Optical Random Riemann Waves in Integrable Turbulence

We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schrödinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we show that the development of IT can be divided into two distinct stages, the initial, prebreaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions.

Figure 1

Numerical simulation of Eq. (1) between $\xi =0$ and $\xi =0.156$ with $\epsilon =0.014$. Time evolution of (a) $\rho$, (b) $r_{1,2}$, and (c),(d) associated PDFs. In (c) the black dashed line represents the exponential distribution $\mathcal{P}(\rho )=e^{-\rho }$ and in (d), it represents the Rayleigh distribution [$\mathcal{P}(x)=2x{e}^{-x^2/2}$]. The PDF (green line) in (c) is obtained at $\xi =1.56$ where the wave system has reached a statistical stationary state.

Figure 2

Experimental setup. Partially coherent light at 1064 nm is injected inside a 1.4-km-long PM fiber in a regime where nonlinear effects strongly dominate linear ones ($L_{NL}=1.3\mathrm{km}$, $L_D=6250\mathrm{km}$, $\epsilon =0.014$). Real-time observation of $\rho (\tau )$ and of $u(\tau )$ at the input and output ends of the PM fiber is achieved by combining a time-division multiplexing technique and a heterodyne measurement (see text).

Figure 3

Experiments. Time evolution of $\rho$, $u$, and of the Riemann invariants $r_{1,2}=u\pm 2\sqrt{\rho }$ at the input end ($\xi =0$) and at the output end ($\xi =0.0156$) of the fiber ($T_0=250\mathrm{ps}$). The black dashed lines represent Riemann invariants calculated from numerical integration of Eqs. (5) while starting from the experiment initial conditions (blue lines).

Figure 4

Experimental PDFs of (a) $\rho$, (b) $r_1/2$. Blue (respectively, yellow) lines represent PDFs at the input (respectively, output) end of the fiber. In (a) the black dashed line represents the exponential distribution and in (b) it represents the Rayleigh distribution.