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 analog 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 three dimensions. 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.

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Nederlandse samenvatting 182 Acknowledgments 183 which is also known as the Ioffe-Regel criterion [40, 44]. The quest for Anderson localization of light continues to receive attention to date [22, 45, 46]. Another well-known interference phenomenon in strongly-scattering random structures is the weak localization or enhanced backscattering of light [47, 48], in analogy to that of electrons [49] Three-dimensional photonic crystals are an important class of ordered nanophotonic structures [12, 17, 50]. In photonic crystals, the refractive index varies spatially with a periodicity on length scales comparable to the wavelength of light. Photonic crystals can be seen as an optical analogue of semiconductors [50]. Interference of waves diffracted by different lattice planes determines the optical modes and dispersion. The periodicity gives rise to Bragg diffractions, which are associated with frequency windows that are forbidden for propagation of light in a certain direction [51]. Such stop gaps have long been known to arise for light in one-dimensionally periodic structures, known as dielectric mirrors or Bragg stacks [52]. A stop gap is associated with propagation along a specific direction. In three-dimensional photonic crystals a stop gap for all directions simultaneously can be achieved, resulting in a so-called photonic band gap [12, 17, 50], in analogy to the electronic band gap in a semiconductor crystal In the finite-difference time-domain (FDTD) method, both time and space are discretized, i.e., all spatial and temporal derivatives in Maxwell's equations are replaced by finite difference quotients [94-96]. In order to ensure a stable numerical result, the time increment ∆t should satisfy the Courant condition V d 3 re −iG•r u k (r) with V the volume of the unit cell. To satisfy the divergence constraint (k + G) • c G = 0, the expansion coefficients are written in terms of two units vectorsê