## Abstract

To maximize the diffraction efficiency of cylinder lenses with high numerical apertures (such as F/0.5 lenses) we use an iterative algorithm to determine the optimum field distribution in the lens plane. The algorithm simulates the free-space propagation between the lens and the focal plane applying the angular spectrum of plane waves. We show that the optimum field distribution in the lens plane is the phase distribution of a converging cylindrical wave-front and an amplitude distribution with Gaussian-profile. The computed results are verified by rigorous calculations, simulating a F/0.5 lens with subwavelength structures.

© 1997 Optical Society of America

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### Equations (10)

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(1)
$$\phi \left(x\right)={k}_{0}n\left(F-\sqrt{{F}^{2}+{x}^{2}}\right),$$
(2)
$${E}_{y}\left(x,z=0\right)=\mid {E}_{y}\left(x,z=0\right)\mid \phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i\phi \left(x\right)\right).$$
(3)
$${E}_{y}\left(x,z=0\right)=\frac{1}{\sqrt{2\pi}}\phantom{\rule{.2em}{0ex}}\int \psi \left({k}_{x}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i{k}_{x}x\right)d{k}_{x}.$$
(4)
$${E}_{y}\left(x,z=-F\right)=\frac{1}{\sqrt{2\pi}}\phantom{\rule{.2em}{0ex}}\int \psi \left({k}_{x}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i{k}_{z}\left(-F\right)\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i{k}_{x}x\right)d{k}_{x}$$
(5)
$${k}_{z}=\sqrt{{k}_{0}^{\phantom{\rule{.2em}{0ex}}2}-{k}_{x}^{\phantom{\rule{.2em}{0ex}}2}}.$$
(6)
$$O(x,\nu )=\{\begin{array}{c}\bullet \nu \phantom{\rule{.8em}{0ex}}\mid x\mid \ge {x}_{0}\phantom{\rule{.8em}{0ex}}0\le \nu \le 1\\ \bullet 1\phantom{\rule{.8em}{0ex}}\mid x\mid <{x}_{0}\phantom{\rule{2.8em}{0ex}}\end{array}\phantom{\rule{2.3em}{0ex}},$$
(7)
$$\mid {E}_{y}\phantom{\rule{.2em}{0ex}}\left(x,z=0\right)\mid =\mathrm{exp}(-\frac{4\phantom{\rule{.2em}{0ex}}\mathrm{ln}\left(2\right){x}^{2}}{{\xi}^{2}}),$$
(8)
$$\u3008{S}_{z}\u3009=\frac{1}{2}\mathrm{RE}\left({E}_{y}{H}_{x}^{\phantom{\rule{.2em}{0ex}}*}\right),$$
(9)
$$\u3008{\overline{S}}_{z}(x,z)\u3009=\frac{\u3008{S}_{z}(x,z)\u3009}{\underset{\mid x\mid \le 20\lambda}{\int}\mathit{dx}{\u3008{S}_{z}\left(x,z=\lambda \right)\u3009}_{\mathrm{inc}}},$$
(10)
$$\eta =\underset{\mid x\mid \le {x}_{0}}{\int}\mathit{dx}\u3008{\overline{S}}_{z}\left(x,z=-F\right)\u3009.$$