Scattering of Electromagnetic Waves Incident Normally on Square Patch-Type FSS Placed at the Interface between Two Dielectric Media

TL;DR

This paper presents an analytical method to calculate how electromagnetic waves scatter when they hit a square patch FSS at the boundary between two dielectric media, providing accurate reflection and transmission estimates.

Contribution

It introduces a new analytical approach for computing scattering matrix elements for square patch FSS at dielectric interfaces, enhancing practical accuracy.

Findings

Analytical expressions for reflection and transmission coefficients are derived.

The method achieves sufficient accuracy for practical applications.

The approach simplifies the analysis of FSS at dielectric interfaces.

Abstract

In this paper an analytical approach for calculating scattering matrix elements for the case of normal incidence of the plane electromagnetic waves on the square patch-type Frequency Selective Surface (FSS), which is placed at the interface between two dielectric media is proposed. Analytical expressions for the reflection and transmission coefficients are shown to be accurate enough for practical purposes.

Tables

Table 1. Table 1: Frequency and wavelength dependencies of the relative deviations δ ​ S 11 𝛿 subscript 𝑆 11 \delta S_{11} , δ ​ S 21 𝛿 subscript 𝑆 21 \delta S_{21} of the reflection and transmission coefficients S 11 subscript 𝑆 11 S_{11} , S 21 subscript 𝑆 21 S_{21} calculated by the analytical expression ( 30 ) from S 11 CST superscript subscript 𝑆 11 CST S_{11}^{\text{CST}} , S 21 CST superscript subscript 𝑆 21 CST S_{21}^{\text{CST}} obtained with high accuracy by 3D electromagnetic simulation with CST MWS for D = 2 𝐷 2 D=2 mm, w = 0.8 ​ D w 0.8 𝐷 {\mathrm{w}}=0.8D , ε 1 = 1 subscript 𝜀 1 1 \varepsilon_{1}=1 , ε 2 = 3 subscript 𝜀 2 3 \varepsilon_{2}=3 ; λ 𝜆 \lambda is the wavelength in the second medium ( ε 2 = 3 subscript 𝜀 2 3 \varepsilon_{2}=3 ). Here δ ​ S i ​ j = | S i ​ j − S i ​ j CST | / | S i ​ j CST | 𝛿 subscript 𝑆 𝑖 𝑗 subscript 𝑆 𝑖 𝑗 superscript subscript 𝑆 𝑖 𝑗 CST superscript subscript 𝑆 𝑖 𝑗 CST \delta S_{ij}=|S_{ij}-S_{ij}^{\text{CST}}|/|S_{ij}^{\text{CST}}| .

Frequency [GHz]	Wavelength [mm]	$\delta S_{11}$	$\delta S_{21}$
2	86.5	$4.1\times {10}^{-4}$	$3.2\times {10}^{-5}$
4	43.3	$1.4\times {10}^{-3}$	$1.2\times {10}^{-4}$
6	28.8	$2.7\times {10}^{-3}$	$2.6\times {10}^{-4}$
8	21.6	$3.8\times {10}^{-3}$	$4.3\times {10}^{-4}$
10	17.3	$4.5\times {10}^{-3}$	$6.0\times {10}^{-4}$
12	14.4	$4.7\times {10}^{-3}$	$7.4\times {10}^{-4}$
14	12.4	$4.3\times {10}^{-3}$	$8.0\times {10}^{-4}$
16	10.8	$3.4\times {10}^{-3}$	$7.5\times {10}^{-4}$

Equations

$$E_x^{\mathrm{inc}}$$

$$H_y^{\mathrm{inc}}$$

$$△E_x=0,△H_y=0,△E_z=0,△H_z=0,$$

$$\displaystyle E_{x}\raisebox{-6.0pt}{$\left|\rule{0.0pt}{12.0pt}\right.$}\vphantom{E}_{\begin{subarray}{1}x=\mathrm{w}/2+\rho,\\ z=0.\end{subarray}}\propto$$

$$\displaystyle E_{x}\raisebox{-6.0pt}{$\left|\rule{0.0pt}{12.0pt}\right.$}\vphantom{E}_{\begin{subarray}{1}y=\mathrm{w}/2+\rho,\\ z=0.\end{subarray}}\propto$$

$$△\tilde{\phi }=0,△=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial z^2},$$

