Arrest of Domain Coarsening via Antiperiodic Regimes in Delay Systems

Motionless domain walls representing connecting temporal or spatial orbits typically exist only for very specific parameters, around the so-called Maxwell point. This report introduces a novel mechanism for stabilizing temporal domain walls away from this peculiar equilibrium, opening up new possibilities to encode information in dynamical systems. It is based on antiperiodic regimes in a delayed system close to a bistable situation, leading to a cancellation of the average drift velocity. The results are demonstrated in a normal form model and experimentally in a laser with optical injection and delayed feedback.

Figure 1

Illustration of dynamics in space-time diagrams. These space-time diagrams are folded over a time $T=\tau +|\eta {|}^{-1}$. (a) Temporal pattern imposed as an initial condition. The $P1$ regime where $\kappa =-1$ leads to the coarsening dynamics of the temporal time trace depicted in (b) and (c) towards the high and low states when $\beta =-10^{-2}$ (b) and $\beta =10^{-2}$ (c), respectively. Other parameters are $\tau ={10}^3$, $\eta =5\times 10^{-2}$, and $\mu =2.5\times 10^{-2}$, or equivalently $a=-5\times 10^{-2}$, $b_1=0$, and $a_1=7.5\times {10}^4$. The $P2$ regime where $\kappa =1$ induces the inversion of the initial condition at each round-trip as exemplified by the checkerboard pattern in (d). It does not coarsen over longer time scales even for very large values of the asymmetry $\beta =0.1$. The time trace is folded over times $T$ in (e) and also $2T$ in (f) for clarity. Parameters are $\eta =-0.2$ and $\mu =-2.5\times 10^{-2}$, or equivalently $a=-0.2$, $b_1=0$, and $a_1=17.25\times {10}^4$. The temporal profile is represented in (g). Notice the large values of $a_1$, signaling that Eq. (3) remains qualitatively valid far from the bifurcation points.

Figure 2

(a) Nonlinear output intensity $I=|E{|}^2$ as a function of the injected field $Y$ in blue. The black dots represents the amplitude of the upper and lower plateau in the $P2$ regime. The space-time diagram of the intensity in panel (b) is folded with a period $T=2\tau +0.1$ and shows the stability of the domain walls, even in the presence of noise, see also panel (c). Parameters are $\alpha =2$, $J=-0.1$, $\mathrm{\Delta }=2.3$, $\tilde{\eta }=0.1$, and $\mathrm{\Omega }=\pi$.

Figure 3

(a) Experimental setup. ML, master laser; PC, polarization controller; VOA; variable optical attenuator; PM, power meter; FP, Fabry-Perot; CIRC, circulator; PIN, 12 GHz photodetector; SCOPE, 13 GHz oscilloscope. (b) LI curve of the SL at $T=293\mathrm{K}$. (c) I/O power relationship of the SL under external injection with $\mathrm{\Delta }f=-10\mathrm{GHz}$ and $I=4.85\mathrm{mA}$.

Figure 4

Temporal trace of the laser intensity (a) and space-time diagram with folding parameter $\tau$ (b) and $2\tau$ (c).

Figure 5

Same as Fig. 4. A new pair of domain wall is nucleated at $n=128$ in panel (a) and remains stable while for different parameters bistable phase coarsening occurs.