Navigating Phase Singularities: A Topological and Fractal Map for Speckle‐Free Holography

ABSTRACT Phase singularities, manifesting as null intensity in optical speckle patterns, fundamentally constrain the uniformity of holographic reconstruction. A comprehensive understanding of their intrinsic physical properties is thus critical for advancing coherent optical systems, especially in the pursuit of speckle‐free holography. Using a singularity‐tracking methodology, we numerically investigate the evolution of these singularities as they propagate from holographic planes into free space. Our findings reveal two primary morphological classes of singularity trajectories—closed loops and linear paths—both exhibiting identical fractal characteristics. The topology invariance observed across these trajectories provides a robust framework for achieving high‐uniformity speckle‐free holography, either by eliminating singularities at the initial phase plane or by guiding them outside the target pattern. Notably, the fractal dimension of the trajectories ranges from 1.0 to 1.6 and demonstrates a negative correlation with the F ‐number of the hologram, exemplifying an optical analogy to a biased Brownian random walk under an attractive potential. This work offers novel statistical and topological insights into holographic speckle fields, paving the way for speckle‑less reconstruction using highly coherent light.