Spatiotemporal Rogue Events in Optical Multiple Filamentation

The transient appearance of bright spots in the beam profile of optical filaments formed in xenon is experimentally investigated. Fluence profiles are recorded with high-speed optical cameras at the kilohertz repetition rate of the laser source. A statistical analysis reveals a thresholdlike appearance of heavy-tailed fluence distributions together with the transition from single to multiple filamentation. The multifilament scenario exhibits near-exponential probability density functions, with extreme events exceeding the significant wave height by more than a factor of 10. The extreme events are isolated in space and in time. The macroscopic origin of these experimentally observed heavy-tail statistics is shown to be local refractive index variations inside the nonlinear medium, induced by multiphoton absorption and subsequent plasma thermalization. Microscopically, mergers between filament strings appear to play a decisive role in the observed rogue wave statistics.

Figure 1

Setup (schematic): Laser: pulsed Ti:sapphire chirped-pulse amplifier, delivering up to 5 mJ pulse energy at 1 kHz repetition rate. Pulse duration: 40 fs. $÷1/÷10$ selectable pulse division. Kerr cell: 1.5 m long cell filled with nonlinear refractive material (Xe, $\mathrm{S}{\mathrm{F}}_6$, water). ATT1: reflective attenuator. ATT2: absorptive attenuator. 2DCAM: Basler A504k, $12\times 12\mu {\mathrm{m}}^2$ pixel size, $1280\times 1024$ pixels, frame size 1.3 megapixels, $15.4\times 12.3{\mathrm{mm}}^2$ detector size. Line CAM: Schäfter & Kirchhoff SK2048DDW, $13\times 300\mu {\mathrm{m}}^2$ pixel size, 26.5 mm width, 2048 pixels. Inset: Example for resulting fluence histogram.

Figure 2

Numerical simulations of the evolution of fluence distributions along the propagation coordinate $z$ during multiple filamentation (for details see the Supplemental Material [20]). (a) Isofluence surfaces, with color coding showing relative phases. (b) Fluence statistics of filament strings, with separated merger events.

Figure 3

Histogram analysis of 1 min line scan camera recordings (60 000 laser shots). Symbols represent the histogram of the accumulated 20 neighboring pixels in the most intense part of the beam profile. Line: Weibull fit. $\beta$ defined according to Eq. (2). Detector fluence calibrated relative to the maximum limiting value. (a) Laser measured directly. $P\ll 8P_{\mathrm{crit}}$ (b) Single filamentation. $P=8P_{\mathrm{crit}}$ (c) Multiple filamentation. $P=19P_{\mathrm{crit}}$. Dashed lines indicate multiples of the significant fluence $F_{1/3}$. Data were taken out of the small marked part of the beam profile in Fig. 4c.

Figure 4

Statistical analysis of 1 min line scan camera recording that includes 60 000 individual scans. Solid orange lines: maximum fluence $F_{\max }$. Blue dots: average fluence $\overline{F}$. Solid black lines: significant fluence $F_{1/3}$. (a) Input profile ($P\ll P_{\mathrm{cr}}$). (b) Profile after single filamentation ($P=8P_{\mathrm{cr}}$). (c)–(d) Profile after multiple filamentation ($P=19$ and $53P_{\mathrm{cr}}$). The region examined in the statistical analysis of Fig. 3c has been marked by a yellow semitransparent rectangle in (c). The sensitivity dip at 3 mm is an artifact of the camera.

Figure 5

(a)–(e) Sequence of 5 multifilament fluence profiles recorded with high-speed two-dimensional camera, with an extreme event appearing at 4.89 s after the start of recording. (f) Fluence observed at position (6 mm, 7 mm).

Figure 6

(a) Fourier transform of line scan camera time series recorded directly at the laser output and after the Kerr cell for the case of multifilamentation. Sampling limit is 500 Hz. (b) Temporal correlation between individual line scan camera frames as a function of temporal delay between recordings in the multifilamentation regime.