A multiresolution study of effective properties of complex electromagnetic systems

A systematic study of the across-scale coupling phenomenology in electromagnetic (EM) scattering problems is addressed using the theory of multiresolution decomposition and orthogonal wavelets. By projecting an integral equation formulation of the scattering problem onto a set of subspaces that constitutes a multiresolution decomposition of L₂ (R), one can derive two coupled formulations. The first governs the macroscale response, and the second governs the microscale response. By substituting the formal solution of the latter in the former, a new self-consistent formulation that governs the macroscale response component is obtained. This formulation is written on a macroscale grid, where the effects of the microscale heterogeneity are expressed via an across-scale coupling operator. This operator can also be interpreted as representing the effective properties of the microstructure. We study the properties of this operator versus the characteristics of the Green function and the microstructure for various electromagnetic problems, using general asymptotic considerations. A specific numerical example of TM scattering from a laminated complex structure is provided