## Abstract

We present a stable and efficient method for the Bloch-mode computation of one-dimensional grating waveguides. The approach uses the Fourier modal method and the **S**-matrix algorithm to remove numerical instabilities. The use of perfectly matched layers provide a high accuracy. Numerical results obtained for different lamellar grating waveguides and for both TE and TM polarizations illustrate the performance of the approach.

© 2002 Optical Society of America

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### Equations (8)

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(1)
$$\frac{{\partial}^{2}{E}_{y}}{\partial {z}^{2}}=-{k}_{0}^{2}\u220a{E}_{y}-\frac{{\partial}^{2}{E}_{y}}{\partial {x}^{2}},$$
(2)
$$\frac{{\partial}^{2}{E}_{y}}{\partial {z}^{2}}=-{k}_{0}^{2}{\mu}_{\mathit{xx}}\u220a{E}_{y}-{\mu}_{\mathit{xx}}\frac{\partial}{\partial x}\left(\frac{1}{{\mu}_{\mathit{zz}}}\frac{\partial {E}_{y}}{\partial x}\right),$$
(3)
$${E}_{y}=\sum _{m=-\infty}^{\infty}{S}_{m}(z)exp(\mathit{jmKx}),$$
(4)
$${S}_{p}(z)=\sum _{m}{W}_{m}\{{u}_{m}exp[-{k}_{0}{\mathrm{\lambda}}_{m}z]+{d}_{m}exp[{k}_{0}{\mathrm{\lambda}}_{m}z]\},$$
(5)
$$\left(\begin{array}{c}{\mathbf{d}}_{2}\\ {\mathbf{u}}_{2}\end{array}\right)=\mathbf{T}\left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{1}\end{array}\right),\hspace{1em}\hspace{1em}\hspace{1em}$$
(6)
$$\left(\begin{array}{c}{\mathbf{d}}_{2}\\ {\mathbf{u}}_{1}\end{array}\right)=\left[\begin{array}{c}{\mathbf{S}}_{11}{\mathbf{S}}_{12}\\ {\mathbf{S}}_{21}{\mathbf{S}}_{22}\end{array}\right]\left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{2}\end{array}\right),$$
(7)
$$\mathbf{T}\left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{1}\end{array}\right)=\beta \left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{1}\end{array}\right),$$
(8)
$$\left[\begin{array}{rr}{\mathbf{S}}_{11}& \mathbf{0}\\ {\mathbf{S}}_{21}& -\mathbf{1}\end{array}\right]\left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{1}\end{array}\right)=\beta \left[\begin{array}{rr}\mathbf{I}& -{\mathbf{S}}_{12}\\ \mathbf{0}& -{\mathbf{S}}_{22}\end{array}\right]\left(\begin{array}{c}{\mathbf{d}}_{1}\\ {\mathbf{u}}_{1}\end{array}\right).$$