光电工程  2019, Vol. 46 Issue (5): 180273      DOI: 10.12086/oee.2019.180273     
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
孙文瀚1,2,3 , 王帅1,2 , 何星1,2 , 陈小君1,2,3 , 许冰1,2     
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
摘要:模式耦合误差常见于横向剪切干涉测量中基于波前梯度数据的模式复原法,其原因是用于表示波前的基函数——Zernike圆多项式的导数不正交。使用一种含有Gram矩阵的矩阵方程进行复原,直接利用Zernike圆多项式m≠0模式角向导数对于权重函数w(ρ) = ρ (极坐标下)的正交性,以及Zernike圆多项式m = 0模式径向导数对于权重函数w(ρ) = ρ(1-ρ2)(极坐标下)的正交性进行复原。该方法无需构造辅助的向量函数,并可得到无耦合的Zernike系数,复原结果表明,模式耦合得到了避免。该方法可推广到环上,得到无耦合的Zernike环多项式系数。
关键词Zernike圆多项式    Zernike环多项式    剪切干涉    波前复原    模式耦合    
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

横向剪切干涉仪[1-4]通过剪切干涉获得的干涉条纹信息,再通过Fourier分析处理得到波前梯度的近似分布,在自适应光学系统中[5],横向剪切干涉仪是一种重要的波前传感器,因其系统简洁稳定而受到广泛应用[6]。从波前梯度信息出发,可通过区域法[4]、模式法[1-3]等方式复原波前信息。模式法将波前展开为正交的基函数的线性组合,因此获取了各阶模式系数也就复原了波前信息。

单位圆上常用的模式为Zernike圆多项式。横向剪切干涉仪中的Zernike模式复原是基于由Rimmer和Wyant建立的关于Zernike各阶模式的超定矩阵方程,称为Rimmer-Wyant方法[1]。通过求超定方程在最小二乘意义下的解,获得各阶Zernike模式的系数,其矩阵方程与Shack-Hartmann波前传感器的模式复原的矩阵方程有相同结构[7-9],区别仅在于方程的向量数目。该方法的缺点是存在模式耦合现象[7,10-14],即,复原模式的截断数J小于实际波前的模式数Mact时,最小二乘法得到的模式系数不准确。但实际波前的模式数不能先验地预测,为了获得准确的Zernike系数,需要设法避免耦合误差。耦合误差的原因本质上是Zernike圆多项式的梯度并不正交[7,10-14],要规避耦合误差需要从正交性的角度进行考虑。

通常考虑的克服耦合误差的方法是构造辅助的向量函数[15],使其正交于Zernike圆多项式的梯度,复原时用梯度数据与向量函数各阶模式联立在圆域求积分,在该积分定义的内积下向量函数各阶模式与Zernike圆多项式各阶相应模式正交,因此可以无耦合地得到模式系数的准确值,即是说,即便复原截断数没有达到实际波前模式数,也可获得正确的复原系数,从而避免耦合现象。合适的向量函数并不唯一,有若干相关文献对此进行了讨论[15-18]

另一种思路是采用梯度正交的模式取代Zernike圆多项式进行复原,Huang等采用了Laplace算子本征函数进行模式复原[12-14],单位圆上该函数为圆谐函数,其径向函数是Bessel函数。由于任何单阶的Bessel函数都是幂函数的无穷级数,各阶Bessel函数与Zernike径向多项式互相都是无穷级数展开。因此,单阶Zernike圆多项式需要用无穷阶圆谐函数展开,并无简单的关系。

本文考虑一种更简洁的方法,当采样点数较高时,直接利用Zernike圆多项式角向导数和径向导数的正交性质,通过Gram矩阵方法,无耦合地获得Zernike系数。该复原不需要构造复杂的辅助向量函数。该方法可方便地推广到环上,得到无耦合的Zernike环多项式系数。

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

文献[19]中提出可用一种含Gram矩阵的矩阵方程进行波前复原,其形式如下:

