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随着离轴TMA系统口径不断增大,重力环境变化和光学元件支撑应力引起的面形误差中以像散与彗差面形误差最为常见,因此讨论反射镜面存在的面形误差,主要为离轴TMA系统三镜上的像散及彗差。对于离轴TMA系统,其次镜(SM)相对于主镜(PM)与三镜(TM)来说口径较小,因此,文中所考虑主要为光阑位置的主镜,以及非光阑位置三镜的面形误差。
当主镜存在面形误差时,入射光束充满光瞳位置,其归一化光瞳矢量与系统一致,如图1(a)所示;当反射镜面位于系统非光阑位置时,各视场入射光束轨迹在三镜上覆盖位置不同,且只覆盖三镜表面的一部分,非光阑位置镜面面形误差对系统出瞳波像差影响与系统视场具有相关性,如图1(b)所示。
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以入射光阑位于主镜的离轴TMA系统为例,当面形误差存在于光阑位置镜面时,所有视场发出的光束在镜面上的轨迹相同充满光瞳,因此主镜作为离轴三反系统的孔径光阑,其光瞳矢量
$ \stackrel{\rightharpoonup }{\;\rho } $ 与系统光瞳矢量$ \stackrel{\rightharpoonup }{\;\rho } $ 一致。考虑到光阑位置存在像散与彗差的情况,它们对应的Fringe Zernike系数分别为$ {C_5}、{C_6} $ (像散)和$ {C_7}、{C_8} $ (彗差),像散面形误差在极坐标系下的面形矢高可由下列Zernike多项式表示:$$ {E_{222}}(\rho ,\theta ) = C_5^{{\rm{PM}}}{\rho ^2}\cos (2\theta ) + C_6^{{\rm{PM}}}{\rho ^2}\sin (2\theta ) $$ (1) 式中:
$C_5^{{\rm{PM}}}$ 与$C_6^{{\rm{PM}}}$ 代表由主镜像散面形误差引起的系统出瞳波像差中第5、6项Fringe Zernike系数变化量。依据矢量乘法法则,公式(1)的矢量形式可写为:$$ \begin{split} {E}_{222}(\stackrel{\rightharpoonup }{\rho })=\left[\begin{array}{c}{C}_{5}^{{\rm{PM}}}\\ {C}_{6}^{{\rm{PM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2} \end{split}$$ (2) 则主镜像散面形误差对系统出瞳像差场贡献量可表示为:
$$ {\stackrel{\rightharpoonup }{C}}_{F222}=(n{'}-n){E}_{222}(\stackrel{\rightharpoonup }{\rho })=(n{'}-n)\left[\begin{array}{c}{C}_{5}^{{\rm{PM}}}\\ {C}_{6}^{{\rm{PM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2} $$ (3) 式中:
$ n' $ 与$ n $ 分别表示反射光线和入射光线所在空间的折射率。同上述像散面形误差推导过程,彗差面形误差矢高如下:$$ {E_{131}}(\rho ,\theta ) = C_7^{{\rm{PM}}}(3{\rho ^3} - 2\rho )\cos \theta + C_8^{{\rm{PM}}}(3{\rho ^3} - 2\rho )\sin \theta $$ (4) $$ {E}_{131}(\stackrel{\rightharpoonup }{\rho })=\left[\begin{array}{c}3{C}_{7}^{{\rm{PM}}}\\ 3{C}_{8}^{{\rm{PM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho }) $$ (5) 主镜彗差面形误差对系统出瞳波像差贡献量的矢量形式可写为:
$$ \begin{split} \begin{array}{l}{\stackrel{\rightharpoonup }{C}}_{F131}= (n{'}-n){E}_{222}(\stackrel{\rightharpoonup }{\rho })=\\ \qquad \quad \, (n{'}-n)\left[\begin{array}{l}3{C}_{7}^{{\rm{PM}}}\\ 3{C}_{8}^{{\rm{PM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho })\end{array} \end{split} $$ (6) 整理公式(3)与公式(6),可得到系统光阑位置像散、彗差面形误差对系统出瞳波像差影响,表示为如下形式:
$$ \left \{\begin{array}{l}{W}_{{\rm{figure}},{\rm{PM}}}^{{\rm{astigmatism}}}=(n{'}-n)\left[\begin{array}{c}{C}_{5}^{{\rm{PM}}}\\ {C}_{6}^{{\rm{PM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2} \\ {W}_{{\rm{figure}},{\rm{PM}}}^{{\rm{coma}}}=(n{'}-n)\left[\begin{array}{c}3{C}_{7}^{{\rm{PM}}}\\ 3{C}_{8}^{{\rm{PM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho }) \end{array}\right. $$ (7) 式中:
$W_{{\rm{figure}},{\rm{PM}}}^{{\rm{astigmatism}}}$ 表示主镜面形误差对系统出瞳像散场的影响;$W_{{\rm{figure}},{\rm{PM}}}^{{\rm{coma}}}$ 表示主镜面形误差对系统出瞳彗差场的影响。 -
由于次镜口径较小,面形加工精度较高,其面形误差对系统出瞳波像差影响忽略不计,文中非光阑位置镜面指离轴TMA系统中口径较大的三镜。当主镜存在面形误差时,入射光束充满光瞳位置,其归一化光瞳矢量与系统一致。但各视场入射光束轨迹在三镜上覆盖位置不同,且只覆盖三镜表面的一部分,如图1(a)所示,非光阑位置镜面面形误差对系统出瞳波像差影响与系统视场具有相关性。离轴三反系统非光阑位置光束轨迹孔径矢量与系统光瞳矢量间的变换关系如图1(b)所示[15-16],其解析表达式如下:
$$ {\stackrel{\rightharpoonup }{\rho }}_{{\rm{Not}}\_{\rm{stop}}}={\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\stackrel{\rightharpoonup }{l} $$ (8) 其中
$$ {\beta _{{\rm{TM}}}} = \frac{r}{R} $$ (9) $$ \stackrel{\rightharpoonup }{l}=\frac{\overline{y}}{R}\stackrel{\rightharpoonup }{H} $$ (10) 式中:
$\;{\beta _{{\rm{TM}}}}$ 表示非光阑位置镜面上光束轨迹相对于三镜镜面口径的缩放因子;$ \stackrel{\rightharpoonup }{l} $ 表示非光阑位置光束轨迹的离心矢量;r表示光束在非光阑面上的轨迹半径;R表示光学系统非光阑位置镜面口径半径;$ \overline y $ 表示视场近轴主光线在非光阑镜面上的高度。值得注意的是,此时的光束轨迹离心矢量不是固定值,其数值随视场变化,上述参数可通过光线追迹获得。公式(8)将非光阑位置光束轨迹与离轴系统光瞳联系起来,代入面形误差解析表达式中可得到非光阑位置面形误差对系统出瞳波像差的解析描述。将公式(8)代入像散面形误差对系统出瞳波像差贡献量表达式(公式(3))中得到:
