﻿ 变焦系统中可调光焦度器件初值分析
 光学仪器  2020, Vol. 42 Issue (4): 25-32 PDF

1. 上海健康医学院 发展规划处，上海 201318;
2. 上海大学 精密机械系，上海 200444

Analysis initial value of zoom system with variable focal power lens
CHENG Hongtao1, LI Hengyu2
1. Department of Development and Plan, Shanghai University of Medicine & Health Sciences, Shanghai 201318, China;
2. School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China
Abstract: In order to realize the analysis of the initial value of the variable focal power lens(VFPLs) for the zoom system, the aberration of the VFPLs for the zoom system is analyzed. The primary aberration of the zoom system is determined according to the aperture angle of the beam and the height of the object. The relation between the shape parameters and the magnification of the zoom system is analyzed by using the theory of thin transparent image difference, and the initial value of the zoom system is determined by using the minimum value of spherical aberration and coma. The results show that the calculation equation of the magnification and shape parameters of the zoom system can reach the requirements of determining the initial value of the VFPLs in the zoom system, and it is simple, fast and reliable, which provides a reference for further computer correction of the aberration of the zoom system.
Key words: zoom system    aberration    variable focal power lens

1 可调光焦度器件变焦系统的像差分析

 图 1 光线通过光学系统图 Figure 1 A paraxial ray passing through an optical system

2 变焦系统中可调光焦度器件初始结构分析 2.1 球差与形状参数、放大率的关系

 图 2 球差示意图 Figure 2 Spherical aberration
 $\begin{split} {T_{SC}} =\;& - \frac{{{y^4}}}{{u'}}\left({G_1}{c^3} - {G_2}{c^2}{c_1} + {G_3}{c^2}v + {G_4}cc_1^2 -\right.\\ & \left. {G_5}c{c_1}v + {G_6}c{v^2}\right) \end{split}$ (1)

${ X} = \dfrac{{{c_1} + {c_2}}}{{{c_1} - {c_2}}}$ $\phi = \left( {n - 1} \right)\left( {{c_1} - {c_2}} \right)$ ，代入式（1）得到

 ${T_{SC}} = {K_S}\left[ {{e_1}{{\left( {X + 1} \right)}^2} + {e_2}(X + 1) + {e_3}} \right]$ (2)

 $\begin{split} &{K_S} = - \frac{{{y^4}}}{{u'}};\\& {e_1} = \frac{1}{4}\frac{{{G_4}{\phi ^3}}}{{{{\left( {n - 1} \right)}^3}}};\\ &{e_2} = - \frac{1}{2}\frac{{{\phi ^2}}}{{{{(n - 1)}^2}}}({G_5}v + {G_2}\frac{\phi }{{n - 1}});\\ &{e_3} = \frac{{{G_1}{\phi ^3}}}{{{{(n - 1)}^3}}} + \frac{{{G_3}{\phi ^2}v}}{{{{(n - 1)}^2}}} + \frac{{{G_6}\phi {v^2}}}{{n - 1}}{\text{。}} \end{split}$

 ${t_{sc}}_X = {e_1}{\left( {X + 1} \right)^2} + {e_2}(X + 1) + {e_3}$ (3)

 $\frac{{{\rm{d}}{t_{sc}}_X}}{{{\rm{d}}X}} = 2{e_1}(X + 1) + {e_2}$ (4)

 $X = - \left(1 + \frac{{{e_2}}}{{2{e_1}}}\right)$ (5)

 ${c_1} = \frac{{\phi n(2n + 1) + 4v({n^2} - 1)}}{{2(n + 2)(n - 1)}}$ (6)
 ${c_2} = \frac{{\phi (2{n^2} - n - 4) + 4v({n^2} - 1)}}{{2(n + 2)(n - 1)}}$ (7)

 $X = \frac{{2(\phi + 2v)({n^2} - 1)}}{{\phi (n + 2)}}$ (8)

（1）物点在无限远

 ${X_{op}} = \frac{{2({n^2} - 1)}}{{(n + 2)}}$ (9)
 ${\left( {\frac{{{c_1}}}{\phi }} \right)_{op}} = \frac{{n(2n + 1)}}{{2(n + 2)(n - 1)}}$ (10)
 ${\left( {\frac{{{c_2}}}{\phi }} \right)_{op}} = \frac{{(2{n^2} - n - 4)}}{{2(n + 2)(n - 1)}}$ (11)
 $\frac{{{t_{sc}}_X}}{{{\phi ^3}}} = \frac{1}{4}\frac{{{G_4}}}{{{{\left( {n - 1} \right)}^3}}}{\left( {X + 1} \right)^2} - \frac{1}{2}\frac{{{G_2}}}{{{{(n - 1)}^3}}}(X + 1) + \frac{{{G_1}}}{{{{(n - 1)}^3}}}$ (12)

 图 3 折射率与形状参数的关系图 Figure 3 Relation between refractive index and shape parameter

 图 4 折射率与 ${({c_1}/\phi )_{op}}$ 的关系图 Figure 4 Relation between refractive index and ${({c_1}/\phi )_{op}}$

 图 5 折射率与 ${({c_2}/\phi )_{op}}$ 的关系图 Figure 5 Relation between refractive index and ${({c_2}/\phi )_{op}}$

 图 6 折射率、形状参数与 ${t_{sc}}_X/{\phi ^3}$ 的函数关系图 Figure 6 Relation of refractive index, shape parameter and ${t_{sc}}_X/{\phi ^3}$

 图 7 不同折射率情况下， ${({t_{sc}}_X/{\phi ^3})_{op}}$ 与X之间的函数关系图 Figure 7 Relation between ${({t_{sc}}_X/{\phi ^3})_{op}}$ and $X$ under different refractive indices

