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待检测非球面是微晶材料离轴抛物面,具体参数如表1所示。
表 1 离轴非球面参数
Table 1. Parameters of off-axis aspheric surface
Parameter Value/mm Radius of curvature R
Rex
Rey
X axis decenter
Y axis decenter3803.552
300
300
0
500使用Zernike标准面型对计算全息图进行优化,图1为计算全息图的设计光路,标准镜采用4D 4 in(1 in=2.54 cm)F#3.3镜头,干涉仪焦点距离CGH前表面640 mm,CGH后表面距离待检测微晶反射镜中心3170 mm。
图 1 CGH离轴非球面检测光路图
Figure 1. Optical path of off-axis aspheric surface interferometric testing by CGH
主区域设计波像差残差RMS优于0.0028λ,见图2。
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对准区域的条纹对比度受反射光栅的衍射效率影响,分别使用相位法和傅里叶变换法对衍射效率模型进行分析。
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使用线性光栅来表示计算全息图的局部模型,如图3所示。
图中,w为光栅周期,w1为光栅顶部宽度,h为刻蚀深度,θi为光栅的入射角度,θm为光栅的出射角。
一个小台阶上的光程差为
$ w_1(\sin {\theta _i} + \sin {\theta _m}) $ ,相邻台阶的光程差为$ w_1(\sin {\theta _i} + \sin {\theta _m}) - h(\cos {\theta _i} + \cos {\theta _m}) $ ,设焦面上存在一点P,只考虑单个台阶P点的复振幅为:$$ E\left( P \right) = A\left( {\frac{{\sin \alpha }}{\alpha }} \right) $$ (1) 式中:
$\alpha = \dfrac{{\pi w_1(\sin {\theta _i} + \sin {\theta _m})}}{\lambda } = \dfrac{\pi }{2}m$ 。相邻台阶,即一个周期的光栅产生的相位差为:
$$ \delta = \frac{{2\pi }}{\lambda }\left[ {w_1\left( {\sin {\theta _i} + \sin {\theta _m}} \right) - h\left( {\cos {\theta _i} + \cos {\theta _m}} \right)} \right] $$ (2) 则一个周期的光栅产生的多光束干涉结果为:
$$ E\left( P \right) = A\left( {\frac{{\sin \alpha }}{\alpha }} \right)\left[ {1 + {{\rm e}^{i\delta }}} \right] = a\left( {\frac{{\sin \alpha }}{\alpha }} \right)\left( {\frac{{\sin \delta }}{{2\sin \dfrac{\delta }{2}}}} \right){{\rm e}^{i\tfrac{\delta }{2}}} $$ (3) 则M个周期的光栅产生的多光束干涉结果为:
$$ E\left( P \right) = A\left( {\dfrac{{\sin \alpha }}{\alpha }} \right)\left( {\dfrac{{\sin \delta }}{{2\sin \dfrac{\delta }{2}}}} \right)\left( {\dfrac{{\sin \dfrac{{M\rho }}{2}}}{{\sin \dfrac{\rho }{2}}}} \right){{\rm e}^{i\tfrac{\delta }{2} + i\left( {M - 1} \right)\tfrac{\rho }{2}}} $$ (4) 式中:
$\rho = \dfrac{{2\pi }}{\lambda }w(\sin {\theta _i} + \sin {\theta _m}) = 2\pi m$ ,m为级次。 -
假设光栅周期为w,非刻蚀区域宽度为w1,占空比
$D =w_1/w$ ,刻蚀深度为h,光线经过光栅反射后,刻蚀区的复振幅为A0,非刻蚀区的复振幅为$ {A_1}\exp ( - i\phi ) $ ,其中振幅大小A1由膜层的反射率决定,$$ \phi = \frac{{4\pi }}{\lambda }h $$ (5) 设刻蚀深度为h,宽度为
$w_1 = w/2$ ,入射角为θi,出射角为θm。$$ \phi = \frac{{2\pi }}{\lambda }\left[ {w_1\left( {\sin {\theta _i} + \sin {\theta _m}} \right) - h\left( {\cos {\theta _i} + \cos {\theta _m}} \right)} \right] $$ (6) 式中:λ为波长,空气折射率设为1。
对于垂直入射的平面波,光栅的反射率函数可以表示为:
$$ u\left( x \right) = {A_0} + \left( {{A_1}{{\rm e}^{ - i\phi }} - {A_0}} \right){\rm rect}\left( {\frac{x}{{w_1}}} \right) *{\rm comb}\left( {\frac{x}{w}} \right)\frac{1}{w} $$ (7) 根据弗朗和费衍射理论,假设在M个周期中,反射后的波前可以表示为:
$$\begin{split} & U\left( \omega \right) = F\left[ {u\left( x \right)} \right] = \\& \left\{ {{A_0}\delta \left( \omega \right) + \left[ {{A_1}\cos \left( { - \phi } \right) - {A_0}} \right]D{\rm sinc}\left( {Dw\omega } \right)\sum\limits_0^M {\delta \left( {\omega - \dfrac{ {\textit{z}}}{w}} \right)} } \right\} + \\& i\left\{ {\left[ {{A_1}\sin \left( { - \phi } \right)} \right]D{\rm sinc}\left( {Dw\omega } \right)\sum\limits_0^M {\delta \left( {\omega - \dfrac{ {\textit{z}}}{w}} \right)} } \right\} = \end{split}$$ $$\begin{split} & \left\{ \begin{array}{l} \left\{ {{A_0} + \left[ {{A_1}\cos \left( { - \phi } \right) - {A_0}} \right]D} \right\} + i\left\{ {\left[ {{A_1}\sin \left( { - \phi } \right)} \right]D} \right\},m = 0 \\ \left\{ {\left[ {{A_1}\cos \left( { - \phi } \right) - {A_0}} \right]D{\rm sinc}\left( {Dm} \right)\left( {\dfrac{{\sin \dfrac{{M\rho }}{2}}}{{\sin \dfrac{\rho }{2}}}} \right)} \right\} +\\ i\left\{ {\left[ {{A_1}\sin \left( { - \phi } \right)} \right]D{\rm sinc}\left( {Dm} \right)\left( {\dfrac{{\sin \dfrac{{M\rho }}{2}}}{{\sin \dfrac{\rho }{2}}}} \right)} \right\} ,m = \pm 1, \pm 2,\cdots \\ \end{array} \right. \end{split}$$ (8) 式中:
