## Abstract

The coupling interactions in electromagnetic scattering from a finite array of two-dimensional identical cavities engraved in a perfectly electric conducting screen covered with stratified dielectric coating are presented using an algorithm based on the hybrid finite element-boundary integral method. The solution to the scattering from a finite array of cavities is approximated using the array factor method, which does not take into account the coupling between the cavities, and is compared to the recently developed finite element-boundary element method to demonstrate the importance of the inclusion of the coupling effect. Dependence of the coupling interactions between the cavities on various parameters such as separation periods, incident angle of the plane-wave excitation, and permittivity and thickness of the dielectric coating is demonstrated quantitatively through several numerical examples.

© 2012 Optical Society of America

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### Equations (19)

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(1)
$$\nabla \xb7(\frac{1}{p(\mathit{\rho})}\nabla {u}_{z}^{t})+q(\mathit{\rho}){k}_{0}^{2}{u}_{z}^{t}=0,$$
(2)
$${\left[\begin{array}{ccc}{M}_{ii}& {M}_{ib}& 0\\ {M}_{bi}& {M}_{bb}& {M}_{bo}\\ 0& {M}_{ob}& {M}_{oo}\end{array}\right]}^{(j)}{\left[\begin{array}{c}{u}_{i}\\ {u}_{b}\\ {u}_{o}\end{array}\right]}^{(j)}={\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]}^{(j)},$$
(3)
$$\left[\begin{array}{cccc}{[M]}^{(1)}& 0& \dots & 0\\ 0& {[M]}^{(2)}& \dots & 0\\ \vdots & \ddots & \vdots \\ 0& 0& \dots & {[M]}^{(J)}\end{array}\right]\left[\begin{array}{c}{[u]}^{(1)}\\ {[u]}^{(2)}\\ \vdots \\ {[u]}^{(J)}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ \vdots \\ 0\end{array}\right].$$
(4)
$${u}_{z}(\mathit{\rho})={u}_{z}^{a}(\mathit{\rho})-{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}{u}_{z}({\mathit{\rho}}^{\prime})\frac{\partial {G}^{e}(\mathit{\rho},{\mathit{\rho}}^{\prime})}{\partial {n}^{\prime}}\mathrm{d}{\mathrm{\Gamma}}^{\prime}$$
(5)
$${u}_{z}(\mathit{\rho})={u}_{z}^{a}(\mathit{\rho})+{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}{G}^{h}(\mathit{\rho},{\mathit{\rho}}^{\prime})\frac{\partial {u}_{z}({\mathit{\rho}}^{\prime})}{\partial {n}^{\prime}}\mathrm{d}{\mathrm{\Gamma}}^{\prime}$$
(6)
$$[{u}_{o}]=[S][{u}_{b}]+[T].$$
(7)
$$\left[\begin{array}{c}{\begin{array}{c}[{u}_{0}]\end{array}}^{(1)}\\ {\begin{array}{c}[{u}_{0}]\end{array}}^{(2)}\\ \vdots \\ {\begin{array}{c}[{u}_{0}]\end{array}}^{(J)}\end{array}\right]=\left[\begin{array}{cccc}{\begin{array}{c}[S]\end{array}}^{(11)}& {\begin{array}{c}[S]\end{array}}^{(12)}& \dots & {\begin{array}{c}[S]\end{array}}^{(1J)}\\ {\begin{array}{c}[S]\end{array}}^{(21)}& {\begin{array}{c}[S]\end{array}}^{(22)}& \dots & {\begin{array}{c}[S]\end{array}}^{(2J)}\\ \vdots & \ddots & \vdots \\ {\begin{array}{c}[S]\end{array}}^{(J1)}& {\begin{array}{c}[S]\end{array}}^{(J2)}& \dots & {\begin{array}{c}[S]\end{array}}^{(JJ)}\end{array}\right]\left[\begin{array}{c}{\begin{array}{c}[{u}_{b}]\end{array}}^{(1)}\\ {\begin{array}{c}[{u}_{b}]\end{array}}^{(2)}\\ \vdots \\ {\begin{array}{c}[{u}_{b}]\end{array}}^{(J)}\end{array}\right]\phantom{\rule{0ex}{0ex}}+\left[\begin{array}{c}{\begin{array}{c}[T]\end{array}}^{(1)}\\ {\begin{array}{c}[T]\end{array}}^{(2)}\\ \vdots \\ {\begin{array}{c}[T]\end{array}}^{(J)}\end{array}\right],$$
