Scattering of electromagnetic waves by many thin cylinders

Electromagnetic wave scattering by many parallel to z-axis, thin, perfectly conducting, circular infinite cylinders is studied asymptotically as a → 0. Let Dm be the cross-section of the mth cylinder, a   be its radius, and xˆm=(xm1,xm2) be its center, 1 ⩽ m ⩽ M, M = M(a  ). It is assumed that the points xˆm are distributed so thatN(Δ)=ln1a∫ΔN(x)dx[1+o(1)],where N(Δ)N(Δ) is the number of points xˆm in an arbitrary open subset Δ of the plane xoy. The function N(x) ⩾ 0 is a given continuous function. An equation for the self-consistent (efficient) field is derived as a → 0. The cylinders are assumed perfectly conducting. A formula is derived for the effective refraction coefficient in the medium in which many cylinders are distributed. These cylinders may model nanowires embedded in the medium. Our result shows how these cylinders influence the refraction coefficient of the medium.