$\rho = \dfrac{{2\pi }}{\lambda }w(\sin {\theta _i} + \sin {\theta _m}) = 2\pi m$ ,m为级次。假设反射率相同A0= A1,且D=0.5,经过运算化简得:
$$\begin{split} & = \left\{ {\left[ {{A_1}\cos \left( { - \dfrac{\phi }{2}} \right)} \right]{\rm sinc}\left( {0.5m} \right)\left( {\dfrac{{\sin \dfrac{{M\rho }}{2}}}{{\sin \dfrac{\rho }{2}}}} \right){{\rm e}^{i\tfrac{\phi }{2} + i\left( {M - 1} \right)\tfrac{\rho }{2}}}} \right\}, \\& m = \pm 1, \pm 2,\cdots. \end{split}$$ (9) 显然与相位法完全相对应,得以证明两种方法的正确性。
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笔者以相位法,通过matlab计算仿真,探究反射率对衍射效率的影响。
$$ I = {\left[ {A\left( {\frac{{\sin 0.5\pi m}}{{0.5\pi m}}} \right)\cos \left( {\frac{\delta }{2}} \right)\left( {\frac{{\sin M\pi m}}{{\sin \pi m}}} \right)} \right]^2} $$ (10) 选择衍射级次为+1级,入射角设为0°,刻蚀深度为h。
计算得此时光强为:
$$ I = r{\left[ {A\left( {\frac{2}{\pi }} \right)} \right]^2} = 0.400 \; 5{A^2}r $$ (11) 式中:r为膜层反射率。
根据对比度计算公式[13]:
$$ ratio = \frac{{AR}}{{AT}} = \sqrt {\frac{I}{{{I_0}}}} = \frac{{2\sqrt r }}{{0.2\pi }} = 3.183\sqrt r $$ (12) $$ contrast = \frac{{2ratio}}{{1 + rati{o^2}}} = \frac{{6.366\sqrt r }}{{1 + 10.131r}} $$ (13) 根据此公式在matlab中绘制r在(40%,50%)范围内的对比度图像,见图4。
图 4 条纹对比度与对准全息反射率的关系
Figure 4. Relationship between reflectivity of alignment holographic and fringe contrast
根据图4得知,在反射率为9.87%时,对比度达到最大值,此时条纹最清晰,方便检测人员对准。
进一步的,笔者将刻蚀深度h设为未知数,希望得到在对比度保持最大值的前提下,刻蚀深度与反射率之间的联系。
修正公式可以获得反射率与刻蚀深度的关系表达式如下:
$$ r = {\left( {\dfrac{1}{{\cos \left( {0.5\pi - \dfrac{{h\pi }}{\lambda }} \right)}}} \right)^2} $$ (14) 对比度为1的时候,反射率和刻蚀深度的关系如图5所示。
图 5 条纹对比度为1的条件下,刻蚀深度与反射率的关系
Figure 5. Relationship between etching depth and reflectivity when fringe contrast maintains 1
由于工艺原因,刻蚀深度为0.5λ时,CGH对刻蚀误差非常敏感,因此在实际加工中,笔者一般选用刻蚀深度为0.3λ~0.4λ[14]。根据图像分析,也非容易界定该范围的反射率值。
Testing of aspheric surface with low reflectivity using hybrid type computer-generated hologram (invited)
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摘要: 计算全息图被广泛应用于高精度离轴非球面面形检测,为提升低反射率材料检测过程中的条纹对比度,提出了一种混合型计算全息图,对准区域使用反射型全息图,基于衍射效率模型给出了最佳对比度对应的膜层反射率,主区域使用位相型全息图实现衍射效率的提升,从而提升检测区域的条纹对比度。通过多步半导体工艺实现混合全息图的加工,并对某微晶材料离轴非球面进行检测,实验结果表明:该混合型计算全息图可以获得清晰的条纹对比度,最终检测精度优于0.02λ (λ=632.8 nm),该混合型计算全息图可广泛应用于低反射率非球面检测。Abstract: Computer-generated hologram (CGH) is widely used in high-precision off-axis aspheric surface testing. In order to improve the fringe contrast in the testing process of materials with low reflectivity, a hybrid type computer-generated hologram is proposed. The alignment area uses a reflective hologram, the optimal contrast corresponding to the film reflectivity is derived based on diffraction efficiency model, the main area applies phase type CGH to achieve diffraction efficiency improvement, thereby improving the fringe contrast of the detection area. The processing of hybrid hologram is realized by multi-step photo lithography process, and the off-axis aspheric surface of glass-ceramics material is tested using the fabricated CGH. The experimental results show that the hybrid type CGH can obtain clear fringe contrast, and the final detection accuracy is better than 0.02λ(λ=632.8 nm). The proposed hybrid type CGH can be widely used in the testing of aspheric surfaces with low reflectivity material.
