We present an analysis of the diffraction effects from a segmented aperture with a very large number of segments-prototype of the next generation of extremely large telescopes. This analysis is based on the point-spread-function analytical calculation for Keck-type hexagonal segmentation geometry. We concentrate on the effects that lead to the appearance of speckles and/or a regular pattern of diffraction peaks. These effects are related to random piston and tip-tilt errors on each segment, gaps between segments, and segment edge distortion. We deliver formulas and the typical numerical values for the Strehl ratio, the relative intensity of higher-order diffraction peaks, and the averaged intensity of speckles associated with each particular case of segmentation error.

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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.