Unified Homogenization of Photonic/Phononic Crystals: Controllable   First-band Negative Refraction
  It is shown, for the first time, that negative refraction with positive phase velocity refraction can be realized (and controlled) over a wide range of frequency on 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 illustrative examples are worked out for each case, revealing a rich body of refractive characteristics of these crystals, and showing by means of these examples that the homogenized effective parameters do yield the exact results, and that the resulting homogenized medium does in fact embody exactly the actual band structure and dispersive properties of the considered phononic/photonic crystal. As a consequence when the homogenized medium is placed in contact with a normal homogeneous half-space, it could display, even on the first pass band, 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. 
Sia Nemat-Nasser
cond-mat.mtrl-sci