陳貞屹，趙文川，張啟燦，漢 語，劉元坤

基于立體相位測量偏折術(shù)的預(yù)應(yīng)力薄鏡面形檢測

陳貞屹1,2，趙文川2，張啟燦1*，漢 語2，劉元坤1

1四川大學(xué)電子信息學(xué)院，四川 成都 610065；2中國科學(xué)院光電技術(shù)研究所，四川 成都 610209

應(yīng)力拋光技術(shù)通過在鏡面上施加預(yù)定載荷，將包括自由曲面在內(nèi)的非球面轉(zhuǎn)化為球面進(jìn)行加工，對加工鏡面的形變進(jìn)行精準(zhǔn)檢測是實現(xiàn)高精度應(yīng)力拋光的關(guān)鍵。利用立體相位測量偏折術(shù)對預(yù)應(yīng)力薄鏡進(jìn)行鏡面面形和形變檢測，獲得被測鏡表面的連續(xù)相位分布，結(jié)合表面法線唯一性與梯度分布積分，最終得到被測鏡的高度分布和面形。模擬了系統(tǒng)誤差成分，同時采用旋轉(zhuǎn)平均法對系統(tǒng)誤差進(jìn)行標(biāo)定去除，保證和提高了測量精度。對一塊口徑320 mm，球面半徑5200 mm的預(yù)應(yīng)力薄鏡面形和變形量進(jìn)行測量，靜態(tài)測量結(jié)果與三坐標(biāo)機(jī)測量結(jié)果對比，動態(tài)應(yīng)變測量結(jié)果與有限元仿真結(jié)果對比，分別一致吻合，表明本文方法具備微米級的測量精度，相比于干涉儀和三坐標(biāo)機(jī)更適用于大面形變化的預(yù)應(yīng)力薄鏡檢測。

光學(xué)面形檢測；立體相位測量偏折術(shù)；系統(tǒng)誤差；預(yù)應(yīng)力拋光

1 引 言

拼接鏡面主動光學(xué)技術(shù)在大口徑望遠(yuǎn)鏡研制過程中的成功應(yīng)用，幫助天文望遠(yuǎn)鏡突破了口徑限制[1]。然而隨著主鏡口徑的不斷增大，構(gòu)成主鏡所需的離軸子鏡數(shù)量也呈幾何量級增長。為實現(xiàn)離軸子鏡的高效率加工，加州大學(xué)Nelson教授等[2-3]提出了應(yīng)力拋光技術(shù)，該技術(shù)能夠?qū)ㄗ杂汕嬖趦?nèi)的非球面轉(zhuǎn)化為球面進(jìn)行加工，與數(shù)控子孔徑加工技術(shù)相比，材料去除效率大幅提高[4]。

應(yīng)力拋光技術(shù)的關(guān)鍵之一是對鏡面變形進(jìn)行精準(zhǔn)測量，現(xiàn)今主要測量手段包括干涉儀和三坐標(biāo)機(jī)。通常，此類被測鏡球面曲率半徑較大，長達(dá)數(shù)米，若使用干涉測量[5]，干涉儀需放在曲率中心處，測量距離也達(dá)到數(shù)米，空氣擾動將大大影響測量結(jié)果；另外，預(yù)應(yīng)力薄鏡的變形量通常都在微米量級以上，超出了干涉儀的動態(tài)范圍而無法測量。三坐標(biāo)機(jī)測量屬于接觸式測量，測頭需要與鏡面接觸，采樣密度低，測量速度慢，無法對面形進(jìn)行實時反饋。

相位測量偏折術(shù)(Phase measuring deflectometry，PMD)基于結(jié)構(gòu)光三維檢測技術(shù)，具有系統(tǒng)簡單、成本低、動態(tài)范圍大、靈敏度高、測量效率高的特點[6-7]，經(jīng)過高精度的系統(tǒng)標(biāo)定與誤差校準(zhǔn)，可以實現(xiàn)高精度的面形測量。同時，通過條紋的相對變形，可以很精確地檢測到面形的變化，也就是說，該技術(shù)可以非常方便容易地測量面形的相對變化，非常適合預(yù)應(yīng)力薄鏡的檢測需求。本文研究將立體相位測量偏折術(shù)運(yùn)用到預(yù)應(yīng)力薄鏡的面形檢測中，同時結(jié)合旋轉(zhuǎn)平均法消除系統(tǒng)誤差影響，實現(xiàn)了對一塊口徑320 mm，球面半徑5200 mm的預(yù)應(yīng)力薄鏡初始狀態(tài)面形和面形變形量的快速測量，結(jié)果與三坐標(biāo)機(jī)以及有限元仿真理論影響力函數(shù)對比一致。