$$ \begin{array}{c}\stackrel{\rightharpoonup }{C}{{'}}_{F222}=(n{'}-n)E{{'}}_{222}({\stackrel{\rightharpoonup }{\rho }}_{{\rm{Not}}\_{\rm{stop}}})=\\ (n{'}-n)E{{'}}_{222}({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H})=\\ (n{'}-n)\left[\begin{array}{c}{C}_{5}^{{\rm{TM}}}\\ {C}_{6}^{{\rm{TM}}}\end{array}\right]\cdot \left(\beta_{{\rm{T M}}} \vec{\rho}+\dfrac{\overline y}{R}\stackrel{\rightharpoonup } {H}\right)^2 =\\ (n{'}-n){\beta }_{{\rm{TM}}}^{2}\left[\begin{array}{c}{C}_{5}^{{\rm{TM}}}\\ {C}_{6}^{{\rm{TM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}+\Delta \end{array} $$ (11) 式中:
$C_5^{{\rm{TM}}}$ 与$C_6^{{\rm{TM}}}$ 表示由三镜像散面形误差引起的系统出瞳波像差第5、6项Fringe Zernike系数相对于设计值的变化量;$ \Delta $ 表示不影响像质的倾斜项与常量项。由公式(11)可见,依据矢量乘法法则展开后,${\left({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\right)}^{2}$ 展开出$ {\stackrel{\rightharpoonup }{\rho }}^{2} $ 项,即非光阑位置像散面形误差只影响系统出瞳波像差中的三阶像散。为得到非光阑位置彗差面形误差对系统出瞳波像差影响的解析表达式,将公式(8)代入彗差面形矢高表达式(公式(6))中,依据矢量乘法法则展开得到下式:
$$ \begin{array}{c}E{{'}}_{131}\left({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\right)= \left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot \left({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\right)\left[\left({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\right)\cdot \left({\beta }_{{\rm{TM}}}\stackrel{\rightharpoonup }{\rho }+\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\right)\right]=\\ \left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot \left[\stackrel{{\rm{coma}}}{\overbrace{{\beta }_{{\rm{TM}}}^{3}\stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho })}}+\stackrel{{\rm{curvature}}}{\overbrace{2{\beta }_{{\rm{TM}}}^{2}\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho })}}+\stackrel{{\rm{astigmatism}}}{\overbrace{2{\beta }_{{\rm{TM}}}^{2}\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}}}+2{\beta }_{{\rm{TM}}}\left({\dfrac{\overline{y}}{R}}\right)^{2}{\stackrel{\rightharpoonup }{H}}^{2}\cdot \stackrel{\rightharpoonup }{\rho }+\left({\dfrac{\overline{y}}{R}}\right)^{3}{\stackrel{\rightharpoonup }{H}}^{2} \cdot \stackrel{\rightharpoonup }{H}\right]=\\ {\beta }_{{\rm{TM}}}^{3}\left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho })+2{\beta }_{{\rm{TM}}}^{2}\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}+\Delta \end{array} $$ (12) 由公式(12)可见,依据矢量乘法法则展开彗差像差表达式后,不仅得到
$ \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho }) $ 项,同时也会展开出$ {\stackrel{\rightharpoonup }{\rho }}^{2} $ 项,即非光阑位置彗差面形误差不仅影响系统出瞳像差场中的三阶彗差,对系统三阶像散也有影响,且其对系统三阶像散场的影响与视场矢量呈线性相关。整理公式(11)与公式(12)可得到系统非光阑位置像散、彗差面形误差对系统出瞳波像差影响,表示为如下形式:
$$\left \{\begin{array}{l}{W}_{{\rm{figure}},{\rm{TM}}}^{{\rm{astigmatism}}}=(n{'}-n){\beta }_{{\rm{TM}}}^{2}\left[\begin{array}{c}{C}_{5}^{{\rm{TM}}}\\ {C}_{6}^{{\rm{TM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}+2(n{'}-n){\beta }_{{\rm{TM}}}^{2}\dfrac{\overline{y}}{R}\stackrel{\rightharpoonup }{H}\left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2} \\ {W}_{{\rm{figure}},{\rm{TM}}}^{{\rm{coma}}}=(n{'}-n){\beta }_{{\rm{TM}}}^{3}\left[\begin{array}{c}3{C}_{7}^{{\rm{TM}}}\\ 3{C}_{8}^{{\rm{TM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho }) \end{array}\right. $$ (13) 式中:
$W_{{\rm{figure,{\rm{TM}}}}}^{{\rm{astigmatism}}}$ 表示非光阑位置面形误差对系统出瞳像散场的贡献量;$W_{{\rm{figure,{\rm{TM}}}}}^{{\rm{coma}}}$ 表示非光阑位置面形误差对系统出瞳彗差场的贡献量。 -
