Open access peer-reviewed chapter - ONLINE FIRST

Diffraction by a Rectangular Hole in a Thick Conducting Screen

By Hirohide Serizawa

Submitted: April 10th 2019Reviewed: August 6th 2019Published: October 10th 2019

DOI: 10.5772/intechopen.89029

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 2a×2bsize rectangular hole in an infinitely large conducting screen with a finite thickness das shown in Figure 1. The screen is made of a PEC, and the center of the upper aperture is chosen as the origin Oof the Cartesian coordinates xyz. The space is filled with an isotropic homogeneous medium with parameters ϵμ. Through this chapter, a harmonic time dependence expjωtis assumed and omitted from the equations. The expression of the incident wave (the angles of incidence are θ0and ϕ0) is given by

Einc=E1iϕ+E2iθexpjkΦincrE1
Hinc=YE1iθE2iϕexpjkΦincrE2

where k=ωϵμ, Y=ϵ/μ, and

iθ=cosθ0cosϕ0ix+cosθ0sinϕ0iysinθ0izE3
iϕ=sinϕ0ix+cosϕ0iyE4
Φincr=xsinθ0cosϕ0+ysinθ0sinϕ0+zcosθ0.E5

In Figure 1, Erefis the reflected wave when the plane z=0is occupied by an infinite conducting screen.

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, Ed2is the transmitted wave.

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

Fz+Fz=aϵm=0n=0mn00AmnEexpjhmnzaBmnEexpjhmnzacos2ξ+1cos2η+1E6
Az+Az=κ2ωm=1n=1AmnMexpjhmnzaBmnMexpjhmnzasin2ξ+1sin2η+1E7
hmn=κ2/22p2/22,κ=ka,p=a/b=1/qE8

where ξ=x/a, η=y/b, and za=z/aare the normalized coordinates.

For the diffracted waves in the half-spaces, we use the x- and y-components of the electric vector potential F(the use of the tangential components to the aperture is indispensable to incorporate the edge condition correctly), and they are given by Fourier spectral representations. The condition that the tangential electric field vanishes on the conducting screen can be satisfied by using the WS integrals [15], and we have

Fxdi=1iaϵm=0n=000Λ2mσαcosαξAmnxiΛ2nτβcosβη+BmnxiΛ2n+1τβsinβη+Λ2m+1σαsinαξCmnxiΛ2nτβcosβη+DmnxiΛ2n+1τβsinβηexp1iζ(αβ)zaiζαβdαdβE9
Fydi=1iaϵm=0n=000Λ2mταcosαξAmnyiΛ2nσβcosβη+BmnyiΛ2n+1σβsinβη+Λ2m+1ταsinαξCmnyiΛ2nσβcosβη+DmnyiΛ2n+1σβsinβηexp1iζ(αβ)zaiζαβdαdβE10
ζαβ=α2+p2β2κ2,za1=za,za2=za+da,da=d/aE11

where Λνx=J+νx/xνand Jnxis the Bessel function of order n. The index icorresponds to the region number (i=1for region I and i=2for region II). The expressions of (9) and (10) are the Kobayashi potentials for the present problem. Parameters σand τare selected so as to incorporate the correct edge property into the electric field [8], and the correct values for the right-angled wedge, which are seemed to be valid for a deep hole, are σ=7/6and τ=1/6=σ1. AmnxiDmnxiand AmnyiDmnyiare determined from the remaining boundary conditions on the aperture.

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:

KAmnstuv+SAmnst±uv1u+vpGAmnstuv+TAmnst±uv1u+vqGBmnstu¯v¯+TBmnst±u¯v¯KBmnstu¯v¯+SBmnst±u¯v¯Xmn±uvYmn±u¯v¯=2jju+vPxΛ2s+uτκaΛ2t+vσκbju¯+v¯q2PyΛ2s+u¯σκaΛ2t+v¯τκb,s=0,1,2,t=0,1,2,E12
uv=00,01,10,11,u¯=1u,v¯=1v.E13

Here, Xmn±uv, Ymn±uv, Px, Py, κa, and κbare defined as follows:

