CN103292738B - A kind of absolute detection method for surface shape error of spherical surface - Google Patents

Abstract

The invention discloses a kind of absolute detection method for surface shape error of spherical surface, utilize tested sphere in the measurement data of confocal position multiple rotary and homocentric translation, adopt the rotation translation algorithm based on Zernike fitting of a polynomial, build the system of equations about tested sphere and reference surface Zernike multinomial coefficient, use least square method to solve Zernike multinomial coefficient, thus obtain the absolute face shape information of tested sphere and reference surface.The method equally also can be used for the detection of plane surface shape error.Because the thinking that the process employs global optimization resolves the multinomial coefficient of tested sphere and reference surface simultaneously, thus more can suppress systematic error and random noise, anti-interference is stronger, has important using value.

Description

A kind of absolute detection method for surface shape error of spherical surface

Technical field

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

Background technology

Along with the development of Modern Optics Technology, the surface testing accuracy requirement of optical element is more and more higher.High-precision surface shape interference detection is focus and the difficulties of optical detection field tests always.Sphere is as element important in optical system, and its accuracy of detection is limited to the face form quality amount of reference surface, and adopts absolute sense technology effectively can be separated the face shape information of reference surface and tested surface.

At present, conventional absolute detection method has two sphere method (K.E.Elssner, R.Burow, J.Grzanna, andR.Spolaczyk, " Absolutesphericitymeasurement, " Appl.Opt.28,4649-4661,1989, B.Truax, " Absoluteinterferometrictestingofsphericalsurfaces, " Proc.SPIE1400, 61-68, 1990), even-odd method (SchreinerR, SchwiderJ, LindleinN, andK.Mantel. " AbsolutetestofthereferencesurfaceofaFizeauinterferometer througheven/odddecompositions ", Appl.Opt, 47, 6134-6141, 2008) and binary channels self-calibration method (JanBurke. " RapidandreliablereferencespherecalibrationforFizeauinter ferometry ", OptLet, 33, 2536-2538, 2008).These methods all include the measurement of opal position, because opal position is insensitive to alignment error, easily introduce astigmatism.Another conventional absolute detection method is random ball standardization (R.E.Parks, C.J.Evans, L.Shao. " Calibrationofinterferometertransmissionspheres ", OpticalFabricationandtestWorkshopOSATechnicalDigestSerie s, 12,80-83,1998).The method is by relatively detecting at a large amount of random site a spherical displacer, and it is average then to carry out data, and the error of spherical displacer goes to zero along with the increase detecting number of times, and average result will mainly reflect the face shape error information of standard lens reference surface.This Method And Principle is simple, but the development of automatic detection device exists certain difficulty, and testing process is more consuming time, and it cannot be demarcated and disperse camera lens.

For the deficiency of above method, the researchist of Germany and Japan proposes the sphere absolute detection method (Bernd based on rotating translation andG ü ntherSeitz, " Interferometrictestingofopticalsurfacesatitscurrentlimit, " Optik.112,392-398,2001; HajimeIchikawaandTakahiroYamamoto. " Apparatusandmethodforwavefrontabsolutecalibrationandmeth odofsynthesizingwavefronts; " U.S.patent5,982,490,9November1999), the method obtains the face shape error of tested surface asymmetrical by processing the detection data angularly rotating tested surface, and obtains the rotational symmetric face shape error of tested surface by the data processed before and after homocentric translation.Because this method exists the theoretical error of kN θ, the face shape error information of tested surface accurately cannot be obtained.The researchist of the U.S. proposes the rotation translation algorithm (JohannesA.SoonsandUlfGriesmann based on the calculating of pixel unknown quantity subsequently, " Absoluteinterferometrictestsofsphericalsurfacesbasedonro tationalandtranslationalshears; " 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 proposes the rotation shifting method (DongqiSu adopting Zernike fitting of a polynomial, ErlongMiao, YongxinSuiandHuaijiangYang, " Absolutesurfacefiguretestingbyshift-rotationmethodusingZ ernikepolynomials, " Opt.Lett.37, 3198-3200, 2012), but the method is based on local optimum, the Zernike multinomial coefficient of tested surface or reference surface can only be calculated during solving equations, therefore it is comparatively responsive to the neighbourhood noise detected in data, when neighbourhood noise is larger, result of calculation easily produces relatively large deviation.

Summary of the invention

In order to solve in high-precision surface shape testing process, the problem that tested surface shape error of spherical surface cannot accurately be separated with reference surface face shape error, the present invention proposes a kind of absolute detection method for surface shape error of spherical surface.

