# 大口径望远镜消旋K镜旋转策略及准直技术研究Rotating Strategy and Alignment Technology of K Mirror in Large Aperture Telescope

DOI: 10.12677/OE.2019.93019, PDF, HTML, XML, 下载: 419  浏览: 906  科研立项经费支持

Abstract: Aiming at the image field rotation of 2 m Alt-az telescope, the K mirror is designed. First, according to optical transfer characteristic of K mirror system, the mathematical matrix model is calculated. The second, based on the method of ray tracing, the transformation relation between incident light coordinate and coude focus coordinate is fitted. The image field rotation of incident light through the light path of coude is analyzed, and the conversion regularity of the image field rotation involving the azimuth axis and pitching axis of telescope is deduced, thus the initial zero and rotation rules of K mirror are obtained. Then, in allusion to the alignment accuracy of K mirror, through the analysis of the periodic variation characteristics of alignment error in the rotation period of K mirror, the alignment adjustment strategy of K mirror is designed basing on the theory of Pascal Spiral, the complex problem of alignment adjustable is transformed into the parameter correction of mathematical equations, and the calculation model of the alignment accuracy and the key parameters in Pascal Spiral equation is established. Finally, the alignment adjustable experiment is carried out. The key parameters in the model are modified repeatedly during the experiment. The alignment error curve keeps getting closer and closer to the optimal value. The correctness and effectiveness of the research is verified from the experimental results.

1. 引言

(a) K mirror in JCMT (b) k mirror in Hooker (c) k mirror in LBT

Figure 1. The K mirror in large aperture telescope

2. K镜光学消旋数学模型

${R}_{k}=I-2{N}_{k}\cdot {N}_{k}^{\text{T}}$ (1)

$\begin{array}{l}{N}_{k1}=\left[-\mathrm{cos}\alpha ,0,-\mathrm{sin}\alpha \right]\\ {N}_{k2}=\left[\mathrm{cos}\beta ,0,\mathrm{sin}\beta \right]\\ {N}_{k3}=\left[-\mathrm{cos}\gamma ,0,\mathrm{sin}\gamma \right]\end{array}$ (2)

${A}^{\prime }={R}_{k3}{R}_{k2}{R}_{k1}A$ (3)

$A=\left[\begin{array}{ccc}\mathrm{cos}\theta & -\mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{x}\\ {A}_{y}\\ {A}_{z}\end{array}\right]$ (4)

K镜中，K1、K2、K3反射镜作用矩阵则相应地变为：

$\begin{array}{l}{N}_{k1}=\left[\begin{array}{ccc}\mathrm{cos}\theta /2& -\mathrm{sin}\theta /2& 0\\ \mathrm{sin}\theta /2& \mathrm{cos}\theta /2& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}-\mathrm{cos}\alpha \\ 0\\ -\mathrm{sin}\alpha \end{array}\right]\\ {N}_{k2}=\left[\begin{array}{ccc}\mathrm{cos}\theta /2& -\mathrm{sin}\theta /2& 0\\ \mathrm{sin}\theta /2& \mathrm{cos}\theta /2& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\mathrm{cos}\beta \\ 0\\ \mathrm{sin}\beta \end{array}\right]\\ {N}_{k3}=\left[\begin{array}{ccc}\mathrm{cos}\theta /2& -\mathrm{sin}\theta /2& 0\\ \mathrm{sin}\theta /2& \mathrm{cos}\theta /2& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}-\mathrm{cos}\gamma \\ 0\\ \mathrm{sin}\gamma \end{array}\right]\end{array}$ (5)

K镜系统内部光学元件的位置关系一般满足一下条件：

$\alpha =\gamma ,\beta =0$ (6)

${A}^{\prime }={R}_{k3}{R}_{k2}{R}_{k1}A=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{x}\\ {A}_{y}\\ {A}_{z}\end{array}\right]$ (7)

Figure 2. Light path of K mirror

${R}_{k3}{R}_{k2}{R}_{k1}=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ (8)

