## Abstract

The common focusing characteristics of a cylindrical microlens with a long focal depth and under a given multiple-wavelength illumination are analyzed based on the boundary element method (BEM). The surface-relief profile of a finite-substrate-thickness microlens with a long focal depth is presented. Its focusing performances, such as the common extended focal depth (CEFD), the spot size, and the diffraction efficiency, are numerically studied in the case of TE polarization. The results show that the CEFD of the microlens increases initially, reaches a peak value, and then decreases with increasing preset focal depth. Two modified profiles of a finite-substrate-thickness cylindrical microlens are proposed for enlarging the CEFD. The rigorous numerical results indicate that the modified surface-relief structures of a cylindrical microlens can successfully modulate the optical field distribution to achieve longer CEFD, higher transverse resolution, and higher diffraction efficiency simultaneously, compared with the prototypical microlens. These investigations may provide useful information for the design and application of micro-optical elements in various multiwavelength optical systems.

© 2007 Optical Society of America

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### Equations (8)

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(1)
$$h(x,{f}_{0},\lambda )=\frac{{[{f}^{2}(x,{f}_{0})+\frac{{n}_{1}\left(\lambda \right)+1}{{n}_{1}\left(\lambda \right)-1}{x}^{2}]}^{1\u22152}-f(x,{f}_{0})}{{n}_{1}\left(\lambda \right)+1}.$$
(2)
$$f(x,{f}_{0})={f}_{0}+\frac{{x}^{2}}{{R}^{2}}\delta f,$$
(3)
$$\overline{h}\left(x\right)=\frac{1}{2}[h(x,{f}_{0},{\lambda}_{1})+h(x,{f}_{0},{\lambda}_{n})].$$
(4)
$$-{\varphi}_{1}\left({r}_{1}\right)+{\int}_{\Gamma}[{\varphi}_{1}\left({r}_{\Gamma}^{\prime}\right)\widehat{n}\bullet \nabla {G}_{1}({r}_{1},{r}_{\Gamma}^{\prime})-{p}_{1}{G}_{1}({r}_{1},{r}_{\Gamma}^{\prime})\widehat{n}\bullet \nabla {\varphi}_{1}\left({r}_{\Gamma}^{\prime}\right)]\mathrm{d}{l}^{\prime}=-{\varphi}^{\mathit{inc}}\left({r}_{1}\right),$$
(5)
$${\varphi}_{2}\left({r}_{2}\right)+{\int}_{\Gamma}[{\varphi}_{2}\left({r}_{\Gamma}^{\prime}\right)\widehat{n}\bullet \nabla {G}_{2}({r}_{2},{r}_{\Gamma}^{\prime})-{p}_{2}{G}_{2}({r}_{2},{r}_{\Gamma}^{\prime})\widehat{n}\bullet \nabla {\varphi}_{2}\left({r}_{\Gamma}^{\prime}\right)]\mathrm{d}{l}^{\prime}=0,$$
(6)
$$(\frac{{\theta}_{\Gamma}}{2\pi}-1)\varphi \left({r}_{\Gamma}\right)+{\int}_{\Gamma}[\varphi \left({r}_{\Gamma}^{\prime}\right)\widehat{n}\bullet \nabla {G}_{1}({r}_{\Gamma},{r}_{\Gamma}^{\prime})-{G}_{1}({r}_{\Gamma},{r}_{\Gamma}^{\prime})\widehat{n}\bullet \nabla \varphi \left({r}_{\Gamma}^{\prime}\right)]\mathrm{d}{l}^{\prime}=-{\varphi}^{\mathit{inc}}\left({r}_{\Gamma}\right),$$
(7)
$$\left(\frac{{\theta}_{\Gamma}}{2\pi}\right)\varphi \left({r}_{\Gamma}\right)+{\int}_{\Gamma}[\varphi \left({r}_{\Gamma}^{\prime}\right)\widehat{n}\bullet \nabla {G}_{2}({r}_{\Gamma},{r}_{\Gamma}^{\prime})-{G}_{2}({r}_{\Gamma},{r}_{\Gamma}^{\prime})\widehat{n}\bullet \nabla \varphi \left({r}_{\Gamma}^{\prime}\right)]\mathrm{d}{l}^{\prime}=0,$$
(8)
$${G}_{i}({r}_{i},{r}_{\Gamma}^{\prime})=(-j\u22154){H}_{0}^{\left(2\right)}\left({k}_{i}\mid {r}_{i}-{r}_{\Gamma}^{\prime}\mid \right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}(i=1,2),$$