Turbulent dynamics of an incoherently pumped passive optical fiber cavity: Quasisolitons, dispersive waves, and extreme events

We study numerically and experimentally the dynamics of an incoherently pumped passive optical fiber ring cavity nearby the zero-dispersion wavelength of the fiber. We show that the cavity exhibits a quasisoliton turbulence dynamics, whose properties are controlled by the degree of coherence of the injected pump wave: As the coherence of the pump is degraded, the cavity exhibits a transition from the quasisoliton condensation regime toward the weakly nonlinear turbulent regime characterized by short-lived rogue wave events. This behavior is reminiscent of the corresponding dynamics obtained in the purely conservative (Hamiltonian) problem. We report experimental results of an all-integrated incoherently pumped fiber cavity that provide some spectral and temporal complementary signatures of the processes predicted numerically.

Figure 1

Numerical simulations of the incoherently pumped passive optical cavity in the high-finesse regime $\mathcal{F}\simeq 500$. (a) “Quasisoliton condensation curve” reporting the intensity maxima of the optical wave vs kinetic energy of the pump $E_F$ once a statistically stationary regime has been reached in the cavity. The left and right columns refer to the points (green, $\diamond )$ reported in (a) for $E_F=1$ and $E_F=40$, respectively. (b), (c) Evolutions of the optical wave spectrum vs number of round trips $m$ corresponding to $E_F=1$ (b), $E_F=40$ (c), and corresponding spectral profiles recorded at $m=5000$ (d), (e). (f), (g) Evolutions of the contributions to the total energy (nonconserved Hamiltonian) $\overline{H}\left(T\right)={\overline{E}}_2+{\overline{E}}_3+\overline{U}$ vs the slow-time $T(=m{t}_R)$, where ${\overline{E}}_2\left(T\right)$ and ${\overline{E}}_3\left(T\right)$ refer to the second- and third-order dispersion contributions to the kinetic energy $\overline{E}$. The dashed dark line in (f) denotes $\sim exp(-t/{\tau }_{\mathrm{inj}})=exp(-m\Gamma )$: The effective quasisoliton lifetimes in the cavity are mainly determined by the injection time (see a zoom in the inset). Parameters are given in the text $({\tau }_0=1.4$ ps, ${\overline{E}}_{2,3}=E_{2,3}/T_0$, $\overline{U}=U/T_0$ where $T_0$ denotes the size of the numerical temporal window $T_0=20{\tau }_0)$.

Figure 2

(a), (b) Intensity PDFs corresponding to a pump kinetic energy $E_F=1$ (a), and $E_F=30$ (b) [see the condensation curve in Fig. 1]. The insets show the corresponding PDFs of the maxima of the intensities. (c), (d) Corresponding space-–time intensity patterns: (c) For $E_F=1$ the cavity field self-organizes into robust and persistent quasisoliton structures [whose typical intensity is denoted by a red circle in (a)] that propagate in the midst of small-scale fluctuations; (d) for $E_F=30$ the cavity field evolves in the weakly nonlinear regime and exhibits short-lived extreme events [see temporal intensity profile in (d), dark line] whose amplitudes deviate from Gaussian statistics [see red circles in (d) and (b)]. Parameters of the simulations are the same as those used in Fig. 1: ${\tau }_0=1.4$ ps, size of the temporal numerical window $T_0=20{\tau }_0$, the vertical coordinate $m$ in (c) and (d) denotes the number of round trips (see Fig. 1).

Figure 3

Numerical simulations of the incoherently pumped passive optical cavity in a low-finesse regime $\mathcal{F}\simeq 22$. (a) Quasisoliton condensation curve reporting the intensity maxima of the optical wave vs kinetic energy of the pump $E_F$ once a statistically stationary regime has been reached. (b), (c) Evolutions of the optical wave spectrum vs number of round trips $m$ corresponding to $E_F=0.33$ (b), $E_F=22.7$ (c), and corresponding spectral profiles recorded at $m=1000$ (d), (e). (f), (g) Evolutions of the contributions to the total energy (nonconserved Hamiltonian) $\overline{H}={\overline{E}}_2+{\overline{E}}_3+\overline{U}$ vs the slow time $T(=m{t}_R)$, where ${\overline{E}}_2\left(T\right)$ and ${\overline{E}}_3\left(T\right)$ refer to the second- and third-order dispersion contributions to the kinetic energy $\overline{E}$. The green diamonds in (a) correspond to the value of $E_F$ used. The dashed dark line in (f) denotes $\sim exp(-t/{\tau }_{\mathrm{inj}})=exp(-m\Gamma )$: the effective quasisoliton lifetimes in the cavity is mainly determined by the injection time (see a zoom in the inset). Parameters are given in the text $({\tau }_0=1.19$ ps, ${\overline{E}}_{2,3}=E_{2,3}/T_0$, $\overline{U}=U/T_0$ where $T_0$ denotes the size of the numerical temporal window $T_0=320{\tau }_0)$.

Figure 4

Experimental setup. ASE: amplified spontaneous emission; PPG: pulse pattern generator; Pol: polarizer; IM: intensity modulator; SS: spectral shaper; SF: spectral filter; EDFA: erbium-doped fiber amplifier; OSA: optical spectrum analyzer; Att: attenuator; PC: polarization controller.

Figure 5

(a) Experimental spectra recorded by varying the wavelength of the incoherent pump injected in the cavity, and (b) corresponding numerical NLS simulations of Eqs. (1) and (2) realized with the experimental parameters. The average power of the incoherent pump is kept fixed to 893 mW, while its spectral bandwidth to 10 GHz (see the text for all experimental parameters of the incoherently pumped passive optical cavity). Note the satisfactory qualitative agreement between the experimental results and the numerical results obtained with the experimental parameters given in Sec. 4.

Figure 6

Experimental (solid blue curve) and numerical (dashed-dotted red curve) spectral profiles of the intracavity optical wave corresponding to an injected incoherent pump of spectral bandwidth 10 GHz (a) and 1 THz (b) for a fixed value of the pump intensity (893 mW): The DW spectral signature that evidences the presence of quasisolitons is reduced as the coherence of the pump is degraded. The spectrum of the injected incoherent pump wave is reported in dashed dark line. The numerical NLS simulations of Eqs. (1) and (2) have been performed with the experimental parameters given in the text (see Sec. 4).

Figure 7

PDFs of the intensity (a), and of the maxima of the intensity (b), computed numerically by solving the NLS equations (1) and (2), with three different pumps of spectral widths 0.01 THz (red), 0.1 THz (green), and 1 THz (blue) (in log scale). The dashed line in (a) stands for Gaussian statistics of the field amplitude, i.e., exponential PDF for the intensity. The incoherence of the pump wave significantly reduces the efficiency of generation of high-amplitude coherent quasisoliton structures.

Figure 8

(a) PDFs of the intensity recorded experimentally with an oscilloscope of spectral bandwidth 36 GHz, for three different pumps of spectral widths 0.01 THz (red), 0.1 THz (green), and 1 THz (blue), with a constant average power of the incoherent pump, 893 mW. (b) Corresponding PDFs obtained by numerical simulations of Eqs. (1) and (2) with the experimental parameters, in which the limited spectral bandwidth of the oscilloscope has been modeled by a square-shaped (super-Gaussian, $n=8)$ spectral filter of 36 GHz. The inset shows a typical temporal intensity profile of the NLS simulation (solid blue curve) and the corresponding smoothed profile (dashed red curve). A qualitative agreement is obtained between the numerical results and the PDFs recorded experimentally, thus confirming that the main impact of pump incoherence is to significantly reduce the efficiency of generation of quasisoliton structures.