$\left[\begin{array}{c}1-2{\mathrm{cos}}^{2}\gamma \\ 1\\ 1-2{\mathrm{sin}}^{2}\gamma \end{array}\right]\left[\begin{array}{c}1-2{\mathrm{cos}}^{2}\beta \\ 1\\ 1-2{\mathrm{sin}}^{2}\beta \end{array}\right]\left[\begin{array}{c}1-2{\mathrm{cos}}^{2}\alpha \\ 1\\ 1-2{\mathrm{sin}}^{2}\alpha \end{array}\right]=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ (9)

$±\left(\alpha -\gamma \right)=\beta$ (10)

3. K镜消旋策略研究

Figure 3. Light path of 2m telescope

2 m望远镜采用地平坐标，A为方位角，从正北向东计量，范围0˚~360˚，对应于方位轴的转动。Z为天顶距，从天顶向前下方计量，范围为0˚~90˚，对应于俯仰轴的转动。定义望远镜主光路入瞳处坐标系为o-xyz，坐标系中角度的方向，迎着光线(逆光)观察逆时针方向为正方向。从坐标系o-xyz中分别发出四条光线x+、x−、y+、y−，沿主光路方向最终汇聚至coude像面，光线坐标系o-xyz经过M3、M4、M5、M6、M7镜之后分别形成局部坐标系o3-x3y3z3、o4-x4y4z4、o5-x5y5z5、o6-x6y6z6、o7-x7y7z7。如图4所示。

Figure 4. COUDE path of 2 m telescope

M3镜固定在2 m主镜之上，其随望远镜方位轴及俯仰轴转动，M4、M5、M6镜安装在望远镜水平轴立柱上，其随望远镜方位轴转动，当望远镜俯仰轴旋转时，其保持固定状态。K镜放置于M6镜之后，因此我们要建立主光路经过M6镜之后的像面旋转与望远镜俯仰轴、方位轴旋转的关系。

$\Delta \theta =\Delta A-\Delta Z$ (11)

$\omega =\frac{\Delta \theta }{\Delta t}=\frac{\Delta A-\Delta Z}{\Delta t}={\omega }_{A}-{\omega }_{Z}$ (12)

Figure 5. Rotation schematic diagram of local coordinate system

${\omega }_{K}=1/2\frac{\Delta \theta }{\Delta t}=1/2\left({\omega }_{A}-{\omega }_{Z}\right)$ (13)

Figure 6. The ray tracing of coude path

4. 准直技术及误差分析

2 m望远镜在跟踪目标过程中，由于俯仰、方位轴的转动，光线在2 m库德光路中会相应的发生旋转，导致望远镜成像焦点处产生像旋。为避免库德焦点处的图像旋转，需要在2 m主光路中引入K镜系统。为保证K镜具有最好的消旋效果，必须要确保K镜光轴、K镜机械回转轴及2 m望远镜主光轴三轴重合，本节对K镜的准直技术及误差分析展开深入研究。

$\Delta {D}_{3}=2\left(\Delta {D}_{1}\mathrm{cos}\theta -\Delta {D}_{2}\right)$ (14)

(a) ∆D1≠0,∆D2≠0 (b) ∆D1 ≠0,∆D2 =0

Figure 7. The alignment error curve of K mirror

Figure 8. The mechanical structure of K mirror

2 m望远镜K镜系统放置于M6镜与M7镜之间。在主光路中放置具有自调焦功能的测微准直望远镜(CZW)，在K镜之后沿主光路分别放置2个CCD，用于记录主光路经过K镜后的位移轨迹，如图9图10所示。

Figure 9. The sketch of alignment adjustment strategy

Figure 10. Alignment adjustable experiment of K mirror

Figure 11. Light-spot recording in the process of alignment calibration (CCD2)

$\sigma ={\mathrm{tan}}^{-1}\left(\frac{\Delta {D}_{3}}{H}\right)=4″$ (14)

5. 结论

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