System of asymptotic integral equations in the problem of permittivity and permeability tensors determination of a volumetric body in a rectangular waveguide

Background. Objective of the work is to study the mathematical model of electromagnetic waves scattering on volumetric anisotropic heterogeneous bodies in a rectangular waveguide. Materials and methods. The initial boundary value problem for Maxwell's equations is reduced using the method of vector potentials to the system of integro-differential equations on heterogeneity area (the falling field is supposed to be harmonically time-dependent). Then the asymptotic equations are derived from the properties of Green's tensor at the infinity. Results. The main lemma about uniform tending to zero on infinity of the tensor Green's function first component is proved. On the basis of the result received in that lemma the asymptotic behavior of all components of the Green's tensor as well as their derivatives of any order are studied. The system of the asymptotic integral equations for definition of tensors of dielectric and magnetic permeabilities of the volumetric body on passing coefficient is obtained. The method of rotations of the volumetric body for definition of all the components of permittivity and permeability tensors is offered. Expressions for the transformed permittivity tensors in the case of arbitrary turns around the coordinate axes are received. Conclusions. The results can be successfully applied to solve the inverse problem of diffraction in a rectangular waveguide.