Impressive laser intensity increase at the trailing stage of femtosecond laser filamentation in air

The longitudinal distribution of the laser peak intensity inside a half meter long femtosecond laser filament in air is studied by measuring the signal ratio of two nitrogen fluorescence lines, 391 nm and 337 nm. The experimental results reveal that laser peak intensity initially remains almost constant (~4.3 × 10 13 W/cm 2 ) inside the filament. However, before the end of the filament, surprisingly the laser intensity undergoes dramatic increase. A maximum intensity as high as 2.8×10 14 W/cm 2 could be reached. The experimental result is unexpected by the conventional intensity clamping scenario, according to which the laser peak intensity would feature low variation along a filament. The experimental result is then interpreted as being due to the generation of a short pulse at trailing stage of the filamentation with reduced diameter. This phenomenon might be of great interest owing to its potential application in high-order-harmonic generation and producing isolated single attosecond laser pulse through simple experimental approach. ©2012 Optical Society of America OCIS codes: (190.7110)Ultrafast non-linear optics; (260.5950) Self-focusing; (300.6410) Spectroscopy, multiphoton. References and links 1. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005). 2. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47– 189 (2007). 3. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). 4. J. Kasparian and J.-P. 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Introduction
In recent years, the filamentation of femtosecond laser pulses has stimulated extensive research interests (for review, see, e.g., [1][2][3][4][5]).Its underlying physical mechanism has been understood as the dynamic counteraction of the optical Kerr effect induced self-focusing and the defocusing effect of the self-generated plasma.Recently, high order kerr effect (HOKE) has been brought forward as an alternative mechanism to balance the self-focusing [6].Though there is still hot debate on this subject [6,7], intensity clamping phenomenon, which is one of the most profound effects during femtosecond filamentation, takes place in both.
The first experimental demonstration of intensity clamping in air was carried out by A. Braun et al [8].They measured the pulse energy contained inside a filament and found it is roughly constant over long propagation distance.Later, J. Kasparian et al. have suggested a clamped intensity of 4 × 10 13 W/cm 2 in air [9].The intensity is in good agreement with the value of 4.5 × 10 13 W/cm 2 interpreted by Lange et al. based on the higher order harmonic spectrum measurement [10].
It has become one of the central concerns in the community that to what extend the clamped intensity could hold constant.Recently, the effect of external focusing condition has been discussed intensively.F. Théberge et.al. have found that the clamped intensity increases with shortening of focal length [11].According to the numerical simulation of P. Kiran et.al., the peak intensity could be as high as 10 15 W/cm 2 under strong geometric focusing with numerical aperture up to 0.1 [12].X. Liu et al. have lowered this value to 5 × 10 14 W/cm 2 by considering the double ionization in the analytical model [13].Furthermore, enhancement of laser intensity inside the filament in air has been observed by using spatiotemporal focusing technique [14].Efforts have also been made by focusing PW laser pulse in air in SIOM [15] and TW laser pulse into argon gas [16].The outcome hints that the laser intensity was increased no more than 30% though the input laser power was increased by two orders of magnitude.
However, the laser peak intensity along a single filament is generally considered to be rather flat due to the intensity clamping phenomenon.In this paper, we experimentally characterize the longitudinal laser intensity evolution of a half meter long filament in air with a newly established experimental approach.The experimental results demonstrate that the laser peak intensity first goes through a plateau region.Impressively, at the last few centimeters of the filamentation zone, the peak intensity is found to increase sharply.The maximum intensity reaches about 6 times higher than that within the plateau region.
The basic principle of our laser peak intensity measurement technique about a filament has been described in [17].It is based on the measurement of the signal ratio of two nitrogen fluorescence lines emitted from the filament in air, namely, 391 nm and 337 nm which are assigned to the second positive band of N 2 (C 3 ∏ u →B 3 ∏ g ) and the first negative band system of , respectively [18][19][20].Because of two distinct excitation mechanisms, the signals of the two fluorescence lines increase with the laser intensity at different orders of nonlinearity.It turns out that the laser peak intensity during filamentation is a simple function of the ratio of the two fluorescence signals and could be simply determined by measuring the ratio.

