# Fractality and topology of optical singularities

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## Abstract

Optical singularities are points in complex scalar and vector fields where a property of the field becomes undefined (singular). In complex scalar fields these are phase singularities and in vector fields they are polarisation singularities. In the former the phase of the field is singular and in the latter it is the polarisation ellipse axes. In three dimensions these singularities are lines and natural light fields are threaded by these lines.

The interference between three, four and five waves is investigated and inequalities are given which establish the topology of the singularity lines in fields composed of four plane waves. Beyond several waves, numerical simulations are used, supported by experiments, to establish that optical singularties in speckle fields have the fractal properties of a Brownian random walk. Approximately 73% of singularity lines percolate random optical fields, the remainder forming closed loops. The statistical results are found to be similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior.

It is also established that a random superposition of plane-waves, such as optical speckle, form singularities which not only map out fractal lines, but create topological features within them. These topological features are rare and include vortex loops which are threaded by infinitely long lines and pairs of loops that form links. Such structures should be not only limited to optical fields but will be present in all systems that can be modeled as random wave superpositions such as those found in cosmic strings and Bose-Einstein condensates.

Also reported are results from experiments that generated compact vortex knots and links in real Gaussian beams. These results were achieved through the use of algebraic knot theory and random search optimisation algorithms.

Finally, polarisation singularity densities are measured experimentally which confirm analytic predictions.