Turbulent Transitions in Optical Wave Propagation

We report the direct observation of the onset of turbulence in propagating one-dimensional optical waves. The transition occurs as the disordered hosting material passes from being linear to one with extreme nonlinearity. As the response grows, increased wave interaction causes a modulational unstable quasihomogeneous flow to be superseded by a chaotic and spatially incoherent one. Statistical analysis of high-resolution wave behavior in the turbulent regime unveils the emergence of concomitant rogue waves. The transition, observed in a photorefractive ferroelectric crystal, introduces a new and rich experimental setting for the study of optical wave turbulence and information transport in conditions dominated by large fluctuations and extreme nonlinearity.

Figure 1

Nonlinear wave propagation in unstable photorefractive ferroelectric crystals. (a) Sketch of the setup geometry adopted. Scale bars for input and output intensity distributions correspond to $20\mu \mathrm{m}$. (b) Characterization of transverse breaking: intensity dependence of the process, with the minimum required voltage and the average time scale $\tau$ providing the formation of periodic structures. Lines are linear fits.

Figure 2

Observation of the turbulent transition in one-dimensional beam dynamics. (a) Detected output spatial intensity distributions as a function of the nonlinearity expressed through the continuous dimensionless control parameter $t/\tau$. (b) Corresponding width of the spatial Fourier spectrum and mean intensity autocorrelation (see main text) increasing the nonlinearity. The red line serves as a guide at the sharp transition signaling the onset of optical turbulence.

Figure 3

Onset of optical turbulence for spatially modulated input waves. (a) Output intensity distribution increasing the nonlinearity $t/\tau$. (b) Input power spectrum for spatially modulated beams (blue line) in comparison with the quasihomogeneous case of Fig. 2 (red line).

Figure 4

Spectral properties of the optical state before and after the turbulent transition. (a),(b) Intensity and spectral sample distributions of single-shot measurements at moderate nonlinearity ($t/\tau \approx 1$) showing the excitation of the spatial frequency $\overline{k_y}=0.05\mu {\mathrm{m}}^{-1}$ (blue line), spectrally broad noise amplification (red line), and simultaneous development of well-defined low and high frequency modes (magenta line). (c) Single-shot disordered intensity distributions detected in the turbulent regime at $t/\tau \simeq 2.5$. (d) Ensemble spectrum of modulational instability before the transition ($t/\tau \approx 1$). (e) Measured wave-turbulent power spectrum (red line) fitted on large spatial scales with the scaling behavior $\propto k_y^{-\gamma }$, $\gamma =0.15\pm 0.01$ (black line).

Figure 5

Spatial rogue wave generation in the turbulent regime. (a) Peak-intensity PDF of localized structures emerging from instability at $t/\tau \approx 1$, experimental counts (blue bars), and the Gaussian trend of the distribution tails (red line). (b) Measured long-tail statistics in optical turbulence at $t/\tau \approx 2.5$ (blue line) and consistent Gaussian exponential scaling for comparison.