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Rigorous coupled wave analysis upgraded with combined boundary conditions method for 2D metamaterials in microwave

Lin Wang, Debao Fang, Haibo Jin, and Jingbo Li

Open Access Open Access

Abstract

We combine rigorous coupled wave analysis (RCWA) and combined boundary conditions method (CBCM) to analyze optical properties of two-dimensional periodic metamaterial grating with arbitrary shapes in microwave domain, and the adaptive spatial resolution (ASR) function is introduced to increase convergence especially for the crossed metamaterial gratings. The numerical results show that RCWA with CBCM can get the correct convergence result and distinguish the contribution of the grating lobes to the total transmission and reflection. The ASR technique improves dramatically the convergence speed of crossed grating. This method provides a more effective computational tool for the design of metamaterial grating in microwave domain.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Two-dimensional periodic metamaterial absorbers have attracted much attention due to their broad application prospects in the fields of radar target stealth, antenna design, and electromagnetic protection [1,2]. Two common numerical techniques, finite difference time domain (FDTD), finite element method (FEM) are used to calculate the optical properties of metamaterial [3–5]. However, they need small enough mesh size to simulate accurate results, and they are time-consuming.

Rigorous coupled wave analysis (RCWA) is a powerful, fast semi-analytical method to numerically investigate the optical properties of periodic gratings in infrared and visible optical domains [6–10]. In the microwave domain, the non-negligible Gibbs phenomenon in analyzing 2D metal-dielectric gratings severely hinders the convergence. This problem in 1D strip gratings was addressed by the combined boundary conditions method (CBCM) [11–13], and the adaptive spatial resolution (ASR) was introduced to increase convergence for CBCM [14,15]. We extend the above CBCM research of 1D strip gratings to simulate 2D periodic metamaterial gratings with arbitrary shapes, and introduce the ASR technique to improve the convergence of crossed metamaterials grating in CBCM.

In this work, the RCWA with CBCM in the Cartesian is presented in detail in Section 2. In Section 3, we introduce a new transformed system with ASR function, and propose the RCWA with CBCM in the transformed system. In Section 4, the numerical results show that RCWA with CBCM can get the correct convergence result and distinguish the contribution of the grating lobes to the total transmission and reflection. This method is perfectly suitable for multi-layer periodic structure calculations.

2. RCWA with CBCM in the Cartesian

2.1 Statement of the problem

First, the relative complex permittivity of the metal in microwave frequency can be written in terms of the metal conductivity $\sigma$, and the free space wavelength $\lambda$ in the SI system [16]: $\varepsilon _{metal} =1-ja\sigma \lambda$, where $a=60\, {\rm S}^{-1}$, the complex permittivity for common metals is as high as $1-j10^{8}$, so common metal in microwave can be regarded as perfectly conducting metal. Note, the error caused by Gibbs phenomenon between the imaginary parts of metal and air, which are $10^8$ and 0, respectively, makes standard RCWA in the Cartesian unable to converge. Second, in actual application, the thickness of the microwave metal layer is much smaller than that of other absorbing material layers. Based on the above two facts, the investigated structure is shown in Fig. 1. Metal grating in microwave frequency can be regarded as a perfectly conducting infinitely thin arbitrary shape metal grating between two free-space regions of zero thickness. We denote by $\Omega _{1}$ the set of areas belonging to the metal and by $\Omega _{2}$ its complementary in the total interface $\Omega {\rm \; =\; }\Omega _{1} \cup \Omega _{2}$. The metal area is defined with period $\Lambda _{x}$ and $\Lambda _{y}$ in the x- and y-directions, respectively, and filling fraction f. The electromagnetic field is defined by the propagation vector k$_{inc}$ and the amplitudes in the s and p directions (i.e., perpendicular and parallel to the plane of incidence, respectively).

 figure: Fig. 1.

Fig. 1. Model of metal grating, metal grating between two free-space regions of zero thickness, $h$ =0

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2.2 Modal equation of RCWA in the Cartesian

For the two free-space regions, $\varepsilon _{xy} =1$ and $\mu _{xy} {\rm =}1$, the modal equation of RCWA in the Cartesian can be deduced:

$$\frac{\mathrm{d}}{\mathrm{d} \tilde{z}}\left[\begin{array}{l} \mathbf{u}_{x} \\ \mathbf{u}_{y} \end{array}\right]=\mathbf{Q}_{x y}\left[\begin{array}{l} \mathbf{s}_{x} \\ \mathbf{s}_{y} \end{array}\right], \frac{\mathrm{d}}{\mathrm{d} \tilde{z}}\left[\begin{array}{l} \mathbf{s}_{x} \\ \mathbf{s}_{y} \end{array}\right]=\mathbf{P}_{x y}\left[\begin{array}{l} \mathbf{u}_{x} \\ \mathbf{u}_{y} \end{array}\right] ,$$
$$\mathbf{Q}_{x y}=\left[\begin{array}{cc} \tilde{\mathbf{K}}_{x} \tilde{\mathbf{K}}_{y} & \mathbf{I}-\tilde{\mathbf{K}}_{x} \tilde{\mathbf{K}}_{x} \\ \tilde{\mathbf{K}}_{y} \tilde{\mathbf{K}}_{y}-\mathbf{I} & -\tilde{\mathbf{K}}_{y} \tilde{\mathbf{K}}_{x} \end{array}\right],$$
$$\mathbf{P}_{x y}=\left[\begin{array}{cc} \tilde{\mathbf{K}}_{x} \tilde{\mathbf{K}}_{y} & \mathbf{I}-\tilde{\mathbf{K}}_{x} \tilde{\mathbf{K}}_{x} \\ \tilde{\mathbf{K}}_{y} \tilde{\mathbf{K}}_{y}-\mathbf{I} & -\tilde{\mathbf{K}}_{y} \tilde{\mathbf{K}}_{x} \end{array}\right],$$
where $\tilde {\mathbf {K}}_{x}$ and $\tilde {\mathbf {K}}_{y}$ are normalized wave vectors $\mathbf {K}_{x} / k_{0}$ and $\mathbf {K}_{y} / k_{0}$, $\tilde {z}=k_{0} z$, $k_{0} {\rm =}\omega \sqrt {\varepsilon _{0} \mu _{0} }$, $\mathbf {K}_{x}=\operatorname {diag}\left (k_{x, m}\right )$, $k_{x,m} =k_{0} \sqrt {\varepsilon _{r,} {}_{ref} } \sin \theta \cos \varphi -m\frac {2\pi }{\Lambda _{x} }$, $\mathbf {K}_{y}=\operatorname {diag}\left (k_{y, n}\right )$, $k_{y, n}=k_{0} \sqrt {\varepsilon _{r, r e f}} \sin \theta \sin \varphi - n \frac {2 \pi }{\Lambda _{\mathrm {y}}}$, m and n are integers. s and u are the 2D Fourier transform coefficients of $\mathbf {E}$ and $\mathbf {H}'$, $\mathbf {H}=j \sqrt {\frac {\varepsilon _{0}}{\mu _{0}}} \mathbf {H}^{\prime }$.

