High Accuracy Nonstandard Finite-Difference Time-Domain Algorithms For Computational Electromagnetics: Applications To Optics And Photonics

We apply nonstandard finite difference (NSFD) models to develop new high accuracy finite-difference time-domain (FDTD) algorithms for computational electromagnetics and optics. The basic FDTD algorithm is simple and easy to program, but the accuracy of conventional FDTD algorithms is low. For step size h, the error is epsilon similar to h(2) and the computational cost is C similar to h(4) in three dimensions. Thus halving h reduces the epsilon by a factor of four, but C rises by a factor of sixteen. Using NSFD models we constructed FDTD algorithms for which epsilon similar to h(6) with about the same computational cost. We introduce NSFD versions of the FDTD algorithm to solve the wave equation, Maxwell's equations (Yee algorithm) and a new version of the Mur absorbing boundary condition. We tested our new algorithms by computing scattering off cylinders and spheres in the Mie regime and comparing with the analytic solutions. The accuracy of the NSFD algorithms is superior to the conventional FDTD ones. We illustrate our methods with some example applications from our current research. Among the topics we cover are propagation in dispersive media and surface plasmons, light propagation in structures with subwavelength features including conducting diffraction gratings and biological structures, and the improvement of light coupling through media interfaces using subwavelength structures.