Creating a large aspheric primary mirror using spherical segments

Abstract

The use of aspheric mirrors is a common practice to design astronomical telescopes with a few optical elements. In the most preferred optical design Ritchey Chretien (RC), both primary and secondary mirrors are hyperboloid. Nowadays large telescopes are being built using small mirror segments, however, making aspheric off-axis mirror segments is still a challenge. We have conducted a study in which, we explored the possibility to mimic an aspheric hyperbolic primary mirror by making use of smaller spherical mirror segments. Three different methods have been used to form a large segmented aspheric primary of nearly 12m aperture. In the first method, fixed ROC(radius of curvature) spherical mirror segments are reconfigured by a piston, tip, and tilt (PTT). In the other two methods, in addition to PTT, ROC of the segments are also varied. We further attempted to reduce the telescope wave-front error by varying the segment size and the F ratio of the primary. We found out that none of these three methods provided acceptable image quality unless we incorporate the warping harness in the segment support. The use of the warping harness emulated by Zernike coefficient correction, remarkably reduced the wave-front error and delivered a decent image quality over a large field of view. In this paper, we present the results of our study on designing an RC type optics for a 12m class optical-NIR(Near Infrared) telescope using spherical mirror segments.

Appendix

The best fit radius of curvature is found for each segment using matrix inversion techniques for a set of (x,y,z) coordinates corresponding to the chosen segment.

$$ S={\sqrt{({{R_{b}^{2}}}-(x-x_{c})^{2}-(y-y_{c})^{2})}+z_{c}} $$

(3)

where x_c, y_c, z_c are the coordinates of the center of the sphere, and R_b is the radius of the best fit sphere. We can form following matrix equations as

$$ A= 2 \begin{bmatrix} \sum\limits_{i=1}^{n}{\frac{x_{i}({x_{i}-\overline{x}})}{n}} &\sum\limits_{i=1}^{n}{\frac{x_{i}({y_{i}-\overline{y}})}{n}} & \sum\limits_{i=1}^{n}{\frac{x_{i}({z_{i}-\overline{z}})}{n}}\\ \sum\limits_{i=1}^{n}{\frac{y_{i}({x_{i}-\overline{x}})}{n}} &\sum\limits_{i=1}^{n}{\frac{y_{i}({y_{i}-\overline{y}})}{n}} & \sum\limits_{i=1}^{n}{\frac{y_{i}({z_{i}-\overline{z}})}{n}}\\ \sum\limits_{i=1}^{n}{\frac{z_{i}({x_{i}-\overline{x}})}{n}} &\sum\limits_{i=1}^{n}{\frac{z_{i}({y_{i}-\overline{y}})}{n}} & \sum\limits_{i=1}^{n}{\frac{z_{i}({z_{i}-\overline{z}})}{n}}\\ \end{bmatrix} $$

(4)

$$ B= \begin{bmatrix} \sum\limits_{i=1}^{n} {\frac{({x_{i}}^{2}+{y_{i}}^{2}+{z_{i}}^{2}).({x_{i}-\overline{x}})}{n}}\\ \sum\limits_{i=1}^{n} {\frac{({x_{i}}^{2}+{y_{i}}^{2}+{z_{i}}^{2}).({y_{i}-\overline{y}})}{n}}\\ \sum\limits_{i=1}^{n} {\frac{({x_{i}}^{2}+{y_{i}}^{2}+{z_{i}}^{2}).({z_{i}-\overline{z}})}{n}}\\ \end{bmatrix} $$

(5)

and

$$ X= \begin{bmatrix} x_{c}\\ y_{c}\\ z_{c}\\ \end{bmatrix} $$

(6)

Where, x_i, y_i, z_i are the data points on the aspheric surface and

$$ \overline{x}=\frac{1}{n}{\sum\limits_{i=1}^{n} x_{i}}, \ \ \ \ \ \overline{y}=\frac{1}{n}{\sum\limits_{i=1}^{n} y_{i}}, \ \ \ \ \ \overline{z}=\frac{1}{n}{\sum\limits_{i=1}^{n} z_{i}} $$

(7)

Therefore, the solution for the center of the best fit sphere can be obtained by matrix inversion as follows

$$ \begin{bmatrix} x_{c}\\ y_{c}\\ z_{c}\\ \end{bmatrix} =(A^{T}.A)^{-1} .A^{T}.B $$

(8)

Once the center of the best fit is known then the best fit radius of curvature R_b can be derived

$$ R_{b}={\sqrt{\frac{{\sum\limits_{i=1}^{n}((x_{i}-x_{c})^{2}+(y_{i}-y_{c})^{2}}+(z_{i}-z_{c})^{2})}{n}}} $$

(9)