By combining the $\mathbf {P}_{xy}$ and $\mathbf {Q}_{xy}$ matrices, the wave equation is written in terms of the E components:

$$\frac{\mathrm{d}^{2}}{\mathrm{~d} \tilde{z}^{2}}\left[\begin{array}{l} \mathbf{s}_{x} \\ \mathbf{s}_{y} \end{array}\right]=\mathbf{\Omega}_{x y}^{2}\left[\begin{array}{l} \mathbf{s}_{x} \\ \mathbf{s}_{y} \end{array}\right]=\mathbf{W}_{x y} \boldsymbol{\beta}_{x y}^{2} \mathbf{W}_{x y}^{{-}1}\left[\begin{array}{l} \mathbf{s}_{x} \\ \mathbf{s}_{y} \end{array}\right] ,$$
where $\mathbf {\Omega }_{x y}^{2}=\mathbf {P}_{xy} \mathbf {Q}_{xy}$, $\mathbf {W}_{xy}$ and $\boldsymbol {\beta }_{xy}^{2}$ are the eigenvectors and eigenvalues matrixes of the matrix $\mathbf {P}_{xy} \mathbf {Q}_{xy}$, respectively. The general solutions of the wave equation for $\mathbf {E}_{xy}$ and $\mathbf {H}'_{xy}$ are:
$$\mathbf{E}_{x y}=\left[\begin{array}{c} \mathbf{s}_{x}(z) \\ \mathbf{s}_{y}(z) \end{array}\right]=\mathbf{W}_{x y} e^{-\boldsymbol{\beta}_{x y} k_{0}\left(z-z_{1}\right)} \mathbf{c}_{x y}^{+}+\mathbf{W}_{x y} e^{\boldsymbol{\beta}_{x y} k_{0}\left(z-z_{2}\right)} \mathbf{c}_{x y}^{-} ,$$
$$\mathbf{H}_{x y}^{\prime}=\left[\begin{array}{c} \mathbf{u}_{x}(z) \\ \mathbf{u}_{y}(z) \end{array}\right]={-}\mathbf{V}_{x y} e^{-\boldsymbol{\beta}_{x y} k_{0}\left(z-z_{1}\right)} \mathbf{c}_{x y}^{+}+\mathbf{V}_{x y} e^{\boldsymbol{\beta}_{x y} k_{0}\left(z-z_{2}\right)} \mathbf{c}_{x y}^{-} ,$$
where $\mathbf {V}_{x y}=\mathbf {P}_{x y}^{-1} \mathbf {W}_{x y} \boldsymbol {\beta }_{x y}=\mathbf {Q}_{x y} \mathbf {W}_{x y} \boldsymbol {\beta }_{x y}^{-1}$, $\mathbf {c}_{xy}^{+}$ and $\mathbf {c}_{xy}^{-}$ are the coefficient vectors of the propagating waves in positive and negative directions, respectively. The layer starts at $z=z_{1}$ and ends at $z=z_{2}$.

2.3 Combined boundary conditions method in the Cartesian

The key of CBCM is matching boundary conditions at the surface of the metal grating. The tangential component of the electric field in the perfect metal area $\Omega _{1}$ must vanish, which results in:

$$E_{1 x}=0, E_{1 y}=0 \quad \forall(x, y) \in \Omega_{1}.$$
$$E_{2 x}=0, E_{2 y}=0 \quad \forall(x, y) \in \Omega_{1}.$$

On the other hand, outside the metal $\Omega _{2}$, the laws of electromagnetics ensure the continuity of the tangential components of both electric field and magnetic field, which leads to:

$$E_{1 x}=E_{2 x}, E_{1 y}=E_{2 y} \quad \forall(x, y) \in \Omega_{2}.$$
$$H_{1 x}^{\prime}=H_{2 x}^{\prime}, H_{1 y}^{\prime}=H_{2 y}^{\prime} \quad \forall(x, y) \in \Omega_{2}.$$

In order to get relations valid on the total area $\Omega$, we combine Eq. (5) and Eq. (6a)

$$\left[\begin{array}{l} E_{1 x} \\ E_{1 y} \end{array}\right]=\left[\begin{array}{l} E_{2 x} \\ E_{2 y} \end{array}\right] \quad \forall(x, y) \in \Omega.$$

Following Montiel and Nevire [11,12], Eq. (5b) and (6b) can also be combined in a single equation that holds for $\forall (x,y)\in \Omega$:

$$\chi(x, y)\left[\begin{array}{l} E_{2 x} \\ E_{2 y} \end{array}\right]+\tau \tilde{\chi}(x, y)\left(\left[\begin{array}{c} H_{1 x}^{\prime} \\ H_{1 y}^{\prime} \end{array}\right]-\left[\begin{array}{c} H_{2 x}^{\prime} \\ H_{2 y}^{\prime} \end{array}\right]\right)=0 \quad \forall(x, y) \in \Omega,$$
where $\tau$ is an arbitrary constant, $\chi (x,y)$ and $\tilde {\chi }(x,y)$ are the characteristic functions of set $\Omega _{1}$ and $\Omega _{2}$, respectively:
$$\chi(x, y)= \begin{cases}1 & \text{ if }(x, y) \in \Omega_{1} \\ 0 & \text{ if }(x, y) \in \Omega_{2}\end{cases},$$
$$\tilde{\chi }(x,y)=1-\chi (x,y) .$$

