Theory of singular phase reconstruction in speckle-field

Abstract

Phase reconstruction under the conditions of strong scintillation is limited by the presence of phase discontinuities (singularities) that accompany the intensity nulls in speckle field. The singularities are hidden for the wave front sensors of conventional adaptive optics systems. Rigorous theory has been developed in the paper to reconstruct the singular phase with allowance for the vortex nature of the phase gradient vector field. Singular functions of the wavefront slope (phase gradient) are exposed to an appropriate regularization; the divergence and rotor of the regularized phase gradient are determined. It has been established that the divergence of the phase gradients contains singularities of the same type as gradient and, therefore, proper phase reconstruction can be executed using the regularized slopes only. Topological transformation of the wave front in the process of appearance and annihilation of the discontinuities is manifested as changing in relief of the regularized rotor, the relief characteristics correspond to the discontinuity parameters. With the use of Pompeiu integral the phase gradient is presented as a sum of divergent and rotor parts added by the term depending on the receiving aperture boundary conditions. The obtained wavefront slopes are unwrapped into the singular phase ignoring the noise and singularities of the phase gradient divergence and rotor. Derived analytical expression connecting the phase reconstruction from the measurements of wavefront slopes. Examples of such a reconstruction are given. The developed approach can be base for creating the discrete phase reconstruction algorithms.