光电工程

Coefficient	PMMA
B₀	2.18645820
B₁	−2.44753480×10⁻⁴
B₂	1.41557870×10⁻²
B₃	−4.43297810×10⁻⁴
B₄	7.76642590×10⁻⁵
B₅	−2.99363820×10⁻⁶

Coefficient	PMMA
B₀	2.18645820
B₁	−2.44753480×10⁻⁴
B₂	1.41557870×10⁻²
B₃	−4.43297810×10⁻⁴
B₄	7.76642590×10⁻⁵
B₅	−2.99363820×10⁻⁶

基于改进DBS算法的三波长消色差透镜设计

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蔡吉祥 ¹ , 胡成宇 ¹ , 李玮 ² , 张婷婷 ¹^,^*^,^*

光电工程 | 科研论文 2026,53(5): 250319-1

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光电工程 |科研论文 2026 , 53 (5) : 250319 -1

基于改进DBS算法的三波长消色差透镜设计

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蔡吉祥¹, 胡成宇¹, 李玮², 张婷婷¹^,^*^,^*

作者信息

¹内蒙古工业大学机械工程学院，内蒙古自治区呼和浩特 010051

²中国科学院大学中国科学院长春光学精密机械与物理研究所，吉林长春 130033

通讯作者:

张婷婷，20230000011@imut.edu.cn

作者简介:

作者贡献

蔡吉祥：研究资料的收集、分析，指导论文选题，论文审核与编辑写作；胡成宇，李玮：论文初稿写作与论文修改；张婷婷：论文审核与编辑写作。

蔡吉祥 (1994-)，男，博士，讲师，硕士研究生导师，2017年获得内蒙古工业大学测控技术与仪器学士学位，2023年获得重庆大学仪器科学与技术博士学位 (中国科学院光电技术研究所联合培养)，主要从事微纳光学-超构表面、光学系统设计的研究。E-mail：jxcai@imut.edu.cn jxcai@imut.edu.cn

张婷婷 (1989-)，女，博士，讲师，硕士研究生导师，2023年获得太原理工大学光学工程博士学位，主要从事光电检测技术的研究。E-mail：20230000011@imut.edu.cn 20230000011@imut.edu.cn

Design of three-wavelength achromatic lens with modified DBS algorithm

Jixiang Cai¹, Chengyu Hu¹, Wei Li², Tingting Zhang¹

Affiliations

¹School of Mechanical Engineering, Inner Mongolia University of Technology, Hohhot, Inner Mongolia Autonomous Region 010051, China

²Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin 130033, China

Corresponding Author:

20230000011@imut.edu.cn

出版时间: 2026-05-26 doi: 10.12086/oee.2026.250319

文章导航

摘要

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传统衍射透镜受限于材料的固有特性，存在聚焦效率低、工作波长单一和色差等问题，常通过多个透镜级联提高成像质量，限制了光学系统的小型化与集成化发展。多台阶微透镜为光学系统的微型化提供了新思路，然而仍受到色差等影响且设计较为复杂。为此，本文提出一种基于改进直接二进制搜索算法的三波长消色差透镜设计方法，在1.064 μm、1.300 μm和1.550 μm三波长下实现了高效率消色差。理论结果表明，优化后的八台阶、十六台阶透镜的平均聚焦效率分别达到90.99%和92.46%，相较于优化前分别提升了13.95%和13.64%。该方法可用于三波长成像系统、光计算与光信息处理和生物医学成像等领域。

关键词

三波长 / 直接二进制搜索算法 / 消色差 / 高效率 / 衍射透镜

Abstract

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Objective

Traditional diffraction lenses can manipulate wavefronts by adjusting the refractive index of materials and surface profiles of devices, which are widely applied in optical imaging. However, such imaging capabilities are usually constrained by limitations of low focusing efficiency, a single operating wavelength, and chromatic aberration. To suppress chromatic aberration and improve image quality, it is commonly used by combining positive and negative refractive index lenses. Nevertheless, this scheme hinders the miniaturization and integration of optical systems. Although monolithic refractive-diffractive hybrid lenses provide an alternative method to enhance system integration for achromatic performance, complex structural design restricts their further applications. Recently, metasurfaces offer another methodology for realization of broadband achromatic, but the high processing precision in micro-nanofabrication is always required. Consequently, there is still a great challenge for designing diffractive lenses with simple structure, low processing difficulty, and achromatism.

