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.

This paper deals with the scattering of a TM plane wave from a periodic grating with single defect, of which position is known. The surface is perfectly conductive and made up with a periodic array of rectangular grooves and a defect where a groove is not formed. The scattered wave above grooves is written as a variation from the diffracted wave for the perfectly periodic case. Then, an integral equation for the scattering amplitude is obtained, which is solved numerically by use of truncation and the iteration method. The differential scattering cross section and the optical theorem are calculated in terms of the scattering amplitude and are illustrated in figures. It is found that incoherent Wood's anomaly appears at critical angles of scattering. The physical mechanisms of Wood's anomaly and incoherent Wood's anomaly are discussed in relation to the guided surface wave excited by the incident plane wave. It is concluded that incoherent Wood's anomaly is caused by the diffraction of the guided surface wave.

This paper deals with the scattering of TE plane wave from a periodic grating with single defect, of which position is known. The surface is perfectly conductive and made up with a periodic array of rectangular grooves and a defect where a groove is not formed. By use of the modal expansion method, the field inside grooves is expressed as a sum of guided modes with unknown amplitudes. The mode amplitudes are regarded as a sum of the base component and the perturbed component due to the defect, where the base component is the solution in case of the perfectly periodic grating. An equation for the base component is obtained in the first step. By use of the base component, a new equation for the perturbed component is derived in the second step. A new representation of the optical theorem, relating the total scattering cross section with the reduction of the scattering amplitude is obtained. Also, a single scattering approximation is proposed to express the scattered field. By use of truncation, we numerically obtain the base component and the perturbed component, in terms of which the total scattering cross section and the differential scattering cross section are calculated and illustrated in figures.

This paper deals with the diffraction of TM plane wave by a perfectly conductive periodic surface. Applying the Rayleigh hypothesis, a linear equation system determining the diffraction amplitudes is derived. The linear equation is formally solved by Cramer's formula. It is then found that, when the angle of incidence becomes a low grazing limit, the amplitude of the specular reflection becomes -1 and any other diffraction amplitudes vanish for any perfectly conductive periodic surfaces with small roughness and gentle slope.

The energy conservation law and the optical theorem in the grating theory are discussed: the energy conservation law states that the incident energy is equal to the sum of diffracted energies and the optical theorem means that the diffraction takes place at the loss of the specularly reflection amplitude. A mathematical relation between the optical theorem and the energy conservation law is given. Some numerical examples are given for a TM plane wave diffraction by a sinusoidal surface.

A periodic approach introduced previously is applied to the TM wave scattering from a finite periodic surface. A mathematical relation is proposed to estimate the scattering amplitude from the diffraction amplitude for the periodic surface, where the periodic surface is defined as a superposition of surface profiles generated by displacing the finite periodic surface by every integer multiple of the period . From numerical examples, it is concluded that the scattering cross section for the finite periodic surface can be well estimated from the diffraction amplitude for a sufficiently large .

As a method of analyzing the wave scattering from a finite periodic surface, this paper introduces a periodic approach. The approach first considers the wave diffraction by a periodic surface that is a superposition of surface profiles generated by displacing the finite periodic surface by every integer multiple of the period . It is pointed out that the Floquet solution for such a periodic case becomes an integral representation of the scattered field from the finite periodic surface when the period goes to infinity. A mathematical relation estimating the scattering amplitude for the finite periodic surface from the diffraction amplitude for the periodic surface is proposed. From some numerical examples, it is concluded that the scattering cross section for the finite periodic surface can be well estimated from the diffraction amplitude for a sufficiently large .