2 基本原理

測量系統(tǒng)如圖1所示，由相機(jī)、被測鏡、顯示屏組成。兩個相機(jī)通過被測鏡的反射，拍攝顯示屏上依次相移的水平和垂直方向正弦條紋，經(jīng)過相移條紋的對應(yīng)相位計算和相位展開后，得到攜帶了被測鏡面形信息的連續(xù)相位分布。按照光線可逆原理，假設(shè)光線從相機(jī)發(fā)出，相機(jī)每個像素點發(fā)出的光線經(jīng)被測面反射后最終入射到顯示屏上。在針孔相機(jī)模型下，由連續(xù)相位分布可以得到相機(jī)每個像素點入射到顯示屏上的坐標(biāo)。

在立體相位測量偏折術(shù)中，Camera1任意像素點1出射到被測鏡上的物點坐標(biāo)可通過假設(shè)來確定。如圖2所示，1,2分別為Camera1和Camera2的相機(jī)光心，實際情況下，1對應(yīng)的物點位置應(yīng)在被測鏡表面的點，1發(fā)出的光線入射到顯示屏上的點為1，Camera2對點的成像點為2，2發(fā)出的光線入射到顯示屏上的點為2。現(xiàn)假設(shè)1點對應(yīng)的物點位置在￠處，則Camera2的成像點由2變?yōu)椤?，經(jīng)過￠點的光線與顯示屏的交點由2變?yōu)椤?。由1，￠，1和￠2，￠，￠2的坐標(biāo)可分別算出Camera1，Camera2中￠點的表面法線1，2。根據(jù)法線唯一性原則[8-9]，如果假設(shè)的點￠在真實物點處，則1應(yīng)該與2相等，否則繼續(xù)假設(shè)物點位置，反復(fù)迭代前述過程，直到1=2為止，由此點的坐標(biāo)便可找到。

圖1 測量系統(tǒng)示意圖

圖2 雙目PMD原理示意圖

理論上對所有點都使用該法線唯一性就可求出整個被測鏡的面形和高度，但由于雙目視覺測量精度對噪聲比較敏感，測得精度有限，計算速度也非常慢[10]。本文將利用假設(shè)點的法線束來獲取梯度分布[10]，結(jié)合一個可靠點的準(zhǔn)確高度，通過Southwell模型區(qū)域波前重構(gòu)積分[11-13]得到初始面形分布，反復(fù)迭代此過程得到被測面真實面形[14]。由于選取的可靠點往往不是被測鏡中心點，導(dǎo)致積分的采樣面坐標(biāo)系與物體坐標(biāo)系間有一定旋轉(zhuǎn)，在得到恢復(fù)的矢高面形后，應(yīng)將面形旋轉(zhuǎn)回物體坐標(biāo)系。

從光強(qiáng)中提取的相位信息除了被測反射鏡的高度調(diào)制相位，還有系統(tǒng)誤差，因此最后恢復(fù)的反射鏡面形也包含了系統(tǒng)誤差。這些誤差來自于條紋屏平面性、相機(jī)噪聲、計算積分時誤差擴(kuò)散和系統(tǒng)標(biāo)定引入的誤差等[15]，其中系統(tǒng)標(biāo)定的誤差是主要影響因素[16]。如何去除這些系統(tǒng)誤差將取決于其組成成份。為了檢驗和分析此系統(tǒng)中標(biāo)定不準(zhǔn)確帶來的系統(tǒng)誤差成份，根據(jù)標(biāo)定的系統(tǒng)位置參數(shù)建立系統(tǒng)模型[17]，并模擬了系統(tǒng)誤差。圖3為標(biāo)定誤差較大時的系統(tǒng)誤差分布，表1為系統(tǒng)誤差的Zernike多項式分解。表1的結(jié)果證明了PMD中系統(tǒng)誤差主要是低階非旋轉(zhuǎn)對稱部分[18]。因此，本文采用旋轉(zhuǎn)平均法[19-21]來標(biāo)定和去除系統(tǒng)誤差，有效保證和提升測量精度。

