Turbulence-Induced Rogue Waves in Kerr Resonators

Spontaneous emergence of self-organized patterns and their bifurcations towards a regime of complex dynamics in nonequilibrium dissipative systems is a paradigm of phase transition. Indeed, the behavior of these patterns in the highly nonlinear regime remains less explored, even in recent high-quality-factor resonators such as Kerr-nonlinear optical ones. Here, we investigate theoretically and experimentally the alteration of the resulting Kerr frequency combs from the weakly to the highly nonlinear regime, in the frameworks of spatiotemporal chaos, and dissipative phase transitions. We reveal the existence of a striking and easily accessible scenario of spatiotemporal chaos, free of cavity solitons, in a monostable operating regime, wherein a transition to amplitude turbulence via spatiotemporal intermittency is evidenced. Moreover, statistics of the light bursts in the resulting turbulent regime unveils the existence of rogue waves as extreme events characterized by long-tail statistics.

Popular Summary

Extreme phenomena such as severe floods, earthquakes, rogue ocean waves, or financial crises feature complex dynamics and abrupt transitions from one state to another that make them difficult to study. One inroad to studying these phase transitions is to investigate the spontaneous emergence of self-organized patterns, a field of exploration well suited to fiber-laser cavities. However, the behavior of these patterns in highly nonlinear situations, in which extreme phenomena arise, remains little explored. We use an optical-fiber ring resonator to experimentally observe and theoretically confirm a possible universal scenario for the onset of rogue waves induced by turbulence.

An optical ring resonator is like a whispering gallery for light, in which light circles around a loop of optical-fiber waveguides. This generates a frequency comb—a source of light whose spectrum has many discrete, equally spaced frequency lines. The continuous input light source is turned into a periodic wave inside the resonator.

By increasing the power of the input, we observe two changes in the spectrum. First, we see intermittent changes in the periodicity that reflect the onset of transient turbulence, as expected. But as the input power increases further, the frequency comb loses all periodicity; we capture the onset of proliferating and sustained turbulence accompanied by flashes of light, heralding rogue optical waves. This tells us that rogue waves may be preceded by the collapse of a periodic system.

Our results highlight how experiments in optics can be used to develop an understanding of nonlinear dynamics in many branches of science in which critical phenomena and extreme events are universal.

Figure 1

Experimental setup. ISO, optical isolator; PC, polarization controller; EOM, electro-optic (intensity) modulator; EDFA, erbium-doped fiber amplifier; OBPF, bandpass optical filter; OC, optical circulator; PD, photodiode; HNLF, highly nonlinear optical fiber; AUTOCO, autocorrelator; OSA, optical spectrum analyzer.

Figure 2

Experimental results. Intracavity spectra recorded for different input pump powers: (a) 0.16 W, (b) 0.24 W, (c) 0.56 W, (d) 0.9 W, (e) 3.1 W, and (f) 20.7 W.

Figure 3

Experimental results. Autocorrelation traces recorded for different input pump powers: (a) 0.16 W, (b) 0.24 W, (c) 0.56 W, (d) 0.9 W, (e) 3.1 W, and (f) 20.7 W.

Figure 4

Numerical results obtained from Eqs. (1). Intracavity temporal profiles and corresponding autocorrelation signals obtained for different pump powers: (a) 0.188 W, (b) 0.24 W, (c) 0.45 W, (d) 0.9 W, (e) 3.1 W, and (f) 12.5 W.

Figure 5

Numerical results obtained from Eqs. (1). Probability density function of the intensity peaks of the circulating intracavity field calculated for different pump powers: (a) 0.45 W, (b) 0.9W, (c) 3.1W, and (d) 12.5 W.

Figure 6

Interpolation of experimental data extracted from the experimental measured spectra.

Figure 7

(a) Experimental (red) and numerical (blue) intracavity spectra recorded for different input pump powers. For clarity, experimental spectra are vertically offset by $-20\mathrm{dB}$. (b) Evolution of the intracavity intensity profile over several round trips, computed numerically from the Lugiato-Lefever equation (2) for the same values of the pump powers in the top panel. The value of $I_s$ is calculated in ratio of the intracavity mean power at the threshold $P_{\mathrm{in}}^{\mathrm{th}}$. The detuning parameter is $\mathrm{\Delta }=0.55$.

Figure 8

(a) Lyapunov spectra computed from the numerical integration of Eq. (2) with increasing pump power. (b) The Lyapunov dimension density dimension. (c) Dependence of the Kaplan-Yorke dimension (blue triangles) and the proportion of extreme events (red squares) on the pump power. (d) The equal-time two-points correlation length with respect to the input parameter. (e) Square of the maximum exponential decay rate of the laminar periods inside a round trip of temporal profiles.

Figure 9

Illustration of the process transforming the temporal trace into black (turbulent) and white (laminar) region. In (a) the hills and valley are detected. (b) The temporal trace is transformed into the value of local hills (blue), valley (red), and peak-to-peak (gray, $\mathrm{\Delta }H=⟨H{⟩}_i-⟨V{⟩}_i$). These local values are given by the moving average over three consecutive values: $⟨V{⟩}_i=(V_{i-1}+2V_i+V_{i+1})/4$. (c) Setting $c$ a detection threshold, the temporal trace is locally transformed into laminar (below threshold, white) or turbulent (below threshold, black). Varying this threshold we can see that the statistics of the black or white region changes. (d) Example of binarized round-trip dynamics increasing (from left to right) the detection threshold.

Figure 10

Log-log plot of the probability density distribution of the laminar duration time (${\tau }_L$) computed from the numerical simulations for $\mathrm{\Delta }=0.5$, taking different values of the detection threshold $c$. The slope $\mu$ of the green lines correspond to the mean value of the best linear fit estimated with the PDF obtained for $2.0\lesssim I_s\lesssim 2.75$. Therefore, we can write that $\mathrm{PDF}(x)=A{x}^{-\mu }$.

Figure 11

(a) Semi-log plot of the probability density distribution of the laminar duration time computed from the numerical simulations for $\mathrm{\Delta }=0.5$, taking different values of the detection threshold $c$. For a given value of the control parameter, the slope $m$ of the green lines decreases as $c$ increases such that $m(I_s,c)=m_0(I_s)e^{-c/c_0}$. Hence, the intrinsic slope $m_0(I_s)$ corresponds to the $y$ intercept as illustrated by (b).

Figure 12

Probability density distribution ($+$) of the laminar duration time computed from the numerical simulations for $\mathrm{\Delta }=0.5$, for $c=0.9$. The solid line corresponds to the fit following the function $P(x)=(A{x}^{-\mu }+B)e^{-mx}$.

Figure 13

(a) Superposition of the round-trip temporal profiles containing an extreme event (gray lines). The blue (red) line corresponds to the function ${|C(\mathrm{\Delta }\tau )|}^2$ obtained from the numerical (experimental spectra) data. Their maxima have been rescaled to the value of the highest peak amplitude. (b) Logarithmic scaled probability density functions of spatiotemporal peaks when increasing the pump power. The green line refers to peaks above twice the characteristic height and the black line is the Rayleigh distribution with unity mean value.