## Abstract

A method for null-testing fast convex aspheric optical surfaces is
presented. The method consists of using a cylindrical screen with a
set of lines drawn on it in such a way that its image, which is formed
by reflection on a perfect surface, yields a perfect square
grid. Departures from this geometry are due to imperfections of the
surface, allowing one to know if the surface is close to the design
shape. Tests conducted with a full hemisphere and with the
parabolic surface of a lens show the feasibility of the
method. Numerical simulations show that it is possible to detect
surface departures as small as 5 µm.

© 2000 Optical Society of America

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### Equations (11)

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(1)
$$z=\frac{{\mathit{cs}}^{2}}{1+{\left[1-\left(k+1\right){c}^{2}{s}^{2}\right]}^{1/2}},$$
(2)
$$\mathrm{\xi}=\frac{a\left(\mathit{bQ}+r\right)-{\left[{a}^{2}{r}^{2}-{\mathrm{\rho}}^{2}b\left(\mathit{bQ}+2r\right)\right]}^{1/2}}{{a}^{2}Q+{\mathrm{\rho}}^{2}}\mathrm{\rho},$$
(3)
$${z}_{2}=\frac{\mathrm{\xi}}{\mathrm{\rho}}a-b,$$
(4)
$${\mathrm{\rho}}_{3}=R,$$
(5)
$${z}_{3}=\frac{a{\mathrm{\xi}}^{2}+a{\left({\mathit{Qz}}_{2}-r\right)}^{2}-2\left({\mathit{Qz}}_{2}-r\right)\left(\mathrm{\rho}\mathrm{\xi}+{\mathit{aQz}}_{2}-\mathit{ar}\right)}{-\mathrm{\rho}{\mathrm{\xi}}^{2}-\mathrm{\rho}{\left({\mathit{Qz}}_{2}-r\right)}^{2}+2\mathrm{\xi}\left(\mathrm{\rho}\mathrm{\xi}+{\mathit{aQz}}_{2}-\mathit{ar}\right)}\times \left(-R+\mathrm{\xi}\right)+{z}_{2},$$
(6)
$$b=\frac{\mathit{aD}}{\sqrt{2}d}-\mathrm{\beta},$$
(7)
$$\mathrm{\beta}=\left\{\begin{array}{ll}\frac{{D}^{2}}{8r}& \mathrm{for}Q=0\\ \frac{r}{Q}\left[1-{\left(1-\frac{{\mathit{QD}}^{2}}{4{r}^{2}}\right)}^{1/2}\right]& \mathrm{for}Q\ne 0\end{array}\right..$$
(8)
$$D=\frac{\sqrt{8}\mathit{ra}}{{\left({d}^{2}+2{a}^{2}\right)}^{1/2}}$$
(9)
$$\mathrm{\rho}={\left({n}^{2}+{m}^{2}\right)}^{1/2}l,tan\mathrm{\varphi}=\frac{m}{n}$$
(10)
$$X=R\mathrm{\varphi},Y={z}_{3}\left(\mathrm{\rho}\right),$$
(11)
$$\mathrm{\Delta}P={\left[{\left(x-x\prime \right)}^{2}+{\left(y-y\prime \right)}^{2}\right]}^{1/2}.$$