CN103292738A - Absolute detection method for surface shape error of spherical surface - Google Patents

Abstract

The invention discloses an absolute detection method for a surface shape error of a spherical surface. The absolute detection method comprises the steps of: establishing an equation system about Zernike multinomial coefficients of a measured spherical surface and a reference surface by utilizing measurement data of multiple rotation and concentric translation of the measured spherical surface at a con-focal position and adopting a Zernike multinomial fitting based rotation translation algorithm; and obtaining the Zernike multinomial coefficient by applying a least square method so as to obtain absolute surface shape information of the measured spherical surface and the reference surface. The absolute detection method is also available for the detection of the surface shape error of a plane. As a global optimized idea is adopted to resolve multinomial coefficients of the measured spherical surface and the reference surface at the same time, the absolute detection method for the surface shape error of the spherical surface can suppress system error and random noise for the better, and achieves stronger interference resistance and important application values.

Description

A kind of sphere face shape error absolute sense method

Technical field

The present invention relates to a kind of absolute sense method of face shape error, belong to advanced optics detection range, particularly high-precision optical mirror plane absolute sense field.

Background technology

Along with the development of Modern Optics Technology, the face shape accuracy of detection of optical element requires more and more higher.It is focus and the difficult point problem of optical detection field tests that high precision face shape is interfered detection always.Sphere is as element important in the optical system, and its accuracy of detection is subject to the face shape quality of reference surface, and the face shape information that adopts the absolute sense technology can effectively separate reference surface and tested surface.

At present, Chang Yong absolute sense method has two sphere methods (K.E.Elssner, R.Burow, J.Grzanna, andR.Spolaczyk, " Absolute sphericity measurement, " Appl.Opt.28,4649-4661,1989; B.Truax, " Absolute interferometric testing of spherical surfaces; " Proc.SPIE1400,61-68,1990), odd even function method (Schreiner R, Schwider J, Lindlein N, and K.Mantel. " Absolute test of the reference surface of a Fizeau interferometer through even/odd decompositions ", Appl.Opt, 47,6134-6141,2008) and binary channels from standardization (Jan Burke. " Rapid and reliable reference sphere calibration for Fizeau interferometry ", Opt Let, 33,2536-2538,2008).These methods all include the measurement of opal position, because the opal position is insensitive to adjusting error, introduce astigmatism easily.Another absolute sense method commonly used is ball standardization (R.E.Parks at random, C.J.Evans, L.Shao. " Calibration of interferometer transmission spheres ", Optical Fabrication and test Workshop OSA Technical Digest Series, 12,80-83,1998).This method is by detecting relatively at a large amount of random sites a spherical displacer, and it is average to carry out data then, and the error of spherical displacer is along with the increase that detects number of times goes to zero, and average result will mainly reflect the face shape error information of standard lens reference surface.This method principle is simple, but there is certain difficulty in the development of robotization pick-up unit, and testing process is more consuming time, and it can't be demarcated and disperse camera lens.

At the deficiency of above method, the researchist of Germany and Japan has proposed the sphere absolute sense method (Bernd based on the rotation translation

And G ü nther Seitz, " Interferometric testing of optical surfaces at its current limit, " Optik.112,392-398,2001; Hajime Ichikawa and Takahiro Yamamoto. " Apparatus and method for wavefront absolute calibration and method of synthesizing wavefronts; " U.S.patent5,982,490,9November1999), this method obtains tested surface by the detection data of handling equal angles rotation tested surface and rotates asymmetrical face shape error, and obtains the rotational symmetric face shape error of tested surface by the data of handling homocentric translation front and back.Because there is the theoretical error of kN θ in this method, can't accurately obtain the face shape error information of tested surface.The researchist of the U.S. proposes rotation translation algorithm (the JohannesA.Soons and Ulf Griesmann based on the calculating of pixel unknown quantity subsequently, " Absolute interferometric tests of spherical surfaces based on rotational and translational shears; " Proc.SPIE8493,84930G, 2012), this method is for the clear aperture of 900 pixels, and the detection data of each position all exist 1.2 * 10 ⁶Individual unknown quantity and 6 * 10 ⁶Individual equation, calculated amount is huge and consuming time.Simultaneously, the researchist of public technology has proposed to adopt rotation translation method (the Dongqi Su of Zernike fitting of a polynomial, Erlong Miao, Yongxin Sui and Huaijiang Yang, " Absolute surface figure testing by shift-rotation method using Zernike polynomials; " Opt.Lett.37,3198-3200,2012), but this method is based on local optimum, can only calculate the Zernike multinomial coefficient of tested surface or reference surface when solving an equation group, so comparatively responsive to the neighbourhood noise that detects in the data, result of calculation is easy to generate than large deviation when neighbourhood noise is big.

