Volume integral equations combined with orthogonality of guided mode and non-guided field are proposed for the TE incidence of two-dimensional optical slab waveguide. The slab waveguide is assumed to satisfy the single mode condition. The formulation of the integral equations are described in detail. The matrix equation obtained by applying the method of moments to the integral equations is shown. Numerical results for step, gap, and grating waveguides are given. They are compared to published papers to validate the proposed method.

New boundary integral equations are proposed for two-port slab waveguides which satisfy single mode condition. The boundary integral equations are combined with the orthogonality of guided mode and non-guided field. They are solved by the standard boundary element method with no use of mode expansion technique. Reflection and transmission coefficients of guided mode are directly determined by the boundary element method. To validate the proposed method, step waveguides for TE wave incidence and triangular rib waveguides for TM wave incidence are investigated by numerical calculations.

A method to reconstruct the surface shape of a scatterer from the relative intensity of the scattered field is proposed. Reconstruction of the scatterer shape has been studied as an inverse problem. An approach that employs boundary-integral equations can determine the scatterer shape with low computation resources and high accuracy. In this method, the reconstruction process is performed so that the error between the measured far field of the sample and the computed far field of the estimated scatterer shape is minimized. The amplitude of the incident wave at the sample is required to compute the scattered field of the estimated shape. However, measurement of the incident wave at the sample (measurement without the sample) is inconvenient, particularly when the output power of the wave source is temporally unstable. In this study, we improve the reconstruction method with boundary-integral equations for practical use and expandability to various types of samples. First, we propose new boundary-integral equations that can reconstruct the sample shape from the relative intensity at a finite distance. The relative intensity is independent from the amplitude of the incident wave, and the reconstruction process can be performed without measuring the incident field. Second, the boundary integral equation for reconstruction is discretized with boundary elements. The boundary elements can flexibly discretize various shapes of samples, and this approach can be applied to various inverse scattering problems. In this paper, we present a few reconstruction processes in numerical simulations. Then, we discuss the reason for slow-convergence conditions and introduce a weighting coefficient to accelerate the convergence. The weighting coefficient depends on the distance between the sample and the observation points. Finally, we derive a formula to obtain an optimum weighting coefficient so that we can reconstruct the surface shape of a scatterer at various distances of the observation points.

This paper presents the method of moments based on electric field integral equation which is capable of solving three-dimensional metallic waveguide problem with no use of another method. Metals are treated as perfectly electric conductor. The integral equation is derived in detail. In order to validate the proposed method, the numerical results are compared with those in a published paper. Three types of waveguide are considered: step discontinuity waveguide, symmetrical resonant iris waveguide, and unsymmetrical resonant iris waveguide. The numerical results are also verified by the law of conservation of energy.

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 numerical investigation revealed the relation between the groove randomness of actual-size diffraction gratings and the diffraction efficiencies. The diffraction gratings we treat in this study have around 10000 grooves. When the illumination wavelength is 600 nm, the entire grating size becomes 16.2 mm. The simulation was performed using the difference-field boundary element method (DFBEM). The DFBEM treats the vectorial field with a small amount of memory resources as independent of the grating size. We firstly describe the applicability of DFBEM to a considerably large-sized structure; regularly aligned grooves and a random shallow-groove structure are calculated by DFBEM and compared with the results given by standard BEM and scalar-wave approximation, respectively. Finally we show the relation between the degree of randomness and the diffraction efficiencies for two orthogonal linear polarizations. The relation provides information for determining the tolerance of fabrication errors in the groove structure and measuring the structural randomness by acquiring the irradiance of the diffracted waves.

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.

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.

The higher-order characteristic basis function method (HO-CBFM) is clearly formulated. HO-CBFM provides results accurately even if a block division is arbitrary. The HO-CBFM combined with a volume integral equation (VIE) is used in the analysis of various antennas in the vicinity of a dielectric object. The results of the numerical analysis show that the HO-CBFM can reduce the CPU time while still achieving the desired accuracy.

The main purpose of this paper is to apply the boundary integral equation (BIE) method to the analysis of spoof localized surface plasmons (spoof LSPs) excited in a perfectly conducting cylinder with longitudinal corrugations. Frequency domain BIE schemes based on electric field integral equation (EFIE), magnetic field integral equation (MFIE) and combined field integral equation (CFIE) formulations are used to solve two-dimensional electromagnetic (EM) problems of scattering from the cylinder illuminated by a transverse electric plane wave. In this approach effects of spoof LSPs are included in the secondary surface current and charge densities resulting from the interaction between the plane wave and the cylinder. Numerical results obtained with the BIE schemes are validated by comparison with that of a recently proposed modal solution based on the metamaterial approximation.

“How to get the original ideas” is the fundamental and critical issue for the researchers in science and technology. In this paper, the author writes his experiences concerning how he could encounter the interesting and original ideas of three research subjects, i.e., the accelerating medium effect, the guided-mode extracted integral equation and the surface plasmon gap waveguide.

This paper deals with an integral equation method for analyzing the diffraction of a transverse magnetic (TM) plane wave by a perfectly conductive periodic surface. In the region below the periodic surface, the extinction theorem holds, and the total field vanishes if the field solution is determined exactly. For an approximate solution, the extinction theorem does not hold but an extinction error field appears. By use of an image Green's function, new formulae are given for the extinction error field and the mean square extinction error (MSEE), which may be useful as a validity criterion. Numerical examples are given to demonstrate that the formulae work practically even at a critical angle of incidence.

