## Abstract

Formulation of the Fourier modal method for multilevel structures with spatially adaptive resolution is presented for TE and TM polarizations using a slightly reformulated representation for the spatial coordinates. Projections to Fourier base in boundary value problem are used allowing extensions to multilayer profiles with differently placed transitions. We evade the eigenvalue problem in homogeneous regions demanded in the original formulation of the Fourier modal method with adaptive spatial resolution.

©2002 Optical Society of America

Full Article |

PDF Article

**OSA Recommended Articles**
### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

### Equations (28)

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

(1)
$${E}_{1}\left(x,z\right)=\mathrm{exp}\left[i\left({\alpha}_{0}x+{r}_{0}z\right)\right]+\sum _{m}{R}_{m}\mathrm{exp}\left[i\left({\alpha}_{m}x-{r}_{m}z\right)\right],$$
(2)
$${E}_{3}\left(x,z\right)=\sum _{m}{T}_{m}\mathrm{exp}\left[\text{i}\left({\alpha}_{m}x+{t}_{m}z\right)\right],$$
(3)
$${r}_{m}=\sqrt{{\left(k{n}_{1}\right)}^{2}-{\alpha}_{m}^{2}},$$
(4)
$${t}_{m}=\sqrt{{\left(k{n}_{3}\right)}^{2}-{\alpha}_{m}^{2}}$$
(5)
$$\eta {R}_{m}=\Re ({r}_{m}/{r}_{0}){\mid {R}_{m}\mid}^{2}$$
(6)
$$\eta {T}_{m}=C\Re ({t}_{m}/{r}_{0}){\mid {T}_{m}\mid}^{2},$$
(7)
$$\frac{\partial}{\partial x}\frac{\partial {E}_{y}}{\partial x}+\frac{\partial}{\partial z}\frac{\partial {E}_{y}}{\partial z}+{k}^{2}{n}^{2}\left(x\right){E}_{y}=0$$
(8)
$${n}^{2}\left(x\right)\left\{\frac{\partial}{\partial x}\left[\frac{1}{{n}^{2}x}\frac{\partial {H}_{y}}{\partial x}\right]+{k}^{2}{H}_{y}\right\}+\frac{\partial}{\partial z}\frac{\partial {H}_{y}}{\partial z}=o$$
(9)
$$\frac{\partial}{\partial x}\to \frac{\partial u}{\partial x}\frac{\partial}{\partial u},$$
(10)
$$f\left(u\right)=\frac{\partial x}{\partial u}.$$
(11)
$$a\left(u\right)={n}^{2}\left(u\right)f\left(u\right),$$
(12)
$$b\left(u\right)=\frac{f\left(u\right)}{{n}^{2}\left(u\right)}.$$
(13)
$${E}_{y}\left(u,z\right)=\mathrm{exp}\left(i\gamma z\right)\sum _{m}{E}_{m}\mathrm{exp}\left(i{\alpha}_{m}u\right),$$
(14)
$${\gamma}^{2}\mathit{E}={f}^{-1}\left[{k}^{2}\text{a}-\alpha {\text{f}}^{-1}\alpha \right],$$
(15)
$${\gamma}^{2}\mathit{H}={\text{b}}^{-1}\left[{k}^{2}\text{f}-\alpha {\text{a}}^{-1}\alpha \right],$$
(16)
$${x}_{l}\left(u\right)={a}_{1}+{a}_{2}u+\frac{{a}_{3}}{2\pi}\mathrm{sin}\left[2\pi \frac{u-{u}_{l-1}}{{u}_{l}-{u}_{l-1}}\right],$$
(17)
$${a}_{1}=\frac{{u}_{l}{x}_{l-1}-{u}_{l-1}{x}_{l}}{{u}_{l}-{u}_{l-1}}$$
(18)
$${a}_{2}=\frac{{x}_{l}-{x}_{l-1}}{{u}_{l}-{u}_{l-1}}$$
(19)
$${a}_{3}=G\left({u}_{l}-{u}_{l-1}\right)-\left({x}_{l}-{x}_{l-1}\right)$$
(20)
$${E}_{y}\left(u,z\right)=\sum _{v=1}^{V}\left\{{A}_{v,j}\mathrm{exp}\left[\text{i}{\gamma}_{v,j}\left(z-{h}_{j}\right)\right]+{B}_{v,j}\mathrm{exp}\left[-\text{i}{\gamma}_{v,j}\left(z-{h}_{j+1}\right)\right]\right\}$$
(20)
$$\times \sum _{m=-M}^{M}{E}_{m,v}^{j}\mathrm{exp}\left[\text{i}{\alpha}_{m}{u}_{j}\left(x\right)\right],$$
(21)
$${E}_{y,j}\left(x,{h}_{j}\right)={E}_{y,j+1}\left(x,{h}_{j}\right)$$
(22)
$$\frac{\partial}{\partial z}\left[{E}_{y,j}\left(x,{h}_{j}\right)\right]=\frac{\partial}{\partial z}\left[{E}_{y,j+1}\left(x,{h}_{j}\right)\right]$$
(23)
$${H}_{y,j}\left(x,{h}_{j}\right)={H}_{y,j+1}\left(x,{h}_{j}\right)$$
(24)
$$\frac{\partial}{\partial z}\left[\frac{1}{{\u220a}_{j}\left(x\right)}{H}_{y,j}\left(x,{h}_{j}\right)\right]=\frac{\partial}{\partial z}\left[\frac{1}{{\u220a}_{j+1}\left(x\right)}{H}_{y,j+1}\left(x,{h}_{j}\right)\right].$$
(25)
$$\mathbf{K}={\left[K\right]}_{p,m}\frac{1}{d}{\int}_{0}^{d}f\left(u\right)\mathrm{exp}\left[-\text{i}{\alpha}_{p}x\left(u\right)+\text{i}{\alpha}_{m}u\right]du,$$
(26)
$${\mathbf{E}}_{j}=\mathbf{K}{\mathbf{E}}^{u},\phantom{\rule{.9em}{0ex}}{\mathbf{H}}_{j}=\mathbf{K}{\mathbf{H}}^{u},\phantom{\rule{.9em}{0ex}}{\mathbf{Q}}_{j}=\mathbf{K}{\mathbf{Q}}^{u}.$$
(27)
$$\left[\begin{array}{cc}{\mathbf{H}}_{j+1}& -{\mathbf{H}}_{j}\\ {\mathbf{Q}}_{j+1}{\mathbf{\Gamma}}_{j+1}& {\mathbf{Q}}_{j}{\mathbf{\Gamma}}_{j}\end{array}\right]\phantom{\rule{.2em}{0ex}}\left[\begin{array}{c}{\mathbf{A}}_{j+1}\\ {\mathbf{B}}_{j}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{H}}_{j+1}{\mathbf{X}}_{j}& -{\mathbf{H}}_{j}{\mathbf{X}}_{j+1}\\ {\mathbf{Q}}_{j+1}{\mathbf{\Gamma}}_{j+1}{\mathbf{X}}_{j}& {\mathbf{Q}}_{j}{\mathbf{\Gamma}}_{j}{\mathbf{X}}_{j+1}\end{array}\right]\phantom{\rule{.2em}{0ex}}\left[\begin{array}{c}{\mathbf{A}}_{j}\\ {\mathbf{B}}_{j+1}\end{array}\right],$$