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A stable Cartesian grid staircase-free finite-difference time-domain formulation for arbitrary material distributions in general geometries is introduced. It is shown that the method exhibits higher accuracy than the classical Yee scheme for complex geometries since the computational representation of physical structures is not of a staircased nature. Furthermore, electromagnetic boundary conditions are correctly enforced. The method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry. Both perfect electric conductors and dielectric structures have been investigated. Numerical results are presented and discussed.Index Terms-Computational models in electromagnetics and optics, finite-difference time-domain methods, numerical solution of partial differential equations, staircase, time-domain solution of Maxwell's equations.

Theoretical and numerical investigations of energy flow in photonic crystal waveguides made of line defects and branching points are presented. It is shown that vortices of energy flow may occur, and the net energy flow along the line defect is described via the effective propagation velocity. Single-mode and multimode operations are studied, and dispersion relations are computed for different waveguide widths. Both strong positive, strong negative, and zero dispersion are possible. It is shown that geometric parameters such as the nature of the lattice, the line defect orientation, the defect width, and the branching-point geometry have a significant influence on the electrodynamics. These are important issues for the fabrication of photonic crystal structures.

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An electromagnetic vector-field model for design of optical components based on the finite-difference time-domain method and radiation integrals is presented. Its ability to predict the optical electromagnetic dynamics in structures with complex material distributions is demonstrated. Theoretical and numerical investigations of finite-length surface-relief structures embedded in polymer dielectric waveguiding materials are presented. The importance of several geometric parameter dependencies is indicated as far-field power distributions are rearranged between diffraction orders. The influences of the variation in grating period, modulation depth, length, and profile are investigated.