Large dynamic cophasing error correction based on improved stochastic parallel gradient descent algorithm

The correction of cophasing errors in segmented mirrors is essential for achieving diffraction-limited performance in large-aperture telescopes. In this paper, we propose an improved stochastic parallel gradient descent (SPGD) algorithm, referred to as cophasing SPGD (CSPGD), specifically designed for large dynamic cophasing error correction. The improvements include the integration of Nesterov momentum and the Adam optimizer to accelerate convergence, along with adaptive gain coefficients to ensure stability. We introduce novel metric functions tailored for significant cophasing errors, which extend the algorithm's capture range. The normalized second moment of the image intensity is utilized for large tip-tilt correction, while the integral of the side lobes of the squared modulation transfer function (MTF) under narrowband and broadband illumination is employed for large piston correction and fine phasing, respectively. Through numerical simulations and experimental validations, the CSPGD algorithm demonstrates superior performance in correcting large piston and tip-tilt errors in segmented mirrors, providing a robust and efficient solution for the cophasing tasks of segmented telescopes.