Strong turbulence causes phase discontinuities known as branch points in an optical field. These discontinuities complicate the phase unwrapping necessary to apply phase corrections onto a deformable mirror in an adaptive optics (AO) system. This paper proposes a non-optimal but effective and implementable phase unwrapping method for optical fields containing branch points. This method first applies a least-squares (LS) unwrapper to the field which isolates and unwraps the LS component of the field. Four modulo-2π-equivalent non-LS components are created by subtracting the LS component from the original field and then restricting the result to differing ranges. 2π phase jumps known as branch cuts are isolated to the non-LS components and the different non-LS realizations have different branch cut placements. The best placement of branch cuts is determined by finding the non-LS realization with the lowest normalized cut length and adding it to the LS component. The result is an unwrapped field which is modulo-2π-equivalent to the original field while minimizing the effect of phase cuts on system performance. This variable-range ‘ϕLS+ϕnon-LS’ unwrapper, is found to outperform other unwrappers designed to work in the presence of branch points at a reasonable computational burden. The effect of improved unwrapping is demonstrated by comparing the performance of a system using a fixed-range ‘ϕLS+ϕnon-LS’ realization unwrapper against the variable-range ‘ϕLS+ϕnon-LS’ unwrapper in a closed-loop simulation. For the 0.5 log-amplitude variance turbulence tested, the system Strehl performance is improved by as much as 41.6 percent at points where fixed-range ‘ϕLS+ϕnon-LS’ unwrappers result in particularly poor branch cut placement. This significant improvement in previously poorly performing regions is particularly important for systems such as laser communications which require minimum Strehl ratios to operate successfully.
Todd M. Venema and Jason D. Schmidt, "Optical phase unwrapping in the presence of branch points," Opt. Express 16, 6985-6998 (2008). https://doi.org/10.1364/OE.16.006985