Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors

Interferometers accurately measure the difference between two wavefronts, one from a reference surface and the other from an unknown surface. If the reference surface is near perfect or is accurately known from some other test, then the shape of the unknown surface can be determined. We investigate the case where neither the reference surface nor the surface under test is well known. By making multiple shear measurements where both surfaces are translated and/or rotated, we obtain sufficient information to reconstruct the figure of both surfaces with a maximum likelihood reconstruction method. The method is demonstrated for the measurement of a 1.6 m flat mirror to 2 nm rms, using a smaller reference mirror that had significant figure error.