A new integral representation for quasi-periodic scattering problems in two dimensions. (English) Zbl 1214.65061

Summary: Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.
In this paper, we bypass both problems by means of 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 expanding the linear system to enforce quasi-periodicity. Summing nearby images directly leaves auxiliary densities that are smooth, hence easily represented in the Fourier domain using Sommerfeld integrals. Wood’s anomalies are handled analytically by deformation of the Sommerfeld contour. The resulting integral equation is of the second kind and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls are handled easily and automatically. We include an implementation and simple code example with a freely-available MATLAB toolbox.


65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q61 Maxwell equations
78A45 Diffraction, scattering
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
76Q05 Hydro- and aero-acoustics
76M15 Boundary element methods applied to problems in fluid mechanics


Full Text: DOI


[1] Alpert, B.K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20, 1551–1584 (1999) · Zbl 0933.41019
[2] Arens, T., Chandler-Wilde, S.N., DeSanto, J.A.: On integral equation and least squares methods for scattering by diffraction gratings. Commun. Comput. Phys. 1, 1010–1042 (2006) · Zbl 1137.78346
[3] Arens, T., Sandfort, K., Schmitt, S., Lechleiter, A.: Analysing Ewald’s method for the evaluation of Green’s functions for periodic media. IMA J. Numer. Anal. (2010, submitted). Available at http://digbib.ubka.uni-karlsruhe.de/volltexte/1000019136 · Zbl 1283.35014
[4] Atwater, H.A., Polman, A.: Plasmonics for improved photovoltaic devices. Nature Mater. 9(3), 205–213 (2010)
[5] Bao, G., Dobson, D.C.: Modeling and optimal design of diffractive optical structures. Surv. Math. Ind. 8, 37–62 (1998) · Zbl 0932.65116
[6] Barnett, A.H., Betcke, T.: An exponentially convergent non-polynomial finite element method for time-harmonic scattering from polygons. SIAM J. Sci. Comput. 32(3), 1417–1441 (2010) · Zbl 1216.65151
[7] Barnett, A.H., Greengard, L.: A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations. J. Comput. Phys. 229, 6898–6914 (2010) · Zbl 1197.78025
[8] Barty, C.P.J., et al.: An overview of LLNL high-energy short-pulse technology for advanced radiography of laser fusion experiments. Nuclear Fusion 44(12), S266 (2004)
[9] Bonnet-BenDhia, A.-S., Starling, F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17, 305–338 (1994) · Zbl 0817.35109
[10] Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983) · Zbl 0522.35001
[11] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 2nd edn. Springer, Berlin (1998) · Zbl 0893.35138
[12] Dienstfrey, A., Hang, F., Huang, J.: Lattice sums and the two-dimensional, periodic Green’s function for the Helmholtz equation. Proc. R. Soc. Lond. A 457, 67–85 (2001) · Zbl 1048.78015
[13] Garabedian, P.R.: Partial Differential Equations. Wiley, New York (1964)
[14] Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice Hall, Englewood Cliffs (1988)
[15] Hale, N., Higham, N.J., Trefethen, L.N.: Computing A, log (A), and related matrix functions by contour integrals. SIAM J. Numer. Anal. 46(5), 2505–2523 (2008) · Zbl 1176.65053
[16] Holter, H., Steyskal, H.: Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays. IEEE Trans. Antennae Propag. 50(12), 1725–1731 (2002)
[17] Horoshenkov, K.V., Chandler-Wilde, S.N.: Efficient calculation of two-dimensional periodic and waveguide acoustic Green’s functions. J. Acoust. Soc. Am. 111, 1610–1622 (2002)
[18] Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998) · Zbl 0114.42903
[19] Joannopoulos, J.D., Johnson, S.G., Meade, R.D., Winn, J.N.: Photonic Crystals: Molding the Flow of Light, 2nd edn. Princeton Univ. Press, Princeton (2008) · Zbl 1144.78303
