Spontaneous emergence of rogue waves in partially coherent waves: A quantitative experimental comparison between hydrodynamics and optics

Rogue waves are extreme and rare fluctuations of the wave field that have been discussed in many physical systems. Their presence substantially influences the statistical properties of a partially coherent wave field, i.e., a wave field characterized by a finite band spectrum with random Fourier phases. Their understanding is fundamental for the design of ships and offshore platforms. In many meteorological conditions waves in the ocean are characterized by the so-called Joint North Sea Wave Project (JONSWAP) spectrum. Here we compare two unique experimental results: the first one has been performed in a 270 m wave tank and the other in optical fibers. In both cases, waves characterized by a JONSWAP spectrum and random Fourier phases have been launched at the input of the experimental device. The quantitative comparison, based on an appropriate scaling of the two experiments, shows a very good agreement between the statistics in hydrodynamics and optics. Spontaneous emergence of heavy tails in the probability density function of the wave amplitude is observed in both systems. The results demonstrate the universal features of rogue waves and provide a fundamental and explicit bridge between two important fields of research. Numerical simulations are also compared with experimental results.

Figure 1

Sketch of the water wave tank used in the experiment. At one end a fully programmable wave maker is placed to generate waves; at the other end an absorbing beach is placed to minimize reflections. Fifteen probes were placed along the center of the tank at distances of $10,30,35,40,45,60,65,70,75,80,85,115,120,160,200$ m from the wave maker; two extra probes were placed transversally at $z=75$ meters and $z=160$ m to verify if any transverse mode developed in the tank.

Figure 2

Optical sampling of the partially coherent wave fluctuating with time (the signal) is achieved from sum frequency generation (SFG). Green pulses are generated at $\lambda =529\mathrm{nm}$ from the interaction of the signal with femtosecond pump pulses inside a ${\chi }^{\left(2\right)}$ crystal. The $140\mathrm{fs}$ pump pulses are emitted by mode-locked laser at ${\lambda }_p=800\mathrm{nm}$. The partially coherent wave is emitted by an ASE source at ${\lambda }_s=1562\mathrm{nm}$ and is amplified by an Erbium fiber amplifier. 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 3

Spectral power density at $z=0$ in the optical fiber, red (gray) line, and in the water tank, green (light gray) line, for run $A$ (a) and $B$ (b). The JONSWAP spectrum (black line) with parameters taken from Table 1 is also shown. The frequency $f^{\prime }$ is normalized by using Eq. (20).

Figure 4

Probability density function of the normalized intensity, $I$, for the experiments in run $B$ ($\epsilon \sim 0.5$ at different nondimensional distances: (a) $z=z^{\prime }/z_{\text{nlin}}=0$, (b) $z\simeq 0.6$, (c) $z\simeq 1.4$, (d) $z\simeq 4$. The green (light gray) line corresponds to the water wave experiment and the red one (gray) to the optical one. The black line corresponds to the exponential distribution.

Figure 5

Fourth-order normalized moment of the wave envelope moment, $\kappa ={\langle \left|A\right|}^4{\rangle /\langle \left|A\right|}^2{\rangle }^2$, as a function of the nondimensional propagation coordinate $z$: experiments and numerical simulations for run $A$ (panel a) and run $B$ (panel b). Optical experiments (red points), hydrodynamical experiments (green triangles), simulations of the NLSE (dashed line), simulations of Euler equation (magenta line).