Unified homogenization of photonic/phononic crystals with first-band negative refraction

Abstract

It is shown, for the first time, that negative refraction with positive phase velocity refraction can be realized (and controlled) on a wide frequency-range of the first (lowest) pass band of simple photonic and phononic crystals.First a unified approach is presented to accurately, efficiently, and uniquely produce the homogenized effective material properties of doubly periodic phononic crystals for anti-plane shear (SH) and photonic crystals for transverse electric (TE) and transverse magnetic (TM) electromagnetic Bloch-form waves in such a manner that they exactly reproduce the band structure of the composite over any desired frequency band. Then the correspondence between phononic and photonic field equations is established and their effective homogenized material parameters are calculated. Finally the actual calculation procedure is detailed and illustrative examples are worked out for each case, revealing a rich body of refractive characteristics of these crystals, and showing by means of illustrative examples that the homogenized effective parameters do in fact yield the exact results. Thus the resulting homogenized solid embodies exactly the actual band structure and dispersive properties of the considered phononic/photonic crystal, which clearly distinguishes it from any anisotropic normal material.When in contact with a normal homogeneous half-space, the homogenized medium would display positive, negative, or even no energy refraction depending on the frequency and wave vector of plane waves incident from the normal homogeneous solid to the interface. Remarkably, using the same volume fraction of the same constituents, it is possible to design a (homogenized) material that would display negative energy refraction with positive phase velocity refraction even on the first (acoustic) pass band. The window of the frequency and the range of wave-vector values at which, say, the negative energy refraction would occur can be controlled by judicious selection and design of the unit cell’s constituent geometry and properties.The unit cell of the crystal may consist of anisotropic constituents of any constant or variable properties that may also admit large discontinuities. For elliptical and rectangular inclusion geometries, analytical expressions are given for the calculation of the key quantities.