite materials. The analytic method developed here is used for bounding passbands and stop bands for photonic and phononic composite materials. The bounds are explicit and include geometric and spectral information on the geometry of the included phase. These parameters 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.
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