Methods

For conventional direct binary search (DBS) algorithm to optimize diffraction lenses, the random initial structures are commonly used, whereas it commonly leads to slow convergence, sensitivity to the starting state. Here, a modified DBS algorithm is proposed for designing three-wavelength multi-order achromatic diffraction microlenses by directly tuning the discrete surface-relief heights. Utilizing the radially symmetric initial height distribution, such approach can reduce the randomness and improve computational stability and corresponding main procedures are as follows: First, the ideal continuous phase distribution of achromatic focusing lens with a parabolic phase profile can be generated; Second, such continuous phase is quantized into discrete phase corresponding to different step-heights for optimizing multilevel diffractive lens by iterative perturbations of local phase; Third, a radially symmetric initial structure determined by modulo function is introduced to replace the conventional random initialization, and then the step heights corresponding to different wavelengths are alternately distributed at distinct positions. Finally, a figure-of-merit (FOM) is adopted to evaluate the performance of the designed lens and guides the iterative updates of the step-height distributions. As a proof of concept, the 8-step and 16-step diffraction lenses (with a diameter, focal length, and numerical aperture of 560 μm, 10 mm, and 0.028, respectively) are designed and analyzed both theoretically and through simulation.

Results and Discussions

For pre-optimization, the average focusing efficiencies of the diffractive lenses with 8 phase levels (L=8) and 16 phase levels (L=16) are 77.04% and 78.82% at the wavelengths of 1.064 μm, 1.300 μm, and 1.550 μm, respectively. Therefore, the average focusing efficiencies of both lenses are less than 79% affected by chromatic aberration in theory. To improve the focusing efficiency and suppress chromatic aberration, a modified DBS algorithm is employed to optimize the step-height distribution. For post-optimization, the focusing efficiencies of diffraction lens (L=8) can reach 87.92%, 94.32%, and 90.73% (the efficiency is 90.99%) at 1.064 μm, 1.30 μm, and 1.55 μm after 4000 iterations, respectively, which is improved 13.95% compared to pre-optimization in theory. When L is changed to 16, the efficiencies exceed 90.01%, 96.63%, and 90.73% (at 1.064 μm, 1.300 μm, and 1.550 μm) respectively, and corresponding average value surpasses 92.46% (the improvement is 13.64% compared to pre-optimization) under the same conditions. These results indicate that the proposed achromatic diffraction lens maintains higher performance with the proposed modified DBS algorithm in theory. To further validate the achromatic performance, numerical simulations are conducted using CST microwave studio. For the diffraction lens with L=8 and 16, the focusing efficiencies at 1.064 μm are 87.17% and 89.53% in simulation, respectively. Moreover, it can be increased to 89.10% and 96.24% at 1.30 μm, while the efficiencies are 89.10% and 89.08% at 1.55 μm, respectively. Therefore, corresponding average efficiencies can reach 90.09% (L=8) and 91.62% (L=16), respectively. Note that the slight difference between theoretical prediction and numerical simulation may be attributed to the ideal assumptions (neglected the interactions between structures) in the theory. Furthermore, the point spread functions (PSFs) at the focal plane for different wavelengths are employed to characterize the achromatic performance of the designed diffraction lens. The simulated full width at half maximums (FWHM) are 25.8 μm, 26.9 μm, and 33.4 μm (L=8) and 26.9 μm, 26.9 μm, and 33.4 μm (L=16) at 1.064 μm, 1.300 μm, and 1.550 μm, respectively, which are agree well with the theoretical results of 21.8 μm, 26.6 μm, and 32.2 μm and close to their diffraction limits of 19.00 μm, 23.22 μm, and 27.69 μm at different wavelengths. Totally, both the 8-step and 16-step diffraction lenses possess an average focusing efficiency of ~79% (at 1.064 μm, 1.30 μm, and 1.55 μm) before optimization. However, the average efficiencies can exceed 90.99% and 92.46%, (improvements of 13.95% and 13.64%), respectively, based on the improved DBS algorithm (after 4000 iterations) in theory. Moreover, it has also demonstrated that the FWHM of the 8-step and 16-step diffraction lenses can reach the diffraction limit at three wavelengths, exhibiting high imaging performance.

