## Abstract

As an extension of the knife-edge test, a noninterferometric method
for inspecting circularly symmetric aspheres is proposed in which the
test surface is illuminated by a spherical wave. When a small
circular stop is placed around the curvature center of the best-fitting
sphere, only rays characteristic for the deviation from a perfect
sphere remain unobstructed. An image of the tested surface is
thereby obtained, showing light and dark regions whose boundaries are
correlated to the surface profile, the stop size, and the stop position
along the symmetry axis. The experiment has been carried out with a
paraboloid.

© 1998 Optical Society of America

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### Equations (12)

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(1)
$$R\left(2\mathrm{\Delta}\mathrm{\varphi}\right)=\left(d/2\right)cos\mathrm{\alpha},$$
(2)
$$\mathrm{\beta}=\mathrm{arctan}\left(\mathrm{d}z/\mathrm{d}r\right).$$
(3)
$$\mathrm{\gamma}=\mathrm{\alpha}-\mathrm{\beta}.$$
(4)
$$R\left(r\right)={\left[1+{\left(\frac{\mathrm{d}z}{\mathrm{d}r}\right)}^{2}\right]}^{3/2}/\frac{{\mathrm{d}}^{2}z}{\mathrm{d}{r}^{2}},$$
(5)
$$\mathrm{\alpha}=\mathrm{arctan}\left[\frac{r}{R\left(0\right)-z\left(r\right)}\right].$$
(6)
$$\angle \mathit{OPQ}=\mathrm{\beta}-\mathrm{\gamma},$$
(7)
$$h=\left(\overline{\mathit{PQ}\prime}-\overline{\mathit{CQ}\prime}-{z}_{s}\right)tan(\angle \mathit{OPQ})=\left[\frac{r}{tan\left(\mathrm{\beta}-\mathrm{\gamma}\right)}-\frac{r}{tan\left(\mathrm{\beta}\right)}-{z}_{s}\right]tan\left(\mathrm{\beta}-\mathrm{\gamma}\right).$$
(8)
$$z\left(r\right)={r}^{2}/\left(4f\right),$$
(9)
$$tan\left(\mathrm{\beta}\right)=r/\left(2f\right),$$
(10)
$$tan\left(\mathrm{\gamma}\right)={r}^{3}/\left(16{f}^{3}+2{\mathit{fr}}^{2}\right).$$
(11)
$$tan\left(\mathrm{\beta}-\mathrm{\gamma}\right)=16{f}^{3}r/\left(32{f}^{4}+4{f}^{2}{r}^{2}+{r}^{4}\right),$$
(12)
$$h=16{f}^{3}r\left(\frac{{r}^{2}}{2f}+\frac{{r}^{4}}{16{f}^{3}}-{z}_{s}\right)/\left(32{f}^{4}+4{f}^{2}{r}^{2}+{r}^{4}\right).$$