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It is shown, for the first time, that negative refraction with positive phasevelocity refraction can be realized (and controlled) over a wide range offrequency on the first (lowest) pass band of simple photonic and phononiccrystals. First a unified approach is presented to accurately, efficiently, anduniquely produce the homogenized effective material properties of doublyperiodic phononic crystals for anti-plane shear (SH) and photonic crystals fortransverse electric (TE) and transverse magnetic (TM) electromagneticBloch-form waves in such a manner that they exactly reproduce the bandstructure of the composite over any desired frequency band. Then thecorrespondence between phononic and photonic field equations is established andtheir effective homogenized material parameters are calculated. Finallyillustrative examples are worked out for each case, revealing a rich body ofrefractive characteristics of these crystals, and showing by means of theseexamples that the homogenized effective parameters do yield the exact results,and that the resulting homogenized medium does in fact embody exactly theactual band structure and dispersive properties of the consideredphononic/photonic crystal. As a consequence when the homogenized medium isplaced in contact with a normal homogeneous half-space, it could display, evenon the first pass band, positive, negative, or even no energy refractiondepending on the frequency and wave vector of plane waves incident from thenormal homogeneous solid to the interface.

Unified Homogenization of Photonic/Phononic Crystals: Controllable First-band Negative Refraction