文中补偿离轴反射系统的目的是提高系统成像质量,满足系统成像性能要求。因此,基于成像质量准则优化系统性能更为合理。上述内容定量化分析了元件位姿调整量与光学表面面形误差对系统出瞳波像差的影响,下面将对系统各位置镜面面形误差与元件位姿间的补偿关系展开深入的研究,构建像差补偿模型。
在系统像差补偿过程中,离轴三反系统出瞳波像差构成可以写为如下形式[12]:
$$ {W_{\rm{{off - axis}}}}{\text{ = }}{W_{{\rm{design}}}} + {W_{{\rm{alignment}}}} + {W_{{\rm{figure}}}} $$ (14) 式中:
${W_{{\rm{design}}}}$ 为光学系统设计时的固定残余像差;${W_{{\rm{alignment}}}}$ 表示由元件位姿调整所引入的像差;${W_{{\rm{figure}}}}$ 表示由反射镜面形误差而引入的像差。在不考虑系统设计产生的残余像差条件下,认为光学系统的像差主要由反射镜面形误差与系统元件位姿调整量产生。由此,为达到像差补偿的目的,将像差补偿模型目标函数写为如下形式:
$$ {W}_{{\rm{alignment}}}+{W}_{{\rm{figure}}}=\stackrel{\rightharpoonup }{0} $$ (15) 依据离轴系统矢量像差理论模型[17],可以得到离轴系统元件位姿调整,与系统出瞳像散场、彗差场的关系如下式所示:
$$ \left\{\begin{array}{l}{W}_{{\rm{alignment}}}^{{\rm{astigmatism}}}=\dfrac{1}{2}(-2\stackrel{\rightharpoonup }{H}{\stackrel{\rightharpoonup }{A}}_{222j}+{\stackrel{\rightharpoonup }{B}}_{222j}^{2})\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}-{\stackrel{\rightharpoonup }{A}}_{131j}\stackrel{\rightharpoonup }{L}\cdot {\stackrel{\rightharpoonup }{\rho }}^{2} \\ {W}_{{\rm{alignment}}}^{{\rm{coma}}}=-{\stackrel{\rightharpoonup }{A}}_{131j}\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho }) \end{array}\right. $$ (16) 式中:
$ {\stackrel{\rightharpoonup }{A}}_{222 j}={\displaystyle \sum _{j}{W}_{222 j}}\stackrel{\rightharpoonup }{{\sigma }_{j}} $ ,$ {\stackrel{\rightharpoonup }{A}}_{131 j}={\displaystyle \sum _{j}{W}_{131 j}}\stackrel{\rightharpoonup }{{\sigma }_{j}} $ ,$ {\stackrel{\rightharpoonup }{B}}_{222 j}^{2}= {\displaystyle \sum _{j}{W}_{222 j}}{\stackrel{\rightharpoonup }{{\sigma }_{j}}}^{2} $ ,分别为系统元件失调所引入的像散及彗差系数;$W_{{\rm{alignment}}}^{{\rm{astigmatism}}}$ 表示系统元件位姿调整对系统出瞳像散场的影响;$W_{{\rm{alignment}}}^{{\rm{coma}}}$ 表示系统元件位姿调整对系统出瞳彗差场的影响。联立公式(7)、公式(15)与公式(16),可列出元件位姿调整量补偿光阑位置面形误差方程组,考虑视场依赖性将其写为矩阵形式,如下所示:
$$ \begin{array}{c}\left[\begin{array}{cccc}\begin{array}{c}{H}_{x}\\ {H}_{y}\end{array}& \begin{array}{c}-{H}_{y}\\ {H}_{x}\end{array}& \begin{array}{c}-{1}/{2}\\ 0\end{array}& \begin{array}{c}0\\ -{1}/{2}\end{array}\end{array}\right]\left[\begin{array}{c}{A}_{222j,x}\\ {A}_{222j,y}\\ {B}_{222j,x}^{2}\\ {B}_{222j,y}^{2}\end{array}\right]+\left[\begin{array}{cc}0& -{L}_{y}\\ {L}_{y}& 0\end{array}\right]\left[\begin{array}{c}{A}_{131j,x}\\ {A}_{131j,y}\end{array}\right]=(n{'}-n)\left[\begin{array}{c}{C}_{5}^{{\rm{PM}}}\\ {C}_{6}^{{\rm{PM}}}\end{array}\right]\cdot {\stackrel{\rightharpoonup }{\rho }}^{2}\\ \left[\begin{array}{c}{A}_{131j,x}\\ {A}_{131j,y}\end{array}\right]=(n{'}-n)\left[\begin{array}{c}3{C}_{7}^{{\rm{PM}}}\\ 3{C}_{8}^{{\rm{PM}}}\end{array}\right]\cdot \stackrel{\rightharpoonup }{\rho }(\stackrel{\rightharpoonup }{\rho }\cdot \stackrel{\rightharpoonup }{\rho })\end{array} $$ (17) 联立公式(13)、公式(15)与公式(16),可列出元件位姿调整量补偿非光阑位置面形误差方程组,如下所示:
$$ \begin{split} & \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{H_x}} \\ {{H_y}} \end{array}}&{\begin{array}{*{20}{c}} { - {H_y}} \\ {{H_x}} \end{array}}&{\begin{array}{*{20}{c}} { - {1}/{2}} \\ 0 \end{array}}&{\begin{array}{*{20}{c}} 0 \\ { - {1}/{2}} \end{array}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{A_{222j,x}}} \\ {{A_{222j,y}}} \\ {B_{222j,x}^2} \\ {B_{222j,y}^2} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&{ - {L_y}} \\ {{L_y}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{A_{131j,x}}} \\ {{A_{131j,y}}} \end{array}} \right]= \\ & (n' - n)\beta _{{\rm{TM}}}^2\left[ {\begin{array}{*{20}{c}} {C_5^{{\rm{TM}}}} \\ {C_6^{{\rm{TM}}}} \end{array}} \right] + 