Intermittency in Integrable Turbulence

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by applying a bandpass frequency optical filter, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian. These phenomena are very well described by numerical simulations of the one-dimensional nonlinear Schrödinger equation.

Figure 1

Experimental setup. Linearly polarized partially coherent light having a Gaussian statistics is launched inside a polarization maintaining (PM) single-mode fiber through a polarizing cube (PC). Mean optical power spectra are measured by using an optical spectrum analyzer (OSA). The light spectrum broadens from 14 to 32 GHz from nonlinear propagation. Time detection of light power fluctuations $P(t)$ is made by using fast photodiodes (PDs) and a 36-GHz oscilloscope. A tunable optical bandpass filter with a narrow bandwidth of 6 GHz is used to analyze fluctuations of output light in sliced parts of the spectrum.

Figure 2

Experiments. Global PDFs measured at the input (black line) and at the output (red line) ends of the PM fiber without using the optical bandpass filter.

Figure 3

Experiments. PDFs measured at the output of the optical bandpass filter by detuning its central frequency by $\sim -5\mathrm{GHz}$ (black full line), $\sim 21\mathrm{GHz}$ (blue dashed line), and $\sim 35\mathrm{GHz}$ (green dotted line) from the center of the output spectrum. Inset: Corresponding power spectra measured at the output of the tunable bandpass optical filter. The $\sim 32$-GHz-wide power spectrum plotted by the red line is measured without bandpass filter at the output of the PM fiber.

Figure 4

Numerical simulations of Eq. (1) between $z=0\mathrm{km}$ and $z=15\mathrm{km}$ with $P_0=205\mathrm{mW}$, $\mathrm{\Delta }{\nu }_0=14\mathrm{GHz}$, ${\beta }_2=+20{\mathrm{ps}}^2{\mathrm{km}}^{-1}$, $\gamma =6.2{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$. (a) Input (red dashed line) and output (blue full line) mean power spectra of the partially coherent wave. (b) Input (red dashed line) and output (blue full line) PDFs for $P(t)=|A(z,t){|}^2$. (c) PDFs for the power fluctuations $P_F(t)$ found at the output of an ideal bandpass filter centered at ${\nu }_0=0\mathrm{GHz}$ (green full line), ${\nu }_0=100\mathrm{GHz}$ (brown dashed line), ${\nu }_0=150\mathrm{GHz}$ (cyan dotted line). Vertical colored lines in (a) indicate the corresponding central frequencies. (d) Kurtosis $\kappa$ of fluctuations at the output of the bandpass frequency filter.