CN111486974A - Reconstruction method of high-order free-form surface wavefront with arbitrary aperture shape - Google Patents

Reconstruction method of high-order free-form surface wavefront with arbitrary aperture shape Download PDF

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CN111486974A
CN111486974A CN202010320641.0A CN202010320641A CN111486974A CN 111486974 A CN111486974 A CN 111486974A CN 202010320641 A CN202010320641 A CN 202010320641A CN 111486974 A CN111486974 A CN 111486974A
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吴珍
魏朝阳
胡晨
万嵩林
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Shanghai Institute of Optics and Fine Mechanics of CAS
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Abstract

A high-order free-form surface wavefront reconstruction method with any aperture shape. The method combines a mode method based on a numerical orthogonal polynomial and a finite difference method based on the polynomial, and can realize the complex free-form surface wavefront reconstruction of any aperture shape. Firstly, aiming at different aperture shapes, a numerical orthogonal polynomial is constructed to be used for fitting wavefront slope information, and polynomial coefficients and first wavefront are obtained. And then linearly combining the polynomial coefficient and the numerical gradient polynomial to obtain the slope of the first reconstructed wavefront, and calculating the slope residual error. And then fitting the slope residual error which cannot be fitted by the polynomial by using a region method based on the polynomial to obtain a second wavefront, wherein the finally obtained reconstructed wavefront is the sum of the two wavefronts. Compared with the existing wavefront reconstruction method, the method has the main advantages that: the method is suitable for the rapid reconstruction of the complex free-form surface wavefront with any aperture, namely wavefront mode information can be obtained, and the wavefront reconstruction precision of the finite-term numerical orthogonal polynomial can be improved.

Description

任意孔径形状高阶自由曲面波前的重构方法Reconstruction method of high-order free-form surface wavefront with arbitrary aperture shape

技术领域technical field

本发明属于光学测量技术领域，特别是一种任意孔径形状由波前斜率信息进行波前重构的混合重构方法，适用于哈特曼波前传感器、条纹反射法等波前梯度信息检测技术。The invention belongs to the technical field of optical measurement, in particular to a hybrid reconstruction method for wavefront reconstruction using wavefront slope information for any aperture shape, and is suitable for wavefront gradient information detection technologies such as Hartmann wavefront sensors and fringe reflection methods. .

背景技术Background technique

光学自由曲面因其具有较大的自由度和非球面度，能提高系统光学性能的同时简单系统结构，在医学成像、航空航天、军事等领域得到了广泛的应用。哈特曼传感器和条纹反射法是两种典型的测量自由曲面波前斜率信息的检测技术，因此，需要波前重构方法将检测得到的斜率信息恢复成波前。波前重构方法一般可以分为区域法和模式法两种。Optical free-form surfaces have been widely used in medical imaging, aerospace, military and other fields because of their large degrees of freedom and asphericity, which can improve the optical performance of the system and at the same time have a simple system structure. The Hartmann sensor and the fringe reflection method are two typical detection techniques for measuring the wavefront slope information of free-form surfaces. Therefore, a wavefront reconstruction method is required to restore the detected slope information into a wavefront. The wavefront reconstruction methods can generally be divided into two types: the regional method and the mode method.

区域法是通过孔径周围点测得的梯度数据估计中间点的高度，利用有限差分法对波前进行估计[请参见文献1:W.H.Southwell,"Wave-front estimation from wave-frontslope measurements,"J.Opt.Soc.Am.70(8),The area method is to estimate the height of the intermediate point from the gradient data measured at the points around the aperture, and use the finite difference method to estimate the wavefront [Please refer to Reference 1:W.H.Southwell,"Wave-front estimation from wave-frontslope measurements,"J. Opt.Soc.Am.70(8),