2\beta _{{\rm{TM}}}^2\dfrac{{\overline y }}{R}\left[ {\begin{array}{*{20}{c}} {{H_x}}&{ - {H_y}} \\ {{H_y}}&{{H_x}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {3C_7^{{\rm{TM}}}} \\ {3C_8^{{\rm{TM}}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{A_{131j,x}}} \\ {{A_{131j,y}}} \end{array}} \right] = (n' - n)\beta _{{\rm{TM}}}^3\left[ {\begin{array}{*{20}{c}} {3C_7^{{\rm{TM}}}} \\ {3C_8^{{\rm{TM}}}} \end{array}} \right] \end{split} $$ (18) 由上述方程组可见,未知量个数为六个,当视场取值点个数大于等于2时,即可通过公式(18)所构成的超定方程组利用最小二乘法对其进行求解[18]。经上述方程组求出未知量
$ {\stackrel{\rightharpoonup }{A}}_{222 j} $ 与$ {\stackrel{\rightharpoonup }{A}}_{131 j} $ 的x方向与y方向的分量,代入公式(19)与公式(20)中确定光学系统中元件的位姿补偿量。需要补充的是,TMA系统中,次镜相对于主镜与三镜口径较小,调整次镜位姿位置较为灵活,工程实现代价最小,因此假设系统中只有次镜配备了调节机构。$$ \left\{\begin{array}{c}{\stackrel{\rightharpoonup }{A}}_{klm,x}^{}={W}_{klm,{\rm{SM}}}^{{\rm{sph}}}{\stackrel{\rightharpoonup }{\sigma }}_{x,{\rm{SM}}}^{{\rm{sph}}}+{W}_{klm,{\rm{SM}}}^{{\rm{asph}}}{\stackrel{\rightharpoonup }{\sigma }}_{x,{\rm{SM}}}^{{\rm{asph}}}\\ {\stackrel{\rightharpoonup }{A}}_{klm,y}^{}={W}_{klm,{\rm{SM}}}^{{\rm{sph}}}{\stackrel{\rightharpoonup }{\sigma }}_{y,{\rm{SM}}}^{{\rm{sph}}}+{W}_{klm,{\rm{SM}}}^{{\rm{asph}}}{\stackrel{\rightharpoonup }{\sigma }}_{y,{\rm{SM}}}^{{\rm{asph}}}\end{array}\right. $$ (19) 式中:
$W_{klm,{\rm{SM}}}^{{\rm{sph}}}$ 、$W_{klm,{\rm{SM}}}^{{\rm{asph}}}$ 为次镜SM的像差系数,对光学系统来说其值为常数;${\stackrel{\rightharpoonup }{\sigma }}_{x,{\rm{SM}}}^{{\rm{sph}}}$ 、${\stackrel{\rightharpoonup }{\sigma }}_{x,{\rm{SM}}}^{{\rm{asph}}}$ 、${\stackrel{\rightharpoonup }{\sigma }}_{y,{\rm{SM}}}^{{\rm{sph}}}$ 、${\stackrel{\rightharpoonup }{\sigma }}_{y,{\rm{SM}}}^{{\rm{asph}}}$ 分别为像差场偏移矢量在x方向与y方向上的分量,其上角标sph与asph分别表示次镜基本球面部分与基本非球面组成部分。通过求解公式(19)可以得到次镜基本球面部分和非球面偏离部分的像差场偏移矢量,代入公式(20)中求解次镜位姿补偿量。$$ \left\{ {\begin{array}{*{20}{l}} {XD{E_{{\rm{SM}}}} = - {u_{pm}}{d_1}\sigma _{{\rm{SM}},x}^{{\rm{asph}}}} \\ {YD{E_{{\rm{SM}}}} = - {u_{pm}}{d_1}\sigma _{{\rm{SM}},y}^{{\rm{asph}}}} \\ {AD{E_{{\rm{SM}}}} = {u_{pm}}(1 + {c_{{\rm{SM}}}}{d_1})\sigma _{{\rm{SM}},y}^{{\rm{sph}}} - {c_{{\rm{SM}}}}YD{E_{{\rm{SM}}}}} \\ {BD{E_{{\rm{SM}}}} = - {u_{pm}}(1 + {c_{{\rm{SM}}}}{d_1})\sigma _{{\rm{SM}},x}^{{\rm{sph}}} + {c_{{\rm{SM}}}}XD{E_{{\rm{SM}}}}} \end{array}} \right. $$ (20) 式中:
$XD{E_{{\rm{SM}}}}$ 、$YD{E_{{\rm{SM}}}}$ 分别表示次镜沿x方向与y方向的偏心位姿补偿量;$AD{E_{{\rm{SM}}}}$ 、$BD{E_{{\rm{SM}}}}$ 分别代表次镜绕坐标轴x与坐标轴y的倾斜位姿补偿量。如上,用于补偿非光阑位置面形误差的次镜位姿补偿量求解完成。 -
为验证上述非光阑位置像差补偿校正模型的有效性,以口径500 mm、焦距6 m、F数为10、光阑为其主镜、视场大小为1°×1°的宽视场离轴TMA系统作为研究对象进行仿真实验,孔径光阑沿y方向偏移−460 mm,其光学结构参数及像差系数如表1和表2所示。为保证补偿结果准确并根据离轴TMA系统视场离轴特点选定五个视场,其中包括四个边缘视场和一个中心视场,五个视场点的坐标为(−0.35°, 0°)、(0.35°, 0°)、(0°, −0.5°)、(−0.35°, −0.8°)、(0.35°, −0.8°),如图2所示。
表 1 离轴TMA望远镜的光学参数
Table 1. Optical parameters of off-axis TMA telescope
Surface Conic constant Radius/mm Thickness/mm PM −0.921 −3600.41 −1551.777 SM −4.828 −910.903 1558.7 TM −0.292 −1219.431 −1533.359 Image − Infinity 表 2 离轴TMA望远镜的次镜像差系数(λ=632.8 nm)
Table 2. Wave aberration coefficients of SM for the off-axis TMA telescope (λ=632.8 nm)
Surface $W_{222,{\rm{SM}}}^{{\rm{sph}}}$/λ $W_{222,{\rm{SM}}}^{{\rm{asph}}}$/λ $W_{131,{\rm{SM}}}^{{\rm{sph}}}$/λ $W_{131,{\rm{SM}}}^{{\rm{asph}}}$/λ SM −28.28 54.21 196.86 224.59 -
为验证文中像差补偿模型的正确性,随机在主镜表面引入一组像散及彗差像差系数,取值如表3所示。将补偿后系统出瞳波像差RMS值作为像差补偿模型是否有效的评价标准。表4给出次镜补偿主镜面形误差的位姿调整量,其中面形误差取值范围参考文献[14],表5给出经次镜补偿前后系统出瞳波像差RMS值的变化。
表 3 离轴TMA系统引入主镜面形误差Zernike系数
Table 3. Introduced Zernike coefficients for figure error on PM of the off-axis TMA system
