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如图 1所示,当一束平行于光轴的平行光正入射到抛物面反射镜表面,抛物面反射镜会对平行光进行聚焦,形成一个聚焦点[15],由于制作工艺的限制,聚焦的平行光不会在该点处形成一个聚焦”点”,而是形成一个聚焦的光斑——弥散斑。通过在焦点处采集光斑图像,利用计算机可以计算出光斑大小,完成弥散斑的测量,通过移动工业相机,可以找到弥散斑半径最小的位置,完成对实际焦距的测量。
对于本项目检测的环形抛物面反射镜,抛物面的顶点不在反射镜底面上,在测量过程中无法直接测得焦距,如图 2所示。根据数量关系求得焦距,抛物面顶点距离反射镜底面距离l,焦平面距离反射镜底面为L,那么反射镜焦距f为:
$ f = L - 1 $
(1) Figure 2. Schematic diagram of the apex of the parabola and the bottom surface of the mirror not overlapping
在一些系统中,会用到光轴和机械轴存在夹角的抛物面反射镜[16],即反射镜的机械轴和光轴存在夹角,就需要对夹角大小进行测量,检测抛物面反射镜是否符合要求,如图 3所示。假设机械轴和光轴之间的夹角为θ,此时平行光的方向平行于机械轴,照射到抛物面反射镜上,平行光会聚在焦平面上,但是偏离光轴和机械轴,此时焦点和抛物面顶点所在的直线与机械轴之间的夹角,利用几何光学原理,可计算得到(2)式,即焦点和抛物面顶点所在的直线与机械轴之间的夹角和机械轴与光轴之间的夹角成两倍的关系。通过计算出焦点和抛物面顶点所在的直线与机械轴之间的夹角,反推出机械轴与光轴之间的夹角,实现对机械轴与光轴之间夹角的测量。
Figure 3. A schematic diagram of a parabolic mirror with an angle between the optical axis and the mechanical axis
$ \alpha = \theta + \theta = 2\theta $
(2) 如何求出焦点和抛物面顶点所在的直线与机械轴之间的夹角是系统设计的关键,本文中提出了一种方法,通过反射镜绕着机械轴旋转,如图 4所示。由于光轴和机械轴不重合,抛物面反射镜聚焦的光斑在机械轴外,通过转动反射镜,聚焦点会绕着机械轴旋转,形成一个以机械轴为圆心,聚焦光斑到机械轴的距离R为半径的圆[17]。
在焦平面出放置相机,随着反射镜的旋转,可以采集一系列的光斑图像,通过计算各光斑中心和光斑中心的空间位置,利用最小二乘法,拟合得到圆的轨迹并计算出半径R,从而求出聚焦光斑到机械轴的距离R,通过几何关系可以得到反射镜机械轴和光轴的夹角,在图 2中,平行于机械轴的平行光经过反射镜聚焦形成的焦点F,焦平面与机械轴的交F′,以及抛物面的顶点O,3个点形成一个三角形,其中FF′=R,OF=f,利用几何关系,可以求出:
$ \theta = \frac{R}{{2f}} $
(3) 式中,R为聚焦光斑到机械轴的距离,可以通过拟合圆求出,f为抛物面反射镜的焦距,在测量弥散斑的过程中可以求出。
在实际测量过程中,很难控制平行光和机械轴平行,因此平行光和机械轴不平行会引入较大的测量误差。本文中提出了一种消除误差的方法,即一个抛物面反射镜在完成测量之后,旋转180°放置在旋转台上再次进行测量,两个结果相加取平均值,减小误差。如图 5a所示,光轴与机械轴夹角为θ,平行光与机械轴夹角为γ,平行光方向和光轴在机械轴的同侧,此时反射镜对平行光聚焦的焦点落在图示“焦点方向”的直线上,焦点方向的直线与机械轴的夹角为(2θ-γ), 反射镜旋转形成的轨迹拟合得到的圆的半径R1为:
$ {R_1} = f \times \left( {2\theta - \gamma } \right) $
(4) 当反射镜旋转180°,如图 5b所示,光轴和平行光方向分列机械轴两侧,此时焦点方向的直线与机械轴的夹角为(2θ+γ), 反射镜自转形成的圆的半径R2为
$ {R_2} = f \times \left( {2\theta + \gamma } \right) $
(5) R1和R2取平均:
$ R = \frac{{{R_1} + {R_2}}}{2} = 2f\theta $
(6) 结合(3)式~(6)式,从而得到:
$ \theta = \frac{{{R_1} + {R_2}}}{{4f}} $
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采用一组标准的环形抛物面反射镜验证测量系统,这组标准抛物面反射镜规格一样,焦距均为41.875mm,光轴与机械轴之间的夹角18′50″,即0.314°。
在旋转台上放置标准的抛物面反射镜,移动移动台,采集焦点附近的一系列的光斑图像,如图 7所示。采集图像的位置距离抛物面顶点的距离为d,单位为mm。通过观察,可明显看出,在采集位置逐渐靠近焦点处时,光斑由环形逐渐聚焦为一个斑,越靠近焦点处,光斑半径越小,光斑越亮。
Figure 7. Spot images with different distances between the acquisition position and the vertex of the parabola
通过对图 7中形成的圆形光斑进行图像处理,求出光斑半径大小,对每个光斑的位置和光斑大小进行拟合,得到光斑最小的位置,即焦距,再将移动台移动到最小位置处,采集图像,得到焦点处弥散斑,如图 8中的图像数据,弥散斑半径最小的位置为41.860mm,即反射镜的焦距为41.860mm,此时得到焦距位置的弥散斑大小为0.363mm。
旋转旋转台,抛物面反射镜绕着机械轴以45°步进旋转,一周采集得到8张图像,将抛物面旋转180°,再次放置在旋转台上采集8张图像,如图 9所示。通过拟合光斑运动轨迹,得到半径R1=0.375mm和R2=0.532mm的两个圆,利用(6)式计算得到θ=0.311°。