$$\tilde{\phi }$$

$$\tilde{\phi }(x,z)=\mathrm{Im}{V}_0(z+ix),$$

$$E_x^{\mathrm{I},\mathrm{II}}(x,z)$$

$$E_z^{\mathrm{I},\mathrm{II}}(x,z)$$

$$V(x+iz)=\mathrm{arcosh}\left(\frac{\cos \left(\pi (x+iz)/D\right)}{\cos \alpha }\right),\alpha =\frac{\pi \mathrm{w}}{2D},$$

$$E_x^{\mathrm{I},\mathrm{II}}(x,z)=\mp \frac{\partial }{\partial x}\mathrm{Im}V(x+iz),\forall z,$$

$$E_z^{\mathrm{I},\mathrm{II}}(x,z)=\mp \frac{\partial }{\partial z}\mathrm{Im}V(x+iz),\forall z.$$

$$\frac{\partial H_y^{\mathrm{I},\mathrm{II}}}{\partial x}=\pm i\omega {\epsilon }_0{\epsilon }_{1,2}\frac{\partial }{\partial z}\mathrm{Im}V(x+iz).$$

$$\frac{\partial H_y^{\mathrm{I},\mathrm{II}}}{\partial x}=\pm i\omega {\epsilon }_0{\epsilon }_{1,2}\frac{\partial }{\partial x}\mathrm{Re}V(x+iz).$$

$$H_y^{\mathrm{I},\mathrm{II}}(x,z)=\pm i\omega {\epsilon }_0\frac{{\epsilon }_1+{\epsilon }_2}{2}\mathrm{Re}V(x+iz).$$

$$\displaystyle H_{y}\raisebox{-2.0pt}{$\left|\rule{0.0pt}{10.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=-0\end{subarray}}=H_{y}\raisebox{-2.0pt}{$\left|\rule{0.0pt}{10.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=+0\end{subarray}}.$$

$$\displaystyle H_{y}^{\mathrm{II,\,III,\,IV}}\raisebox{-2.0pt}{$\left|\rule{0.0pt}{10.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=\mp\,0\end{subarray}}=0.$$

$$\displaystyle\langle H_{y}\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\dfrac{1}{D^{2}}\int\limits_{-\mathrm{w}/2}^{D-\mathrm{w}/2}\,\int\limits_{-\mathrm{w}/2}^{D-\mathrm{w}/2}H_{y}\raisebox{-2.0pt}{$\left|\rule{0.0pt}{10.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=\mp\,0\end{subarray}}\,dx\,dy=\dfrac{\mathrm{w}}{D^{2}}\int\limits_{-\mathrm{w}/2}^{\mathrm{w}/2}H_{y}^{\mathrm{I}}\raisebox{-3.0pt}{$\left|\rule{0.0pt}{11.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=\mp\,0\end{subarray}}\,dx=\mp\,i\,\omega\,\varepsilon_{0}\,\dfrac{\varepsilon_{1}+\varepsilon_{2}}{2}{\mathrm{w}}/D\ln\cos\alpha,\quad$$

$$\displaystyle\int\limits_{-\mathrm{w}/2}^{\mathrm{w}/2}\mathop{\mathrm{Re}}\nolimits V(x+iz)\raisebox{-3.0pt}{$\left|\rule{0.0pt}{11.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}\vphantom{z}\\ z=\mp\,0\end{subarray}}\,dx=\mathop{\mathrm{Re}}\nolimits\int\limits_{-\mathrm{w}/2}^{\mathrm{w}/2}\operatorname{arcosh}\left(\dfrac{\cos\left(\pi x/D\right)}{\cos\alpha}\right)\,dx=\dfrac{D}{\pi}\mathop{\mathrm{Re}}\nolimits\int\limits_{-\alpha}^{\alpha}\operatorname{arcosh}\left(\dfrac{\cos t}{\cos\alpha}\right)dt=$$

$$=-D\ln \cos \alpha .$$

$$\displaystyle H_{y}\raisebox{-6.0pt}{$\left|\rule{0.0pt}{12.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}y=\mathrm{w}/2-\rho,\\ z=0.\end{subarray}}\propto H_{y}\raisebox{-9.0pt}{$\left|\rule{0.0pt}{16.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}y=\mathrm{w}/2+\rho\cos\varphi,\\ z=\rho\sin\varphi,\\ z\neq 0.\end{subarray}}\propto\rho^{-1/2},\qquad\rho\to+0,$$

