  The concept of the Brillouin zone (BZ) in relation to a photonic crystal
fabricated in an optically anisotropic material is explored both experimentally
and theoretically. In experiment, we used femtosecond laser pulses to excite
THz polaritons and image their propagation in lithium niobate and lithium
tantalate photonic crystal (PhC) slabs. We directly measured the dispersion
relation inside PhCs and observed that the lowest bandgap expected to form at
the BZ boundary forms inside the BZ in the anisotropic lithium niobate PhC. Our
analysis shows that in an anisotropic material the BZ - defined as the
Wigner-Seitz cell in the reciprocal lattice - is no longer bounded by Bragg
planes and thus does not conform to the original definition of the BZ by
Brillouin. We construct an alternative Brillouin zone defined by Bragg planes
and show its utility in identifying features of the dispersion bands. We show
that for an anisotropic 2D PhC without dispersion, the Bragg plane BZ can be
constructed by applying the Wigner-Seitz method to a stretched or compressed
reciprocal lattice. We also show that in the presence of the dispersion in the
underlying material or in a slab waveguide, the Bragg planes are generally
represented by curved surfaces rather than planes. The concept of constructing
a BZ with Bragg planes should prove useful in understanding the formation of
dispersion bands in anisotropic PhCs and in selectively tailoring their optical
properties.