Conclusions

Here, a new methodology is proposed for achieving three-wavelength achromatism, and may offer advantages in reduction of device complexity, requirements of machining accuracy, and the manufacturing costs. These results may be further extended to diverse applications, such as three-wavelength imaging systems, optical computing, lightweight collimators, optical communications, biomedical imaging, and laser processing.

Key words

three-wavelength / direct binary search algorithm / achromatism / high focusing efficiency / diffraction lens

引用本文

蔡吉祥, 胡成宇, 李玮, 张婷婷. 基于改进DBS算法的三波长消色差透镜设计. 光电工程, 2026 , 53 (5) : 250319 -1 . DOI: 10.12086/oee.2026.250319

Jixiang Cai, Chengyu Hu, Wei Li, Tingting Zhang. Design of three-wavelength achromatic lens with modified DBS algorithm[J]. Opto-Electronic Engineering, 2026 , 53 (5) : 250319 -1 . DOI: 10.12086/oee.2026.250319

正文

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1　引　言

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传统折射透镜系统 (如显微镜和望远镜)长期受限于光学材料色散和光学器件结构缺陷引起的色差问题，限制了其成像质量的提高^[1]。为降低色差影响，常基于互补色散特性^[2-3]，通过传统折射透镜与衍射透镜混合实现色差补偿^[4-6]，然而，此类折衍混合透镜系统所需要透镜的数量与消色差的波数成正比^[7]，在一定程度上制约了光学系统集成化发展。单片式折衍混合透镜^[8-10]采用一侧为折射面，另外一侧为衍射面的设计，可在消色差的同时有效减少该功能学元件的数量，但是整体结构较复杂，限制了其广泛应用。超构表面^[11-13]可在亚波长尺度实现对光波相位、偏振、振幅和频率等参量的灵活调控，已实现宽带消色差能力^[14-15]，广泛用于偏振成像^[16-18]、聚焦透镜^[19]、电磁隐身^[20-22]和负折射材料^[23-24]等领域，但是，其加工精度要求较高，工艺复杂度远超传统衍射元件^[25-26]。目前，主要采用结构较简单且加工难度较低的衍射透镜消除色差。

计算机技术的快速发展为消色差透镜设计提供了新途径。2021年，中国科学院光电技术研究所巩畅畅等^[27]设计了一种分区式消色差衍射透镜，通过RGB三色波长对应各自的同心环带，实现了衍射透镜结构的简化，平均效率低于40%。次年，苏州大学马顾等^[28]设计了一种大孔径、高效率消色差衍射透镜 (phase-coded diffractive lens, PCDL)，在50 nm的带宽和16°视场的范围内实现了衍射极限成像。2025年，李奥玲等^[29]设计了两种多波长混合折射透镜，聚焦效率分别为74.64%和86.03%。2021年，郑高等^[30]利用材料低色散特性和四重对称结构在可见光波段设计了一种聚焦效率接近71%宽带消色差超透镜。综合而言，最近设计的消色差透镜仍存在效率提升有限或结构设计较复杂的问题。为此，本文提出了基于改进直接二进制搜索算法的多台阶衍射透镜设计方法，在1.064 μm、1.300 μm与1.550 μm三波长实现了平均超过90%的聚焦效率，在光学效率、器件集成化与加工工艺实现较好平衡。

本文采用改进直接二进制搜索算法，对微结构高度进行迭代优化，设计了一种近红外三波长消色差多台阶衍射透镜。该透镜在1.064 μm、1.300 μm和1.550 μm三波长下实现了高效率消色差，优化后的八台阶和十六台阶一维透镜的仿真聚焦效率分别达到 90.09%和91.62%，均表现出高效消色差性能，且分辨率接近衍射极限，各波长归一化强度响应得到有效优化，抑制了色差的影响。该设计方法可降低成像系统的结构复杂度，为光学领域的创新提供新的思路和方向。