Summary of the invention

In order to solve in the high precision face shape testing process, tested sphere face shape error and the problem that the reference surface face shape error can't accurately separate the present invention proposes a kind of sphere face shape error absolute sense method.

In order to realize above-mentioned purpose, a kind of sphere face shape error absolute sense method provided by the invention, the step of described sphere face shape error absolute sense is as follows:

Step S1: utilize interferometer, choose standard mirror that the F number is complementary and tested sphere is carried out face shape detect, the face shape that obtains initial position is detected data T ₁(x, y) and be expressed as follows:

T ₁(x，y)＝W(x，y)+RS(x，y) (1)

In the formula 1: (x y) is rectangular coordinate system on the charge-coupled image sensor, and x, y represent the coordinate points in the rectangular coordinate system; W (x, y) and RS (x y) represents the face shape error of tested sphere and reference surface respectively;

Step S2: keep the interferometer system parameter constant, tested sphere is rotated an angle delta θ around the interferometer system optical axis, obtain to rotate the tested sphere detection data T at an angle place ₂(x, y) and be expressed as follows:

T ₂(ρ，θ)＝W(ρ，θ+Δθ)+RS(ρ，θ) (2)

In the formula 2: the tested sphere face shape error after W (ρ, θ+Δ θ) expression rotation one angle, ((ρ θ) is (x, y) Dui Ying polar coordinate system to RS for ρ, the θ) face shape error of expression reference surface; θ represents the angle coordinate, and Δ θ is the anglec of rotation, and ρ represents radial coordinate;

Step S3: keep the interferometer system parameter constant, again with the θ of tested sphere edge with respect to initial position ₁The homocentric translation certain distance of direction Δ s, the tested sphere that obtains homocentric translation one distance detects data T ₃(x, y) and be expressed as follows:

T ₃(x, y)=W (x+sx, y+sy)+(x is y) in (3) formula 3: sx=Δ scos θ for RS ₁, sy=Δ ssin θ ₁, s represents homocentric translational movement; Sx and sy represent tested sphere respectively along the translational movement of X and Y-direction, and W (x+sx, y+sy) the tested sphere of expression is respectively the face shape error of sx and sy along the translational movement of X and Y-direction; The following expression of definition operator Γ (Δ θ, Δ s):

$\left\{\begin{array}{c}\mathrm{\Δs}\≠0,\mathrm{\Γ}(\mathrm{\Δ\θ},\mathrm{\Δs})\·W(x,y)=W(x+\mathrm{sx},y+\mathrm{sy}),\mathrm{sx}=\mathrm{\Δs}\·\cos \mathrm{\Δ\θ},\mathrm{sy}=\mathrm{\Δs}\·\sin \mathrm{\Δ\θ}\\ \mathrm{\Δs}=0,\mathrm{\Γ}(\mathrm{\Δ\θ},0)\·W(\mathrm{\ρ},\mathrm{\θ})=W(\mathrm{\ρ},\mathrm{\θ}+\mathrm{\Δ\θ}),\end{array}\right.---\left(4\right)$

Thereby above-mentioned testing result T ₁(x, y), T ₂(x, y), T ₃(x, the form that y) is expressed as:

T _θ，s(x，y)＝Γ(θ，s)·W(x，y)+RS(x，y) (5)

Γ (θ, s) expression rotation and homocentric translation that tested sphere is carried out, T _{θ, s}(x, y) corresponding face shape is detected data after the tested sphere rotation of the expression translation;

Step S4: utilize the matrix operation instrument, aforesaid equation (5) is rewritten as matrix form:

$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\·\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)---\left(6\right)$

In the formula 6:

${\left(A_{11}\right)}_{i,i^{\′}}=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}{Z}_i(x,y)\·Z_{i^{\′}}(x,y),$

${\left(A_{22}\right)}_{i,i^{\′}}=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}[\mathrm{\Γ}(\mathrm{\θ},s)\·Z_i(x,y)]\·[\mathrm{\Γ}(\mathrm{\θ},s)\·Z_{i^{\′}}(x,y)],$

${\left(A_{12}\right)}_{i,i^{\′}}=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}{Z}_i(x,y)\·[\mathrm{\Γ}(\mathrm{\θ},s)\·Z_{i^{\′}}(x,y)],$

