Exact equations for averaged electromagnetic field and its fluctuations at wave multiple scattering by plane periodic array of magnetic microelements.

Electromagnetic wave (EM) multiple scattering by a plane periodic array of magnetic microelements in free space is considered analytically by natural subdividing of the EM wave into the averaged and fluctuation components. Each magnetic element is characterized by magnetic susceptibility tensor and shape. An exact Dyson integral equation is derived for the magnetic field Floquet-Bloch amplitude in-plane averaged over an array unit cell. The mass operator of the Dyson equation is expressed via the T-scattering operator of the array unit cell that satisfies a type of the Lippmann-Schwinger equation. We showed that magnetic field fluctuations are generated by the Bragg-Laue diffraction of an averaged magnetic field on the periodic array and are described inside the array as waves propagating with the Laue wave vectors equal to the difference between the in-plane wave vector of the incident magnetic field and the reciprocal lattice wave vector. We derived, for the first time, an exact quadrature to calculate magnetic field fluctuations from their averaged value. These general results are illustrated by a simple Born approximation. In particular, we revealed a mechanism of discrete waveguide excitation by an incident plane EM wave via the averaged EM wave Brag-Laue diffraction on the magnetic microelement array in the quasi-static approach when the wavelength of incident EM is much larger than the sizes of magnetic elements and periods of the array. The mode energy excitation coefficient at normal incidence of the plane EM wave on the array is evaluated.