By projecting Eq. (7) and (8) onto the exp(-jK${}_{x}$x)exp(-jK${}_{y}$y) basis, we obtain the following matrix equation:

$$\left[\begin{array}{l} \mathbf{c}_{1}^{+} \\ \mathbf{c}_{1}^{-} \end{array}\right]=\mathbf{T}_{\mathrm{CBCM}}\left[\begin{array}{l} \mathbf{c}_{2}^{+} \\ \mathbf{c}_{2}^{-} \end{array}\right],$$
$$\mathbf{T}_{\mathrm{CBCM}}=\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\tau \tilde{\mathbf{Q}} \mathbf{V}_{0} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{0} \end{array}\right]\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\tau \tilde{\mathbf{Q}} \mathbf{V}_{0}-\mathbf{Q} \mathbf{W}_{0} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{0}-\mathbf{Q} \mathbf{W}_{0} \end{array}\right],$$
where $\mathbf {W}_{0}$ and $\mathbf {V}_{0}$ are eigen-modes for free space in Eq. (4), $\mathbf {Q}=\left [\begin {array}{cc}\left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ] & \mathbf {0} \\ \mathbf {0} & \left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ]\end {array}\right ]$, $\left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ]$ is ‘block-Toeplitz Toeplitz-block’ (BTTB) matrix whose elements are
$$\left[\kern-0.15em\left[ \chi \right]\kern-0.15em\right]=\left[\kern-0.15em\left[ \chi_{m-p, n-q} \right]\kern-0.15em\right]=\frac{1}{\Lambda_{x} \Lambda_{y}} \int_{\Lambda_{x}} \int_{\Lambda_{y}} \chi(x, y) e^{{-}j(m-p) \frac{2 \pi}{\Lambda_{x}} x} e^{{-}j(n-q) \frac{2 \pi}{\Lambda_{y}} y} \mathrm{d} x\mathrm{d} y,$$
and $\tilde {\mathbf {Q}}=\left [\begin {array}{cc}\left [\kern -0.15em\left [ \tilde {\chi } \right ]\kern -0.15em\right ] & \mathbf {0} \\ \mathbf {0} & \left [\kern -0.15em\left [ \tilde {\chi } \right ]\kern -0.15em\right ]\end {array}\right ]$, $\left [\kern -0.15em\left [ \tilde {\chi } \right ]\kern -0.15em\right ]=\mathbf {I}-\left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ]$.

Then CBCM successfully applied into the standard transfer matrix(T-matrix) method. The S matrix can be obtained by shifting the terms of the T matrix in Eq. (11):

$$\left[\begin{array}{c} \mathbf{c}_{1}^{-} \\ \mathbf{c}_{2}^{+} \end{array}\right]=\mathbf{S}_{\mathrm{CBCM}}\left[\begin{array}{l} \mathbf{c}_{1}^{+} \\ \mathbf{c}_{2}^{-} \end{array}\right],$$
$$\mathbf{S}_{\mathrm{CBCM}}=\left[\begin{array}{cc} \mathbf{W}_{0} & -\mathbf{W}_{0} \\ \tau \tilde{\mathbf{Q}} \mathbf{V}_{0} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{0}+\mathbf{Q} \mathbf{W}_{0} \end{array}\right]^{{-}1}\left[\begin{array}{cc} -\mathbf{W}_{0} & \mathbf{W}_{0} \\ \tau \tilde{\mathbf{Q}} \mathbf{V}_{0} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{0}-\mathbf{Q} \mathbf{W}_{0} \end{array}\right].$$

For S matrix, the combination with other layers can be through the common Redheffer Star Product and final reflection and transmission coefficients can be obtained [17]. Metal gratings of arbitrary shape in microwave calculated by RCWA with CBCM in the Cartesian only need calculate the fast Fourier transform in the Eq. (13) for the corresponding shape, and the obtained S matrix can directly combine with the modal solutions of the standard RCWA of other layers in the Cartesian.

3. RCWA with CBCM and ASR in the transformed system

3.1 Coordinate transformation

The electric field of the metal region $\Omega _{1}$ is null, while the electric field in other region $\Omega _{2}$ is not null, which will inevitably lead to the electric field being a discontinuous function. The Gibbs phenomenon of the Fourier transform of the discontinuous function E may lead to slower convergence. This problem can however be solved by introducing the concept of adaptive spatial resolution (ASR). In this technique, a new orthogonal coordinate system (u, v, z) is transformed from the Cartesian system (x, y, z), which will increase the spatial resolution around the discontinuous function. The transformation equation is chosen according to Vallius et al [18].

$$x(u)=a_{1x} +a_{2x} u+\frac{a_{3x} }{2\pi } \sin \left(2\pi \frac{u-u_{l-1} }{u_{l} -u_{l-1} } \right) ,$$
where $a_{1x} =\frac {u_{l} x_{l-1} -u_{l-1} x_{l} }{u_{l} -u_{l-1} }$, $a_{2x} =\frac {x_{l} -x_{l-1} }{u_{l} -u_{l-1} }$, $a_{3x} =G\left (u_{l} -u_{l-1} \right )-\left (x_{l} -x_{l-1} \right )$, and G is an almost zero constant, here chosen to be G = 0.001. The choice of jumping point is based on the following formula:
$$\Delta u_{k} =\frac{\sqrt[{3}]{\Delta x_{k} } }{\sqrt[{3}]{\sum _{l}\Delta x_{l} } } ,$$
where $\Delta u_{k} =u_{k} -u_{k-1}$, $\Delta x_{k} =x_{k} -x_{k-1}$. This chosen equation will make the third derivative of x(u) continuous, which can further increase the convergence speed of the Fourier series in the transformed space [19]. Figure 2(a) depicts the plot of the ASR function according to the jump point selection in Eq. (17). The similar relationship between y and v is defined by substituting y and v for x and u , respectively.

 figure: Fig. 2.

Fig. 2. (a) Graph of x(u) and $y(v)$ function, grid point and metal region distribution in (b) xy and (c) uv space

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After the above coordinate transformation in x and y, the distribution diagram of metal region is shown in Fig. 2. Note that in the uv space, the metal domain $\Omega _{1}$ must be obtained from the result of the coordinate transformation. The Fourier transform in the uv space is equal to taking points uniformly in the uv space, which is equivalent to increasing spatial resolution around the discontinuities in the xy space, so the influence of the Gibbs phenomenon can be suppressed.

3.2 Modal equation of RCWA in the transformed system

For the two free-space regions, $\varepsilon _{uv} =1$ and $\mu _{uv} {\rm =}1$, the modal equation of RCWA in the transformed system can be deduced:

$$\frac{\mathrm{d}}{\mathrm{d} \tilde{z}}\left[\begin{array}{l} \mathbf{s}_{u} \\ \mathbf{s}_{v} \end{array}\right]=\mathbf{P}_{u v}\left[\begin{array}{l} \mathbf{u}_{u} \\ \mathbf{u}_{v} \end{array}\right], \frac{\mathrm{d}}{\mathrm{d} \tilde{z}}\left[\begin{array}{l} \mathbf{u}_{u} \\ \mathbf{u}_{v} \end{array}\right]=\mathbf{Q}_{u v}\left[\begin{array}{l} \mathbf{s}_{u} \\ \mathbf{s}_{v} \end{array}\right],$$
$$\mathbf{P}_{u v}=\left[\begin{array}{cc} \tilde{\mathbf{K}}_{u} \left[\kern-0.15em\left[ \varepsilon^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{v} & \left\lceil \left\lfloor \mu^{22} \right\rfloor \right\rceil -\tilde{\mathbf{K}}_{u} \left[\kern-0.15em\left[ \varepsilon^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{u} \\ \tilde{\mathbf{K}}_{v} \left[\kern-0.15em\left[ \varepsilon^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{v}-\left\lfloor\left[\mu^{11}\right\rceil\right\rfloor & -\tilde{\mathbf{K}}_{v} \left[\kern-0.15em\left[ \varepsilon^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{u} \end{array}\right],$$
$$\mathbf{Q}_{u v}=\left[\begin{array}{cc} \tilde{\mathbf{K}}_{u} \left[\kern-0.15em\left[ \mu^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{v} & \left\lceil \left\lfloor \varepsilon^{22} \right\rfloor \right\rceil-\tilde{\mathbf{K}}_{u} \left[\kern-0.15em\left[ \mu^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{u} \\ \tilde{\mathbf{K}}_{v} \left[\kern-0.15em\left[ \mu^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{v}-\left\lfloor\left \lceil \varepsilon^{11}\right\rceil\right\rfloor & -\tilde{\mathbf{K}}_{v} \left[\kern-0.15em\left[ \mu^{33} \right]\kern-0.15em\right]^{{-}1} \tilde{\mathbf{K}}_{u} \end{array}\right],$$
where $\varepsilon ^{11} =\varepsilon _{uv} \frac {g(v)}{f(u)}$, $\varepsilon ^{22} =\varepsilon _{uv} \frac {f(u)}{g(v)}$, $\varepsilon ^{33} =\varepsilon _{uv} f(u)g(v)$, $\mu ^{11} =\mu _{uv} \frac {g(v)}{f(u)}$, $\mu ^{22} =\mu _{uv} \frac {f(u)}{g(v)}$ and $\mu ^{33} =\mu _{uv} f(u)g(v)$, $f(u)=\frac {\partial x}{\partial u} =\frac {\partial x(u)}{\partial u}$, $g(v)=\frac {\partial y}{\partial v} =\frac {\partial y(v)}{\partial v}$. The $\left \lfloor \left \lceil \right \rceil \right \rfloor$ and $\left \lceil \left \lfloor \right \rfloor \right \rceil$ symbols correspond to the application of the Fourier factorization rules as discussed in detail in Ref. [20]. $\tilde {\mathbf {K}}_{u}$ and $\tilde {\mathbf {K}}_{v}$ are normalized wave vectors $\mathbf {K}_{u} /k_{0}$ and $\mathbf {K}_{v} /k_{0}$, $\mathbf {K} _{u} {\rm =diag(}k_{u,p} {\rm )}$ , $k_{u,p} =k_{0} \sqrt {\varepsilon _{r,} {}_{ref} } \sin \theta \cos \varphi -p\frac {2\pi }{\Lambda _{u} }$, $\mathbf {K} _{v} {\rm =diag(}k_{v,q} {\rm )}$, $\mathbf {K}_{v,q} =k_{0} \sqrt {\varepsilon _{r,} {}_{ref} } \sin \theta \sin \varphi -q\frac {2\pi }{\Lambda _{v} }$, s and u are the 2D Fourier transform coefficients of $\mathbf {E}$ and $\mathbf {H}'$.

By combining the $\mathbf {P}_{uv}$ and $\mathbf {Q}_{uv}$ matrices, the wave equation is written in terms of the E components:

$$\frac{\mathrm{d}^{2}}{\mathrm{~d} \tilde{z}^{2}}\left[\begin{array}{l} \mathbf{s}_{u} \\ \mathbf{s}_{v} \end{array}\right]=\mathbf{\Omega}_{u v}^{2}\left[\begin{array}{l} \mathbf{s}_{u} \\ \mathbf{s}_{v} \end{array}\right]=\mathbf{W}_{u v} \boldsymbol{\beta}_{u v}^{2} \mathbf{W}_{u v}^{{-}1}\left[\begin{array}{l} \mathbf{s}_{u} \\ \mathbf{s}_{v} \end{array}\right],$$
where $\boldsymbol {\Omega }_{u v}^{2}=\mathbf {P}_{uv}\mathbf {Q}_{uv}$. $\mathbf {W}_{uv}$ and $\boldsymbol {\beta }_{uv}^{2}$ are the eigenvectors and eigenvalues matrixes of the matrix $\mathbf {P}_{uv}\mathbf {Q}_{uv}$, respectively. The general solutions of the wave equation for $\mathbf {E}_{uv}$ and $\mathbf {H}'_{uv}$ are:
$$\mathbf{E}_{u v}=\left[\begin{array}{c} \mathbf{s}_{u}(z) \\ \mathbf{s}_{v}(z) \end{array}\right]=\mathbf{W}_{u v} e^{-\boldsymbol{\beta}_{u v} k_{0}\left(z-z_{1}\right)} \mathbf{c}_{u v}^{+}+\mathbf{W}_{u v} e^{\boldsymbol{\beta}_{u v} k_{0}\left(z-z_{2}\right)} \mathbf{c}_{u v}^{-},$$
$$\mathbf{H}_{u v}^{\prime}=\left[\begin{array}{l} \mathbf{u}_{u}(z) \\ \mathbf{u}_{v}(z) \end{array}\right]={-}\mathbf{V}_{u v} e^{-\boldsymbol{\beta}_{u v} k_{0}\left(z-z_{1}\right)} \mathbf{c}_{u v}^{+}+\mathbf{V}_{u v} e^{\boldsymbol{\beta}_{u v} k_{0}\left(z-z_{2}\right)} \mathbf{c}_{u v}^{-},$$
where $\mathbf {V}_{u v}=\mathbf {P}_{u v}^{-1} \mathbf {W}_{u v} \boldsymbol {\beta }_{u v}=\mathbf {Q}_{u v} \mathbf {W}_{u v} \boldsymbol {\beta }_{u v}^{-1}$, $\mathbf {c}_{uv}^{+}$ and $\mathbf {c}_{uv}^{-}$ are the coefficient vectors of the propagating waves in positive and negative directions, respectively. The layer starts at $z=z_{1}$ and ends at $z=z_{2}$.