Experiment and results
The experimental setup is illustrated in Fig. 1.A femtosecond laser pulse was sent by an inverse telescope consisting of a convex lens (f = 40 cm) and a concave lens (f = −20 cm) in ambient air.At the output of the concave lens, the laser beam diameter was about 3 mm at 1/e 2 level.The pulse duration was 45 fs (full width at half maximum, FWHM).The central wavelength of the laser pulse was 800 nm, and the laser energy was 2 mJ/pulse.Filamentation thus took place starting from a distance approximately z = 90 cm with respect to the concave lens.Typical laser profiles recorded by a piece of burn paper are demonstrated in inset (a) for three propagation distances, z = 95 cm, z = 110 cm, z = 130 cm.These positions correspond to the beginning, the middle and the end of the filamentation region, respectively.Inset (a) of Fig. 1 confirms a single filament was created in the experiment.Perpendicular to the propagation direction, the "round" end of a round-to-line fiber bundle was placed close to the filament.The fiber bundle consists of 19 individual solarization-resistant fibers with 100 µm diameter core.The distance between the fiber end and the filament was about 5 mm.The "line" end of the fiber bundle was attached to the input port of a cryogenically-cooled CCD equipped spectrometer as indicated by Fig. 1.The inset (b) of Fig. 1 displays the end views of the round and line ends of the fiber bundle.The fluorescence spectra emitted from the filament was measured by scanning the "round" end of the fiber bundle along the propagation axis.As reported previously [17,18], nitrogen fluorescence lines could be clearly distinguished, while the contribution of the plasma continuum to the spectrum was essentially negligible.
Figure 2 shows the longitudinal distributions of three fluorescence lines -337 nm (black line), 357 nm (red line) and 391 nm (blue line), respectively.The distributions are all normalized to their signals at distance z = 101 cm.The error bars are deduced from the rootmean-square fluctuations of 20 repeated measurements.The measurement error mainly originated from the input laser energy fluctuation, which influenced the subtle propagation dynamic of the filamentation process and introduced the variation of the fluorescence signal [21].In Fig. 2, the plots of 337 nm and 357 nm overlaps over the entire propagations distance.However, it is not the case for the 391 nm signal.As indicated by the blue solid triangles in Fig. 2, the longitudinal distribution of 391 nm initially follows closely 337 nm signal until z = 109 cm.It then increases from the trend of the 337nm and sharply increases towards the end of the filament, implying increased signal ratio between 391 nm and 337 nm.This phenomenon was further highlighted in Fig. 3 by the black line (left label), which shows the ratio of the two nitrogen fluorescence lines, i.e.R = S 391nm /S 337nm , as a function of the propagation distance.R remains almost constant between z = 95 cm and z = 110 cm.On further propagation, R increases and reaches its maximum at z = 131 cm.It has been addressed in our previous study that the 391nm signal is proportional to the number of ions in the excited state through inner valence electron ionization [18].While the fluorescence emission associated with the second positive band system of N 2 (C 3 ∏ u →B 3 ∏ g ), such as 337 nm and 357 nm, is proportional to the total number of ions [14,17].Since 337 nm and 357 nm lines originate from the same excitation mechanism, they depend on the laser intensity in the same way, giving rise to the same normalized signal as shown in Fig. 2. The ratio R between the two fluorescence lines is written as [17]: In Eq. ( 1), I represents the pulse peak intensity, n1, n2 denote the effective orders of nonlinearity of ionizing the molecule into the excited or ground state ion.a and b are proportionality constants.Equation (1) could be further written in a simplified form: ( 1) , where α, β and m are constants that could be determined experimentally.Indeed, an empirical formula has been established [17]: By implementing Eq. ( 3), the laser peak intensity could be obtained by the value of R. Consequently, the red line in Fig. 3 (right label) show the laser peak intensity evolution as a function of propagation calculated according