${\left(A_{21}\right)}_{i,i^{\′}}=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}[\mathrm{\Γ}(\mathrm{\θ},s)\·Z_i(x,y)]\·Z_{i^{\′}}(x,y),$

${\left(B_1\right)}_i=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}{Z}_i(x,y)\·T_{\mathrm{\θ},s}(x,y),$

${\left(B_2\right)}_i=\underset{\mathrm{\θ},s}{\mathrm{\Σ}}\underset{x,y}{\mathrm{\Σ}}[\mathrm{\Γ}(\mathrm{\θ},s)\·Z_i(x,y)]\·T_{\mathrm{\θ},s}(x,y),$

Z _i(x y) is i item Zernike polynomial expression; W _a=[a ₁, a ₂..., a _n] ^TBe the Zernike multinomial coefficient of tested sphere, RS _c=[c ₁, c ₂..., c _n] ^TBe the Zernike multinomial coefficient of reference surface, T is transposition, i=1, and 2 ... n, n are Zernike polynomial expression item number, adopt least square method, separate this equation, obtain:

$\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)={\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)}^{-1}\&CenterDot;\left(\begin{array}{c}{B}_1\\ {\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HDA00003413145200011.png,HDA00003413145200021.png"\; state="\{\{state\}\}">B</figure-callout>}}_2\end{array}\right);$

A ₁₁, A ₁₂, A ₂₂, A ₂₁B ₁, B ₂Deng parameter, just the process variable of introducing for convenience of calculation does not have concrete physical significance;

Step S5: the face shape error of tested sphere

The face shape error of reference surface

A1 _,a ₂, a _nBe every Zernike multinomial coefficient of tested sphere, c ₁c ₂, c _nBe every Zernike multinomial coefficient of reference surface, it is respectively corresponding Zernike multinomial coefficient, a _i=a ₁, a ₂..., a _n, c _i=c ₁, c ₂..., c _n

Wherein, the anglec of rotation of described tested sphere and homocentric translational movement size require to select according to accuracy of detection.

Wherein, described tested sphere wheel measuring and homocentric translation measurement number of times are optional, comprise 1 rotation and 1 homocentric translation at least.

Wherein, described tested sphere and reference surface face shape error are represented with the Zernike polynomial expression.

Wherein, when making up the matrix equation formula, comprise the Zernike multinomial coefficient of tested sphere and reference surface simultaneously, when resolving matrix equation, calculate the Zernike multinomial coefficient of tested sphere and reference surface simultaneously.

Wherein, described absolute sense method is applied to the detection of plane surface shape error.

The present invention's advantage compared with prior art is:

1) the sphere face shape error absolute sense method of the present invention's proposition is based on the Zernike fitting of a polynomial, and principle is clear, and method is simply effective.

2) the sphere face shape error absolute sense method that proposes of the present invention is the thinking that has adopted global optimization, calculates the Zernike multinomial coefficient of tested sphere and reference surface simultaneously, more can suppress systematic error and neighbourhood noise, and anti-interference is stronger.

3) the sphere face shape error absolute sense method of the present invention's proposition need not the measurement of opal position, can demarcate and disperse camera lens, and the form that more can lack optical cavity detects the sphere of long radius-of-curvature, and versatility is good.

Description of drawings

Horizontal pick-up unit synoptic diagram when Fig. 1 is concave mirror for tested sphere among the present invention.

Fig. 2 is for being rotated the pick-up unit synoptic diagram of measurement to tested sphere under the vertical condition among the present invention.

Fig. 3 is for carrying out the pick-up unit synoptic diagram of homocentric translation measurement to tested sphere under the vertical condition among the present invention.

Fig. 4 disperses the horizontal pick-up unit synoptic diagram that camera lens detects the tested sphere of long radius-of-curvature for using among the present invention.

Fig. 5 illustrates the process flow diagram of sphere face shape error absolute sense method of the present invention.

Embodiment

Below in conjunction with accompanying drawing and embodiment the present invention is described.

Horizontal pick-up unit when illustrating as Fig. 1 that tested sphere is concave mirror among the present invention, comprising: the measured lens 5 after standard mirror 1, reference surface 2, tested sphere 3, measured lens 4, the homocentric translation, the focus of standard mirror 1 overlaps with the center of curvature of tested sphere 3, the two is in homocentric position, its homocentric position with standard mirror 1, tested sphere 3 system for winding optical axises rotation back concerned remain unchanged, when tested sphere 3 was done homocentric translation along a certain direction, tested sphere 3 also remained unchanged with the homocentric position relation of standard mirror 1.XYZ among the figure is the rectangular coordinate system about tested sphere 3 positions.

