1. Introduction

Transport of high-peak power femtosecond radiation through atmospheric air has been of great interest due to atmospheric remote sensing with supercontinuum [1], discharge control [2, 3], breakdown spectroscopy, lasing in sky [4]. Most of the long-range experiments in air were performed using TW-power systems with the central wavelength of 800 nm (Ti:Sapphire master oscillator) [4, 5], some experiments employ joule-energy picosecond pulses at 1053 nm from Nd:glass systems [6, 7]. One of the objectives of the experiments with TW-power femtosecond pulses was to deliver uninterrupted plasma channel far through the atmosphere [8]. To avoid the gaps between the plasma spots, which arise inevitably in the long-range filamentation in air [9], the amplitude [10–11] or phase [13–15]regularization of multiple filaments might be applied.

Employing ultraviolet (UV) 248-nm beam allows one to create wide transverse area of the low-density plasma and ~100 times decrease the critical power for self-focusing as compared with 800-nm pulse [16–18]. The focusing-refocusing cycles leading to the gaps between the plasma spots in 800-nm filament are less prononunced with 248-nm one due to less spectral broadening and suppressed conical emission [18–20]. So, filament of 248-nm pulse can be considered as a competitor to 800-nm one for producing plasma and light channels in air [21]. Delivery of the UV radiation over long distance [12] is important for UV photochemistry of the atmosphere. Halogen species and chlorine oxides, which play a significant role in the ozone depletion, absorb at 248 nm [22]. UV pulses can be used for water pollution remote monitoring [23].

In this paper we apply the technique of the formation of femtosecond UV filaments arrays [12] to separate the particular filament and to study its evolution up to filamentation end. Different sizes of the mesh cell are explored and one particular is found to be the optimum and to correspond exactly to a single filament formation per a cell. These single filaments form an array of longitudinally uninterrupted light channels with the length not less than 15 meters in air.

2. Experiment and results

We used hybrid Ti:Sa/KrF sub-picosecond 248-nm laser facility to study multiple filamentation along ~100-m propagation path in the corridor [Fig. 1(a)]. The laser pulses are produced by the Ti:Sa laser facility Start 248M (“Avesta-Project”) with the wavelength 744 nm, energy up to 4 mJ, and duration 90 fs. Third harmonic (248 nm, 300 μJ, 90 fs) of these pulses is amplified by the two-cascade e-beam-pumped wide-aperture excimer KrF system [24] up to 0.2–0.5 J for the duration of (870 ± 50) fs, measured with XeF autocorrelator [25]. The collimated beam with the size of 10 cm propagated freely or was transmitted through a mesh with three sizes of square cells: 4.5, 11 and 17 mm [Fig. 1(b)]. These cells were cut out to reveal the optimum cell size.

 

Fig. 1 (a) Experimental setup for corridor measurements of regularized and stochastic 248- nm filamentation. (a, inset) Registration system on the mobile table. (b) Mesh with different cells: 4.5 × 4.5 mm² (~4 P_cr), 11 × 11 mm² (~20 P_cr), and 17 × 17 mm² (~60 P_cr). (c) Calibration curve. (d, e) Filamentation patterns before and after calibration, respectively

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To compare regularized and stochastic filamentation we measured the transverse fluence distribution along the ~100-m path inside the building with the step of ~5 m [inset to Fig. 1(a)]. High-intensity UV radiation hits K8 glass plate and initiates its luminescence. The objective lens translates the luminescence image to Ophir Spiricon SP620U CCD-camera. The luminescence L depends nonlinearly on the incident UV fluence F and reveals essential pedestal [Fig. 1(d)]. The incident fluence in arbitrary units can be reconstructed from the glass luminescence based on the calibration curve obtained from a separate experiment. In this experiment the 3^rd harmonic (248 nm) of Ti:Sa pulses amplified in the discharge-pumped KrF system Lambda Physik EMG TMSC 150 were stretched up to 870 fs by a fused silica plate and focused by a 1-m lens onto K8 glass placed into the geometrical focus. At this position the beam diameter on the glass surface was (240 ± 50) μm for the incident energies of 1–250 μJ. So, we can assume that in this experiment the average fluence F = W/S is proportional to energy W measured by a calorimeter. The luminescence measured as in the corridor experiment is related to the laser energy by the power law W(L) ∝ L^3.5 [Fig. 1(c)], and so is the incident fluence: F(L) ∝ L^3.5. Using this relation, we recalculated luminescence to fluence [cf. Figs. 1(d) and 1(e)]. The recalculated images have the increased contrast and are closer to the actual UV fluence distribution.

