Localized standing waves induced by spatiotemporal forcing

Particle-type solutions are observed in out-of-equilibrium systems. These states can be motionless, oscillatory, or propagative depending on the injection and dissipation of energy. We investigate a family of localized standing waves based on a liquid-crystal light valve with spatiotemporal modulated optical feedback. These states are nonlinear waves in which energy concentrates in a localized and oscillatory manner. The organization of the family of solutions is characterized as a function of the applied voltage. Close to the reorientation transition, an amplitude equation allows us to elucidate the origin of these localized states and establish their bifurcation diagram. Theoretical findings are in qualitative agreement with experimental observations. Our results open the possibility of manipulating localized states induced by light, which can be used to expand and improve the storage and manipulation of information.

Figure 1

Experimental localized standing wave state. (a) Schematic representation of a liquid-crystal light valve (LCLV) with optical spatiotemporal modulated feedback. SLM accounts for the spatial light modulator, M are mirrors, PSB is the polarized beam splitter, and $V_0$ is the driven voltage applied to LCLV. O is an optical objective, and FB is a fiber bundle. (b) shows an experimental 3D-spatial view of a localized standing wave and (c) shows a spatiotemporal evolution for the same structure. The color scale shows the intensity of the detected light. This standing wave was obtained at $V_0=2.800V_{\text{rms}}$.

Figure 2

(a) Bifurcation diagram of localized waves. Average light intensity over a cycle $I$ as a function of the driven voltage $V_0$. H and SW account for uniform states and homogeneous standing waves (at $I_1=0.056\text{mW}$), by increasing (I) or decreasing (D) the voltage in the experiment. LS $n\text{−}m$ accounts for the localized standing wave with $n$ and $m$ bump in an oscillation cycle. (b) Spatiotemporal evolution, initial ($t_1$) and final ($t_2$) states for different localized standing waves. Left and right panels account for localized standing wave 3-3 and 1-2, respectively. (c) Experimental expansion, $V_0=2.878V_{\text{rms}}$ (left), and contraction, $V_0=2.700V_{\text{rms}}$ (right), of the localized structure. The top and bottom panels are instantaneous profiles and spatiotemporal diagrams, respectively. The arrows indicate the propagation of the upper standing wave.

Figure 3

Localized standing waves obtained by numerical simulations of the bistable model Eq. (7) with $\mu =1.0, \lambda =0.035, \omega =0.53$, and $\gamma =0.5$. (a) and (b) Localized standing wave profiles and spatiotemporal diagram between 1-2 and 2-3 bumps in an oscillation cycle. The middle panels account for the spatiotemporal evolution of the localized standing waves. $\delta$ and $\mathrm{\Delta }$ account for the position and width of localized standing waves. (c) Bifurcation diagram of model Eq. (7) with $\mu =1.0, \lambda =0.07, \omega =4$, and $\gamma =2$. Total area $‖\mathrm{\Theta }\left(t\right)‖=\int \mathrm{\Theta }(x,t)dx/L$ as a function of the bifurcation parameter $\eta$. $L$ is the system size. ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ account for the upper and lower standing extended waves, respectively. $\text{LS}n\text{−}m$ accounts for the localized standing wave with $n$ and $m$ bump in an oscillation cycle.

Figure 4

Destabilization of localized standing waves. Numerical expansion, $\eta =-0.04$ (a), and contraction, $\eta =0.04$ (b), of the localized standing wave of model Eq. (1) by $\mu =1.0, \lambda =0.035, \omega =0.53$, and $\gamma =0.5$.