3.3 Combined boundary conditions method in the transformed system

The last step of the computation consists in matching boundary conditions at the surface of the metal grating. The boundary conditions are

$$E_{1 u}=0, E_{1 v}=0 \quad \forall(u, v) \in \Omega_{1}.$$
$$E_{2u} =0, E_{2v} =0 \quad \forall (u,v)\in \Omega _{1} .$$
$$E_{1u} =E_{2u}, E_{1v} =E_{2v} \quad \forall (u,v)\in \Omega _{2} .$$
$$H'_{1u} =H'_{2u}, H'_{1v} =H'_{2v} \quad \forall (u,v)\in \Omega _{2} .$$

Equation (22a) – Eq. (22c) are easily combined:

$$\left[\begin{array}{c} {E_{1u} } \\ {E_{1v} } \end{array}\right]{\rm =}\left[\begin{array}{c} {E_{2u} } \\ {E_{2v} } \end{array}\right] \quad \forall (u,v)\in \Omega.$$

Equation (22b) and (22d) can also be combined:

$$\chi (u,v)\left[\begin{array}{c} {E_{2u} } \\ {E_{2v} } \end{array}\right]+\tau \tilde{\chi }(u,v)\left(\left[\begin{array}{c} {H'_{1u} } \\ {H'_{1v} } \end{array}\right]-\left[\begin{array}{c} {H'_{2u} } \\ {H'_{2v} } \end{array}\right]\right)=0 \quad \forall (u,v)\in \Omega,$$
where $\tau$ is an arbitrary constant, $\chi (u,v)$ and $\tilde {\chi }(u,v)$ are the characteristic functions of set $\Omega _{1}$ and $\Omega _{2}$ in uv space, respectively:
$$\chi (u,v)=\left\{\begin{array}{ll} {1} & {{\rm \; if\; }(u,v)\in \Omega _{1} } \\ {0} & {{\rm \; if\; }(u,v)\in \Omega _{2} } \end{array}\right. ,$$
$$\tilde{\chi }(u,v)=1-\chi (u,v) .$$

By projecting Eq. (23) and (24) onto the exp(-jK${}_{u}$u) exp(-jK${}_{v}$v) basis, we obtain the following matrix equation:

$$\left[\begin{array}{cc} \mathbf{W}_{u} & \mathbf{W}_{u} \\ -\tau \tilde{\mathbf{Q}} \mathbf{V}_{u} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{u} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{1}^{+} \\ \mathbf{c}_{1}^{-} \end{array}\right]=\left[\begin{array}{cc} \mathbf{W}_{u} & \mathbf{W}_{u} \\ -\tau \tilde{\mathbf{Q}} \mathbf{V}_{u}-\mathbf{Q} \mathbf{W}_{u} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{u}-\mathbf{Q} \mathbf{W}_{u} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{2}^{+} \\ \mathbf{c}_{2}^{-} \end{array}\right],$$
where $\mathbf {W}_{u}$ and $\mathbf {V}_{u}$ are eigen-modes for free space in Eq. (21), $\mathbf {Q}=\left [\begin {array}{cc}\left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ] & \mathbf {0} \\ \mathbf {0} & \left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ]\end {array}\right ]$, $\left [\kern -0.15em\left [ \tilde {\chi } \right ]\kern -0.15em\right ]=\mathbf {I}-\left [\kern -0.15em\left [ \chi \right ]\kern -0.15em\right ]$. Note that Eq. (27) is established in the transformed system, so it cannot be directly combined with modal solutions of other layer in the Cartesian. We previously established the conversion relation of the RCWA modal solution in the Cartesian system and the transformed system in Ref. [21] to solve this problem. The transformation matrix T is:
$$\mathbf{T}=\left[\begin{array}{cc} \mathbf{T}_{x} & \mathbf{0} \\ \mathbf{0} & \mathbf{T}_{y} \end{array}\right],$$
where $\left [\mathbf {T}_{x}\right ]_{p q, m n}=\frac {1}{\Lambda _{x} \Lambda _{y}} \int _{\Lambda _{x}} \int _{\Lambda _{y}} g(v) e^{-j\left [k_{x, p} u+k_{y, q} v\right ]+j\left [k_{x, m} x(u)+k_{y, n} v(v)\right ]} \mathrm {d} u\mathrm {d} v$, $\left [\mathbf {T}_{y}\right ]_{p q, m n}=\frac {1}{\Lambda _{x} \Lambda _{y}} \int _{\Lambda _{x}} \int _{\Lambda _{y}} f(u) e^{-j\left [k_{x, p} u+k_{y, q} v\right ]+j\left [k_{x, m} x(u)+k_{y, n} y(v)\right ]} \mathrm {d} u \mathrm {d} v$.

Two free-space layers of zero thickness are added on the upper and lower sides to use the above conversion relation. On the lower side in the Cartesian, the combined boundary is

$$\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\mathbf{V}_{0} & \mathbf{V}_{0} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{0}^{+} \\ \mathbf{c}_{0}^{-} \end{array}\right]=\left[\begin{array}{cc} \mathbf{T} \mathbf{W}_{u} & \mathbf{T} \mathbf{W}_{u} \\ -\mathbf{T} \mathbf{V}_{u} & \mathbf{T} \mathbf{V}_{u} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{1}^{+} \\ \mathbf{c}_{1}^{-} \end{array}\right].$$

On the upper side in the Cartesian, the combined boundary is

$$\left[\begin{array}{cc} \mathbf{T} \mathbf{W}_{u} & \mathbf{T} \mathbf{W}_{u} \\ -\mathbf{T} \mathbf{V}_{u} & \mathbf{T V _ { u }} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{2}^{+} \\ \mathbf{c}_{2}^{-} \end{array}\right]=\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\mathbf{V}_{0} & \mathbf{V}_{0} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{3}^{+} \\ \mathbf{c}_{3}^{-} \end{array}\right].$$

Equation (27), (29) and (30) are easily combined:

$$\left[\begin{array}{c} \mathbf{c}_{0}^{+} \\ \mathbf{c}_{0}^{-} \end{array}\right]=\mathbf{T}_{\mathrm{CBCM}-\mathrm{ASR}}\left[\begin{array}{l} \mathbf{c}_{3}^{+} \\ \mathbf{c}_{3}^{-} \end{array}\right],$$
$$\begin{aligned} & \mathbf{T}_{\mathrm{CBCM}-\mathrm{ASR}}=\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\mathbf{V}_{0} & \mathbf{V}_{0} \end{array}\right]^{{-}1}\left[\begin{array}{cc} \mathbf{T} \mathbf{W}_{u} & \mathbf{T} \mathbf{W}_{u} \\ -\mathbf{T V} & \mathbf{T} \mathbf{V}_{u} \end{array}\right]\left[\begin{array}{cc} \mathbf{W}_{u} & \mathbf{W}_{u} \\ -\tau \tilde{\mathbf{Q}} \mathbf{V}_{u} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{u} \end{array}\right]^{{-}1} \\ & \cdot\left[\begin{array}{cc} \mathbf{W}_{u} & \mathbf{W}_{u} \\ -\tau \mathbf{Q}_{u}-\mathbf{Q} \mathbf{W}_{u} & \tau \tilde{\mathbf{Q}} \mathbf{V}_{u}-\mathbf{Q} \mathbf{W}_{u} \end{array}\right]\left[\begin{array}{cc} \mathbf{T} \mathbf{W}_{u} & \mathbf{T} \mathbf{W}_{u} \\ -\mathbf{T} \mathbf{V}_{u} & \mathbf{T} \mathbf{V}_{u} \end{array}\right]^{{-}1}\left[\begin{array}{cc} \mathbf{W}_{0} & \mathbf{W}_{0} \\ -\mathbf{V}_{0} & \mathbf{V}_{0} \end{array}\right]. \end{aligned}$$

