## Abstract

We demonstrate that the beam-propagation method can be employed to calculate the electric-field amplitude inside and outside a grating structure excited by an arbitrary incident field. We establish the accuracy of the method by comparing our results for constant-period gratings with results obtained from other theoretical methods. Subsequently, we analyze thick-focusing gratings with the beam-propagation method.

© 1982 Optical Society of America

Full Article |

PDF Article
### Equations (7)

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

(1)
$${\nabla}^{2}E+\frac{{\omega}^{2}}{{c}^{2}}{n}^{2}(x,y,z)E=0,$$
(2)
$$n(\overline{x})=n+\delta n(\overline{x}),$$
(3)
$$E(x,y,\mathrm{\Delta}z)=\text{exp}\left[-i\mathrm{\Delta}z{({{\nabla}_{\perp}}^{2}+{\left(\frac{\omega}{c}\right)}^{2}{(n+\delta n)}^{2})}^{1/2}\right]\hspace{0.17em}E(x,y,0).$$
(4)
$${\left[{{\nabla}_{\perp}}^{2}+{\left(\frac{\omega}{c}\right)}^{2}{(n+\delta n)}^{2}\right]}^{1/2}~{\left[{{\nabla}_{\perp}}^{2}+{\left(\frac{\omega}{c}\right)}^{2}{n}^{2}\right]}^{1/2}+\frac{\omega}{c}\delta n.$$
(5)
$${({{\nabla}_{\perp}}^{2}+{k}^{2})}^{1/2}=\frac{{{\nabla}_{\perp}}^{2}}{{({{\nabla}_{\perp}}^{2}+{k}^{2})}^{1/2}+k}+k.$$
(6)
$$E(x,y,z)=\text{exp}(-ikz)\u220a(x,y,z),$$
(7)
$$\begin{array}{l}\u220a(x,y,\mathrm{\Delta}z)=\text{exp}\hspace{0.17em}\left\{-i\frac{\mathrm{\Delta}z}{2}\hspace{0.17em}\left[\frac{{{\nabla}_{\perp}}^{2}}{{({{\nabla}_{\perp}}^{2}+{k}^{2})}^{1/2}+k}\right]\right\}\times \text{exp}\hspace{0.17em}\left(-i\mathrm{\Delta}zk\frac{\delta n}{n}\right)\\ \times \hspace{0.17em}\text{exp}\left\{-i\frac{\mathrm{\Delta}z}{2}\hspace{0.17em}\left[\frac{{{\nabla}_{\perp}}^{2}}{{({{\nabla}_{\perp}}^{2}+{k}^{2})}^{1/2}+k}\right]\right\}\u220a(x,y,0)+0(\mathrm{\Delta}{z}^{3}).\end{array}$$