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Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium

Published online by Cambridge University Press:  24 October 2008

David Colton  and
Lassi Päivärinta
David Colton
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, U.S.A.
Lassi Päivärinta
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland
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Abstract

We consider the scattering of time harmonic electromagnetic waves by an inhomogeneous medium of compact support. It is first shown that the set of far field patterns of the electric fields corresponding to incident plane waves propagating in arbitrary directions is complete in the space of square-integrable tangential vector fields defined on the unit sphere. We then show that under certain conditions the electric far field patterns satisfy an integral identity involving the unique solution of a new class of boundary value problems for Maxwell's equations called the interior transmission problem for electromagnetic waves. Finally, it is indicated how this integral identity can be used to formulate an optimization scheme yielding an optimal solution of the inverse scattering problem for electromagnetic waves.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 561 - 575
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Blöhbaum, J.. On two methods for solving the inverse scattering problem for electromagnetic waves. Technical report, University of Delaware (1987).Google Scholar
[2]Colton, D.. Dense sets and far field patterns for acoustic waves in an inhomogeneous medium. Proc. Edinburgh Math. Soc. (2). to appear.Google Scholar
[3]Colton, D. and Kirsch, A.. Dense sets and far field patterns in acoustic wave propagation. SIAM J. Math. Anal. 15 (1984). 996–1006.CrossRefGoogle Scholar
[4]Colton, D. and Kress, R.. Integral Equation Methods in Scattering Theory (Wiley, 1983).Google Scholar
[5]Colton, D. and Kress, R.. Dense sets and far field patterns in electromagnetic wave propagation. SIAM J. Math. Anal. 16 (1985). 1049–1060.CrossRefGoogle Scholar
[6]Colton, D. and Monk, P.. A novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region. SIAM J. Appl. Math. 45 (1985), 1039–1053.CrossRefGoogle Scholar
[7]Colton, D. and Monk, P.. A novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region. II. SIAM J. Appl. Math. 46 (1986), 506–523.CrossRefGoogle Scholar
[8]Colton, D. and Monk, P.. The numerical solution of the three dimensional inverse scattering problem for time harmonic acoustic waves. SIAM J. Sci. Statist. Comput. 8 (1987), 278–291.CrossRefGoogle Scholar
[9]Colton, D. and Monk, P.. The inverse scattering problem for time harmonic acoustic waves in a penetrable medium. Quart. J. Mech. Appl. Math. 40 (1987), 189–212.CrossRefGoogle Scholar
[10]Colton, D. and Monk, P.. The inverse scattering problem for time harmonic acoustic waves in an inhomogeneous medium. Quart. J. Mech. Appl. Math., to appear.Google Scholar
[11]Kirsch, A.. The denseness of the far field patterns for the transmission problem. IMA J. Appl. Math. 37 (1986). 213–225.CrossRefGoogle Scholar
[12]Kirsch, A.. Properties of far field operators in acoustic scattering, to appear.Google Scholar
[13]Müller, C.. Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).CrossRefGoogle Scholar
[14]Päivärinta, L. and Somersalo, E.. The uniqueness of the one dimensional electromagnetic inversion with bounded potentials. J. Math. Anal. Appl., to appear.Google Scholar
[15]Somersalo, E.. One dimensional electromagnetic reflection problem. Ann. Acad. Sci. Fenn. Ser AI Math. Dissertationes 60 (1986), 1–31.Google Scholar
[16]Wilde, P. and Willers, A.. Dense sets and far field patterns for the vector Helmholtz equation under transmission boundary conditions. SIAM J. Math. Anal., to appear.Google Scholar