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  1. Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A 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 for any fixed quasi-momentum. 
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  2. We provide a platform to examine the effect of inclusion geometry on three-dimensional metamaterial crystals to tune frequency-dependent effective properties for control of leading order dispersive behaviour. The crystal is non-magnetic and made from all dielectric components. The design provides novel dispersive properties using subwavelength resonances controlled by the geometry of the media. We numerically calculate the effective tensors of the metamaterial to identify frequency intervals where the metamaterial exhibits band gaps as well as intervals of normal dispersion and double negative dispersion. The frequency intervals can be explicitly controlled by adjusting the geometry and placement of the dielectric inclusions within the period cell of the crystal. 
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  3. We explain the Lorentz resonances in plasmonic crystals that consist of two-dimensional nano-dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a non-magnetic bulk dielectric. We identify local geometric resonances on the length scale of the small-scale period. From a materials perspective, the graphene surface exhibits a dispersive surface conductance captured by the Drude model. Together these phenomena conspire to generate Lorentz resonances at frequencies controlled by the surface geometry and the surface conductance. The Lorentz resonances found in the frequency response of the effective dielectric tensor of the bulk metamaterial are shown to be given by an explicit formula, in which material properties and geometric resonances are decoupled. This formula is rigorous and obtained directly from corrector fields describing local electrostatic fields inside the heterogeneous structure. Our analytical findings can serve as an efficient computational tool to describe the general frequency dependence of periodic optical devices. As a concrete example, we investigate two prototypical geometries composed of nanotubes and nanoribbons. 
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  4. Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A 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 for any fixed quasi-momentum. 
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  5. Analytic representation formulas and power series are developed to describe the band struc- ture inside periodic elastic crystals made from high contrast inclusions. We use source free modes associated with structural spectra to represent the solution operator of the Lam ́e system inside phononic crystals. Convergent power series for the Bloch wave spectrum are obtained using the representation formulas. An explicit bound on the convergence radius is given through the structural spectra of the inclusion array and the Dirichlet spectra of the inclusions. Suffi- cient conditions for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum are identified. A condition is found that is sufficient for the emergence of band gaps. 
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  6. We construct metamaterials from sub-wavelength nonmagnetic resonators and consider the refraction of incoming signals traveling from free space into the metamaterial. We show that the direction of the transmitted signal is a function of its center frequency and bandwidth. The directionality of the transmitted signal and its frequency dependence is shown to be explicitly controlled by sub-wavelength resonances that can be calculated from the geometry of the sub-wavelength scatters. We outline how to construct a medium with both positive and negative index properties across different frequency bands in the near infrared and optical regime. 
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