圖3 模擬系統(tǒng)誤差分布

3 實 驗

實驗裝置由2個CCD相機(jī)(PointGrey, GS3-U3-28S5M-C，分辨率為1920 pixels′1440 pixels，像素尺寸為4.54 μm)、一個80寸(1寸=2.54 cm)液晶顯示屏(Sharp, LCD-80X818A，分辨率為3840 pixels′2160 pixels)、被測彎月型薄鏡及其支撐裝置和計算機(jī)構(gòu)成，如圖4所示。所使用的2個CCD相機(jī)分上下放置在顯示器同一側(cè)，取下方相機(jī)為主相機(jī)。被測預(yù)應(yīng)力薄鏡系統(tǒng)如圖5(a)所示，口徑為320 mm，其球面半徑為5200 mm，被固定在一個轉(zhuǎn)臺上，可靈活旋轉(zhuǎn)。薄鏡后表面中心有一個固定支撐點，外沿均勻分布6個機(jī)電式力促動器(分別記為1,2,3,4,5,6號電機(jī))，如圖5(b)所示。被測薄鏡位置距離顯示屏約2000 mm左右。

測量前首先進(jìn)行系統(tǒng)標(biāo)定，先標(biāo)定2個相機(jī)的內(nèi)參[22-23]并計算彼此相對位置[24]，再用標(biāo)準(zhǔn)靶面當(dāng)作反射鏡標(biāo)定屏幕與主相機(jī)的相對位置[24-25]。圖6是系統(tǒng)位置參數(shù)標(biāo)定結(jié)果示意圖，1號靶面即為標(biāo)定相機(jī)與屏幕關(guān)系的標(biāo)定靶，標(biāo)定雙目系統(tǒng)的靶面位置在圖中已省略。

圖4 測量裝置圖

表1 系統(tǒng)誤差的Zernike系數(shù)

圖5 被測預(yù)應(yīng)力薄鏡。(a) 薄鏡旋轉(zhuǎn)支撐結(jié)構(gòu)側(cè)視圖；(b) 薄鏡背面支撐點位置分布示意圖

圖6 系統(tǒng)位置參數(shù)標(biāo)定結(jié)果示意圖

3.1 初始狀態(tài)面形測量

測量中采用了時間相位展開算法[26-27]進(jìn)行相位計算，拍攝變形條紋時在顯示器上分別用不同頻率、水平和豎直不同方向上的各3組條紋圖相移4次，拍攝耗時約40 s，圖7為上下相機(jī)拍攝的一幀圖像。使用了旋轉(zhuǎn)平均法進(jìn)行系統(tǒng)誤差標(biāo)定，每隔60°轉(zhuǎn)動薄鏡一次，一共轉(zhuǎn)動6次完成一周360°的測量，在每個旋轉(zhuǎn)位置上分別測量3次，求平均值作為該旋轉(zhuǎn)度數(shù)位置下的測量面形，6個度數(shù)下測量的矢高如圖8所示。

圖9為旋轉(zhuǎn)平均法求出的系統(tǒng)誤差，圖10為去系統(tǒng)誤差、平移、傾斜和旋轉(zhuǎn)對稱項后的鏡面面形，其PV=4.53 μm，RMS=0.754 μm。圖11為三坐標(biāo)機(jī)的測量數(shù)據(jù)除去平移、傾斜、旋轉(zhuǎn)對稱部分后的面形誤差分布(PV=7.879 μm, RMS=1.163 μm)。對比兩圖可以看出，本文方法的測量結(jié)果與三坐標(biāo)機(jī)檢測結(jié)果分布大體一致。需要說明的是，在進(jìn)行三坐標(biāo)測量時，預(yù)應(yīng)力薄鏡鏡面豎直向上，而在本系統(tǒng)測量中，預(yù)應(yīng)力薄鏡鏡面是水平方向。

3.2 變形量測量

預(yù)應(yīng)力薄鏡背面的力促動電機(jī)可對鏡體施加雙向載荷，除了每個電機(jī)獨(dú)立工作對鏡面產(chǎn)生變形以外，6個電機(jī)還可以同時施加不同的校正力，產(chǎn)生球差、像散、慧差、三葉草像差。現(xiàn)將6個力促動電機(jī)分別單獨(dú)施加5 N的力向外頂，通過對變形前后兩次面形測量結(jié)果相減，得到各電機(jī)單獨(dú)加力后的鏡面變形量。圖12分別是ANSYS仿真的1~6號促動器施加5 N外力的理論影響力函數(shù)分布，圖13為實測結(jié)果。圖14為理論分析與實際測量的PV、RMS折線圖。

圖7 顯示豎直方向相移條紋時拍攝的條紋圖。(a) 下相機(jī)圖拍攝；(b) 上相機(jī)圖拍攝

圖8 6個旋轉(zhuǎn)角度下的薄鏡表面矢高測量結(jié)果。(a) 0°；(b) 60°；(c) 120°；(d) 180°；(e) 240°；(f) 300°

