Generation of rotary beams by interaction of moving solitons in nonlocal media

We report that nonlocal nonlinear media allow a controlled interaction among coherent solitons with a relative tilt previously imposed, resulting in the generation of rotary self-trapped beams. We demonstrate that two initially separated fundamental solitons can interact, generating stable rotating dipoles for a continuum interval of relative tilt values. Surprisingly, we find that for a higher number of initial solitons launched and after some emission of radiation waves, the initial self-trapped structures can decay into rotating dipole solitons. The normalized orbital angular momentum of these rotating dipoles can be controlled by adjusting the initial tilt.

Figure 1

Propagation dynamics of two solitons with the following parameters: (a) $\lambda =10,\mu =0.06,\left(x_{o,}{y}_o\right)=(-2,1.4)$, and $L=15$, (b) $\lambda =50,\mu =0.5,\left(x_{o,}{y}_o\right)=(-3,1)$, and $L=17$, and (c) and (d) $\lambda =100,\mu =3,\left(x_{o,}{y}_o\right)=(-2,1)$, and $L=12.$ In (d), there is an input noise of 10%. The profiles are shown in an $x-y$ box of $L\times L$. (See the Supplemental Material [27] to observe propagation.)

Figure 2

(a) Existence domain of the dipole solitons showing the averaged power beam $\overline{P}$. (b) Averaged fractional spin beam $\overline{S}$ of the dipole solitons. (c) Evolution of the power beam $P$ in propagation for $\lambda =75$. (d) Evolution of the fractional spin beam $S$ for $\lambda =75$.

Figure 3

Propagation dynamics of multiple solitons with the parameters $\lambda =15,\mu =1$, and $R=10$ for (a) $N=3$, (b) $N=4$, (c) $N=5$, (d) $N=6$, and (e) $N=8$. The profiles are shown in a $x-y$ box of $L\times L$, where $L=25$ for the first two columns and $L=10$ for the last two columns. (f) Existence domain for the generation of rotating dipoles from $N=5$ (lowest power beam $\overline{P})$ to $N=8$ (highest power beam $\overline{P})$. (g) Three-dimensional representation for propagation dynamics when $N=8$. (See Supplemental Material [27] to observe propagation.)

Figure 4

Evolution of (a) power beam $P$ and (b) spin beam $S$ when $N=4$. For $\lambda =10,\mu =0.25$ and $R=4$, and for $\lambda =50,\mu =0.1$ and $R=3.$ Variation of (c) power beam $P$ and (d) spin beam $S$ when $N=8$. In both cases $\lambda =10,\mu =1$, and $R=9.$

Figure 5

Evolution of six $\left(N=6\right)$ fundamental tilted solitons in nonlocal media with a Gaussian [(a) and (c)] and exponential [(b) and (d)] response function $\mathcal{N}\left(r\right)$. (a) $\mu =2.3$, (b) $\mu =2.3$, (c) $\mu =2.7$, and (d) $\mu =2.7.$ In all cases $R=10$ and $P=220.69$ for each fundamental soliton. The profiles are shown in an $x-y$ box of $L\times L$, where $L=24$ for the first two columns and $L=12$ for the last two columns.

Figure 6

Evolution of four fundamental solitons $(N=4)$ in Gaussian [(a) and (c)] and exponential [(b) and (d)] nonlocal response, where two of the solitons have been initially moved by an angular displacement given by ${\Phi }_{\epsilon }$. (a) ${\Phi }_{\epsilon }=\pi /12$, (b) ${\Phi }_{\epsilon }=\pi /12$, (c) ${\Phi }_{\epsilon }=\pi /18$, and (d)${\Phi }_{\epsilon }=\pi /18.$ For each fundamental soliton, $P=220.69$ and $\mu =0.5$. In all cases $R=10$. The profiles are shown in an $x-y$ box of $L\times L$, where $L=24$ for the first column and $L=12$ for the other three columns.