Boundary Integral Methods for Periodic Scattering Problems

Part of the International Mathematical Series book series (IMAT, volume 12)


The paper is devoted to the scattering of a plane wave obliquelyilluminating a periodic surface. Integral equation methods lead to a systemof singular integral equations over the profile. Using boundary integral techniques, we study the equivalence of these equations to the electromagneticformulation, the existence and uniqueness of solutions under general assumptions on the permittivity and permeability of the materials. In particular,new results for materials with negative permittivity or permeability are established.


Integral Equation Singular Integral Equation Boundary Integral Equation Helmholtz Equation Fredholm Operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Elschner, J.: The double layer potential operator over polyhedral domains. I. Solvability in weighted Sobolev spaces. Appl. Anal. 45, 117–134 (1992)MathSciNetMATHGoogle Scholar
  2. 2.
    Elschner, J., Hinder, R., Penzel, F., Schmidt, G.: Existence, uniqueness and regularity for solutions of the conical diffraction problem. Math. Mod. Meth. Appl. Sci. 10, 317–341 (2000)MathSciNetMATHGoogle Scholar
  3. 3.
    Elschner, J., Schmidt, G.: Diffraction in Periodic structures and optimal design of binary gratings I. Direct problems and gradient formulas. Math. Meth. Appl. Sci. 21, 1297–1342 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Friedman, A.: Mathematics in Industrial Problems, Part 7. Springer, Berlin (1995)MATHGoogle Scholar
  5. 5.
    Goray, L.I., Sadov, S. Yu.: Numerical modelling of coated gratings in sensitive cases. OSA Diffractive Optics Micro-Optics 75, 365–379 (2002)Google Scholar
  6. 6.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Berlin (2008)MATHGoogle Scholar
  7. 7.
    Kresin, G.I., Mazya, V.G.: The norm and the essential norm of the double layer elastic and hydrodynamic potentials in the space of continuous functions. Math. Meth. Appl. Sci. 18, 1095–1131 (1995)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Mazya, V.G.: The integral equations of potential theory in domains with piecewise smooth boundary (Russian). Usp. Mat. Nauk 68, 229–230 (1981)MathSciNetGoogle Scholar
  9. 9.
    Maz'ya, V.G.: Boundary integral equations (Russian). In: Current Problems in Mathematics. Fundamental Directions. Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI, Moscow 27, 131–228 (1988); English transl.: Analysis IV. Encyclop. Math. Sci. 27, pp. 127–222. Springer, Berlin (1991)Google Scholar
  10. 10.
    Maz'ya, V., Shaposhnikova, T.: Higher regularity in the layer potential theory for Lipschitz domains. Indiana Univ. Math. J. 54, 99–142 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Maz'ya V. Solov'ev, A.: L p-theory of boundary integral equations on a contour with outward peak. Int. Equ. Oper. Th. 32, 75–100 (1998)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Maystre, D.: Integral methods. In: [14], 63–100Google Scholar
  13. 13.
    Muskhelishvili, N.I.: Singular Integral Equations, P. Noordhoff, Groningen, (1953)MATHGoogle Scholar
  14. 14.
    Petit, R. (Ed.): Electromagnetic theory of gratings. Topics in Current Physics, 22. Springer, Berlin (1980)Google Scholar
  15. 15.
    Petit R., Zolla, F.: The method of fictitious source as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts. PIE Proc. 2532, 374–385 (1997)Google Scholar
  16. 16.
    Pomp, A.: The integral method for coated gratings: computational cost. J. Mod. Optics 38, 109–120 (1991)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Popov, E., Bozhkov, B., Maystre, D., Hoose J.: Integral methods for echelles covered with lossless or absorbing thin dielectric layers. Appl. Optics 38, 47–55 (1999)CrossRefGoogle Scholar
  18. 18.
    Prössdorf, S.: Linear integral equations. Analysis, IV. Encyclop. Math. Sci. 27, pp. 1–125. Springer, Berlin (1991)Google Scholar
  19. 19.
    Schmidt, G.: Integral equations for conical diffraction by coated gratings. WIAS Preprint No. 1296 (2008) [To appear in J. Int. Equ. Appl.]Google Scholar
  20. 20.
    Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of ε and µ (Russian). Usp. Fiz. Nauk. 92, 517–526 (1967)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Weierstrass Institute of Applied Analysis and StochasticsBerlinGermany

Personalised recommendations