## Abstract

We present our analysis of a zoom system based on the variable focal power lens, and we demonstrate how our analysis can be used in zoom system design. The transverse magnification is considered as an independent first-order optics control parameter in the zoom system. The zoom system equations are established through the use of matrix optics. Formulas related to the zoom principles and performance of such optical systems are derived, and numerical and theoretical values are compared using examples.

© 2015 Optical Society of America

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### Equations (31)

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(1)
$$S=\left[\begin{array}{cc}1-{d}_{2}{\varphi}_{a}& {d}_{2}\\ {d}_{2}{\varphi}_{a}{\varphi}_{b}-{\varphi}_{a}-{\varphi}_{b}& 1-{d}_{2}{\varphi}_{b}\end{array}\right].$$
(2)
$$\varphi =\frac{1}{{f}^{\prime}}={\varphi}_{a}+{\varphi}_{b}-{d}_{2}{\varphi}_{a}{\varphi}_{b}.$$
(3)
$$m=\frac{1}{({d}_{2}{\varphi}_{a}{\varphi}_{b}-{\varphi}_{a}-{\varphi}_{b})\times {d}_{1}+1-{d}_{2}{\varphi}_{b}},$$
(4)
$${d}_{3}=\frac{1}{{d}_{2}{\varphi}_{a}{\varphi}_{b}-{\varphi}_{a}-{\varphi}_{b}}[m-(1-{d}_{2}{\varphi}_{a})],$$
(5)
$${d}_{1}+{d}_{2}+{d}_{3}=d.$$
(6)
$${\varphi}_{a}=\frac{d+({d}_{1}+{d}_{2})(m-1)}{m{d}_{1}{d}_{2}},$$
(7)
$${\varphi}_{b}=\frac{d+{d}_{1}(m-1)}{{d}_{2}(d-{d}_{1}-{d}_{2})}.$$
(8)
$${\varphi}_{a}=\alpha +\beta /m,$$
(9)
$${\varphi}_{b}=\chi +\eta m,$$
(10)
$$\begin{array}{c}\alpha =\frac{{d}_{1}+{d}_{2}}{{d}_{1}{d}_{2}},\\ \beta =\frac{d-{d}_{1}-{d}_{2}}{{d}_{1}{d}_{2}},\\ \chi =\frac{d-{d}_{1}}{{d}_{2}(d-{d}_{1}-{d}_{2})},\\ \eta =\frac{{d}_{1}}{{d}_{2}(d-{d}_{1}-{d}_{2})}.\end{array}$$
(10)
$$\mathbf{N}={\mathbf{T}}_{\mathbf{j}-1}{\mathbf{R}}_{\mathbf{j}-1}\cdots {\mathbf{T}}_{1}{\mathbf{R}}_{1}{\mathbf{T}}_{0},$$
(12)
$${\mathbf{T}}_{\mathbf{j}}=\left[\begin{array}{cc}1& {d}_{j+1}\\ 0& 1\end{array}\right],\phantom{\rule{0.2em}{0ex}}{\mathbf{R}}_{\mathbf{j}}=\left[\begin{array}{cc}1& 0\\ -{\varphi}_{j}& 1\end{array}\right].$$
(11)
$$\left[\begin{array}{cc}{N}_{11}& {N}_{12}\\ {N}_{21}& {N}_{22}\end{array}\right].$$
(12)
$$\left|\frac{{R}_{1}}{{N}_{12}}\right|=\left|\frac{{R}_{a}}{{d}_{1}}\right|,$$
(13)
$$\left|\frac{{R}_{2}}{{N}_{12}}\right|=\left|\frac{{R}_{px}}{(1-{d}_{x}{\varphi}_{a}){d}_{1}+{d}_{x}}\right|,$$
(14)
$$\left|\frac{{R}_{3}}{{N}_{12}}\right|=\left|\frac{{R}_{b}}{(1-{d}_{2}{\varphi}_{a}){d}_{1}+{d}_{2}}\right|.$$
(15)
$${\overrightarrow{Q}}_{1}=\left[\begin{array}{cc}1-{d}_{x}{\varphi}_{a}& {d}_{x}\\ -{\varphi}_{a}& 1\end{array}\right].$$
(16)
$${l}_{E}=\frac{{d}_{x}}{1-{d}_{x}{\varphi}_{a}},$$
(17)
$${R}_{E}=\frac{{R}_{Px}}{1-{d}_{x}{\varphi}_{a}}.$$
(18)
$${\overrightarrow{Q}}_{2}=\left[\begin{array}{cc}1& {d}_{2}-{d}_{x}\\ -{\varphi}_{b}& 1-{\varphi}_{b}({d}_{2}-{d}_{x})\end{array}\right].$$
(19)
$${l}_{{E}^{\prime}}=-\frac{{d}_{2}-{d}_{x}}{1-{\varphi}_{b}({d}_{2}-{d}_{x})},$$
(20)
$${R}_{{E}^{\prime}}=\frac{{R}_{Px}}{1-{\varphi}_{b}({d}_{2}-{d}_{x})}.$$
(21)
$$\frac{D}{{f}^{\prime}}=\frac{2{R}_{E}}{{\varphi}^{-1}}=\frac{2{R}_{Px}}{1-{d}_{x}{\varphi}_{a}}\times ({\varphi}_{a}+{\varphi}_{b}-{d}_{2}{\varphi}_{a}{\varphi}_{b}),$$
(22)
$$\frac{D}{{f}^{\prime}}=\frac{2{R}_{Px}}{1-{d}_{x}(\alpha +\beta /m)}\times [\alpha +\beta /m+\chi +\eta m-{d}_{2}(\alpha +\beta /m)(\chi +\eta m)].$$
(23)
$${R}_{Px}=\frac{1}{2}\left(\frac{D}{{f}^{\prime}}\right)\frac{(Am+B)}{(C{m}^{2}+Em+F)},$$
(26)
$$\begin{array}{c}A=1-{d}_{x}\alpha ,\\ B=-{d}_{x}\beta ,\\ C=\eta (1-{d}_{2}\alpha ),\\ E=\alpha +\chi -{d}_{2}(\alpha \chi +\beta \eta ),\\ F=\beta (1-{d}_{2}\chi ).\end{array}$$
(24)
$$\alpha =\chi =3/20,\phantom{\rule{0.2em}{0ex}}\beta =\eta =1/10.$$
(25)
$${\varphi}_{a}=\frac{3}{20}+\frac{1}{10m},$$
(26)
$${\varphi}_{b}=\frac{3}{20}+\frac{m}{10}.$$
(27)
$${\varphi}_{ab}=-\frac{2{m}^{2}+m+2}{40m}.$$
(28)
$${R}_{Px}=-\frac{5(m-2)}{2(2{m}^{2}+m+2)}.$$