A Third Order Numerical Method for Doubly Periodic Electromegnetic Scattering
Abstract
We here developed a third-order accurate numerical method for scattering of 3D electromagnetic
waves by doubly periodic structures. The method is an intuitively simple numerical
scheme based on a boundary integral formulation. It involves smoothing the singular
Green's functions in the integrands and finding correction terms to
the resulting smooth integrals. The analytical method is based on the singular integral
methods of J. Thomas Beale, while the scattering problem is motivated by the 2D work
of Stephanos Venakides, Mansoor Haider, and Stephen Shipman. The 3D problem was done
with boundary element methods by Andrew Barnes. We present a method that is both more
straightforward and more accurate. In solving these problems, we have used the M\"{u}ller
integral equation formulation of Maxwell's equations, since it is a Fredholm integral
equation of the second kind and is well-posed. M\"{u}ller derived his equations for
the case of a compact scatterer. We outline the derivation and adapt it to a periodic
scatterer. The periodic Green's functions found in the integral equation contain singularities
which make it difficult to evaluate them numerically with accuracy. These functions
are also very time consuming to evaluate numerically. We use Ewald splitting to represent
these functions in a way that can be computed rapidly.We present a method of smoothing
the singularity of the Green's function while maintaining its periodicity. We do local
analysis of the singularity in order to identify and eliminate the largest sources
of error introduced by this smoothing. We prove that with our derived correction terms,
we can replace the singular integrals with smooth integrals and only introduce a error
that is third order in the grid spacing size. The derivation of the correction terms
involves transforming to principal directions using concepts from differential geometry.
The correction terms are necessarily invariant under this transformation and depend
on geometric properties of the scatterer such as the mean curvature and the differential
of the Gauss map. Able to evaluate the integrals to a higher order, we implement a
\mbox{GMRES} algorithm to approximate solutions of the integral equation. From these
solutions, M\"{u}ller's equations allow us to compute the scattered fields and transmission
coefficients. We have also developed acceleration techniques that allow for more efficient
computation.We provide results for various scatterers, including a test case for which
exact solutions are known. The implemented method does indeed converge with third
order accuracy. We present results for which the method successfully resolves Wood's
anomaly resonances in transmission.
Type
DissertationDepartment
MathematicsPermalink
https://hdl.handle.net/10161/383Citation
Nicholas, Michael J (2007). A Third Order Numerical Method for Doubly Periodic Electromegnetic Scattering. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/383.Collections
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