Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all positive wave numbers.
Keywords: biperiodic scattering, uniqueness, electromagnetic waves
@article{M2AN_2013__47_4_1167_0, author = {Lechleiter, Armin and Nguyen, Dinh-Liem}, title = {On uniqueness in electromagnetic scattering from biperiodic structures}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1167--1184}, publisher = {EDP-Sciences}, volume = {47}, number = {4}, year = {2013}, doi = {10.1051/m2an/2012063}, mrnumber = {3082293}, zbl = {1282.78022}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012063/} }
TY - JOUR AU - Lechleiter, Armin AU - Nguyen, Dinh-Liem TI - On uniqueness in electromagnetic scattering from biperiodic structures JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1167 EP - 1184 VL - 47 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012063/ DO - 10.1051/m2an/2012063 LA - en ID - M2AN_2013__47_4_1167_0 ER -
%0 Journal Article %A Lechleiter, Armin %A Nguyen, Dinh-Liem %T On uniqueness in electromagnetic scattering from biperiodic structures %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1167-1184 %V 47 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012063/ %R 10.1051/m2an/2012063 %G en %F M2AN_2013__47_4_1167_0
Lechleiter, Armin; Nguyen, Dinh-Liem. On uniqueness in electromagnetic scattering from biperiodic structures. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 4, pp. 1167-1184. doi : 10.1051/m2an/2012063. http://www.numdam.org/articles/10.1051/m2an/2012063/
[1] Electromagnetic waves in periodic media, in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, Newark, DE. SIAM, Philadelphia (1993) 1-9. | MR | Zbl
,[2] A quasi-periodic boundary value problem for the laplacian and the continuation of its resolvent. Proc. Royal Soc. Edinburgh 82 (1979) 251-272. | MR | Zbl
,[3] Scattering by biperiodic layered media: The integral equation approach.Habilitation Thesis, Universität Karlsruhe (2010).
,[4] Variational approximation of Maxwell's equations in biperiodic structures. SIAM J. Appl. Math. 57 (1997) 364-381. | MR | Zbl
,[5] Mathematical modeling in optical science. SIAM Frontiers Appl. Math. SIAM, Philadelphia (2001). | MR | Zbl
, and ,[6] On the scattering by a biperiodic structure. Proc. Amer. Math. Soc. 128 (2000) 2715-2723. | MR | Zbl
and ,[7] Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17 (1994) 305-338. | MR | Zbl
and ,[8] Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal. 37 (2005) 598-618. | MR | Zbl
and ,[9] Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. http://hal.archives-ouvertes.fr/hal-00453934/.
, and ,[10] The time-harmonic Maxwell's equations in a doubly periodic structure. J. Math. Anal. Appl. 166 (1992) 507-528. | MR | Zbl
and ,[11] A variational method for electromagnetic diffraction in biperiodic structures. Math. Model. Numer. Anal. 28 (1994) 419-439. | Numdam | MR | Zbl
,[12] Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal. 43 (2011) 2418-2433. | MR | Zbl
and ,[13] Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK (2000). | MR | Zbl
,[14] Finite Element Methods for Maxwell's Equations. Oxford Science Publications, Oxford (2003). | Zbl
,[15] Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Zeitschrift 46 (1940) 635-636. Doi: 10.1007/BF01181459. | JFM | MR | Zbl
,[16] On the diffraction by biperiodic anisotropic structures. Appl. Anal. 82 (2003) 75-92. | MR | Zbl
,[17] Scattering Theory for Diffraction Gratings. Appl. Math. Sci. Springer-Verlag 46 (1984). | MR | Zbl
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