## Abstract

The coordinate transformation method (C method) with adaptive spatial resolution and the Fourier modal method (FMM) are compared in the case of conducting discontinuous multilevel gratings in TM polarization. A procedure permitting analysis of such gratings more efficiently with the C method than with the FMM is presented. The C method is observed to converge more rapidly than the FMM, whose instabilities are shown to harm the convergence in the aforementioned case.

© 2002 Optical Society of America

Full Article |

PDF Article
### Equations (27)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\frac{{\partial}^{2}}{\partial {x}^{2}}{H}_{y}(x,z)+\frac{{\partial}^{2}}{\partial {z}^{2}}{H}_{y}(x,z)+{k}^{2}{n}^{2}{H}_{y}(x,z)=0,$$
(4)
$$\frac{\mathrm{d}x}{\mathrm{d}u}=f,$$
(5)
$$\frac{\mathrm{d}a}{\mathrm{d}u}=h,$$
(6)
$$\frac{\partial}{\partial x}=\frac{1}{f}\frac{\partial}{\partial u}-\frac{1}{f}h\frac{\partial}{\partial v},$$
(7)
$$\frac{\partial}{\partial z}=\frac{\partial}{\partial v}.$$
(8)
$$\frac{1}{f}\left(h\frac{1}{f}h+f\right)\frac{{\partial}^{2}}{\partial {v}^{2}}{H}_{y}(u,v)-\frac{1}{f}\left(h\frac{1}{f}\frac{\partial}{\partial u}+\frac{\partial}{\partial u}\frac{1}{f}h\right)\frac{\partial}{\partial v}{H}_{y}(u,v)+\frac{1}{f}\frac{\partial}{\partial u}\frac{1}{f}\frac{\partial}{\partial u}{H}_{y}(u,v)+{k}^{2}{n}^{2}{H}_{y}(u,v)=0.$$
(9)
$${H}_{y}(u,v)=\sum _{m}{H}_{m}exp[\mathrm{i}({\alpha}_{m}u+\gamma v)],$$
(10)
$${H}_{y}^{\prime}(u,v)=\frac{1}{\mathrm{i}}\frac{\partial}{\partial v}{H}_{y}(u,v),$$
(11)
$$\left[\begin{array}{cc}-\mathit{\alpha}{\mathbf{f}}^{-1}\mathit{\alpha}+\mathbf{f}{k}^{2}{n}^{2}& \mathbf{0}\\ \mathbf{0}& \mathbf{I}\end{array}\right]\left[\begin{array}{c}\mathbf{H}\\ {\mathbf{H}}^{\prime}\end{array}\right]=\gamma \left[\begin{array}{cc}-{\mathbf{hf}}^{-1}\mathit{\alpha}-\mathit{\alpha}{\mathbf{f}}^{-1}\mathbf{h}& {\mathbf{hf}}^{-1}\mathbf{h}+\mathbf{f}\\ \mathbf{I}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathbf{H}\\ {\mathbf{H}}^{\prime}\end{array}\right],$$
(12)
$${\mathbf{L}}_{\mathrm{A}}^{-1}{\mathbf{L}}_{\mathrm{B}}\tilde{\mathbf{H}}=\frac{1}{\gamma}\tilde{\mathbf{H}},$$
(13)
$${H}_{y,1}(u,v)=\sum _{l}{A}_{l}\sum _{m}{H}_{\mathit{ml},1}^{+}exp[\mathrm{i}({\alpha}_{m}u+{\gamma}_{l,1}^{+}v)]+\sum _{l}{R}_{l}\sum _{m}{H}_{\mathit{ml},1}^{-}exp[\mathrm{i}({\alpha}_{m}u+{\gamma}_{l,1}^{-}v)],$$
(14)
$${H}_{y,2}(u,v)=\sum _{l}{T}_{l}\sum _{m}{H}_{\mathit{ml},2}^{+}exp[\mathrm{i}({\alpha}_{m}u+{\gamma}_{l,2}^{+}v)],$$
(15)
$$\sum _{l}{A}_{l}{H}_{\mathit{ml},1}^{+}+\sum _{l}{R}_{l}{H}_{\mathit{ml},1}^{-}=\sum _{l}{T}_{l}{H}_{\mathit{ml},2}^{+}.$$
(16)
$$\mathbf{t}=f\stackrel{\u02c6}{\mathbf{x}}+h\stackrel{\u02c6}{\mathbf{z}},$$
(17)
$${E}_{\mathrm{t}}=\mathbf{t}\xb7\mathbf{E}={\mathit{fE}}_{x}+{\mathit{hE}}_{z}.$$
(18)
$${E}_{x}=\frac{1}{{n}^{2}}\frac{{Z}_{0}}{\mathrm{i}}\frac{\partial}{\partial v}{H}_{y},$$
(19)
$${E}_{z}=-\frac{1}{{n}^{2}}\frac{{Z}_{0}}{\mathrm{i}}\frac{1}{f}\left(\frac{\partial}{\partial u}{H}_{y}-h\frac{\partial}{\partial v}{H}_{y}\right),$$
(20)
$$\left[\begin{array}{cc}{\mathbf{H}}_{1}^{+}& {\mathbf{H}}_{1}^{-}\\ {\mathbf{G}}_{1}^{+}& {\mathbf{G}}_{1}^{-}\end{array}\right]\left[\begin{array}{c}\mathbf{A}\\ \mathbf{R}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{H}}_{2}^{+}& {\mathbf{H}}_{2}^{-}\\ {\mathbf{G}}_{2}^{+}& {\mathbf{G}}_{2}^{-}\end{array}\right]\left[\begin{array}{c}\mathbf{T}\\ \mathbf{0}\end{array}\right],$$
(21)
$${\mathbf{G}}^{\pm}=\frac{1}{{n}^{2}}[{\mathbf{hf}}^{-1}\mathit{\alpha}-(\mathbf{f}+{\mathbf{hf}}^{-1}\mathbf{h})]{\mathbf{H}}^{\pm}{\mathbf{\Gamma}}^{\pm},$$
(22)
$${\eta}_{l}^{R}=\mathfrak{R}\{{\gamma}_{l,1}^{-}/{\gamma}_{0}\}|{R}_{l}{|}^{2}$$
(23)
$${\eta}_{l}^{T}=C\mathfrak{R}\{{\gamma}_{l,2}^{+}/{\gamma}_{0}\}|{T}_{l}{|}^{2},$$
(24)
$${x}_{j}(u)={F}_{j}(u)={a}_{1}+{a}_{2}u+\frac{{a}_{3}}{2\pi}sin\left(2\pi \frac{u-{u}_{j-1}}{{u}_{j}-{u}_{j-1}}\right).$$
(25)
$${a}_{1}=\frac{{u}_{j}{x}_{j-1}-{u}_{j-1}{x}_{j}}{{u}_{j}-{u}_{j-1}},$$
(26)
$${a}_{2}=\frac{{x}_{j}-{x}_{j-1}}{{u}_{j}-{u}_{j-1}},$$
(27)
$${a}_{3}=G({u}_{j}-{u}_{j-1})-({x}_{j}-{x}_{j-1}),$$