## Abstract

In order to accurately analyze and design the transmittance characteristic of a diffraction phase grating, the validity of both the scalar diffraction theory and the effective medium theory is quantitatively evaluated by the comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by the rigorous vector electromagnetic theory. The effect of surface profile parameters, including the normalized period, the normalized depth, and the fill factor for the precision of the simplified methods is determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light, the scalar diffraction theory is useful to estimate the transmittance of the phase grating. When the fill factor approaches 0.5, the error of the scalar method is minimized, and the scalar theory is accurate even at the grating period of two wavelengths. The transmittance characteristic as a function of the normalized period is strongly influenced by the grating duty cycle, but the diffraction performance on the normalized depth is independent of the fill factor of the grating. Additionally, the effective medium theory is accurate for evaluating the diffraction efficiency within an error of less than around 1% when no higher-order diffraction waves appear and only the zero-order waves exist. The precision of the effective medium theory for calculating transmittance properties as a function of the normalized period, the normalized groove depth, and the polarization state of incident light is insensitive to the fill factor of the phase grating.

© 2011 Optical Society of America

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### Equations (11)

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(1)
$$t(x)=\{\begin{array}{c}({n}_{g}\mathrm{cos}{\theta}_{g}/{n}_{0}\mathrm{cos}{\theta}_{0}{)}^{1/2}\tau ({\theta}_{0})\mathrm{exp}(i{\phi}_{a}),0<x<f\mathrm{\Lambda}\\ ({n}_{g}\mathrm{cos}{\theta}_{g}/{n}_{0}\mathrm{cos}{\theta}_{0}{)}^{1/2}\tau ({\theta}_{0})\mathrm{exp}(i{\phi}_{b}),f\mathrm{\Lambda}<x<\mathrm{\Lambda}\end{array},$$
(2)
$$\eta (\lambda )=|\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}t(x)\mathrm{exp}(-2\pi imx/\mathrm{\Lambda})\mathrm{d}x{|}^{2},$$
(3)
$${\eta}_{0}=({n}_{g}\mathrm{cos}{\theta}_{g}/{n}_{0}\mathrm{cos}{\theta}_{0}){\tau}^{2}({\theta}_{0})[1-2f(1-f)(1-\mathrm{cos}\mathrm{\Delta}\phi )],$$
(4)
$${\eta}_{m\ne 0}=({n}_{g}\mathrm{cos}{\theta}_{g}/{n}_{0}\mathrm{cos}{\theta}_{0}){\tau}^{2}({\theta}_{0})[(1/{m}^{2}{\pi}^{2})(1-\mathrm{cos}2m\pi f)(1-\mathrm{cos}\mathrm{\Delta}\phi )].$$
(5)
$${n}_{\mathrm{TE}}^{(2)}=[({n}_{\mathrm{TE}}^{(0)}{)}^{2}+\frac{1}{3}(\frac{\mathrm{\Lambda}}{\lambda}{)}^{2}{\pi}^{2}{f}^{2}(1-f{)}^{2}({n}_{g}^{2}-{n}_{0}^{2}{)}^{2}{]}^{1/2},$$
(6)
$${n}_{\mathrm{TM}}^{(2)}=[({n}_{\mathrm{TM}}^{(0)}{)}^{2}+\frac{1}{3}(\frac{\mathrm{\Lambda}}{\lambda}{)}^{2}{\pi}^{2}{f}^{2}(1-f{)}^{2}\phantom{\rule{0ex}{0ex}}(\frac{1}{{n}_{g}^{2}}-\frac{1}{{n}_{0}^{2}}{)}^{2}({n}_{\mathrm{TM}}^{(0)}{)}^{6}({n}_{\mathrm{TE}}^{(0)}{)}^{2}{]}^{1/2},$$
(7)
$${n}_{\mathrm{TE}}^{(0)}=[(1-f){n}_{0}^{2}+f{n}_{g}^{2}{]}^{1/2},$$
(8)
$${n}_{\mathrm{TM}}^{(0)}=[(1-f)/{n}_{0}^{2}+f/{n}_{g}^{2}{]}^{-1/2},$$
(9)
$$\left[\begin{array}{c}B\\ C\end{array}\right]=\left\{\prod _{q=1}^{N}\right[\begin{array}{cc}\mathrm{cos}{\delta}_{q}& (i\mathrm{sin}{\delta}_{q})/{\eta}_{q}\\ i{\eta}_{q}\mathrm{sin}{\delta}_{q}& \mathrm{cos}{\delta}_{q}\end{array}\left]\right\}\left[\begin{array}{c}1\\ {\eta}_{s}\end{array}\right],$$
(10)
$${n}_{0}\mathrm{sin}{\theta}_{0}=n(q)\mathrm{sin}{\theta}_{q}={n}_{s}\mathrm{sin}{\theta}_{s}.$$
(11)
$$R=\left(\frac{{\eta}_{0}-Y}{{\eta}_{0}+Y}\right)(\frac{{\eta}_{0}-Y}{{\eta}_{0}+Y}{)}^{*},$$