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

其中:${(\frac{{\partial W}}{{\partial \rho }})_{{\rm{meas}}}}$为波前径向导数的测量值,${Z_j}$表示Zernike模式,${\alpha _j}$为待求模式系数。G是一个Gram矩阵,其元素由待求向量(即待求函数,此处为Zernike模式的径向导数)的内积${G_{jj'}}$组成:

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

其中$w(\rho )$为一合适的权重函数,当内积取为欧式内积时,$w(\rho ) = \rho $。由代数定理[20],Gram矩阵为非奇异矩阵当且仅当组成Gram矩阵的向量线性无关。由于此时组成Gram矩阵的向量$\left\{ {\frac{{\partial Z}}{{\partial \rho }}} \right\}$线性无关,不论权重函数$w(\rho )$如何选取,一定会给出唯一的解。

文献[19]中讨论到,如果组成Gram矩阵的向量关于权重函数$w(\rho )$不正交,那么当复原模式的截断数J小于实际波前的模式数Mact时,Gram矩阵方法不会给出准确值,这是Gram矩阵方法中的“耦合”现象。但如果组成Gram矩阵的向量关于式(2)中的权重函数$w(\rho )$正交,则G成为对角矩阵,“耦合”现象消失[19]

这意味着,假如采用欧式内积构成的Gram矩阵进行Zernike模式复原,由于Zernike圆多项式的径向导数互相线性无关,Gram矩阵方法会给出唯一解,但由于Zernike圆多项式的径向导数在欧式内积下并不完全正交,依然会出现“耦合”误差。因此,需要考虑利用角向导数的正交性。

2.2 Zernike圆多项式角向导数的正交性

本文中Zernike圆多项式定义如下[21]

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

其中:$n = 0,1,2, \ldots $$m = 0, \pm 1, \pm 2, \ldots $$n - m$为偶数。注意上式的归一化因子与文献[21]中不完全相同。当方位频率m > 0时,Zernike径向多项式$R_n^m(\rho )$

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

由Zernike圆多项式的定义不难得到Zernike圆多项式的角向导数:

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

由此不难得到Zernike圆多项式的角向导数的正交性:

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

因此,由Zernike圆多项式的角向导数和波前的角向导数建立的Gram矩阵如下:

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

其中内积取为欧式内积,即w(ρ) = ρ。由Zernike角向导数的正交关系,可得到无耦合的Zernike复原系数。

但该方法需要补充考虑的一点是,m = 0的Zernike模式角向导数为零,无法用Gram矩阵方法测量,需要用m = 0模式径向导数的Gram矩阵单独复原,但这些模式在欧式内积下并不正交,需要对内积做出改变,下面我们来证明,m = 0的Zernike模式的径向导数在极坐标中关于权重函数$w(\rho ) = \rho (1 - {\rho ^2})$正交。

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)

其中n只能取偶数,${G_n}(p,q,\rho )$为次数为n的Jacobi多项式,pq为权重函数参数且满足q > 0,p-q > -1,并有正交关系[23-24]

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

其中${N_n}(p,q)$为归一化系数:

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

式中$\Gamma $表示熟知的Gamma函数。

由Jacobi多项式的微分公式[25]可得:

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

由式(10)中的Jacobi多项式的正交关系可得到如下的正交关系:

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

整理可得Zernike模式m = 0项径向多项式的导数的正交性:

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

考虑到三角函数系的正交性,可得(归一化的)Zernike圆多项式中m = 0的项径向导数的正交性:

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

从上式可看出,在权重函数$w(\rho ) = \rho (1 - {\rho ^2})$下,Zernike圆多项式中m = 0的项径向导数互相正交,且与m≠0项的径向导数正交。所以可利用Gram矩阵方程对Zernike圆多项式中m = 0的项进行复原,其中内积加权重函数$w(\rho ) = \rho (1 - {\rho ^2})$,由此可得到m = 0的项的无耦合的Zernike复原系数。

