## Abstract

The range of validity and the accuracy of scalar diffraction theory for periodic diffractive phase elements (DPE’s) is evaluated by a comparison of diffraction efficiencies predicted from scalar theory to exact results calculated with a rigorous electromagnetic theory. The effects of DPE parameters (depth, feature size, period, index of refraction, angle of incidence, fill factor, and number of binary levels) on the accuracy of scalar diffraction theory is determined. It is found that, in general, the error of scalar theory is significant (*∊* > ±5%) when the feature size is less than 14 wavelengths (*s* < 14λ). The error is minimized when the fill factor approaches 50%, even for small feature sizes (*s* = 2λ); for elements with an overall fill factor of 50% the larger period of the DPE replaces the smaller feature size as the condition of validity for scalar diffraction theory. For an 8-level DPE of refractive index 1.5 analyzed at normal incidence the error of the scalar analysis is greater than ±5% when the period is less than 20 wavelengths (Λ < 20λ). The accuracy of the scalar treatment degrades as either the index of refraction, the depth, the number of binary levels, or the angle of incidence is increased. The conclusions are, in general, applicable to nonperiodic as well as other periodic (trapezoidal, two-dimensional) structures.

© 1994 Optical Society of America

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### Equations (16)

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(1)
$$s=\begin{array}{cc}\{\begin{array}{l}(1-f)\mathrm{\Lambda}\\ f\mathrm{\Lambda}\end{array}& \begin{array}{l}s\ge 0.5\\ s<0.5\end{array}\end{array}.$$
(2)
$$\mathrm{\Delta}\varphi ={k}_{0}d({n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0}),$$
(3)
$${\eta}_{p}={\tau}^{2}({\theta}_{0})4{f}^{2}{Sinc}^{2}(fp){cos}^{2}\left(\frac{1}{2}{k}_{0}\mathrm{\Delta}\varphi +\frac{\pi}{2}p\right),$$
(4)
$${d}_{\text{opt}}=\begin{array}{cc}\{\begin{array}{l}\frac{1}{2}\frac{\mathrm{\lambda}}{({n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0})}\\ \frac{\mathrm{\lambda}}{({n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0})}\end{array}& \begin{array}{l}p\hspace{0.17em}\text{odd}\\ p\hspace{0.17em}\text{even}\end{array}\end{array}.$$
(5)
$${\eta}_{p}={\tau}^{2}({\theta}_{0})4{f}^{2}{Sinc}^{2}(pf).$$
(6)
$$\%\text{Error}=\frac{{\eta}_{\text{RCWT}}-{\eta}_{\text{ST}}}{{\eta}_{\text{RCWT}}}\times 100\%.$$
(7)
$${\eta}_{p}={\tau}^{2}({\theta}_{0}){Sinc}^{2}\left(\frac{p}{N}\right),$$
(8)
$${d}_{\text{opt}}=\left(\frac{N-1}{N}\right)\frac{p{\mathrm{\lambda}}_{0}}{{n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0}}$$
(9)
$$\begin{array}{cc}\hfill \mathrm{\Delta}\varphi & ={k}_{0}d\left(\frac{{n}_{2}}{cos{{\theta}_{0}}^{\prime}}-{n}_{1}tan{{\theta}_{0}}^{\prime}sin{\theta}_{0}+\frac{{n}_{1}}{cos{{\theta}_{0}}^{\prime}}-{n}_{2}tan{\theta}_{0}sin{{\theta}_{0}}^{\prime}\right)\hfill \\ \hfill & ={k}_{0}d({n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0}).\hfill \end{array}$$
(10)
$${t}_{\text{uc}}(x)=\tau ({\theta}_{0})\hspace{0.17em}\left[\text{\u2211}_{i=1}^{2}\text{rect}\left(\frac{x}{{w}_{i}\mathrm{\Lambda}}\right)exp(-ji\mathrm{\Delta}\varphi )\delta (x-{s}_{i}\mathrm{\Lambda})\right],$$
(11)
$$t(x)={t}_{\text{uc}}(x)\text{\u2211}_{p=-\infty}^{\infty}\delta (x-p\mathrm{\Lambda}).$$
(12)
$$T({f}_{x})=\tau ({\theta}_{0})\hspace{0.17em}\left\{\text{\u2211}_{i=1}^{2}{w}_{i}\mathrm{\Lambda}Sinc({w}_{i}\mathrm{\Lambda}{f}_{x})\times exp[-j(i\mathrm{\Delta}\varphi +2\pi {s}_{i}\mathrm{\Lambda}{f}_{x})]\right\}\frac{1}{\mathrm{\Lambda}}\text{\u2211}_{p=-\infty}^{\infty}\delta \left({f}_{x}-\frac{p}{\mathrm{\Lambda}}\right).$$
(13)
$${U}^{+}({f}_{x})=\tau ({\theta}_{0})\hspace{0.17em}\left\{\text{\u2211}_{i=1}^{2}{w}_{i}Sinc({w}_{i}p)exp[-j(i\mathrm{\Delta}\varphi +2\pi {s}_{i}p)]\right\}\times \text{\u2211}_{p=-\infty}^{\infty}\delta \left({f}_{x}+{k}_{0}{n}_{1}sin{\theta}_{0}-\frac{P}{\mathrm{\Lambda}}\right).$$
(14)
$${\eta}_{p}={\tau}^{2}({\theta}_{0}){\left|\text{\u2211}_{i=1}^{2}{w}_{i}Sinc({w}_{i}p)\times exp[-ji({k}_{0}d({n}_{1}cos{\theta}_{0}-{n}_{2}cos{{\theta}_{0}}^{\prime})+p\pi ]\right|}^{2}.$$
(15)
$$\mathrm{\Delta}\varphi ={k}_{0}\frac{d}{N-1}({n}_{2}cos{{\theta}_{0}}^{\prime}-{n}_{1}cos{\theta}_{0}).$$
(16)
$${\eta}_{p}={\tau}^{2}({\theta}_{0}){Sinc}^{2}\left(\frac{p}{N}\right)\frac{1}{{N}^{2}}{\left|\text{\u2211}_{i=1}^{N}exp\left\{-ji\left[\frac{{k}_{0}d}{N-1}({n}_{1}cos{\theta}_{0}-{n}_{2}cos{{\theta}_{0}}^{\prime})+\frac{p2\pi}{N}\right]\right\}\right|}^{2}.$$