$C_5^{{\rm{PM}}}/\lambda$ $C_6^{{\rm{PM}}}/\lambda$ $C_7^{{\rm{PM}}}/\lambda$ $C_8^{{\rm{PM}}}/\lambda$ 0.5 0.6 0.4 0.5 表 4 补偿离轴TMA系统主镜面形误差次镜调整量
Table 4. The adjustment of SM in compensating PM figure error
$XD{E_{{\rm{SM}}} }/{\rm{mm}}$ $YD{E_{{\rm{SM}}} }/{\rm{mm}}$ $AD{E_{ {\rm{SM} } } }/ (^\circ)$ $BD{E_{ {\rm{SM} } } }/ (^\circ)$ −0.0584 −0.061 −0.0074 0.0077 表 5 补偿面形误差前后系统出瞳RMS值
Table 5. RMS value of system before and after compensation of surface figure error
RMS/λ F1 F2 F3 F4 F5 Before compensation 0.1864 0.1772 0.1624 0.1682 0.1649 After compensation 0.0851 0.0783 0.0812 0.0642 0.0816 由表5可以看出,经次镜位姿补偿后,光阑位置存在像散与彗差的系统出瞳RMS值有效下降至系统设计值。相应状态下,离轴TMA系统补偿前后系统像散场、彗差场及出瞳波像差分布状态的全视场图如图3、图4和图5所示,系统像散场与彗差场经补偿后均靠近系统设计状态。
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随机在三镜表面引入一组像散及彗差像差系数,取值如表6所示,由像差补偿模型计算所得次镜相应调整量如表7所示。
表 6 离轴TMA系统引入三镜面形误差Zernike系数
Table 6. Introduced Zernike coefficients for figure error on TM of the off-axis TMA system
$C_5^{{\rm{TM}}}/\lambda$ $C_6^{{\rm{TM}}}/\lambda$ $C_7^{{\rm{TM}}}/\lambda$ $C_8^{{\rm{TM}}}/\lambda$ 0.1 −0.08 0.05 0.05 表 7 补偿离轴TMA系统三镜面形误差次镜调整量
Table 7. The adjustment of SM in compensating TM figure error of the off-axis TMA system
$XD{E_{{\rm{SM}}} }/{\rm{mm} }$ $YD{E_{{\rm{SM}}} }/{\rm{mm} }$ $AD{E_{ {\rm{SM} } } }/(^\circ)$ $BD{E_{ {\rm{SM} } } }/ (^\circ)$ −0.083 −0.046 0.035 −0.067 表 8 补偿面形误差前后系统出瞳RMS值
Table 8. RMS value of system before and after compensation of surface figure error
RMS/λ F1 F2 F3 F4 F5 Before compensation 0.3175 0.3561 0.2749 0.2344 0.2795 After compensation 0.1092 0.1041 0.0940 0.1388 0.1038 由表8可以看出,经次镜补偿后,三镜带有面形误差的系统出瞳RMS值补偿至设计值。图6、图7和图8为相应状态下,补偿前后系统像散场、彗差场及系统出瞳波像差分布状态的全视场图。
如图6、图7和图8所示,系统像散场与彗差场以及出瞳波像差经补偿后均靠近系统设计状态。由以上实验结果可以看出,在数量级接近的情况下,三镜面形误差对系统出瞳影响比主镜更大,且由图5(a)与图8(a)可以看出,当三镜与主镜存在同样类型的面形误差时,其对系统出瞳像差场分布影响不同,三镜面形误差对系统出瞳像差场分布的影响与视场相关。为验证像差补偿模型的有效性,进行蒙特卡洛仿真实验。选取如表9所示的面形误差范围作为扰动量区间,在扰动区间内,随机选取100组不同的扰动量作为三镜面形误差值。系统经补偿后,五个视场出瞳波像差RMS值如图9所示。可以看出,经像差补偿模型所得的次镜位姿调整量对非光阑位置镜面存在面形误差的系统进行补偿之后,各视场的出瞳波像差RMS值均得到了有效校正。
由上述实验结果及全视场图可知,像差补偿模型的补偿能力出现不能完全补偿面形误差对系统出瞳波像差影响的情况。出现上述补偿能力有限情况的原因可能是:(1)面形误差所引入的高阶像差未考虑至像差补偿模型中,系统高阶像差场未得到完全补偿;(2)次镜的补偿能力有限,仅依靠次镜的补偿维度难以完全补偿三镜面形误差对系统出瞳波像差的影响。考虑增加系统位姿补偿维度,在三镜上安装位姿调整装置。
表 9 三镜面形误差x/y向像散与x/y向彗差系数阈值列表
Table 9. The range of x/y astigmatism and x/y coma coefficient of TM figure error
$C_5^{{\rm{TM}}}/\lambda$ $C_6^{{\rm{TM}}}/\lambda$ $C_7^{{\rm{TM}}}/\lambda$ $C_8^{{\rm{TM}}}/\lambda$ ±0.03 ±0.03 ±0.03 ±0.03 图 9 五个视场补偿前后及标称状态下系统的出瞳波像差RMS值。(a)F1视场;(b) F2视场;(c) F3视场;(d) F4视场;(e)F5视场。蓝色点表示三镜存在面形误差时的系统出瞳波像差RMS值,粉色点表示补偿后的系统出瞳波像差RMS值,绿色点表示标称状态下系统出瞳波像差RMS值
Figure 9. The values of RMS before and after compensation of the five fields of view and the nominal state. (a) F1; (b) F2; (c) F3; (d) F4; (e) F5. Blue dots indicate the value of RMS when the TM surface figure error exists, pink dots indicate the value of RMS after compensation, and green dots indicate the RMS of the system in the nominal state
Compensation mechanism of primary mirror and the third mirror figure error of off-axis three-mirror telescope
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摘要: 为保证大口径离轴三反消像散(Three-Mirror Anastigmat,TMA)光学系统在轨成像质量,探明离轴TMA系统中次镜位姿与主镜及三镜面形误差补偿机理,以矢量像差理论为基础,用Zernike多项式表述离轴TMA系统镜面面形误差,并对系统镜面面形误差进行解析。通过分析发现,位于非光阑位置三阶彗差经光瞳坐标变换衍生出与视场线性相关像散;提出结合失调离轴系统矢量像差校正解析式,以系统出瞳波像差RMS值为评价标准,构建离轴TMA系统像差补偿模型,利用次镜位姿对主镜及三镜存在面形误差的离轴TMA系统进行补偿。仿真实验表明:系统主镜存在0.5λ像散与彗差时,所构建像差补偿模型可将系统出瞳波像差由0.18λ补偿至0.08λ;系统三镜存在0.05λ像散与彗差时,可将出瞳波像差由0.3λ补偿至0.1λ,且当三镜面形误差在(−0.03λ,0.03λ)范围内时,可将系统各视场RMS值补偿至系统设计值,使系统成像质量满足要求,为大口径反射式空间望远镜在轨主动装调提供进一步理论指导。Abstract:
