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.

Figure 1

Topology in the tangle. (a) shows a projection of vortex lines in a cubic volume of side length $3\Lambda$. [$\Lambda$ is coherence length of the field distribution defined by Eq. (1)]. Sections of infinite lines are shown in gray, loops in black and the two highlighted (linked) loops in red and green (b) shows a view of the vortex link.

Figure 2

Linked vortex loops. (a) Shows a linked pair of vortex loops, $\mathbf{a}$, $\mathbf{b}$ in a simulated speckle field. (b) Shows the chord angle ${\theta }_{\mathbf{a},\mathbf{b}}(i,j)$ pairs of points on the loops projected into viewing plane. The singularities in the chord angle correspond to signed crossings with values $s$ and $p$ given in the table.

Figure 3

A Log-log plot of the radius of gyration $R_g$ against loop length, $L$. For illustrative purposes, three loops of different lengths are shown. The red line is a fit to the straight line section (0.2–2.0) on the log $L$ scale and has a gradient of 0.52. Links between two loops were only found for $\log (L)>0.2$.

Figure 4

A plot of the probability $P$ of loops of a given length $L$, not to be threaded by vortex lines of any type (green circles) or linked to another loop (orange squares). The red lines show fit to straight lines indicating an exponential decay lengths of approximately $30\Lambda$ and $185\Lambda$ for not being threaded by a vortex line and vortex loop, respectively.

Figure 5

A three component link found within our simulated speckle field. Within a total calculated volume of approximately $1.5\times {10}^5{\Lambda }^3$ only 10 such features were found.