998-1006(1980)和文献2:L.Huang,J.Xue,B.Gao,C.Zuo,and M.Idir,"Splinebased least squares integration for two-dimensional shape or wavefrontreconstruction,"Opt.Lasers Eng.91,221-226(2017).]。区域法虽然可以实现任意孔径形状自由曲面的高精度波前重构，但因其原理为有限差分法，只能得到离散波前，不能将波前以表达式的形式呈现，因此其无法得到波前模式特征，进而无法得到波前初级像差。998-1006 (1980) and Reference 2: L. Huang, J. Xue, B. Gao, C. Zuo, and M. Idir, "Splinebased least squares integration for two-dimensional shape or wavefront reconstruction," Opt. Lasers Eng. 91, 221-226 (2017).]. Although the area method can realize high-precision wavefront reconstruction of free-form surfaces with arbitrary aperture shapes, because its principle is the finite difference method, only discrete wavefronts can be obtained, and wavefronts cannot be represented in the form of expressions, so it cannot obtain wavefronts. Therefore, the wavefront primary aberration cannot be obtained.

模式法是将波前用一组正交多项式线性表示，而相应的多项式系数可以得到波前的模式特征，与光学设计中的像差对应。但是，模式法中的多项式仅在特定孔径形状内正交，不具有一般性，对于任意孔径形状，多项式不正交，会导致模式系数偶尔和噪声传递[参见文献3:J.Herrmann,"Cross coupling and aliasing in modal wave-frontestimation,"J.Opt.Soc.Am.71,989-992(1981)]。基于此，针对任意孔径离散数据，构造数值化正交多项式用于任意孔径形状内的斜率信息拟合[参见文献4:J.F.Ye,Z.S.Gao,S.Wang,J.L.Cheng,W.Wang,and W.Q.Sun,"Comparative assessment of orthogonalpolynomials for wavefront reconstruction over the square aperture,"J.Opt.Soc.Am.A.31(10),2304-2311(2014)]。但是当自由曲面的局部具有较大梯度时，要达到纳米级重构精度，需要上百甚至上千项高阶多项式，而高阶多项式的表达式非常复杂，这大大增加了计算难度和时间。因此，如何提高有限多项式的波前重构精度是一个问题，特别是针对那些有局部陡峭或急剧变化的波前。In the mode method, the wavefront is linearly represented by a set of orthogonal polynomials, and the corresponding polynomial coefficients can obtain the mode characteristics of the wavefront, which correspond to the aberrations in the optical design. However, the polynomials in the mode method are only orthogonal within a specific aperture shape and are not general, and for arbitrary aperture shapes, the polynomials are not orthogonal, resulting in occasional mode coefficients and noise transfer [see Reference 3: J. Herrmann, "Cross" Coupling and aliasing in modal wave-frontestimation, "J. Opt. Soc. Am. 71, 989-992 (1981)]. Based on this, for discrete data of arbitrary apertures, a numerical orthogonal polynomial is constructed for slope information fitting in arbitrary aperture shapes [see Reference 4: J.F.Ye, Z.S.Gao, S.Wang, J.L.Cheng, W.Wang, and W.Q. Sun, "Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture," J.Opt.Soc.Am.A.31(10), 2304-2311(2014)]. However, when the free-form surface has a large gradient locally, hundreds or even thousands of high-order polynomials are required to achieve nanoscale reconstruction accuracy, and the expressions of high-order polynomials are very complicated, which greatly increases the computational difficulty and time. Therefore, how to improve the wavefront reconstruction accuracy of finite polynomials is a problem, especially for those wavefronts with local steep or sharp changes.

发明内容SUMMARY OF THE INVENTION

为了提高有限多项式的波前重构精度，特别是针对那些有局部陡峭或急剧变化的任意孔径波前。本发明提供一种任意孔径形状的高阶自由曲面波前的重构方法。In order to improve the wavefront reconstruction accuracy of finite polynomials, especially for those arbitrary aperture wavefronts with local steep or sharp changes. The present invention provides a method for reconstructing high-order free-form surface wavefronts with arbitrary aperture shapes.

