﻿ 横向剪切干涉测量中一种获得无耦合Zernike系数的模式复原方法
 光电工程  2019, Vol. 46 Issue (5): 180273      DOI: 10.12086/oee.2019.180273

1. 中国科学院光电技术研究所，四川 成都 610209;
2. 中国科学院自适应光学重点实验室，四川 成都 610209;
3. 中国科学院大学，北京 100049

Modal wavefront reconstruction to obtain Zernike coefficient with no cross coupling in lateral shearing measurement
Sun Wenhan1,2,3, Wang Shuai1,2, He Xing1,2, Chen Xiaojun1,2,3, Xu Bing1,2
1. Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
2. Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
3. University of the Chinese Academy of Sciences, Beijing 100049, China
Abstract: Modal cross coupling frequently occurs in modal approaches from wavefront gradient data such as lateral shearing measurement through Zernike circle polynomials, since the gradients of Zernike circle polynomials are not orthogonal. We use a modal approaches incorporating the Gram matrix, using the orthogonality of angular derivative of m≠0 modes with respect to weight function w(ρ) = ρ (polar coordinates), and the orthogonality of radial derivative of m = 0 modes with respect to weight function w(ρ) = ρ(1-ρ2) (polar coordinates). The Gram matrix method needs no auxiliary vector functions. The Zernike coefficients can be obtained with no modal cross coupling. The simulation results are given, which indicate that the modal cross coupling is avoided by using Gram matrix method. This method can be easily extended to annulus, and the coefficients of Zernike annular polynomials with no modal cross coupling can be obtained.
Keywords: Zernike circle polynomials    Zernike annular polynomials    shearing interferometer    wavefront reconstruction    modal cross coupling

1 引言

2 基于Gram矩阵方程的Zernike模式复原方法 2.1 Gram矩阵方法

 $\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \rho }})} \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \rho }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right] = {\boldsymbol{G}} \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (1)

 ${G_{jj'}} = \int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_0^1 {w(\rho )\frac{{\partial {Z_j}}}{{\partial \rho }}\frac{{\partial {Z_{j'}}}}{{\partial \rho }}{\rm{d}}\rho {\rm{d}}\theta } } ,$ (2)

2.2 Zernike圆多项式角向导数的正交性

 $Z_n^m(\rho ,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\sqrt {\frac{{n + 1}}{{\rm{ \mathit{ π} }}}} R_n^{\left| m \right|}(\rho )\sqrt 2 \cos (\left| m \right|\theta ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m > 0)} \\ {\sqrt {\frac{{n + 1}}{{\rm{ \mathit{ π} }}}} R_n^{\left| m \right|}(\rho )\sqrt 2 \sin (\left| m \right|\theta ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m < 0)} \\ {\sqrt {\frac{{n + 1}}{{\rm{ \mathit{ π} }}}} R_n^0(\rho )\;\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m = 0)} \end{array}} \right.,$ (3)

 $R_n^m(\rho ) = \sum\limits_{s = 0}^{(n - m)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s!(\frac{{n - m}}{2} - s)!(\frac{{n + m}}{2} - s)!}}} {\rho ^{n - 2s}}。$ (4)

Zernike圆多项式模式间有正交关系：

 $\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_0^1 {\rho Z_n^m(\rho ,\theta )Z_{n'}^{m'}(\rho ,\theta ){\rm{d}}\rho {\rm{d}}\theta } } = {\delta _{mm'}}{\delta _{nn'}}。$ (5)

 $\frac{{\partial Z_n^m(\rho ,\theta )}}{{\partial \theta }} = \left\{ \begin{gathered} \sqrt {\frac{{n + 1}}{{\rm{ \mathit{ π} }}}} R_n^{\left| m \right|}(\rho )\sqrt 2 \cos (\left| m \right|\theta ) \cdot ( - \left| m \right|)\;(m > 0) \\ \sqrt {\frac{{n + 1}}{{\rm{ \mathit{ π} }}}} R_n^{\left| m \right|}(\rho )\sqrt 2 \sin (\left| m \right|\theta ) \cdot {\kern 1pt} ( - \left| m \right|)\;(m < 0) \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(m = 0) \\ \end{gathered} \right.,$ (6)

 $\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_0^1 {\rho \frac{\partial }{{\partial \theta }}Z_n^m(\rho ,\theta )\frac{\partial }{{\partial \theta }}Z_{n'}^{m'}(\rho ,\theta ){\rm{d}}\rho {\rm{d}}\theta } } = {m^2}{\delta _{mm'}}{\delta _{nn'}}。$ (7)

 $\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \theta }})} \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \theta }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] = {\bf{G}} \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right]\\= \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (8)

2.3 Zernike圆多项式中m=0的项径向导数的正交性

Zernike圆多项式中m = 0的项可表示为[22-23]

 $R_n^0(\rho ){\kern 1pt} = {( - 1)^{\frac{1}{2}n}}{G_{\frac{1}{2}n}}(1,1,{\rho ^2}),$ (9)

 $\int\limits_0^1 {{\rho ^{q - 1}}{{(1 - \rho )}^{p - q}}{G_n}(p,q,\rho ){G_{n'}}(p,q,\rho ){\rm{d}}\rho } \\ = {N_n}(p,q){\delta _{nn'}},$ (10)

 ${N_n}(p,q) = \frac{{{{[\Gamma (q)]}^2}\Gamma (p + n - q + 1)}}{{\Gamma (p + n)\Gamma (q + n)}}\frac{{n!}}{{p + 2n}},$ (11)