Under the vertical condition tested sphere 3 is rotated the pick-up unit of measurement as shown in Figure 2 among the present invention, after optical axis rotated an angle delta θ, tested sphere 3 still kept homocentric position relation with standard mirror 1 with tested sphere 3.XYZ among the figure is the rectangular coordinate system about reference surface 2 and tested sphere 3 positions.

As Fig. 3 the pick-up unit synoptic diagram that under the vertical condition tested sphere 3 is carried out homocentric translation measurement among the present invention is shown, tested sphere 3 is rotated an angle θ around the interferometer system optical axis ₁, and behind the homocentric translation distance Δ s, itself and standard mirror 1 still keep homocentric position relation.

The inventive method utilizes tested sphere 3 repeatedly to rotate measurement data with homocentric translation in confocal position, employing is based on the rotation translation algorithm of Zernike fitting of a polynomial, structure is about the equation of the Zernike multinomial coefficient of tested sphere 3 and reference surface 2, use least square method to solve the Zernike multinomial coefficient, thereby obtain the absolute face shape information of tested sphere 3 and reference surface 2.Because this method has adopted the thinking of global optimization to resolve the multinomial coefficient of tested sphere and reference surface simultaneously, thereby more can suppress systematic error and random noise, anti-interference is stronger, has important use and is worth.Illustrate as Fig. 5 and to utilize device of the present invention to realize that to sphere face shape error absolute sense the detection step of its method is as follows:

Step S1: as shown in Figure 1, the 1 pair of tested sphere 3 of standard mirror that adopts the F number to be complementary carries out face shape and detects, and the face shape that obtains initial position is detected data T ₁(x y) (removes constant term, inclination and out of focus), (x y) is rectangular coordinate system on the charge coupled device ccd,

T ₁(x，y)＝W(x，y)+RS(x，y)，

W (x, y) and RS (x y) is respectively the face shape error of tested sphere 3 and reference surface 2, and x, y represent the coordinate points in the rectangular coordinate system;

Step S2: the detection signal that among the present invention tested sphere 3 is rotated measurement is shown as Fig. 2, keep the interferometer system parameter constant, around the interferometer system optical axis Δ θ (size optional) that turns an angle, the tested sphere 3 that obtains herein detects data T with tested sphere 3 ₂(x y) (removes constant term, inclination and out of focus), i.e. T ₂(ρ, θ)=W (ρ, θ+Δ θ)+RS (ρ, θ), the tested sphere face shape error after W (ρ, θ+Δ θ) expression rotation one angle wherein, ((ρ θ) be the polar coordinate system of correspondence to RS for ρ, the θ) face shape error of expression reference surface; θ represents the angle coordinate, and Δ θ is the anglec of rotation, and ρ represents radial coordinate;

Step S3: among the present invention tested sphere is carried out the detection signal of homocentric translation measurement as shown in Figure 3, keep the interferometer system parameter constant, again with the θ of tested sphere 3 edges with respect to initial position ₁The homocentric translation one distance, delta s of (the initial position size is optional) direction (size is optional), the tested sphere 3 that obtains homocentric translation one distance detects data T ₃(x y) (removes constant term, inclination and out of focus), i.e. T ₃(x, y)=W (x+sx, y+sy)+RS (x, y), sx=Δ scos θ ₁, sy=Δ ssin θ ₁, sx=Δ scos θ ₁, sy=Δ ssin θ ₁, s represents homocentric translational movement; Sx and sy represent tested sphere respectively along the translational movement of X and Y-direction, and W (x+sx, y+sy) the tested sphere of expression is respectively the face shape error of sx and sy along the translational movement of X and Y-direction; Definition operator Γ (θ ₁, Δ s) and following expression:

Tested sphere 3 wheel measurings and homocentric translation measurement number of times are optional, comprise 1 rotation and 1 homocentric translation at least, thus the general type that above-mentioned testing result all can be expressed as:

T _θ，s(x，y)＝Γ(θ，s)·W(x，y)+RS(x，y)；

Γ (θ, s) expression rotation and homocentric translation that tested sphere is carried out, T _{θ, s}(x, y) corresponding face shape is detected data after the tested sphere rotation of the expression translation;

Step S4: utilize the matrix operation instrument, aforesaid equation is rewritten as matrix form:

