Anomalous interaction of nonlocal solitons in media with competing nonlinearities

We theoretically investigate properties of individual bright spatial solitons and their interaction in nonlocal media with competing focusing and defocusing nonlinearities. We consider the general case with both nonlinear responses characterized by different strengths and degrees of nonlocality. We employ a variational approach to analytically describe soliton properties. In particular, we prove analytically that the interplay of focusing and defocusing nonlocal nonlinearities leads to attraction or repulsion of solitons depending on their separation distance. We then study the propagation and interaction of solitons using numerical simulations of the full model of beam propagation. The numerical simulations fully confirm our analytical results.

Figure 1

Various profiles of the nonlocal response function $R\left(x\right)$ as a function of material parameters: (a) ${\alpha }_1=20$, ${\alpha }_2=-1$, ${\sigma }_1=2$, and ${\sigma }_2=1$; (b) ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$; (c) ${\alpha }_1=2$, ${\alpha }_2=-2$, ${\sigma }_1=1$, and ${\sigma }_2=3$; and (d) ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=10$, and ${\sigma }_2=1$.

Figure 2

Effective potential [Eq. (8)] for the beam power $P=100$ and material parameters corresponding to (a) ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$ and (b) ${\alpha }_1=2$, ${\alpha }_2=-2$, ${\sigma }_1=1$, and ${\sigma }_2=3$.

Figure 3

(a) Normalized intensity of stationary soliton with power $P=100$ propagated stably over 20 diffraction lengths. (b) Intensity (blue) and nonlinearity-induced index (red) profiles. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 4

(a), (b) Evolution of beam width: the initial beam width $W$ exceeding the value corresponding to the maximum of an effective potential in Fig. 2b; (b) zoom of graph (a). (c) Initial and (d) final transverse intensity and refractive index profiles. Here the initial power of the beam $P=100$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-2$, ${\sigma }_1=1$, and ${\sigma }_2=3$.

Figure 5

(a) Effective potential $V$ for in-phase (dashed red line) and out-of-phase (solid blue line) interacting solitons corresponding to the parameters $P=50$, ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$. (b) Zoom from (a).

Figure 6

Intensity (solid red line) and nonlinear refractive index change (dashed blue line) of two out-of-phase solitons with spatial separation of (a) 0.8${\xi }_t$ and (b) 1.5${\xi }_t$, where ${\xi }_t\approx 0.44$. The parameters are $P=50$, ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 7

(a) Effective potential $V\left(\xi \right)$ for in-phase (dashed red line) and out-of-phase (solid blue line) interacting solitons corresponding to the parameters $P=50$, ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=2$, and ${\sigma }_2=1$. (b) Zoom from (a).

Figure 8

Properties of effective potential $V\left(\xi \right)$ for (a) in-phase and (b) out-of-phase interacting solitons. Blue (i) indicates an always attractive potential, while red (ii) indicates that the potential is either attractive or repulsive depending on the spatial separation. The parameters are $P=10$, ${\alpha }_1=2$, and ${\alpha }_2=-1$.

Figure 9

(a) Transition separation as a function of power ($P\in [10,500]$) for $\phi =0$ (solid blue line) and $\phi =\pi$ (dashed red line). The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$. (b) Transition separation as a function of width of nonlocal nonlinearities for $\phi =\pi$ and $P=50$. The medium parameters are ${\alpha }_1=2$ and ${\alpha }_2=-1$.

Figure 10

Trajectories of in-phase interacting solitons with power $P=50$ and initial separation (a) $\xi ={\xi }_t$ and (b) $\xi =1.02{\xi }_t$, where ${\xi }_t\approx 1.11$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 11

Trajectories of interacting out-of-phase solitons with power $P=50$ and initial separation (a) $\xi ={\xi }_t$ and (b) $\xi =1.02{\xi }_t$, where ${\xi }_t\approx 1.07$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 12

Trajectories of interacting in-phase solitons with power $P=50$ and initial separation (a) $\xi ={\xi }_t\approx 1.11$ and (b) $\xi ={\xi }_t\approx 1.07$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 13

(a) Intensity profile of the dipole-mode nonlocal soliton (or bound state of two out-of-phase solitons). Separation between constituent solitons ${\xi }_s\approx 0.44$. (b) Dynamics of the dipole soliton initially perturbed by increasing the separation between solitons to $\xi =1.1{\xi }_s$. The parameters are $P=50$, ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 14

(a) Effective potential of incoherently interacting solitons for $P=50$, ${\alpha }_1=2$, and ${\alpha }_2=-1$. Dashed red (solid blue) curve corresponds to ${\sigma }_1=1,{\sigma }_2=2$ (${\sigma }_1=2,{\sigma }_2=1$). (b) Properties of the potential for ${\alpha }_1>\left|{\alpha }_2\right|$. Blue (i) indicates an always attractive potential, while red (ii) indicates that the potential is either attractive or repulsive depending on the spatial separation of the solitons.

Figure 15

Intensity (solid red line) and nonlinear refractive index change (dashed blue line) of two incoherent solitons with spatial separation of (a) 0.8${\xi }_t$ and (b) 1.5${\xi }_t$, where ${\xi }_t\approx 0.44$. The parameters are $P=50$, ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 16

Trajectories of incoherently interacting solitons for ${\alpha }_1>\left|{\alpha }_2\right|$ with power $P=50$ and initial separation (a) $\xi ={\xi }_t$ and (b) $\xi =1.02{\xi }_t$, where ${\xi }_t\approx 1.09$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 17

Trajectories of incoherently interacting solitons for ${\alpha }_1<\left|{\alpha }_2\right|$ with power $P=50$ and initial separation (a) $\xi ={\xi }_t$ and (b) $\xi =1.02{\xi }_t$, where ${\xi }_t\approx 0.97$. The medium parameters are ${\alpha }_1=1$, ${\alpha }_2=-2$, ${\sigma }_1=1$, and ${\sigma }_2=3$.

Figure 18

Extended trajectories of incoherently colliding solitons from Fig. 16. The power $P=50$ and initial separation $\xi ={\xi }_t$, where ${\xi }_t\approx 1.09$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$. (b) Normalized input ($z/L_D=0$) (red line) and output ($z/L_D=50$) (blue line) intensity profiles.

Figure 19

Trajectories of colliding mutually incoherent solitons with power $P=50$ and initial separation $\xi =0.75{\xi }_t$, where ${\xi }_t\approx 1.09$. Initial velocity (a) $B=1.10$ and (b) $B=1.21$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.

Figure 20

Trajectories of interacting mutually incoherent solitons with power $P=50$ and initial separation $\xi =1.25{\xi }_t$, where ${\xi }_t\approx 1.09$. Initial velocity (a) $B=-0.63$ and (b) $B=-0.66$. The medium parameters are ${\alpha }_1=2$, ${\alpha }_2=-1$, ${\sigma }_1=1$, and ${\sigma }_2=2$.