 $\frac{{{\rm{d}}{\kern 1pt} }}{{{\rm{d}}\rho }}[R_n^0(\rho )] = {( - 1)^{\frac{1}{2}n + 1}}\frac{{n(n + 2)}}{2}\rho {G_{\frac{1}{2}n - 1}}(3,2,{\rho ^2})。$ (12)

 $\int\limits_0^1 {(1 - {\rho ^2})\frac{{{\rm{d}}[R_n^0(\rho )]{\kern 1pt} }}{{{\rm{d}}\rho }}\frac{{{\rm{d}}[R_{n'}^0(\rho )]{\kern 1pt} }}{{{\rm{d}}\rho }}{\rm{d}}({\rho ^2})} \\ = {N_{\frac{1}{2}n - 1}}(3,2)\frac{{{n^2}{{(n + 2)}^2}}}{4}{\delta _{nn'}}。$ (13)

 $\int\limits_0^1 {\rho (1 - {\rho ^2})\frac{{{\rm{d}}[R_n^0(\rho )]{\kern 1pt} }}{{{\rm{d}}\rho }}\frac{{{\rm{d}}[R_{n'}^0(\rho )]{\kern 1pt} }}{{{\rm{d}}\rho }}{\rm{d}}\rho } \\ = \frac{{n(n + 2)}}{{2(n + 1)}}{\delta _{nn'}}。$ (14)

 $\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_0^1 {\rho (1 - {\rho ^2})\frac{\partial }{{\partial \rho }}Z_n^0(\rho ,\theta )\frac{\partial }{{\partial \rho }}Z_{n'}^{m'}(\rho ,\theta ){\rm{d}}\rho {\rm{d}}\theta } } \\ = n(n + 2){\delta _{0m'}}{\delta _{nn'}}。$ (15)

2.4 复原流程

 $R_n^m(\rho ;\varepsilon ) = N_n^m\left[ {R_n^m(\rho ) - \sum\limits_{i = 1}^{(n - m)/2} {\frac{{2(n - 2i + 1)}}{{1 - {\varepsilon ^2}}}R_{n - 2i}^m(\rho ;\varepsilon )} } \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \cdot \int\limits_0^1 {\rho R_n^m(\rho )} R_{n - 2i}^m(\rho ;\varepsilon ){\rm{d}}\rho } \right],$ (25)

Zernike环多项式模式间有正交关系：

 $\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_\varepsilon ^1 {\rho Z_n^m(\rho ,\theta ;\varepsilon )Z_{n'}^{m'}(\rho ,\theta ;\varepsilon ){\rm{d}}\rho {\rm{d}}\theta } } = {\delta _{mm'}}{\delta _{nn'}}。$ (26)

 $\frac{{\partial Z_n^m(\rho ,\theta ;\varepsilon )}}{{\partial \theta }}\\ = \left\{ {\begin{array}{*{20}{c}} {\sqrt {\frac{{n + 1}}{{{\rm{ \mathit{ π} }}(1 - {\varepsilon ^2})}}} R_n^{\left| m \right|}(\rho ;\varepsilon )\sqrt 2 \cos (\left| m \right|\theta ) \cdot ( - \left| m \right|){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m > 0)} \\ {\sqrt {\frac{{n + 1}}{{{\rm{ \mathit{ π} }}(1 - {\varepsilon ^2})}}} R_n^{\left| m \right|}(\rho ;\varepsilon )\sqrt 2 \sin (\left| m \right|\theta ) \cdot {\kern 1pt} ( - \left| m \right|){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m < 0)} \\ {0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m = 0)} \end{array}} \right.,$ (27)

 $\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_\varepsilon ^1 {\rho \frac{{\partial Z_n^m(\rho ,\theta ;\varepsilon )}}{{\partial \theta }}\frac{{\partial Z_{n'}^{m'}(\rho ,\theta ;\varepsilon )}}{{\partial \theta }}{\rm{d}}\rho {\rm{d}}\theta } } = {m^2}{\delta _{mm'}}{\delta _{nn'}}。$ (28)

 $\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \theta }})} \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \theta }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] = {\boldsymbol{G}} \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (29)

4.2 Zernike环多项式中m=0的项径向导数的正交性

Zernike环多项式中m = 0的项的径向多项式有显表达式[26]:

 $R_n^0(\rho ;\varepsilon ) = R_n^0\left( {\sqrt {\frac{{{\rho ^2} - {\varepsilon ^2}}}{{1 - {\varepsilon ^2}}}} } \right)。$ (30)

 ${(\frac{{\partial W}}{{\partial \rho }})_{{\rm{meas}}}}(\rho ,\theta ) = {(\frac{{\partial W}}{{\partial x}})_{{\rm{meas}}}}\cos \theta + {(\frac{{\partial W}}{{\partial y}})_{{\rm{meas}}}}\sin \theta ,$ (36)
 ${(\frac{{\partial W}}{{\partial \theta }})_{{\rm{meas}}}}(\rho ,\theta ) = - {(\frac{{\partial W}}{{\partial x}})_{{\rm{meas}}}}\rho \sin \theta + {(\frac{{\partial W}}{{\partial y}})_{{\rm{meas}}}}\rho \cos \theta ,$ (37)

 $\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \theta }})} \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \theta }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (38)

 $\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \rho }})} \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \rho }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right]。$ (39)

4.4 复原结果

 图 6 波前复原误差和残差λr与截断数J的关系 Fig. 6 Reconstruction error and remaining error λr versus truncation of reconstruction modes J

 图 7 波前复原结果。(a) 原始波前；(b) 复原波前；(c) 复原残留误差 Fig. 7 Reconstruction results. (a) Original wavefront; (b) Reconstructed wavefront; (c) Residual error
5 结论

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