2.4 复原流程

下面总结一下复原流程:首先,从波前的梯度数据${(\frac{{\partial W}}{{\partial x}})_{{\rm{meas}}}}$${(\frac{{\partial W}}{{\partial y}})_{{\rm{meas}}}}$,通过如下公式转换得到波前的径向导数和角向导数:

${(\frac{{\partial W}}{{\partial \rho }})_{{\rm{meas}}}}(\rho ,\theta ) = {(\frac{{\partial W}}{{\partial x}})_{{\rm{meas}}}}\cos \theta + {(\frac{{\partial W}}{{\partial y}})_{{\rm{meas}}}}\sin \theta ,$ (16)
${(\frac{{\partial W}}{{\partial \theta }})_{{\rm{meas}}}}(\rho ,\theta ) = - {(\frac{{\partial W}}{{\partial x}})_{{\rm{meas}}}}\rho \sin \theta + {(\frac{{\partial W}}{{\partial y}})_{{\rm{meas}}}}\rho \cos \theta ,$ (17)

然后,用矩阵方程:

$\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \theta }})} \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \theta }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \theta }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_1}}}{{\partial \theta }})}&{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_2}}}{{\partial \theta }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \theta }},\frac{{\partial {Z_J}}}{{\partial \theta }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (18)

复原m≠0项的Zernike系数,权重函数取w(ρ) = ρ,在直角坐标系做数值积分时,考虑到极坐标和直角坐标转换的Jacobi行列式,权重函数w(ρ) = 1。

最后,用矩阵方程:

$\left[ {\begin{array}{*{20}{c}} {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_1}}}{{\partial \rho }})} \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_2}}}{{\partial \rho }})} \\ \vdots \\ {({{(\frac{{\partial W}}{{\partial \rho }})}_{{\rm{meas}}}},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_1}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ {(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_2}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \\ \vdots & \vdots & \ddots & \vdots \\ {(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_1}}}{{\partial \rho }})}&{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_2}}}{{\partial \rho }})}& \cdots &{(\frac{{\partial {Z_J}}}{{\partial \rho }},\frac{{\partial {Z_J}}}{{\partial \rho }})} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\alpha _2}} \\ \vdots \\ {{\alpha _J}} \end{array}} \right],$ (19)

复原m = 0项的Zernike系数(活塞项除外),权重函数取$w(\rho ) = \rho (1 - {\rho ^2})$,在直角坐标系做数值积分时,考虑到极坐标和直角坐标转换的Jacobi行列式,权重函数$w(\rho ) = 1 - {\rho ^2}$

3 复原结果

由于在式(3)中采用了归一化的Zernike模式,当复原截断数J大于等于实际波前模式数Mact时,文献[14]中定义的波前复原误差$\lambda $可化简为

$\lambda = \frac{{\sum\nolimits_1^{{M_{{\rm{act}}}}} {(\alpha _i^2 - \beta _i^2)} + \sum\nolimits_{{M_{{\rm{act}}}} + 1}^J {\alpha _i^2} }}{{\sum\nolimits_1^{{M_{{\rm{act}}}}} {\beta _i^2} }},$ (20)

其中:${\alpha _i}$是复原系数,${\beta _i}$是实际波前系数。

当复原截断数J小于等于实际波前模式数Mact时,文献中定义的波前复原误差$\lambda $可化简为

$\lambda = {\lambda _{\rm{r}}} + {\lambda _{\rm{t}}} = \frac{{\sum\nolimits_1^J {(\alpha _i^2 - \beta _i^2) + \sum\nolimits_{J + 1}^{{M_{{\rm{act}}}}} {\alpha _i^2} } }}{{\sum\nolimits_1^{{M_{{\rm{act}}}}} {\beta _i^2} }},$ (21)

其中λr为残差:

