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检测大口径凸非球面中心时,由于中心非球面度较小,采用非零位检测的方法,用干涉仪直接进行测量,通过标准镜将干涉仪出射的平面波转化为球面波,近似沿法线方向入射到被检面上,光线近似原路返回;检测其边缘时,由于其偏离量较大,采用零位检测的方法,利用CGH补偿元件,将干涉仪出射的波前转换为与被检非球面的理论形状一致的非球面波,光线沿法线入射并原路返回,与参考波面形成干涉条纹。最后,利用综合优化的子孔径拼接算法进行 全口径拼接计算,即可获得大口径凸球面全口径的面形误差分布。检测流程如图1所示。
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在规划子孔径大小的过程中,要保证规划的子孔径能够对全面形口径实现覆盖,各相邻子孔径的重叠区域面积一般大于子孔径面积的1/4[13-14]。
在中心子孔径的测量过程中应选取适当的标准镜用干涉仪直接检测。选取标准镜时,应满足F#≥R#,f>R (F#=f/D,R#=R/d),其中F#为标准镜的F数,R#为中心子孔径的R数,f为标准镜焦距,D为标准镜口径,R为中心子孔径顶点的曲率半径,d为中心子孔径口径。根据光斑大小确定中心子孔径的大小。
而外圈子孔径的大小取决于CGH的设计,根据CGH主区域的大小以及其与被检镜之间的距离,通过光线追迹的方式得到被检镜上的光斑大小,从而确定外圈子孔径的大小;再根据子孔径规划的原理确定外圈子孔径的个数。
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干涉仪直接测量大口径凸非球面中心的示意图如图2所示,它由干涉仪、标准球面镜、待测大口径凸非球面以及调整机构组成。干涉仪对准被检镜中心,直接测量获得中心子孔径的面形相位。
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根据待测大口径凸非球面的参数设计外圈子孔径所需的CGH。CGH一般分为三个区域[15-16]:主区域即检测区域,用于检测被检镜的面形;对准区域用于干涉仪和CGH之间的对准;基准区域用于CGH和被检镜之间的对准。设计主区域时,应使检测光路沿原路返回,直到波像差最小,其衍射图样条纹密度应满足现有的CGH加工工艺制作条件。
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用CGH补偿法检测外圈的示意图如图3所示,由干涉仪、CGH、待检镜及调整装置组成。固定CGH的位置,通过旋转调整装置来实现对待检镜外圈不同子孔径的检测。
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测量中心子孔径时,由于是球面标准镜直接检测非球面,因此入射光线并不是沿法线入射到被检非球面,这时会引入相应的非共路误差。这部分误差是由非球面与最接近球面之间的偏差引起的,由二者法向相减后得到,并在拼接算法中去除。
在拼接外圈子孔径的过程中,各子孔径之间有存在着一定的重叠区域,通过其相位数据求解相邻子孔径之间的相对误差。由于外圈子孔径采用的是零位检测的方式,因此子孔径之间的相对失调量会引起平移、倾斜和离焦。在多个子孔径进行拼接的过程中,为了便于定位和测量,选定其中一个子孔径作为基准,该子孔径即为基准子孔径。假设基准子孔径的相位分布为
${w_0}$ ,则其他子孔径相位分布与其关系为:$$ \begin{split} & {w_0} = {w_1} + {p_1} + {a_1}{x_1} + {b_1}{y_1} + {c_1}(x_1^2 + y_1^2) = \\& {w_2} + {p_2} + {a_2}{x_2} + {b_2}{y_2} + {c_{_2}}(x_2^2 + y_2^2) \cdots =\\& {w_{M - 1}} + {p_{M - 1}} + {a_{M - 1}}{x_{M - 1}} + {b_{M - 1}}{y_{M - 1}}+ \\& {c_{_{M - 1}}}(x_{M - 1}^2 + y_{M - 1}^2) \end{split} $$ (1) 式中:
${a_i}$ 、${b_i}$ 、${c_i}$ 和$ {p_i} $ 分别为轴外子孔径相对基准子孔径沿x、y方向的倾斜系数、离焦系数和平移系数。再利用最小二乘拟合,使重叠区域相位差的平方和值最小[17]。可得:$$ \begin{split} S =& \sum\limits_{i = 1}^n {{{[{w_1} + {a_1}x + {b_1}y + {c_1}({x^2} + {y^2}) + {p_1} - {w_0}]}^2}} +\\& \sum\limits_{j = 1}^{M - 2} {\sum\limits_{i = 1}^n {\{ [{w_{j + 1}} + {a_{j + 1}}x + {b_{j + 1}}y + {c_{j + 1}}({x^2} + {y^2}) + {p_{j + 1}}]} } - \\& [{w_j} + {a_j}x + {b_j}y + {c_j}({x^2} + {y^2}) + {p_j}]{\} ^2} = \min \\[-10pt] \end{split} $$ (2) 利用最小二乘法对各个系数分别求偏导并令其值为零,可得:
$$ \left\{ {\begin{array}{*{20}{c}} {\dfrac{{\partial S}}{{\partial {a_j}}} = 0} \\ {\dfrac{{\partial S}}{{\partial {b_j}}} = 0} \\ {\dfrac{{\partial S}}{{\partial {c_j}}} = 0} \\ {\dfrac{{\partial S}}{{\partial {p_j}}} = 0} \end{array}} \right. $$ (3) 得到各子孔径相对基准子孔径的最佳拼接因子,从而消除调整误差,完成拼接[15] 。
Sub-aperture stiching and CGH mixed compensation for the testing of large convex asphere (invited)
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摘要: 为了实现大口径凸非球面的高精度检测,提出了将子孔径拼接检测法和计算全息补偿检测法相结合的检测方法。由于其中心的非球面度较小,采用球面波直接检测;而外圈的非球面度较大,采用子孔径拼接和计算全息混合补偿的方法进行测量,再通过拼接算法将中心检测数据和外圈检测数据进行拼接从而得到全口径面形。结合实例对一块口径为540 mm的大口径凸非球面进行测量,并将检测结果与Luphoscan 检测结果进行对比,两种方法检测面形残差的RMS值为0.019λ,自检验子孔径与拼接结果点对点相减后的RMS值为0.017λ。结果表明该方法能够实现大口径凸非球面的高精度检测。Abstract: In order to achieve high-precision testing of large convex asphere, a testing method combining sub-aperture stitching and computer generated hologram compensation is proposed. Because the asphericity of the center is small, the direct testing method of spherical wave is used; while the asphericity of the outer ring is large, the method of sub-aperture stitching and computer generated hologram (CGH) mixed compensation is used for measurement. Then, the center testing data and the outer ring testing data are stitched by the stitching algorithm to obtain the full-aperture surface shape. Combined with an example, a large convex asphere with a diameter of 540 mm is measured. The test results were compared with the Luphoscan testing results. The residual error of the two methods to test the RMS value of the surface is 0.019λ, and RMS value after subtracting the self-test aperture and stitching result point-to-point is 0.017λ. The results show that the method can achieve high-precision testing of large convex asphere.
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