Enhancing optical extreme events through input wave disorder

We demonstrate how the emergence of extreme events strongly depends on the correlation length of the input field distribution. Observing the behavior of optical waves in turbulent photorefractive propagation with partially incoherent excitations, we find that rogue waves are strongly enhanced for a characteristic input correlation scale. Waveform analysis identifies this scale with a characteristic peak-intensity-independent wave size, suggesting a general role played by saturation in the nonlinear response in rogue phenomena.

Figure 1

Partially incoherent beams in photorefractive ferroelectric crystals. (a) Sketch of the experimental setup with lenses ($f1=f2=50$ mm), adjustable glass diffuser $D$ (average particle size of $2\mu \mathrm{m}$), long-working-distance objective $\mathrm{OBJ}$ ($\mathrm{NA}=0.55$) and KLTN sample. (b)–(c) Input and output intensities with the corresponding spatial autocorrelation function $g\left(\mathrm{\Delta }r\right)$ for two different positions of the scatterer. $\sigma$ indicates the output autocorrelation length and $S$ is the corresponding input source size. Scale bars correspond to $30\mu \mathrm{m}$.

Figure 2

Scale-dependent behavior of the intensity statistics. (a) PDF measured in linear conditions ($E=0,P=400\mu \mathrm{W}$) for beams with different coherent lengths expressed through the input parameter $S$. (b) Corresponding distributions for nonlinear propagation ($E=2$ kV/cm, $P=400\mu \mathrm{W}$), showing a long-tail behavior depending on the specific correlation length, with a large enhancement in rogue-wave appearance for $S=220\mu \mathrm{m}$. Suppression of the tail occurs for highly incoherent fields ($S=500\mu \mathrm{m}$).

Figure 3

Unveiling optical rogue waveforms. (a) Detected transverse width $\mathrm{\Delta }X$ and peak intensity $I_P$ of extreme events for data at different applied fields. (b) High-resolution spatial intensity distributions containing localized abnormal waveforms. Red curves are $x$-profiles along the dotted lines.

Figure 4

Evidence of a typical scale in rogue waveforms. Measured extreme event widths at different coherence lengths $\sigma$ (dashed lines). The two scales are resonant for $S=220\mu \mathrm{m}$, where a large increase results in the probability of rogue-wave appearance [see main text and Fig. 2]. The diagram on the right illustrates how results imply the presence of a typical spatial scale for rogue events.

Figure 5

Possible mechanism underlying the appearance of extreme intensity fluctuations. Existence curve of nonequilibrium solitons in saturable nonlinearities (red line) on which the waveform $w\left(\xi \right)$ is schematically shown. Arrows indicate the magnitude of width and amplitude fluctuations in the gray region, which is in proximity of the localized, self-trapped, wave solution. For comparison with Figs. 4 and 2, input correlation lengths used in experiments are also reported.