Figure 1

(Color online) Schematic of the experimental setup. BS: beam splitter; CCD: camera; FF: Fourier filter; L: lens; M: mirror; MO: microscope objective; PH: pinhole; SLM: spatial light modulator.Reuse & Permissions

Figure 2

(Color online) Sketch of the Fourier image and numerically calculated lattice intensity and refractive index profiles for the symmetric hexagonal lattice (top panels) and the stretched lattice (bottom panels). The lattice beams in Fourier space are indicated by dots forming an equilateral triangle for the unstretched lattice and an isosceles triangle for the stretched lattice. The refractive index profiles are shown for focusing (left) and defocusing (right) nonlinearities.Reuse & Permissions

Figure 3

(Color online) (a) An input single-charge vortex beam; (b) the beam profile and phase at the output crystal face for low input intensity; (c) output for high input intensity. In both cases we see the breakup of the single-charge vortex. Here and below in experimental figures: left panels show intensity; right panels reveal phase structure; circles indicate positions of vortices with charge $m=+1$ (red) and $m=-1$ (blue).Reuse & Permissions

Figure 4

(Color online) The same as in Fig. 3, but for the case when the input beam (a) has a double-charge vortex phase. (b) Output at the crystal face demonstrates discrete diffraction for low power and (c) discrete double-charge vortex generation at high power.Reuse & Permissions

Figure 5

Numerical simulation of a single-charge vortex $\left(E_{\text{ext}}=2.5\text{kV}/\text{cm},I_{\text{latt}}={\left|A_{\text{latt}}\right|}^2=1,\beta =3\right)$. (a) Initial vortex beam profile; (b) beam profile at $z=20\text{mm}$ for low input power; (c) beam profile at $z=20\text{mm}$ for high input power; (d) high-power output at $z=280\text{mm}$. Top panels: intensity; bottom panels: phase.Reuse & Permissions

Figure 6

Numerical simulation of a double-charge vortex $\left(E_{\text{ext}}=2.5\text{kV}/\text{cm},I_{\text{latt}}={\left|A_{\text{latt}}\right|}^2=1,\beta =3\right)$. (a) Initial vortex beam profile; (b) beam profile at $z=20\text{mm}$ for low input power; (c) beam profile at $z=20\text{mm}$ for high input power; (d) high-power output at $z=280\text{mm}$. Top panels: intensity; bottom panels: phase.Reuse & Permissions

Figure 7

(Color online) The hexagonal cell is approximated by a contour shown in (a). The images below this show the relative phases for the (c) double- and (e) single-charge vortices distinguishing stable (with ${\gamma }_j<0$ for all $j$) and unstable (with ${\gamma }_j>0$) solutions. For $C=1$ the phases of adjacent excited nodes are equidistant and all are a distance of $2\pi /3$ or $\pi /3$ from one another. These are the isotropic double- and single-charge vortices. For smaller $C$, the relative phase of ${\theta }_2$ and ${\theta }_3$ (shown in black) becomes smaller for the double-charge (c) solution [and larger for the single charge (e)] and when $\left|{\theta }_2-{\theta }_3\right|<(>)\pi /2$, for $C<C_{\text{cr}}=0.708$, one corresponding eigendirection becomes unstable (stable). This can be observed in (d) and (f), in which the six eigenvalues of the linearization matrix $\mathcal{M}$ are presented as a function of $C$. When they are all negative the solution is stable close to the anticontinuum limit. Notice for the $S=2$ solution (d) the smallest magnitude one becomes positive for $C<C_{\text{cr}}$, leading to instability. (b) The bifurcation of the relevant eigenvalue of the $S=2$ vortex through the origin is represented by the maximum real part of the linearization spectrum, $\text{max}\left[\text{Re}\left(\lambda \right)\right]$ as a function of both the anisotropy parameter, $C$, as well as the coupling $\epsilon$. Notice the critical point (in $C$) shifts only very slightly from the first-order prediction for $0<\epsilon <0.1$.Reuse & Permissions

Figure 8

Numerical simulation of a single-charge vortex for defocusing nonlinearity $\left(E_{\text{ext}}=-2.5\text{kV}/\text{cm},I_{\text{latt}}={\left|A_{\text{latt}}\right|}^2=4,\beta =2\right)$. (a) Initial vortex beam profile; (b) beam profile at $z=20\text{mm}$ for low input power; (c) beam profile at $z=20\text{mm}$ for high input power; (d) high-power output at $z=280\text{mm}$. Top panels: intensity; bottom panels: phase.Reuse & Permissions

Figure 9

Numerical simulation of a double-charge vortex for defocusing nonlinearity $\left(E_{\text{ext}}=-2.5\text{kV}/\text{cm},I_{\text{latt}}={\left|A_{\text{latt}}\right|}^2=4,\beta =2\right)$. (a) Initial vortex beam profile; (b) beam profile at $z=20\text{mm}$ for low input power; (c) beam profile at $z=20\text{mm}$ for high input power; (d) high-power output at $z=280\text{mm}$. Top panels: intensity; bottom panels: phase.Reuse & Permissions

Figure 10

(Color online) (a) An input single-charge vortex beam in the defocusing regime; (b) the beam profile and phase at the output crystal face for low input intensity; (c) output for high input intensity showing generation of a stable single-charge vortex.Reuse & Permissions

Figure 11

(Color online) (a) An input double-charge vortex beam in the defocusing regime; (b) the beam profile and phase at the output crystal face for low input intensity; (c) output for high input intensity showing instability of the double-charge vortex.Reuse & Permissions