本发明的技术解决方案如下：The technical solution of the present invention is as follows:

一种任意孔径形状的高阶自由曲面波前的重构方法，包括以下步骤：A method for reconstructing high-order free-form surface wavefronts with arbitrary aperture shapes, comprising the following steps:

1)通过条纹反射法或哈特曼传感器检测技术得到自由曲面的波前斜率和对应的位置信息，针对任意孔径上的离散梯度数据点，以Zernike多项式为基底，通过数值正交变化法构造梯度正交多项式,F＝ZBT，其中F代表数值化正交多项式，Z代表Zernike多项式，通过Cholesky分解法求得线性组合系数矩阵B,对矩阵F求导得到▽F，并正交化处理得到数值化正交梯度多项式

Figure BDA0002461262550000021
即
Figure BDA0002461262550000022
矩阵C的求法和B一致；1) The wavefront slope and corresponding position information of the free-form surface are obtained by the fringe reflection method or the Hartmann sensor detection technology. For discrete gradient data points on any aperture, the Zernike polynomial is used as the base, and the gradient is constructed by the numerical orthogonal change method. Orthogonal polynomial, F=ZB T , where F represents a numerical orthogonal polynomial, and Z represents a Zernike polynomial. The linear combination coefficient matrix B is obtained by the Cholesky decomposition method, and the matrix F is derived to obtain ▽F, and orthogonalized to obtain Numerical Orthogonal Gradient Polynomial
Figure BDA0002461262550000021
which is
Figure BDA0002461262550000022
The calculation method of matrix C is consistent with B;

2)用数值化正交梯度多项式拟合波前斜率，即

Figure BDA0002461262550000023
其中，
Figure BDA0002461262550000024
W'是波前斜率，α＝[α1 α2 … αn]T,
Figure BDA0002461262550000031
α表示数值化正交梯度多项式系数，n表示数值化正交梯度多项式项数，
Figure BDA0002461262550000032
表示
Figure BDA0002461262550000033
的转置矩阵，(x,y)为波前斜率对应的坐标，波前斜率还可以表示成W'＝▽FCTα，对该式积分：W＝FCTα，得到第一次拟合波前W和对应的模式系数CTα；2) Fit the wavefront slope with a numerical orthogonal gradient polynomial, namely
Figure BDA0002461262550000023
in,
Figure BDA0002461262550000024
W' is the wavefront slope, α=[α 1 α 2 … α n ] T ,
Figure BDA0002461262550000031
α represents the numerical orthogonal gradient polynomial coefficient, n represents the number of numerical orthogonal gradient polynomial terms,
Figure BDA0002461262550000032
express
Figure BDA0002461262550000033
The transposed matrix of , (x, y) is the coordinate corresponding to the slope of the wavefront. The slope of the wavefront can also be expressed as W'=▽FC T α. Integrate this formula: W=FC T α to obtain the first fitting the wavefront W and the corresponding mode coefficient C T α;

3)第一次拟合得到的波前斜率表示成：

Figure BDA0002461262550000034
其中，
Figure BDA0002461262550000035
是用数值化正交梯度多项式拟合斜率得到的系数矩阵，
Figure BDA0002461262550000036
通过原始斜率和第一次拟合的波前斜率相减得到波前的斜率残差
Figure BDA0002461262550000037
其中w'＝[wx wy]T，其中wx和wy分别表示x和y方向的斜率残差；3) The wavefront slope obtained by the first fitting is expressed as:
Figure BDA0002461262550000034
in,
Figure BDA0002461262550000035
is the coefficient matrix obtained by fitting the slopes with numerically orthogonal gradient polynomials,
Figure BDA0002461262550000036
The slope residual of the wavefront is obtained by subtracting the original slope and the first fitted wavefront slope
Figure BDA0002461262550000037
where w'=[w x w y ] T , where w x and w y represent the slope residuals in the x and y directions, respectively;

4)将斜率残差w'用三次多项式的差分法拟合如下：

Figure BDA0002461262550000038
Figure BDA0002461262550000039
4) Fit the slope residual w' with the difference method of the cubic polynomial as follows:
Figure BDA0002461262550000038
Figure BDA0002461262550000039

