Дослідження властивостей розв’язків задачі електромагнітного розсіяння на сукупності включень малого розміру

Abstract

The problem of scattering of the electromagnetic (EM) waves by many small impedance bodies (particles), embedded in a homogeneous medium is studied. Physical properties of the particles are described by their boundary impedance. The boundary integral equation is obtained for the effective EM field in the limiting medium for the case if radius of particles tends to zero and number of particles tends to infinity by suitable rate. The medium, created by the embedding of the small particles, has new physical properties. Although scattering of EM waves by small bodies has a long history, the obtained results are new and useful in applications because EM wave scattering in nanostructures and small dust particles in the air are examples of the problem to which our approach can be applied. The developed previously Mie theory deals with scattering by a sphere, not necessarily small, and gives the solution to scattering problem in terms of the series in spherical harmonics. If the sphere is small, then the first term in the Mie series yields the main part of solution. The proposed approach is applicable only to small particles; it is development of ideas proposed earlier for the scattering of acoustic waves. However, the scattering of EM waves brought new technical difficulties. These difficulties come from the vector nature of boundary condition. The particles in our approach can be of arbitrary shape. The solution of initial EM wave scattering problem is reduced to solving a linear algebraic system. This system is not obtained by a discretization of some boundary integral equation, and it has a clear physical meaning. Its limiting form yields an integro-differential equation for the limiting effective field in the medium where the small particles are embedded. The new analytical-numerical method for solving the scattering problem of electromagnetic waves on the set of small particles has been developed. Investigation of properties of the solutions to problem depending on the parameters of medium, size of particles and their impedance has been carried out. The numerical results allowed to establish the correctness of assumption about property of divergence of the tangential component of electric field on the particle’s surface, which was used essentially for obtaining the asymptotic solution. The numerical results testify that the relative error of the obtained numerical solution, while compare it with the similar solution obtained by some complicate procedure, does not exceed of several percents. Розглянуто задачу розсіяння електромагнітних хвиль малими імпедансними тілами (включеннями) у однорідному середовищі. Фізичні властивості включень описуються їх поверхневим імпедансом. Отримано граничне інтегральне рівняння для ефективного електромагнітного поля при a→0 , де a – характерний розмір включень, а M(a) – кількість включень. Проведено дослідження властивостей розв’язків задачі залежно від параметрів середовища, розміру включень та їх поверхневого імпедансу. Наведено результати числового моделювання.