Integral equations for conical diffraction by coated grating. (English) Zbl 1241.78015

The author studies a scattering problem for time-harmonic electromagnetic plane waves by gratings, which are media with particular symmetry properties: they are constant with respect to a given Cartesian coordinate and periodic with respect to another Cartesian coordinate. In particular, coated gratings are given by two non-intersecting interfaces that separate different homogeneous media. For this kind of structure, oblique incident plane waves are considered. The scattering problem is formulated as a two-dimensional boundary value problem for the Helmholtz equation, and it is also formulated as a system of suitable integral equations. The analytic properties of the corresponding integral operators are analyzed, and it is shown the integral equation formulation is equivalent to the differential equation formulation. For the integral equation formulation, existence and uniqueness results are provided.


78A45 Diffraction, scattering
45F15 Systems of singular linear integral equations
Full Text: DOI


[1] T. Arens, S.N. Chandler-Wilde and J.A. DeSanto, On integral equation and least squares methods for scattering by diffraction gratings , Comm. Comp. Phys. 1 (2006), 1010-1042. · Zbl 1137.78346
[2] X. Chen and A. Friedman, Maxwell’s equation in a periodic structure , Trans. Amer. Math. Soc. 323 (1991), 465-507. · Zbl 0727.35131
[3] M. Costabel, Boundary integral operators on Lipschitz domains : Elementary results , SIAM J. Math. Anal. 19 (1988), 613-626. · Zbl 0644.35037
[4] —, Some historical remarks on the positivity of boundary integral operators , in Boundary element analysis - Mathematical aspects and applications , M. Schanz and O. Steinbach, eds., LN Appl. Comp. Mech. 29 , Springer, Berlin, 2007 · Zbl 1298.47060
[5] M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems , J. Math. Anal. Appl. 106 (1985), 367-413. · Zbl 0597.35021
[6] J. Elschner, R. Hinder, F. Penzel and G. Schmidt, Existence, uniqueness and regularity for solutions of the conical diffraction problem , Math. Models Methods Appl. Sci. 10 (2000), 317-341. · Zbl 1010.78008
[7] J. Elschner and M. Yamamoto, An inverse problem in periodic diffraction optics : Reconstruction of Lipschitz grating profiles , Appl. Anal. 81 (2002), 1307-1328. · Zbl 1028.78008
[8] L.I. Goray and S.Yu. Sadov, Numerical modelling of coated gratings in sensitive cases , in
[9] C.M. Linton, The Green’s function for the two-dimensional Helmholtz equation in periodic domains , J. Eng. Math. 33 (1998), 377-402. · Zbl 0922.76274
[10] D. Maystre, Integral methods , in Electromagnetic theory of gratings , R. Petit, ed., Topics Current Physics 22 , Springer-Verlag, Berlin, 1980, 63-100.
[11] N.I. Muskhelishvili, Singular integral equations , P. Noordhoff, Groningen, 1953. · Zbl 0051.33203
[12] J.C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations , SIAM J. Math. Anal. 22 (1991), 1679-1701. · Zbl 0756.35004
[13] R. Petit, ed., Electromagnetic theory of gratings , Topics in Current Physics 22 , Springer-Verlag, Berlin, 1980.
[14] A. Pomp, The integral method for coated gratings: computational cost , J. Mod. Optics 38 (1991), 109-120. · Zbl 0941.78532
[15] E. Popov, B. Bozhkov, D. Maystre and J. Hoose, Integral methods for echelles covered with lossless or absorbing thin dielectric layers , Applied Optics 38 (1999), 47-55.
[16] A. Rathsfeld, G. Schmidt and B.H. Kleemann, On a fast integral equation method for diffraction gratings , Comm. Comp. Phys. 1 (2006), 984-1009. · Zbl 1116.78032
[17] J. Saranen and G. Vainikko, Periodic integral and pseudodifferential equations with numerical approximation , Springer, Berlin, 2002. · Zbl 0991.65125
[18] O. Steinbach and W.L. Wendland, On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries , J. Math. Anal. Appl. 262 (2001), 733-748. · Zbl 0998.35014
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.