${\lambda _{\rm{r}}} = \frac{{\sum\nolimits_1^J {(\alpha _i^2 - \beta _i^2)} }}{{\sum\nolimits_1^{{M_{{\rm{act}}}}} {\beta _i^2} }}。$ (22)

当采样率高到模式混淆可以忽略时,λr可以衡量模式耦合,原因是这时λr反映了复原截断数J以下系数的复原准确程度。当截断数J变化时,如λr始终保持很小,则表明耦合误差消除。

λt为截断误差:

${\lambda _{\rm{t}}} = \frac{{\sum\nolimits_{J + 1}^{{M_{{\rm{act}}}}} {\alpha _i^2} }}{{\sum\nolimits_1^{{M_{{\rm{act}}}}} {\beta _i^2} }}。$ (23)

原始波前设为前21阶Zernike圆多项式(不含活塞项),实际波前模式系数设为${\beta _2} = {\beta _3} = \cdots = {\beta _{21}} = 1$。根据剪切干涉测量的典型采样点数目,取采样点数π×1002≈3.1×104

图 1显示了运用本文所述式(18)与式(19)的Gram矩阵方法时,波前复原误差λ和残差λr与截断数J的关系。λ的变化曲线表明,如同上文讨论的,当J大于等于实际波前模式数Mact = 21时,Gram矩阵方法可以给出准确的复原。λr的变化曲线表明,当J小于等于实际波前模式数Mact = 21时,残差λr很小,波前复原误差几乎都来自截断误差,耦合误差得到消除。

图 1 波前复原误差λ和残差λr与截断数J的关系,运用本文所述式(18)与式(19)的Gram矩阵方法 Fig. 1 Reconstruction error λ and remaining error λr versus truncation of reconstruction modes J, by Gram matrix method in Eq.(18) and Eq.(19)

图 2显示了运用式(1)Gram矩阵方法时,波前复原误差λ和残差λr与截断数J的关系,其中内积取为欧式内积,即权重函数w(ρ) = ρλ的变化曲线表明,如同上文讨论的,当J大于等于实际波前模式数Mact = 21时,Gram矩阵方法可以给出准确的复原。λr的变化曲线表明,当J小于等于实际波前模式数Mact = 21时,出现较大的残差,这是文献[19]中讨论到的Gram矩阵方法中的“耦合”现象。图 1图 2的复原结果的对比表明,可通过利用Zernike圆多项式m≠0项角向导数的正交性,和Zernike圆多项式m = 0项径向导数的加权正交性,无耦合地复原模式系数。

图 2 波前复原误差λ和残差λr与截断数J的关系,运用式(1) Gram矩阵方法,权重函数w(ρ) = ρ (极坐标) Fig. 2 Reconstruction error λ and remaining error λr versus truncation of reconstruction modes J, by Gram matrix method in Eq.(1), w(ρ) = ρ (polar coordinate)

图 3显示了运用本文所述Gram矩阵方法,当截断数J达到实际波前模式数时波前重构结果。

图 3 波前复原结果。(a)原始波前;(b)式(18)复原的m≠0项的波前;(c)式(19)复原的m = 0项的波前;(d)复原波前;(e)复原残留误差 Fig. 3 Reconstruction results. (a) Original wavefront; (b) Reconstructed wavefront by Eq.(18), including m≠0 modes; (c) Reconstructed wavefront by Eq.(19), including m = 0 modes; (d) Reconstructed wavefront; (e) Residual error

下面考虑一个实测例子,应用四波横向剪切干涉系统检测一实际波前,该波前经过带有球差的像差板调制,由于实际复原口径比像差板略小,存在一个标度变换,复原区域的波前应当是一个Zernike球差$Z_4^0$、离焦$Z_2^0$与活塞项$Z_0^0$的线性组合。由图 4所示的数值化的干涉条纹经Fourier分析处理得到波前梯度的近似分布,再用本文所述方法进行模式复原。