$$\displaystyle H_{y}\raisebox{-6.0pt}{$\left|\rule{0.0pt}{12.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}x=\mathrm{w}/2-\rho,\\ z=0.\end{subarray}}\propto H_{y}\raisebox{-9.0pt}{$\left|\rule{0.0pt}{16.0pt}\right.$}\vphantom{H}_{\begin{subarray}{1}x=\mathrm{w}/2+\rho\cos\varphi,\\ z=\rho\sin\varphi,\\ z\neq 0.\end{subarray}}\propto\rho^{1/2},\qquad\rho\to+0.$$

$$△\psi =0,△=\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2},$$

$$\psi$$

$$\psi (y,z)=\mathrm{Im}{\Psi }_0(z+iy),$$

$$H_y^{\mathrm{I},\mathrm{IV}}(y,z)$$

$$\Psi (y+iz)=\arcsin \left(\frac{\sin \left(\pi (y+iz)/D\right)}{\sin \alpha }\right),\alpha =\frac{\pi \mathrm{w}}{2D},$$

$$H_y(x,y,z)=C\mathrm{Re}V(x)\frac{\partial }{\partial y}\mathrm{Re}\Psi (y+iz),z\ge 0,$$

$$H_y(x,y,z)=\pm C\mathrm{Re}V(x)\frac{\partial }{\partial y}\mathrm{Re}\Psi (y+iz),\forall z.$$

$$⟨H_y⟩=\pm C\frac{1}{D^2}\mathrm{Re}\Psi (y\mp i0)|\cdot \underset{-\mathrm{w}/2}{\overset{\mathrm{w}/2}{\int }}\mathrm{Re}V(x)dx=\mp C\frac{\pi }{D}\ln \cos \alpha ,$$

$$\displaystyle\int\limits_{-\mathrm{w}/2}^{\mathrm{w}/2}\mathop{\mathrm{Re}}\nolimits V(x)\,dx=-D\ln\cos\alpha,\quad\mathop{\mathrm{Re}}\nolimits\Psi(y\mp\,i\,0)\raisebox{-0.4pt}{$\left|\rule{0.0pt}{22.98138pt}\right.$}_{-\mathrm{w}/2}^{\mathrm{w}/2}=\mathop{\mathrm{Re}}\nolimits\bigl{[}\arcsin(1\mp\,i\,0)-\arcsin(-1\mp\,i\,0)\bigr{]}=\pi.$$

$$C=i\omega {\epsilon }_0\frac{{\epsilon }_1+{\epsilon }_2}{2}\frac{\mathrm{w}}{\pi },$$

Full text

Keywords: Electromagnetic waves; Maxwell equations; Frequency Selective Surface; Patch-Type FSS; Edge Condition; Meixner's Condition;Scattering matrix

1 Introduction

At present, analytical and numerical methods [3, 4, 6, 7, 11, 12, 13, 14, 15, 5, 8, 9, 10, 16, 17] are widely used to solve problems of electromagnetic wave scattering by frequency selective surfaces (FSS).

In this paper an analytical approach for calculating scattering matrix elements for the case of normal incidence of plane electromagnetic waves on a square patch-type FSS from both sides, which is placed at an interface between two dielectric media with different dielectric permittivities, is proposed. Simple analytical expressions for the elements of the scattering matrix are derived under quasi-static assumption.

A comparison of frequency dependencies of the reflection and transmission coefficients calculated analytically by the derived formulae and computed numerically by 3D electromagnetic simulation is carried out. Approximation error estimates of these analytical expressions are given.

2 Analytical solution

2.1 Electromagnetic field

Let us consider electromagnetic oscillations near the patch-type FSS (see Figure  1) excited by two plane waves incident normally from both sides:

[TABLE]

where $E_{x}^{\mathrm{inc}},H_{y}^{\mathrm{inc}}$ are component of electric and magnetic fields of the incident waves, respectively; ${k_{1,\,2}}=\omega\sqrt{\mathstrut\varepsilon_{0}\,\varepsilon_{1,2}\,\mu_{0}}$ are wave numbers, ${Z_{1,\,2}}=Z_{0}/\sqrt{\mathstrut\varepsilon_{1,\,2}}$ are characteristic impedance, $\varepsilon_{1,\,2}$ are relative dielectric permittivities of medium 1 ($z<0$) and medium 2 ($z>0$), respectively separated by the FSS plane ($z=0$); $Z_{0}=\sqrt{\mathstrut\mu_{0}/\varepsilon_{0}}$; $\varepsilon_{0},\mu_{0}$ are absolute dielectric permittivity and absolute magnetic permeability of free space.

