Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T20:50:28.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse Scattering Problem for the Maxwell’sEquations

Published online by Cambridge University Press:  28 January 2013

A. G. Ramm
Get access

Abstract

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of theMaxwell’s system to a new Fredholm second-kind integral equation with a scalarweakly singular kernel is given for electromagnetic (EM) wave scattering. Thisequation allows one to derive a formula for the scattering amplitude in which only ascalar function is present. If this function is small (an assumption that validates aBorn-type approximation), then formulas for the solution to the inverse problem areobtained from the scattering data: the complex permittivityϵ′(x) in a bounded regionD ⊂ R3 is found from the scattering amplitudeA(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and allβ,α ∈ S2, whereS2 is the unit sphere in R3,ϵ0 and μ0 are constantpermittivity and magnetic permeability in the exterior regionD′ = R3\D. The novel pointsin this paper include: i) A reduction of the inverse problem for vectorEM waves to a vector integral equation with scalar kernelwithout any symmetry assumptions on the scatterer, ii) A derivation of thescalar integral equation of the first kind for solving the inversescattering problem, and iii) Presenting formulas for solving this scalar integralequation. The problem of solving this integral equation is an ill-posed one. A method fora stable solution of this problem is given.

Keywords

Electromagnetic wavesMaxwell’s equationswave scatteringinverse scattering
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 200 - 206
Copyright
© EDP Sciences, 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

L. Landau, E. Lifschitz, L. Pitaevskii. Electrodynamics of continuous medium. Pergamon Press, Oxford, 1984.
L. Tsang, J. Kong, K.Ding. Scattering of electromagnetic waves : Theories and applications. Wiley, New York, 2000.
C. Müller. Grundprobleme der mathematischen Theorie electromagnetischer Schwingungen. Springer-Verlag, Berlin, 1957.
A. G. Ramm. Wave scattering by small bodies of arbitrary shapes. World Sci. Publishers, Singapore, 2005.
A. G. Ramm. Scattering by obstacles. D.Reidel, Dordrecht, 1986.
A. G. Ramm. Inverse problems. Springer, New York, 2005.
Ramm, A. G.. Many-body wave scattering by small bodies and applications. J. Math. Phys., 48, No. 10, (2007), 103511. CrossRefGoogle Scholar
Ramm, A. G.. Electromagnetic wave scattering by small bodies. Phys. Lett. A, 372/23, (2008), 4298–4306. CrossRefGoogle Scholar
Ramm, A. G.. Wave scattering by many small particles embedded in a medium. Phys. Lett. A, 372/17, (2008), 3064–3070. CrossRefGoogle Scholar
Ramm, A. G.. Wave scattering by many small bodies and creating materials with a desired refraction coefficient. Afrika Matematika, 22, No. 1, (2011), 33–55. CrossRefGoogle Scholar
Ramm, A. G.. Inversion of the Radon transform with incomplete data. Math. Methods in the Appl.Sci., 15, No. 3, (1992), 159–166. CrossRefGoogle Scholar
A. G. Ramm, A.I. Katsevich. The Radon transform and local tomography. CRC Press, Boca Raton, 1996.