Localized light orbitals: Basis states for three-dimensional photonic crystal microscale circuits

We demonstrate the utility of three-dimensional (3D) optical Wannier functions (WF's) for quantitative description of electromagnetic wave localization and propagation in 3D photonic band gap (PBG) microcircuits. Using these localized "optical orbitals" we reconstruct electromagnetic waveguiding in bulk two-dimensional (2D) and 3D PBG materials, 2D-3D PBG heterostructures composed of 3D PBG structures inserted with 2D microchip layers, and 2D membrane photonic crystals. In 3D photonic crystal circuits, the expansion of electromagnetic fields with typically less than 20 maximally localized WF's (MLWF's) simplifies the calculation of electromagnetic phenomena of spectral bandwidth surrounding the PBG and improves computational efficiency, compared to the plane wave expansion and the finite-difference time-domain methods. The MLWF's are defined by a unitary transformation on the extended Bloch mode basis, chosen to minimize a wave function delocalization functional, while retaining symmetries of the underlying Bloch modes. We introduce an effective approach to constructing modified MLWF's in 3D architectures involving surface polarization charges and electromagnetic field discontinuities. These modified optical orbitals are also corrected for Gibbs phenomena arising near sharp dielectric interfaces. We demonstrate the accuracy of our localized light orbital method for recapturing TM and TE modes in idealized 2D photonic crystals and mixed polarization waveguide modes in 3D architectures.