The field components of incident waves (1) are homogeneous in the plane of the FSS ($E_{x}^{\mathrm{inc}},H_{y}^{\mathrm{inc}}$ do not depend on $x,y$), hence according to Floquet's principle [1] the components $E_{x},H_{y},E_{z},H_{z}$ of the excited near-field of the FSS are periodic functions with respect to $x,y$ with the period equal to the FSS unit cell size $D$. Therefore restrict ourself to consideration of the unit cell (Figure  1 (b)). Here $D$ is the unit cell size, $\mathrm{w}$ is the square patch width, $D-\mathrm{w}$ is the gap between patches.

Under the quasi-static condition ($D\ll\lambda$, where $\lambda$ is a wavelength) the Helmholtz's equation is approximated by the Laplace's equation:

[TABLE]

where $\triangle=\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}$.

We find a solution of (2) as a linear combination of two particular solutions. Since any function can be written as a sum of even and odd functions, we will find the component $E_{x}$ for the first particular solution as an even function with respect to $z$, for the second one – as an odd function. Other components $H_{y},E_{z},H_{z}$ of these particular solutions are either even or odd functions with respect to $z$.

2.1.1 The first particular solution

For the first particular solution of (2) we will find the component $E_{x}$ as an even function with respect to $z$. It follows from the time-harmonic Maxwell's equations that $E_{x}\propto(\operatorname{\mathbf{rot}}{\mathbf{H}})_{x}$, $E_{z}\propto(\operatorname{\mathbf{rot}}{\mathbf{H}})_{z}$. Hence, $H_{y}$, $E_{z}$ is odd, $H_{z}$ is even with respect to $z$.

Asymptotic behavior of $E_{x},E_{z}$ near edges is defined by the following edge conditions:

[TABLE]

where $\rho>0$ is the distance between the observation point $(x,y,z)$ and the nearest edge, $\rho$ is sufficiently small; distances to another edges are assumed to be much larger than $\rho$; $\varphi$ is the polar angle, $-\pi<\varphi<\pi,\;\varphi\neq 0$; the sign $\propto$ means asymptotic proportionality at $\rho\to+0$ ($\Theta$-notation).

The edge conditions (3) are the strengthening of Meixner's ones [2, 3]. Note that exact solution for one-dimensional planar array FSS formed by infinitesimally thin parallel perfectly conducting strips [6] satisfies (3).

It follows from (3) that in the regions I ($z\neq 0$) and II ($\forall z$) (see above 1(b)) near the edges ($\rho$ is sufficiently small): $\left|\dfrac{\partial^{2}E_{x}}{\partial x^{2}}\right|\gg\left|\dfrac{\partial^{2}E_{x}}{\partial y^{2}}\right|$. Hence, in the Laplace's equation (2) the second derivative with respect to $y$ can be neglected. Therefore, we will obtain an analytical expression for $E_{x}$, in the regions I, II under the assumption of independence on $y$.

At a sufficiently small distance from the edge $x=\mathrm{w}/2$ (so that distances to the another edges $y=\pm\,\mathrm{w}/2$ are much larger than distance to $x=\mathrm{w}/2$) the regions I and II can be approximately regarded as infinite along the y-axis (I - an infinitesimally thin perfectly conducting strip, II - a dielectric medium). In this case the incident waves (1) are E-polarized and electro-quasi-static approximation of Maxwell equations can be used, i.e. ${\mathbf{E}}=-\nabla\widetilde{\varphi}$, where $\widetilde{\varphi}$ is the electric scalar potential. Due to the symmetry of ${\mathbf{E}}$ with respect to the planes $x=0$, $z=0$ it is sufficient to restrict the computational domain to $\{(x,z):\quad 0\leq x\leq D/2,\;z\geq 0\}$. The corresponding Laplace's equation for the electric potential is:

[TABLE]

where $\widetilde{\varphi}_{1},\;\widetilde{\varphi}_{2}$ are some given constants.

The exact solution of (4) obtained by analogy with [6] by using the conformal mapping method, up to a constant multiplier, is:

[TABLE]

where $V_{0}(z+ix)=\operatorname{arcosh}\left(\dfrac{\cosh\left(\pi(z+ix)/D\right)}{\cos\alpha}\right),\quad\alpha=\dfrac{\pi\mathrm{w}}{2D}$.

