Aspherical wavefront measurements: Shack-Hartmann numerical and practical experiments

We consider an application of the original Hartmann method to bundles of rays generated by a Shack-Hartmann analyser. Absolute Shack-Hartmann measurements of converging wavefronts with the nominal method of collimating optics, used to locate the real image of a pupil on a microlens array, are not applicable when the wavefront asphericity is so strong that real subimages produced by individual lenslets of the array are no longer simultaneously focused at a common plane. As examples of strongly aspherical wavefronts we consider reflected beams obtained when testing large aspherical mirrors at their centre of curvature. Analytic formulae are applied to several instances and a ray-tracing program for a large-diameter strongly paraboloidal liquid mirror suggests that the Shack-Hartmann method could, however, be used by combining several cross sections of interlaced rays located downstream from the microlens array. In order to estimate how precisely subbundles of rays may be reconstructed from several cross sections we performed a small-scale experiment to measure an aspherical wavefront departing by more than from a best-fit sphere. A microlens array samples 2000 subareas per pupil. Eleven cross sections, corresponding to as many real and virtual subbundles of rays, are obtained upstream and downstream from an array using a relay optics to give enlarged real images on photographic film. We measured 57 subbundles and verified the straight line propagation of light to within a precision on negatives corresponding to a local 45 nm wavefront uncertainty. The uncertainty value for calibration using additional cross sections upstream and downstream from the microlens array amounts to 8 nm. We conclude from these numerical and practical experiments that the Shack-Hartmann method may be modified in order to measure strongly aspherical wavefronts, including reflected wavefronts obtained from centre-of-curvature testing for large aspheric mirrors.