Calibration of symmetric and non-symmetric errors for interferometry of ultra-precise imaging systems

The azimuthal Zernike coefficients for shells of Zernike functions with shell numbers n<N may be determined by making measurements at N equally spaced rotational positions. However, these measurements do not determine the coefficients of any of the purely radial Zernike functions. Label the circle that the azimuthal Zernikes are measured in as circle A. Suppose that the azimuthal Zernike coefficients for n<N are also measured in a smaller circle B which is inside circle A but offset so that it is tangent to circle A and so that it has the center of circle A just inside its circular boundary. The diameter of circle B is thus only slightly larger than half the diameter of circle A. From these two sets of measurements, all the Zernike coefficients may be determined for n<N. However, there are usually unknown small rigid body motions of the optic between measurements. Then all the Zernike coefficients for n<N except for piston, tilts, and focus may be determined. We describe the exact mathematical algorithm that does this and describe an interferometer which measures the complete wavefront from pinholes in pinhole aligners. These pinhole aligners are self-contained units which include a fiber optic, focusing optics, and a 'pinhole mirror'. These pinhole aligners can then be used in another interferometer so that its errors would then be known. Physically, the measurements in circles A and B are accomplished by rotating each pinhole aligner about an aligned axis, then about an oblique axis. Absolute measurement accuracies better than 0.2 nm were achieved