2　基本原理

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2.1　传统衍射微透镜

目前，多数介电材料的折射率随波长增大而减小，传统折射透镜的焦距随波长的增加而增大，从而产生色差，如图1 (a)所示。传统衍射透镜通常被认为是衍射光学元件^[31-33] (diffractive optical element, DOE)，其典型结构通常一侧为平面，另外一侧为不同高度的同心圆环浮雕结构，如图1 (b)所示。衍射透镜的色差通常由透镜表面微结构高度的差异所致，使不同波长的光在空间上分离，常通过优化其结构高度抑制色差。DOE通常采用抛物线的相位模型进行相位调制，其数学表达式为

$ \varphi_1(r)=\dfrac{2\text{π}}{\lambda}\left(f-\sqrt{r^2+f^2}\right), $

(1)

式中：λ是入射光波长；r表示衍射透镜半径；f为衍射透镜的焦距。传统衍射透镜通常会将上述相位限制在0~2π区间，并进行离散相位梯度处理。由式 (1)可知，相位分布与λ相关，这导致传统衍射透镜的工作带宽有限，并受色差影响。图1(c)所示的一维多台阶衍射透镜在一定程度上可提高聚焦效率，为消色差透镜设计提供了新方案。

2.2　DBS算法相位优化

目前常用于消色差衍射透镜优化算法常采用 Gerchberg-Saxton (GS)算法^[34]、遗传算法^[35-36]、深度学习算法^[37-39]和直接二进制 (direct binary search, DBS)算法^[40-41]。GS算法通常需要填零减少采样间隔，导致计算量增大，且输出平面的非信号区有明显的斑纹^[34]；遗传算法需大量预设参数，尤其针对多变量问题，在全局范围内搜索速度较慢，收敛效率较低^[42]；深度学习算法需大量的训练数据集，且实现起来具有一定的挑战性^{[37, 43]}。因此，本文基于式 (1)结合DBS算法，将多台阶衍射透镜的相位量化过程体现为离散的台阶高度变化，并通过迭代扰动局部区域的相位状态优化多台阶衍射透镜，此相位延迟模型可表述为

$ \varphi_i=k_0\left(n(\lambda)-1\right)\cdot h_i, $

(2)

式中：

$ {k}_{0}=2{\text{π}} /\lambda $

为自由空间的波矢；n (λ)是波长为λ对应的折射率；h_i表示第i级台阶的高度。基于高度与相位的定量关系，DBS算法能筛选出最优台阶高度分布h_i，抑制色差。此时，衍射透镜的台阶高度为

$ h(\lambda)=\dfrac{\lambda}{2\text{π}(n(\lambda)-1)}\varphi_i. $

(3)

传统DBS算法常需多次生成随机初始结构实现高度优化，降低了计算效率，并且台阶高度不具备旋转对称性，难以推广至二维多台阶衍射元件。针对传统DBS算法受随机结构影响、效率低、耗时严重等问题^[44]，本文提出了一种采用径向对称初始结构的改进DBS算法，其计算方法公式为

$ {h}_{{\mathrm{chushi}}}=\begin{cases} {\rm{Mod}}\left[\dfrac{{ (m\cdot d)}^{2}}{\text{3}f (n ({\lambda }_{1})-1)},{h}_{\max } ({\lambda }_{1})\right],\begin{array}{lll}m & {\mathrm{is}} & \text{0}\end{array}\\{\rm{Mod}}\left[\dfrac{{ (m\cdot d)}^{2}}{\text{3}f (n ({\lambda }_{2})-1)},{h}_{\max } ({\lambda }_{2})\right],\begin{array}{lll}m & {\mathrm{is}} & \text{1}\end{array}\\{\rm{Mod}}\left[\dfrac{{ (m\cdot d)}^{2}}{\text{3}f (n ({\lambda }_{3})-1)},{h}_{\max } ({\lambda }_{\text{3}})\right],\begin{array}{lll}m & {\mathrm{is}} & \text{2}\end{array}\end{cases}, $

