Filamentation of femtosecond light pulses in the air: Turbulent cells versus long-range clusters

The filamentation of ultrashort pulses in air is investigated theoretically and experimentally. From the theoretical point of view, beam propagation is shown to be driven by the interplay between random nucleation of small-scale cells and relaxation to long waveguides. After a transient stage along which they vary in location and in amplitude, filaments triggered by an isotropic noise are confined into distinct clusters, called “optical pillars,” whose evolution can be approximated by an averaged-in-time two-dimensional (2D) model derived from the standard propagation equations for ultrashort pulses. Results from this model are compared with space- and time-resolved numerical simulations. From the experimental point of view, similar clusters of filaments emerge from the defects of initial beam profiles delivered by the Teramobile laser facility. Qualitative features in the evolution of the filament patterns are reproduced by the 2D reduced model.

Figure 1

(a) Soliton power vs $\Lambda$, (b) soliton amplitude vs $\Lambda$, (c) soliton width vs $\Lambda$, and (d) eigenvalues of the internal modes $\delta \varphi$ vs $\Lambda$. The dashed line marks the maximum value of $\lambda$ for which discrete (localized) perturbative modes exist, i.e., $\lambda ⩽\Lambda$.

Figure 2

Isointensity plots $\left[I_{\mathrm{iso}}=0.5\right]$ of the fusion of conservative solitons $\left(\nu =0\right)$ with individual power $P_{\mathrm{fil}}$ and separation distance $\Delta$. The insets show the radial shape of the respective soliton. (a) $P_{\mathrm{fil}}=1.02P_c\left(\Lambda =0.137\right)$ and $\Delta =15$. (b) $P_{\mathrm{fil}}=1.92P_c\left(\Lambda =0.254\right)$ and $\Delta =15$. (c) $P_{\mathrm{fil}}=3.84P_c\left(\Lambda =3.02\right)$ and $\Delta =15$. (d) Same parameters as in (b), but with $\Delta =20$.

Figure 3

(a) Decrease of soliton power $P_s$ vs $Z$, $\nu =0.154$, for solitons with $P_s=1.92P_c\left(\Lambda =0.254\right)$, $P_s=3.84P_c\left(\Lambda =3.02\right)$, and $P_s=7.56P_c\left(\Lambda =3.32\right)$. (b) Same as in Fig. 2(d), but with $\nu =0.154$. In the region $100<z<150$ preceding the fusion event, beam components slightly diffract with an intensity going below the selected isointensity level $\left[I_{\mathrm{iso}}=0.5\right]$.

Figure 4

Isointensity patterns $\left(I_{\mathrm{iso}}\approx {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2\right)$ of filamentary structures described by Eq. (3) and created from a SG beam with (a) $N=3∕2$, $P_{\mathrm{in}}=20.5P_{\mathrm{cr}}$ and $w_0=1$ mm, and (b) $N=2$, $P_{\mathrm{in}}=88P_{\mathrm{cr}}$ and $w_0=2\mathrm{mm}$.

Figure 5

(a) (Color online) Plasma strings $\left[{\max }_t\rho (x,y,z,t)⩾{10}^{15}{\mathrm{cm}}^{-3}\right]$ of the $20.5P_{\mathrm{cr}}$ SG beam used in Fig. 4a. (b) ${\mid \mathcal{E}\mid }^2$ vs $(x,0,t)$ for the same beam. (c) Plasma strings from the $88P_{\mathrm{cr}}$ SG beam of Fig. 4b.

Figure 6

(Color) Shot-to-shot fluctuations in the filamentation pattern of the $10\text{−}\mathrm{Hz}$ rated Teramobile laser delivering $230\text{−}\mathrm{mJ}$ pulses with (a) $600\mathrm{fs}$ duration $\left(P_{\mathrm{in}}\simeq 120P_{\mathrm{cr}}\right)$ at $z=40\mathrm{m}$, and (b) $100\mathrm{fs}$ duration $\left(P_{\mathrm{in}}\simeq 700P_{\mathrm{cr}}\right)$ at $z=35\mathrm{m}$.

Figure 7

(Color) Filamentation patterns (a) produced experimentally for the $120P_{\mathrm{cr}}$ beam at $z=1,30,40$, and $55\mathrm{m}$. (b) Numerical computations of the same beam from Eq. (3). Maximum intensity is limited to twice the input intensity. In this figure as well as in Figs. 8, 9, the image scale is about $1.5w_0\times 1.5w_0$ ($w_0=$input beam waist), for both the experimental and numerical snapshots.

Figure 8

(Color) Filamentation patterns (a) produced experimentally for the $700P_{\mathrm{cr}}$ beam at $z=30$ and $50\mathrm{m}$. (b) Numerical computations of the same beam from Eq. (3). Labels (1)–(3) indicate beam zones discussed in the text.

Figure 9

(Color) Filamentation patterns of the $1000P_{\mathrm{cr}}$ beam delivered by the Teramobile at different propagation distances: (a) Experimental transverse distributions. (b) Image plots from numerical computations performed with Eq. (3).

Figure 10

(Color online) Intensity vs $(x,y)$ for the beams shown in Fig. 7b (top row) and Fig. 9b (bottom row).

Figure 11

(Color) Three “robust” filamentary structures propagating over $\sim 8\mathrm{m}$ from the focal point $\left(f\simeq 40\mathrm{m}\right)$ of a converging beam with $760P_{\mathrm{cr}}$ delivered by the Teramobile laser system.

Figure 12

Propagation of the same beam as in Fig. 11, numerically computed from Eq. (3) with a digitized data file of the input beam intensity profile affected by a spatially parabolic phase. Three filaments, identified by the labels 1, 2, and 3, can develop long sequences $\left(⩾2\mathrm{m}\right)$ after the focus. Although partly disconnected over $\sim 10\mathrm{m}$, their strong directivity yields the appearance of quasicontinuous strings of light. The numerous filaments occurring at $z<40\mathrm{m}$ are not visible in Fig. 11 due to the strong parallax in the beam imaging.

Figure 13

Underresolved $\left(3\mathrm{D}+1\right)$ numerical results with spatial stepsize $\Delta x=\Delta y\simeq 100\mu \mathrm{m}$ for the multiple filamentation patterns shown in Figs. 5c, 8.