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摘要:
Zernike多项式因其正交性和旋转不变性，被广泛应用于光学表面的表征和优化中。它能够有效地减少拟合误差，并通过少量的系数实现对复杂曲面的高精度描述，有助于提升光学系统的成像质量和简化性能分析。本文首先对自由曲面描述方法进行了概括，包括全局描述与局部描述两种方式；随后讨论了国内外关于Zernike多项式在曲面表征中的研究进展，探讨了Zernike多项式在曲面表征中的实际应用；最后展望了Zernike多项式在曲面表征领域的发展前景。
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关键词:
- Zernike多项式 /
- 自由曲面 /
- 曲面表征 /
- 光学设计
Abstract:Zernike polynomials, due to their orthogonality and rotational invariance, are widely used in the characterization and optimization of optical surfaces. They can effectively reduce fitting errors and provide high-precision descriptions of complex surfaces with only a few coefficients, contributing to improved imaging quality and simplified performance analysis in optical systems. This paper provides an overview of freeform surface description methods, including both global and local approaches. It discusses the research progress on Zernike polynomials in surface characterization, both domestically and internationally, explores their practical applications in this field, and finally anticipates the future prospects of Zernike polynomials in surface characterization.
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Key words:
- Zernike polynomials /
- freeform surface /
- surface characterization /
- optical design
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Overview: This manuscript offers a detailed review of the advancements in the application of Zernike polynomials for surface characterization in optics. Zernike polynomials, known for their orthogonality and rotational invariance, have been extensively utilized in the characterization and optimization of optical surfaces. This paper first provides an overview of freeform surface description methods and discusses significant progress in the research of Zernike polynomials both domestically and internationally, exploring their practical applications in surface characterization. Finally, it anticipates the future prospects of Zernike polynomials in the field of surface characterization. Freeform surfaces, due to their non-rotational symmetric properties, present significant challenges for traditional optical design. Zernike polynomials offer a precise method to describe these complex surfaces, significantly improving the imaging quality and overall performance of optical systems. The paper highlights the application of Zernike polynomials in optical wavefront analysis and wavefront distortion, emphasizing their ability to decompose complex surfaces into independent components, reducing redundancy, and simplifying error analysis and characterization processes. International research has expanded the application range of Zernike polynomials beyond circular apertures, addressing limitations in traditional methods. These studies have explored applications, such as wavefront reconstruction and optical surface characterization. By developing orthogonal polynomials for elliptical, rectangular, and square apertures, researchers have significantly broadened the applicability of Zernike polynomials. Domestically, Chinese researchers have made significant contributions by generating orthogonal polynomials for non-circular apertures and studying the impact of sampling points on fitting accuracy. This research has enhanced the precision of Zernike polynomials in surface characterization, with applications including wavefront reconstruction and wavefront analysis, demonstrating the versatility and accuracy of Zernike polynomials in various optical design tasks. Compared with other characterization methods such as Q-type polynomials and XY polynomials, Zernike polynomials stand out for their high precision and flexibility, though combining these methods can further enhance the accuracy and efficiency of surface characterization. The manuscript suggests future research should focus on continued theoretical advancements, improved computational efficiency, and integration with other polynomial methods. These improvements will expand the application range and precision of Zernike polynomials in optical system design and optimization, driving progress in optical technology. Continued research and innovation in this field will further enhance the accuracy and efficiency of surface characterization, making Zernike polynomials an increasingly important tool in the advancement of optical systems.
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图 2 像差及其对应的Zernike多项式波前图
Figure 2. Wavefront plots of aberrations and their corresponding Zernike polynomials
表 1 常见表征函数优劣性比较
Table 1. Comparison of common characterization functions: advantages and disadvantages
曲面表征函数与方法 优势 缺点 Zernike多项式 正交，与经典相差一一对应，解析波前等 对局部凸起难以精确表征，高阶项计算复杂 Q型正交多项式 正交，加工可控性强，适合表征全局特性等 高阶计算复杂，尚未广泛集成到计算机软件当中 XY多项式 设计自由度高，适合表征全局特性等 非正交，不可解析波前 径向基函数 局部表征能力强 非正交 NURBS函数 局部表征能力强，构造灵活 非正交，构造复杂 
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表 2 Zernike多项式前9项与像差对应关系[5]
Table 2. Correspondence between the first 9 Zernike polynomials and optical aberrations[5]
项数 Zernike多项式 像差 Z1 1 平移 Z2 $ \rho \mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right) $ X轴倾斜 Z3 $ \rho \mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right) $ Y轴倾斜 Z4 $ {\rho }^{2}\mathrm{c}\mathrm{o}\mathrm{s}2\theta $ 初级像散 ($ 0\text{° }\text{或 90}\text{°}\text{轴)} $ Z5 $ 2{\rho }^{2}-1 $ 离焦 Z6 $ {\rho }^{2}\mathrm{s}\mathrm{i}\mathrm{n}2\theta $ 初级像散 ($ \pm {45}^{\circ } $轴) Z7 $ {\rho }^{3}\mathrm{c}\mathrm{o}\mathrm{s}3\theta $ 初级三叶草 (X-轴) Z8 $ \left(3{\rho }^{3}-2\rho \right)\mathrm{c}\mathrm{o}\mathrm{s}\theta $ 初级慧差 (X-轴) Z9 $ \left(3{\rho }^{3}-2\rho \right)\mathrm{s}\mathrm{i}\mathrm{n}\theta $ 初级慧差 (Y-轴)
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