Focusing of light in a three-dimensional cubic photonic crystal

The dielectric properties of photonic crystals are, in general, anisotropic spatially. This means that the constant-frequency surfaces of photons in wave-vector space are deformed from spherical shapes, leading to the photon focusing effects which are significantly enhanced at frequencies close to the photonic band gaps (PBG's). We find that the associated group-velocity surfaces of photons in real space are folded near PBG's and this gives rise to the appearance of photon caustics. We explicitly show these effects by illustrating theoretical images of photons transmitted through a photonic crystal. The calculation is made with the plane-wave-expansion method and numerical examples are given for a three-dimensional cubic lattice consisting of air and silicon.