In order to realize above-mentioned object, a kind of absolute detection method for surface shape error of spherical surface provided by the invention, the step of described surface shape error of spherical surface absolute sense is as follows:

Step S1: utilize interferometer, chooses the standard mirror that F number matches and carries out surface testing to tested sphere, obtain the surface testing data T of initial position ₁(x, y) is also expressed as follows:

T ₁(x，y)＝W(x，y)+RS(x，y)(1)

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

Step S2: keep interferometer system parameter constant, tested sphere is rotated an angle delta θ around interferometer system optical axis, the tested sphere obtaining rotation one angle place detects data T ₂(x, y) is also expressed as follows:

T ₂(ρ，θ)＝W(ρ，θ+Δθ)+RS(ρ，θ)(2)

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

Step S3: keep interferometer system parameter constant, then by tested sphere along the θ relative to initial position ₁direction homocentric translation certain distance Δ s, the tested sphere obtaining homocentric translation one distance detects data T ₃(x, y) is also expressed as follows:

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

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

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

T _θ，s(x，y)＝Γ(θ，s)·W(x，y)+RS(x，y)(5)

Γ (θ, s) represents the rotation and homocentric translation carry out tested sphere, T _{θ, s}(x, y) be corresponding surface testing data after representing tested spherical rotary translation;

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

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

In 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-th Zernike polynomial expression; W _a=[a ₁, a ₂..., a _n] ^tfor the Zernike multinomial coefficient of tested sphere, RS _c=[c ₁, c ₂..., c _n] ^tfor the Zernike multinomial coefficient of reference surface, T is transposition, i=1,2 ... n, n are Zernike polynomial expression item number, adopt least square method, separate this equation, obtain:

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

A ₁₁, A ₁₂, A ₂₂, A ₂₁b ₁, B ₂deng parameter, just in order to the process variable that convenience of calculation is introduced, there is no concrete physical significance;

Step S5: the face shape error of tested sphere the face shape error of reference surface a1 _,a ₂, a _nfor every Zernike multinomial coefficient of tested sphere, c ₁c ₂, c _nfor 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 spherical rotary measure and homocentric translation measurement number of times optional, at least comprise 1 rotation and 1 homocentric translation.

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

Wherein, when building matrix equation, 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 detection method, is applied to the detection of plane surface shape error.

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

1) absolute detection method for surface shape error of spherical surface of the present invention's proposition is based on Zernike fitting of a polynomial, and clear principle, method is simply effective.

2) absolute detection method for surface shape error of spherical surface that the present invention proposes is the thinking that have employed global optimization, and calculate the Zernike multinomial coefficient of tested sphere and reference surface, more can suppress systematic error and neighbourhood noise, anti-interference is stronger simultaneously.

3) absolute detection method for surface shape error of spherical surface that proposes of the present invention, without the need to the measurement of opal position, can demarcate and disperse camera lens, and more the form of short optical cavity can detect the sphere of long radius-of-curvature, versatility is good.

Accompanying drawing explanation

Fig. 1 is horizontal type detecting device schematic diagram when tested sphere is concave mirror in the present invention.

Fig. 2 is the pick-up unit schematic diagram under vertical condition, tested sphere being carried out to wheel measuring in the present invention.

Fig. 3 is the pick-up unit schematic diagram under vertical condition, tested sphere being carried out to homocentric translation measurement in the present invention.

Fig. 4 is with the horizontal type detecting device schematic diagram dispersed camera lens and detect the tested sphere of long radius-of-curvature in the present invention.

Fig. 5 illustrates the process flow diagram of absolute detection method for surface shape error of spherical surface of the present invention.

Embodiment

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

As horizontal type detecting device when Fig. 1 illustrates that in the present invention, tested sphere is concave mirror, comprising: the measured lens 5 after standard mirror 1, reference surface 2, tested sphere 3, measured lens 4, homocentric translation, the focus of standard mirror 1 overlaps with the center of curvature of tested sphere 3, the two is in homocentric position, after tested sphere 3 system for winding optical axis being rotated, the homocentric position relationship of itself and standard mirror 1 remains unchanged, when tested sphere 3 is done homocentric translation along a direction, tested sphere 3 also remains unchanged with the homocentric position relationship of standard mirror 1.XYZ in figure is the rectangular coordinate system about tested sphere 3 position.

Under vertical condition, tested sphere 3 is carried out to the pick-up unit of wheel measuring as shown in Figure 2 in the present invention, by tested sphere 3 after optical axis rotates an angle delta θ, tested sphere 3 still keeps homocentric position relationship with standard mirror 1.XYZ in figure is the rectangular coordinate system about reference surface 2 and tested sphere 3 position.

As Fig. 3 illustrates the pick-up unit schematic diagram under vertical condition, tested sphere 3 being carried out to homocentric translation measurement in the present invention, tested sphere 3 is rotated an angle θ around interferometer system optical axis ₁, and after homocentric translation distance Δ s, itself and standard mirror 1 still keep homocentric position relationship.