其中，a,b代表斜率残差点的横纵坐标，

Figure BDA00024612625500000310
k＝1,2,3,是多项式拟合系数，Δx＝xa,b+1-xa,b,Δy＝ya+1,b-ya,b是斜率残差点之间的距离；求得多项式拟合系数后，由有限差分原理得到：Among them, a, b represent the abscissa and ordinate of the slope residual point,
Figure BDA00024612625500000310
k=1,2,3, is the polynomial fitting coefficient, Δx=x a,b+1 -x a,b ,Δy=y a+1,b -y a,b is the distance between the slope residual points; After the polynomial fitting coefficient is obtained, it can be obtained by the finite difference principle:

Figure BDA00024612625500000312
Figure BDA00024612625500000312

上述式子可写成Dxw＝Gx，Dyw＝Gy，进一步写成Dw＝G，其中D是由-1,0,1组成的稀疏矩阵，

Figure BDA00024612625500000313
w是第二次重构波前；The above formula can be written as D x w = G x , D y w = G y , and further written as Dw = G, where D is a sparse matrix consisting of -1, 0, 1,
Figure BDA00024612625500000313
w is the second reconstructed wavefront;

5)最后重构的波前是两次重构波前之和，即W+w。5) The final reconstructed wavefront is the sum of the two reconstructed wavefronts, namely W+w.

本发明的技术效果如下：The technical effect of the present invention is as follows:

本发明适用于任意孔径的复杂自由曲面波前的快速重构，既可以得到波前模式信息，又能提高有限项数值化正交多项式的波前重构精度。The invention is suitable for rapid reconstruction of complex free-form surface wavefronts with arbitrary apertures, and can not only obtain wavefront mode information, but also improve the wavefront reconstruction accuracy of finite-term numerical orthogonal polynomials.

附图说明Description of drawings

图1是是本发明任意孔径形状的高阶自由曲面波前的重构方法的流程图。FIG. 1 is a flow chart of a method for reconstructing a high-order free-form surface wavefront with an arbitrary aperture shape according to the present invention.

具体实施方式Detailed ways

下面结合实施流程图对本发明进行详细说明，但不应以此限制本发明的保护范围。The present invention will be described in detail below in conjunction with the implementation flow chart, but the protection scope of the present invention should not be limited by this.

如图1所示，本发明一种任意孔径形状的高阶自由曲面波前的重构方法，包括以下步骤：As shown in FIG. 1 , a method for reconstructing a high-order free-form surface wavefront with an arbitrary aperture shape of the present invention includes the following steps:

1)通过条纹反射法或哈特曼传感器检测技术得到自由曲面的波前斜率和对应的位置信息，针对任意孔径上的离散梯度数据点，以Zernike多项式为基底，通过数值正交变化法构造梯度正交多项式,F＝ZBT，其中F代表数值化正交多项式，Z代表Zernike多项式，通过Cholesky分解法求得线性组合系数矩阵B,对矩阵F求导得到▽F，并正交化处理得到数值化正交梯度多项式

Figure BDA0002461262550000041
即
Figure BDA0002461262550000042
矩阵C的求法和B一致；1) The wavefront slope and corresponding position information of the free-form surface are obtained by the fringe reflection method or the Hartmann sensor detection technology. For discrete gradient data points on any aperture, the Zernike polynomial is used as the base, and the gradient is constructed by the numerical orthogonal change method. Orthogonal polynomial, F=ZB T , where F represents a numerical orthogonal polynomial, and Z represents a Zernike polynomial. The linear combination coefficient matrix B is obtained by the Cholesky decomposition method, and the matrix F is derived to obtain ▽F, and orthogonalized to obtain Numerical Orthogonal Gradient Polynomial
Figure BDA0002461262550000041
which is
Figure BDA0002461262550000042
The calculation method of matrix C is consistent with B;