图 4 数值化的干涉条纹 Fig. 4 Numerical interferogram

尽管实验涉及的像差板阶数只到4阶,但只需进行这样的复原,一次复原截断的阶数为4阶(Zernike球差),一次复原截断的阶数为2阶(Zernike离焦),然后对比两次复原时Zernike离焦项$Z_2^0$的系数,即可说明本文的问题。如果两次复原系数差别很小,说明耦合得到避免,如果复原系数差别较大,说明存在耦合。

图 5显示了运用式(19)和式(1)Gram矩阵方法时,Zernike离焦项$Z_2^0$的复原波前在截断数为4和2时的变化关系。可以看出,运用本文所述式(19)复原时,Zernike离焦项$Z_2^0$的复原结果在截断数变化时差别很小,耦合得到避免,而采用式(1)复原时,当截断数小于实际波前阶数时,存在明显的耦合现象。

图 5 波前复原结果。运用本文所述式(19)的 Gram 矩阵方法。(a) Zernike 离焦项 Z20 的复原波前,截断数 J=4;(b) Zernike 离焦项 Z20 的复原波前,截断数 J=2;(c) 复原残留误差原。运用式(1)的 Gram 矩阵方法:(d) Zernike 离焦项 Z20 的复原波前,截断数 J=4;(e) Zernike 离焦项 Z20 的复原波前,截断数 J=2;(f) 复原残留误差原 Fig. 5 Reconstruction results by Gram matrix method in Eq.(19). (a) Reconstructed wavefront of Z20 , where truncation number J=4; (b) Reconstructed wavefront of Z20 , where truncation number J=2; (c) Residual error. Reconstruction results by Gram matrix method in Eq.(1); (d) Reconstructed wavefront of Z20 , where truncation number J=4; (e) Reconstructed wavefront of Z20 , where truncation number J=2; (f) Residual error
4 环上推广

本节考虑上述Gram矩阵方法在环上的推广,采用在环上正交的Zernike环多项式展开波前函数,并试图利用正交性得到无耦合的Zernike环多项式系数。

4.1 Zernike环多项式角向导数的正交性

本文中Zernike环多项式定义在外环半径为1,内环半径为ε的环上,定义如下[26]

$Z_n^m(\rho ,\theta ;\varepsilon ) = \left\{ {\begin{array}{*{20}{c}} {\sqrt {\frac{{n + 1}}{{{\rm{ \mathit{ π} }}(1 - {\varepsilon ^2})}}} R_n^{\left| m \right|}(\rho ;\varepsilon )\sqrt 2 \cos (\left| m \right|\theta ){\kern 1pt} {\kern 1pt} (m > 0)} \\ {\sqrt {\frac{{n + 1}}{{{\rm{ \mathit{ π} }}(1 - {\varepsilon ^2})}}} R_n^{\left| m \right|}(\rho ;\varepsilon )\sqrt 2 \sin (\left| m \right|\theta ){\kern 1pt} {\kern 1pt} {\kern 1pt} (m < 0)} \\ {\sqrt {\frac{{n + 1}}{{{\rm{ \mathit{ π} }}(1 - {\varepsilon ^2})}}} R_n^0(\rho ;\varepsilon ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (m = 0)} \end{array}} \right.,$ (24)

其中:n = 0, 1, 2…;$m = 0, \pm 1, \pm 2, \ldots $$n - m$为偶数。注意上式的归一化因子与文献[26]中不完全相同。当方位频率m > 0时,Zernike环多项式的径向多项式$R_n^m(\rho ;\varepsilon )$由对Zernike圆多项式的径向多项式$R_n^m(\rho )$做Gram–Schmidt正交化,递归地给出,一般并无统一的幂函数显表达式。$R_n^m(\rho ;\varepsilon )$的递归表达式如下:

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

其中:$N_n^m$为每次Gram–Schmidt正交化步骤中的归一化因子。

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)

由Zernike环多项式的定义不难得到Zernike环多项式的角向导数:

$\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)

