2015/03/02 by А. Г. Рамм, Alexander G. Ramm, Ramm, Alexander G.
Engineering · Mathematics · Physics and Astronomy · #78A25 #78A45 #Composite Material Mechanics #Electromagnetic Scattering and Analysis #FOS: Physical sciences #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #Optics (physics.optics) #msc:78A25 #msc:78A45 #physics.optics
paper · pdf · doi:10.48550/arxiv.1503.00639
arxiv created 2015/03/02 · openalex publication_date 2015/03/02 · arxiv updated 2015/03/03 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
An explicit formula is derived for the electromagnetic (EM) field scattered by one small impedance particle D of an arbitrary shape. If a is the characteristic size of the particle, λ is the wavelength, a<<λ and ζ is the boundary impedance of D, [N,[E,N]]=ζ[N,H] on S, where S is the surface of the particle, N is the unit outer normal to S, and E, H is the EM field, then the scattered field is Esc=[∇ g(x,x1), Q]. Here g(x,y)=\fraceik|x-y|4π|x-y|, k is the wave number, x1∈ D is an arbitrary point, and Q=-(ζ|S|)/(iωμ)τ∇ × E0, where E0 is the incident field, |S| is the area of S, ω is the frequency, μ is the magnetic permeability of the space exterior to D, and τ is a tensor which is calculated explicitly. The scattered field is O(|ζ| a2)>> O(a3) as a→ 0 when λ is fixed and ζ does not depend on a. Thus, |Esc| is much larger than the classical value O(a3) for the field scattered by a small particle. It is proved that the effective field in the medium, in which many small particles are embedded, has a limit as a→ 0 and the number M=M(a) of the particles tends to ∞ at a suitable rate. Thislimit solves a linear integral equation. The refraction coefficient of the limiting medium is calculated analytically. This yields a recipe for creating materials with a desired refraction coefficient.