  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.