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Key words:
- CGH /
- low reflectivity /
- diffraction efficiency /
- fringe contrast
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表 1 离轴非球面参数
Table 1. Parameters of off-axis aspheric surface
Parameter Value/mm Radius of curvature R
Rex
Rey
X axis decenter
Y axis decenter3803.552
300
300
0
500 -
[1] Gao Songtao, Wu Dongcheng, Yu Changsong. Computer-generated hologram design for high-precision aspherical surface testing [J]. Laser & Optoelectronics Progress, 2016, 53(9): 090901. (in Chinese) [2] Qiu Hongwei. Research on ultra-precise aspheric surface testing based on computer-generated hologram [D]. Beijing: University of Chinese Academy of Sciences, 2021. (in Chinese) [3] Zhou Wengcai, Xu Feng, Wei Xiaoxiao. System design of computed graphic holograph in testing the freeform surface [J]. Optics & Optoelectronic Technologty, 2016, 14(1): 69-73. (in Chinese) [4] Zhu Deyan, Zhang Xuejun. Design of high-precision phase computer-generated hologram [J]. Acta Optica Sinca, 2015, 35(7): 0712002. (in Chinese) doi: 10.3788/AOS201535.0712002 [5] Liang Zijian, Yang Yongyin, Zhao Hongxiang, et al. Advances in research and applications of optical aspheric surface metrology [J]. Chinese Optics, 2022, 15(2): 161-186. (in Chinese) doi: 10.37188/CO.2021-0143 [6] Yan Lisong. Research on the algorithm testing optical mirror by sub-aperture stitching interferometry[D]. Beijing: University of Chinese Academy of Sciences, 2015. (in Chinese) [7] Zhou Ping, Burge J H. Fabrication error analysis and experimental demonstration for computer-generated holograms [J]. Appl Opt, 2007, 46(5): 657-663. doi: 10.1364/AO.46.000657 [8] Zhao Chunyu, Burge James H. Optical testing with computer generated holograms: comprehensive error analysis [C]//Pro-ceedings of SPIE, 2013, 8838: 88380 H. [9] Li Fazhi, Luo Xiao, Zhao Jingli, et al. Test of off-axis aspheric surfaces with CGH [J]. Optics and Precision Engineering, 2011, 19(4): 709-716. (in Chinese) doi: 10.3788/OPE.20111904.0709 [10] Li Shijie, Zhang Jin, Liu Weiguo, et al. Measurement investigation of an off-axis aspheric surface via a hybrid compensation method [J]. Appl Opt, 2018, 57(28): 8220-8227. doi: 10.1364/AO.57.008220 [11] Wang Ruoqiu, Zhang Zhiyu, Bai Yingying, et al. Scanning ion beam etching: A method for the fabrication of computer-generated hologram with nanometric accuracy for aspherical testing [J]. Optics and Lasers in Engineering, 2021, 139(22): 106503. [12] Bai Yingying, Zhang Zhiyu, Chen Tianbao, et al. Evaluation of wavefront aberrations induced by overlay errors in stitching computer-generated holograms [J]. Optics and Lasers in Engineering, 2022, 152: 106944. doi: 10.1016/j.optlaseng.2021.106944 [13] Zhu Zijue. Research on optical fiber interference sensing and demodulation method based on fringe contrast measurement [D]. Wuhan: Huazhong University of Science and Technology, 2021. (in Chinese) [14] Zhou Ping, Burge J H. Optimal design of computer-generated holograms to minimize sensitivity to fabrication errors [J]. Opt Express, 2007, 15(23): 15410-15417. doi: 10.1364/OE.15.015410