Nonlinear Localization of Dissipative Modulation Instability

Modulation instability (MI) in the presence of noise typically leads to an irreversible and complete disintegration of a plane wave background. Here we report on experiments performed in a coherently driven nonlinear optical resonator that demonstrate nonlinear localization of dissipative MI: formation of persisting domains of MI-driven spatiotemporal chaos surrounded by a stable quasi-plane-wave background. The persisting localization ensues from a combination of bistability and complex spatiotemporal nonlinear dynamics that together permit a locally induced domain of MI to be pinned by a shallow modulation on the plane wave background. We further show that the localized domains of spatiotemporal chaos can be individually addressed—turned on and off at will—and we explore their transport behavior as the strength of the pinning is controlled. Our results reveal new fundamental dynamics at the interface of front dynamics and MI, and offer a route for tailored patterns of noiselike bursts of light.

Figure 1

(a) Intensity levels of the steady-state plane wave solutions of Eq. (1) for a detuning $\mathrm{\Delta }=9$ as a function of the driving intensity $X$. (b) Results obtained by numerically integrating Eq. (1) with the split-step Fourier method with a homogeneous driving intensity $X=75$. The initial condition corresponds to the low-level homogeneous state superimposed with a localized perturbation, and gives rise to a uniformly expanding domain due to MI. (c) Numerical simulation results, showing how the expansion of the MI-induced domain can be arrested by a modulated driving field ($X_0=60$, $\omega =2\pi /50$, $ϵ=0.25$). Top panels in (b) and (c) highlight the driving profile $X(\tau )$ while black and blue horizontal curves in (c) show $X_s=53$ and $X_c=40$ (values pertinent to $\mathrm{\Delta }=9$), respectively.

Figure 2

(a) Schematic diagram of the experimental setup. Amplitude modulator (AM), Erbium-doped fibre amplifier (EDFA), oscilloscope (osc.), optical spectrum analyzer (OSA), photodetector (PD). (b) Illustration of a nonlinearly tilted cavity resonance. Stable plane wave state and persisting spatiotemporal chaos can coexist for detunings slightly above the beginning of plane wave bistability (magenta shaded region). (c) Experimental measurement of the hysteresis around the up-switching point. Solid curves show the mean intensity of the intracavity field as the pump-cavity detuning is increased (red curve) or decreased (blue curve). Shaded regions correspond to one-quarter of the standard deviation of the recorded intensity, highlighting the incoherent (coherent) character of the state born from MI (plane wave). (d),(e) Concatenation of oscilloscope traces, showing how a perturbation pulse permits the deterministic excitation of a localized domain. In (d) the driving field exhibits no modulation and the MI domain expands uniformly. In (e), a modulation with $ϵ=0.25$ applied on the driving field arrests the expansion, localizing the spatiotemporal complexity.

Figure 3

(a) Experimental results, showing how a dip perturbation permits the deterministic erasure of a chaotic domain. (b) Numerical simulation results corresponding to the experiments shown in (a). The simulation results were obtained from integration of Eq. (1) and subsequently convolved with an 80 ps detector response function to facilitate comparison with our experiments. (c) Experimental demonstration of individual and parallel addressing of three localized MI domains. Solid green circles (black crosses) highlight roundtrips and locations where pulse (dip) perturbations are added on the driving field. Color bar for all panels is the same as in Figs. 2 and 2. Note the different axis limits in (a),(b), and (c).

Figure 4

(a)–(c), Experimental results, showing the emergence of localized domains born from MI for three different modulation frequencies applied on the driving field: (a), 0.96; (b), 1.9; (c), 2.9 GHz. For each case, the modulation depth $ϵ\approx 0.25$. The top panels show oscilloscope recordings of the driving field intensity profiles. (d) Experimentally observed discrete transport of domains, realized with an intermediate modulation depth $ϵ\approx 0.20$ and a modulation frequency of 2.7 GHz. (e) Numerical simulation results corresponding to experiments in (d). Color bar for all panels is the same as in Figs. 2 and 2.