James B. Cole

A high-accuracy realization of the Yee algorithm using non-standard finite differences

New nonstandard second-order finite differences (FDs) are introduced, which when substituted into the Yee algorithm, reduce the solution error by a factor of 10/sup -4/ on a coarse computational grid. Using /spl lambda//h (grid spacings per wavelength)=8, one achieves the same accuracy as the standard Yee algorithm does at /spl lambda//h=1140. In addition, greater algorithmic stability allows a reduction in the number of iterations needed to solve a problem.

High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications

We previously described a high-accuracy version of the Yee algorithm that uses second-order nonstandard finite differences (NSFDs) and demonstrated its accuracy numerically. We now prove that at fixed frequency and grid spacing h, the leading error term is O(h/sup 6/) versus O(h/sup 2/) for the ordinary Yee algorithm with standard finite differences (SFDs). We numerically verify the superior accuracy of the NSFD algorithm by simulating near-field Mie scattering on a coarse grid and comparing with the SFD one and with analytical solutions. We present an updated stability analysis and show that the maximum time step for the NSFD algorithm is 20% longer than the SFD time step in two dimensions, and 36% longer in three dimensions. Finally, parameters that were previously given numerically are now analytically defined.

Simulation of whispering gallery modes in the Mie regime using the nonstandard finite-difference time domain algorithm

The nonstandard (NS) finite-difference time domain (FDTD) algorithm has proved be remarkably accurate on a coarse numerical grid, but the well-known resonances called whispering gallery modes (WGMs) in the Mie regime are very sensitive to the scatterer representation on the computational grid, and a very large number of time steps are needed to correctly calculate the modes because the electromagnetic field outside the scatterer is weakly coupled to the inside. Using the NS-FDTD algorithm on a coarse grid, we were able to accurately simulate the WGMs of dielectric cylinders in the Mie regime.

High-accuracy FDTD solution of the absorbing wave equation, and conducting Maxwell's equations based on a nonstandard finite-difference model

We previously introduced high-accuracy finite-difference time-domain (FDTD) algorithms based on nonstandard finite differences (NSFD) to solve the nonabsorbing wave equation and the nonconducting Maxwell equations. We now extend our methodology to the absorbing wave equation and the conducting Maxwell equations. We first derive an exact NSFD model of the one-dimensional wave equation, and extend it to construct high-accuracy FDTD algorithms to solve the absorbing wave equation, and the conducting Maxwells Equations in two and three dimensions. For grid spacing h, and wavelength /spl lambda/, the NSFD solution error is /spl epsiv//spl sim/(h//spl lambda/)/sup 6/ compared with (h//spl lambda/)/sup 2/ for ordinary FDTD algorithms using second-order central finite-differences. This high accuracy is achieved not by using higher-order finite differences but by exploiting the analytical properties of the decaying-harmonic solution basis functions. Besides higher accuracy, in the NSFD algorithms the maximum time step can be somewhat longer than for the ordinary second-order FDTD algorithms.

Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm

Propagation properties of optical pulses in a two-dimensional photonic crystal with a straight waveguide structure imbedded were examined using a high accuracy finite-difference time-domain (FDTD) algorithm based on nonstandard finite differences. A tunable and significantly large group velocity dispersion was found even for photonic crystal structures as small as 10 unit cells long. Detailed calculations indicated that a very small photonic crystal with an imbedded waveguide can be used to control pulse dispersion, i.e., a just 25 μm long photonic crystal with waveguide can compress a 1% up-chirped pulse to the Fourier transform limit. Further, our FDTD calculations showed excellent agreement with the prediction of photonic band calculations on infinite structures.

Simulation of subwavelength metallic gratings using a new implementation of the recursive convolution finite-difference time-domain algorithm.

We introduce a new implementation of the finite-difference time-domain (FDTD) algorithm with recursive convolution (RC) for first-order Drude metals. We implemented RC for both Maxwells equations for light polarized in the plane of incidence (TM mode) and the wave equation for light polarized normal to the plane of incidence (TE mode). We computed the Drude parameters at each wavelength using the measured value of the dielectric constant as a function of the spatial and temporal discretization to ensure both the accuracy of the material model and algorithm stability. For the TE mode, where Maxwells equations reduce to the wave equation (even in a region of nonuniform permittivity) we introduced a wave equation formulation of RC-FDTD. This greatly reduces the computational cost. We used our methods to compute the diffraction characteristics of metallic gratings in the visible wavelength band and compared our results with frequency-domain calculations.

Applications of Nonstandard Finite Difference Models to Computational Electromagnetics

We introduce an exact nonstandard finite-difference model of the one-dimensional absorbing wave equation, and use it to develop a high accuracy version of the finite-difference time-domain (FDTD) algorithm to solve the absorbing wave equation and the conducting Maxwells equations in two and three dimensions. For grid spacing h, and wavelength λ, the solution error of the ordinary FDTD algorithm is because it uses second-order central finite difference approximations. By exploiting the analytical properties of decaying-harmonic solution basis functions in a nonstandard finite difference model we reduce the error to without using higher order finite differences. We have verified the accuracy of the algorithms by comparing with analytic solutions of near-field Mie scattering, and have used them to investigate the optical properties of subwavelength conductive diffraction gratings.

Simulation of light propagation in two-dimensional photonic crystals with a point defect by a high-accuracy finite-difference time-domain method

A high-accuracy finite-difference time-domain method based on what are called nonstandard finite differences was used to simulate optical propagation in a two-dimensional photonic crystal with a point defect. We used a photonic crystal consisting of a triangular lattice of air columns embedded in a high-refractive index medium. We found that the transmittance spectrum has four peaks in the photonic band-gap region, and that these peaks correspond to the resonant energies of light localized at the point defect. For a point defect consisting of an air hole with a radius smaller than that of the air holes of the photonic crystal, these peaks shift to higher energy. The peak shift of the resonant mode that is associated with the electric field concentrated about the center of the point defect is larger than the peak shift of the other modes.

Advances in finite-difference time-domain calculation methods

Although the finite-difference time-domain (FDTD) method was developed in the 1960s, beginning with Yee’s famous algorithm [1], and many advances have been made since then, FDTD is still an active field of research.

Cylindrical invisibility cloak based on photonic crystal layers that permits communication with the outside

Invisibility cloaks designed by transformation optics include a perfect shield, which exclude electromagnetic fields from the cloaked region. Due to the shield, observers inside the cloak cannot see the outside. We propose a cloak that permits communication with the outside, based on a layered photonic crystal (PC) structure. The PC acts as an effective shield in the reflection bandgap, leaving the transmission band available for communication with the outside. A procedure to design an infinitely long cylindrical cloak consisting of concentric layers of dielectric and metal is given. For the proposed structure, the performance of cloaking in the reflection band and of communication in the transmission band is computed.