Communications in Mathematical Sciences

Volume 6 (2008)

Number 3

A higher order numerical method for 3-d double periodic electromagnetic scattering problem

Pages: 669 – 694



Michael J. Nicholas


We develop a method for 3D doubly periodic electromagnetic scattering. We adapt the Müller integral equation formulation of Maxwell's equations to the periodic problem, since it is a Fredholm equation of the second kind. We use Ewald splitting to efficiently calculate the periodic Green's functions. The approach is to regularize the singular Green's functions and to compute integrals with a trapezoidal sum. Through asymptotic analysis near the singular point, we are able to identify the largest part of the smoothing error and to subtract it out. The result is a method that is third order in the grid spacing size. We present 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.


periodic electromagnetic scattering; periodic Helmholtz Green's function; singular integrals; Ewald summation; photonic crystals; Wood's anomaly

2010 Mathematics Subject Classification

00A69, 32A55, 74J20

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