基于权重函数优化的渐进多焦点镜片设计 下载: 578次
Progressive addition lenses (PALs) are commonly used to correct presbyopia and prevent pseudomyopia and other visual accommodation disorders in adolescents. Compared with conventional single-vision lenses, bifocal lenses, and trifocal lenses, the focal length of a PAL changes gently from far to near, and the lens has no obvious boundary of imaging distortion. The shape is well-designed, and it only needs one pair of glasses to look far and closely. At present, the design methods of progressive freeform surfaces are mainly divided into two types: direct method and indirect method. The direct method directly calculates the sag distribution of the entire surface based on the mean curvature. However, this method may make peripheral astigmatism become two times the addition power (ADD), which will affect the visual imaging effect. By constructing the model of mean curvature and principal curvature difference and solving a specific high-order partial differential equation to minimize it, the indirect design method obtains the sag value of the PAL indirectly, which can reduce peripheral astigmatism more effectively while ensuring the lens's photo focal quality. At present, the research on the design of the PAL indirect method mainly focuses on surface optimization, and there is less research on the optimization of the weight function design algorithm. Therefore, based on the existing research, this paper realizes the optimization of the PAL by optimizing the weight function and the mean curvature distribution in the construction of the minimization model algorithm, so as to explore their influence on the optical performance optimization of PALs.
According to a PAL minimization model algorithm in the present studies, two groups of weight functions and mean curvature distributions were first designed, and three groups of free-form PALs with different weight functions and mean curvature distributions were calculated respectively. In the next step, an optimization method was proposed to reset the weight distribution of the lenses according to the conformity matrix, and two groups of PALs with different optimization degrees were recalculated under the same parameters. Then, five groups of lenses were machined and evaluated by free-form surface machine tools and testing instruments, and the effects of the optimization method in this paper on the optical power, astigmatism, and other optical properties of PALs were analyzed. In addition, issues such as wearing personalization and comfort were considered.
The channel length can be shortened, and the ADD can be increased by changing the mean curvature distribution to a circular design. The difference in the shape, area, and weight value of the weight function will affect the sizes of the distance area and the near area. The optical power of the distance area is also improved, and the astigmatism is significantly reduced (Table 2). Lenses after recalculation of weights according to the power conformity matrix are further optimized, and the higher weight value and the more concentrated weight distribution make the optical power error of the lenses become almost zero. The weight function combines the advantages of progressive surfaces with hard and soft designs, and the change of astigmatism from the distance area to the blending area increases slowly, which can reduce the imaging distortion when the line of sight switches from left to right. Maximum astigmatism in the blending area decreased to 1.75 D (Fig. 10). In addition, it has a relatively wide and small astigmatism channel, which can reduce the imaging distortion when the line of sight is from far to near. The machining results are basically consistent with the simulation results. This optimization method can optimize the optical properties of free-form PALs.
This paper mainly explores the influence of weight function and mean curvature distribution on PAL design. Five groups of PALs were calculated by designing five groups of different weight functions and mean curvature distributions. The machining and measurement results show that the weight function and mean curvature distribution jointly affect the optical properties of the lenses. The different shapes, areas, and weight values of the weight function can improve the optical performance of the lens in different aspects. The combination of reasonable mean curvature distribution can improve the power accuracy of the PALs. The weight function reset according to the focal power conformity can reduce the lens photometric error and make the astigmatism of the blending area change more gently, so as to bring a better visual experience for wearers. This paper also considers the flexibility of the design under different refractive states, which can meet the individual needs of different wearers. The research results of this paper can be further deepened, and more different groups of weight function and mean curvature distribution can be designed for comparison, so as to achieve a better optimization effect on the optical performance of the lens.
