Vortex Solitons in Twisted Circular Waveguide Arrays

We address the formation of topological states in twisted circular waveguide arrays and find that twisting leads to important differences of the fundamental properties of new vortex solitons with opposite topological charges that arise in the nonlinear regime. We find that such system features the rare property that clockwise and counterclockwise vortex states are nonequivalent. Focusing on arrays with ${\mathcal{C}}_{6v}$ discrete rotation symmetry, we find that a longitudinal twist stabilizes the vortex solitons with the lowest topological charges $m=\pm 1$, which are always unstable in untwisted arrays with the same symmetry. Twisting also leads to the appearance of instability domains for otherwise stable solitons with $m=\pm 2$ and generates vortex modes with topological charges $m=\pm 3$ that are forbidden in untwisted arrays. By and large, we establish a rigorous relation between the discrete rotation symmetry of the array, its twist direction, and the possible soliton topological charges.

Figure 1

Angular Bloch modes ${\psi }_m$ as representations of the point group ${\mathcal{C}}_6$ for exemplary array with $N=6$. They are described as the roots of unity $e^{im\pi /3}$ in terms of their OAPM $m$. (a) For $\omega =0$ (nonrotating case) the $m=\pm 1,\pm 2$ modes form degenerate doublets due to time reversal symmetry. (b) For $\omega \ne 0$, time reversal is broken, which now connects an $m$ mode with its partner $-m$ with opposite value of $\omega$. A frequency value $-\omega$ is represented by a red cross whereas a frequency value $\omega$ is indicated by a green dot.

Figure 2

Exact numerically calculated eigenvalues (lines) and their analytical approximation (dots) for linear modes of the rotating structure with $N=6$ waveguides vs $\omega$ and examples of modes ($|\psi |$ distributions and $\arg (\psi )$ distributions in the insets) with different $m$, $\omega$ values. Open blue circles indicate crossings of the horizontal dashed line $b=2.3$ with $m=\pm 2$ mode families that occur at different frequencies.

Figure 3

Power (left) and maximal perturbation growth rate (right) versus rotation frequency $\omega$ for $m=\pm 1$ solitons at $b=2.8$ (a) and (b), $m=\pm 2$ solitons at $b=2.3$ (c) and (d), and $m=3$ solitons at $b=2.3$ and $b=2.8$ (e) and (f). Stable branches are shown black, unstable ones are shown red. Dots correspond to solitons, whose propagation dynamics is depicted in Fig. 5.

Figure 4

(a) Power vs propagation constant $b$ for $m=\pm 2$ solitons at $\omega =0.2$. (b) Maximal perturbation growth rate vs $b$ for all solitons with $m=\pm 1,\pm 2,+3$ at $\omega =0.2$.

Figure 5

Propagation of stable vortex solitons with $m=-1$, $b=2.8$, $\omega =0.38$ (a) and $m=-2$, $b=2.3$, $\omega =0.38$ (c), and decay of the unstable states with $m=+1$, $b=2.8$, $\omega =0.38$ (b), and $m=+3$, $b=2.8$, $\omega =0.54$ (d).