This paper deals with the diffraction of a monochromatic plane wave by a periodic grating. We discuss a problem how to obtain a numerical diffraction efficiency (NDE) satisfying the reciprocity theorem for diffraction efficiencies, because diffraction efficiencies are the subject of the diffraction theories. First, this paper introduces a new formula that decomposes an NDE into two components: the even component and the odd one. The former satisfies the reciprocity theorem for diffraction efficiencies, but the latter does not. Therefore, the even component of an NDE becomes an answer to our problem. On the other hand, the odd component of an NDE represents an unwanted error. Using such the decomposition formula, we then obtain another new formula that decomposes the conventional energy error into two components. One is the energy error made by even components of NDE's. The other is the energy error constructed by unwanted odd ones and it may be used as a reciprocity criterion of a numerical solution. This decomposition formula shows a drawback of the conventional energy balance. The total energy error is newly introduced as a more strict condition for a desirable solution. We point out theoretically that the reciprocal wave solution, an approximate solution satisfying the reciprocity for wave fields, gives another solution to our problem. Numerical examples are given for the diffraction of a TM plane wave by a very rough periodic surface with perfect conductivity. In the case of a numerical solution by the image integral equation of the second kind, we found that the energy error is much reduced by use of the even component of an NDE as an approximate diffraction efficiency or by use of a reciprocal wave solution.

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.

When a monochromatic electromagnetic plane wave is incident on an infinitely extending surface with the translation invariance property, a curious phenomenon often takes place at a low grazing angle of incidence, at which the total wave field vanishes and a dark shadow appears. This paper looks for physical and mathematical reasons why such a shadow occurs. Three cases are considered: wave reflection by a flat interface between two media, diffraction by a periodic surface, and scattering from a homogeneous random surface. Then, it is found that, when a translation invariant surface does not support guided waves (eigen functions) propagating with real propagation constants, such the shadow always takes place, because the primary excitation disappears at a low grazing angle of incidence. At the same time, a shadow form of solution is proposed. Further, several open problems are given for future works.

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.

This paper deals with the scattering and diffraction of a plane wave by a randomly rough half-plane by three tools: the small perturbation method, the Wiener-Hopf technique and a group theoretic consideration based on the shift-invariance of a homogeneous random surface. For a slightly rough case, the scattered wavefield is obtained up to the second-order perturbation with respect to the small roughness parameter and represented by a sum of the Fresnel integrals with complex arguments, integrals along the steepest descent path and branch-cut integrals, which are evaluated numerically. For a Gaussian roughness spectrum, intensities of the coherent and incoherent waves are calculated in the region near the edge and illustrated in figures, in terms of which several characteristics of scattering and diffraction are discussed.

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 a new formulation for the diffraction of a plane wave by a periodic grating. As a simple example, the diffraction of a transverse magnetic wave by a perfectly conductive periodic array of rectangular grooves is discussed. On the basis of a shadow hypothesis such that no diffraction takes place and only the reflection occurs with the reflection coefficient -1 at a low grazing limit of incident angle, this paper proposes the scattering factor as a new concept. In terms of the scattering factor, several new formulas on the diffraction amplitude, the diffraction efficiency and the optical theorem are obtained. It is newly found that the scattering factor is an even function due to the reciprocity. The diffraction efficiency is defined for a propagating incident wave as well as an evanescent incident wave. Then, it is theoretically found that the 0th order diffraction efficiency becomes unity and any other order diffraction efficiencies vanish when a real angle of incidence becomes low grazing. Numerical examples of the scattering factor and diffraction efficiency are illustrated in figures.

This paper deals with a probabilistic formulation of the diffraction and scattering of a plane wave from a periodic surface randomly deformed by a binary sequence. The scattered wave is shown to have a stochastic Floquet's form, that is a product of a periodic stationary random function and an exponential phase factor. Such a periodic stationary random function is then represented in terms of a harmonic series representation similar to Fourier series, where `Fourier coefficients' are mutually correlated stationary processes rather than constants. The mutually correlated stationary processes are written by binary orthogonal functionals with unknown binary kernels. When the surface deformations are small compared with wavelength, an approximate solution is obtained for low-order binary kernels, from which the scattering cross section, coherently diffracted power and the optical theorem are numerically calculated and are illustrated in figures.

This paper studies the scattering of a TM plane wave from a perfectly conductive sinusoidal surface with finite extent by the small perturbation method. We obtain the first and second order perturbed solutions explicitly, in terms of which the differential scattering cross section and the total scattering cross section per unit surface are calculated and are illustrated in figures. By comparison with results by a numerical method, it is concluded that the perturbed solution is reasonable even for a critical angle of incidence if the surface is small in roughness and gentle in slope and if the corrugation width is less than certain value. A brief discussion is given on multiple scattering effects.