Suppose the $\mathbf {T}_{{\rm CBCM}-{\rm ASR}}$ matrix consists of ABCD elements:

$$\left[\begin{array}{c} \mathbf{c}_{0}^{+} \\ \mathbf{c}_{0}^{-} \end{array}\right]=\left[\begin{array}{ll} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array}\right]\left[\begin{array}{l} \mathbf{c}_{3}^{+} \\ \mathbf{c}_{3}^{-} \end{array}\right].$$

S matrix can be obtained by shifting the terms of the $\mathbf {T}_{{\rm CBCM}-{\rm ASR}}$ matrix:

$$\left[\begin{array}{c} \mathbf{c}_{0}^{-} \\ \mathbf{c}_{3}^{+} \end{array}\right]=\mathbf{S}_{\mathrm{CBCM}-\mathrm{ASR}}\left[\begin{array}{l} \mathbf{c}_{0}^{+} \\ \mathbf{c}_{3}^{-} \end{array}\right],$$
$$\mathbf{S}_{\mathrm{CBCM}-\mathrm{ASR}}=\left[\begin{array}{cc} \mathbf{C A}^{{-}1} & \mathbf{D}-\mathbf{C A}^{{-}1} \mathbf{B} \\ \mathbf{A}^{{-}1} & -\mathbf{A}^{{-}1} \mathbf{B} \end{array}\right].$$

4. Numerical validation and application

As a first example, an L-shaped metal patch in Fig. 2(b) for microwave response is considered to research the effect of $\tau$ values on convergence. The grating parameters are as follows, $\Lambda _{x} =\Lambda _{y} =30 \, {\rm mm}$, $L_{1}=15 \, {\rm mm}$, $L_{2}=10 \, {\rm mm}$ , $L_{3}=5 \, {\rm mm}$ and TE or TM polarisation. The total transmission of L-shaped metal patch at 16 GHz is plotted versus the truncation order in Fig. 3. The value of $\tau$ has a decisive influence on the convergence for the CBCM method with TE polarisation in Fig. 3(a) and TM polarisation in Fig. 3(c). The curves corresponding to 1i converge quickly to the stable results 0.818 and 0.783 for TE and TM polarisation, respectively, while results for 370 and 110i do not converge even with truncation order up to 25. The CBCM-ASR results with TE polarisation in Fig. 3(b) and TM polarisation in Fig. 3(d) show better convergence than CBCM results with the same truncation order, and among these results, $\tau {\rm \; =\; }1i$ also shows the best convergence and the convergence values are 0.815 and 0.788 for TE and TM polarizations, respectively. Since E in CBCM is a discontinuous function, the Gibbs phenomenon in the Fourier domain leads to slower convergence, while the introduction of the ASR function can suppress the influence of Gibbs phenomenon, so CBCM-ASR method exhibits better convergence than CBCM and $\tau {\rm \; =\; }1i$ is chosen for the next calculation. Note, the condition number of the conversion matrix T increases exponentially with the truncation order, and $\mathbf {T}_{{\rm CBCM}-{\rm ASR}}$ in Eq. (32) is much ill-conditioned, so the CBCM-ASR method can only calculate the result with truncation order up to 10 for this L-shaped structure due to the limited computer calculation precision.

 figure: Fig. 3.

Fig. 3. Convergence curves of total transmission at 16 GHz for L-shaped metal patch with different $\tau$ calculated by (a) CBCM, TE polarisation, (b) CBCM-ASR, TE polarisation, (c) CBCM, TM polarisation and (d) CBCM-ASR, TM polarisation. Program language: MATLAB, double precision.

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As second example, a simple square metal patch in microwave is considered to verify the effectiveness of CBCM and CBCM-ASR. The grating parameters are as follows, $\Lambda _{x} =\Lambda _{y} =30 \, {\rm mm}$ and $f=0.5$. The zeroth-order and total reflection and transmission of the square metal patch for normal incidence are illustrated in Fig. 4(a). Clearly, the zeroth-order reflection and transmission of CBCM and High Frequency Structure Simulator (HFSS) results show consistency, and the total reflection and transmission of CBCM-ASR, adaptive spatial resolution with factorization rules (ASR-FR) in Ref. [21] results also show consistency. The total reflection and transmission of CBCM-ASR with N = M = 5 show better convergence than those of CBCM with N = M = 10, so the introduction of ASR function for CBCM can highly improve the convergence. Furthermore, the zeroth-order and total reflection and transmission coincide only at frequencies below 10 GHz and differ greatly at frequencies above 10 GHz. The difference between zeroth-order and total transmission and reflection of CBCM-ASR is shown in Fig. 4(b). Periodically structured gratings can be essentially regarded as frequency selective surfaces (FSS), FSS will diffract an applied wave into discrete directions (or called gratings lobes) if the frequency is higher than $f_{c}$. Assuming the FSS is operated in air for normal incidence, this cutoff condition is $f_{c} =\frac {c}{\Lambda }$, for this example, $f_{c} =10{\rm \; GHz}$, so the difference between zeroth order and total transmission and reflection of CBCM-ASR is caused by gratings lobes.

 figure: Fig. 4.

Fig. 4. (a) Zeroth-order and total transmission/reflection and (b) difference between zeroth-order and total transmission/reflection of square metal patch with CBCM-ASR at 2-18 GHz for normal incidence, CBCM with N = M =10, CBCM-ASR with N = M = 5, ASR-FR with N = M = 8 in Ref. [21]

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As another example, a simple circular metal patch in microwave is considered. The grating parameters are as follows, $\Lambda _{x} =\Lambda _{y} =30 \, {\rm mm}$ and $f=0.5$. The zeroth-order and total reflection and transmission of the circular metal patch for normal incidence are displayed in Fig. 5(a). Clearly, the zeroth-order reflection and transmission of CBCM and HFSS results show consistency, while the zeroth-order and total reflection and transmission of CBCM coincide only at frequencies below 10 GHz and differ greatly at frequencies above 10 GHz. The cutoff condition $f_c$ for this structure is also $10{\rm \; GHz}$, so the difference between zeroth-order and total reflection and transmission in Fig. 5(b) is also caused by the grating lobes, and the CBCM method can correctly distinguish the contribution of the grating lobe to the overall reflection and transmission.

 figure: Fig. 5.

