Surface characterization using Zernike polynomials

Jingyuan Liang; Xiwen Li; Chenghu Ke; Xizheng Ke

doi:10.12086/oee.2024.240163

Journals >Opto-Electronic Engineering >Volume 51 >Issue 11 >Page 240163-1 > Article

Opto-Electronic Engineering
Vol. 51, Issue 11, 240163-1 (2024)

Surface characterization using Zernike polynomials

Jingyuan Liang¹, Xiwen Li¹, Chenghu Ke², and Xizheng Ke^1,3,*

Author Affiliations

¹School of Automation and Information Engineering, Xi'an University of Technology, Xi’an, Shaanxi 710048, China

²School of Information Engineering, Xi’an University, Xi’an, Shaanxi 710048, China

³Shaanxi Civil-Military Integration Key Laboratory of Intelligence Collaborative Networks, Xi’an, Shaanxi 710048, China

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DOI: 10.12086/oee.2024.240163 Cite this Article

Jingyuan Liang, Xiwen Li, Chenghu Ke, Xizheng Ke. Surface characterization using Zernike polynomials[J]. Opto-Electronic Engineering, 2024, 51(11): 240163-1 Copy Citation Text

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Fig. 1. 3D plots of the first 20 Zernike polynomials

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Fig. 2. Wavefront plots of aberrations and their corresponding Zernike polynomials

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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

Table 1. Comparison of common characterization functions: advantages and disadvantages

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项数	Zernike多项式	像差
Z1	1	平移
Z2	$ρ cos (θ)$	X轴倾斜
Z3	$ρ sin (θ)$	Y轴倾斜
Z4	$ρ^{2} cos 2 θ$	初级像散 ( $0^{\circ} 或 90^{\circ} 轴)$
Z5	$2 ρ^{2} - 1$	离焦
Z6	$ρ^{2} sin 2 θ$	初级像散 ( $\pm 45^{\circ}$ 轴)
Z7	$ρ^{3} cos 3 θ$	初级三叶草 (X-轴)
Z8	$(3 ρ^{3} - 2 ρ) cos θ$	初级慧差 (X-轴)
Z9	$(3 ρ^{3} - 2 ρ) sin θ$	初级慧差 (Y-轴)

Table 2. Correspondence between the first 9 Zernike polynomials and optical aberrations^[5]

Jingyuan Liang, Xiwen Li, Chenghu Ke, Xizheng Ke. Surface characterization using Zernike polynomials[J]. Opto-Electronic Engineering, 2024, 51(11): 240163-1

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