Computational nanophotonics in 3D

Abstract

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