Carl A. Bauer

A more accurate, stable, FDTD algorithm for electromagnetics in anisotropic dielectrics

A more accurate, stable, finite-difference time-domain (FDTD) algorithm is developed for simulating Maxwell@?s equations with isotropic or anisotropic dielectric materials. This algorithm is in many cases more accurate than previous algorithms (G.R. Werner et al., 2007 [5]; A.F. Oskooi et al., 2009 [7]), and it remedies a defect that causes instability with high dielectric contrast (usually for @e@?10) with either isotropic or anisotropic dielectrics. Ultimately this algorithm has first-order error (in the grid cell size) when the dielectric boundaries are sharp, due to field discontinuities at the dielectric interface. Accurate treatment of the discontinuities, in the limit of infinite wavelength, leads to an asymmetric, unstable update (C.A. Bauer et. al., 2011 [6]), but the symmetrized version of the latter is stable and more accurate than other FDTD methods. The convergence of field values supports the hypothesis that global first-order error can be achieved by second-order error in bulk material with zero-order error on the surface. This latter point is extremely important for any applications measuring surface fields.

A second-order 3D electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a Yee mesh

A new frequency-domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a Yee mesh with second-order error in resonant frequencies. The algorithm is systematically derived using the finite integration formulation of Maxwells equations on the Yee mesh. Second-order convergence of the error in resonant frequencies is achieved by guaranteeing first-order error on dielectric boundaries and second-order error in bulk (possibly anisotropic) regions. Convergence studies, conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant, both show second-order convergence of frequency error; the convergence is sufficiently smooth that Richardson extrapolation yields roughly third-order convergence. The convergence of electric fields near the dielectric interface for the analytic problem is also presented.

Truncated photonic crystal cavities with optimized mode confinement

Optimization of a truncated, dielectric photonic crystal cavity leads to configurations that are far from truncated crystal cavities, and which have significantly better radiation confinement. Starting from a two-dimensional truncated photonic crystal cavity with optimal Q-factor, moving the rods from the lattice positions can increase the Q-factor by orders of magnitude, e.g., from 130 to 11 000 for a cavity constructed from 18 rods. In the process, parity symmetry breaking occurs. Achieving the same Q-factor with a regular lattice requires 60 rods. Therefore, using optimized irregular structures for photonic cavities can greatly reduce material requirements and device size.

A fast multigrid-based electromagnetic eigensolver for curved metal boundaries on the Yee mesh

For embedded boundary electromagnetics using the Dey-Mittra (Dey and Mittra, 1997) [1] algorithm, a special grad-div matrix constructed in this work allows use of multigrid methods for efficient inversion of Maxwells curl-curl matrix. Efficient curl-curl inversions are demonstrated within a shift-and-invert Krylov-subspace eigensolver (open-sourced at [ofortt]https://github.com/bauerca/maxwell[cfortt]) on the spherical cavity and the 9-cell TESLA superconducting accelerator cavity. The accuracy of the Dey-Mittra algorithm is also examined: frequencies converge with second-order error, and surface fields are found to converge with nearly second-order error. In agreement with previous work (Nieter et al., 2009) [2], neglecting some boundary-cut cell faces (as is required in the time domain for numerical stability) reduces frequency convergence to first-order and surface-field convergence to zeroth-order (i.e. surface fields do not converge). Additionally and importantly, neglecting faces can reduce accuracy by an order of magnitude at low resolutions.

Origin and reduction of wakefields in photonic crystal accelerator cavities

Photonic crystal (PhC) defect cavities that support an accelerating mode tend to trap unwanted higher-order modes (HOMs) corresponding to zero-group-velocity PhC lattice modes at the top of the bandgap. The effect is explained quite generally from photonic band and perturbation theoretical arguments. Transverse wakefields resulting from this effect are observed in a hybrid dielectric PhC accelerating cavity based on a triangular lattice of sapphire rods. These wakefields are, on average, an order of magnitude higher than those in the waveguide-damped Compact Linear Collider (CLIC) copper cavities. The avoidance of translational symmetry (and, thus, the bandgap concept) can dramatically improve HOM damping in PhC-based structures.