$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\&CenterDot;\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)=\left(\begin{array}{c}{B}_1\\ {\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HDA00003413145200011.png,HDA00003413145200021.png"\; state="\{\{state\}\}">B</figure-callout>}}_2\end{array}\right),$ Wherein,

${\left(A_{11}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;Z_{i^{\&prime;}}(x,y),$ Be i item Zernike polynomial expression,

${\left(A_{22}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_{i^{\&prime;}}(x,y)],$

${\left(A_{12}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_{i^{\&prime;}}(x,y)],$

${\left(A_{21}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;Z_{i^{\&prime;}}(x,y),$

${\left(B_1\right)}_i=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;T_{\mathrm{\&theta;},s}(x,y),$

${\left(B_2\right)}_i=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;T_{\mathrm{\&theta;},s}(x,y),$

Z _i(x y) is i item Zernike polynomial expression; W _a=[a ₁, a ₂..., a _n] ^TBe the Zernike multinomial coefficient of tested sphere, RS _c=[c ₁, c ₂..., c _n] ^TBe the Zernike multinomial coefficient of reference surface, T is transposition, i=1, and 2 ... n, n are Zernike polynomial expression item number, adopt least square method, separate this equation, obtain:

$\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)={\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)}^{-1}\&CenterDot;\left(\begin{array}{c}{B}_1\\ {\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HDA00003413145200011.png,HDA00003413145200021.png"\; state="\{\{state\}\}">B</figure-callout>}}_2\end{array}\right);$

A ₁₁, A ₁₂, A ₂₂, A ₂₁B ₁, B ₂Deng parameter, just the process variable of introducing for convenience of calculation does not have concrete physical significance;

Step S5: the face shape error of tested sphere 3

The face shape error of reference surface 2 a ₁, a ₂, a _nBe every Zernike multinomial coefficient of tested sphere, c ₁c ₂, c _nBe every Zernike multinomial coefficient of reference surface, it is respectively corresponding Zernike multinomial coefficient, a _i=a ₁, a ₂, a _n, c _i=c ₁, c ₂, c _n

Absolute sense method of the present invention can be applicable to disperse camera lens 6 and detects the signal that the tested sphere 7 of demarcating long radius-of-curvature detects, as shown in Figure 4.

Same absolute sense method of the present invention also can be applicable in the detection of plane surface shape error.

The above; only be the embodiment among the present invention; but protection scope of the present invention is not limited thereto, and any part of people in the disclosed technical scope of the present invention of being familiar with this technology revised or replaced, and all should be encompassed in of the present invention comprising within the scope.

Claims (6)

1. sphere face shape error absolute sense method is characterized in that comprising following steps:

Step S1: utilize interferometer, choose standard mirror that the F number is complementary and tested sphere is carried out face shape detect, the face shape that obtains initial position is detected data T ₁(x, y) and be expressed as follows:

T ₁(x，y)＝W(x，y)+RS(x，y) (1)

In the formula 1: (x y) is rectangular coordinate system on the charge-coupled image sensor, and x, y represent the coordinate points in the rectangular coordinate system; W (x, y) and RS (x y) represents the face shape error of tested sphere and reference surface respectively;

Step S2: keep the interferometer system parameter constant, tested sphere is rotated an angle delta θ around the interferometer system optical axis, obtain to rotate the tested sphere detection data T at an angle place ₂(x, y) and be expressed as follows:

T ₂(ρ，θ)＝W(ρ，θ+Δθ)+RS(ρ，θ) (2)

In the formula 2: the tested sphere face shape error after W (ρ, θ+Δ θ) expression rotation one angle, ((ρ θ) is (x, y) Dui Ying polar coordinate system to RS for ρ, the θ) face shape error of expression reference surface; θ represents the angle coordinate, and Δ θ is the anglec of rotation, and ρ represents radial coordinate;

Step S3: keep the interferometer system parameter constant, again with the θ of tested sphere edge with respect to initial position ₁The homocentric translation certain distance of direction Δ s, the tested sphere that obtains homocentric translation one distance detects data T ₃(x, y) and be expressed as follows:

T ₃(x，y)＝W(x+sx，y+sy)+RS(x，y) (3)

In the formula 3: sx=Δ scos θ ₁, sy=Δ ssin θ ₁, s represents homocentric translational movement; Sx and sy represent tested sphere respectively along the translational movement of X and Y-direction, and W (x+sx, y+sy) the tested sphere of expression is respectively the face shape error of sx and sy along the translational movement of X and Y-direction; The following expression of definition operator Γ (Δ θ, Δ s):

