## Abstract

The zero-gravity surface figure of optics used in spaceborne astronomical instruments must be known to high accuracy, but earthbound metrology is typically corrupted by gravity sag. Generally, inference of the zero-gravity surface figure from a measurement made under normal gravity requires finite-element analysis (FEA), and for accurate results the mount forces must be well characterized. We describe how to infer the zero-gravity surface figure very precisely using the alternative classical technique of averaging pairs of measurements made with the direction of gravity reversed. We show that mount forces as well as gravity must be reversed between the two measurements and discuss how the St. Venant principle determines when a reversed mount force may be considered to be applied at the same place in the two orientations. Our approach requires no finite-element modeling and no detailed knowledge of mount forces other than the fact that they reverse and are applied at the same point in each orientation. If mount schemes are suitably chosen, zero-gravity optical surfaces may be inferred much more simply and more accurately than with FEA.

© 2007 Optical Society of America

Full Article |

PDF Article
### Equations (54)

Equations on this page are rendered with MathJax. Learn more.

(1)
\sim 37\text{\hspace{0.17em}}\mathrm{nm}
(3)
\sim 8\text{\hspace{0.17em}}\mathrm{nm}
(4)
\sim 343\text{\hspace{0.17em}}\mathrm{mm}
(5)
\sim 2.2\text{\hspace{0.17em} m}
(6)
6.3\text{\hspace{0.17em}}\mathrm{nm}
(8)
3.875\text{\hspace{0.17em}}\mathrm{kg}
(9)
41.9\text{\hspace{0.17em}}\mathrm{kg}\text{\hspace{0.17em}}{\text{m}}^{-2}
(10)
100\text{\hspace{0.17em}}\mathrm{nm}
(11)
20\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(12)
\sim 6.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(13)
83\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(14)
\sim 11.5\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(20)
$${x}_{i}^{1g}={x}_{i}^{0g}+\delta {x}_{i}\text{.}$$
(21)
\left({f}^{g}\right)
(22)
\left({\text{f}}^{m}\right)
(23)
$${\mathit{A}}_{ij}\xb7\delta {x}_{j}={\text{f}}_{i}={\text{f}}_{i}^{g}+{\text{f}}_{i}^{m}\text{,}$$
(25)
\delta {x}_{i}\prime
(26)
$${\mathit{A}}_{ij}\xb7\delta {x}_{j}\prime ={\text{f}}_{i}\prime =-{\text{f}}_{i}^{g}-{\text{f}}_{i}^{m}\text{,}$$
(27)
$$\delta {x}_{j}\prime =-\delta {x}_{j}\text{\hspace{1em} for \hspace{0.17em}}\mathrm{all}\text{\hspace{0.17em}}j\text{.}$$
(28)
$$\frac{1}{2}\left({x}_{i}^{1g}+{x}_{i}^{1g}\prime \right)=\frac{1}{2}\left({x}_{i}^{0g}+\delta {x}_{i}+{x}_{i}^{0g}+\delta {x}_{i}\prime \right)={x}_{i}^{0g}\text{.}$$
(30)
6.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(31)
12.0\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(32)
14.9\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(33)
5.7\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(34)
1.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(35)
\sim 4.9\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(36)
1.5\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(38)
6.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(39)
\sim 6\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(40)
3.8\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(41)
6.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(42)
6.3\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(43)
343\text{\hspace{0.17em}}\mathrm{mm}
(44)
\sim 2.2\text{\hspace{0.17em} m}
(45)
20\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(46)
100\text{\hspace{0.17em}}\mathrm{nm}
(47)
14\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(48)
\sim 6.3\text{\hspace{0.17em}}\mathrm{nm}
(49)
11.5\text{\hspace{0.17em}}\mathrm{nm}
(50)
0.0003\text{\hspace{0.17em}}\mathrm{nm}
(51)
5.7\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}
(52)
35\text{\hspace{0.17em}}\mathrm{nm}
(53)
1.5\text{\hspace{0.17em}}\mathrm{nm}
(54)
3.8\text{\hspace{0.17em}}\mathrm{nm}\text{\hspace{0.17em}}\mathrm{rms}