Laser energy deposition with ring-Airy beams beyond kilometer range in the atmosphere

Controlled deposition of high laser-power density at remote distances still remains a challenge. Previous experimental works using terawatt peak-power laser systems to initiate filamentation at distances of hundreds or even thousands of meters in the atmosphere have documented limited control on laser energy deposition due to the effects of diffraction and turbulence. In this work, we demonstrate a promising scenario for the projection of high power densities at kilometric distances in air, which requires multiple-gigawatt laser pulses reshaped into ring-Airy beams. We show that a power distribution over a relatively large primary ring accompanied by the inward power flux characterizing a ring-Airy beam limits the occurrence of nonlinear effects to the vicinity of the focal point. Close to focus, self-focusing sets in, which accelerates the convergence process of the beam. We quantify the influence of self-focusing on the focal shift and propose an empirical law that fits our numerical simulation results for the position of the nonlinear focus as a function of the input power and the apodization factor of the ring-Airy beam. We show that once the intensity of the beam exceeds the ionization threshold of air, a short filament is generated whose power content rather accurately corresponds to the critical power for self-focusing, independent of the input parameters.

Figure 1

Normalized intensity plot of a finite-energy ring-Airy beam, with $w_0=4$ mm, $r_0=60$ mm, $\alpha =0.05$, and $z=0$.

Figure 2

Linear propagation of a ring-Airy beam with $P_{\text{in}}\le P_{\text{cr}}$. Evolution of (a) the normalized intensity profile, (b) the power content in the primary ring (red dashed curve) and the on-axis intensity (blue solid curve), (c) the normalized intensity profile, and (d) the energy current density around the autofocus. The energy current is defined as $j_{\perp }(z,r)=\frac{1}{2i}[{\mathcal{E}}^{*}(z,r)\partial \mathcal{E}/\partial r-\mathcal{E}(z,r)\partial {\mathcal{E}}^{*}/\partial r]={\left|\mathcal{E}(z,r)\right|}^2\partial \mathrm{\Phi }/\partial r$ for $\mathcal{E}(z,r)=\left|\mathcal{E}\right(z,r\left)\right|exp\left[i\mathrm{\Phi }\right(z,r\left)\right]$; negative (red) and positive (blue) values represent inward and outward flux, respectively [44].

Figure 3

Ring-Airy beam propagation for input powers up to 70 GW. Evolution of (a) the amplitude, (b) the energy current density (red $=$ inward, blue $=$ outward), and (c) on-axis intensity of the field as a function of propagation distance $z$.

Figure 4

(a) Maximum intensity (blue curve with circles) reached during forward propagation and peak intensity contrast (red curve with diamonds) as a function of input power $P_{\text{in}}$. For comparison, the peak intensity contrast for linear propagation is also plotted with red stars. (b) Nonlinear focus shift (red curve and circles) and position of maximum intensity (blue curve and stars) as a function of the input power; the black solid line denotes the fitting by Eq. (6) with $\alpha =0.05$; the yellow dashed line represents the fitting directly using the expression in Ref. [5]. See the Appendix for fitting with other truncation factors. (c) Fitting parameters ${\xi }_{\alpha }$ (blue curve and stars) and ${\zeta }_{\alpha }$ (red curve and circles) as a function of the truncation factor $\alpha$.

Figure 5

(a) Detailed view of the intensity profile $I(r,z)$ near the nonlinear focus ($z\sim 973$ m) when $P_{\text{in}}=50$ GW. Power contained in a beam section with a radius of (b) $100\mu \mathrm{m}$, corresponding to the filament, (c) 0.5 mm, and (d) 1 mm as a function of propagation distance for different input powers. The value of the critical power $P_{\text{cr}}\sim 3.2$ GW is also plotted as a gray dashed line in (b)–(d). The power $P_{r1}$ is calculated using Eq. (5) by replacing the upper limit of the integral with $r_1$ = 0.1, 0.5, 1.0 mm.

Figure 6

(a)–(e) Simulation results for different truncation factors $\alpha =0.03–0.07$. The notation for each line is the same as in Fig. 4.