圖9 系統(tǒng)誤差分布

圖10 去除系統(tǒng)誤差后的鏡面面形

圖11 三坐標(biāo)機(jī)測量的薄鏡面形結(jié)果

從圖中可以看出，理論上單個電機(jī)施加5 N的力能造成鏡面約1.68 μm的變形量，與實際測量結(jié)果一致，幅度和分布也一致。各電機(jī)分布中心處的低點是因為鏡面后表面中心為固定支撐點時，邊緣促動電機(jī)外頂造成的；加力電機(jī)另一邊的鏡面突起也是由于中心被固定后，另一邊被作用點一同帶起。

接著對預(yù)應(yīng)力薄鏡的像差變形進(jìn)行了測量，四種像差的理論目標(biāo)校正函數(shù)如圖15所示，對應(yīng)變形后的檢測結(jié)果如圖16所示，表2為理論目標(biāo)校正函數(shù)與實際變形檢測結(jié)果的PV、RMS對比。圖表展示的結(jié)果表明，4種像差對應(yīng)薄鏡變形的本文檢測結(jié)果和相應(yīng)的目標(biāo)校正函數(shù)是一致的。

4 結(jié) 論

本文充分利用偏折術(shù)測量速度快、動態(tài)范圍大和靈敏度高的特點，采用立體相位測量偏折術(shù)，對預(yù)應(yīng)力薄鏡的初始面形和應(yīng)變量進(jìn)行了檢測。與三坐標(biāo)機(jī)檢測方法相比，該方法結(jié)構(gòu)簡單，可以一次完成非接觸全場測量，可以對面形變形進(jìn)行快速反饋。與干涉方法相比，該方法有更大的測量動態(tài)范圍，受環(huán)境振動和氣流影響更小。這些特點使該方法非常適用于大口徑、大變形量的自由面應(yīng)力薄鏡檢測。文中模擬了系統(tǒng)誤差成分，并通過旋轉(zhuǎn)平均去除系統(tǒng)誤差，保證和提高了檢測精度。本文檢測結(jié)果與三坐標(biāo)機(jī)檢測結(jié)果的細(xì)微差別主要是被測鏡體放置方式的不同，導(dǎo)致重力影響的不同而引起的。實驗表明，本文方法的檢測結(jié)果與三坐標(biāo)機(jī)測量結(jié)果，以及促動器的理論目標(biāo)函數(shù)結(jié)果相一致。

圖12 各個促動電機(jī)單獨(dú)施力的理論影響力函數(shù)。(a) 1號；(b) 2號；(c) 3號；(d) 4號；(e) 5號；(f) 6號

圖13 各個促動電機(jī)單獨(dú)施力的變形量測量結(jié)果。(a) 1號；(b) 2號；(c) 3號；(d) 4號；(e) 5號；(f) 6號

圖14 電機(jī)理論變形與實測變形對比。(a) PV；(b) RMS

圖15 4種像差的理論目標(biāo)校正函數(shù)。(a) 球差；(b) 像散；(c) 慧差；(d) 三葉草像差

圖16 4種像差的變形量實測結(jié)果。(a) 球差；(b) 像散；(c) 慧差；(d) 三葉草像差

表2 像差理論值與實際測量值對比

本文拓展了相位測量偏折術(shù)的應(yīng)用領(lǐng)域，將其應(yīng)用于預(yù)應(yīng)力薄面形檢測中，下一步可對鏡面面形進(jìn)行實時監(jiān)測，對力促動器的性能和效果進(jìn)行實時測量反饋，提高校正的動態(tài)能力。另外，在后續(xù)工作中，可在標(biāo)定顯示屏與主相機(jī)位置關(guān)系時使用平面性更好的標(biāo)準(zhǔn)反射鏡，或者用三坐標(biāo)機(jī)先對顯示屏的平面性進(jìn)行檢測，消除非平面性引入的誤差，以進(jìn)一步提高測量精度。

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Shape measurement of stressed mirror based on stereoscopic phase measuring deflectometry

Chen Zhenyi1,2, Zhao Wenchuan2, Zhang Qican1*, Han Yu2, Liu Yuankun1

1School of Electronics and Information Engineering, Sichuan University, Chengdu, Sichuan 610065, China;2Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China

Measuring device setup

Overview:Stressed polishing technology was firstly proposed by Jerry E.Nelson in 1980, considered as an effective method of aspheric fabrication. According to the calculated results of elastic mechanics, this method exerts an external force on the mirror to form a deformation which opposite to the desired deformation from spherical surface to off-axis aspheric surface. Under the state of deformation, the mirror is polished into sphere surface. After removing the external force, the required off-axis aspheric surface can be obtained. Because aspheric fabrication is converted to spherical fabrication, tools with large diameter can be used and efficiency is greatly improved.

