Nonlocal Stabilization of Nonlinear Beams in a Self-Focusing Atomic Vapor

We show that ballistic transport of optically excited atoms in an atomic vapor provides a nonlocal nonlinearity which stabilizes the propagation of vortex beams and higher order modes in the presence of a self-focusing nonlinearity. Numerical experiments demonstrate stable propagation of higher order nonlinear states (dipole, vortices, and rotating azimuthons) over a hundred diffraction lengths, before dissipation leads to decay of these structures.

Figure 1

Characteristic length scales ${\ell }_c^{(g-g)}$, ${\ell }_c^{(g-e)}$, and ${\ell }_d$ in a Rb vapor cell as a function of temperature. The inset shows the logarithm of the ballistic response function $\sim ({\ell }_d/r){\int }_0^{\infty }d\xi e^{-r/({\ell }_d\xi )}{e}^{-{\xi }^2}$ (solid red line) and the response function $K_0(r/{\ell }_d)$ for a 2D diffusive nonlocal equation [8] (dashed blue line) as a function of $r/{\ell }_d$. The response functions have been scaled to be equal at $r={\ell }_d$.

Figure 2

(a) Saturation function $f_{\mathrm{sat}}=\mathrm{Re}[Z(a+i{b}_I)]-\mathrm{Re}[Z(a+ib)]$ for $a=4$ and $b=0.0083$. (b) Nonlocal single charged vortex mode with power $P\simeq 0.4\mathrm{W}$ (red line and red axis). The solid blue line shows the nonlocal nonlinear index ${\chi }_{\mathrm{nl}}^{\prime }$ computed from Eq. (5), the dashed blue line the local one computed from Eq. (2a) (blue axis).

Figure 3

(a) Nonlocal (solid lines) and local (dashed lines) dissipative propagation of the single charged vortex mode with input power 0.4 W. The blue lines and blue axis show the beam power, the red lines and red axis the maximal intensity versus propagation distance. (b) Intensity and phase distribution of the nonlocal single charged vortex at input $z=0$ and at $z=160z_d$ just before it decays. (c) Maximal intensity (red) and FWHM (green) of the nonlocal single charged vortex as a function of beam power. The solid lines are computed upon propagation, the diamonds from stationary numerical solutions of the conservative problem. (d) Intensity and phase distribution of the local single charged vortex at $z=0$ and at $z=15z_d$ when it decays. (e) Intensity and phase distribution of the nonlocal dipole mode at $z=0$ and at $z=160z_d$ just before it decays. (f) Same for the nonlocal double-charged vortex at $z=0$ and at $z=100z_d$.

Figure 4

Dissipative propagation of a dipole azimuthon (upper row) and quadrupole azimuthon (lower row). In both cases the input power was about 0.4 W and the modulation depth $\kappa \simeq 0.5$.