(8)
$$\left[\begin{array}{cccc}{[{M}^{\prime}]}^{(1)}& {[C]}^{(12)}& \dots & {[C]}^{(1J)}\\ {[C]}^{(21)}& {[{M}^{\prime}]}^{(2)}& \dots & {[C]}^{(2J)}\\ \vdots & \ddots & \vdots \\ {[C]}^{(J1)}& {[C]}^{(J2)}& \dots & {[{M}^{\prime}]}^{(J)}\end{array}\right]\left[\begin{array}{c}{[{u}^{\prime}]}^{(1)}\\ {[{u}^{\prime}]}^{(2)}\\ \vdots \\ {[{u}^{\prime}]}^{(J)}\end{array}\right]=\left[\begin{array}{c}{[F]}^{(1)}\\ {[F]}^{(2)}\\ \vdots \\ {[F]}^{(J)}\end{array}\right],$$
(9)
$${[C]}^{(ij)}=\left[\begin{array}{cc}0& 0\\ 0& {[{M}_{bo}]}^{(i)}{[S]}^{(ij)}\end{array}\right].$$
(10)
$${[{M}^{\prime}]}^{(j)}=\left[\begin{array}{cc}{[{M}_{ii}]}^{(j)}& {[{M}_{ib}]}^{(j)}\\ {[{M}_{bi}]}^{(j)}& {[{M}_{bb}]}^{(j)}+{[{M}_{bo}]}^{(j)}{[S]}^{(jj)}\end{array}\right],$$
(11)
$${[F]}^{(j)}=\left[\begin{array}{c}0\\ -{[{M}_{bo}]}^{(j)}{[T]}^{(j)}\end{array}\right],$$
(12)
$${[{u}^{\prime}]}^{(j)}=\left[\begin{array}{c}{[{u}_{i}]}^{(j)}\\ {[{u}_{b}]}^{(j)}\end{array}\right],$$
(13)
$$E(\mathit{\rho})=-2\widehat{z}{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}({E}_{z}({\mathit{\rho}}^{\prime}{)}_{{|}_{{y}^{\prime}=0}})\frac{\partial}{\partial y}{G}^{\mathrm{HSD}}(\mathit{\rho},{\mathit{\rho}}^{\prime})\mathrm{d}{x}^{\prime},$$
(14)
$$H(\mathit{\rho})=+2\widehat{z}{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}\left(\frac{\partial}{\partial {y}^{\prime}}{H({\mathit{\rho}}^{\prime})}_{{|}_{{y}^{\prime}=0}}\right){G}^{\mathrm{HSD}}(\mathit{\rho},{\mathit{\rho}}^{\prime})\mathrm{d}{x}^{\prime},$$
(15)
$${\stackrel{\sim}{G}}^{\mathrm{HSD}}(y,{y}^{\prime},{k}_{x})=\frac{{e}^{-j{k}_{{y}_{0}}y}{e}^{j{k}_{{y}_{1}}{y}^{\prime}}}{j({k}_{{y}_{1}}+{k}_{{y}_{0}}){e}^{j({k}_{{y}_{1}}-{k}_{{y}_{0}}){t}_{s}}},\phantom{\rule{2em}{0ex}}{t}_{s}<y.$$
(16)
$${\stackrel{\sim}{G}}^{\mathrm{HSD}}(y,{y}^{\prime},{k}_{x})=\sum _{q=1}^{Q}{\alpha}_{q}{e}^{{S}_{q}{k}_{{y}_{0}}}\frac{{e}^{-j{k}_{{y}_{0}}y}}{j{k}_{{y}_{0}}},\phantom{\rule{2em}{0ex}}{t}_{s}<y.$$
(17)
$${G}^{\mathrm{HSD}}(\mathit{\rho},{\mathit{\rho}}^{\prime})=\frac{-j}{4}\sum _{q=1}^{Q}{\alpha}_{q}{H}_{0}^{2}({k}_{0}{R}_{q}),$$
(18)
$$E(\mathit{\rho})=\phantom{\rule{0ex}{0ex}}\widehat{z}{k}_{0}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi \left|{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}({E}_{z}({\mathit{\rho}}^{\prime}{)}_{{|}_{{y}^{\prime}=0}})\left(\sum _{q=1}^{Q}{\alpha}_{q}{e}^{-j{k}_{0}{R}_{q}}\right)\mathrm{d}{x}^{\prime}\right|,$$
(19)
$$H(\mathit{\rho})=\phantom{\rule{0ex}{0ex}}\widehat{z}\frac{1}{\sqrt{{k}_{0}}}|{\int}_{\sum _{j=1}^{J}{{\mathrm{\Gamma}}_{B}}_{j}}(\frac{\partial}{\partial {y}^{\prime}}H({\mathit{\rho}}^{\prime}{)}_{{|}_{{y}^{\prime}=0}})\left(\sum _{q=1}^{Q}{\alpha}_{q}{e}^{-j{k}_{0}{R}_{q}}\right)\mathrm{d}{x}^{\prime}|,$$