Stable higher-charge discrete vortices in hexagonal optical lattices

We show that double-charge discrete optical vortices may be completely stable in hexagonal photonic lattices where single-charge vortices always exhibit dynamical instabilities. Even when unstable the double-charge vortices typically have a much weaker instability than the single-charge vortices, and thus their breakup occurs at longer propagation distances.

Figure 1

(Color online) Example of an always unstable single-charge discrete optical vortex for $\beta =-0.76$ (marked by a circle in Fig. 2). (a) Intensity (top) and phase (bottom); (b) real (top) and imaginary (bottom) components; (c) absolute value of the corresponding Fourier transforms; (d) spectrum of the linearized equation displaying the linear instability of the configuration due to the presence of positive real parts in the eigenvalues $\lambda$ in the spectrum.

Figure 2

(Color online) Family of single-charge vortices vs propagation constant $\beta$. Top: maximum real part of the linear stability spectrum. Bottom: power $P={\int }_{\infty }^{\infty }{U}^{2}dxdy$. The circle corresponds to the discrete vortex given in Fig. 1. The dashed line indicates another unstable branch which, for larger $\beta$, bifurcates into different configurations.

Figure 3

(Color online) Examples of (a) stable and (b) unstable double-charge discrete optical vortices for $\beta =-0.76$ and $-0.96$, respectively (marked, respectively, by the circle and square in Fig. 4). The layout of the panels is the same as in Fig. 1.

Figure 4

(Color online) Family of double-charge vortices vs propagation constant $\beta$. Top: maximum real part of linear stability spectrum (when nonzero, this denotes instability). Bottom: power $P={\int }_{\infty }^{\infty }{U}^{2}dxdy$. The circle and square correspond to the stable and unstable discrete vortex configurations shown in Figs. 3a, 3b, respectively. The dashed line indicates an unstable branch which, for larger $\beta$, bifurcates into different configurations.

Figure 5

(Color online) Top three panels (a) depict the evolution of a stable double-charge vortex configuration after a random perturbation with amplitude 5% of the initial amplitude. Bottom three panels (b) show the evolution of a single-charge vortex configuration. In both cases $\beta =-0.7$. The color bar on the right provides a scale of the intensity [note that the intensities of the single-charge vortex are lower relative to (a) initially and saturated on this scale after breakup].

Figure 6

(Color online) Same set of panels as in Fig. 1 except for a stable double-charge vortex in a honeycomb lattice.

Figure 7

(Color online) (a) Input beam intensity profile relative to the lattice (position of lattice intensity maxima are shown as rings). The intensity is given by the color bar on the immediate right. Appearance of the beam at $z=60\mathrm{mm}$ with different initial vortex phases (intensity not to scale of color bar): (b) single-charge vortex, (c) double-charge vortex. Top panels: intensities; bottom panels: phase.