This paper proposes a further improved technique on the stochastic functional approach for randomly rough surface scattering. The original improved technique has been established in the previous paper [Waves in Random and Complex Media, vol.19, no.2, pp.181-215, 2009] as a novel numerical-analytical method for a Wiener analysis. By deriving modified hierarchy equations based on the diagonal approximation solution of random wavefields for a TM plane wave incidence or even for a TE plane wave incidence under large roughness, large slope or low grazing incidence, such a further improved technique can provide a large reduction of required computational resources, in comparison with the original improved technique. This paper shows that numerical solutions satisfy the optical theorem with very good accuracy, by using small computational resources.

This paper deals with the singular behavior of the diffraction of transverse magnetic (TM) waves by a perfectly conductive triangular periodic surface at a low grazing limit of incidence. The wave field above the highest excursion of the surface is represented as a sum of Floquet modes with modified diffraction amplitudes, whereas the wave field inside a triangular groove is written as a sum of guided modes with unknown mode amplitudes. Then, two sets of equations are derived for such amplitudes. From the equation sets, all the amplitudes are analytically shown to vanish at a low grazing limit of incidence. From this fact, it is concluded analytically that no diffraction takes place and only reflection occurs at a low grazing limit of incidence for any period length and any triangle height. This theoretical result is verified by a numerical example.

This paper proposes a novel image integral equation of the first type (IIE-1) for a TE plane wave scattering from periodic rough surfaces with perfect conductivity by means of the method of image Green's function. Since such an IIE-1 is valid for any incident wavenumbers including the critical wavenumbers, the analytical properties of the scattered wavefield can be generally and rigorously discussed. This paper firstly points out that the branch point singularity of the bare propagator inevitably appears on the incident wavenumber characteristics of the scattered wavefield and its related quantities just at the critical wavenumbers. By applying a quadrature method, the IIE-1 becomes a matrix equation to be numerically solved. For a periodic rough surface, several properties of the scattering are shown in figures as functions of the incident wavenumbers. It is then confirmed that the branch point singularity clearly appears in the numerical solution. Moreover, it is shown that the proposed IIE-1 gives a numerical solution satisfying sufficiently the optical theorem even for the critical wavenumbers.

This paper deals with the scattering of a cylindrical wave by a perfectly conductive periodic surface. This problem is equivalent to finding the Green's function G(x,z|xs,zs), where (x,z) and (xs,zs) are the observation and radiation source positions above the periodic surface, respectively. It is widely known that the Green's function satisfies the reciprocity: G(x,z|xs,zs)=G(xs,zs|x,z), where G(xs,zs|x,z) is named the reversal Green's function in this paper. So far, there is no numerical method to synthesize the Green's function with the reciprocal property in the grating theory. By combining the shadow theory, the reciprocity theorem for scattering factors and the average filter introduced previously, this paper gives a new numerical method to synthesize the Green's function with reciprocal property. The reciprocity means that any properties of the Green's function can be obtained from the reversal Green's function. Taking this fact, this paper obtains several new formulae on the radiation and scattering from the reversal Green's function, such as a spectral representation of the Green's function, an asymptotic expression of the Green's function in the far region, the angular distribution of radiation power, the total power of radiation and the relative error of power balance. These formulae are simple and easy to use. Numerical examples are given for a very rough periodic surface. Several properties of the radiation and scattering are calculated for a transverse magnetic (TM) case and illustrated in figures.

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.

This paper newly proposes a fast computation technique on the method of image Green's function for p-characteristic calculations, when a plane wave with the transverse wavenumber p is incident on a periodic rough surface having perfect conductivity. In the computation of p-characteristics, based on a spectral domain periodicity of the periodic image Green's function, the image integral equation for a given incidence p maintains the same form for other particular incidences except for the excitation term. By means of a quadrature method, such image integral equations lead to matrix equations. Once the first given matrix equation is performed by a solution procedure as calculations of its matrix elements and its inverse matrix, the other matrix equations for other particular incidences no longer need such a solution procedure. Thus, the total CPU time for the computation of p-characteristics is largely reduced in complex shaped surface cases, huge roughness cases or large period cases.

This paper studies scattering and diffraction of a TE plane wave from a periodic surface with semi-infinite extent. By use of a combination of the Wiener-Hopf technique and a perturbation method, a concrete representation of the wavefield is explicitly obtained in terms of a sum of two types of Fourier integrals. It is then found that effects of surface roughness mainly appear on the illuminated side, but weakly on the shadow side. Moreover, ripples on the angular distribution of the first-order scattering in the shadow side are newly found as interference between a cylindrical wave radiated from the edge and an inhomogeneous plane wave supported by the periodic surface.

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.

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 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 studies reflection and transmission of a TE plane wave from a two-dimensional random slab with statistically anisotropic fluctuation by means of the stochastic functional approach. By starting with a representation of the random wavefield presented in the previous paper [IEICE Trans. Electron., vol.E92-C, no.1, pp.77-84, Jan. 2009], a solution algorithm of the multiple renormalized mass operator is newly shown even for anisotropic fluctuation. The multiple renormalized mass operator, the first-order incoherent scattering cross section and the optical theorem are numerically calculated and illustrated in figures. The relation between statistical properties and anisotropic fluctuation is discussed.

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 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.

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.

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.