## Abstract

Rigorous coupled-wave theory of diffraction by dielectric gratings is extended to cover *E*-mode polarization and losses. Unlike in the *H*-mode-polarization case, it is shown that, in the *E*-mode case, direct coupling exists between all diffracted orders rather than just between adjacent orders.

© 1983 Optical Society of America

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### Equations (16)

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(1)
$$\u220a(x,z)={\widehat{\u220a}}_{0}+{\u220a}_{1}cos[K(xsin\varphi +zcos\varphi )],$$
(2)
$${\widehat{\u220a}}_{0}={\u220a}_{0}-j{\sigma}_{0}/\omega {\varepsilon}_{0},$$
(3)
$${\nabla}^{2}\u0112+\nabla \left(\u0112\cdot \frac{\nabla \u220a}{\u220a}\right)+{k}^{2}\u220a(x,z)\u0112=0,$$
(4)
$${\nabla}^{2}\overline{H}+\frac{\nabla \u220a}{\u220a}\times \nabla \times \overline{H}+{k}^{2}\u220a(x,z)\overline{H}=0,$$
(5)
$${\nabla}^{2}E+{k}^{2}\u220a(x,z)E=0.$$
(6)
$${\nabla}^{2}H-\left(\frac{\nabla \u220a}{\u220a}\cdot \nabla \right)H+{k}^{2}\u220a(x,z)H=0.$$
(7)
$$E(x,z)=\text{\u2211}_{i=-\infty}^{+\infty}{S}_{i}(z)exp(-j{\overline{\sigma}}_{i}\cdot \overline{r}),$$
(8)
$$\frac{1}{2{\pi}^{2}}\frac{{\text{d}}^{2}{S}_{i}(z)}{\text{d}{z}^{2}}-j\frac{2}{\pi}\left[\frac{{({\widehat{\u220a}}_{0}-{\u220a}_{I}sin{\theta}^{\prime})}^{1/2}}{\mathrm{\lambda}}-\frac{icos\varphi}{\mathrm{\Lambda}}\right]\frac{\text{d}{S}_{i}(z)}{\text{d}z}+\frac{2i(m-i)}{{\mathrm{\Lambda}}^{2}}{S}_{i}(z)+\frac{{\u220a}_{1}}{{\mathrm{\lambda}}^{2}}[{S}_{i+1}(z)+{S}_{i-1}(z)]=0,$$
(9)
$$m\equiv 2(\mathrm{\Lambda}/\mathrm{\lambda})[{{\u220a}_{I}}^{1/2}sin\varphi sin{\theta}^{\prime}+{({\widehat{\u220a}}_{0}-{\u220a}_{I}{sin}^{2}{\theta}^{\prime})}^{1/2}cos\varphi ].$$
(10)
$$-\left(\frac{\nabla \u220a}{\u220a}\cdot \nabla \right)H=\frac{{\u220a}_{1}sin(\overline{K}\cdot \overline{r})}{{\u220a}_{0}+{\u220a}_{1}cos(\overline{K}\cdot \overline{r})}\left(sin\varphi \frac{\partial H}{\partial x}+cos\varphi \frac{\partial H}{\partial z}\right)\frac{2\pi}{\mathrm{\Lambda}},$$
(11)
$$\frac{{\u220a}_{1}sin(\overline{K}\cdot \overline{r})}{{\u220a}_{0}+{\u220a}_{1}cos(\overline{K}\cdot \overline{r})}=-j\text{\u2211}_{h=-\infty}^{+\infty}{A}_{h}exp(jh\overline{K}\cdot \overline{r}),$$
(12)
$$H(x,z)=\text{\u2211}_{i=-\infty}^{+\infty}{U}_{i}(z)exp(-j{\overline{\sigma}}_{i}.\overline{r}),$$
(13)
$$\begin{array}{l}\frac{1}{2{\pi}^{2}}\frac{{\text{d}}^{2}{U}_{i}(z)}{\text{d}{z}^{2}}-j\frac{2}{\pi}\left[\frac{{({\widehat{\u220a}}_{0}-{\u220a}_{I}sin{\theta}^{\prime})}^{1/2}}{\mathrm{\lambda}}-\frac{icos\varphi}{\mathrm{\Lambda}}\right]\frac{\text{d}{U}_{i}(z)}{\text{d}z}\hfill \\ -j\frac{cos\varphi}{\pi \mathrm{\Lambda}}\text{\u2211}_{h}{A}_{h}\frac{\text{d}{U}_{i-h}(z)}{\text{d}z}+\frac{2i(m-i)}{{\mathrm{\Lambda}}^{2}}{U}_{i}(z)\hfill \\ +\frac{{\u220a}_{1}}{{\mathrm{\lambda}}^{2}}[{U}_{i+1}(z)+{U}_{i-1}(z)]+\frac{2}{{\mathrm{\Lambda}}^{2}}\text{\u2211}_{h}\left(i-h-\frac{m}{2}\right)\hfill \\ \times {A}_{h}{U}_{i-h}(z)=0.\hfill \end{array}$$
(14)
$${H}_{x}=(-j/\omega \mu )\frac{\partial}{\partial z}\text{\u2211}_{i=-\infty}^{+\infty}{S}_{i}(z)exp(-j{\overline{\sigma}}_{i}\cdot \overline{r}).$$
(15)
$$\frac{1}{\u220a(x,z)}=\text{\u2211}_{h=-\infty}^{+\infty}{G}_{h}exp(jh\overline{K}\cdot \overline{r}),$$
(16)
$${E}_{x}=(j/\omega {\varepsilon}_{0})\text{\u2211}_{i=-\infty}^{+\infty}exp(-j{\overline{\sigma}}_{i}\cdot \overline{r})\times \text{\u2211}_{h=-\infty}^{+\infty}{G}_{h}\left[\frac{\text{d}{U}_{i-h}(z)}{\text{d}z}-j({\overline{\sigma}}_{i-h}\cdot \widehat{z}){U}_{i-h}(z)\right].$$