Evolution of vortex density in a non-diffracting speckle field with its continuous phase removed

We break down the equilibrium state and the diffraction invariant property of a non-diffracting speckle field by removing its continuous part of the phase while leaving all vortices behind. During the propagation of such a phase corrected non-diffracting speckle field, the vortex density drops down to a minimum value and then comes back to an equilibrium value which is even higher than the initial one. Before the phase corrected field returns back to its new equilibrium state, another least-squares phase removal will be applied, at the position where there is a minimum vortex density, to further remove vortices from the speckle field. Such a process of removing least-squares phase and propagating the phase corrected field over a distance can be repeated to eliminate most of optical vortices. Statistical results show that most of optical vortices can be removed from a non-diffracting speckle field. Finally, a semi-plane wave without optical vortices can be obtained from an initial non-diffracting speckle field with multiple steps of least-square phase correction.