1. Introduction

For more than a decade now the field of femtosecond laser filamentation in transparent media has proven very rich both in terms of basic physics as well as in applications [1]. Filaments are dynamically self-trapped laser beams that propagate over extended distances well beyond the Rayleigh length. A number of physical effects, linear and nonlinear, are involved in the process including the Kerr effect, multiphoton ionization, nonlinear losses, dispersion and diffraction. The main characteristic of filaments is their high intensity along the propagation path reaching in air ~5·10¹³ W/cm². Due to this high intensity a number of interesting effects and nonlinear processes are linked to the filament, such as the generation of long connected plasma strings, extended spectral broadening (supercontinuum), and nonlinear wave mixing.

One of the most interesting nonlinear phenomena that can occur during the filamentation is harmonic generation. In particular, third-harmonic (TH) generation has drawn considerable interest in the last decade with air being the most popular medium for harmonic generation [2–9]. An attractive feature of TH generation using filamentation of high intensity IR femtosecond laser pulses is that ultra-short pulses can be generated in the UV wavelength range [2]. So far in air, TH conversion efficiency up to 0.2% has been reported [3, 5, 6]. Besides air, generation of third-harmonic wave through filamentation has been observed in noble gases [4, 10–12], methane [13], and various liquids [14].

Regarding the spatio-spectral characteristics of the TH beam, it has been demonstrated both numerically [8, 15, 16] and experimentally [7–9] that TH generation during IR femtosecond laser pulse filamentation in air results in both on-axis and off-axis (conical emission) TH components. Axial TH emission starts at pump powers below the critical power for self-focusing in air and it grows as the pump energy is increased despite the fact that the phase-matching condition between the fundamental and TH waves on the propagation axis cannot be satisfied. To explain this, it has been proposed that some sort of quasi-phase-matching could be achieved due to a nonlinear phase-locking mechanism between the fundamental and TH pulses in the filament [6, 7]. On the other hand, generation of the off-axis TH component due to phase-matching between the fundamental and TH waves starts at pump powers around the critical power. In the far field, the off-axis component appears as a ring whose diameter corresponds to a phase-matching cone half-angle of about 6 mrad [7, 15]. At pump powers above the critical power for self-focusing, most of the TH energy is concentrated in the ring pattern.

In this work, we report the first to our knowledge experimental results on the reshaping of the typical far-field ring profile of TH wave generated within an IR light filament in air into a Bessel-like far-field distribution when a plasma string is placed perpendicularly into the path of the filament. A significant (more than two orders of magnitude) enhancement of TH pulse energy has also been observed under these conditions.

2. Experimental setup

The experiments were conducted using a Ti:Sapphire chirped-pulse amplification laser system supplying 35 fs-long (FWHM) transform limited near-infrared pulses at a central wavelength of 800 nm and capable of producing pulses with energies up to 30 mJ at 50 Hz repetition rate. The experimental setup is shown in Fig. 1. First, the fundamental laser beam was split into two arms, Pump and Signal, using a beamsplitter. Laser pulses in the Signal arm with energies up to 3mJ were focused using a 40 cm-focal-length lens L2, and created a light filament in air with a length about 2 cm. The Signal arm beam was apertured with a circular iris 7 mm in diameter in order to obtain a long filament. Pulses in the Pump arm with energies up to 5 mJ were focused perpendicular to the Signal beam with another lens L1 (focal length 10 cm) to produce a dense plasma channel in air which was indicated by the appearance of an approximately 2 mm-long spark. The position of L1 was adjusted to allow the light filament in the Signal arm to cross the central part of the spark. Similarly, to optimize energy and spatial profile of the generated TH wave, we varied the position of the Pump plasma string along the filament by changing the position of L2. The delay between the pulses in the two arms was adjusted with a delay line.

After the filament, the far-field spatial patterns of the TH and fundamental waves in the Signal arm were visualized using a paper screen (not shown) where the images of produced fluorescence were captured with a digital camera. Also, for more accurate measurements of the far-field intensity distribution of the generated third harmonic, the Signal beam was projected onto a 12-bit UV sensitive linear charge-coupled device (CCD) camera, preceded by appropriate interference filters at 266 nm in order to isolate the TH beam. For the TH energy measurements, a fused silica prism P was used as a dispersive element to separate the TH from the fundamental, and total TH energy was measured with a calibrated photodetector D as shown in Fig. 1.

 

Fig. 1. Schematic of the experimental setup. Here: BS, beamsplitter; L1, L2, lenses; P, prism; D, photodetector.

