Computer Science NASC Seminar
Diffraction Gratings and Photonic Crystals: New Integral Representations for Periodic Scattering and Eigenvalue Problems
Alex Barnett, Dartmouth College
January 28, 2011
Warren Weaver Hall, Room 1302
251 Mercer Street
New York, NY, 10012-1110
Spring 2011 NASC Seminars Calendar
Many numerical problems arising in modern photonic and electromagnetic
applications involve the interaction of linear waves with periodic,
piecewise-homogeneous media. Boundary integral equations are an
efficient approach to solving such boundary-value problems with
high-order convergence. In the case of plane-wave scattering from an
array (grating), the standard way to periodize is then to replace the
free-space Green's function kernel with its quasi-periodic cousin.
However, a major drawback is that the quasi-periodic Green's function
fails to exist for parameter families known as Wood's anomalies, even
though the underlying scattering problem remains well-posed.
We bypass this problem with a new integral representation that relies
on the *free-space* Green's function alone, adding auxiliary layer
potentials on the boundary of the unit cell strip, while enforcing
quasi-periodicity with an expanded linear system. The result is a 2nd
kind scheme that achieves spectral accuracy, is immune to Wood's
anomalies, avoids lattice sums, and reuses existing scattering codes.
A doubly-periodic version provides similar benefits for the robust
solution of the eigenvalue (band structure) problem for Bloch waves in
a photonic crystal. We show two-dimensional examples achieving 10-digit
accuracy with only a couple of hundred unknowns.
Joint work with Leslie Greengard (NYU).