光子学报  2017, Vol. 46 Issue (4): 0422002  DOI: 10.3788/gzxb20174604.0422002 0

### 引用本文

LIU Zhi-ying, TIAN Yu, LI Dan, XUE Chang-xi, LI Chuang. Analysis of Relationship between Surface Error and Optical Power of PAL[J]. Acta Photonica Sinica, 2017, 46(4): 0422002. DOI: 10.3788/gzxb20174604.0422002.

### 文章历史

(长春理工大学 光电工程学院, 长春 130022)

Analysis of Relationship between Surface Error and Optical Power of PAL
LIU Zhi-ying, TIAN Yu, LI Dan, XUE Chang-xi, LI Chuang
(Changchun University of Science and Technology, School of Electro-Optical Engineering, Changchun 130022, China)
Foundation item: The National Natural Science Foundation of China (No.11474037)
Abstract: The progressive addition lenses have been used widely applied, because its advantages can meet the requirements of distant and near vision at both time. The basic structure, design and evaluation method of progressive addition lenses are introduced in this paper. The relationship between surface error and optical power of progressive addition lenses is analyzed. The relationship equations are derived from the surface equation and sphere equation. Based on the national glasses fabrication standard that the difference between practical optical power and nominal designed value should be less than 0.1D during the lens effective area, A progressive addition lenses example with 2.0D addition (6.0D~8.0D) was designed, evaluated and surface error tolerance analyzed. After fabrication with given surface error, the optical power difference is calculated correspondingly. It is shown that the simulation result is consistent with the solution of relationship equations. The relationship model provides theoretical basis for the surface precision during lens fabrication process.
Key words: Progressive addition lenses    Optical power distribution    Relationship model    Surface error    Surface equation    Spherical power equation    Power addition
OCIS Codes: 220.4610;220.1000;330.7326;350.4800;240.5770
0 引言

1 渐进多焦点镜片的基本结构

 图 1 渐进多焦点镜片的正面投影图 Fig.1 Front projection drawing of PAL
2 渐进多焦点镜片的设计与评价 2.1 渐进多焦点镜片的设计

 $\frac{1}{{r\left( u \right)}} = \frac{1}{{{r_{\rm{D}}}}} + \left[ {\frac{1}{{{r_{\rm{R}}}}} - \frac{1}{{{r_{\rm{D}}}}}} \right]\sum\limits_{n = m}^{m + l - 1} {{C_n}} {\left[ {u + L} \right]^n}$ (1)
 图 2 渐进多焦点镜片的坐标系 Fig.2 Coordinate of PAL

 $z = z\left( {x,y} \right) = \zeta - {\left[ {r{{\left( u \right)}^2} - {{\left( {x - \xi } \right)}^2} - {{\left( {y - \eta } \right)}^2}} \right]^{1/2}}$ (2)

2.2 渐进多焦点镜片的评价

 $\left\{ {\begin{array}{*{20}{l}} {P = 1000\left( {n - 1} \right)\frac{{{k_1} + {k_2}}}{2} = 1000\left( {n - 1} \right)H}\\ {C = 1000\left( {n - 1} \right)\left| {{k_1} - {k_2}} \right| = 2000\left( {n - 1} \right)\sqrt {{H^2} - K} } \end{array}} \right.$ (3)

 ${g^4}{k^2} + g\left[ {2pqs - \left( {1 + {p^2}} \right)t - \left( {1 + {q^2}} \right)r} \right]k + \left( {rt - {s^2}} \right) = 0$ (4)

HK分别称为该点的平均曲率和高斯曲率，表达式为

 $H = \frac{{{k_1} + {k_2}}}{2} = \frac{{\left( {1 + {p^2}} \right)t + \left( {1 + {q^2}} \right)r - 2pqs}}{{2{g^3}}},K = {k_1}{k_2} = \frac{{rt - {s^2}}}{{{g^4}}}.$

 $\begin{array}{l} p = \frac{{\partial z(x,y)}}{{\partial x}},q = \frac{{\partial z(x,y)}}{{\partial y}},r = \frac{{\partial z(x,y)}}{{\partial {x^2}}},\\ s = \frac{{{\partial ^2}z(x,y)}}{{\partial x\partial y}},t = \frac{{{\partial ^2}z(x,y)}}{{\partial {y^2}}},g = \sqrt {1 + {p^2} + {q^2}} \end{array}$ (5)

3 渐进多焦点镜片面形的公差分析 3.1 渐进多焦点镜片的公差分析方法

 $\Delta z = p\Delta x$ (6)

 $b = \frac{{\partial P}}{{\partial x}}$ (7)

 $\Delta P = b\Delta x$ (8)

 $\Delta P = \frac{b}{p}\Delta z$ (9)

 $\Delta z = \frac{{0.1p}}{b}$ (10)

3.2 实例面形误差分析

 图 3 等球面度曲线 Fig.3 Spherical power contour map
 图 4 等柱面度曲线 Fig.4 Cylinder power contour map
 图 5 球面度图 Fig.5 Spherical power
 图 6 柱面度图 Fig.6 Cylinder power
3.3 面形误差分析结果

 图 7 X方向的面形误差分布 Fig.7 Surface error distribution along with X-direction
 图 8 Y方向的面形误差分布 Fig.8 Surface error distribution along with Y-direction

3.4 实验验证

 图 9 实例镜片加工实验 Fig.9 PAL Example fabrication
 图 10 面形误差0.006 mm时光焦度分布变化 Fig.10 Optical power distribution difference with given surface error
4 结论