1 引言
渐进多焦点镜片(PAL)常被用来矫正老花眼,也可用于预防青少年假性近视和其他视力调节障碍[1]。与普通的单光镜、双焦点镜、三焦点镜相比,渐进多焦点镜片焦距从远用到近用变化平缓,镜片上没有明显的成像畸变分界线,外形设计更加美观,而且看远看近只需要一副眼镜。
目前,渐进自由曲面设计方法主要分为直接法和间接法两种[2]。直接法根据平均曲率直接计算出整个曲面的矢高分布,苏州大学唐运海团队[3]利用平均曲率流优化镜片光学性能,又于2017年使用拉普拉斯方程进行研究以满足不同佩戴者对镜片的个性化需求[4]。2022年张海平等[5]将非球面方程与渐进多焦点镜片设计结合,优化镜片像散;但这种设计方法可能致使周边散光接近2倍的加光度,影响视觉成像效果[6]。1998年Loos等[7]提出了间接设计方法,通过构建平均曲率和主曲率差的模型并求解特定的高阶偏微分方程,间接获得渐进多焦点自由镜片的矢高值,在保证镜片光焦度情况下更有效地减少周边像散。2004年Wang等[2]在Loos的最小化模型基础上选择了有限元的方法求解面型,并且指出需要进一步研究最小化模型中的权重函数设计。2006年Steele等[8]使用迭代求解泊松方程的方法计算渐进镜片矢高。2012年Hsu等[9]利用B样条的局部控制能力来描述自由曲面。2014年Jiang等[10]提出了一种变分差分法的数值方法,以高效地求解泛函。2016年Li等[11]在间接设计方法的基础上,使用Zernike多项式来拟合自由曲面以提高加工精度。
以上研究表明渐进多焦点自由曲面镜片的间接法设计具有明显优势。目前,关于渐进镜片间接法设计的研究多集中在面型优化方面,对于权重函数设计算法的优化研究较少。因此,本文在已有的研究基础上[12-13],通过优化构建最小化模型算法中的权重函数来实现渐进多焦点镜片的优化,设计对比了两组权重函数以及两组平均曲率分布,以探究它们对渐进多焦点自由曲面光学性能优化的影响。
2 基本原理
渐进镜片主要分为4个区域,远用区、近用区、中间区(加光通道)和像散区(周边区域),如
理想的渐进多焦点自由曲面镜片应该具有平滑的渐进光焦度分布,并且没有额外的像散。然而,只有镜片表面为平面或球面时,整个表面上才有可能达到零像散,但单光镜无法满足规定的光焦度分布。因此,要达到理想的光焦度分布和不需要的像散的最优解,需要构建最小化镜片质量函数模型
式中:
3 不同曲率分布和权重函数设计
3.1 曲率分布
本文提出两种不同的平均曲率分布设计。根据镜片的直径、基弯以及光焦度直接计算得出镜片后表面平均曲率分布
图 2. 两种平均曲率分布。(a)间接法平均曲率分布;(b)直接法平均曲率分布
Fig. 2. Two mean curvature distributions. (a) Mean curvature distribution of indirect method; (b) mean curvature distribution of direct method
3.2 基于边界曲线的权重函数设计
为了满足不同佩戴者的要求,给定0.5 D(1 D=1 m-1)等散光边界曲线的函数表达,设计了一种不同于传统的权重函数
式中:
图 3. 硬设计和软设计权重函数分布。(a)硬设计权重函数;(b)软设计权重函数
Fig. 3. Weight function distributions of hard design and soft design. (a) Hard design weight function; (b) soft design weight function
权重函数
依据
3.3 重建权重分布
为了更好地优化渐进多焦点自由曲面镜片的光学性能,本节提出一种新的优化方法,以达到重建镜片面型的目的。以3.1节中镜片1的设计方法为基础,得到直径为
对于镜片上任意一个网格格点
设计光焦度与配戴者处方光焦度之间的差值为
符合度定义为设计值与处方值之差和人眼视力矫正之间的关系,用
对应镜片上
符合度权重函数分布如
图 5. 重建权重函数分布。(a)符合度权重函数分布;(b)优化重建权重函数分布
Fig. 5. Reconstruction weight function distributions. (a) Conformity weight function distribution; (b) optimized reconstruction weight function distribution
为了优化权重值位于0到1之间的不规则部分,将
4 分析与讨论
4.1 仿真实例
镜片的基础参数如
表 1. 镜片参数
Table 1. Lens parameters
|
对镜片1和镜片5进行仿真设计和对比分析,光焦度与像散仿真结果如图
图 6. 镜片1和5的光焦度仿真结果。(a)镜片1的光焦度;(b)镜片5的光焦度
Fig. 6. Simulation results of optical power of lenses 1 and 5. (a) Optical power of lens 1; (b) optical power of lens 5
