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June  2015, 8(3): 435-473. doi: 10.3934/dcdss.2015.8.435

Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures

 1 Berlin Mathematical School, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received  November 2013 Revised  March 2014 Published  October 2014

This paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a $2\pi$-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
Citation: Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1995. [2] H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure, Inverse Problems, 11 (1995), 823-833. doi: 10.1088/0266-5611/11/4/013. [3] T. Arens, Scattering by Biperiodic Layered Media: The Integral Equation Approach, habilitation thesis, Universität Karlsruhe in Karlsruhe, 2010. [4] G. Bao and D. C. Dobson, On the scattering by a biperiodic structure, Proc. AMS, 128 (2000), 2715-2723. doi: 10.1090/S0002-9939-00-05509-X. [5] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations I. An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci., 24 (2001), 9-30. doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. [6] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci., 24 (2001), 31-48. doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X. [7] A. Buffa, M. Costabel and C. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains, Numer. Math., 92 (2002), 679-710. doi: 10.1007/s002110100372. [8] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845-867. doi: 10.1016/S0022-247X(02)00455-9. [9] A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math., 95 (2003), 459-485. doi: 10.1007/s00211-002-0407-z. [10] B. Bugert, On Integral Equation Methods for Electromagnetic Scattering by Biperiodic Structures, PhD thesis, Technische Universität Berlin in Berlin, 2014. [11] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013. [12] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626. doi: 10.1137/0519043. [13] M. Costabel and F. Le Louër, On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., 71 (2011), 635-656. doi: 10.1137/090779462. [14] D. C. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, Anal. Appl., 166 (1992), 507-528. doi: 10.1016/0022-247X(92)90312-2. [15] D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, Math. Anal. Numer., 28 (1994), 419-439. [16] 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. doi: 10.1142/S0218202500000197. [17] V. Yu. Gotlib, Solutions of the Helmholtz equation, concentrated near a plane periodic boundary, J. Math. Sci., 102 (2000), 4188-4194. doi: 10.1007/BF02673850. [18] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008. doi: 10.1007/978-3-540-68545-6. [19] G. Hu and A. Rathsfeld, Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings, IMA Journal of Applied Mathematics, (2014), 1-25. doi: 10.1093/imamat/hxt054. [20] I. V. Kamotski and S. A. Nazarov, The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain, Journal of Mathematical Sciences, 111 (1988), 3657-3664. doi: 10.1023/A:1016377707919. [21] R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math., 48 (1988), 307-325. doi: 10.1137/0148016. [22] A. Lechleiter and D.-L. Nguyen, On uniqueness in electromagnetic scattering from biperiodic structures, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1167-1184. doi: 10.1051/m2an/2012063. [23] D. Maystre, Integral methods, in Electromagnetic Theory of Gratings, (ed. R. Petit), Springer, 1980, 63-100. [24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [25] J.-C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Appl. Math., 22 (1991), 1679-1701. doi: 10.1137/0522104. [26] G. Schmidt, Boundary integral methods for periodic scattering problems, in Around the Research of Vladimir Maz'ya II, (ed. A. Laptev), 12 of International Mathematical Series, Springer, 2010, 337-364. doi: 10.1007/978-1-4419-1343-2_16. [27] G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equations approach, Applications of Mathematics, 58 (2013), 279-307. doi: 10.1007/s10492-013-0014-6. [28] O. Steinbach and M. Windisch, Modified combined field integral equations for electromagnetic scattering, SIAM J. Numer. Anal., 47 (2009), 1149-1167. doi: 10.1137/070698063. [29] K. Yosida, Functional Analysis, Springer, 5th edition, 1980.

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References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 1995. [2] H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure, Inverse Problems, 11 (1995), 823-833. doi: 10.1088/0266-5611/11/4/013. [3] T. Arens, Scattering by Biperiodic Layered Media: The Integral Equation Approach, habilitation thesis, Universität Karlsruhe in Karlsruhe, 2010. [4] G. Bao and D. C. Dobson, On the scattering by a biperiodic structure, Proc. AMS, 128 (2000), 2715-2723. doi: 10.1090/S0002-9939-00-05509-X. [5] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations I. An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci., 24 (2001), 9-30. doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. [6] A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci., 24 (2001), 31-48. doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X. [7] A. Buffa, M. Costabel and C. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains, Numer. Math., 92 (2002), 679-710. doi: 10.1007/s002110100372. [8] A. Buffa, M. Costabel and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845-867. doi: 10.1016/S0022-247X(02)00455-9. [9] A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math., 95 (2003), 459-485. doi: 10.1007/s00211-002-0407-z. [10] B. Bugert, On Integral Equation Methods for Electromagnetic Scattering by Biperiodic Structures, PhD thesis, Technische Universität Berlin in Berlin, 2014. [11] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013. [12] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626. doi: 10.1137/0519043. [13] M. Costabel and F. Le Louër, On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., 71 (2011), 635-656. doi: 10.1137/090779462. [14] D. C. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, Anal. Appl., 166 (1992), 507-528. doi: 10.1016/0022-247X(92)90312-2. [15] D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, Math. Anal. Numer., 28 (1994), 419-439. [16] 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. doi: 10.1142/S0218202500000197. [17] V. Yu. Gotlib, Solutions of the Helmholtz equation, concentrated near a plane periodic boundary, J. Math. Sci., 102 (2000), 4188-4194. doi: 10.1007/BF02673850. [18] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, 2008. doi: 10.1007/978-3-540-68545-6. [19] G. Hu and A. Rathsfeld, Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings, IMA Journal of Applied Mathematics, (2014), 1-25. doi: 10.1093/imamat/hxt054. [20] I. V. Kamotski and S. A. Nazarov, The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain, Journal of Mathematical Sciences, 111 (1988), 3657-3664. doi: 10.1023/A:1016377707919. [21] R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math., 48 (1988), 307-325. doi: 10.1137/0148016. [22] A. Lechleiter and D.-L. Nguyen, On uniqueness in electromagnetic scattering from biperiodic structures, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1167-1184. doi: 10.1051/m2an/2012063. [23] D. Maystre, Integral methods, in Electromagnetic Theory of Gratings, (ed. R. Petit), Springer, 1980, 63-100. [24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [25] J.-C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Appl. Math., 22 (1991), 1679-1701. doi: 10.1137/0522104. [26] G. Schmidt, Boundary integral methods for periodic scattering problems, in Around the Research of Vladimir Maz'ya II, (ed. A. Laptev), 12 of International Mathematical Series, Springer, 2010, 337-364. doi: 10.1007/978-1-4419-1343-2_16. [27] G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equations approach, Applications of Mathematics, 58 (2013), 279-307. doi: 10.1007/s10492-013-0014-6. [28] O. Steinbach and M. Windisch, Modified combined field integral equations for electromagnetic scattering, SIAM J. Numer. Anal., 47 (2009), 1149-1167. doi: 10.1137/070698063. [29] K. Yosida, Functional Analysis, Springer, 5th edition, 1980.
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