Dynamics of filament formation in a Kerr medium

We have studied the large-scale beam breakup and filamentation of femtosecond pulses in a Kerr medium. We have experimentally monitored the formation of stable light filaments, conical emission, and interactions between filaments. Three major stages lead to the formation of stable light filaments: First the beam breaks up into a pattern of connected lines (constellation), then filaments form on the constellations, and finally the filaments release a fraction of their energy through conical emission. We observed a phase transition to a faster filamentation rate at the onset of conical emission. We attribute this to the interaction of conical emissions with the constellation which creates additional filaments. Numerical simulations show good agreement with the experimental results.

Figure 1

Experimental setup. A femtosecond pulse traverses a glass cell filled with carbon disulfide. The beam profile at the output of the cell is imaged on a CCD camera. A magnet inside the cell is moved to adjust the liquid level, which determines the propagation distance.

Figure 2

Beam profile and 2D fast Fourier transform (FFT) before and after traversing $10\mathrm{mm}$ of $\mathrm{C}{\mathrm{S}}_2$. (a) Input beam profile. (b) Beam profile after traversing $10\mathrm{mm}$ of $\mathrm{C}{\mathrm{S}}_2$. (c) and (d) are the 2D FFT of (a) and (b), respectively. (e) and (f) are the input and output beam profiles for a modulated beam.

Figure 3

Beam breakup and filamentation as a function of propagation distance (left to right) and energy (top to bottom). $E=\left(\mathrm{a}\right)$ 0.70, (b) 0.86, (c) 1.0, and (d) $1.1\mathrm{mJ}$. The size of each image is $0.5\times 0.5{\mathrm{mm}}^2$.

Figure 4

The amplitude of the input beam (a) was measured and used as the input of the numerical simulation. There is good agreement between the experimental (b) and simulated (c) output beam profile. The images are $0.5\times 0.5{\mathrm{mm}}^2$.

Figure 5

Numerical simulation of beam propagation. A constellation and filaments are generated from an input beam with random amplitude noise. The images show the beam profile for propagation distances of 0, 6, 8, 10, and $12\mathrm{mm}$.

Figure 6

Filament statistics as a function of input pulse energy. (a) Number of filaments (filled circles) and conical emissions (CE) (squares). (b) Mean size. (c) Mean intensity. A transition in the filamentation rate is observed at $0.5\mathrm{mJ}$.

Figure 7

Conical emission and seeding of new filaments. (a) For an input energy of $0.54\mathrm{mJ}$ a single filament is observed. (b) When the energy is increased to $0.64\mathrm{mJ}$ the filament emits conical waves before reaching the output face of the nonlinear material. (c) and (d) For higher pulse energies of 0.74 (c) and $0.86\mathrm{mJ}$ (d), new filaments are formed at the intersection of the conical waves with constellation.

Figure 8

Behavior of individual filaments. (a) Experimental observation of conical emission. (b) Fusion of two filaments. (c) Splitting. (d) Fusion followed by conical emission. The step in propagation distance is $0.50\mathrm{mm}$ for (a) and $0.25\mathrm{mm}$ for (b), (c), and (d).

Figure 9

Simulation for the propagation of a single filament over $10\mathrm{mm}$. The intensity of the filament is plotted as a function of radius (vertical scale) and propagation distance (horizontal scale). The filament is observed to periodically re-focus.

Figure 10

Simulated filament propagation in the presence of a noisy background. (a)–(c) The filament focuses and emits conical waves. (d) and (e) The conical waves seed the formation of new filaments on the constellation. (f)–(h) The original filament is attracted and fuses with a new filament. The propagation distance from (a) through (h) is from $7.5\text{to}11\mathrm{mm}$ in steps of $0.5\mathrm{mm}$.