由此不难得到Zernike环多项式的角向导数的正交性:

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

因此,由Zernike环多项式的角向导数和波前的角向导数建立的Gram矩阵如下:

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

其中内积取为欧式内积,即在极坐标下w(ρ) = ρ。由Zernike角向导数的正交关系,可得到无耦合的Zernike复原系数。

但该方法需要补充考虑的一点是,m = 0的Zernike模式角向导数为零,无法用Gram矩阵方法测量,需要用m = 0模式径向导数的Gram矩阵单独复原,但这些模式在欧式内积下并不正交,需要对内积做出改变,接下来将证明,m = 0的Zernike环多项式的径向导数在极坐标中关于权重函数$w(\rho ) = ({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})/\rho {\kern 1pt} $正交。

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)

令:$\kappa = \frac{{{\rho ^2} - {\varepsilon ^2}}}{{1 - {\varepsilon ^2}}}$$\gamma = \frac{{1 - {\rho ^2}}}{{1 - {\varepsilon ^2}}}$。将上式代入式(9),可得:

$R_n^0(\rho ;\varepsilon ){\kern 1pt} = {( - 1)^{\frac{n}{2}}}{G_{\frac{n}{2}}}(1,1,\kappa )。$ (31)

由Jacobi多项式的微分公式[20]可得:

$\frac{{{\rm{d}}[R_n^0(\rho ;\varepsilon )]{\kern 1pt} }}{{{\rm{d}}\rho }}\\ = {( - 1)^{\frac{1}{2}n + 1}}\frac{{n(n + 2)}}{2}\frac{\rho }{{1 - {\varepsilon ^2}}}{G_{\frac{1}{2}n - 1}}(3,2,\kappa )。$ (32)

由式(10)中的Jacobi多项式的正交关系可得到如下的正交关系:

$\int\limits_0^1 {\kappa \cdot \gamma \cdot {G_{\frac{1}{2}n - 1}}(3,2,\kappa ){G_{\frac{1}{2}n' - 1}}(3,2,\kappa ){\rm{d}}\kappa } = {N_{\frac{1}{2}n - 1}}(3,2){\delta _{nn'}}。$ (33)

将式(32)代入式(33)并整理,注意自变量的变换造成积分区域的变换,可得Zernike环多项式m = 0项径向多项式的导数的正交性:

$\int\limits_\varepsilon ^1 {\frac{{({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})}}{\rho }\frac{{{\rm{d}}[R_n^0(\rho ;\varepsilon )]{\kern 1pt} }}{{{\rm{d}}\rho }}\frac{{{\rm{d}}[R_{n'}^0(\rho ;\varepsilon )]{\kern 1pt} }}{{{\rm{d}}\rho }}{\rm{d}}\rho } \\ = (1 - {\varepsilon ^2})\frac{{n(n + 2)}}{{2(n + 1)}}{\delta _{nn'}}。$ (34)

考虑到三角函数系的正交性,可得(归一化的)Zernike环多项式中m = 0的项径向导数的正交性:

$\int\limits_0^{2{\rm{ \mathit{ π} }}} {\int\limits_\varepsilon ^1 {\frac{{({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})}}{\rho }\frac{\partial }{{\partial \rho }}Z_n^0(\rho ,\theta ;\varepsilon )\frac{\partial }{{\partial \rho }}Z_{n'}^{m'}(\rho ,\theta ;\varepsilon ){\rm{d}}\rho {\rm{d}}\theta } } \\ = n(n + 2){\delta _{0m'}}{\delta _{nn'}}。$ (35)

式(35)可知,在权重函数$w(\rho ) = ({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})/\rho {\kern 1pt} $下,Zernike环多项式中m = 0的项径向导数互相正交,且与m≠0项的径向导数正交。所以可利用Gram矩阵方程对Zernike环多项式中m = 0的项进行复原,其中内积加权重函数$w(\rho ) = ({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})/\rho $,由此可得到m = 0的项的无耦合的Zernike环多项式复原系数。