Hence,

[TABLE]

since $V_{0}(z+ix)=\overline{V(x+iz)}$, where

[TABLE]

Since in the first particular solution $E_{x}$ must be even function with respect to $z$, we construct an even extension of $E_{x}$ to the half-space $z<0$:

[TABLE]

where the upper sign ("-") corresponds to the half-space $z\leq 0$, the lower sign ("+") corresponds to the half-space $z\geq 0$; on the plane $z=0$ both expressions coincide. Here $\mp$ is due to the oddness of the function $\dfrac{\partial}{\partial x}\mathop{\mathrm{Im}}\nolimits V(x+iz)$ with respect to $z$.

Thus, $E_{x}^{\mathrm{I,\,II}}$ satisfies the edge condition (3a) and approximately satisfies (due to neglecting the second derivative with respect to $y$) the Laplace's equation in three dimensions (2).

In analogy with (7), we derive from (5) that:

[TABLE]

The $z$-component of the time-harmonic Maxwell's equations: $(\operatorname{\mathbf{rot}}{\mathbf{H}})_{z}=-i\,\omega\,\varepsilon_{0}\,\varepsilon_{1,\,2}E_{z},\;\forall z$, therefore

[TABLE]

The function $V(x+iz)$ defined by (6) is an analytic function in the regions under consideration ($-\mathrm{w}/2<x<D-\mathrm{w}/2,\,z<0$ and $-\mathrm{w}/2<x<D-\mathrm{w}/2,z>0$) (in each half-spaces $z<0$, $z>0$ separately). Hence, its real and imaginary parts, considered as functions of two real variables, satisfy the Cauchy–Riemann equations in these regions, so that

[TABLE]

Integrating (9) with respect to $x$ yields

[TABLE]

Here an additive function of integration is defined so that $H_{y}$ is odd function with respect to $z$, as the first particular solution requires.

It follows from boundary conditions for the electromagnetic field at the planar interface between two dielectric media that $H_{y}$ in the regions II, III, IV is continuous across the interface ($z=0$):

[TABLE]

Continuity and oddness of $H_{y}$ with respect to $z$ results in

[TABLE]

Note that $H_{y}^{\mathrm{II}}$ defined by (10) satisfies the interface condition (11).

It follows from (11), (10), (6) that

[TABLE]

since

[TABLE]

Let us obtain an analytical expression for $E_{x}$ in the regions I ($\forall z$), IV ($\forall z$). For this purpose, we first derive analytical expressions for $H_{y}$.

Asymptotic behavior of $H_{y}$ near the edge $y=\mathrm{w}/2$ is defined by:

[TABLE]

The edge conditions (13) are the strengthening of Meixner's ones [2, 3]. Note that exact solution for one-dimensional planar array FSS formed by infinitesimally thin parallel perfectly conducting strips [6] satisfies (13).

It follows from (13) that in the Laplace's equation (2) in the regions I ($\forall z$) and IV ($z\neq 0$) near the edges ($\rho$ is sufficiently small): $\left|\dfrac{\partial^{2}H_{y}}{\partial y^{2}}\right|\gg\left|\dfrac{\partial^{2}H_{y}}{\partial x^{2}}\right|$.

Hence, dependence of $H_{y}$ on $x$ is weak (compared with dependence on $y$) in the regions I, IV ($\forall z$) (here, at $z=0$, consistently with (11), $H_{y}^{\mathrm{IV}}=0$).

At a sufficiently small distance from the edge $y=\mathrm{w}/2$ (so that distances to the another edges $x=\pm\,\mathrm{w}/2$ are much larger than distance to $y=\mathrm{w}/2$) the regions I and IV can be approximately regarded as infinite along the x-axis (I - an infinitesimally thin perfectly conducting strip, II - a dielectric medium).

In this case the incident waves (1) are H-polarized and magneto-quasi-static approximation of Maxwell equations can be used, i.e. ${\mathbf{H}}=-\nabla\psi$, where $\psi$ is the magnetic scalar potential. Due to the symmetry of ${\mathbf{H}}$ with respect to the planes $y=0$, $z=0$ it is sufficient to restrict the computational domain to $\{(y,z):\quad 0\leq y\leq D/2,\;z\geq 0\}$. The corresponding Laplace's equation for the magnetic potential is:

[TABLE]

where $\psi_{1},\;\psi_{2}$ are some given constants.