图 7. 镜片1和5的像散仿真结果。(a)镜片1的像散;(b)镜片5的像散
Fig. 7. Simulation results of astigmatism for lenses 1 and 5. (a) Astigmatism of lens 1; (b) astigmatism of lens 5
在
从
4.2 加工实例
为了更好地说明权重函数和曲率分布对渐进镜片光学性能的影响,使用自由曲面机床(明月镜片股份有限公司)对这5组镜片进行加工,得到镜片实物,如
使用VM2000自由曲面检测仪器对这5组镜片进行测量[19],得到镜片的光焦度分布与像散分布如
图 9. 镜片光焦度测量结果。(a)镜片1的光焦度;(b)镜片2的光焦度;(c)镜片3的光焦度;(d)镜片4的光焦度;(e)镜片5的光焦度
Fig. 9. Measurement results of lens optical power. (a) Optical power of lens 1; (b) optical power of lens 2; (c) optical power of lens 3; (d) optical power of lens 4; (e) optical power of lens 5
图 10. 镜片像散测量结果。(a)镜片1的像散;(b)镜片2的像散;(c)镜片3的像散;(d)镜片4的像散;(e)镜片5的像散
Fig. 10. Measurement results of lens astigmatism. (a) Astigmatism of lens 1; (b) astigmatism of lens 2; (c) astigmatism of lens 3; (d) astigmatism of lens 4; (e) astigmatism of lens 5
使用焦度计测量这5组镜片的实际远用区球镜度(SPH)、像散(CYL)以及ADD,镜片的实际远用区、近用区及通道宽度如
表 2. 渐进镜片测量光度值及区域宽度
Table 2. Measured photometric values and zone width of PALs
|
综合
镜片2为直接法圆形设计,通道长度缩短,因此加光度ADD达到设计要求。镜片2近用区面积略缩小,远用区面积增大,像散明显减小。远用区光焦度也有所改善,与设计值误差减小,符合国家标准。
镜片3结合了硬设计渐进曲面与软设计渐进曲面的优点,因此远用区面积略有减小。镜片3远用区光焦度及像散都得到了优化,加光度与设计值之间的误差也减小了。从
镜片4为根据符合度矩阵重新计算权重后的镜片。远用区光焦度大于国家标准0.03 D[18],散光值也偏高,整体上对镜片1的算法实现了一定优化。远近用区面积与重置前比有所增大,但通道宽度过窄,周边区域最大散光大于1倍ADD,因此需要进行进一步优化。
镜片5为优化镜片4符合度矩阵后重置的镜片。更高的权重值以及更集中的权重分布使镜片处方与设计值的光焦度差几乎为0。镜片5与镜片2结合后,远用区到周边区域的散光变化增加缓慢,出现了一定的过渡,视线左右切换时的成像畸变减小。周边区域最大散光减小到1.75 D。远近用区面积虽略有减小,但也减小了相对较宽,散光较小,视线由远及近时的成像畸变减小。加工结果与仿真结果基本一致,此种优化方法能够实现渐进多焦点自由曲面镜片光学性能优化的目的。
5 结论
本文主要探究权重函数和平均曲率分布对渐进多焦点镜片设计的影响。通过设计5组不同的权重函数和平均曲率分布,计算出5组渐进多焦点自由曲面面型。镜片的加工和测量结果表明了权重函数和曲率分布共同影响镜片的光学性能。权重函数的形状、面积以及权重值的不同可对镜片光学性能带来不同方面的优化。合理的平均曲率分布可以提升渐进自由曲面的光焦度精准度。利用光焦度符合度重置后的权重函数可减小镜片光度误差,且使周边区域像散更加平缓地变化,以为佩戴者带来更好的视觉体验。本文还考虑了不同屈光状态下设计的灵活性,以满足不同佩戴者的个性化需求。在本文研究基础上,未来可以设计不同组的权重函数和平均曲率分布来进行对比研究,以得到光学性能更优越的渐进多焦点镜片。
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Article Outline
王亚琼, 项华中, 詹小蝶, 张云进, 丁琦慧, 张欣, 郑刚, 陈家璧, 王成, 张大伟, 庄松林. 基于权重函数优化的渐进多焦点镜片设计[J]. 光学学报, 2023, 43(8): 0822028. Yaqiong Wang, Huazhong Xiang, Xiaodie Zhan, Yunjin Zhang, Qihui Ding, Xin Zhang, Gang Zheng, Jiabi Chen, Cheng Wang, Dawei Zhang, Songlin Zhuang. Design of Progressive Addition Lens Based on Weight Function Optimization[J]. Acta Optica Sinica, 2023, 43(8): 0822028.