Light-driven liquid-crystalline nonlinear oscillator under optical periodic forcing

An all-optically driven strategy to govern a liquid-crystalline collective molecular nonlinear oscillator is discussed. It does not require external feedback of any kind while the oscillator and a time-dependent perturbation both are sustained by incident light. Various dynamical regimes such as frequency-locked, quasiperiodic, forced, and chaotic are observed, in agreement with a theoretical approach developed in the limit of the plane-wave approximation.

Figure 1

Experimental setup. Multiline ${\mathrm{Ar}}^{+}$, argon-ion laser; DP, dispersion prism; $M$, mirror; $\lambda ∕2$, half-wave plate; $P$, polarizer; $R\text{−}\lambda ∕2$, rotating half-wave plate; BS, beam splitter; $\lambda ∕4$, quarter-wave plate; $L$, lens; NLC, film of nematic LC; $d$, diaphragm; IF, interferential filter for $514.5\mathrm{nm}$; $D_i$, photodiodes. (a),(b) Linear and circular geometries. ${\mathbf{k}}_0$ and $\mathbf{E}$: incident wave vector and electric field.

Figure 2

Maps of the dynamical regimes near the 2:1 entrainment region in linear and circular geometries. (a),$\left({\mathrm{a}}^{\prime }\right)$ $L_{p∕q}$, frequency-locked $(f=f_F∕p)$; $F$, forced $(f=f_F)$; QP, quasiperiodic; PD, region of period doubling [$f=f_F∕\left(2n\right)$, $n=2,3,\dots$]; $C$, chaos. The director dynamics on the unit sphere is sketched below (b),$\left({\mathrm{b}}^{\prime }\right)$ and above (c),$\left({\mathrm{c}}^{\prime }\right)$, the secondary Hopf bifurcation.

Figure 3

Demonstration of QP, $L_{2∕1}$, and $L_{3∕2}$ regimes in linear geometry for $\delta \rho ∕{\rho }_0=0.05$ and $\alpha =3.5°$. Gray (black) curves: experimental (calculated) power spectra of $I_{\mathrm{c}}$ $\left(\Delta \right)$. (a) Right part of the $L_{2∕1}$ tongue at $f_F∕f_N=2.37$, (b) inside the $L_{2∕1}$ tongue at $f_F∕f_N=2$, (c) left part of the $L_{2∕1}$ tongue at $f_F∕f_N=1.62$, and (d) inside the $L_{3∕2}$ tongue at $f_F∕f_N=1.5$. For QP states (a) and (c) the frequency peaks (1–5) refer to ($f_N$, $f_F-f_N$, $2f_N$, $f_F$, $f_N+f_F$), respectively.

Figure 4

Director trajectories in the $({\varphi }_1,{\varphi }_2,{\theta }_1)$ $\left[\right({\Theta }_1,{\Theta }_2,{\Phi }_1\left)\right]$ phase space for linear [circular] geometry illustrating the transition to chaos at $\delta \rho ∕{\rho }_0=0.3$ $[\delta \rho ∕{\rho }_0=0.27]$ for the same parameters as for the maps shown in Fig. 2. (a),$\left({\mathrm{a}}^{\prime }\right)$ $f_F∕f_N=1.5$ $\left[1.767\right]$ (limit cycle in the $L_{2∕1}$ regime); (b),$\left({\mathrm{b}}^{\prime }\right)$ $f_F∕f_N=1.35$ $\left[1.742\right]$ (limit cycle after the first period doubling); (c),$\left({\mathrm{c}}^{\prime }\right)$ $f_F∕f_N=1.27$ $\left[1.691\right]$ (limit cycle after the second period doubling); (d),$\left({\mathrm{d}}^{\prime }\right)$ $f_F∕f_N=1.18$ $\left[1.632\right]$ (chaotic attractor).

Figure 5

Demonstration of the sequence $L\to \mathrm{PD}\to C$ (circular geometry). Power spectra of $I_{\mathrm{c}}$ (experiment) and $\Delta$ (calculated) are shown at $\delta \rho ∕{\rho }_0=0.22$. (a),$\left({\mathrm{a}}^{\prime }\right)$ Locked state at $f_F∕f_N\simeq 2.05$ and $f_F∕f_N=2.06$; (b),$\left({\mathrm{b}}^{\prime }\right)$ state within PD region at $f_F∕f_N\simeq 1.6$ and $f_F∕f_N=1.65$; (c),$\left({\mathrm{c}}^{\prime }\right)$ chaotic state at $f_F∕f_N\simeq 1.45$ and $f_F∕f_N=1.58$.

Figure 6

Window of regular dynamics in the chaotic region for $\delta \rho ∕{\rho }_0=0.22$ in the circular case. (a) $f_F∕f_N=1.525$ (chaotic state), (b) $f_F∕f_N=1.5195$ (periodic state with $f=f_F∕3$), (c) $f_F∕f_N=1.5184$ (periodic state with $f=f_F∕6$), (d) $f_F∕f_N=1.5182$ (periodic state with $f=f_F∕12$), and (e) $f_F∕f_N=1.5165$ (chaotic state).