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Chandler-Wilde, Simon N.; Zhang, Bo (1999)
Publisher: Society for Industrial and Applied Mathematics
Languages: English
Types: Article
We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations.
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    • [1] G. Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal. 32 (1995), pp. 1155-1169.
    • [2] G. Bao, D. C. Dobson, and J. A. Cox, Mathematical studies in rigorous grating theory, J. Opt. Soc. Amer. A 12 (1995), pp. 1029-1042.
    • [3] A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem, Math. Methods Appl. Sci., 17 (1994), pp. 305-338.
    • [4] S. N. Chandler-Wilde, Boundary value problems for the Helmholtz equation in a half-plane, in Proceedings, Third Int. Conf. on Mathematical and Numerical Aspects of Wave Propagation, G. Cohen, L. Halpern, and P. Joly, eds., Proceedings Appl. Math 50, SIAM, Philadelphia, PA, 1995, pp. 188-197.
    • [5] S. N. Chandler-Wilde, The impedance boundary value problem for the Helmholtz equation in a half-plane, Math. Methods Appl. Sci., 20 (1997), pp. 813-840.
    • [6] S. N. Chandler-Wilde and C. R. Ross, Scattering by rough surfaces: The Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane, Math. Methods Appl. Sci., 19 (1996), pp. 959-976.
    • [7] S. N. Chandler-Wilde and B. Zhang, Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. Roy. Soc. London Ser. A, 454 (1998), pp. 519-542.
    • [8] S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), pp. 1774-1790.
    • [9] S. N. Chandler-Wilde and B. Zhang, On the solvability of a class of second kind integral equations on unbounded domains, J. Math. Anal. Appl., 214 (1997), pp. 482-502.
    • [10] S. N. Chandler-Wilde and B. Zhang, A Generalized Collectively Compact Operator Theory with an Application to Integral Equations on Unbounded Domains, in preparation.
    • [11] X. Chen and A. Friedman, Maxwell's equations in a periodic structure, Trans. Amer. Math. Soc., 323 (1991), pp. 465-507.
    • [12] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley, New York, 1983.
    • [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer-Verlag, Berlin, 1998.
    • [14] C. Macaskill and P. Cao, A new treatment of rough surface scattering, Proc. Roy. Soc. London Ser A, 452 (1996), pp. 2593-2612.
    • [15] J. A. DeSanto and P. A. Martin, On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface, J. Acoust. Soc. Am. 102 (1997), pp. 67-77.
    • [16] C. L. DeVito, Functional Analysis, Academic Press, New York, 1978.
    • [17] D. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl. 166 (1992), pp. 507-528.
    • [18] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983.
    • [19] A. Kirsch, Diffraction by periodic structures, in Inverse Problems in Mathematical Physics, L. Paivarinta and E. Somersalo, eds., Lecture Notes in Phys. 422, Springer-Verlag, 1993, pp. 87-102.
    • [20] A. Kirsch, An inverse problem for periodic structures, in Inverse Scattering and Potential Problems in Mathematical Physics, R.E. Kleinman, R. Kress, and E. Martensen, eds., Peter Lang, Frankfurt, 1995, pp. 75-93.
    • [21] D. A. Kapp and G. S. Brown, A new numerical method for rough surface scattering calculations, IEEE Trans. Antennas and Propagation, 44 (1996), pp. 711-721.
    • [22] J. C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Math. Anal. 22 (1991), pp. 1679- 1701.
    • [23] R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980.
    • [24] H. W. Schu¨rmann, V. S. Serov, and Yu. V. Shestopalov, On the theory of TE-polarized waves in a linear three-layer structure, Electromagnetic Waves and Electronic Systems, 1 (1997), pp. 49-59.
    • [25] B. Strycharz, An acoustic scattering problem for a periodic, inhomogeneous media, Math. Methods Appl. Sci., 21 (1998), pp. 969-983.
    • [26] L. Tsang, C. H. Chan, K. Pak, and H. Sangani, Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method, IEEE Trans. Antennas and Propagation, 43 (1995), pp. 851-859.
    • [27] G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin, 1993.
    • [28] G. Vainikko, Fast Solvers of the Lippmann-Schwinger Equation, Research reports A387, Institute of Mathematics, Helsinki University of Technology, Finland 1997.
    • [29] R. L. Wagner, J. M. Song, and W. C. Chew, Monte-Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces, IEEE Trans. Antennas and Propagation, 45 (1997), pp. 235-246.
    • [30] Y. Xu, Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity, IMA J. Appl. Math., 54 (1995), pp. 9-29.
    • [31] Bo Zhang and S. N. Chandler-Wilde, Acoustic scattering by an inhomogeneous layer on a rigid plate, SIAM J. Appl. Math. 58 (1998), pp. 1931-1950.
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