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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An inverse problem for scattering
by a doubly periodic structure

Authors: Gang Bao and Zhengfang Zhou
Journal: Trans. Amer. Math. Soc. 350 (1998), 4089-4103
MSC (1991): Primary 35R30; Secondary 35P15, 78A45
MathSciNet review: 1487607
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Abstract | References | Similar Articles | Additional Information

Abstract: Consider scattering of electromagnetic waves by a doubly periodic structure $S=\{x_3=f(x_1, x_2)\}$ with $f(x_1+n_1\Lambda _1, x_2+n_2\Lambda _2)=f(x_1, x_2)$ for integers $n_1$, $n_2$. Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at $x_3=b$ for some large $b$. To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain $\Omega$:

\begin{displaymath}\left\{ \begin{array}{l} - \triangle u = \lambda u \quad \text{in} \quad \Omega, \\ \nabla \cdot u = 0 \quad \text{in} \quad \Omega, \\ n \times u = 0 \quad \text{on} \quad \partial \Omega. \end{array} \right. \end{displaymath}

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  • 1. T. Abboud, Formulation variationnelle des équations de Maxwell dans un réseau bipéiodique de R$^3$, C. R. Acad. Sci. Paris, Série I, t. 317 (1993), 245-248. MR 94f:78002
  • 2. T. Abboud and J. C. Nédélec, Electromagnetic waves in an inhomogeneous medium, J. Math. Anal. Appl. 164 (1992), 40-58. MR 93a:78005
  • 3. H. Ammari, Théorèmes d'unicité pour un problème inverse dans une structure bipériodique, C. R. Acad. Sci. Paris, Série I, t. 320 (1995), 815-820. MR 96a:78007
  • 4. G. Bao, A uniqueness theorem for an inverse problem in periodic diffractive optics, Inverse Problems 10 (1994), 335-340. MR 95c:35263
  • 5. G. Bao, Variational approximation of Maxwell's equations in biperiodic structures, SIAM J Appl. Math. 57 (1997), 364-381. MR 97m:65199
  • 6. G. Bao and A. Friedman, Inverse problems for scattering by periodic structures, Arch. Rational Mech. Anal. 132 (1995), 49-72. MR 96i:35131
  • 7. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, N.Y., 1992. MR 93j:35124
  • 8. D. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, RAIRO Modél. Math. Anal. Numér., 28 (1994), 419-439. MR 95m:78017
  • 9. D. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl., 166 (1992), 507-528. MR 92m:78015
  • 10. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, New York, 1986. MR 88b:65129
  • 11. L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal. 5 (1960), 286-292. MR 22:8198
  • 12. Electromagnetic Theory of Gratings, Topics in Current Physics, Vol. 22, edited by R. Petit, Springer-Verlag, Heidelberg, 1980. MR 82a:78001

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Additional Information

Gang Bao
Affiliation: Department of Mathematics University of Florida Gainesville, Florida 32611

Zhengfang Zhou
Affiliation: Department of Mathematics Michigan State University East Lansing, Michigan 48824

Keywords: Uniqueness and stability for an inverse diffraction problem, estimation of eigenvalues, scattering by doubly periodic structures
Received by editor(s): September 18, 1996
Additional Notes: The research of the first author was partially supported by NSF grant DMS 95-01099, NSF University-Industry Cooperative Research Programs grant DMS 97-05139, and a Research Development Award (University of Florida)
Article copyright: © Copyright 1998 American Mathematical Society