Xmn±uv=Xmn1uv±Xmn2uv,Ymn±uv=Ymn1uv±Ymn2uvE14
Xmni00=Amnxi,Xmni01=Bmnxi,Xmni10=Cmnxi,Xmni11=DmnxiE15
Ymni00=Amnyi,Ymni01=Bmnyi,Ymni10=Cmnyi,Ymni11=DmnyiE16
Px=κE1cosθ0cosϕ0+E2sinϕ0,κa=κsinθ0cosϕ0E17
Py=κE1cosθ0sinϕ0E2cosϕ0,κb=sinθ0sinϕ0.E18

In (12), KA,Band GA,Bare the double infinite integrals of four Bessel functions, and they are given by

KAmnstuv=00κ2α2ζαβΛ2m+uσαΛ2s+uταΛ2n+vτβΛ2t+vσβdαdβE19
KBmnstuv=00q2κ2β2ζαβΛ2m+uταΛ2s+uσαΛ2n+vσβΛ2t+vτβdαdβE20
GAmnstuv=00αβζαβΛ2m+1uταΛ2s+uταΛ2n+1vσβΛ2t+vσβdαdβE21
GBmnstuv=00αβζαβΛ2m+1uσαΛ2s+uσαΛ2n+1vτβΛ2t+vτβdαdβE22

and SA,Band TA,Bare the double infinite series of four Bessel functions and they are given by

SAmnst±uv=π2m=0n=0Γ2m+1+u,2n+v±1+δ0,2n+vκ22m+1+u2π2Λ2m+uσ2m+1+u2π×Λ2s+uτ2m+1+u2πΛ2n+vτ2n+v2πΛ2t+vσ2n+v2πE23
SBmnst±uv=π2m=0n=0Γ2m+u,2n+1+v±1+δ0,2m+uq2κ22n+1+v2π2Λ2m+uτ2m+u2π×Λ2s+uσ2m+u2πΛ2n+vσ2n+1+v2πΛ2t+vτ2n+1+v2πE24
TAmnst±uv=π2m=0n=0Γ2m+1+u,2n+v±2m+1+u2π2n+v2πΛ2m+1uτ2m+1+u2π×Λ2s+uτ2m+1+u2πΛ2n+1vσ2n+v2πΛ2t+vσ2n+v2πE25
TBmnst±uv=π2m=0n=0Γ2m+u,2n+1+v±2m+u2π2n+1+v2πΛ2m+1uσ2m+u2π×Λ2s+uσ2m+u2πΛ2n+1vτ2n+1+v2πΛ2t+vτ2n+1+v2πE26

where δmnis the Kronecker delta and

Γmn±=1γmn1expγmnda1±expγmnda,γmn=22+p222κ2=jhmn.E27

In Eqs. (19)(26), parameters σand τare determined by considering the edge property of the magnetic field [8]. When the hole is deep, we can use the edge condition of the right-angled wedge, and for this case the values of σ=1/6=τand τ=5/6=σ+1can be selected.

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

Fxdi=1iπqa2ϵ2expjkrrm=0n=0Λ2mσκxAmnxiΛ2nτκy+jBmnxi×Λ2n+1τκy+Λ2m+1σκxjCmnxiΛ2nτκyDmnxiΛ2n+1τκyE28
Fydi=1iπqa2ϵ2expjkrrm=0n=0Λ2mτκxAmnyiΛ2nσκy+jBmnyi×Λ2n+1σκy+Λ2m+1τκxjCmnyiΛ2nσκyDmnyiΛ2n+1σκyE29
κx=κsinθcosϕ,κy=sinθsinϕ.E30