2)用数值化正交梯度多项式拟合波前斜率，即

Figure BDA0002461262550000043
其中，
Figure BDA0002461262550000044
W'是波前斜率，α＝[α1 α2 … αn]T,
Figure BDA0002461262550000045
α表示数值化正交梯度多项式系数，n表示数值化正交梯度多项式项数，
Figure BDA0002461262550000046
表示
Figure BDA0002461262550000047
的转置矩阵，(x,y)为波前斜率对应的坐标，波前斜率还可以表示成W'＝▽FCTα，对该式积分：W＝FCTα，得到第一次拟合波前W和对应的模式系数CTα；2) Fit the wavefront slope with a numerical orthogonal gradient polynomial, namely
Figure BDA0002461262550000043
in,
Figure BDA0002461262550000044
W' is the wavefront slope, α=[α 1 α 2 … α n ] T ,
Figure BDA0002461262550000045
α represents the numerical orthogonal gradient polynomial coefficient, n represents the number of numerical orthogonal gradient polynomial terms,
Figure BDA0002461262550000046
express
Figure BDA0002461262550000047
The transposed matrix of , (x, y) is the coordinate corresponding to the slope of the wavefront. The slope of the wavefront can also be expressed as W'=▽FC T α. Integrate this formula: W=FC T α to obtain the first fitting the wavefront W and the corresponding mode coefficient C T α;

3)第一次拟合得到的波前斜率表示成：

Figure BDA0002461262550000048
其中，
Figure BDA0002461262550000049
是用数值化正交梯度多项式拟合斜率得到的系数矩阵，
Figure BDA0002461262550000051
通过原始斜率和第一次拟合的波前斜率相减得到波前的斜率残差
Figure BDA0002461262550000052
其中w'＝[wx wy]T，其中wx和wy分别表示x和y方向的斜率残差；3) The wavefront slope obtained by the first fitting is expressed as:
Figure BDA0002461262550000048
in,
Figure BDA0002461262550000049
is the coefficient matrix obtained by fitting the slopes with numerically orthogonal gradient polynomials,
Figure BDA0002461262550000051
The slope residual of the wavefront is obtained by subtracting the original slope and the first fitted wavefront slope
Figure BDA0002461262550000052
where w'=[w x w y ] T , where w x and w y represent the slope residuals in the x and y directions, respectively;

4)将斜率残差w'用三次多项式的差分法拟合如下：

Figure BDA0002461262550000053
Figure BDA0002461262550000054
4) Fit the slope residual w' with the difference method of the cubic polynomial as follows:
Figure BDA0002461262550000053
Figure BDA0002461262550000054

其中，a,b代表斜率残差点的横纵坐标，

Figure BDA0002461262550000055
k＝1,2,3,是多项式拟合系数，Δx＝xa,b+1-xa,b,Δy＝ya+1,b-ya,b是斜率残差点之间的距离；求得多项式拟合系数后，由有限差分原理得到：Among them, a, b represent the abscissa and ordinate of the slope residual point,
Figure BDA0002461262550000055
k=1,2,3, is the polynomial fitting coefficient, Δx=x a,b+1 -x a,b ,Δy=y a+1,b -y a,b is the distance between the slope residual points; After the polynomial fitting coefficient is obtained, it is obtained by the finite difference principle:

Figure BDA0002461262550000056
Figure BDA0002461262550000056

Figure BDA0002461262550000057
Figure BDA0002461262550000057

上述式子可写成Dxw＝Gx，Dyw＝Gy，进一步写成Dw＝G，其中D是由-1,0,1组成的稀疏矩阵，

Figure BDA0002461262550000058
w是第二次重构波前；The above formula can be written as D x w = G x , D y w = G y , and further written as Dw = G, where D is a sparse matrix consisting of -1, 0, 1,
Figure BDA0002461262550000058
w is the second reconstructed wavefront;

5)最后重构的波前是两次重构波前之和，即W+w。5) The final reconstructed wavefront is the sum of the two reconstructed wavefronts, namely W+w.