通过实验测量其它的标准抛物面反射镜,测量数据如表 1所示。
Table 1. Measurement data and results
number focal length/mm diffuse spot radius/mm R1/mm R2/mm R/mm θ/(°) 1 41.860 0.181 0.375 0.532 0.454 0.311 2 41.868 0.176 0.395 0.506 0.451 0.309 3 41.865 0.178 0.380 0.575 0.468 0.320 4 41.885 0.180 0.347 0.560 0.490 0.335 5 41.882 0.176 0.383 0.609 0.449 0.307 6 41.873 0.181 0.302 0.596 0.442 0.302 7 41.879 0.178 0.398 0.509 0.454 0.311 theoretical 41.875 < 0.2 0.459 0.459 0.459 0.314 data maximum 0.015 0.031 0.021 error error rate/% 0.03 6.7 6.6 分析测量数据可知,该测量系统的焦距测量,相对误差为0.03%;测量的弥散斑半径大小也符合测量要求,抛物面反射镜光轴和机械轴的夹角相对误差为6.6%,认为测量结果正确,符合测量要求。
关于检测抛物面反射镜质量的技术研究
Technical research on testing the quality of parabolic mirror
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摘要: 为了对小尺寸抛物面反射镜进行质量检测, 采用平行光聚焦的方法测量焦距, 旋转被测反射镜的方法测量偏心角, 再结合图像处理的方法测量弥散斑大小, 并提出了一种减小测量误差的方法; 对这些测量方法进行了理论分析和实验验证, 得到了待测反射镜的焦距、偏心角和弥散斑大小。结果表明, 焦距的相对误差在0.1%以内, 偏心角误差在7%以内, 弥散斑大小小于0.2mm, 测量结果可靠。该方案设计为其它小口径的非球面的检测提供了很好的思路, 并且为减小测量误差、提高测量精度提供了很好的研究方法。Abstract: In order to inspect the quality of the small-size parabolic mirrors, the focal length was measured by the parallel light focusing method, the eccentric angle was measured by rotating the measured mirror.The size of the dispersion spot was measured by combing with the image processing method, and a reduction measurement was proposed.Theoretical analysis and experimental verification of these measurement methods were carried out.The focal length, eccentric angle and size of the diffuse spot of the mirror under test were obtained. The results show that the relative error of the focal length is within 0.1%, the error of the eccentric angle is within 7%, the size of the dispersion spot is less than 0.2mm, and the measurement result is reliable. The design of this scheme provides a good idea for the detection of other small-caliber aspheric surfaces, and provides a good research method for reducing measurement errors and improving measurement accuracy.
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Table 1. Measurement data and results
number focal length/mm diffuse spot radius/mm R1/mm R2/mm R/mm θ/(°) 1 41.860 0.181 0.375 0.532 0.454 0.311 2 41.868 0.176 0.395 0.506 0.451 0.309 3 41.865 0.178 0.380 0.575 0.468 0.320 4 41.885 0.180 0.347 0.560 0.490 0.335 5 41.882 0.176 0.383 0.609 0.449 0.307 6 41.873 0.181 0.302 0.596 0.442 0.302 7 41.879 0.178 0.398 0.509 0.454 0.311 theoretical 41.875 < 0.2 0.459 0.459 0.459 0.314 data maximum 0.015 0.031 0.021 error error rate/% 0.03 6.7 6.6 -
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