## Abstract

Diffraction properties of dielectric surface-relief gratings were investigated by solving Maxwell’s equations numerically using the differential method. The diffraction efficiency of a grating with a groove depth about twice as deep as the grating period is comparable with the efficiency of a volume phase grating. Dielectric surface-relief gratings interferometrically recorded in photoresist can possess a high diffraction efficiency of up to 94% (throughput efficiency 85%). Calculated results were also found in good agreement with the experimental data.

© 1984 Optical Society of America

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### Equations (15)

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(1)
$$\begin{array}{c}\mathrm{\Delta}{\mathbf{E}}_{z}(x,y)+{k}^{2}{\mathbf{E}}_{z}(x,y)=0,\\ \text{where}\hspace{0.17em}{k}^{2}={\left(\frac{2\pi}{\mathrm{\lambda}}n\right)}^{2}.\end{array}$$
(2)
$${\mathbf{E}}_{z}(x,y)=\sum _{m=-\infty}^{\infty}{E}_{m}(y)\hspace{0.17em}\text{exp}(j{\alpha}_{m}X).$$
(3)
$${\mathbf{k}}^{2}(x,y)=\sum _{m=-\infty}^{\infty}{({\mathbf{k}}^{2})}_{m}\hspace{0.17em}\text{exp}(jm\mathbf{K}x),$$
(4)
$$\frac{{d}^{2}}{d{y}^{2}}{E}_{m}={\alpha}_{m}^{2}{E}_{m}-\sum _{p=-\infty}^{\infty}{({\mathbf{k}}^{2})}_{m-p}\xb7{E}_{p}.$$
(5)
$$\frac{\partial}{\partial x}\left[\frac{1}{{k}^{2}}\xb7\frac{\partial {H}_{z}(x,y)}{\partial x}\right]+\frac{\partial}{\partial y}\left[\frac{1}{{k}^{2}}\xb7\frac{\partial {H}_{z}(x,y)}{\partial y}\right]+{H}_{z}(x,y)=0.$$
(6)
$$\tilde{E}(x,y)=\frac{1}{{\mathbf{k}}^{2}(x,y)}\xb7\frac{\partial}{\partial y}{H}_{z}(x,y),$$
(7)
$$\frac{\partial}{\partial y}\tilde{E}(x,y)=-\frac{\partial}{\partial x}\left[\frac{1}{{\mathbf{k}}^{2}}\xb7\frac{\partial}{\partial x}{H}_{z}(x,y)\right]-{H}_{z}(x,y).$$
(8)
$$\frac{1}{{\mathbf{k}}^{2}(x,y)}=\sum _{m=-\infty}^{\infty}{\left(\frac{1}{{\mathbf{k}}^{2}}\right)}_{m}\hspace{0.17em}\text{exp}(jmkx),$$
(9)
$${H}_{z}(x,y)=\sum _{m=-\infty}^{\infty}{H}_{m}(y)\hspace{0.17em}\text{exp}(j{\alpha}_{m}x),$$
(10)
$$\tilde{E}(x,y)=\sum _{m=-\infty}^{\infty}{\tilde{E}}_{m}(y)\hspace{0.17em}\text{exp}(j{\alpha}_{m}x).$$
(11)
$$\frac{d}{dy}{H}_{m}(y)=\sum _{p=-\infty}^{\infty}{({\mathbf{k}}^{2})}_{m-p}{E}_{p}(y),$$
(12)
$$\frac{d}{dy}{\tilde{E}}_{m}(y)={\alpha}_{m}\sum _{p=-\infty}^{\infty}\left[{\alpha}_{p}\xb7{\left(\frac{1}{{\mathbf{k}}^{2}}\right)}_{m-p}\xb7{H}_{p}(y)\right]-{H}_{m}(y).$$
(13)
$$\eta =\frac{(\text{diffracted}\hspace{0.17em}\text{wave}\hspace{0.17em}\text{power})}{(\text{incident}\hspace{0.17em}\text{wave}\hspace{0.17em}\text{power})}.$$
(14)
$${\theta}_{\text{B}}={\text{sin}}^{-1}(\mathrm{\lambda}/2d).$$
(15)
$${\eta}_{t1}={I}_{1}/{I}_{0},$$