(4)

式中：Mod (a,b)为取余函数；f为衍射透镜的焦距；n (λ₁)、n (λ₂)和n (λ₃)是不同波长下材料的折射率；m是台阶的序号；d是台阶的宽度；h_max (λ)是不同波长下相移为2π时的台阶高度。取余函数将不同波长的台阶高度交替分布在不同位置，每个波长在相应位置上可获得有效相位调制。同时，所设计的衍射透镜具有径向对称的台阶高度分布，有助于提高DBS算法的计算效率。

$ h_{\max}(\lambda)=\dfrac{2\text{π}}{k_0(n(\lambda)-1)}. $

(5)

为评价所设计衍射透镜的性能，采用式 (6)所示的评价函数 (figure-of-merit，FOM)进行量化评估：

$ FOM={\left(\displaystyle\sum_{i=1}^M\omega_i\mu_i\right)}/{M}-\alpha\left({\displaystyle\sum_{i=1}^M\omega_i\varepsilon_i}\right)/{M}, $

(6)

式中：第一项表示M个波长的平均加权效率；第二项为M个波长的模拟函数和目标函数的加权归一化绝对差值。为平衡两项的权重，引入调节系数α (α =10)。同时，通过加权系数ω_i调控不同波长的权重占比。为进一步评估效率，引入效率μ和归一化(normalized)绝对差值ε^[39]：

$ \mu_j=\dfrac{\displaystyle\int_{x_{\min}}^{x_{\max}}I_j(x)\cdot T_j(x)\mathrm{d}x}{\displaystyle\int_{x_{\min}}^{x_{\max}}I_j(x)\mathrm{d}x}, $

(7)

$ \varepsilon_j=\dfrac{\displaystyle\int_{x_{\min}}^{x_{\max}}\left|normalize(I_j(x))-T_j(x)\right|\mathrm{d}x}{\displaystyle\int_{x_{\min}}^{x_{\max}}\mathrm{d}x}, $

(8)

$ T_j(x)=\exp\left\{-\dfrac{\left[x-(x_{\max}+x_{\min})/2\right]^2}{\left(\lambda_j/(4\cdot NA)\right)^2}\right\}, $

(9)

式中：

$ {I}_{j} (x)=|{U}_{j} (x,y){|}^{2} $

为第j个波长在像面的模拟光强分布；

$ {T}_{j}\left (x\right) $

是第j个波长的观测平面上聚焦点扩散函数 (PSF)的一阶近似值，以

$ (x\mathrm{_{min}}+x_{\mathrm{max}})/2 $

为中心的高斯函数，是评估衍射透镜消色差能力的关键指标。基于上述原理，本文改进的DBS算法流程如图2所示。

3　衍射透镜设计

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在近红外波段，1.064 μm、1.300 μm和1.550 μm波长的激光广泛应用于生物医学和光纤通信领域。本文设计的近红外三波长消色差衍射透镜的直径为560 μm，由宽度为2 μm的同心台阶构成，每个台阶的高度在0~3 μm范围内以0.1 μm为特征尺寸变化，共31级，NA = 0.028，焦距f =10 mm。在近红外范围，PMMA在入射光波长<2.8 μm时，保持高透过率，且熔点为150 ℃，便于制造。凭借PMMA具有低熔点、透光率高，线性膨胀系数低等优势，且可兼容光刻技术快速加工掩模板，实现低成本和大规模生产，常用作玻璃的替代材料。PMMA材料色散系数及折射率公式如表1和式 (10)所示：

$ n^2(\lambda)=B_0+B_1\lambda^2+B_2\lambda^{-2}+B_3\lambda^{-4}+B_4\lambda^{-6}+B_5\lambda^{-8}, $

(10)

式中：n (λ)为PMMA在波长λ处的折射率；λ为入射光波长。依据表1和式 (10)，PMMA在波长1.064 μm、1.300 μm和1.550 μm时的折射率分别为1.4827、1.4813和1.4804。

