Topology of light's darkness

We numerically study the topology of optical vortex lines (nodal lines) in volumes of optical speckle, modeled as superpositions of random plane waves. It is known that the vortex lines may be infinitely long, or form closed loops. Loops are occasionally threaded by infinite lines, or linked with other loops. We find the probability of a loop not being threaded decays exponentially with the length of the loop. This behavior has a similarity to scaling laws studied in superfluid turbulence, and polymers modeled as random walks.