Math. Model. Nat. Phenom.
Volume 8, Number 1, 2013Harmonic analysis
|Page(s)||200 - 206|
|Published online||28 January 2013|
Inverse Scattering Problem for the Maxwell’s Equations
Mathematics Department, Kansas State
University, Manhattan, KS
∗ Corresponding author. E-mail: firstname.lastname@example.org
Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of the Maxwell’s system to a new Fredholm second-kind integral equation with a scalar weakly singular kernel is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity ϵ′(x) in a bounded region D ⊂ R3 is found from the scattering amplitude A(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and all β,α ∈ S2, where S2 is the unit sphere in R3, ϵ0 and μ0 are constant permittivity and magnetic permeability in the exterior region D′ = R3\D. The novel points in this paper include: i) A reduction of the inverse problem for vector EM waves to a vector integral equation with scalar kernel without any symmetry assumptions on the scatterer, ii) A derivation of the scalar integral equation of the first kind for solving the inverse scattering problem, and iii) Presenting formulas for solving this scalar integral equation. The problem of solving this integral equation is an ill-posed one. A method for a stable solution of this problem is given.
Mathematics Subject Classification: 35J10 / 70F10 / 74J25 / 81U40 / 81V05
Key words: Electromagnetic waves / Maxwell’s equations / wave scattering / inverse scattering
© EDP Sciences, 2013
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.