（2）物点在有限远

 $\frac{1}{{s\phi }} = \frac{m}{{m - 1}}$ (13)

v代入式（13）得到

 $v = \frac{{m\phi }}{{m - 1}}$ (14)

 ${t_{sc}}_X = {e_4}{\left( {X + 1} \right)^2} + {e_5}(X + 1) + {e_6}$ (15)

 $\begin{split} & {e_4} = \frac{1}{8}\frac{{(n + 2)}}{{n{{\left( {n - 1} \right)}^2}}}{\phi ^3};\\ & {e_5} = - \frac{{{\phi ^3}}}{{{{(n - 1)}^2}}}\left[ {\frac{{(2n + 1)}}{4} + \frac{{m({n^2} - 1)}}{{n(m - 1)}}} \right];\\ & {e_6} = \frac{{{n^2}{\phi ^3}}}{{2{{(n - 1)}^2}}} + \frac{{m(3n + 1){\phi ^3}}}{{2(n - 1)(m - 1)}} + \frac{{(3n + 2){\phi ^3}{m^2}}}{{2n{{(m - 1)}^2}}}{\text{。}} \end{split}$

 ${X_{ops}} = \frac{{2(1 - {n^2})\left( {m + 1} \right)}}{{(n + 2)(m - 1)}}$ (16)
 ${\left( {\frac{{{c_1}}}{\phi }} \right)_{op}} = \frac{{2{n^2}(1 + m) + n(1 - m) - 4m}}{{2(n + 2)(n - 1)(1 - m)}}$ (17)
 ${\left( {\frac{{{c_2}}}{\phi }} \right)_{op}} = \frac{{2{n^2}(1 + m) - n(1 - m) - 4}}{{2(n + 2)(n - 1)(1 - m)}}$ (18)
 ${\left( {\frac{{{t_{sc}}_X}}{{{\phi ^3}}}} \right)_{op}} = \frac{{n\left[ {(4n - 1){m^2} - (4{n^2} + 2)m + 4n - 1} \right]}}{{8{{(n - 1)}^2}{{(m - 1)}^2}(n + 2)}}$ (19)

 图 8 ${X_{ops}}$ 与放大率的函数曲线 Figure 8 Relation between ${X_{ops}}$ and magnification

 图 9 ${({c_1}/\phi )_{op}}$ 与放大率的函数曲线 Figure 9 Relation between ${({c_1}/\phi )_{op}}$ and magnification

 图 10 ${({c_2}/\phi )_{op}}$ 与放大率的函数曲线 Figure 10 Relation between ${({c_2}/\phi )_{op}}$ and magnification

 图 11 ${({t_{sc}}_X/{\phi ^3})_{op}}$ 与放大率的函数曲线 Figure 11 Relation between ${({t_{sc}}_X/{\phi ^3})_{op}}$ and magnification

2.2 彗差与形状参数、放大率的关系

 图 12 彗差示意图 Figure 12 Coma
 $CC = h'{y^2}\left( { - \frac{1}{4}{G_5}c{c_1} + {G_7}cv + {G_8}{c^2}} \right)$ (20)

 ${G_7} = \left( {2n + 1} \right)\left( {n - 1} \right)/2n,{G_8} = n\left( {n - 1} \right)/2{\text{。}}$

 ${c_X} = - \frac{1}{4}{G_5}c{c_1} + {G_7}cv + {G_8}{c^2}$ (21)

 $\frac{{{c_X}}}{{{\phi ^2}}} = - \frac{1}{4}\frac{{(n + 1)(X + 1)}}{{n(n - 1)}} + \frac{1}{2}\frac{{(2n + 1)m}}{{n(1 - m)}} + \frac{1}{2}\frac{n}{{n - 1}}$ (22)

 图 13 $\dfrac{{{c_X}}}{{{\phi ^2}}}$ 、 $\dfrac{{ts{c_X}}}{{{\phi ^3}}}$ 与 $X$ 的关系图 Figure 13 Relation of $\dfrac{{{c_X}}}{{{\phi ^2}}}$ , $\dfrac{{ts{c_X}}}{{{\phi ^3}}}$ and $X$

 ${X_{opc}} = \frac{{(n - 1)(2n + 1)(1 + m)}}{{(n + 1)(1 - m)}}$ (23)

 $\frac{\;\;{{X_{opc}}}\;\;}{{\dfrac{{1 + m}}{{1 - m}}}} = \frac{{(n - 1)(2n + 1)}}{{(n + 1)}}$ (24)

 $\frac{\;\;{{X_{ops}}}\;\;}{{\dfrac{{1 + m}}{{1 - m}}}} = \dfrac{{2({n^2} - 1)}}{{(n + 2)}}$ (25)

 曲线1—最优彗差时${X_{opc}}/\left( {\dfrac{{1 + m}}{{1 - m}}} \right)$的取值；曲线2—最优球差时${X_{ops}}/\left( {\dfrac{{1 + m}}{{1 - m}}} \right)$的取值；曲线3—曲线1与曲线2之间的差值 图 14 最优值的比较图 Figure 14 Comparison of the optimal values

 ${X_{opc}} = 0.969\;231 \times \frac{{1 + m}}{{1 - m}}$ (26)
 ${X_{ops}} = 0.866\;667 \times \frac{{1 + m}}{{1 - m}}$ (27)

 曲线1—最优彗差时${X_{opc}}$的取值；曲线2—最优球差时${X_{ops}}$的取值 图 15 ${X_{op}}$ 与放大率之间的关系图 Figure 15 Relation between ${X_{op}}$ and magnification
3 结　论

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