Segmented mirror phasing using the focal-plane intensity.

We develop a method for subaperture piston phase retrieval in a telescope using a segmented primary mirror. We assume that the mirror subapertures are arranged on a two-dimensional lattice, and in addition, the separate subaperture point-spread functions are focused and overlapped on the focal plane. Therefore, the residual errors are the subaperture piston phase errors, represented as a phasor, a unit modulus complex number, for each subaperture. Under these conditions, we find considerable simplicity in the calculated optical transfer function (OTF) at special subaperture lattice spatial frequencies. We then construct a phasor-based error function based on the modulus squared of the difference between the measured OTF and the calculated OTF. The remaining steps in our piston phase retrieval algorithms are developed by calculating the error-function variation, with respect to each phasor element. The resulting equations for the error gradient are then used iteratively, in a phasor-based algorithm, to find the minimum of the error function. In the applications, we simulate photon-noise-limited piston retrieval for a segmented primary with 18 hexagonal subapertures. When we invoke phase diversity, the piston retrievals prove unique and accurate.