In the scattering problem of periodic gratings, at a low grazing limit of incidence, the incident plane wave is completely cancelled by the reflected wave, and the total wave field vanishes and physically becomes a dark shadow. This problem has received much interest recently. Nakayama et al. have proposed “the shadow theory”. The theory was first applied to the diffraction by perfectly conductive gratings as an example, where a new description and a physical mean at a low grazing limit of incidence for the gratings have been discussed. In this paper, the shadow theory is applied to the analyses of multilayered dielectric periodic gratings, and is shown to be valid on the basis of the behavior of electromagnetic waves through the matrix eigenvalue problem. Then, the representation of field distributions is demonstrated for the cases that the eigenvalues degenerate in the middle regions of multilayered gratings in addition to at a low grazing limit of incidence and some numerical examples are given.

The printed dipole on a semi-infinite substrate is investigated. The solution is based on the moment method in the Fourier transform domain. We analyze far-field and near-field radiation patterns for a printed dipole. Therefore, we make radiation fields clear.