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 an integral method analyzing the diffraction of a transverse electric (TE) wave by a perfectly conductive periodic surface. The conventional integral method fails to work for a critical angle of incidence. To overcome such a drawback, this paper applies the method of image Green's function. We newly obtain an image integral equation for the basic surface current in the TE case. The integral equation is solved numerically for a very rough sinusoidal surface. Then, it is found that a reliable solution can be obtained for any real angle of incidence including a critical angle.

This paper deals with the diffraction of a transverse magnetic (TM) plane wave by a perfectly conductive periodic surface by an integral method. However, it is known that a conventional integral method does not work for a critical angle of incidence, because of divergence of a periodic Green's function (integral kernel). To overcome such a divergence difficulty, we introduce an image Green's function which is physically defined as a field radiated from an infinite phased array of dipoles. By use of the image Green's function, it is newly shown that the diffracted field is represented as a sum of radiation from the periodic surface and its image surface. Then, this paper obtains a new image integral equation for the basic surface current, which is solved numerically. A numerical result is illustrated for a very rough sinusoidal surface. Then, it is concluded that the method of image Green's function works practically even at a critical angle of incidence.

In this study, we demonstrate an acceleration of flexible generalized minimal residual algorithm (FGMRES) implemented with the method of moments and the fast multipole method (FMM), based on a combined tangential formulation. For the implementation of the FGMRES incorporated with the FMM concept, we propose a new definition of the truncation number for the FMM operator within the inner solver. The proposed truncation number provides an optimal variable preconditioner by controlling the accuracy and computational cost of the inner iteration. Moreover, to further accelerate the convergence, we introduce the concept of a multistage preconditioner. Numerical experiments reveal that our new version of FGMRES, based on the proposed truncation number for the inner solver and the multistage preconditioner, achieves outstanding acceleration of the convergence for large-scale and practical electromagnetic scattering and radiation problems with several levels of geometrical complexity.

In this paper, the performance of the induced dimension reduction (IDR) method implemented along with the method of moments (MoM) is described. The MoM is based on a combined field integral equation for solving large-scale electromagnetic scattering problems involving conducting objects. The IDR method is one of Krylov subspace methods. This method was initially developed by Peter Sonneveld in 1979; it was subsequently generalized to the IDR(s) method. The method has recently attracted considerable attention in the field of computational physics. However, the performance of the IDR(s) has hardly been studied or practiced for electromagnetic wave problems. In this study, the performance of the IDR(s) is investigated and clarified by comparing the convergence property and memory requirement of the IDR(s) with those of other representative Krylov solvers such as biconjugate gradient (BiCG) methods and generalized minimal residual algorithm (GMRES). Numerical experiments reveal that the characteristics of the IDR(s) against the parameter s strongly depend on the geometry of the problem; in a problem with a complex geometry, s should be set to an adequately small value in order to avoid the "spurious convergence" which is a problem that the IDR(s) inherently holds. As for the convergence behavior, we observe that the IDR(s) has a better convergence ability than GPBiCG and GMRES(m) in a variety of problems with different complexities. Furthermore, we also confirm the IDR(s)'s inherent advantage in terms of the memory requirements over GMRES(m).

The scattering of a plane wave from the end-face of a three-dimensional waveguide system composed of a large number of cores is treated by the volume integral equation for the electric field and the first order term of a perturbation solution for TE and TM wave incidence is analytically derived. The far scattered field does not almost depend on the polarization of an incident wave and the angle dependence is described as the Fourier transform of the incident field in the cross section of cores. To clarify the dependence of the scattering pattern on the arrangement of cores some numerical examples are shown.

This paper presents flexible inner-outer Krylov subspace methods, which are implemented using the fast multipole method (FMM) for solving scattering problems with mixed dielectric and conducting object. The flexible Krylov subspace methods refer to a class of methods that accept variable preconditioning. To obtain the maximum efficiency of the inner-outer methods, it is desirable to compute the inner iterations with the least possible effort. Hence, generally, inaccurate matrix-vector multiplication (MVM) is performed in the inner solver within a short computation time. This is realized by using a particular feature of the multipole techniques. The accuracy and computational cost of the FMM can be controlled by appropriately selecting the truncation number, which indicates the number of multipoles used to express far-field interactions. On the basis of the abovementioned fact, we construct a less-accurate but much cheaper version of the FMM by intentionally setting the truncation number to a sufficiently low value, and then use it for the computation of inaccurate MVM in the inner solver. However, there exists no definite rule for determining the suitable level of accuracy for the FMM within the inner solver. The main focus of this study is to clarify the relationship between the overall efficiency of the flexible inner-outer Krylov solver and the accuracy of the FMM within the inner solver. Numerical experiments reveal that there exits an optimal accuracy level for the FMM within the inner solver, and that a moderately accurate FMM operator serves as the optimal preconditioner.

An integral equation approach with a new solution procedure using moment method (MoM) is applied for the computation of coupled currents on the surface of a printed dipole antenna and inside its high-permittivity three-dimensional dielectric substrate. The main purpose of this study is to validate the accuracy and reliability of the previously proposed MoM procedure by authors for the solution of a coupled volume-surface integral equations system. In continuation of the recent works of authors, a mixed-domain MoM expansion using Legendre polynomial basis function and cubic geometric modeling are adopted to solve the tensor-volume integral equation. In mixed-domain MoM, a combination of entire-domain and sub-domain basis functions, including three-dimensional Legnedre polynomial basis functions with different degrees is utilized for field expansion inside dielectric substrate. In addition, the conventional Rao-Wilton-Glisson (RWG) basis function is employed for electric current expansion over the printed structure. The accuracy of the proposed approach is verified through a comparison with the MoM solutions based on the spectral domain Green's function for infinitely large substrate and the results of FDTD method.