$\left\{\begin{array}{c}\mathrm{\&Delta;s}\&NotEqual;0,\mathrm{\&Gamma;}(\mathrm{\&Delta;\&theta;},\mathrm{\&Delta;s})\&CenterDot;W(x,y)=W(x+\mathrm{sx},y+\mathrm{sy}),\mathrm{sx}=\mathrm{\&Delta;s}\&CenterDot;\cos \mathrm{\&Delta;\&theta;},\mathrm{sy}=\mathrm{\&Delta;s}\&CenterDot;\sin \mathrm{\&Delta;\&theta;}\\ \mathrm{\&Delta;s}=0,\mathrm{\&Gamma;}(\mathrm{\&Delta;\&theta;},0)\&CenterDot;W(\mathrm{\&rho;},\mathrm{\&theta;})=W(\mathrm{\&rho;},\mathrm{\&theta;}+\mathrm{\&Delta;\&theta;}),\end{array}\right.---\left(4\right)$

Thereby above-mentioned testing result T ₁(x, y), T ₂(x, y), T ₃(x, the form that y) is expressed as:

T _θ，s(x，y)＝Γ(θ，s)·W(x，y)+RS(x，y) (5)

Γ (θ, s) expression rotation and homocentric translation that tested sphere is carried out, T _{θ, s}(x, y) corresponding face shape is detected data after the tested sphere rotation of the expression translation;

Step S4: utilize the matrix operation instrument, aforesaid equation (5) is rewritten as matrix form:

$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)\&CenterDot;\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)---\left(6\right)$

In the formula 6:

${\left(A_{11}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;Z_{i^{\&prime;}}(x,y),$

${\left(A_{22}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_{i^{\&prime;}}(x,y)],$

${\left(A_{12}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_{i^{\&prime;}}(x,y)],$

${\left(A_{21}\right)}_{i,i^{\&prime;}}=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;Z_{i^{\&prime;}}(x,y),$

${\left(B_1\right)}_i=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}{Z}_i(x,y)\&CenterDot;T_{\mathrm{\&theta;},s}(x,y),$

${\left(B_2\right)}_i=\underset{\mathrm{\&theta;},s}{\mathrm{\&Sigma;}}\underset{x,y}{\mathrm{\&Sigma;}}[\mathrm{\&Gamma;}(\mathrm{\&theta;},s)\&CenterDot;Z_i(x,y)]\&CenterDot;T_{\mathrm{\&theta;},s}(x,y),$

Z _i(x y) is i item Zernike polynomial expression; W _a=[ _a1, a ₂..., a _n] ^TBe the Zernike multinomial coefficient of tested sphere, RS _c=[c ₁, c ₂..., c _n] ^TBe the Zernike multinomial coefficient of reference surface, T is transposition, i=1, and 2 ... n, n are Zernike polynomial expression item number, adopt least square method, separate this equation, obtain:

$\left(\begin{array}{c}{W}_a\\ {\mathrm{RS}}_c\end{array}\right)={\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)}^{-1}\&CenterDot;\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right);$

A ₁₁, A ₁₂, A ₂₂, A ₂₁B ₁, B ₂Deng parameter, just the process variable of introducing for convenience of calculation does not have concrete physical significance;

Step S5: the face shape error of tested sphere

The face shape error of reference surface

a ₁, a ₂..., a _nBe every Zernike multinomial coefficient of tested sphere, c ₁c ₂, c _nBe every Zernike multinomial coefficient of reference surface, it is respectively corresponding Zernike multinomial coefficient, a _i=a ₁, a ₂..., a _n, c _i=c ₁, c ₂..., c _n

2. sphere face shape error absolute sense method according to claim 1 is characterized in that, the anglec of rotation of described tested sphere and homocentric translational movement size require to select according to accuracy of detection.

3. sphere face shape error absolute sense method according to claim 1 is characterized in that, described tested sphere wheel measuring and homocentric translation measurement number of times are optional, comprises 1 rotation and 1 homocentric translation at least.

4. sphere face shape error absolute sense method according to claim 1 is characterized in that described tested sphere and reference surface face shape error are represented with the Zernike polynomial expression.

5. sphere face shape error absolute sense method according to claim 1, it is characterized in that, when making up the matrix equation formula, the Zernike multinomial coefficient that comprises tested sphere and reference surface simultaneously, when resolving matrix equation, calculate the Zernike multinomial coefficient of tested sphere and reference surface simultaneously.

6. according to each the described absolute sense method among the claim 1-5, be applied to the detection of plane surface shape error.

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