Fig. 5. (a) Zeroth-order and total transmission/reflection and (b) difference between zeroth-order and total transmission/reflection of circular metal patch with CBCM at 2-18 GHz for normal incidence, CBCM with N = M =10

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The CBCM for metamaterials are easily integrated into the previously published Global ETM-NV algorithm [22] to optimize broadband absorbing structures. A 3-layer periodic stepped structures with frequency selective surface (FSS) are designed based on a reduced graphene (rGO) and thermoplastic polyurethane (TPU) composite material prepared in our group. Figure 6 shows the relative permittivity of the rGO/TPU composite material.

 figure: Fig. 6.

Fig. 6. Relative permittivity of rGO/TPU composite material.

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Figure 7(b) shows the CBCM result in good consistence with the calculation result of HFSS. Compared with the structure without the metamaterial layer, the result with FSS layer shows better absorbing effect. This will provide guidance for designing absorbing structures.

 figure: Fig. 7.

Fig. 7. (a) 3-layer rGO/TPU grating slab with FSS. $L_{x}$= 10 mm, $L_{1}$= 8 mm, $L_{2}$= 6 mm, $\phi$ = 4 mm, $h_{1}$= 6 mm, $h_{2}$= 6 mm, $h_{3}$= 3 mm. (b) Reflection of 3-layer dielectric grating slab with and without FSS for normal incidence. CBCM with N = M = 6.

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5. Conclusion

RCWA with CBCM is proposed to analyze optical properties of two-dimensional periodic metamaterial grating with arbitrary shapes in microwave domain, and the ASR function is introduced to increase convergence especially for the crossed metamaterial gratings. Numerically, RCWA with CBCM can get the correct convergence result and distinguish the contribution of the grating lobes to the total transmission and reflection. The ASR technique can improve dramatically the convergence speed of crossed grating. This method will provide a more effective computational tool for the design of metamaterial grating in microwave domain.

Funding

National Natural Science Foundation of China (51772029, 51972029).

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 51972029,51772029.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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7. Y. Khorrami, D. Fathi, and R. C. Rumpf, “Guided-mode resonance filter optimal inverse design using one-and two-dimensional grating,” J. Opt. Soc. Am. B 37(2), 425–432 (2020). [CrossRef]  

8. Y. Khorrami, D. Fathi, A. Khavasi, and R. C. Rumpf, “Dynamical control of multilayer spacetime structures using extended fourier modal method,” IEEE Photonics J. 13(6), 1–10 (2021). [CrossRef]  

9. Y. H. Ko, N. Razmjooei, H. Hemmati, and R. Magnusson, “Perfectly-reflecting guided-mode-resonant photonic lattices possessing mie modal memory,” Opt. Express 29(17), 26971–26982 (2021). [CrossRef]  

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12. F. Montiel and M. Neviere, “Perfectly conducting gratings: a new approach using infinitely thin strips,” Opt. Commun. 144(1-3), 82–88 (1997). [CrossRef]  

13. B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165(1-3), 1–6 (1999). [CrossRef]  

14. G. Granet and B. Guizal, “Analysis of strip gratings using a parametric modal method by fourier expansions,” Opt. Commun. 255(1-3), 1–11 (2005). [CrossRef]  

15. G. Granet, F. Tinasoa, and K. Raniriharinosy, “Analysis of conical diffraction by curved strip gratings by means of the c-method and the combined boundary conditions method,” Prog. Electromagn. Res. M 17, 183–196 (2011). [CrossRef]  

16. S. Peng and C. Shiao, “Scattering of plane waves by metallic gratings,” in 1994 IEEE MTT-S International Microwave Symposium Digest (Cat. No. 94CH3389-4), (IEEE, 1994), pp. 879–882.

17. R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011). [CrossRef]  

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21. L. Wang, D. B. Fang, H. B. Jin, and J. B. Li, “2D rigorous coupled wave analysis with adaptive spatial resolution for multilayer periodic structure,” Opt. Express 30(12), 21295–21308 (2022). [CrossRef]  

22. L. Wang, D. B. Fang, H. B. Jin, and J. B. Li, “A fast modified global rigorous coupled wave analysis for multilayer 2D periodic radar absorber,” IEEE T. Antenn. Propag., submitted for publication.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Model of metal grating, metal grating between two free-space regions of zero thickness, $h$ =0

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Fig. 2.
Fig. 2. (a) Graph of x(u) and $y(v)$ function, grid point and metal region distribution in (b) xy and (c) uv space

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Fig. 3.
Fig. 3. Convergence curves of total transmission at 16 GHz for L-shaped metal patch with different $\tau$ calculated by (a) CBCM, TE polarisation, (b) CBCM-ASR, TE polarisation, (c) CBCM, TM polarisation and (d) CBCM-ASR, TM polarisation. Program language: MATLAB, double precision.

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Fig. 4.
Fig. 4. (a) Zeroth-order and total transmission/reflection and (b) difference between zeroth-order and total transmission/reflection of square metal patch with CBCM-ASR at 2-18 GHz for normal incidence, CBCM with N = M =10, CBCM-ASR with N = M = 5, ASR-FR with N = M = 8 in Ref. [21]

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Fig. 5.
Fig. 5. (a) Zeroth-order and total transmission/reflection and (b) difference between zeroth-order and total transmission/reflection of circular metal patch with CBCM at 2-18 GHz for normal incidence, CBCM with N = M =10

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Fig. 6.
Fig. 6. Relative permittivity of rGO/TPU composite material.

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Fig. 7.
Fig. 7. (a) 3-layer rGO/TPU grating slab with FSS. $L_{x}$= 10 mm, $L_{1}$= 8 mm, $L_{2}$= 6 mm, $\phi$ = 4 mm, $h_{1}$= 6 mm, $h_{2}$= 6 mm, $h_{3}$= 3 mm. (b) Reflection of 3-layer dielectric grating slab with and without FSS for normal incidence. CBCM with N = M = 6.

