## Abstract

We report on a numerical analysis method for diffractive optical elements that consist of features ranging from subwavelength to more than $10\lambda $. The essence of the method is treating local structures of the optical elements as diffraction gratings. It is shown that the method can provide accuracy of results comparable with fully electromagnetic treatments in much shorter time. The theory and results are explained assuming micro-Fresnel lenses with one-dimensional structures for investigating polarization properties.

© 2009 Optical Society of America

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### Equations (13)

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(1)
$$\varphi \left(x\right)=l\Phi ,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{0.3em}{0ex}}-(l+1)\Phi <\mathrm{mod}[{\varphi}_{0}\left(x\right),2\pi ]\u2a7d-l\Phi ,$$
(2)
$${\varphi}_{0}\left(x\right)=2\pi {n}_{2}(f-\sqrt{{f}^{2}+{x}^{2}})\lambda $$
(3)
$${M}_{L}=\mathrm{int}\left[(\sqrt{{f}^{2}+{r}^{2}}-f)\lambda \right]+1,$$
(4)
$${x}_{m}=\sqrt{{m}^{2}{\lambda}^{2}+2mLf\lambda}L.$$
(5)
$$d\left(n\right)={x}_{P+L}-{x}_{P},$$
(6)
$$h\left(n\right)=\frac{\lambda}{|{n}_{2}-{n}_{1}|}(\mathrm{int}\left[\frac{m-1}{L}\right]-\frac{m}{L}+1).$$
(7)
$${h}^{\prime}=\lambda \u2215L|{n}_{2}-{n}_{1}|$$
(8)
$$u(n,0)=\sum _{q}{T}_{q}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(\mathrm{i}{\alpha}_{q}\overline{x}\right),$$
(9)
$$u(n,0)\leftarrow \{\begin{array}{cc}u(n,0)\sqrt{{n}_{2}\u2215{n}_{1}},& \text{for}\phantom{\rule{0.3em}{0ex}}\mathrm{TE}\\ u(n,0)\sqrt{{n}_{1}\u2215{n}_{2}},& \text{for}\phantom{\rule{0.3em}{0ex}}\mathrm{TM}\end{array}\phantom{\}}.$$
(10)
$${A}_{j}\left(0\right)=\mathrm{DFT}\left\{u(n,0)\right\}.$$
(11)
$${A}_{j}\left(z\right)={A}_{j}\left(0\right)\mathrm{exp}\left[\mathrm{i}(2\pi {n}_{2}\u2215\lambda )z\sqrt{1-{({\nu}_{j}\lambda \u2215{n}_{2})}^{2}}\right],$$
(12)
$$u(n,z)=\mathrm{IDFT}\left\{{A}_{j}\left(z\right)\right\},$$
(13)
$$I\left(n\right)={\left|u(n,z)\right|}^{2}.$$