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3. Results and discussion

First, fluorescence images produced by the TH and fundamental beams on the screen, placed at a distance 1.4 m beyond the filament position, at several values of the relative delay τ between pulses of the two beams were recorded. The Pump and Signal pulse energies were 0.5 mJ and 0.6 mJ, respectively. Figure 2(a) shows spatial distributions of the TH (blue ring) and of the fundamental beams (red central spot) for negative values of τ corresponding to the situation when the Signal pulse arrives at the crossing point well before the Pump pulse. We defined a zero delay point as a delay at which the effect of the Pump pulse on the Signal pulse propagation dynamics becomes noticeable. As one would expect, TH generation is not affected by the presence of the Pump for negative τ, and its far-field profile exhibits a characteristic ring pattern [7, 9]. When t becomes positive, the plasma string generated by the Pump pulse in the Signal beam path changes the TH generation dynamics dramatically, transforming the TH far-field profile into a bright central spot surrounded by rings (Bessel-like shape) as shown in Fig. 2(b). It is worth noting that in the latter case the camera exposure time was reduced by 50 times and still resulted in saturation of the TH central spot indicating a very strong increase of the TH intensity. At small positive delays up to 150 fs, TH beam profile was asymmetric (not shown), and it acquired radial symmetry at τ = 200 fs which corresponds to the overlap of intensity maxima of the two pulses at the center of the Signal filament (200 fs is equivalent to 60 μm of propagation in air, and the laser filament diameter in air is approximately twice that distance [1]). For longer delays up to 40 ps - the maximum range of our delay line – TH distribution remained symmetric while its energy monotonically decreased with an increase of the delay. In all cases, the maximum of TH generation efficiency was observed when the Pump plasma string crossed the central part of the Signal filament where the light intensity was the highest. The polarization of TH emission in both cases was linear and parallel to the polarization of the Signal beam.

 

Fig. 2. Far-field images of the TH (blue ring) and of the fundamental (red central spot) beams taken with a digital camera on a paper screen placed at 1.4 m from the filament for (a) negative τ; (b) τ =200 fs. In (a), the fundamental beam spot is partially shadowed by the TH ring. (c) Azimuthally averaged TH intensity profiles, versus divergence angle, measured with a linear CCD, dashed and solid curves for (a) and (b), respectively.

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Clearly, the centro-symmetric shape of the TH cannot be explained considering only the Pump cylindrical plasma string and would suggest that the Signal filament beam profile plays an important role. By Fourier transforming the far-field Bessel-like TH pattern, one can retrieve its near-field profile, which consists of a central spot surrounded by a single ring. This kind of profile is known to exist in filamentation produced by circularly apertured Gaussian beams (see for instance Ref. 17), as the one used in our experiments. To further confirm this scenario, we repeated the experiments without using the iris aperture on the Signal arm, which resulted in a simple Gaussian filament mode. The TH obtained in the latter case showed only an on-axis component without any surrounding rings.

To find the far-field divergence and relative intensities of the generated third-harmonic beams, we recorded their intensity distributions at various distances from the filament using a linear CCD. Dashed and solid curves in Fig. 2(c) represent the one-dimensional azimuthally averaged TH intensity profiles versus divergence angle which correspond to Figs. 2(a) and 2(b), respectively. It is clear that the insertion of the Pump plasma string dramatically reshapes the intensity distribution in a way that the axial emission now dominates. The divergence of the high-intensity central lobe (~1 mrad half-angle) is 8 times smaller than that of the TH ring of Fig. 2(a) (~8 mrad half-angle).

Total energy of the generated TH wave as a function of the delay τ is presented in Fig. 3. Here, plots for two different Pump pulse energies, 0.5 mJ and 2 mJ, are shown. For each energy, TH signal was recorded first for collinearly polarized Pump and Signal waves (open triangles and open circles for 0.5 mJ and 2 mJ Pump, respectively), and then for cross-polarized beams (solid triangles and solid circles for 0.5 mJ and 2 mJ Pump, respectively). Regardless of the Pump energy, similar behavior was observed. TH signal grew fast with an increase of the delay reaching a maximum at τ ≈ 250 fs, and stabilized at τ ≈ 500 fs. At longer delays (see inset in Fig. 3), the TH energy exhibited a slow monotonic decay well described by a (1+βt)^-1 function with decay constant β ≈ 0.02 ps^-1 which is in good agreement with that measured by Tzortzakis et al. in Ref. 18 for free-electron density decay in a femtosecond laser induced plasma channel in air. This is a strong indication that TH generation efficiency is correlated to free-electron density in the Pump plasma string.

 

Fig. 3. TH energy as a function of the delay between the Pump and the Signal pulses. Two lower curves: Pump energy 0.5 mJ, open and solid triangles for co-polarized and cross-polarized beams, respectively. Two upper curves: Pump energy 2 mJ, open and solid circles for co-polarized and cross-polarized beams, respectively. Inset: the same plot with the delay scale extended to 40 ps.

