Optical Rogue Waves in Integrable Turbulence

We report optical experiments allowing us to investigate integrable turbulence in the focusing regime of the one-dimensional nonlinear Schrödinger equation (1D NLSE). In analogy with broad spectrum excitation of a one-dimensional water tank, we launch random initial waves in a single mode optical fiber. Using an original optical sampling setup, we measure precisely the probability density function of optical power of the partially coherent waves rapidly fluctuating with time. The probability density function is found to evolve from the normal law to a strong heavy-tailed distribution, thus revealing the formation of rogue waves in integrable turbulence. Numerical simulations of 1D NLSE with stochastic initial conditions quantitatively reproduce the experiments. Our numerical investigations suggest that the statistical features experimentally observed rely on the stochastic generation of coherent analytic solutions of 1D NLSE.

Figure 1

Measurement of the statistics of random light. (a) Principle. Optical sampling of the partially coherent wave fluctuating with time (the signal) is achieved from SFG. Blue pulses are generated at $\lambda =457\mathrm{nm}$ from the interaction of the signal with femtosecond pump pulses inside a ${\chi }^{(2)}$ crystal. (b) Setup. The 140 fs pump pulses are emitted by mode-locked laser at ${\lambda }_p=800\mathrm{nm}$. The partially coherent wave is emitted by a ytterbium fiber laser at ${\lambda }_s=1064\mathrm{nm}$. Statistics of partially coherent light is measured from the SFG process either directly at the output of the laser or after propagation inside an optical fiber.

Figure 2

Experiments. (a) Pump pulses (black line) and samples of fiber laser fluctuations (SFG pulses, red line). (b) Statistics of fluctuations at the output of the fiber laser. Probability density function (PDF) of normalized optical power $P/⟨P⟩$ (red line). The PDF is computed from SFG peak powers plotted in (a). Normalized exponential distribution $\mathrm{PDF}[P/⟨P⟩]=\exp (-P/⟨P⟩)$ (black line) (c),(d) Focusing propagation. Optical power spectra (c) and PDFs (d) of the partially coherent wave at the input (red line) and output (green line) of the fiber. Experiments are plotted with solid lines [(c),(d)] and numerical simulations are plotted with circles (c).

Figure 3

Numerical simulations. (a). PDF of optical power at a propagation length $z=0\mathrm{m}$ (solid red line), $z=15\mathrm{m}$ (solid green line), and $z=100\mathrm{m}$ (stationary state, black squares). Experimental PDF ($z=15\mathrm{m}$) of Fig. 2 is displayed for comparison (dashed green line). (b),(c) Typical temporal optical power fluctuations along the nonlinear propagations. Green line in (c): fluctuations at $z=90\mathrm{m}$. Inset in (b): enlarged view of an intense power fluctuation superimposed on the profile of a Peregrine soliton (blue circle) determined from a best-fit procedure.