Two-Dimensional Front Dynamics and Spatial Solitons in a Nonlinear Optical System

Two-dimensional fronts and coarsening dynamics with a $t^{1/2}$ power law are analyzed experimentally and theoretically in a nonlinear optical system of a sodium vapor cell with single-mirror feedback. Modifications of the $t^{1/2}$ power law are observed in the vicinity of a modulational instability leading to the formation of spatial solitons of different sizes. The experimental and numerical observations give direct evidence for the locking of fronts as the mechanism of soliton formation. A phenomenological equation for the dynamics of the domain radius explains the observed behavior.

Figure 1

Schematic experimental setup. $\lambda /8$ retardation plate ($\lambda /8$), mirror (M, $R\approx 0.99$), linear polarizer (LP), CCD camera (integration time $1\mu \mathrm{s}$), Acousto-optic modulator (AOM). The principal axes of the two $\lambda /8$-plate are perpendicular to each other so that the polarization state of the light field after transmission through the cell is reestablished before detection. An “analyzer” LP in a rotary stage is used to measure the polarization of the light field and is typically adjusted to suppress the background state.

Figure 2

The squared radius of a contracting circular domain versus time. Solid squares (open circles) denote experimental (numerical) measurements with a straight line linear fit. The insets are snapshots of the beam shape in a gray scale (“bright” denoting high intensity). The background state has been suppressed by a linear polarizer. Experimental parameters: mirror distance $d=112\mathrm{mm}$; buffer gas pressure $p_{N_2}=300\mathrm{mbar}$; cell temperature $T=346.0°\mathrm{C}$, laser detuning $\Delta =15.8\mathrm{GHz}$, laser power $P_{\mathrm{in}}=90\mathrm{mW}$, laser beam radius $w_0=1.89\mathrm{mm}$, rotation of the slow axis of the $\lambda /8$ plate with respect to input polarization $\Phi =0°0^{\prime }$. Numerical parameters: $d=120\mathrm{mm}$, ground state relaxation rate $\gamma =200{\mathrm{s}}^{-1}$, relaxation rate of optical coherences ${\Gamma }_2=9.94\times {10}^9{\mathrm{s}}^{-1}$, $\Delta =11.9\mathrm{GHz}$, diffusion constant $D=255{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, particle density $N=4.65\times {10}^{19}{\mathrm{m}}^{-3}$, length of heated zone $L=15\mathrm{mm}$, mirror reflectivity $R=99.5\%$, pump rate $P_0=50\times {10}^3{\mathrm{s}}^{-1}$, $\Phi =0°0^{\prime }$.

Figure 3

Coefficients ${\gamma }_i$ and ${\gamma }_c$ versus the wave plate rotation angle $\rho$. Solid squares are experimental results, open circles have been obtained by numerical simulations.

Figure 4

Coefficient ${\gamma }_c$ as a function of the input power (experiment) and pump rate (simulations). Here, experimentally, $T=354.7°\mathrm{C}$, $\Delta =17.5\mathrm{GHz}$.

Figure 5

Contraction of a circular domain close to the locking region ($P_{\mathrm{in}}=190\mathrm{mW}$). Insets show images of the domain at the plateaus of the contraction curve.

Figure 6

Regions of existence of dissipative solitons of different orders [from first to fifth order: squares (red), circles (green), up-triangles (magenta), down-triangles (cyan), diamonds (black)]. (a) Experiment; (b) Newton method; (c) analytic solution of Eq. (2). Here in (a) $T=360.2°\mathrm{C}$ and $\Delta =16.3\mathrm{GHz}$. In (b) $w_0=1.89\mathrm{mm}$.