In the above expressions, θand ϕare the spherical coordinate angles (the angles of diffraction) shown in Figure 1, and ris the distance from the center point of each aperture to the observation point, that is, r=x2+y2+z2for region I and r=x2+y2+z+d2for region II (the origin of the spherical coordinate system for region II is selected at the center of the lower aperture). Far electromagnetic fields are obtained from E=ϵ1×Fand H=Yir×E(iris a unit vector of a radial direction). The transmission coefficient Tis defined by the ratio of the total radiated power WTinto the z<ddark space to the power of the incident plane wave Wion the aperture; T=WT/Wi. WTcan be obtained by integrating the radiated power over the lower hemisphere, and we use the Gauss-Legendre quadrature in practical computation. Wiis analytically calculated, and the result is Wi=4abYcosθ0for Einc=1. For oblique incidence, Tcosθ0, which is the same as the transmission coefficient normalized by the normal incident power Wiθ0=0, is useful to evaluate the actual transmitted power into region II.

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

Exdi=ΓσΓτ22σ+τ1ξ2τ121η2σ12m=0n=01m+nΓ2m+1Γ2m+2τC2mτξ×AmnyiΓ2n+1Γ2n+2σC2nση+BmnyiΓ2n+2Γ2n+2σ+1C2n+1ση+Γ2m+2Γ2m+2τ+1×C2m+1τξCmnyiΓ2n+1Γ2n+2σC2nση+DmnyiΓ2n+2Γ2n+2σ+1C2n+1σηE31
Eydi=ΓσΓτ22σ+τ1ξ2σ121η2τ12m=0n=01m+nΓ2m+1Γ2m+2σC2mσξ×AmnxiΓ2n+1Γ2n+2τC2nτη+BmnxiΓ2n+2Γ2n+2τ+1C2n+1τη+Γ2m+2Γ2m+2σ+1×C2m+1σξCmnxiΓ2n+1Γ2n+2τC2nτη+DmnxiΓ2n+2Γ2n+2τ+1C2n+1τηE32

where Cαxare the Gegenbauer polynomials and Γxis the gamma function.

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 αspace is divided into several subdomains. The infinite integrals over the subdomains outside a very large radius are transformed into more simple forms by applying asymptotic expansions of the Bessel functions (the integrands of the infinite integrals are represented by some elementary functions), and the infinite integrals of the elementary functions are analytically carried out. The finite integrals of the Bessel functions are numerically performed by applying the Gauss-Legendre quadrature scheme (examples of the treatment are shown in [10, 11, 12]). For the double infinite series, the asymptotic expansions are applied to the summands (details are discussed in [13, 14, 15]). Once the expansion coefficients are determined by solving the matrix equations, we can readily obtain the physical quantities from their concise expressions derived in the previous section. Unless otherwise noted, the edge condition of the right-angled wedge (σ=7/6, τ=1/6, σ=1/6, and τ=5/6) is used, and Einc=1is selected in all of the following calculations. In practical computation, the matrix size is truncated to 2nk2×2nk2, where nk1is the maximum value of indices m, n, s, and tin (12). By changing the value of nk, we can numerically verify the convergence of the solution.

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 Tas a function of the nkvalue for square holes of 2a=λ/2(ka=π/2), 2a=λ(ka=π), and 2a=3λ(ka=3π) (λis the wavelength) at θ0=0, 45 degrees (ϕ0=0). The subscripts Eand Hdenote the E-polarization (E1=1, E2=0) and H-polarization (E1=0,E2=1), respectively. For the normal incidence on the square hole, THis the same as TE. From the figure, we see that the convergence is very rapid, and, for example, the results of d/a=0.5at θ0=ϕ0=0degrees have fully converged to four decimal places when nk=4for ka=π/21010, nk=5for ka=π1010, and nk=7for ka=3π1010. By the way, the summands of the series of (23)-(26) have singular points at γmn=0that arise for ka=π/2(2a=λ/2), (=1,2,3,). We can remove this singularity appropriately in the numerical calculation, but we here avoided the singularity by slightly shifting the kavalue by ±1010. The results for ka=π/2±1010have completely agreed for eight decimal places, and similar situations had held for other aperture sizes related with the singular points. Thus, this treatment is also applied for calculating the other physical quantities.