[20] Kelzenberg, M.D., Boettcher, S.W., Petykiewicz, J.A., Turner-Evans, D.B., Putnam, M.C., Warren, E.L., Spurgeon, J.M., Briggs, R.M., Lewis, N.S., Atwater, H.A.: Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications. Nature Mater. 9(3), 239–244 (2010)
[21] Kress, R.: Boundary integral equations in time-harmonic acoustic scattering. Math. Comput. Model. 15, 229–243 (1991) · Zbl 0731.76077
[22] Kress, R.: Numerical Analysis. Graduate Texts in Mathematics, vol. 181. Springer, Berlin (1998) · Zbl 0913.65001
[23] Kurkcu, H., Reitich, F.: Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies. J. Comput. Phys. 228, 75–95 (2009) · Zbl 1157.65067
[24] Li, L., Chandezon, J., Granet, G., Plumey, J.P.: Rigorous and efficient grating-analysis method made easy for optical engineers. Appl. Opt. 38(2), 304–313 (1999)
[25] Linton, C.M.: The Green’s function for the two-dimensional Helmholtz equation in periodic domains. J. Eng. Math. 33, 377–402 (1998) · Zbl 0922.76274
[26] Linton, C.M.: Lattice sums for the Helmholtz equation. SIAM Rev. 52(4), 630–674 (2010). doi: 10.1137/09075130X · Zbl 1208.78016
[27] Linton, C.M., Thompson, I.: Resonant effects in scattering by periodic arrays. Wave Motion 44, 165–175 (2007) · Zbl 1231.76275
[28] McPhedran, R.C., Nicorovici, N.A., Botten, L.C., Grubits, K.A.: Lattice sums for gratings and arrays. J. Math. Phys. 41, 7808–7816 (2000) · Zbl 0980.35007
[29] Mikhlin, S.G.: Integral Equations, 2nd edn. MacMillan, New York (1964) · Zbl 0117.31902
[30] Model, R., Rathsfeld, A., Gross, H., Wurm, M., Bodermann, B.: A scatterometry inverse problem in optical mask metrology. J. Phys., Conf. Ser. 135, 012,071 (2008)
[31] Moroz, A.: Exponentially convergent lattice sums. Opt. Lett. 26, 1119–21 (2001)
[32] Morse, P., Feshbach, H.: Methods of Theoretical Physics, vol. 1. McGraw-Hill, New York (1953) · Zbl 0051.40603
[33] Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin (1969) · Zbl 0181.57203
[34] Nédélec, J.C., Starling, F.: Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations. SIAM J. Math. Anal. 22(6), 1679–1701 (1991) · Zbl 0756.35004
[35] Nicholas, M.J.: A higher order numerical method for 3-D doubly periodic electromagnetic scattering problems. Commun. Math. Sci. 6(3), 669–694 (2008) · Zbl 1168.78003
[36] Otani, Y., Nishimura, N.: A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals. J. Comput. Phys. 227, 4630–4652 (2008) · Zbl 1206.78084
[37] Peter, M.A., Meylan, M.H., Linton, C.M.: Water-wave scattering by a periodic array of arbitrary bodies. J. Fluid Mech. 548, 237–256 (2006)
[38] Petit, R. (ed.): Electromagnetic Theory of Gratings, Topics in Current Physics, vol. 22. Springer, Heidelberg (1980)
[39] Rokhlin, V.: Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 5, 257–272 (1983) · Zbl 0522.73022
[40] Shipman, S.: Resonant scattering by open periodic waveguides. In: Progress in Computational Physics (PiCP), vol. 1, pp. 7–50. Bentham Science Publishers, Dubai (2010)
[41] Shipman, S., Venakides, S.: Resonance and bound states in photonic crystal slabs. SIAM J. Appl. Math. 64, 322–342 (2003) · Zbl 1034.78010
[42] Taflove, A.: Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, Norwood (1995) · Zbl 0840.65126
[43] Venakides, S., Haider, M.A., Papanicolaou, V.: Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures. SIAM J. Appl. Math. 60, 1686–1706 (2000) · Zbl 0973.78028
[44] Weideman, J.A.C.: Numerical integration of periodic functions: a few examples. Am. Math. Mont. 109(1), 21–36 (2002) · Zbl 1022.65027
[45] Wojcik, G.L.J.M. Jr., Marx, E., Davidson, M.P.: Numerical reference models for optical metrology simulation. In: SPIE Microlithography 92: IC Metrology, Inspection, and Process Control VI, vol. 1673-06 (1992)
[46] Wood, R.W.: On a remarkable case of uneven distribution of light in a diffraction grating spectrum. Philos. Mag. 4, 396–408 (1902)
[47] Zhang, B., Chandler-Wilde, S.N.: Integral equation methods for scattering by infinite rough surfaces. Math. Methods Appl. Sci. 26, 463–488 (2003) · Zbl 1016.78006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.