Scaling laws for laser-induced filamentation

Despite all the complexity of the underlying nonlinear physics, the filamentation of ultrashort optical field wave forms is shown to obey a set of physically transparent scaling laws. This scaling is applicable within a remarkably broad range of laser powers, pulse widths, gas pressures, and propagation paths, suggesting specific recipes for the power scaling of filamentation-based pulse compression.

Figure 1

The maps of (a), (b) on-axis field intensity ($\times {10}^{12}$ W/cm²) and (c), (d) on-axis spectral intensity (arb. units) for laser pulses with (a), (c) ${\tau }_0=30$ fs, $a=0.5$ mm in an initially collimated beam, and $P=3P_{cr}$ and (b), (d) ${\tau }_0=115$ fs, $a=2.5$ mm, $f=0.02L_D=0.5$ m, and $P=4.6P_{cr}$ in (a), (c) argon and (b), (d) krypton at a gas pressure of 1 bar.

Figure 2

The on-axis field intensity (a), (b), ionization degree $Z=\frac{N_e}{N_{at}}$ (c), (d), and the on-axis pulse FWHM duration (e), (f) for a laser pulse propagating in argon ${\tau }_0=30$ fs in an initially collimated beam with $a=0.1$ mm, $L_D=4$ cm, $P=0.7$ GW, $p=25$ bar (blue dashed line); $a=0.5$ mm, $L_D=1$ m, $P=18$ GW, $p=1$ bar (green solid line); and $a=1.9$ mm, $L_D=14$ m, $P=250$ GW, $p=0.07$ bar (red dash–dotted line) with (a), (c), (e) and without (b), (d), (f) avalanche ionization.

Figure 3

The on-axis field intensity (a), (b), ionization degree $Z=\frac{N_e}{N_{at}}$ (c), (d), and the on-axis pulse FWHM duration (e), (f) for a laser pulse propagating in krypton with ${\tau }_0=115$ fs; $a=0.83$ mm, $L_D=2.7$ m, $f=5.5$ cm, $P=1.6$ GW, $p=10$ bar (blue dashed line); $a=2.5$ mm, $L_D=24$ m, $f=0.5$ m, $P=15$ GW, $p=1$ bar (green solid line); and $a=8.3$ mm, $L_D=270$ m, $f=5.5$ m, $P$ = 160 GW, $p=0.1$ bar (red dash-dotted line) with (a), (c), (e) and without (b), (d), (f) avalanche ionization.

Figure 4

The on-axis field intensity ($\times {10}^{12}$ W/cm${}^2$) as a function of time $t$ and propagation coordinate $z$ in helium at $p=0.02$ bar, $a=2.9$ mm, $P=1.4$ TW (0.8 $P_{cr}$), ${\tau }_0=30$ fs, ${\lambda }_0=800$ nm, and $f=0.4L_D=9.5$ m (a), $0.6L_D=14.3$ m (b), $0.9L_D=21.4$ m (c), and $2L_D=47.6$ m (d).

Figure 5

The on-axis field intensity ($\times {10}^{12}$ W/cm${}^2$) as a function of time $t$ and propagation coordinate $z$ in krypton at $p=1$ bar, $P=4.4$ GW (4.6 $P_{cr}$), ${\tau }_0=115$ fs, ${\lambda }_0=800$ nm, $f=0.5$ m, and (a) $a=1.0$ mm, (b) $a=1.5$ mm, (c) $a=2.5$ mm, (d) $a=3.5$ mm.