## Abstract

A new hybrid method for the analysis of diffractive optical elements, which combines fully vectorial and scalar theories, is presented. It is suitable for use with elements of arbitrary large zone, even when the local feature size is of the order of the wavelength. To assess its applicability, we have performed cross-checking tests. The model is shown to accurately predict many optical properties of diffractive optical elements based on two-dimensional artificial dielectrics, like the useful energy diffracted into the order of interest or the deterministic loss into high diffraction orders for an illumination with a wavelength different from the design wavelength or for highly oblique incidence.

© 2007 Optical Society of America

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### Equations (12)

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(1)
$$\mathrm{F}:\psi \to g=\mathrm{F}\left(\psi \right)(\equiv 2\pi ),$$
(2)
$$\left(\begin{array}{c}\nu \\ \mu \end{array}\right)=\mathbf{T}(x,y)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right).$$
(3)
$$\left(\begin{array}{c}\nu \\ \mu \end{array}\right)=\mathbf{T}\left(\psi \right)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)=\left(\begin{array}{cc}{t}_{11}\left(\psi \right)& {t}_{12}\left(\psi \right)\\ {t}_{21}\left(\psi \right)& {t}_{22}\left(\psi \right)\end{array}\right)\left(\begin{array}{c}\alpha \\ \beta \end{array}\right),$$
(4)
$$\mathbf{T}\left(\psi \right)=\sum _{m=-\infty}^{m=\infty}{\mathbf{C}}^{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(jm\psi \right),$$
(5)
$${\mathbf{C}}^{m}=\left(\begin{array}{cc}{c}_{11}^{m}& {c}_{12}^{m}\\ {c}_{21}^{m}& {c}_{22}^{m}\end{array}\right)=\frac{1}{2\pi}{\int}_{0}^{2\pi}\mathbf{T}\left(\psi \right)\mathrm{exp}(-jm\psi )\mathrm{d}\psi .$$
(6)
$${\eta}_{m}^{\mathrm{TE}}=({\mid {c}_{21}^{m}\mid}^{2}+{\mid {c}_{11}^{m}\mid}^{2}),$$
(7)
$${\eta}_{m}^{\mathrm{TM}}=({\mid {c}_{22}^{m}\mid}^{2}+{\mid {c}_{12}^{m}\mid}^{2})$$
(8)
$$\sum _{m}{\eta}_{m}^{\mathrm{TE}}=\frac{1}{2\pi}{\int}_{0}^{2\pi}({\mid {t}_{21}\left(\psi \right)\mid}^{2}+{\mid {t}_{11}\left(\psi \right)\mid}^{2})\mathrm{d}\psi $$
(9)
$${c}_{i,j}^{m}=\frac{1}{2\pi}{\int}_{0}^{2\pi}{t}_{i,j}\left(\psi \right)\mathrm{exp}(-jm\psi )\mathrm{d}\psi .$$
(10)
$${\int}_{0}^{2\pi}{t}_{i,j}\left(\psi \right)\mathrm{exp}\left(\kappa \psi \right)\mathrm{d}\psi =\sum _{p=1}^{M}{G}_{p}\left({\psi}_{p}\right)\mathrm{exp}\left(\kappa {\psi}_{p}\right)-{G}_{p}\left({\psi}_{p-1}\right)\mathrm{exp}\left(\kappa {\psi}_{p-1}\right),$$
(11)
$${G}_{p}\left(\psi \right)={({a}_{p}\u2215\kappa )}^{2}{\psi}^{2}+({b}_{p}\u2215\kappa -2{a}_{p}\u2215{\kappa}^{2})\psi +(2{a}_{p}\u2215{\kappa}^{3}-{b}_{p}\u2215{\kappa}^{2}+{c}_{p}\u2215\kappa ).$$
(12)
$${\eta}_{m}={\mathrm{sinc}}^{2}({\lambda}_{0}\u2215\lambda -m),$$