## Abstract

This study presents a 50X five-group zoom lens design with two moving groups and one focus tunable lens (FTL) by applying Gaussian brackets and lens modules. After qualitative analysis of the paraxial properties, the initial structure is obtained by solving the power equations consisting of five groups by numerical analysis method. The optimized nearly 60X lens has a focal length of 5.1-300mm and f-number of 1.8-5.0 in wide and tele positions, with a total length of 160mm. The lens achieves a maximum FOV of 63° at short focal length. The tolerance analysis indicates that the design is suitable for mass production. The method eases the difficulty in initial structure design of the zoom lens with high zoom ratio, improves its optical design efficiency, and can be applied to the fields such as customs security cameras, military detecting devices, machine vision and industrial inspection, etc.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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### Equations (10)

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(1)
$$\begin{array}{c}K=[{k}_{1},-{z}_{1},{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4},{k}_{5}],\\ K{z}_{5}=[{k}_{1},-{z}_{1},{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4}],\end{array}$$
(2)
$$[{a}_{1},\dots {a}_{k}]={a}_{1}[{a}_{2},\dots {a}_{k}]+[{a}_{3},\dots {a}_{k}],$$
(3)
$$\begin{array}{l}K={k}_{1}[-{z}_{1},{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4},{k}_{5}]+[{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4},{k}_{5}],\\ K{z}_{5}={k}_{1}[-{z}_{1},{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4}]+[{k}_{2},-{z}_{2},{k}_{3},-{z}_{3},{k}_{4},-{z}_{4}],\end{array}$$
(4)
$$\begin{array}{c}K=({k}_{1}+{k}_{2}-{k}_{1}{z}_{1}{k}_{2}){E}_{1}+(1-{k}_{1}{z}_{1}){E}_{2},\\ K{z}_{5}=({k}_{1}+{k}_{2}-{k}_{1}{z}_{1}{k}_{2}){E}_{3}+(1-{k}_{1}{z}_{1}){E}_{4},\end{array}$$
(5)
$$\begin{array}{l}\begin{array}{l}{E}_{1}=-{z}_{2}{k}_{3}{z}_{3}{k}_{4}{z}_{4}{k}_{5}+{z}_{2}{k}_{3}{z}_{3}{k}_{4}+{z}_{2}{k}_{4}{z}_{4}{k}_{5}+{z}_{3}{k}_{4}{z}_{4}{k}_{5}+{z}_{2}{k}_{3}{z}_{3}{k}_{5}+{z}_{2}{k}_{3}{z}_{4}{k}_{5}-\\ {z}_{2}{k}_{4}-{z}_{3}{k}_{4}-{z}_{2}{k}_{5}-{z}_{3}{k}_{5}-{z}_{2}{k}_{3}-{z}_{4}{k}_{5}+1,\end{array}\\ {E}_{2}={k}_{3}{z}_{3}{k}_{4}{z}_{4}{k}_{5}-{k}_{3}{z}_{3}{k}_{4}-{k}_{3}{z}_{3}{k}_{5}-{k}_{3}{z}_{4}{k}_{5}-{k}_{4}{z}_{4}{k}_{5}+{k}_{3}+{k}_{4}+{k}_{5},\\ {E}_{3}=-{z}_{2}{k}_{3}{z}_{3}{k}_{4}{z}_{4}+{z}_{2}{k}_{3}{z}_{3}+{z}_{2}{k}_{3}{z}_{4}+{z}_{2}{k}_{4}{z}_{4}+{z}_{3}{k}_{4}{z}_{4}-{z}_{2}-{z}_{3}-{z}_{4},\\ {E}_{4}={k}_{3}{z}_{3}{k}_{4}{z}_{4}-{k}_{3}{z}_{3}-{k}_{3}{z}_{4}-{k}_{4}{z}_{4}+1.\end{array}$$
(6)
$$\begin{array}{l}a=\frac{K-\frac{(K{z}_{5}{E}_{1}-K{E}_{3}){E}_{2}}{{E}_{1}{E}_{4}-{E}_{2}{E}_{3}}}{{E}_{1}},\\ b=\frac{K{z}_{5}{E}_{1}-K{E}_{3}}{{E}_{1}{E}_{4}-{E}_{2}{E}_{3}}.\end{array}$$
(7)
$$\begin{array}{l}{k}_{1}=\frac{1-b}{{z}_{1}},\\ {k}_{2}=\frac{a\times {z}_{1}-1+b}{b\times {z}_{1}}.\end{array}$$
(8)
$$\begin{array}{l}Eq1={k}_{1w}-{k}_{1m},\\ Eq2={k}_{1w}-{k}_{1t},\\ Eq3={k}_{2w}-{k}_{2m},\\ Eq4={k}_{2w}-{k}_{2t}.\end{array}$$
(9)
$$\frac{1}{2\times 2\mu m}=250lp/mm.$$
(10)
$${f}_{sensor}\ge 2\times {f}_{lens}.$$