## Abstract

The phenomenon of diffraction by a rectangular hole in a thick conducting screen is investigated for various structural parameters (aperture sizes, aspect ratios, and screen thicknesses) and some incident angles by making use of the exact solution based on the Kobayashi potential (KP) when an electromagnetic (EM) plane wave with any polarization is impinged on the aperture. Since the KP method yields an eigenfunction expansion of the present geometry, the solution satisfies the proper edge condition as well as all the boundary conditions, and therefore we can obtain highly accurate and fast-convergent results. Many numerical results, which are useful for scientists and engineers, are provided for various physical quantities, such as the far-field diffracted pattern, transmission coefficients (normalized transmitted power), and aperture electric field distributions, and by using these numerical results, we examine the convergent property of the KP solution and discuss the effect of the hole size and shape, screen thickness, and incident polarization on the transmission property of the rectangular hole.

### Keywords

- electromagnetic wave
- exact solution
- rectangular hole
- thick screen
- diffraction
- transmission
- aperture field

## 1. Introduction

A rectangular hole in a conducting screen with nonzero thickness is a fundamental structure and its diffraction property of an electromagnetic (EM) wave is of great importance in many fields of engineering and physics. For example, the knowledge of EM wave transmission through and radiation from the rectangular aperture with a finite thickness helps to design an aperture-type antenna like a slot antenna and to solve the problems of electromagnetic compatibility (EMC)/electromagnetic interference (EMI). The first theoretical work on this problem (three-dimensional problem) was made by Jin and Volakis by combining the finite element and the boundary integral methods [1]. From then, this problem has been solved by a variety of methods, such as the Fourier transform and mode-matching technique [2], a technique based on a perturbation method and modified Green’s functions [3] and so on. In the field of physics, the extraordinary optical transmission (EOT) phenomena for subwavelength apertures have attracted considerable attention, and the study of the optical properties of holes in metallic films has become extremely active [4]. Garcia-Vidal et al. have applied the method based on the mode-expansion and mode-matching technique with some approximations to examine the transmission property of a single rectangular hole in a screen made of a perfect electric conductor (PEC) [5] and of a real metal (with surface impedance) [6]. A similar modal method (but without any approximation in the formulation) was used by Brok and Urbach to calculate the transmission through holes [7]. Many methods have thus been used to solve the EM wave diffraction by a rectangular hole in a thick screen, but all the methods mentioned above do not consider the field singularity at aperture edges in the formulation (e.g., EM fields near the edge of a deep hole are considered to behave like those around the right-angled wedge). It is well known that the incorporation of the proper edge condition [8] into the field is effective in obtaining a highly accurate and faster convergent solution, and the method that can take into account the edge property is required to obtain the exact solution. The method of the Kobayashi potential (KP) [9] is a rigorous technique for solving mixed boundary value problems and has been successfully applied to many three-dimensional wave scattering and radiation problems to obtain exact solutions [10, 11, 12, 13, 14]. The KP method uses the discontinuous properties of the Weber-Schafheitlin (WS) integrals to satisfy a part of the boundary conditions, and, at this step, the required edge condition can be incorporated into the solution. Serizawa and Hongo applied the KP method to the problem of diffraction of an EM plane wave by a rectangular hole in a PEC screen with a finite thickness and derived the exact solution of the diffracted wave that satisfy the proper edge condition as well as all the boundary conditions [15]. By using the derived KP solution, the physical quantities such as the transmission coefficient, far diffracted fields, and power flow around the hole have been calculated for small and large apertures [16, 17, 18], but the numerical results for apertures greater than the half-wavelength in [16] contain errors because of the mistake in the calculation code of double infinite series.

In this chapter, using the KP solution, we provide many numerical results of various physical quantities useful for scientists and engineers, and we investigate the convergent property of the KP solution and the transmission property of the rectangular hole in the thick conducting screen for various structural parameters (aperture sizes, aspect ratios, and screen thicknesses) and some incident angles (all the numerical results in this chapter are newly calculated by using the improved calculation code that can give more accurate results than the previous code).

## 2. Formulation

Consider an EM plane wave diffracted by a

where

In Figure 1,

### 2.1 Field expression in each region

To solve the problem, we split the space into three regions I, II, and WG, denoting the upper and lower half-spaces, and waveguide region, respectively. Needless to say,

The fields in region WG are represented by a linear combination of the TE- and TM-modes, and the axial components of the electric and magnetic vector potentials are given by

where

For the diffracted waves in the half-spaces, we use the

where

### 2.2 Matrix equation

We enforce the remaining boundary conditions that the tangential EM fields are continuous on the aperture and the resultant equations are projected into functional spaces of trigonometric functions and Gegenbauer polynomials by using their orthogonality (for details, see [15]). After some lengthy analysis, we have matrix equations for the expansion coefficients:

Here,

In (12),

and

where

In Eqs. (19)–(26), parameters

### 2.3 Physical quantities

The expression of the far-field is obtained by applying the stationary phase method to (9) and (10), and we have

In the above expressions,

The expression of the aperture electric field is obtained by differentiating (9) and (10) with respect to

where

## 3. Numerical results and discussion

To obtain the numerical results for the physical quantities of interest, the matrix Eq. (12) must be solved. The matrix elements consist of double infinite integrals and double infinite series that converge rather slowly, and we can compute them with the desired accuracy by applying the Hongo method [10]. For the double infinite integrals, the full range of integration that is equal to the first quadrant of the

### 3.1 Convergence of the KP solution

We first calculated the transmission coefficient for various aperture sizes, aspect ratios, screen thicknesses, and incident angles. Figure 2 shows examples of the transmission coefficient

The convergent property of the KP solution was also examined for other physical quantities. Figures 3–5 show the far-field patterns of three kinds of screen thicknesses for various sizes of square apertures (

### 3.2 Transmission properties

We next investigate the effect of the hole size and shape, screen thickness, and polarization on the transmission properties. Some results of the transmission coefficient are compared with those by other methods (Jin and Volakis [1], Brok and Urbach [7]). Unless otherwise noted, the following computations are made for

where

The transmission coefficient is also calculated as a function of the normalized thickness

Finally, we calculate the transmission coefficient

## 4. Conclusions

We have rigorously studied the diffraction of an electromagnetic plane wave by a rectangular hole in a perfectly conducting screen with a finite thickness by applying the KP method. The transmission coefficient, far-field pattern, and aperture electric field distribution were calculated to show the convergence of the solution. The numerical results of the transmission coefficient were also presented as functions of the normalized thickness

## Acknowledgments

This work was partly supported by JSPS KAKENHI Grant Number 25390159 and 17K05150.