Objective When the large aperture off-axis three-mirror anastigmat (TMA) is launched to space, surface degradation appear on the the optical surface of its components due to gravity unloading, which will affect the imaging quality of the system. In order to ensure the imaging quality of the large aperture space reflecting telescope in orbit, it is necessary to explore the surface figure error compensation mechanism of the position of optical elements. Then the compensation mechanism of the secondary mirror position for the primary mirror and the third-mirror shape of the off-axis TMA system was investigated. So that the space telescope can actively use the element pose adjustment to compensate the impact of surface figure degradation on the imaging quality of the system. Methods In order to analyze the progressive compensation mechanism of the surface figure error, the compensation mechanism and compensation amount are defined and calculated based on the nodal aberration theory(NAT). Firstly, the Zernike polynomial vector form is used to describe the surface figure error of the off-axis TMA system based on the vector multiplication rule, and its derived aberration distribution is analyzed. Different from the aberration characteristics of the position of the primary mirror at the stop, the third mirror in the non-stop position of each field of view on the surface of the beam trajectory is different (Fig.1). Therefore, when compensating for the surface figure error of the third mirror, the situation of each field of view is different. This is also the focus of the investigation on the analysis and discussion of using pupil position transformation and least square method to solve the problem. Then a vector aberration correction model is proposed and an aberration compensation model of off-axis TMA system is constructed. In order to objectively evaluate the imaging quality of the imaging system, the exit pupil wave aberration RMS value is taken as the evaluation standard, and the secondary mirror adjustment with small aperture and the highest sensitivity in the TMA system is used to compensate the system exit pupil wave aberration with surface figure errors in the primary mirror and the third mirror. Results and Discussions Simulation experiments show that when the primary mirror of the system has 0.5λ astigmatism and coma, the constructed aberration compensation model can compensate the exit pupil wave aberration RMS value from 0.18λ to 0.08λ (Tab.5). When 0.05λ astigmatism and coma exist on the system's third mirror, the exit pupil wave aberration RMS value can be compensated from 0.3λ to 0.1λ (Tab.8). In order to verify the applicability of the aberration compensation model, Monte Carlo experiment was carried out, which proved that when the third-mirror figure error (astigmatism and coma) was within the range of (−0.03λ, 0.03λ), the RMS value of each field of view of the system could be compensated to the design value of the system (Fig.9). Conclusions A portable surface figure error compensation model of the TMA system is designed. It can compensate the RMS value of the TMA system with 0.5λ in the primary mirror and 0.05λ in the third mirror respectively to the nominal state. Through analysis, it is found that the third-order coma in the non-stop position is derived