  We explore the unconventional propagation of light in a three-dimensional
(3D) superlattice of coupled resonant cavities in a 3D photonic band gap
crystal. Such a 3D cavity superlattice is the photonic analogue of the Anderson
model for spins and electrons in the limit of zero disorder. Using the
plane-wave expansion method, we calculate the dispersion relations of the 3D
cavity superlattice with the cubic inverse woodpile structure that reveal five
coupled-cavity bands, typical of quadrupole-like resonances. For three out of
five bands, we observe that the dispersion bandwidth is significantly larger in
the $(k_x, k_z)$-diagonal directions than in other directions. To explain the
directionality of the dispersion bandwidth, we employ the tight-binding method
from which we derive coupling coefficients in 3D. For all converged
coupled-cavity bands, we find that light hops predominantly in a few
high-symmetry directions including the Cartesian $(x, y, z)$ directions,
therefore we propose the name "Cartesian light". Such 3D Cartesian hopping of
light in a band gap yields propagation as superlattice Bloch modes that differ
fundamentally from the conventional 3D spatially-extended Bloch wave
propagation in crystals, from light tunneling through a band gap, from
coupled-resonator optical waveguiding, and also from light diffusing at the
edge of a gap.