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Equations (44)

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d d z ~ [ u x u y ] = Q x y [ s x s y ] , d d z ~ [ s x s y ] = P x y [ u x u y ] ,
Q x y = [ K ~ x K ~ y I − K ~ x K ~ x K ~ y K ~ y − I − K ~ y K ~ x ] ,
P x y = [ K ~ x K ~ y I − K ~ x K ~ x K ~ y K ~ y − I − K ~ y K ~ x ] ,
d 2   d z ~ 2 [ s x s y ] = Ω x y 2 [ s x s y ] = W x y β x y 2 W x y − 1 [ s x s y ] ,
E x y = [ s x ( z ) s y ( z ) ] = W x y e − β x y k 0 ( z − z 1 ) c x y + + W x y e β x y k 0 ( z − z 2 ) c x y − ,
H x y ′ = [ u x ( z ) u y ( z ) ] = − V x y e − β x y k 0 ( z − z 1 ) c x y + + V x y e β x y k 0 ( z − z 2 ) c x y − ,
E 1 x = 0 , E 1 y = 0 ∀ ( x , y ) ∈ Ω 1 .
E 2 x = 0 , E 2 y = 0 ∀ ( x , y ) ∈ Ω 1 .
E 1 x = E 2 x , E 1 y = E 2 y ∀ ( x , y ) ∈ Ω 2 .
H 1 x ′ = H 2 x ′ , H 1 y ′ = H 2 y ′ ∀ ( x , y ) ∈ Ω 2 .
[ E 1 x E 1 y ] = [ E 2 x E 2 y ] ∀ ( x , y ) ∈ Ω .
χ ( x , y ) [ E 2 x E 2 y ] + τ χ ~ ( x , y ) ( [ H 1 x ′ H 1 y ′ ] − [ H 2 x ′ H 2 y ′ ] ) = 0 ∀ ( x , y ) ∈ Ω ,
χ ( x , y ) = { 1  if  ( x , y ) ∈ Ω 1 0  if  ( x , y ) ∈ Ω 2 ,
χ ~ ( x , y ) = 1 − χ ( x , y ) .
[ c 1 + c 1 − ] = T C B C M [ c 2 + c 2 − ] ,
T C B C M = [ W 0 W 0 − τ Q ~ V 0 τ Q ~ V 0 ] [ W 0 W 0 − τ Q ~ V 0 − Q W 0 τ Q ~ V 0 − Q W 0 ] ,
[ [ χ ] ] = [ [ χ m − p , n − q ] ] = 1 Λ x Λ y ∫ Λ x ∫ Λ y χ ( x , y ) e − j ( m − p ) 2 π Λ x x e − j ( n − q ) 2 π Λ y y d x d y ,
[ c 1 − c 2 + ] = S C B C M [ c 1 + c 2 − ] ,
S C B C M = [ W 0 − W 0 τ Q ~ V 0 τ Q ~ V 0 + Q W 0 ] − 1 [ − W 0 W 0 τ Q ~ V 0 τ Q ~ V 0 − Q W 0 ] .
x ( u ) = a 1 x + a 2 x u + a 3 x 2 π sin ⁡ ( 2 π u − u l − 1 u l − u l − 1 ) ,
Δ u k = Δ x k 3 ∑ l Δ x l 3 ,
d d z ~ [ s u s v ] = P u v [ u u u v ] , d d z ~ [ u u u v ] = Q u v [ s u s v ] ,
P u v = [ K ~ u [ [ ε 33 ] ] − 1 K ~ v ⌈ ⌊ μ 22 ⌋ ⌉ − K ~ u [ [ ε 33 ] ] − 1 K ~ u K ~ v [ [ ε 33 ] ] − 1 K ~ v − ⌊ [ μ 11 ⌉ ⌋ − K ~ v [ [ ε 33 ] ] − 1 K ~ u ] ,
Q u v = [ K ~ u [ [ μ 33 ] ] − 1 K ~ v ⌈ ⌊ ε 22 ⌋ ⌉ − K ~ u [ [ μ 33 ] ] − 1 K ~ u K ~ v [ [ μ 33 ] ] − 1 K ~ v − ⌊ ⌈ ε 11 ⌉ ⌋ − K ~ v [ [ μ 33 ] ] − 1 K ~ u ] ,
d 2   d z ~ 2 [ s u s v ] = Ω u v 2 [ s u s v ] = W u v β u v 2 W u v − 1 [ s u s v ] ,
E u v = [ s u ( z ) s v ( z ) ] = W u v e − β u v k 0 ( z − z 1 ) c u v + + W u v e β u v k 0 ( z − z 2 ) c u v − ,
H u v ′ = [ u u ( z ) u v ( z ) ] = − V u v e − β u v k 0 ( z − z 1 ) c u v + + V u v e β u v k 0 ( z − z 2 ) c u v − ,
E 1 u = 0 , E 1 v = 0 ∀ ( u , v ) ∈ Ω 1 .
E 2 u = 0 , E 2 v = 0 ∀ ( u , v ) ∈ Ω 1 .
E 1 u = E 2 u , E 1 v = E 2 v ∀ ( u , v ) ∈ Ω 2 .
H 1 u ′ = H 2 u ′ , H 1 v ′ = H 2 v ′ ∀ ( u , v ) ∈ Ω 2 .
[ E 1 u E 1 v ] = [ E 2 u E 2 v ] ∀ ( u , v ) ∈ Ω .
χ ( u , v ) [ E 2 u E 2 v ] + τ χ ~ ( u , v ) ( [ H 1 u ′ H 1 v ′ ] − [ H 2 u ′ H 2 v ′ ] ) = 0 ∀ ( u , v ) ∈ Ω ,
χ ( u , v ) = { 1 i f ( u , v ) ∈ Ω 1 0 i f ( u , v ) ∈ Ω 2 ,
χ ~ ( u , v ) = 1 − χ ( u , v ) .
[ W u W u − τ Q ~ V u τ Q ~ V u ] [ c 1 + c 1 − ] = [ W u W u − τ Q ~ V u − Q W u τ Q ~ V u − Q W u ] [ c 2 + c 2 − ] ,
T = [ T x 0 0 T y ] ,
[ W 0 W 0 − V 0 V 0 ] [ c 0 + c 0 − ] = [ T W u T W u − T V u T V u ] [ c 1 + c 1 − ] .
[ T W u T W u − T V u T V u ] [ c 2 + c 2 − ] = [ W 0 W 0 − V 0 V 0 ] [ c 3 + c 3 − ] .
[ c 0 + c 0 − ] = T C B C M − A S R [ c 3 + c 3 − ] ,
T C B C M − A S R = [ W 0 W 0 − V 0 V 0 ] − 1 [ T W u T W u − T V T V u ] [ W u W u − τ Q ~ V u τ Q ~ V u ] − 1 ⋅ [ W u W u − τ Q u − Q W u τ Q ~ V u − Q W u ] [ T W u T W u − T V u T V u ] − 1 [ W 0 W 0 − V 0 V 0 ] .
[ c 0 + c 0 − ] = [ A B C D ] [ c 3 + c 3 − ] .
[ c 0 − c 3 + ] = S C B C M − A S R [ c 0 + c 3 − ] ,
S C B C M − A S R = [ C A − 1 D − C A − 1 B A − 1 − A − 1 B ] .
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