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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1167-1184},
doi = {10.1051/m2an/2012063},
mrnumber = {3082293},
zbl = {1282.78022},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1167-1184. doi : 10.1051/m2an/2012063. http://gdmltest.u-ga.fr/item/M2AN_2013__47_4_1167_0/
[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 1227824 | Zbl 0815.35106
[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 532908 | Zbl 0402.35033
[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 1438758 | Zbl 0872.65108
[5] , and , Mathematical modeling in optical science. SIAM Frontiers Appl. Math. SIAM, Philadelphia (2001). | MR 1831328 | Zbl 0964.00050
[6] and , On the scattering by a biperiodic structure. Proc. Amer. Math. Soc. 128 (2000) 2715-2723. | MR 1694448 | Zbl 1025.78007
[7] and , Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17 (1994) 305-338. | MR 1273315 | Zbl 0817.35109
[8] and , Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal. 37 (2005) 598-618. | MR 2176117 | Zbl 1127.35030
[9] , and , Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. http://hal.archives-ouvertes.fr/hal-00453934/.
[10] and , The time-harmonic Maxwell's equations in a doubly periodic structure. J. Math. Anal. Appl. 166 (1992) 507-528. | MR 1160941 | Zbl 0759.35046
[11] , A variational method for electromagnetic diffraction in biperiodic structures. Math. Model. Numer. Anal. 28 (1994) 419-439. | Numdam | MR 1288506 | Zbl 0820.65087
[12] and , Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal. 43 (2011) 2418-2433. | MR 2861668 | Zbl 1233.35182
[13] , Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK (2000). | MR 1742312 | Zbl 0948.35001
[14] , Finite Element Methods for Maxwell's Equations. Oxford Science Publications, Oxford (2003). | Zbl 1024.78009
[15] , Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Zeitschrift 46 (1940) 635-636. Doi: 10.1007/BF01181459. | JFM 66.0460.01 | MR 2456 | Zbl 0023.04204
[16] , On the diffraction by biperiodic anisotropic structures. Appl. Anal. 82 (2003) 75-92. | MR 1961652 | Zbl 1037.35086
[17] , Scattering Theory for Diffraction Gratings. Appl. Math. Sci. Springer-Verlag 46 (1984). | MR 725334 | Zbl 0541.76001