from the linear correlation astigmatism with the field of view by optical symmetry coordinate transformation. The astigmatism and coma distribution rules can be verified during the analysis of the surface error of each position of the system, which provides a theoretical reference and basis for other types of aberrations and further theoretical guidance for the active in-orbit installation of large aperture reflecting space telescopes. It provides the basic theory and framework for constructing the surface figure error compensation model of the primary mirror and the third-mirror of off-axis TMA system. -
图 9 五个视场补偿前后及标称状态下系统的出瞳波像差RMS值。(a)F1视场;(b) F2视场;(c) F3视场;(d) F4视场;(e)F5视场。蓝色点表示三镜存在面形误差时的系统出瞳波像差RMS值,粉色点表示补偿后的系统出瞳波像差RMS值,绿色点表示标称状态下系统出瞳波像差RMS值
Figure 9. The values of RMS before and after compensation of the five fields of view and the nominal state. (a) F1; (b) F2; (c) F3; (d) F4; (e) F5. Blue dots indicate the value of RMS when the TM surface figure error exists, pink dots indicate the value of RMS after compensation, and green dots indicate the RMS of the system in the nominal state
表 1 离轴TMA望远镜的光学参数
Table 1. Optical parameters of off-axis TMA telescope
Surface Conic constant Radius/mm Thickness/mm PM −0.921 −3600.41 −1551.777 SM −4.828 −910.903 1558.7 TM −0.292 −1219.431 −1533.359 Image − Infinity 表 2 离轴TMA望远镜的次镜像差系数(λ=632.8 nm)
Table 2. Wave aberration coefficients of SM for the off-axis TMA telescope (λ=632.8 nm)
Surface $W_{222,{\rm{SM}}}^{{\rm{sph}}}$ /λ$W_{222,{\rm{SM}}}^{{\rm{asph}}}$ /λ$W_{131,{\rm{SM}}}^{{\rm{sph}}}$ /λ$W_{131,{\rm{SM}}}^{{\rm{asph}}}$ /λSM −28.28 54.21 196.86 224.59 表 3 离轴TMA系统引入主镜面形误差Zernike系数
Table 3. Introduced Zernike coefficients for figure error on PM of the off-axis TMA system
$C_5^{{\rm{PM}}}/\lambda$ $C_6^{{\rm{PM}}}/\lambda$ $C_7^{{\rm{PM}}}/\lambda$ $C_8^{{\rm{PM}}}/\lambda$ 0.5 0.6 0.4 0.5 表 4 补偿离轴TMA系统主镜面形误差次镜调整量
Table 4. The adjustment of SM in compensating PM figure error
$XD{E_{{\rm{SM}}} }/{\rm{mm}}$ $YD{E_{{\rm{SM}}} }/{\rm{mm}}$ $AD{E_{ {\rm{SM} } } }/ (^\circ)$ $BD{E_{ {\rm{SM} } } }/ (^\circ)$ −0.0584 −0.061 −0.0074 0.0077 表 5 补偿面形误差前后系统出瞳RMS值
Table 5. RMS value of system before and after compensation of surface figure error
RMS/λ F1 F2 F3 F4 F5 Before compensation 0.1864 0.1772 0.1624 0.1682 0.1649 After compensation 0.0851 0.0783 0.0812 0.0642 0.0816 表 6 离轴TMA系统引入三镜面形误差Zernike系数
Table 6. Introduced Zernike coefficients for figure error on TM of the off-axis TMA system
$C_5^{{\rm{TM}}}/\lambda$ $C_6^{{\rm{TM}}}/\lambda$ $C_7^{{\rm{TM}}}/\lambda$ $C_8^{{\rm{TM}}}/\lambda$ 0.1 −0.08 0.05 0.05 表 7 补偿离轴TMA系统三镜面形误差次镜调整量
Table 7. The adjustment of SM in compensating TM figure error of the off-axis TMA system
$XD{E_{{\rm{SM}}} }/{\rm{mm} }$ $YD{E_{{\rm{SM}}} }/{\rm{mm} }$ $AD{E_{ {\rm{SM} } } }/(^\circ)$ $BD{E_{ {\rm{SM} } } }/ (^\circ)$ −0.083 −0.046 0.035 −0.067 表 8 补偿面形误差前后系统出瞳RMS值
Table 8. RMS value of system before and after compensation of surface figure error
RMS/λ F1 F2 F3 F4 F5 Before compensation 0.3175 0.3561 0.2749 0.2344 0.2795 After compensation 0.1092 0.1041 0.0940 0.1388 0.1038 表 9 三镜面形误差x/y向像散与x/y向彗差系数阈值列表
Table 9. The range of x/y astigmatism and x/y coma coefficient of TM figure error
$C_5^{{\rm{TM}}}/\lambda$ $C_6^{{\rm{TM}}}/\lambda$ $C_7^{{\rm{TM}}}/\lambda$ $C_8^{{\rm{TM}}}/\lambda$ ±0.03 ±0.03 ±0.03 ±0.03 -
