Mach-Zehnder interferometer for piston and tip-tilt sensing in segmented telescopes: theory and analytical treatment.

A study is presented of a Mach-Zehnder interferometer for the measurement of phasing errors of the type found in segmented telescopes. We show that with a pinhole much larger than the Airy disk and an optical path difference between the arms equal to a quarter of the wavelength, the interferometric signal is related to the second derivative of the wave front. In this condition the signal is produced mostly by the segmentation errors and is marginally sensitive to other aberrations including atmospheric turbulence. The signal has distinguishable symmetric and antisymmetric properties that are related to segment aberrations. We suggest using the antisymmetric component of the signal to retrieve piston, tip, and tilt. The symmetric component of the signal serves as an estimate of the measurement error. In this way we proceed with a study of the errors associated with the misalignment of the interferometer, the segment edge imperfections, and the nonaveraged atmospheric perturbations. The entire study is performed on a theoretical basis, and numerical simulations are used to cross check the analytical results.