Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-30T19:12:35.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On uniqueness in electromagnetic scattering from biperiodicstructures

Published online by Cambridge University Press:  17 June 2013

Armin Lechleiter  and
Dinh-Liem Nguyen
Armin Lechleiter
Affiliation:
Center for Industrial Mathematics, University of Bremen, 28359 Bremen, Germany. . [email protected]
Dinh-Liem Nguyen
Affiliation:
DEFI, INRIA Saclay–Ile-de-France and Ecole Polytechnique, 91128 Palaiseau, France. ; [email protected]
Get access

Abstract

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectricstructure mounted on a perfectly conducting plate in three dimensions. Given thatuniqueness of solution holds, existence of solution follows from a well-known Fredholmframework for the variational formulation of the problem in a suitable Sobolev space. Inthis paper, we derive a Rellich identity for a solution to this variational problem undersuitable smoothness conditions on the material parameter. Under additional non-trappingassumptions on the material parameter, this identity allows us to establish uniqueness ofsolution for all positive wave numbers.

Keywords

Biperiodic scatteringuniquenesselectromagnetic waves
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1167 - 1184
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Alber, H., A quasi-periodic boundary value problem for the laplacian and the continuation of its resolvent. Proc. Royal Soc. Edinburgh 82 (1979) 251–272. Google Scholar
T. Arens, Scattering by biperiodic layered media: The integral equation approach.Habilitation Thesis, Universität Karlsruhe (2010).
Bao, G., Variational approximation of Maxwell’s equations in biperiodic structures. SIAM J. Appl. Math. 57 (1997) 364–381. Google Scholar
G. Bao, L. Cowsar and W. Masters, Mathematical modeling in optical science. SIAM Frontiers Appl. Math. SIAM, Philadelphia (2001).
Bao, G. and Dobson, D.C., On the scattering by a biperiodic structure. Proc. Amer. Math. Soc. 128 (2000) 2715–2723. Google Scholar
Bonnet-Bendhia, A.-S. and Starling, F., Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17 (1994) 305–338. Google Scholar
Chandler-Wilde, S.N. and Monk, P., Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal. 37 (2005) 598–618. Google Scholar
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/.
Dobson, D. and Friedman, A., The time-harmonic Maxwell’s equations in a doubly periodic structure. J. Math. Anal. Appl. 166 (1992) 507–528. Google Scholar
Dobson, D.C., A variational method for electromagnetic diffraction in biperiodic structures. Math. Model. Numer. Anal. 28 (1994) 419–439. Google Scholar
Haddar, H. and Lechleiter, A., Electromagnetic wave scattering from rough penetrable layers. SIAM J. Math. Anal. 43 (2011) 2418–2433. Google Scholar
W. McLean, Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge, UK (2000).
P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford Science Publications, Oxford (2003).
Rellich, F., Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral. Math. Zeitschrift 46 (1940) 635–636. Doi: 10.1007/BF01181459. Google Scholar
Schmidt, G., On the diffraction by biperiodic anisotropic structures. Appl. Anal. 82 (2003) 75–92. Google Scholar
Wilcox, C., Scattering Theory for Diffraction Gratings. Appl. Math. Sci. Springer-Verlag 46 (1984). Google Scholar