## Abstract

A multiple-scattering technique that was recently developed to evaluate wave diffraction by two superposed gratings is extended to situations in which there is an arbitrary number of gratings. In this approach the diffraction process can be represented in terms of a flow graph that serves as a template to construct algorithms for calculating the intensity of any diffracted order. We show that such calculations do not require a large computer memory if they are implemented by judiciously tracking the relevant diffracted order throughout the flow paths. Using two types of typical grating structures as examples, we also investigate the effect of the relative grating phase on the diffraction efficiency. We thus find that the multiple-scattering analysis can readily identify those grating structures that are sensitive to the relative phase relationship.

© 1993 Optical Society of America

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### Equations (17)

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(1)
$$\begin{array}{l}\u220a\left(r\right)={\u220a}_{0}\left[1+g\left(r\right)\right],\\ \phantom{\rule{0.7em}{0ex}}g\left(r\right)={\displaystyle \sum _{\mu =1}^{N}{M}_{\mu}\phantom{\rule{0.2em}{0ex}}\text{cos}\left({K}_{\mu}\xb7r+{\theta}_{\mu}\right)}.\end{array}$$
(2)
$$\left({\nabla}^{2}+{k}_{0}^{2}\right)E\left(r\right)=-{k}_{0}^{2}g\left(r\right)E\left(r\right).$$
(3)
$$E\left(r\right)={E}^{\left(0\right)}\left(r\right)+{k}_{0}^{2}{\displaystyle \int G\left(r,{r}_{i}\right)g\left({r}_{i}\right)E\left({r}_{i}\right)\mathrm{d}{r}_{i}},$$
(4)
$$E\left(r\right)={\displaystyle \sum _{m=0}^{\infty}{\displaystyle \sum _{\nu \left(m\right)}{E}_{n\left(m\right)}}},$$
(5)
$$\begin{array}{l}n\left(m+1\right)\\ =\left[{n}_{1}\left(m+1\right),{n}_{2}\left(m+1\right),\dots ,{n}_{N}\left(m+1\right)\right]\\ =\{\begin{array}{r}\hfill \left[{n}_{1}\left(m+1\right)+1,{n}_{2}\left(m+1\right),\dots ,{n}_{N}\left(m+1\right)\right]\\ \hfill \left[{n}_{1}\left(m+1\right)-1,{n}_{2}\left(m+1\right),\dots ,{n}_{N}\left(m+1\right)\right]\\ \hfill \text{for}\phantom{\rule{0.2em}{0ex}}\mu \left(m+1\right)=1\\ \hfill \left[{n}_{1}\left(m+1\right),{n}_{2}\left(m+1\right)+1,\dots ,{n}_{N}\left(m+1\right)\right]\\ \hfill \left[{n}_{1}\left(m+1\right),{n}_{2}\left(m+1\right)-1,\dots ,{n}_{N}\left(m+1\right)\right]\\ \hfill \text{for}\phantom{\rule{0.2em}{0ex}}\mu \left(m+1\right)=2\\ \hfill \vdots \\ \hfill \left[{n}_{1}\left(m+1\right),{n}_{2}\left(m+1\right),\dots ,{n}_{N}\left(m+1\right)+1\right]\\ \hfill \left[{n}_{1}\left(m+1\right),{n}_{2}\left(m+1\right),\dots ,{n}_{N}\left(m+1\right)-1\right]\\ \hfill \text{for}\phantom{\rule{0.2em}{0ex}}\mu \left(m+1\right)=N\end{array},\end{array}$$
(6)
$${E}_{n\left(m\right)}={A}_{n\left(m\right)}{\Psi}_{n\left(m\right)}\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i{\overline{k}}_{n\left(m\right)}\xb7r\right),$$
(7)
$${\overline{k}}_{n\left(m\right)}=\stackrel{\u02c6}{x}{u}_{n\left(m\right)}+\stackrel{\u02c6}{z}{w}_{n\left(m\right)},$$
(8)
$${u}_{n\left(m\right)}={u}_{0}+{n}_{1}\left(m\right){U}_{1}+{n}_{2}\left(m\right){U}_{2}+\cdots +{n}_{N}\left(m\right){U}_{N},$$
(9)
$${w}_{n\left(m\right)}={\left[{k}_{0}^{2}-{u}_{n\left(m\right)}^{2}\right]}^{1/2}.$$
(10)
$${A}_{n\left(m\right)}={\displaystyle \prod _{q=1}^{m}{\alpha}_{n\left(q\right)}}={\displaystyle \prod _{q=1}^{m}\frac{{k}_{0}^{2}{M}_{\mu \left(q\right)}{z}_{0}}{4{\omega}_{n\left(q\right)}}\text{exp}\left[ir\left(q\right){\theta}_{\mu \left(q\right)}\right]},$$
(11)
$${\Psi}_{n\left(m\right)}={L}^{-1}{\left\{\frac{{i}^{m}}{s\left(s+i{\Delta}_{n\left(m\right)}\right){\displaystyle \prod _{q=1}^{m-1}\left[s+i\left({\Delta}_{n\left(m\right)}-{\Delta}_{n\left(q\right)}\right)\right]}}\right\}}_{\zeta =1},$$
(12)
$${\Delta}_{n\left(q\right)}=\left[{w}_{n\left(q\right)}-{\upsilon}_{n\left(q\right)}\right]{z}_{0},$$
(13)
$${\upsilon}_{n\left(q\right)}={\upsilon}_{0}+{n}_{1}\left(q\right){V}_{1}+{n}_{2}\left(q\right){V}_{2}+\cdots +{n}_{N}\left(q\right){V}_{N},$$
(14)
$${E}_{{n}_{1},{n}_{2},\dots ,{n}_{N}}={\displaystyle \sum _{m=0}^{\infty}{\displaystyle \sum _{\nu \left(m\right)}{E}_{{n}_{1},{n}_{2},\dots ,{n}_{N}}^{\left(m\right)}}},$$
(15)
$${E}^{\left(m\right)}={\displaystyle \sum _{\nu \left(m\right)}{E}_{n\left(m\right)}}\le \left|{E}^{\left(m\right)}\right|={\displaystyle \sum _{\mu \left(m\right)}\left|{E}_{n\left(m\right)}\right|}.$$
(16)
$${\left|{E}^{\left(m\right)}\right|}_{\text{max}}=\frac{{\left(2N{\alpha}_{\text{max}}\right)}^{m}}{m!}.$$
(17)
$$\frac{{\left|{E}^{\left(m+1\right)}\right|}_{\text{max}}}{{\left|{E}^{\left(m\right)}\right|}_{\text{max}}}=\frac{\frac{{\left(2N{\alpha}_{\text{max}}\right)}^{m+1}}{\left(m+1\right)!}}{\frac{{\left(2N{\alpha}_{\text{max}}\right)}^{m}}{m!}}=\frac{2N{\alpha}_{\text{max}}}{m+1},$$