以上所述的具体实施例，对本发明的目的、技术方案和有益效果进行了进一步详细说明，应理解的是，以上所述仅为本发明的具体实施例而已，并不用于限制本发明，凡在本发明的精神和原则之内，所做的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。The specific embodiments described above further describe the purpose, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above-mentioned specific embodiments are only specific embodiments of the present invention, and are not intended to limit the present invention. Within the spirit and principle of the present invention, any modifications, equivalent replacements, improvements, etc. made should be included within the protection scope of the present invention.

本发明适用于任意孔径的复杂自由曲面波前的快速重构，既可以得到波前模式信息，又能提高有限项数值化正交多项式的波前重构精度。The invention is suitable for rapid reconstruction of complex free-form surface wavefronts with arbitrary apertures, and can not only obtain wavefront mode information, but also improve the wavefront reconstruction accuracy of finite-term numerical orthogonal polynomials.

Claims (1)

1. A reconstruction method of high-order free-form surface wave front with any aperture shape is characterized by comprising the following steps:
1) obtaining the wave front slope of the free-form surface and corresponding position information by a fringe reflection method or a Hartmann sensor detection technology, constructing a gradient orthogonal polynomial by a numerical value orthogonal variation method by taking a Zernike polynomial as a substrate according to discrete gradient data points on any aperture, wherein F is ZBTWherein F represents a numerical orthogonal polynomial, Z represents a Zernike polynomial, a linear combination coefficient matrix B is obtained by a Cholesky decomposition method, and the matrix F is derived to obtain
Figure FDA0002461262540000011
And orthogonalizing to obtain a numerical orthogonal gradient polynomial
Figure FDA0002461262540000012
Namely, it is
Figure FDA0002461262540000013
The solving method of the matrix C is consistent with that of the matrix B;
2) fitting the wavefront slope with a numerical orthogonal gradient polynomial, i.e.
Figure FDA0002461262540000014
Wherein,
Figure FDA0002461262540000015
w' is wavefront slope, α ═ α1α2…αn]T,
Figure FDA0002461262540000016
α denotes the coefficients of a numerical orthogonal gradient polynomial, n denotes the terms of a numerical orthogonal gradient polynomial,
Figure FDA0002461262540000017
to represent
Figure FDA0002461262540000018
The (x, y) is the coordinate corresponding to the wave front slope, and the wave front slope can also be expressed as
Figure FDA0002461262540000019
Integrating the equation: w ═ FCTα, obtaining a first fitting wavefront W and a corresponding mode coefficient CTα；
3) The wavefront slope from the first fit is expressed as:
Figure FDA00024612625400000110
wherein,
Figure FDA00024612625400000111
is a coefficient matrix obtained by fitting a slope with a numerical orthogonal gradient polynomial,
Figure FDA00024612625400000112
obtaining the slope residual error of the wave front by subtracting the original slope and the first-fit wave front slope
Figure FDA00024612625400000113
Wherein w ═ wxwy]TWherein w isxAnd wyRepresenting slope residuals in the x and y directions, respectively;
4) the slope residual w' is fitted with the difference method of cubic polynomial as follows:
Figure FDA00024612625400000114
Figure FDA00024612625400000115
wherein a and b represent the horizontal and vertical coordinates of the slope residual point,
Figure FDA0002461262540000021
is a polynomial fitting coefficient, Δ x ═ xa,b+1-xa,b,Δy＝ya+1,b-ya,bIs the distance between the slope residual points; after solving the polynomial fitting coefficient, obtaining the following parameters by the finite difference principle:
Figure FDA0002461262540000022
Figure FDA0002461262540000023
the above formula can be written as Dxw＝Gx，Dyw＝GyFurther written as Dw ═ G, where D is a sparse matrix consisting of-1, 0,1,
Figure FDA0002461262540000024
w is the second reconstructed wavefront;
5) the final reconstructed wavefront is the sum of the two reconstructed wavefronts, i.e., W + W.
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