The exact solution of (14) obtained by analogy with [6] by using the conformal mapping method, up to a constant multiplier, is:

[TABLE]

where $\Psi_{0}(z+iy)=\operatorname{arsinh}\left(\dfrac{\sinh\left(\pi(z+iy)/D\right)}{\sin\alpha}\right),\quad\alpha=\dfrac{\pi\mathrm{w}}{2D}$.

Hence,

[TABLE]

since $\Psi_{0}(z+iy)=i\,\overline{\Psi(y+iz)}$, where

[TABLE]

Thus, $H_{y}^{\mathrm{I,\,IV}}(y,z)$ defined by (15) satisfies only the edge condition (13a) and exactly satisfies the corresponding two-dimensional Laplace's equation ((2) without the second derivative with respect to $x$).

As we have remarked, dependence of $H_{y}$ on $x$ is weak in the regions I ($\forall z$), IV ($\forall z$), hence $H_{y}$ can be constructed by combining (15) and (10) in a manner similar to [7].

[TABLE]

where $C$ is an undetermined constant multiplier.

Since in the first particular solution $H_{y}$ must be odd function with respect to $z$, we construct an odd extension of $H_{y}$ to the half-space $z<0$:

[TABLE]

Here $\pm$ is due to the evenness of the function $\dfrac{\partial}{\partial y}\mathop{\mathrm{Re}}\nolimits\Psi(y+iz)$ with respect to $z$. Furthermore, $H_{y}(x,y,z)$ satisfies the condition (11).

Thus, $H_{y}(x,y,z)$ defined by (17) in the regions I ($\forall z$), IV ($\forall z$) is valid also in II ($z=0$), III ($z=0$) consistently with (11). Here, $H_{y}(x,y,z)$, in contrast to $H_{y}^{\mathrm{I,\,IV}}(y,z)$, satisfies both (13a) and (13b) edge conditions and approximately satisfies (due to the smallness of the second derivative with respect to $x$) the Laplace's equation in three dimensions (2).

To determine the constant $C$, calculate $\langle H_{y}\rangle$ again:

[TABLE]

since

[TABLE]

Comparing (18) with (12), we obtain

[TABLE]

then (17) takes the form:

[TABLE]

The $y$-component of the time-harmonic Maxwell's equations: $(\operatorname{\mathbf{rot}}{\mathbf{E}})_{y}=i\,\omega\mu_{0}H_{y},\;\forall z$, therefore

[TABLE]

The function $\Psi(y+iz)$ defined by (16) is an analytic function in the regions under consideration. Hence, its real and imaginary parts, considered as functions of two real variables, satisfy the Cauchy–Riemann equations, so that

[TABLE]

Integrating (20) with respect to $z$ yields in the regions I ($\forall z$), IV ($\forall z$)

[TABLE]

Here an additive function of integration is defined from interface conditions for the electromagnetic field.

It follows from (7), (21) that

[TABLE]

since

[TABLE]

[TABLE]

[TABLE]

where $s=t-\pi/2$, $\beta=\pi/2-\alpha$.

2.1.2 The second particular solution

Components of the second particular solution of (2) are supplementary to ones of the first particular solution, i.e. $E_{x}$, $H_{z}$ are odd functions, $H_{y}$, $E_{z}$ are even functions with respect to $z$. The simplest nontrivial solution of (2) satisfying these requirements is

[TABLE]

where $H_{0}\neq 0$ is some constant.

Thus, electric and magnetic field components $\widetilde{E}_{x},\widetilde{H}_{y}$, which are a linear combination of the first and second particular solutions with the coefficients $c_{1}$, $c_{2}$, have the following average values:

[TABLE]

respectively, where $\langle E_{x}\rangle,\langle H_{y}\rangle$ are defined by (22), (12).

2.2 Scattering matrix

The considered patch-type FSS can be modelled as a two-port network: some domains to the left ($z<0$) and right ($z>0$) of the FSS plane ($z=0$) is viewed as port 1 and 2 respectively. The relationship between the incident and scattered waves is described by scattering matrix $\mathbf{S}$:

[TABLE]

where $\mathbf{a}=(a_{1},a_{2})^{\intercal}$, $\mathbf{b}=(b_{1},b_{2})^{\intercal}$.

