This paper deals with the scattering of a transverse magnetic (TM) plane wave from a perfectly conductive surface with a finite periodic array of rectangular grooves. By use of the method in a previous paper [IEICE TRANS. ELECTRON. VOL.E90-C, no.4, pp.903-906, APRIL 2007], the total scattering cross section is numerically calculated for several different numbers of grooves at a low grazing angle of incidence. It is newly found that, when the corrugation width becomes thousands times of wavelength, the total scattering cross section slightly depends on the groove depth and the period, and becomes almost proportional to square root of the corrugation width with a small correction.

A transverse magnetic (TM) plane wave is diffracted by a periodic surface into discrete directions. However, only the reflection and no diffraction take place when the angle of incidence becomes a low grazing limit. On the other hand, the scattering occurs even at such a limit, if the periodic surface is finite in extent. To solve such contradiction, this paper deals with the scattering from a perfectly conductive sinusoidal surface with finite extent. By the undersampling approximation introduced previously, the total scattering cross section is numerically calculated against the angle of incidence for several corrugation widths up to more than 104 times of wavelength. It is then found that the total scattering cross section is linearly proportional to the corrugation width in general. But an exception takes place at a low grazing limit of incidence, where the total scattering cross section increases almost proportional to the square root of the corrugation width. This suggests that, when the corrugation width goes to infinity, the total scattering cross section diverges and the total scattering cross section per unit surface vanishes at a low grazing limit of incidence. Then, it is concluded that, at a low grazing limit of incidence, no diffraction takes place by a periodic surface with infinite extent and the scattering occurs from a periodic surface with finite extent.

This paper deals with the scattering of a transverse magnetic (TM) plane wave from a perfectly conductive sinusoidal surface with finite extent. By use of the undersampling approximation and a rectangular pulse approximation, an asymptotic formula for the total scattering cross section at a low grazing limit of incident angle is obtained explicitly under conditions such that the surface is small in roughness and slope, and the corrugation width is sufficiently large. The formula shows that the total scattering cross section is proportional to the square root of the corrugation width but does not depend on the surface period and surface roughness. When the corrugation width is not large, however, the scattered wave can be obtained by a single scattering approximation, which gives the total scattering cross section proportional to the corrugation width and the Rayleigh slope parameter. From the asymptotic formula and the single scattering solution, a transition point is defined explicitly. By comparison with numerical results, it is concluded that the asymptotic formula is fairly accurate when the corrugation width is much larger than the transition point.

This paper deals with the scattering of transverse magnetic (TM) plane wave by a perfectly conductive surface made up of a periodic array of finite number of rectangular grooves. By the modal expansion method, the total scattering cross section pc is numerically calculated for several different numbers of grooves. It is then found that, when the groove depth is less than wavelenght, the total scattering cross section pc increases linearly proportional to the corrugation width W. But an exception takes place at a low grazing angle of incidence, where pc is proportional to Wα and the exponent α is less than 1. From these facts, it is concluded that the total scattering cross section pc must diverge but pc/W the total scattering cross section per unit surface must vanish at a low grazing limit when the number of grooves goes to infinity.

This paper deals with the scattering of a TM plane wave from a perfectly conductive sinusoidal surface with finite extent. For comparison, however, we briefly discuss the diffraction by the sinusoidal surface with infinite extent, where we use the concept of the total diffraction cross section per unit surface introduced previously. To solve a case where the sinusoidal corrugation width is much wider than wave length, we propose an undersampling approximation as a new numerical technique. For a small rough case, the total scattering cross section is calculated against the angle of incidence for several different corrugation widths. Then we find remarkable results, which are roughly summarized as follows. When the angle of incidence is apparently different from critical angles and diffraction beams are all scattered into non-grazing directions, the total scattering cross section increases proportional to the corrugation width and hence the total scattering cross section per unit surface (the ratio of the total scattering cross section to the corrugation width) becomes almost constant, which is nearly equal to the total diffraction cross section per unit surface in case of the sinusoidal surface with infinite extent. When the angle of incidence is critical and one of the diffraction beams is scattered into a grazing direction, the total scattering cross section per unit surface strongly depends on the corrugation width and approximately approaches to the total diffraction cross section per unit surface as the corrugation width gets wide.