Cascaded Generation of Multiply Charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium

Using an example of a Raman active medium we describe how a common nonlinear process of four-wave mixing can be used to induce strong coupling between the spatial and temporal degrees of freedom in optical waves. This coupling produces several unexpected effects. Amongst those are cascaded excitation of multiply charged optical vortices, spatial focusing in a nonlinearly defocusing medium, and generation of helically shaped spatiotemporal optical solitons.

Figure 1

The smaller panels show the transverse profiles of the sideband phases [$\arg \mathbf{(}{E}_n(x,y)\mathbf{)}$]. The larger panel shows the $x$ distribution of intensities, $|E_n(x,y=13){|}^2$. The propagation distance is $z=0.2$. The window size in each plot is $25\times 25$ dimensionless units ( $\simeq 600\times 600\mu \mathrm{m}$). The other parameters are: $\mu =5$, $j=-1$, $p_2=0.005$, $N=23$, $A_0=\sqrt{2}$, $A_1=1$, $w_0=2\sqrt{5}$, $w_1=\sqrt{10}$.

Figure 2

Intensity distribution of the total field $|E_{\mathrm{tot}}{|}^2$ after different propagation distances $z$ for $t=0$. Azimuthal compression is seen for $z=20$ and subsequent recompression for $z=97$. The nonlinearity is defocusing, $j=-1$. $w_0=20\sqrt{5}$, $w_1=10\sqrt{10}$. The other parameters are the same as in Fig. 1. The window size is $250\times 250$ dimensionless units.

Figure 3

Spectral (a) and temporal (b) profiles of the total field $z=10$ (black lines) and $z=20$ (red lines). To plot (a),(b) we used the cut through the helix taken at the point of maximal intensity $|E_{\mathrm{tot}}{|}^2$. Other parameters are the same as in Fig. 2. (c) Isointensity ($x$, $y$, $t$)-plot at 50% of maximum for $z=20$.

Figure 4

(a) Transverse profiles of $|E_n(x,y=13){|}^2$ for the multifrequency soliton with $\mu =1$, $j=1$, $p_2=0.005$. Soliton parameters are: ${\kappa }_1=-0.5$, ${\kappa }_2=0$. (b) Corresponding evolution of the total intensity $|E_{\mathrm{tot}}{|}^2$ along $z$ with 1% noise in the initial conditions. Isointensity plot at 80% of maximum. (c) The same as (b), but for $\mu =5$. (d) Isointensity plot at 20% of maximum for the case of the initial excitation with two beams. $j=1$ and the other parameters as in Fig. 1. In (b),(c) and (d) the total intensity is evaluated for $t=0$.