Self-focusing dynamics of ultraintense accelerating Airy waveforms in water

We report experiments and numerical simulations on the self-focusing and filamentation of ultraintense femtosecond Airy waveforms in water. The accelerating property of Airy waveforms results in the generation of distinct features in the angularly resolved spectra of forward-propagating supercontinuum emission. Fitting these features with appropriate phase-matching conditions allows for the quantification of the propagation history of the waveforms.

Figure 1

Numerically simulated $k$-$\omega$ spectrum generated through filamentation of a 35-fs-long Gaussian pulse with a 1-$\mu$J pulse energy, under the experimental conditions described in the text. Data are shown on a log scale. The pulse propagation dynamics involves two pulse-splitting events. Solid black lines are drawn according to the phase-matching condition, (1), for the two generated pairs of subpulses.

Figure 2

Experimentally observed $k$-$\omega$ spectrum for the case corresponding to the simulation shown in Fig. 1. Solid lines are exact replicas of the fitting curves reproduced here from the numerical simulation in Fig. 1 without any adjustments. The dashed line shows the fit according to Eq. (1), in which the velocity of the leading subpulse is set to $(1+0.0010)c/n_g$, which is different from the correct velocities of both leading subpulses generated upon propagation ($c$ is the speed of light in vacuum and $n_g$ is the linear group refractive index of water). The latter fit is clearly off the experimentally recorded branches corresponding to the two leading subpulses generated in this case.

Figure 3

Numerically simulated $k$-$\omega$ spectra for cases of Airy pulses with 1.6 $\mu$J of pulse energy and with a positive (top) and a negative (bottom) cubic spectral phase of 120 000 fs${}^3$. Insets: The corresponding orientations of the waveforms. Spectra are plotted on a log scale. For the positive phase case, the phase-matching curves for the three splits of the dominant intensity feature of the waveform are shown by solid lines. The dashed line shows the phase-matching curve for the split of the secondary feature of the pulse. For the negative phase case, the solid lines show the phase-matching curves for the first two splits of the dominant intensity feature of the pulse and for the only split of the secondary feature. The dashed lines show the phase-matching curves for the third split of the main intensity feature.

Figure 4

Experimentally observed $k$-$\omega$ spectra for the cases of Airy pulses corresponding to the numerical simulations shown in Fig. 3. The top figure corresponds to a positive, and the bottom figure to a negative, cubic spectral phase of 120 000 fs${}^3$. Lines are exact replicas of the corresponding fitting curves from the numerical simulation plots shown in Fig. 3.