The key to achieve high precision of stressed polishing is to test the deformation of mirror with high precision. However, the main methods of surface measurement nowadays are interferometer and CMM. If interferometer is used, its dynamic range can only support the detection below micron deformation. If CMM is used, probe may scratch the mirror surface, and the detection tempo is very slow. Furthermore, interferometer and CMM are both expensive and complex equipments.

So, aimed at stressed polishing above micron deformation, stereoscopic phase measuring deflectometry was used to test its surface topography and deformation. It is low cost and convenient technique and only screen, camera, and computer were needed when implemented. More importantly, characteristics such as high dynamic range, full-field three-dimensional measurement and excellent performance in medium and high frequency were brought in, which are very suitable for the test of stressed mirror.

When measuring, firstly calculating the unwrapped phase distribution through CCD cameras, then calculating the height of a specific point on the measured surface using normal consistency constraint, and finally the full-field height distribution was obtained by Southwell gradient integral algorithm. To improve the measuring accuracy, composition of systematic errors were simulated, proved that it mainly includes low-order non-rotational symmetry items. According to simulating results, errors were calibrated and removed by-step averaging method to get a absolute surface topography.

In this paper, the absolute surface topography and the deformation of a stressed mirror with a diameter of 320 mm, spherical radius of 5200 mm were measured. The measuring results were consistent with the corresponding result of CMM and finite element simulation, indicating that this proposed method is on the level of micron in terms of accuracy and more suitable for the test of stressed mirror compared with interferometer and CMM.

Citation: Chen Z Y, Zhao W C, Zhang Q C,Shape measurement of stressed mirror based on stereoscopic phase measuring deflectometry[J]., 2020, 47(8): 190435

Shape measurement of stressed mirror based on stereoscopic phase measuring deflectometry

Chen Zhenyi1,2, Zhao Wenchuan2, Zhang Qican1*, Han Yu2, Liu Yuankun1

1School of Electronics and Information Engineering, Sichuan University, Chengdu, Sichuan 610065, China;2Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China

Stressed polishing technology transforms aspheric fabrication into spherical fabrication by applying predetermined loads on the surface of the mirror. The key to achieve high precision of stressed polishing is to test the surface deformation with high precision. Stereoscopic phase measuring deflectometry was used to test the surface topography and the deformation of stressed mirror. After obtained unwrapped phase distribution, and combined with normal consistency constraint and gradient integral algorithm, the height distribution was finally obtained. Composition of systematic errors were simulated. Also, the errors were calibrated and removed by-step averaging method in this system, which improved the measuring precision. In this paper, the surface topography and the deformation of a stressed mirror with a diameter of 320 mm, spherical radius of 5200 mm were measured. The measuring results were consistent with the corresponding result of CMM and finite element simulation, indicating that this proposed method is on the level of micron in terms of accuracy and more suitable for the test of stressed mirror compared with interferometer and CMM.

optical testing; stereoscopic phase measuring deflectometry; systematic error; stressed mirror polishing

TN247；TH741

A

10.12086/oee.2020.190435

: Chen Z Y, Zhao W C, Zhang Q C,. Shape measurement of stressed mirror based on stereoscopic phase measuring deflectometry[J]., 2020,47(8): 190435

陳貞屹，趙文川，張啟燦，等. 基于立體相位測量偏折術(shù)的預(yù)應(yīng)力薄鏡面形檢測[J]. 光電工程，2020，47(8): 190435

Supported by NationalNaturalScienceFoundation of China (61675141, 61505216) and The Youth Innovation Promotion Association, Chinese Academy of Sciences

* E-mail: zqc@scu.edu.cn

2019-07-23；

2019-10-22

國家自然科學(xué)基金資助項目(61675141，61505216)；中國科學(xué)院青年創(chuàng)新促進(jìn)會資助

陳貞屹(1995-)，男，碩士研究生，主要從事光學(xué)元件三維形貌測量的研究。E-mail：13628045693@163.com

張啟燦(1974-)，男，博士，教授，主要從事三維傳感、動態(tài)三維測量等研究。E-mail：zqc@scu.edu.cn