This paper proposes a second order auto-regressive equation with discrete-valued random coefficients. The auto-regressive equation transforms an independent stochastic sequence into a binary sequence, which is a special case of a stationary Markov chain. The power spectrum, correlation function and the transition probability are explicitly obtained in terms of the random coefficients. Some computer results are illustrated in figures.

The Communications Research Laboratory (CRL), the National Astronomical Observatory (NAO), the Institute of Space and Astronoutical Science (ISAS), and the Telecommunication Network Laboratory Group of Nippon Telegraph and Telephone Corporation (NTT) have developed a very-long-baseline-connected-interferometry array, maximum baseline-length was 208 km, using a high-speed asynchronous transfer mode (ATM) network with an AAL1 that corresponds to the constant bit-rate protocol. The very long baseline interferometry (VLBI) observed data is transmitted through a 2.488-Gbps [STM-16/OC-48] ATM network instead of being recorded onto magnetic tape. By combining antennas via a high-speed ATM network, a highly-sensitive virtual (radio) telescope system was realized. The system was composed of two real-time VLBI networks: the Key-Stone-Project (KSP) network of CRL (which is used for measuring crustal deformation in the Tokyo metropolitan area), and the OLIVE (optically linked VLBI experiment) network of NAO and ISAS which is used for astronomy (space-VLBI). These networks operated in cooperation with NTT. In order to realize a virtual telescope, the acquired VLBI data were corrected via the ATM networks and were synthesized using the VLBI technique. The cross-correlation processing and data observation were done simultaneously in this system and radio flares on the weak radio source (HR1099) were detected.

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.

In the theory of periodic gratings, there is no method to make up a numerical solution that satisfies the reciprocity so far. On the basis of the shadow theory, however, this paper proposes a new method to obtain a numerical solution that satisfies the reciprocity. The shadow thoery states that, by the reciprocity, the $m$th order scattering factor is an even function with respect to a symmetrical axis depending on the order $m$ of diffraction. However, a scattering factor obtained numerically becomes an even function only approximately, but not accurately. It can be decomposed to even and odd components, where an odd component represents an error with respect to the reciprocity and can be removed by the average filter. Using even components, a numerical solution that satisfies the reciprocity is obtained. Numerical examples are given for the diffraction of a transverse magnetic (TM) plane wave by a very rough periodic surface with perfect conductivity. It is then found that, by use of the average filter, the energy error is much reduced in some case.

This paper deals with the wave scattering from a periodic surface with finite extent. Modifying a spectral formalism, we find that the spectral amplitude of the scattered wave can be determined by the surface field on only the corrugated part of the surface. The surface field on such a corrugated part is then expanded into Fourier series with unknown Fourier coefficients. A matrix equation for the Fourier coefficients is obtained and is solved numerically for a sinusoidally corrugated surface. Then, the angular distribution of the scattering, the relative power of each diffraction beam and the optical theorem are calculated and illustrated in figures. Also, the relative powers of diffraction are calculated against the angle of incidence for a periodic surface with infinite extent. By comparing a finite periodic case with an infinite periodic case, it is pointed out that relative powers of diffraction beam are much similar in these of diffraction for the infinite periodic case.

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.

The diffraction of a transverse magnetic (TM) plane wave by a perfectly conductive surface made up of a periodic array of rectangular grooves is studied by the modal expansion method. It is found theoretically that the reflection coefficient approaches -1 but no diffraction takes place when the angle of incidence reaches a low grazing limit. Such singular behavior is shown analytically to hold for any finite values of the period, groove depth and groove width and is then demonstrated by numerical examples.

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.

In the theory of diffraction gratings, the conventional integral method is considered as a powerful tool of numerical analysis. But it fails to work at a critical angle of incidence, because a periodic Green's function (integral kernel) diverges. This problem was resolved by the image integral equation in a previous paper. Newly introducing the reflection extinction theorem, this paper derives the image extinction theorem and the image integral equation. Then, it is concluded that the image integral equation is made up of two physical processes: the image surface radiates a reflected plane wave, whereas the periodic surface radiates the diffracted wave.

This paper deals with reflection and transmission of a TE plane wave from a one-dimensional random slab with slanted fluctuation by means of the stochastic functional approach. By starting with a generalized representation of the random wavefield from a two-dimensional random slab, and by using a manner for slanted anisotropic fluctuation, the corresponding random wavefield representation and its statistical quantities for one-dimensional cases are newly derived. The first-order incoherent scattering cross section is numerically calculated and illustrated in figures.