On uniqueness in electromagnetic scattering from biperiodic structures
Armin Lechleiter; Dinh-Liem Nguyen
- Volume: 47, Issue: 4, page 1167-1184
- ISSN: 0764-583X
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topLechleiter, Armin, and Nguyen, Dinh-Liem. "On uniqueness in electromagnetic scattering from biperiodic structures." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.4 (2013): 1167-1184. <http://eudml.org/doc/273267>.
@article{Lechleiter2013,
abstract = {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.},
author = {Lechleiter, Armin, Nguyen, Dinh-Liem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {biperiodic scattering; uniqueness; electromagnetic waves; electromagnetic scattering; biperiodic dielectric structure, variational formulation; Rellich identity},
language = {eng},
number = {4},
pages = {1167-1184},
publisher = {EDP-Sciences},
title = {On uniqueness in electromagnetic scattering from biperiodic structures},
url = {http://eudml.org/doc/273267},
volume = {47},
year = {2013},
}
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 - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 4
SP - 1167
EP - 1184
AB - 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.
LA - eng
KW - biperiodic scattering; uniqueness; electromagnetic waves; electromagnetic scattering; biperiodic dielectric structure, variational formulation; Rellich identity
UR - http://eudml.org/doc/273267
ER -
References
top- [1] T. Abboud, Electromagnetic waves in periodic media, in Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, Newark, DE. SIAM, Philadelphia (1993) 1–9. Zbl0815.35106MR1227824
- [2] H. Alber, A quasi-periodic boundary value problem for the laplacian and the continuation of its resolvent. Proc. Royal Soc. Edinburgh82 (1979) 251–272. Zbl0402.35033MR532908
- [3] T. Arens, Scattering by biperiodic layered media: The integral equation approach.Habilitation Thesis, Universität Karlsruhe (2010).
- [4] G. Bao, Variational approximation of Maxwell’s equations in biperiodic structures. SIAM J. Appl. Math.57 (1997) 364–381. Zbl0872.65108MR1438758
- [5] G. Bao, L. Cowsar and W. Masters, Mathematical modeling in optical science. SIAM Frontiers Appl. Math. SIAM, Philadelphia (2001). Zbl0964.00050MR1831328
- [6] G. Bao and D.C. Dobson, On the scattering by a biperiodic structure. Proc. Amer. Math. Soc.128 (2000) 2715–2723. Zbl1025.78007MR1694448
- [7] A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci.17 (1994) 305–338. Zbl0817.35109MR1273315
- [8] S.N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal.37 (2005) 598–618. Zbl1127.35030MR2176117
- [9] M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. http://hal.archives-ouvertes.fr/hal-00453934/.
- [10] D. Dobson and A. Friedman, The time-harmonic Maxwell’s equations in a doubly periodic structure. J. Math. Anal. Appl.166 (1992) 507–528. Zbl0759.35046MR1160941
- [11] D.C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures. Math. Model. Numer. Anal.28 (1994) 419–439. Zbl0820.65087MR1288506
- [12] H. Haddar and A. Lechleiter, Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal.43 (2011) 2418–2433. Zbl1233.35182MR2861668
- [13] W. McLean, Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK (2000). Zbl0948.35001MR1742312
- [14] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford Science Publications, Oxford (2003). Zbl1024.78009
- [15] F. Rellich, Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Zeitschrift 46 (1940) 635–636. Doi: 10.1007/BF01181459. Zbl0023.04204MR2456JFM66.0460.01
- [16] G. Schmidt, On the diffraction by biperiodic anisotropic structures. Appl. Anal.82 (2003) 75–92. Zbl1037.35086MR1961652
- [17] C. Wilcox, Scattering Theory for Diffraction Gratings. Appl. Math. Sci. Springer-Verlag 46 (1984). Zbl0541.76001MR725334
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