The convergent property of the KP solution was also examined for other physical quantities. Figures 35 show the far-field patterns of three kinds of screen thicknesses for various sizes of square apertures (ka=π/21010(small), 2π1010, and 5π1010(very large)) at the normal incidence. The patterns are computed with Pθ=limr4πr2Edi2/Einc2, and they are given for different nkvalues. It can be seen from the figure that the plotted lines converge with relatively small value of nk(nk=2for ka=π/21010, nk=4for ka=2π1010, and nk=9for ka=5π1010). Figures 6 and 7 show the amplitude distributions of aperture electric field Ey(the co-polarized component) for thin (d/a=0.001) and thick (d/a=0.5) screen cases, respectively. The rate of convergence of thick case is very fast but is very slow for thin case, and we think this is caused by using the edge condition of the right-angled wedge (in spite of the use of the right-angled wedge condition, the convergence of the far-field is very fast as shown in Figure 3. This is due to the fact that the far-field has a stationary nature, that is, it changes little for small changes in field distribution). We also calculate Eyfor thin screen case by applying the edge condition of zero-thickness plate, and the results corresponding to Figure 6 are given in Figure 8. Needless to say, the rate of convergence of the aperture field under the zero-thickness condition is much faster than that of the right-angled wedge condition, especially as the aperture is smaller.

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 nk=6. Figure 9 shows the transmission coefficient TEfor square (b/a=1) and rectangular (b/a=0.2) holes less than the half-wavelength as a function of the normalized thickness d/a(in this case, all the waveguide modes become evanescent modes). It can be seen from the figure that the transmitted power rapidly decreases as the screen’s thickness becomes large except for ka=π/21010, and this is due to the attenuation characteristic of the evanescent wave. However, rather strong transmission is observed for the elongated aperture (b/a=0.2) when the value of d/ais small in spite of the fact that there is no propagating wave in the hole. This is considered to be closely related with the field singularity at long edges in addition to the small attenuation of the evanescent modes. By fitting asymptotic lines to TEcurves in Figure 9 for the part where the value of d/ais large, we found an approximate expression for predicting the transmitted power, and it is given by

TE=T0exp2daπ22ka2E33

where T0is a coefficient depending on kaand is shown in Figure 10.

The transmission coefficient is also calculated as a function of the normalized thickness d/afor various aperture sizes around the half-wavelength (kaπ/2), and the results are shown in Figures 11 and 12 (our results completely agree with those by Brok and Urbach [7]). It can be seen from the figures that the TEcurves for apertures larger than the half-wavelength have oscillation, and this is due to the resonance of the propagating waveguide modes. The results of the total transmitted power normalized by Wiθ0=0are shown in Figures 13 and 14 for various aperture sizes and shapes, and the curves of the transmitted power of the large aperture are complex (we think this is also due to the resonance of the waveguide modes).

Finally, we calculate the transmission coefficient TEas functions of the normalized aperture length 2a/λfor a constant bvalue (2b=0.1λ) and of the aspect ratio b/afor a constant hole area (2a×2b=π/42), and the results are given in Figures 15 and 16, respectively. In obtaining these results, the value of nkwas chosen to be 5. In Figure 15, all the waveguide modes for ka<π/2become evanescent waves, but the strong power transmission is observed, and the position of the peak point approaches to ka=π/2(cutoff frequency) as the screen thickness becomes large. For ka>π/2, we can find that the TEcurves have oscillations that are caused by the resonance of the lowest propagating mode in the waveguide. In Figure 16, b/a=0.25is a value related with the cutoff frequency, and all the waveguide modes become evanescent waves for b/a>0.25. The similar situations of Figure 15 can be also seen in this case.

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 d/ato show the EM transmission property of the various sized holes. It was found from the results that the convergence of the KP solution is very rapid (for thin screen case, the edge condition of the zero-thickness plate is effective) and the transmitted power is strongly affected by the aperture size and shape, screen’s thickness, incident angles, and polarization.

Acknowledgments

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

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Hirohide Serizawa (October 10th 2019). Diffraction by a Rectangular Hole in a Thick Conducting Screen [Online First], IntechOpen, DOI: 10.5772/intechopen.89029. Available from: