(3+1)-dimensional superluminal spatiotemporal optical solitons and vortices at weak light level

A scheme is proposed to produce (3+1)-dimensional superluminal spatiotemporal optical solitons and vortices in a coherent atomic system working in an active Raman gain regime. It is shown that the evolution of the envelope of a signal field obeys a modified (3+1)-dimensional nonlinear Schrödinger equation, which includes dispersion, diffraction, and Kerr nonlinearity. Various solutions of light bullets, light vortices, light-bullet trains, and light-vortex trains are presented, which display many interesting characters, including superluminal propagating velocity and extremely low generating power. In addition, they can be easily manipulated in a controllable way. Stabilization of such high-dimensional superluminal light bullets and vortices can be realized using the trapping potential formed by an additional far-detuned laser field.

Figure 1

(a) Energy-level diagram and excitation scheme of the lifetime-broadened four-state atomic system interacting with a strong CW pump field (with half Rabi frequency ${\Omega }_p$), a weak pulsed signal field (with half Rabi frequency ${\Omega }_s$), and a strong CW control field (with half Rabi frequency ${\Omega }_c$). ${\Delta }_3,$ ${\Delta }_2$, and ${\Delta }_4$ are one-photon, two-photon, and three-photon detunings, respectively. (b) Possible arrangement of beam geometry. ${\omega }_p$, ${\omega }_s$, ${\omega }_c$, and ${\omega }_L$ are angular frequencies of the pump, signal, control, and far-detuned laser fields, respectively.

Figure 2

The linear dispersion relation $K$ of the signal field as functions of dimensionless frequency $\omega /{\gamma }_{2,4}$. Panels (a) and (b) correspond to the absence (${\Omega }_c=0$) and the presence (${\Omega }_c=5\times {10}^7$ Hz) of the control field, respectively. In both panels, the solid and dashed-dotted lines denote, respectively, the real part Re$\left(K\right)$ and the negative imaginary part $-$Im($K$) of $K$.

Figure 3

(a) and (b) Isosurface ($\left|\psi \right|=0.01$) plots of LBs in the self-focusing case ($g_{11}=1$, $g_{12}=1$) for $(c_0,\mu )=(1.8,1.4)$ and $(c_0,\mu )=(2.3,2.7)$, respectively. The external trapping potential contributed by the far-detuned laser field is a zero-order (i.e., $l=0$) Bessel function. (c) Signal-field power $P$ as a function of $\mu$ and $c_0$. (d) The real part of maximum eigenvalue, i.e., Re($\lambda$), as a function of $\mu$ and $c_0$ obtained by solving the eigenvalue problem Eq. (14). In panels (c) and (d), the dotted-dashed, dashed, and solid lines are for $c_0=1.8,2.3$, and $2.8$, respectively.

Figure 4

Evolution of $\left|u\right(x=0,y=0,z,t/{\tau }_0\left)\right|$ as a function of $t/{\tau }_0$ and $z$ by numerically solving Eq. (11). Solid red line and dashed blue line are for $z=0$ and 0.5 cm,respectively.

Figure 5

Isosurface plot (a) with $\left|\psi \right|=0.01$ and 3D shaded surface (i.e., $\tau =0$) plot (b) in the self-focusing case (i.e., $g_{11}=1$, $g_{12}=1$) for $(c_0,\mu )=(2.5,0.9)$. The trapping potential contributed by the far-detuned laser field is a zero-order (i.e., $l=0$) Bessel function. (c) Signal-field power $P$ as a function of $\mu$ and $c_0$. (d) Real part of maximum eigenvalue, i.e., Re($\lambda$), as a function of $\mu$ and $c_0$ obtained by solving the eigenvalue problem Eq. (14). In panels (c) and (d), the dotted-dashed, dashed, and solid lines are for $c_0=1.5,2.5$, and 3.5, respectively.

Figure 6

Isosurface plots for the evolution of a light vortex with $\left|u\right|=0.1$ for $s=0.0$, 3.0, and 6.0, respectively. The initial condition is taken as that given in Fig. 5a.

Figure 7

(a) Isosurface plot of the zero-order LB train with $\left|\psi \right|=0.1$ in the self-defocusing case (i.e., $g_{11}=-1$, $g_{12}=1$) for $(c_0,\mu )=(3,2)$. (b) Signal-field power $P$ as a function of $\mu$ and $c_0$. (c) Real part of maximum eigenvalue, i.e., Re($\lambda$), as a function of $\mu$ and $c_0$ obtained by solving the eigenvalue problem Eq. (14). In panels (b) and (c), the solid, dashed, and dotted-dashed lines are for $c_0=2.3,2.6$, and 3.0, respectively.

Figure 8

Isosurface plots for the evolution of the LB train with $\left|u\right|=0.1$ for $s$ $=$ 0.0, 1.5, 4.5 based on the results by solving Eq. (11). The initial condition is taken as that given in Fig. 7a.

Figure 9

(a) Isosurface plot of the zero-order ($l=0$) light-vortex train with $\left|\psi \right|=0.3$ in the self-defocusing case (i.e., $g_{11}=-1$, $g_{12}=1$) for $(c_0,\mu )=(3,1)$, and $\tau$ from $-$3.5 to 3.5. (b) Signal-field power $P$ as a function of $\mu$ and $c_0$. (c) Real part of maximum eigenvalue, i.e., Re($\lambda$), as a function of $\mu$ and $c_0$ obtained by solving the eigenvalue problem Eq. (14). In panels (b) and (c), the solid, dashed, and dotted-dashed lines are for $c_0=2.5,2.7$, and 3.0, respectively.

Figure 10

Isosurface plots for the evolution of the light-vortex train with $\left|u\right|=0.3$ for $s$ $=$ 0.0, 5.0, 10.0 based on solving Eq. (11). The initial condition is taken as that given in Fig. 9a.