4.3 复原流程

复原流程与2.4节基本一致:首先,从波前的梯度数据${(\partial W/\partial x)_{{\rm{meas}}}}$${(\partial W/\partial y)_{{\rm{meas}}}}$,通过如下式转换得到波前的径向导数和角向导数:

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

然后,用矩阵方程

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

复原m≠0项的Zernike环多项式系数,权重函数取w(ρ) = ρ,在直角坐标系做数值积分时,考虑到极坐标和直角坐标转换的Jacobi行列式,权重函数w(ρ) = 1。

最后,用矩阵方程

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

复原m = 0项的Zernike环多项式系数(活塞项除外),权重函数取$w(\rho ) = ({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})/\rho $,在直角坐标系做数值积分时,考虑到极坐标和直角坐标转换的Jacobi行列式,权重函数$w(\rho ) = ({\rho ^2} - {\varepsilon ^2})(1 - {\rho ^2})/{\rho ^2}$

4.4 复原结果

原始波前设为前21阶Zernike环多项式(不含活塞项),外环半径为1,内环半径ε = 0.3,实际波前模式系数设为β2 = 1,β3 = 2,β4 = 3,…,β21 = 20。根据剪切干涉测量的典型采样点数目,取采样点数π×1002-π×302≈2.8×104

图 6显示了运用式(38)与式(39)的Gram矩阵方法时,波前复原误差λ和残差λr与截断数J的关系。λ的变化曲线表明,如同上文讨论的,当J大于等于实际波前模式数Mact = 21时,Gram矩阵方法可以给出准确的复原。λr的变化曲线表明,当J小于等于实际波前模式数Mact = 21时,残差λr很小,波前复原误差几乎都来自截断误差,耦合误差得到消除。

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显示了运用本文所述Gram矩阵方法, 当截断数J达到实际波前模式数时波前重构结果。

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

基于Gram矩阵方法的模式复原可能存在模式耦合误差,但当构成Gram矩阵的向量互相正交时,Gram矩阵对角化,即可使复原系数不受截断数目及高阶模式影响,从而消除模式耦合。我们利用了Zernike圆多项式m≠0模式角向导数的正交性,并证明Zernike圆多项式m = 0模式径向导数在加入额外权重下的正交性,分别构成Gram矩阵方法中的复原矩阵方程,得到了一种简便的,从波前梯度信息无耦合复原Zernike系数的方法。该方法可推广到环上,得到无耦合的Zernike环多项式系数。