The peak power of our UV pulse is ~230 GW, while the critical power at 248 nm is P_cr = 40–120 MW [16, 18]. The critical power P_cr depends on the pulse duration. Adapting the method of [26], we experimentally obtained P_cr = (75 ± 10) MW [25] twice higher than in [16] for 100-ps 266-nm pulse. The essential drop of the critical power in the UV as compared with the infrared follows from the squared wavelength scaling factor as well as the nonlinear Kerr coefficient increase [17]. If no mesh is inserted into the beam, the filaments are seeded by the amplifier output window imperfections and undergo clusterization [24, 27] (Fig. 2, stochastic). The cell size is critical for successful filament regularization and should contain ~3 P_cr [28].

 

Fig. 2 (a–f) Experimentally measured filamentation patterns at different propagation distances for the regularized (upper row) and stochastic (lower row) cases. (g) The change with the distance of the average number of filaments within a small-size cell

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Based on the measurements of the critical power P_cr for 870-fs pulse, we have chosen three cell sizes: 4.5 mm (~4 P_cr), 11 mm (~20 P_cr) and 17 mm (~60 P_cr) [Fig. 1(b)]. We observed the regularized (one filament per a small-size cell) and semi-regularized filamentation regimes (Fig. 2, regularized, mid- and large-size cells). A pattern within the large cell matches the stochastic filamentation pattern (cf. with stochastic patterns in Fig. 2). By the first position available for registration, the UV beam has already travelled for 22.2 m along the inclined path in air and the filaments are well developed [Fig. 2(a)]. Multiple filamentation persists for the following 30 m, so that by the distance of 53.9 m only few weak “hot spots” can be detected. The fluence distributions in Fig. 2 were taken for the same mesh position and approximately the same geometrical position of the registration table relatively to the bunch of UV filaments.

Hence, we were able to calculate the average number of filaments per any area selected in the image. For the smallest 4.5-mm cell a single filament per a cell was detected with a 5-m interval. Since this single filament was reproducible from shot to shot in the transverse space we can state that it is the same UV light filament extended for ~15 m at least [4 measurements with 5 m interval, see Fig. 2(g)]. Indeed, there are no physical factors that lead to the filament interruption and subsequent reconstruction within this distance. Focusing-refocusing cycles are suppressed in the UV filamentation [18]. Due to periodic spatial arrangement and flat-top shape of the initial UV beam, the average flow of energy between the cells is zero. Since the fluence decrease is reasonably weak from 22.2 to 37.5 m [Figs. 2(a)–2(d)] within the smallest 4.5-mm cells, we can state that the regularized light filaments were longitudinally uninterrupted for 15 m.

We counted the average number of filaments distributed over the beam area of 11 × 11 mm² (four small-size cells or one mid-size cell) or 17 × 17 mm² (one large-size cell) in Figs. 2(a)–2(f). The best regularization is supported by four small cells confidently revealing four filaments in the propagation region 22.2–33.3 m; stronger fluctuation in the number of filaments was found in the mid-size cells [Fig. 3(a)]. In the stochastic case note the highest number of filaments at 22.2 m and the maximum rate for filament number decrease by the distance of 27.5 m [Fig. 3(b)].

 

Fig. 3 The experimentally obtained change with the distance of the average number of filaments regularized (a) by the small- and mid-size cells of the mesh in the area 11 × 11 mm² and (b) by the largest cell and stochastic filamentation in the area 17 × 17 mm²

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3. Numerical simulations

Simulations of the filamentation in small- and mid-size cells were done in the coordinates (t, x, y, z) [29] using the nonlinear envelope equation (NEE) [5, Eq. (73)]. The temporal dynamics of the pulse during multiple filamentation is important for our simulations since the newly born filaments are delayed in time relatively to the primary ones [4]. Due to the extraordinary beam size of ~25 smallest cells, we used periodic boundary conditions in the transverse (x, y) plane according to Ref. [10]. The period is the cell size d, the barrier between cells has a width g [Fig. 4(c)]. We include into NEE the 2^nd and 3^rd orders of air dispersion (k₂ = 48.8 fs²/m and k₃ = 6.92 fs³/m). The rate equation for the plasma density accounts for the field ionization only. The oscillation energy of an electron in 248-nm 100-TW/cm² field is ~0.5 eV, so ~25 collisions are required to achieve the ionization potential of O₂ and N₂. The collisional frequency is ~1 THz. Thus, the avalanche ionization can be neglected even for the few picosecond pulses. The initial radiation inside the cell has the apodized square transverse and Gaussian temporal profile.

