Real part of full field due to an incident plane wave (in direction shown by arrow) hitting an infinite array of dielectrics of index n=1.5, in TM polarization, with a period of roughly 8 wavelengths. Three periods are shown. The computation was done to 14 digit accuracy on a laptop in a time of 2.6 secs to solve for the coefficients, and 47 secs to evaluate the field on a grid of 30000 points. The code to do this using the MPSpack toolbox for MATLAB is 12 lines long. Notice the focusing by the obstacle, and the strong wave traveling along the grating.

We present a new method for periodic scattering problems using 2nd kind boundary integral equations and the free-space Green's function, adding extra degrees of freedom on the unit cell walls, and imposing quasi-periodicity in an expanded linear system. This avoids the problems with the standard approach, namely: the quasi-periodic Green's function is slow to evaluate, it diverges at Wood's anomalies, and it does not handle large aspect ratios well. Our new approach cures all three issues, and can easily be `wrapped' around an existing code (e.g. fast-multipole acceleration) for the isolated-obstacle scattering problem.

Here is the preprint for the 2D case, submitted Aug 2010, which will appear in BIT Numerical Mathematics for their 50th anniversary issue.

Here are the latest talk slides: "A new integral representation for quasi-periodic scattering problems in two dimensions", (PDF 2.5MB). Given at

• Frontiers in Computational and Applied Mathematics, NJIT, Newark, NJ, 21-23 May 2010
• BIT 50 - Trends in Numerical Computing, Lund, Sweden, 17-20 June 2010
• Integral Equation Methods, Fast Algorithms and Applications, IMA hot topics workshop, 2-5 August 2010

Here are some movies:

• contour for the Sommerfeld integrals, moving as a function of incident angle. The contour deforms to stay away from poles in the integrand, and corrects for this analytically.
• scattered field, moving as a function of incident angle. Note the `kinks' as Wood's anomalies are passed; these are real and not numerical artifacts.
• band structure for a 2D dielectric photonic crystal, spinning in 3D. This goes with our related band structure paper in J. Comput. Phys.

We thank Stephen Shipman and Simon Chandler-Wilde for invaluable discussions. The work of AHB was supported by NSF grant DMS-0811005, and by the Class of 1962 Fellowship at Dartmouth College. The work of LG was supported by the Department of Energy under contract DEFG0288ER25053 and by AFOSR under MURI grant FA9550-06-1-0337.

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