Written by Toshimi Taki

Bill Kelley proposed a simple method to make paraboloidal mirror by pulling the center of a spherical mirror ( Sky & Telescope, June 1992, pp.684-686 ). His method is one of active optics which is applied to modern giant telescopes.

I was interested in the method and evaluated the effectiveness of the method analytically. The objective of my study is to evaluate the accuracy of Kelley's method and to find the limitation of the method. I analyzed Kelley's method by finite element analysis which I used for the mirror cell study ( Sky & Telescope, April 1996, pp.75-77 ). The FEA can take account of thickness variation due to the concave shape of mirror and shear deformation of mirror.

Figure 1 shows the schematic of the method. A spherical mirror sits on a ring shaped carpet and the mirror is pulled back by a bolt epoxyed on the center of the mirror. You can tune the pulling force to get the sharpest image of a star while observing.

Figure 1. Kelly's Method

I put assumptions listed below in the analysis.

(1) Center obstruction by diagonal mirror is assumed 20% of the diameter of the mirror.

(2) Material of mirror is Pyrex. Young's modulus is 7600kgf/mm^2 and Poisson's ratio is 0.20 .

(3) Mirror is supported at the perimeter of the mirror.

Figure 2 and 3 show the relationship between pulling force and "surface" RMS error. Marechal's criterion ( 1/27-wave RMS on the surface ) which is a measure of "diffraction-limited" optics is also shown in the figure to evaluate the effectiveness of Kelley's method. (See "Star Testing Astronomical Telescope," p.7 by Suiter for the Marechal's criterion.)

Figure 2

Figure 3

Figure 4

Suppose you have a spherical mirror of 200mm, f/7, 7.5% thickness ratio. See figure 2. Before tuning, the mirror has 3.4E-4 *200 = 0.068-wave (1/15-wave) RMS error. If you tune the pulling force, you will get superb optics of 0.8E-4 *200 = 0.016-wave (1/63-wave) RMS error. The optimum pulling force is 7.8E-5 *200*200 = 3.1kgf (6.9lbs). The tolerance of pulling force is 1.6kgf to 4.7kgf (4.0E-5 *200*200 and 1.17E-4 *200*200). If the mirror has 10% thickness ratio (see figure 3), the optimum pulling force increases to 1.92E-4 *200*200 = 7.7kgf (17lbs). The optimum force is approximately proportional to the cube of the thickness ratio.

You can find in figure 2 that undeformed spherical mirror of 100mm diameter is diffraction-limited if the f/number is slower than seven (7).

Figure 4 shows the relationship between f/number and surface RMS error when the pulling force is optimized. You easily understand from the figure that smaller and slower mirrors are suitable to Kelley's method. The applicability of Kelley's method is summarized as ;

100mm mirror ---- f/4.3 and slower

150mm mirror ---- f/4.9 and slower

200mm mirror ---- f/5.4 and slower

My analysis shows that Kelley's method is very effective as Kelly wrote in his article.