Here $a_{1},a_{2}$ are the normalized complex amplitudes of the incident waves and $b_{1},b_{2}$ are ones of the scattered waves at port 1 and 2 respectively [9]:

[TABLE]

It follows from (26a) that

[TABLE]

where the upper and lower signs correspond to the port 1 and 2 respectively; $H_{1,\,2}^{\mathrm{inc}}$, $H_{1,\,2}^{\mathrm{sct}}$ are the complex amplitudes of the magnetic fields of the incident (1f) and scattered waves respectively.

Boundary conditions for the electromagnetic field at ports 1 and 2 are of the form

[TABLE]

where $\langle\widetilde{E}_{x}\rangle,\langle\widetilde{H}_{y}\rangle$ are defined by (24).

Substituting (24), (26) into (27) gives

[TABLE]

where $\langle E_{x}\rangle,\langle H_{y}\rangle$ are defined by (22), (12).

Let us rewrite the system of linear algebraic equations (28) in the form (25). Denote $\widehat{Z}=\langle E_{x}\rangle/\langle H_{y}\rangle$, where $\langle E_{x}\rangle$, $\langle H_{y}\rangle$ are evaluated, for the sake of definiteness, at port 1. Then

[TABLE]

Substituting $Z=i\,X$ into (29) and using (22), (12), we obtain

[TABLE]

where $\alpha=\dfrac{\pi\mathrm{w}}{2D},\;\;Z_{0}=\sqrt{\mathstrut\mu_{0}/\varepsilon_{0}}\,.$

Electromagnetic wave propagation can be described using a transmission line equivalent circuit model. Then $Z$ is the normalized electrical impedance of the unit cell of the patch-type FSS, X is the normalized electrical reactance, the first term in (30b) is inductive reactance, the second one is capacitive reactance.

3 Numerical solution and comparison of the results

Let us estimate the approximation error of the formulae derived in this paper. For this purpose, the absolute values of the reflection coefficient $S_{11}$ and the transmission coefficient $S_{21}$ calculated by the analytical expression (30), have been compared with ones computed numerically with high accuracy by 3D electromagnetic simulation with CST MWS. Note that $S_{21}=S_{12}$ and $|S_{11}|=|S_{22}|$.

Figure 2 shows frequency dependencies of $|S_{11}|,|S_{21}|$ of the electromagnetic waves incident normally on the patch-type FSS with the fixed period $D$ and the variable relative width of the patch ${\mathrm{w}}/D$.

The reflection coefficients increase and the transmission coefficients decrease with increasing the relative width of the patch, as displayed in Figure 2.

Figure 3 shows frequency dependencies of $|S_{11}|$ of the electromagnetic waves incident normally on the patch-type FSS with the fixed gap between the patches $D-{\mathrm{w}}$ and the variable period $D$.

The reflection coefficients increase with increasing the patch-type FSS period for the fixed gap between the patches, as displayed in Figure 3.

Recall that the approximate analytical solution has been obtained under $\lambda\gg D$, therefore, as expected, the approximation error for $\lambda=6D$ is smaller than for $\lambda=3D$, as shown in Figures 2, 3. Here $\lambda$ is the wavelength in the second medium ($\varepsilon_{2}=3$).

Table 1 presents the estimation of the relative approximation error of the analytical expressions for different frequencies.

The relative deviations $\delta S_{11}$ and $\delta S_{21}$ does not exceed $0.5\>\%$ and $0.08\>\%$ respectively in the frequency range up to $\widehat{f}=$ 16 GHz, i.e. in the wavelength range down to 10.8 mm ($\lambda>6D$), as shown in Table 1.

4 Conclusion

Thus, in this paper the simple, but quite accurate analytical expressions for the elements of the scattering matrix have been derived under the quasi-static assumption for the case of normal incidence of the plane electromagnetic waves on the square patch-type FSS from both sides, which is placed at the interface between two dielectric media with the different dielectric permittivities.

The comparison of frequency dependencies of the reflection and transmission coefficients calculated analytically by the derived formulae and computed numerically with high accuracy by 3D electromagnetic simulation with CST MWS has shown good agreement between both approaches. Numerical results have demonstrated that the formulae obtained in this paper are accurate enough for practical purposes in their applicability domain.

The derived analytical expressions can be used in design of multi-layer patch-type FSS structures. They can help to analytically optimize the FSS structure parameters and hence avoid extensive numerical simulations, and therefore reduce computational costs. Desired reflective properties of such structures can be achieved by varying both the relative width of the patch ${\mathrm{w}}/D$ and the FSS period $D$.