to Eq. ( 3).Within the plateau region, the laser peak intensity is about 4.3 × 10 13 W/cm 2 during the filamentation process.An intensity peak appears and lasts for a few centimeters when the propagation distance exceeds z = 125 cm.The maximum laser intensity achieved is about 2.8 × 10 14 W/cm 2 , roughly 6 times that within the intensity plateau.It is worth mentioning that in Fig. 3 the error bars are estimated from the measurement error of fluorescence signals shown in Fig. 2. When considering the fitting uncertainty of the fitting parameters, namely, α, β and m [17], an additional 10% uncertainty was introduced into the laser intensity calculation.This corresponds approximately to the maximum fluctuation of our laser pulse during filamentation.Furthermore, the validity of the implemented values of α, β and m were ensured up to 1 × 10 14 W/cm 2 as we have previously reported [17].Beyond this range, the precision of Eq. ( 3) would need further confirmation, particularly when the laser intensity is so high that the neutral depletion during ionization is no longer negligible.Nevertheless, these would only influence the specific value of laser intensity at z ≈130 cm shown in Fig. 3, which firmly demonstrates an impressive laser intensity increase at the trailing stage of femtosecond laser filamentation.The experimental result of Fig. 3 is unexpected from the point of view of the conventional intensity clamping scenario, according to which the laser peak intensity would feature low variation along the filament [8].However, the result provides strong evidence on the theoretical prediction proposed in Ref [22].recently, where an intensity spike, having intensity exceeding the clamping intensity by a factor of 3 has been noticed at the trailing  [22], the experimentally observed sharp intensity increase shown in Fig. 3 could be attributed to the re-focusing of the reservoir energy at the end of the filament.In this case, when the inward energy flow to the axis is faster than the energy divergence caused by the plasma due to a shock formation [22], higher laser intensity could be obtained.It is worth mentioning that the high intensity is only achieved in a very confined spatio-temporal area.For example, in argon gas the duration of the created short pulse could be sub-cycle and the diameter is only a few microns, which is around one order of magnitude less than the diameter within the plateau region [22].Indeed, our experimental result of Fig. 2 also implies the shrinking of the pulse.Within the intensity plateau, the laser peak intensity and diameter are almost constant.Therefore, the gradual decline of the fluorescence signal within this region (Fig. 2) could be associated with the pulse shortening as pointed out in [23].However, within the high intensity zone, both the spatial and temporal sizes of the pulse are strongly reduced according to [23].Taking into account that the detected fluorescence signal has been integrated over time and space, it may cancel the signal increase of 337 nm and 357 nm given by the laser peak intensity enhancement.The cancelling effect might not be very pronounced for 391nm signal since its strength depends on the laser peak intensity with higher order nonlinearity than that of 337 nm.

Summary
In conclusion, we have traced the longitudinal laser intensity distribution of a half meter long femtosecond laser filament in air.The laser intensity is interpreted by the signal ratio of two nitrogen fluorescence lines, 391nm and 337 nm.It reveals that the laser intensity is constrained as a constant initially.However, at the end of a long distance propagation, the laser intensity may go up to as high as 2.8 × 10 14 W/cm 2 , and last for a few centimeter.The increased intensity is explained by the appearance of an intense short pulse with a reduced diameter at the trailing edge of the filament under the combined contribution of re-focusing and shock wave formation.The generated short pulse could be of great interest for highorder-harmonics generation, producing isolated single attosecond laser pulse with simplified experimental system [24,25].

Fig. 1 .
Fig. 1.Experimental setup.Inset (a) laser pattern recorded by burn paper at z = 95 cm, z = 110 cm and z = 130 cm, with respect to the convex lens.Inset (b) end views of the fiber bundle.