为验证改进DBS算法的可行性，基于矢量衍射理论计算焦深方向上的理论强度分布^[46]。此外，利用商用电磁仿真软件CST Microwave Studio中通过有限积分技术模拟消色差衍射透镜的光学特性，在x和z方向均采用开放边界条件，y轴方向上采用周期边界条件，平面波激励源为沿z传播的x线偏振光。为提高工作效率缩短计算时间，在距离衍射透镜出口1 μm出提取电场分布，并基于矢量角谱理论计算后续光场。

3.1　理想相位分布与量化阶数对比分析

改进的 DBS算法通过优化衍射透镜的台阶高度，并生成对应的相位轮廓。图3(a-c)和图3(d-f)分别为16台阶衍射透镜，波长为1.064 μm、1.300 μm和1.550 μm的理想相位分布与优化后的相位分布，且在中心位置截取的相位调制曲线如图3(g-i)所示。

为进一步分析改进DBS算法的优化结果，优化高度对比与相位误差分析结果如图4所示。其中，图4(a,b)为优化后8台阶和16台阶衍射透镜高度图。在1.064 μm、1.300 μm和1.550 μm波长处，L=8 (L=16)优化后的相位曲线与理想相位曲线之间的最大允许误差 (maximum permissible errors, MPE)分别为10.85° (8.61°)、11.25° (11.14°)和10.24° (10.21°)，平均值为10.78° (9.98°)，如图4(c)所示。改进DBS算法优化后的相位分布曲线与理论结果基本一致，误差较小，基本满足设计要求。

优化前八台阶 (L=8)和十六台阶 (L=16)的衍射透镜焦深强度分布如图5所示。L=8时，波长为1.064 μm、1.30 μm和1.55 μm时 (图5(a-c))，衍射透镜的聚焦效率分别为61.50%、82.55%和87.08%，平均聚焦效率为77.04%。聚焦效率定义为焦斑3倍半高全宽 (full width at half maximum, FWHM)范围内的光功率与器件的传输光功率之比^[47]。当L=16时 (图5(d-f))，波长为1.064 μm、1.30 μm和1.55 μm时，聚焦效率分别为66.85%、82.94%和86.68%，平均值为78.82%。受色差等影响，两个衍射透镜的平均聚焦效率均小于79%。

为提高衍射透镜的聚焦效率并抑制色差，以上述透镜为基础，采用改进DBS算法优化其结构高度。在不同波长下，随迭代次数变化的聚焦效率分布如图6所示。当L=8 (图6(a))，迭代次数为4000次，波长为1.064 μm、1.300 μm和1.550 μm时，相应的优化后的衍射透镜聚焦效率分别为87.92%、94.32%和90.73%，其平均值达到90.99%，相较优化前提高了13.95%。同样条件下，L=16时(图6(b))，优化后的衍射透镜在不同波长的光入射时，对应的聚焦效率分别达到90.01%、96.63%和90.73% (平均值为92.46%，相较于优化前提高了13.64%)。由于一维衍射透镜结构间的相互作用相较于二维透镜更小，聚焦效率相对偏高。此外，近期研究人员设计了相关高效率消色差衍射透镜，例如Banerji科研团队设计的短波红外波段多级衍射透镜 (multi-level diffractive lens, MDL)通过直接二进制搜索 (DBS)算法，可将点扩散函数 (point spread function, PSF)能量集中于主瓣，能有效抑制高阶衍射能量^[48-50]，模拟聚焦效率可达到91%。此外，迭代次数接近 4000时，两个衍射透镜的平均聚焦效率趋于稳定 (L=8，接近91%；L=16，接近92%)，因此，后续优化过程中，迭代次数均设为4000次。