 

Fig. 4 (a–d) Fluence channels obtained from simulations. (a, b) small-size cell (1 × 1 mm², 4 P_cr), (c, d) mid-size cell (2 × 2 mm², 20 P_cr). (a, c) the transverse projection of the fluence channels (b, d), respectively. (e) Experimentally measured fluence distribution in good qualitative agreement with the results of numerical simulations

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The filamentation simulations require fine transversal and temporal resolution chosen as Δx = Δy ≈ 2 μm, Δt = 1.5 fs. This limits the cell size d and pulse duration in the simulations for the reasonable computation time. Also, the total computation time is inversely proportional to adaptive propagation step Δz ∝ I₀/I_m(z), where I_m is a peak intensity at given z, I₀ ≡ I_m(z = 0). Hence it is reasonable to decrease the computation time by increasing I₀ while preserving n₂I₀d² ∝ P/P_cr, where P is a pulse peak power. So we decreased the cell size and pulse duration by a factor of ~5: duration 200 fs FWe⁻¹M (2¹¹ grid points), cell sizes d = 1 and 2 mm (2⁹ × 2⁹ and 2¹⁰ × 2¹⁰ grid points), barrier g = 0.15 mm, as well as Kerr coefficient by a factor of ~2 (to n₂ = 5 × 10⁻¹⁹ cm²/W, cf. with n₂ = 8−12 × 10⁻¹⁹ cm²/W [17, 18]), and chose initial intensity I₀ = 0.2 TW/cm² so P/P_cr ≈ 4 (20) for small- (mid-) size cell. The propagation step Δz is about 2 cm at z = 0. Nevertheless, the computation time was 6 weeks on a workstation with four Intel Xeon E7-4870 processors (paralleled to 16 threads). Thus, our rescaling keeps the square beam shape and peak power exceed over critical similar to experiment, what ensures the reproduction of the most general behavior features of filamenting 248-nm beam.

The correctly chosen size of the small cell containing 4 P_cr ensures predefined formation of filaments in agreement with the experiment (cf. Figs. 4(a) and 4(b) with Fig. 4(e), small cells). The projection of the fluence channels [Fig. 4(b)] onto the transverse plane [Fig. 4(a)] proves their unchangeable position in (x, y) space for the distance along which these channels survive.

The mid-size cell contains ~20 P_cr. Multiple filaments are initiated in the angles of the square cell so that four parent filaments appear at z = 0.75 m [Fig. 4(c), 4(d)]. The rings outgoing from them interfere inside the cell and initiate the perturbations for the new filaments [Fig. 4(d)]. Following the mesh cell symmetry they appear at the sides of the square similar to the initial cell [Figs. 5(e), 5(f)]. Through successive formation of perturbations through the ring interference, the sides of the square formed by the filaments decrease with propagation in the experiment and the simulations [Figs. 4(d), 5(c), 5(d)]. The filament-to-filament distance in the small-size cell remains constant (the black curves in Figs. 5(c), 5(d) stay parallel to the propagation axis). This confirms the existence of robust uninterrupted light channels with the same transverse position along the propagation direction resulting from the regularized filamentation.

 

Fig. 5 (a, b) Experimental and (e, f) numerical fluence patterns [replicas from Figs. 2(a), 2(e), 4(d)]. (c, d) Decrease of the distance between the “corner” filaments within a mid-size cell (red) and the fixed distance between filaments in neighbouring small-size cells (black)

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4. Conclusions

In conclusion, we have shown that sub-terawatt peak power pulse centered at 248 nm forms multiple filaments in the same positions of the transverse beam plane for more than 15 m of propagation in air. Their reproducible transverse positions result from insertion into ~10-cm beam the mesh with the optimum 4.5 × 4.5 mm² cells containing ~4 critical powers for self-focusing in air. 3D+time numerical simulations show that the beam regularized with the mesh cells of the optimum size produces uninterrupted ultraviolet light channels with fixed positions in the transverse space similar to the experiment. Unoptimized mesh cell size containing ~20 critical powers for self-focusing leads to the shift of the initially formed filaments to the cell center and destruction of uninterrupted propagation in both the experiment and the simulations.