Rigid body motion tracking in axial shift mapping for measuring astronomical X-ray mirror figure

Future X-ray telescopes will require measuring thousands of grazing-incidence mirrors with nanometer accuracy and high throughput. Fizeau interferometry will require accurately separating the reference surface from the surface under test, a challenging proposition for optical surfaces that are highly off-axis conics. We present a lateral shifting interferometry approach called Axial Shift Mapping (ASM), in which we measure the surface under test at multiple positions and numerically separate the reference and test surfaces. In this paper, we estimate the measurement accuracy that would be required for an exemplary 0.5 arcsecond half-power diameter angular resolution telescope. We mathematically describe the procedure for using ASM, plus a lateral shift, to extract the most important quantities for X-ray telescope mirrors: axial profile (including the quadratic component) and cone angle variation. We also conclude that the radius and cone angle of a mirror are unknowable with ASM alone.