Optical wall dynamics induced by coexistence of monostable and bistable spatial regions

When nonequilibrium extended homogeneous systems exhibit multistability, it leads to the presence of domain walls between the existing equilibria. Depending on the stability of the steady states, the dynamics differs. Here, we consider the interface dynamics in the case of a spatially inhomogeneous system, namely, an optical system where the control parameter is spatially Gaussian. Then interfaces connect the monostable and the bistable nonuniform states that are associated with two distinct spatial regions. The coexistence of these two regions of different stability induces relaxation dynamics and the propagation of a wall with a time-dependent speed. We emphasize analytically these two dynamical behaviors using a generic bistable model. Experimentally, an inhomogeneous Gaussian light beam traveling through either a dye-doped liquid crystal cell or a Kerr cavity depicts these behaviors, in agreement with the theoretical predictions.

Figure 1

(a–e) Different configurations of the potential $V(u,x)$ [Eq. (6)] versus the spatially dependent bifurcation parameter $\eta$: (a) $\eta <{\eta }_{\mathrm{down}}$, (b) ${\eta }_{\mathrm{down}}\le \eta <0$, (c) $\eta =0$, (d) $0<\eta \le {\eta }_{\mathrm{up}}$, and (e) ${\eta }_{\mathrm{up}}<\eta$. (f) Spatial profile of $\eta \left(x\right)$, together with corresponding operating regimes (monostable or bistable). (g–k) Spatiotemporal evolution diagrams of $u\left(x\right)$ associated with the potentials $V(u,x)$ (a–e). (i) Spatiotemporal evolution diagrams of $u\left(x\right)$ at the Maxwell point, associated with the potentials $V(u,x)$ (c). (m) Hysteresis cycle for the order parameter $u$ versus $\eta$. ${\eta }_0=1,{\eta }_M=0.5$.

Figure 2

Numerical spatiotemporal front propagation in the single monostable regime. From model Eq. (1) with ${\eta }_M=0.5,w=400$, (a) ${\eta }_0=0.89$, (b) ${\eta }_0=0.90$, and (c) ${\eta }_0=0.92$. Initial condition $u(x,t=t_0)=-1$.

Figure 3

Temporal evolution of $u(x=0,t)$ from the ghost point, $u\approx -1$, to a stationary point of the monostable regime obtained from Fig. 2. ${\eta }_0=0.9,{\eta }_M=0.5,u\left(t=0\right)=-0.9$. $•$ Numerical simulations, $—$ analytical prediction for $u_1\left(t\right)$ from formula (9), and $—$ analytical prediction for $u_2\left(t\right)$ from formula (12).

Figure 4

Evolution of the time spent in the bottleneck versus input energy. (a) Numerical simulations; $•$ numerical values; and $—$ analytical prediction of nondimensional $\tau$, measured in units of numerical simulation following relation (11); $\eta \left(x=0\right)={\eta }_0-{\eta }_{\mathrm{Max}}$; ${\eta }_0=0.9,{\eta }_M=0.5,w=200,u\left(t=0\right)=-0.9$. (b) Experiments on the transition from nematic to liquid phase in dye-doped nematic liquid crystals [37]; $⧫$ experimental values of $T$ in seconds and $—$ fit on experimental points using the relation $T=m\tau$, where $m=1.28$ is a multiplying dimension factor and $\tau$ comes from formula (11).

Figure 5

1D and 2D numerical simulations of Eqs. (1) and (21) with a parabolic forcing. (a) Spatiotemporal diagram for the 1D system, Eq. (1). (b) Front trajectory in the 1D system: $⧫$ numerical points, $—$ analytical prediction from Eq. (19). (c) Spatiotemporal diagram for the 2D system, Eq. (21). (d) Front trajectory in the 2D system: $⧫$ numerical points, $—$ analytical prediction from Eq. (19). (e) Spatiotemporal diagram of the relaxation phenomenon for the 2D system with (f) the front position. (g) Numerical profile of $u\left(r,t\right)$ at $——t=0$, $•—t=5$, $‐‐‐t=20$, $—t=600$. The parameters used for both numerical simulations are ${\eta }_M=0.5$, $w=250$, and ${\eta }_0=1$.

Figure 6

Numerical comparison between the relaxation and front propagation characteristic times versus the bifurcation parameter ${\eta }_0$, Eq. (21); ${\eta }_M=0.5,w=250$.

Figure 7

Schematic sketch of the experimental setup of a nematic-isotropic transition induced by photoisomerization. DDLC: dye-doped liquid crystal illuminated by a vertically polarized laser beam ($P_{\parallel }$), PL: linear polarizer following the $y$ axis ($P_{\perp }$), L: imaging lens, and CDD: charge-coupled device camera.

Figure 8

Experimental nematic-isotropic front propagation. (a,b) 1D experimental snapshots for $P_0=2.2W,w_x=3.4\text{mm}$, and $w_y=0.37\text{mm}$ at 13 and 80 s, respectively. (c) Spatiotemporal diagram of the intensity profile along the dotted line on the (a) snapshot. (d–g) 2D experimental temporal sequence of snapshots for $P_0=350\text{mW}$ and $w_x=w_y=3.4\text{mm}$ at 55, 65, 125, and 500 s, respectively.

Figure 9

Temporal wall location evolution in 1D configuration $x_0\left(t\right):\blacksquare$ experimental points, $—$ fit with hyperbolic tangent expression (19).

Figure 10

One-dimensional spatiotemporal diagram of the experimental two-dimensional front propagation for three different power inputs: (a) $P_0=283\text{mW}$, (b) $P_0=300\text{mW,}$ and (c) $P_0=317\text{mW}$.

Figure 11

(a) Experimental spatiotemporal diagram on Figs. 8–8 taken along one black disk diameter. (b) Wall location evolution: $\blacksquare$ experimental points, $—$ fit with hyperbolic tangent expression (19).

Figure 12

Kerr cavity submitted to an inhomogeneous Gaussian light beam. (a) Schematic representation of a Kerr cavity. $M_1$ and $M_2$ are the cavity mirrors, and LC is the liquid crystal slice. (c) Experimental and (d) numerical spatiotemporal response to a step function of the input intensity from the lower (dark area) to the upper (white area) branch of the bistable cycle. Experiments $I_0=433\text{W}{\text{cm}}^{-2},d=-5\text{mm},\phi =-0.6$ rad, $w_x=1400\mu \text{m},w_y=100\mu \text{m}$, $R_1=81.8\%;R_2=81.4\%$. (b,d) Numerical simulation of model Eq. (26) with $E_0=1.9,\mathrm{\Delta }=3.0$, $\alpha =0.001,w_x=1400\mu \text{m},w_y=100\mu \text{m}$, $\varepsilon =0.4$. (d) Wall location evolution: $⧫$ and dashed curve account for experimental measurements and hyperbolic tangent fit, formula (19). The experimental fit parameters are $a=308\mu \mathrm{m}$, $b=0.069$, and $t_0=35.2$ s.