为分析加工误差对消色差衍射透镜的影响，将消色差衍射透镜引入加工误差 (每个台阶以0.2 μm的高度间隔变化至0.8 μm)，如图7所示。当波长为1.064 μm，误差高度在0~0.8 μm时，8台阶和16台阶的聚焦效率下降幅度较大，聚焦效率分别为67.1%和65.31%，与理想无误差的聚焦效率87.92% (L=8)和90.0% (L=16)分别下降了20.82%和24.69%。相比之下，波长为1.300 μm和1.550 μm时，加工误差的引入对衍射透镜聚焦效率影响较小，8台阶和16台阶的平均聚焦效率约为85%，衍射透镜的聚焦效率下降低于10%。8台阶和16台阶三波长平均聚焦效率~80%，相较于理想无误差降低~12%。本文设计的消色差衍射透镜引入加工误差下仍保持较好的性能，具有一定实际应用的可行性。

3.2　理论与仿真对比分析

优化后的一维8台阶消色差衍射透镜的理论和仿真归一化强度分布如图8所示。衍射透镜的聚焦效率在1.064 μm、1.300 μm和1.550 μm波长处，理论结果分别达到87.92%、94.32%、90.73% (平均值为90.99%)，如图8(a-c)所示。图8(d-f)的仿真结果表明，聚焦效率分别为87.17% (1.064 μm)、94.00% (1.300 μm)和89.10% (1.550 μm)，平均值为90.09%。仿真与理论结果基本一致，且均在焦平面位置 (f=10 mm)聚焦，表明基于改进DBS算法设计的一维衍射透镜具备高效消色差性能，两者之间的差异可能由于理论计算基于完美条件，未考虑结构间的相互作用影响^[51]。

为进一步验证其高效消色差特性，设计了图9所示的一维16台阶消色差衍射透镜。当1.064 μm波长的光入射时，聚焦效率的理论和仿真分别为90.01%和89.53%，如图9(a,d)所示。当入射光波长变为1.30 μm时，聚焦效率分别为96.63% (理论)和96.24% (仿真) (图9(b,c))。波长为λ=1.55 μm时，图9(c,f)所示的理论和仿真效率分别为90.73%和89.08%。不同波长光入射时，该透镜的理论和仿真平均聚焦效率为92.46%和91.62%。相较于图8所示的8台阶透镜，16台阶透镜的平均效率提高～1.5%，同样具备高效消色差性能，但制造难度相对更大。

图10为8台阶 (L=8)和16台阶 (L=16)消色差衍射透镜的焦平面处的理论 (蓝色虚线)和仿真 (红色虚线)归一化强度曲线。如图10(a-c)所示，波长为1.064 μm、1.300 μm和1.550 μm时，理论FWHM分别为21.8 μm、26.6 μm和32.2 μm，仿真为25.8 μm、26.9 μm和33.4 μm。当L=16时，三波长对应的仿真FWHM为26.9 μm、26.9 μm和33.4 μm，而理论结果不变，如图10(d-f)所示。上述理论和仿真结果在不同波长处，均接近其衍射极限 19.00 μm、23.22 μm和27.69 μm (

$ \lambda/(2NA) $

，NA=0.028)^[52]，在消色差的同时，具备较好的成像性能。

4　结　论

收起

本文设计了一种基于改进DBS算法的一维多台阶消色差衍射透镜。在1.064 μm、1.300 μm和1.550 μm入射波长下，本文设计的八台阶和十六台阶消色差衍射透镜理论平均聚焦效率分别为90.99%和92.46%，仿真平均聚焦效率分别为90.09%和91.62%，具备高效消色差性能，且成像分辨率接近衍射极限。本文设计的多台阶消色差衍射透镜为三波长消色差问题提供了新的思路与方向，有望应用于三波长成像系统、光信息处理、轻量化准直器、光通信等领域。

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2026年第53卷第5期

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doi: 10.12086/oee.2026.250319

接收时间：2025-10-28
首发时间：2026-10-22
出版时间：2026-05-26

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收稿日期：2025-10-28
修回日期：2026-01-27
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¹内蒙古工业大学机械工程学院，内蒙古自治区呼和浩特 010051

²中国科学院大学中国科学院长春光学精密机械与物理研究所，吉林长春 130033

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张婷婷，20230000011@imut.edu.cn

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2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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