参考文献
[1]
Rimmer M P. Method for evaluating lateral shearing interferograms[J]. Applied Optics, 1974, 13(3): 623-629. [Crossref]
[2]
Harbers G, Kunst P J, Leibbrandt G W R. Analysis of lateral shearing interferograms by use of Zernike polynomials[J]. Applied Optics, 1996, 35(31): 6162-6172. [Crossref]
[3]
Shen W, Chang M W, Wan D S. Zernike polynomial fitting of lateral shearing interferometry[J]. Optical Engineering, 1997, 36(36): 905-913. [Crossref]
[4]
Hunt B R. Matrix formulation of the reconstruction of phase values from phase differences[J]. Journal of the Optical Society of America, 1979, 69(3): 393-399. [Crossref]
[5]
Jiang W H. Overview of adaptive optics development[J]. Opto-Electronic Engineering, 2018, 45(3): 170489.
姜文汉. 自适应光学发展综述[J]. 光电工程, 2018, 45(3): 170489 [Crossref]
[6]
Tyson R. Principles of Adaptive Optics[M]. 3rd ed. London: CRC Press, 2010: 111-176.
[7]
Zhang Q, Jiang W H, Xu B. Reconstruction of turbulent optical wavefront realized by Zernike polynomial[J]. Opto-Electronic Engineering, 1998, 25(6): 15-19.
张强, 姜文汉, 许冰. 利用Zernike多项式对湍流波前进行波前重构[J]. 光电工程, 1998, 25(6): 15-19 [Crossref]
[8]
Xian H, Li H G, Jiang W H, et al. Measurement of the wavefront phase of a laser beam with Hartmann-Shack sensor[J]. Opto-Electronic Engineering, 1995, 22(2): 38-45.
鲜浩, 李华贵, 姜文汉, 等. 用Hartmann-Shack传感器测量激光束的波前相位[J]. 光电工程, 1995, 22(2): 38-45 [Crossref]
[9]
Zhang R, Yang J S, Tian Y, et al. Wavefront phase recovery from the plenoptic camera[J]. Opto-Electronic Engineering, 2013, 40(2): 32-39.
张锐, 杨金生, 田雨, 等. 焦面哈特曼传感器波前相位复原[J]. 光电工程, 2013, 40(2): 32-39 [Crossref]
[10]
Cubalchini R. Modal wave-front estimation from phase derivative measurements[J]. Journal of the Optical Society of America, 1979, 69(7): 972-977. [Crossref]
[11]
Herrmann J. Cross coupling and aliasing in modal wave-front estimation[J]. Journal of the Optical Society of America, 1981, 71(8): 989-992. [Crossref]
[12]
Huang S Y, Xi F J, Liu C H, et al. Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing[J]. Optics Communications, 2011, 284(12): 2781-2783. [Crossref]
[13]
Huang S Y, Xi F J, Liu C H, et al. Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation[J]. Journal of the Optical Society of America A, 2012, 29(4): 513-520. [Crossref]
[14]
Huang S Y, Yu N, Xi F J, et al. Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian[J]. Optics Communications, 2013, 288: 7-12. [Crossref]
[15]
Gavrielides A. Vector polynomials orthogonal to the gradient of Zernike polynomials[J]. Optics Letters, 1982, 7(11): 526-528. [Crossref]
[16]
Zhao C Y, Burge J H. Orthonormal vector polynomials in a unit circle, Part Ⅰ: basis set derived from gradients of Zernike polynomials[J]. Optics Express, 2007, 15(26): 18014-18024. [Crossref]
[17]
Zhao C Y, Burge J H. Orthonormal vector polynomials in a unit circle, Part Ⅱ: completing the basis set[J]. Optics Express, 2008, 16(9): 6586-6591. [Crossref]
[18]
Mahajan V N, Acosta E. Vector polynomials for direct analysis of circular wavefront slope data[J]. Journal of the Optical Society of America A, 2017, 34(10): 1908-1913. [Crossref]
[19]
Sun W H, Wang S, He X, et al. Jacobi circle and annular polynomials: modal wavefront reconstruction from wavefront gradient[J]. Journal of the Optical Society of America A, 2018, 35(7): 1140-1148. [Crossref]
[20]
Horn R A, Johnson C R. Matrix Analysis[M]. Cambridge: Cambridge University Press, 1990: 407.
[21]
Mahajan V N. Zernike circle polynomials and optical aberrations of systems with circular pupils[J]. Applied Optics, 1994, 33(34): 8121-8124. [Crossref]
[22]
Zernike F. Diffraction theory of the knife-edge test and its improved form, the phase-contrast method[J]. Monthly Notices of the Royal Astronomical Society, 2002, 94(2): 377-384. [Crossref]
[23]
Born M, Wolf E. Principles of Optics[M]. 7th ed. Cambridge: Cambridge University Press, 1999: 905-910.
[24]
Wang Z X, Guo D R. Special Functions[M]. Singapore: World Scientific, 1989: 139.
[25]
Andrews G E, Askey R, Roy R. Special Functions[M]. Cambridge: Cambridge University Press, 1999: 94.
[26]
Mahajan V N. Zernike annular polynomials and optical aberrations of systems with annular pupils[J]. Applied Optics, 1994, 33(34): 8125-8127. [Crossref]