Self-Organized Vortex and Antivortex Patterns in Laser Arrays

Recently, it was shown that dissipatively coupled laser arrays simulate the classical $XY$ model. We show that phase locking of laser arrays can give rise to the spontaneous formation of vortex and antivortex phase patterns that are analogous to topological defects of the $XY$ model. These patterns are stable although their formation is less likely in comparison to the ground-state lasing mode. In addition, we show that small ratios of photon to gain lifetime destabilize vortex and antivortex phase patterns. These findings are useful for studying topological effects in optics as well as for designing laser array devices.

Figure 1

Schematic of the equilibrium phase patterns of a dissipatively coupled laser array. (a) The ground state. (b) A vortex state. Here, the arrows represent the phase of each element.

Figure 2

Equilibrium phase patterns of an array of dissipatively coupled lasers arranged on a $16\times 16$ square lattice. (a)–(d) The ground state, vortex, antivortex, and bound vortex-antivortex states for the case of attractive coupling ($\kappa >0$), associated with a ferromagnetic spin system [first row], and repulsive coupling ($\kappa <0$), associated with an antiferromagnetic spin system [second row]. The $XY$ energy levels [given by Eq. (1)] associated with these phase patterns are shown on top of the panels. Here, $g_0=30$, $\kappa =-1$ for the top row and $\kappa =1$ for the bottom row.

Figure 3

The $XY$ energy distribution associated with the equilibrium phase patterns of a $16\times 16$ rectangular lattice laser array for (a) ferromagnetic, and (b) antiferromagnetic coupling. The histograms are produced based on 1000 simulations with initial phases drawn randomly with uniform distribution from the range $[0,2\pi ]$. All parameters are the same as in Fig. 2. The dashed line indicates the tight-binding energy of a single vortex ($\pi \kappa \ln L$).

Figure 4

The steady-state phase pattern of a laser array arranged on a triangular lattice. Here, $\kappa =-1$ and $g_0=30$. We have the ground state in (a), a single vortex in (b), and a single antivortex in (c).

Figure 5

The $XY$ energy distribution associated with the equilibrium phase patterns of a triangular lattice arrangement of lasers with (a) ferromagnetic ($\kappa <0$), (b) antiferromagnetic ($\kappa >0$) coupling. The lattice size and array parameters are the same as in Fig. 4. The histograms are produced based on 1000 simulations with initial phases drawn randomly with uniform distribution from the range $[0,2\pi ]$.

Figure 6

The transient dynamics of a $16\times 16$ laser array. (a) Snapshots of the phase pattern at intermediate times. (b) Amplitudes and phases of the array elements. In (b), the dashed lines show the time instants associated with the snapshots in (a). All parameters are the same as in Fig. 2.

Figure 7

The dynamics of a $6\times 6$ array for (a) ${\tau }_p/{\tau }_g=10$ (b) 1, and (c) 0.1. In all cases, $\kappa =-1$ and $g_0=30$.

Figure 8

The $XY$ energy distribution associated with the equilibrium phase pattern of laser arrays governed by the dynamical equations, Eqs. (4a), for different photon to gain lifetime ratios. Top: the square lattice for (a) ${\tau }_p/{\tau }_g=10$, (b) 1.0, and (c) 0.01. Bottom: the triangular lattice for (a) ${\tau }_p/{\tau }_g=10$, (b) 1.0, and (c) 0.01. Here, $\kappa =-1$ and $g_0=30$, while the lattice size is the same as in Figs. 2 and 4 for the top and bottom rows, respectively.

Figure 9

The real part of the Jacobian matrix eigenvalues versus the photon to gain lifetime ratio ${\tau }_p/{\tau }_g$. These eigenvalues represent the linear stability test for a vortex pattern (shown as inset) in a $6\times 6$ rectangular lattice array. The largest eigenvalue is highlighted in red.

Figure 10

The formation of a vortex state in an array of solid-state lasers with forced boundary conditions. Here, the lasers located on the lattice boundary are driven by a master laser such that their phase is forced to undergo a gradual full-circle clockwise rotation. The dashed line shows the time instant that the seeding of the boundary elements is turned on. For the bulk elements, Eqs. (4a) are simulated with the following parameters: ${\tau }_p=4\mathrm{ps}$ , ${\tau }_g=2\mathrm{ns}$, $\kappa =0.12$, $g_0=30$, while for the boundary elements a drive term $d\exp (i{\theta }_i)$ is added to Eq. (4a), where $d=0.8$ and ${\theta }_i$ is taken such that its integration around the boundary becomes $2\pi$.

Figure 11

The far-field radiation array factor $f(\theta ,\varphi )$ associated with the steady-state patterns of Fig. 2. The insets depict the radiation intensity and phase patterns at a plane parallel to the surface of the laser array at $z_0=7{\lambda }_0$. For calculating the array factor the center-to-center separation of neighbor lasers is assumed to be ${\lambda }_0$.