The inventive method utilizes tested sphere 3 in the measurement data of confocal position multiple rotary and homocentric translation, adopt the rotation translation algorithm based on Zernike fitting of a polynomial, build the equation of the Zernike multinomial coefficient about tested sphere 3 and reference surface 2, use least square method to solve Zernike multinomial coefficient, thus obtain the absolute face shape information of tested sphere 3 and reference surface 2.Because the thinking that the process employs global optimization resolves the multinomial coefficient of tested sphere and reference surface simultaneously, thus more can suppress systematic error and random noise, anti-interference is stronger, has important using value.Utilize device of the present invention to realize surface shape error of spherical surface absolute sense as Fig. 5 illustrates, the detecting step of its method is as follows:

Step S1: as shown in Figure 1, the standard mirror 1 adopting F number to match carries out surface testing to tested sphere 3, obtains the surface testing data T of initial position ₁(x, y) (removing constant term, inclination and out of focus), (x, y) is the rectangular coordinate system on 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 rectangular coordinate system;

Step S2: the detection signal in the present invention, tested sphere 3 being carried out to wheel measuring as shown in Figure 2, keep interferometer system parameter constant, to be turned an angle Δ θ (size is optional) around interferometer system optical axis by tested sphere 3, the tested sphere 3 obtained herein detects data T ₂(x, y) (removing constant term, inclination and out of focus), i.e. T ₂(ρ, θ)=W (ρ, θ+Δ θ)+RS (ρ, θ), wherein W (ρ, θ+Δ θ) represents the tested surface shape error of spherical surface after rotation one angle, RS (ρ, θ) represent the face shape error of reference surface, (ρ, θ) is corresponding polar coordinate system; θ represents angle coordinate, and Δ θ is the anglec of rotation, and ρ represents radial coordinate;

Step S3: the detection signal in the present invention, tested sphere being carried out to homocentric translation measurement as shown in Figure 3, keeps interferometer system parameter constant, then by tested sphere 3 along the θ relative to initial position ₁the homocentric translation one in (initial position size is optional) direction distance, delta s (size is optional), the tested sphere 3 obtaining homocentric translation one distance detects data T ₃(x, y) (removing 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 represents the translational movement of tested sphere along X and Y-direction respectively, and W (x+sx, y+sy) represents that tested sphere 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) represent as follows:

Tested sphere 3 wheel measuring and homocentric translation measurement number of times optional, at least comprise 1 time and rotate and 1 homocentric translation, 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) represents the rotation and homocentric translation carry out tested sphere, T _{θ, s}(x, y) be corresponding surface testing data after representing tested spherical rotary translation;

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

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

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

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

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

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

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

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

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

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

A ₁₁, A ₁₂, A ₂₂, A ₂₁b ₁, B ₂deng parameter, just in order to the process variable that convenience of calculation is introduced, there is no 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 _nfor every Zernike multinomial coefficient of tested sphere, c ₁c ₂, c _nfor 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 detection method of the present invention can be applicable to disperse camera lens 6 and detects the signal that the tested sphere 7 demarcating long radius-of-curvature carries out detecting, as shown in Figure 4.

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

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

Claims (6)

1. an absolute detection method for surface shape error of spherical surface, is characterized in that comprising following steps:

Step S1: utilize interferometer, chooses the standard mirror that F number matches and carries out surface testing to tested sphere, obtain the surface testing data T of initial position ₁(x, _y) and be expressed as follows:

T ₁(x，y)＝W(x，y)+RS(x，y)(1)

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

Step S2: keep interferometer system parameter constant, tested sphere is rotated an angle delta θ around interferometer system optical axis, the tested sphere obtaining rotation one angle place detects data T ₂(x, y) is also expressed as follows:

T ₂(ρ，θ)＝W(ρ，θ+Δθ)+RS(ρ，θ)(2)

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

Step S3: keep interferometer system parameter constant, then by tested sphere along the θ relative to initial position ₁direction homocentric translation certain distance Δ s, the tested sphere obtaining homocentric translation one distance detects data T ₃(x, y) is also expressed as follows:

T ₃(x，y)＝W(x+sx，y+sy)+RS(x，y)(3)

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

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

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

T _θ，s(x，y)＝Γ(θ，s)·W(x，y)+RS(x，y)(5)

Γ (θ, s) represents the rotation and homocentric translation carry out tested sphere, T _{θ, s}(x, y) be corresponding surface testing data after representing tested spherical rotary translation;

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

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

In formula 6:

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

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

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

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

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

Z _i(x, y) is i-th Zernike polynomial expression; W _a=[a _1,, a ₂..., a _n] ^tfor the Zernike multinomial coefficient of tested sphere, RS _c=[c ₁, c ₂..., c _n] ^tfor the Zernike multinomial coefficient of reference surface, T is transposition, i=1,2 ... n, n are Zernike polynomial expression item number, adopt least square method, separate this equation, obtain:

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

A ₁₁, A ₁₂, A ₂₂, A ₂₁, B ₁, B ₂parameter, just in order to the process variable that convenience of calculation is introduced, does not have concrete physical significance;

Step S5: the face shape error of tested sphere the face shape error of reference surface for every Zernike multinomial coefficient of tested sphere, c ₁, c ₂..., c _nfor 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. absolute detection method for surface shape error of spherical surface 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. absolute detection method for surface shape error of spherical surface according to claim 1, is characterized in that, described tested spherical rotary measure and homocentric translation measurement number of times optional, at least comprise 1 rotation and 1 homocentric translation.

4. absolute detection method for surface shape error of spherical surface according to claim 1, is characterized in that, described tested sphere and reference surface face shape error represent with Zernike polynomial expression.

5. absolute detection method for surface shape error of spherical surface according to claim 1, it is characterized in that, when building matrix equation, 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.

6. the absolute detection method according to any one in claim 1-5, is applied to the detection of plane surface shape error.