Adaptive optics wave function reconstruction and phase unwrapping when branch points are present

An algorithm is presented for reconstruction of adaptive optics wavefront sensor data which produces results that contain the part of the wavefront distortion associated with the hidden phase as well as the scalar phase part that is captured by an ordinary least-mean-square-error reconstructor. The algorithm uses a multigrid formulation and multiplication of complex phasors representing the measured phase differences to reconstruct the distorted wave function. The algorithm is formulated to work with Hudgin-geometry data, but in the appendix a variant of this algorithm is described that allows operation with Fried-geometry data. Also an algorithm is presented that generates a phase function from the reconstructed wave function, a phase function that has its branch cuts placed so that the 2 π discontinuities of the branch cut occur where the optical intensity is much lower than the average intensity. The reconstructor algorithm is formulated as a noise-variance-weighted reconstructor. It is found that the algorithm's noise gain is only slightly greater than that of a noise-variance-weighted least-mean-square-error reconstructor so long as the noise variance for the input phase difference measurement data is less than about 0.25 rad 2.