[1] Zhang Liang, An Yuan, Jin Guang. Optical design of the uncoaxial three mirror system with wide field of view and long focal length [J]. Infrared and Laser Engineering, 2007, 36(2): 278-280. (in Chinese) doi: 10.3969/j.issn.1007-2276.2007.02.034 [2] Song Yanfeng, Shao Xiaopeng, Xu Jun. Off-axis three-mirror reflective optical system [J]. Infrared and Laser Engineering, 2008, 37(4): 706-709. (in Chinese) [3] Li Huan, Xiang Yang. Optical design of off-axis three -mirror telescope systems of imaging spectrometers [J]. Infrared and Laser Engineering, 2009, 38(3): 500-504. (in Chinese) [4] Shack R V, Thompson K. Influence of alignment errors of a telescope system on its aberration field [C]//Proceedings of SPIE, 1980, 251: 146-153. [5] Thompson K P, Schmid T, Cakmakci O. Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry [J]. Journal of the Optical Society of America A, 2009, 26(6): 1503-1517. doi: 10.1364/JOSAA.26.001503 [6] Thompson K P. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: The comatic aberrations [J]. Journal of the Optical Society of America A, 2010, 27(6): 1490-1504. doi: 10.1364/JOSAA.27.001490 [7] Thompson K P. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: The astigmatic aberrations [J]. Journal of the Optical Society of America A, 2011, 28(5): 821-836. doi: 10.1364/JOSAA.28.000821 [8] Thompson K P. The astigmatic aberration field in active primary mirror astronomical telescopes [J]. Adv Opt Technol, 2013, 29(2): 89-95. [9] Gray R W, Rolland J P. Wavefront aberration function in terms of R. V. Shack’s vector product and Zernike polynomial vectors [J]. Applied Optics, 2015, 32(10): 1836-1847. [10] Schmid T, Rolland J P, Rakich A, et al. Separation of the effects of astigmatic figure error from misalignments using nodal aberration theory (NAT) [J]. Optics Express, 2010, 18(3): 17433-17447. [11] Fuerschbach K, Rolland J P, Thompson K P, et al. Theory of aberration fields for general optical systems with freeform surfaces [J]. Optics Express, 2014, 22(22): 26585-26606. doi: 10.1364/OE.22.026585 [12] Zhang X B, Xu S Y, Ma H C, et al. Compensation for the perturbed three mirror anastigmatic telescope based on nodal aberration theory [J]. Optics Express, 2017, 25(11): 12867-12883. [13] Pang Zhihai, Fan Xuewu, Chen Qinfang, et al. Influence of surface-profile error of larger mirror on aberrations characteristics of optical system [J]. Acta Optica Sinica, 2013, 33(4): 0422022. (in Chinese) [14] Ju G H, Yan C X, Gu Z Y, et al. Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory [J]. Applied Optics, 2016, 55(13): 3373-3385. doi: 10.1364/AO.55.003373 [15] Fuerschbach K, Rolland J P, Thompson K P, et al. Extending nodal aberration theory to include mount-induced aberrations with application to freeform surfaces [J]. Optics Express, 2012, 20(18): 20139-20155. doi: 10.1364/OE.20.020139 [16] Dai Yijie, Liu Yingli, Shen Fanqi, et al. Calculation of aberration fields for freeform imaging systems using field-dependent footprints on local tangent planes [J]. Applied Optics, 2022, 61(32): 9576-9582. [17] Wang J X, He X, Luo J, et al. Alignment algorithm of nonsymmetric off-axis reflective astronomical telescopes based on the modified third-order nodal aberration theory [J]. Optics Express, 2022, 30(8): 13159-13183. doi: 10.1364/OE.453005 [18] Wang J X, He X, Zhang X H, et al. Misalignment algorithm of a wide-field survey telescope based on third-order quadratic nodal aberration theory [J]. Journal of Astronomical Telescopes, Instruments, and Systems, 2021, 7(4): 049003.