Generation of Caustics and Rogue Waves from Nonlinear Instability

Caustics are phenomena in which nature concentrates the energy of waves and may exhibit rogue-type behavior. Although they are known mostly in optics, caustics are intrinsic to all wave phenomena. As we demonstrate in this Letter, the formation of caustics and consequently rogue events in linear systems requires strong phase fluctuations. We show that nonlinear phase shifts can generate sharp caustics from even small fluctuations. Moreover, in that the wave amplitude increases dramatically in caustics, nonlinearity is usually inevitable. We perform an experiment in an optical system with Kerr nonlinearity, simulate the results based on the nonlinear Schrödinger equation, and achieve perfect agreement. As the same theoretical framework is used to describe other wave systems such as large-scale water waves, our results may also aid the understanding of ocean phenomena.

Figure 1

Scheme of the experimental setup. The SLM imprints a random phase mask (upper-left inset) onto the transverse profile of a cw laser beam. The first imaging system images the SLM onto the plane shown by the dashed line (SLM plane). At this point, the transverse intensity distribution of the beam follows the Gaussian profile of the input laser. An intensity pattern develops upon propagation. The distance $l$, after which the sharpest pattern is formed, depends on the amplitude $\mathrm{\Delta }$ of the phase modulation. Another imaging system is used to image the pattern plane (dotted line) onto the CCD camera. The upper-right inset shows an example pattern generated from $\mathrm{\Delta }=8\pi$. To study nonlinear propagation, the Rb cell is placed in the end of the propagation before the pattern plane.

Figure 2

Generation of caustics upon linear propagation. (a)–(c) Examples of patterns formed after propagation in free space from different phase masks with amplitudes $\mathrm{\Delta }=2\pi$, $8\pi$, and $16\pi$, respectively. We see that sharp caustics are formed only under strong phase modulation. (d) Intensity distributions for the patterns generated upon propagation through free space from three different phase amplitudes $\mathrm{\Delta }$. The $C$ parameter from the fit function characterizes the heavy-tailed behavior; the lower the $C$ parameter, the longer the tail of the distribution. Thus, sharp caustics are distinguished by their heavy-tailed statistics.

Figure 3

Generation of caustics upon nonlinear propagation. (a) Real and imaginary parts of the total susceptibility of Rb vapor. $\mathrm{Re}\chi$ and thus the refractive index increase with the intensity, indicating nonlinear focusing. (b)–(d) Caustic patterns generated from the same phase masks as in Fig. 2, but after the nonlinear propagation in Rb. In contrast to the linear case shown in Fig. 2, even small phase modulations, with the aid of nonlinear focusing, can concentrate light into sharp caustics. (e) Intensity distributions of the nonlinear caustic patterns generated from three different phase amplitudes $\mathrm{\Delta }$.

Figure 4

Intensity patterns and scintillation indices from the computer simulation. (a)–(c) Patterns obtained from the numerical simulation, which show excellent agreement with the experimental results shown in Figs. 2 and 3. (d) Scintillation indices ${\beta }^2$, averaged over 1000 patterns, calculated from our numerical simulation, as a function of the propagation distance from the entrance of the Rb cell up to 100 mm after the cell where partial speckles are formed. The gray area indicates the 7.5-cm-long Rb cell for the nonlinear cases, and the shaded regions show one standard deviation from the mean. The nonlinear patterns have larger scintillation indices compared to the corresponding linear cases. A scintillation index greater than unity indicates the presence of sharp caustics.