The fractal shape of speckled darkness

Propagating three-dimensional speckle fields are threaded by random networks of nodal lines (optical vortices). We review our recent numerical superpositions of simulations of random plane waves modelling speckle (O'Holleran et al. Phys. Rev. Lett. in press), in which the nodal lines and loops were found to have the fractal structure of brownian random walks. We discuss this result, and its comparison with the discrete vortices of the Z3 lattice model for cosmic strings. We argue that the scaling depends on the geometry of small vortex loops and avoided crossings. The analytic statistics of these events, along with related singularities are discussed, and the densities of vorticity-vanishing points and anisotropy C lines are found explicitly.