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It is clear that Kerr and other intensity related contributions to the TH signal are limited to the delays comparable with ~500 fs needed for the Pump pulse to cross the Signal filament core. This is confirmed by the fact that TH maxima in Fig. 3 are well pronounced for co-polarized beams whereas for cross-polarized geometry they are barely visible. Such difference is attributed to higher intensities produced due to interference of the two pulses in the co-polarized case. On the other hand, for τ > 500 fs, the enhancement of the TH signal occurs solely due to presence of the plasma string generated by the Pump pulse, and compared to the negative-τ case, it represents approximately 50- and 70-fold increase in TH energy for 0.5 mJ and 2 mJ Pump, respectively.

 

Fig. 4. (a) Measured dependence of TH signal on Pump energy for Signal pulse energy 0.15 mJ (squares), 0.7 mJ (circles), and 1.5 mJ (triangles). (b) TH energy versus Signal pulse energy in the absence of Pump (triangles) and at Pump energy 2 mJ/pulse (circles). The ratio of these two curves represents the TH enhancement factor shown with open circles in (b).

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The dependence of the TH signal on Pump energy is shown in Fig. 4(a) for three different Signal pulse energies: 0.15, 0.7, and 1.5 mJ. The delay τ was set at 1 ps to make sure that any Kerr/intensity contribution to the generated TH is avoided. Figure 4(a) shows that regardless of the Signal energy, TH energy reached saturation at the same (approximately 1 mJ) Pump energy. This behavior is correlated with the appearance of saturation in the plasma string electron density to values close to the gas density ~2.5·10¹⁹ cm^-3 confirmed by independent electrical conductivity [19] and in-line holographic measurements [20] that we conducted, and will be discussed in a following publication.

Since the increase of Pump energy beyond 1 mJ/pulse results in saturation of TH energy, we wanted to find optimal conditions in terms of TH generation efficiency. For that purpose, we measured TH energy as a function of Signal pulse energy for a constant Pump of 2 mJ/pulse, and conducted similar measurements in the absence of the Pump beam. The corresponding experimental results, along with the TH enhancement factor calculated as a ratio of the TH signal in the presence of Pump to that without Pump beam, are presented in Fig. 4(b). They show that for Signal energies up to 0.6 mJ/pulse we have about two-orders of magnitude enhancement of TH energy while for 2.5 mJ/pulse it is increased by mere 50%. Hence, TH generation in the presence of the Pump plasma string has a particular advantage at low Signal pulse energies up to 0.6 mJ/pulse enabling us to obtain both the much higher energy and the better collimated TH beam.

The correlation of the TH emission with plasma attributes such as saturation and decay rate is a strong indication that plasma presence is responsible for the significant enhancement of TH conversion efficiency. Although further exploration is needed to establish a clear view on the physics behind the observed effect, we can propose several candidate mechanisms that could lead to this interesting behavior. It is clear that the presence of a thin (< 100 μm) plasma channel perpendicular to the filament propagation leads to a significant and abrupt change of the optical properties of the medium. As it was recently shown by numerical simulations [16], the TH generation arises from the total material polarization response, i.e. this process can be viewed as a first order scattering process of the incident field from the refractive index changes induced by the intense propagating pulse. Thus, the TH beam is practically decoupled from the core filament after being generated by the nonlinear processes. Likewise, the enhanced TH generation in our case can be the result of an effective increase of the nonlinearity that is induced either by the nonlinear properties of plasma [21] or by an interface effect [22]. More specifically, the plasma can be viewed as a nonlinear medium [21] at the intensity levels achieved in the filament core (~ 10¹³ W/cm²) if we take into account the interaction of the strong laser field with electrons under the perturbation of a strongly nonlinear ionic potential. This model predicts an effective enhancement of the TH generation efficiency which is proportional to the square of plasma density. On the other hand, a significant enhancement of the generated TH has been reported [22] when an ultrafast laser beam is focused on interfaces. The effective nonlinear coefficient for the case of an air/glass interface [22] is by at least 3 orders of magnitude higher than that of bulk glass or air. This enhancement was attributed to either an interface induced nonlinear susceptibility affected by the field gradient [22] or to a competition [23] between the group and phase velocity mismatch that is sensitive only to interfaces.

4. Conclusion

In conclusion, we have shown that the characteristic ring-shaped far-field intensity pattern of third-harmonic wave generated through IR filamentation in air is transformed into a Bessel-like distribution when a plasma string is created by another synchronized femtosecond pulse orthogonally to the filament path. Under these conditions, two-order of magnitude enhancement of TH energy has been demonstrated, which, combined with a dramatic reshaping of TH far-field profile with a significant fraction of its total energy contained in the high-intensity axial component, can prove very useful in applications. The proposed technique is quite versatile since in principle the plasma string can also be generated by other means, e.g. using a synchronized electric discharge between two electrodes.

The measured dependence of TH energy on the relative delay between the two pulses has clearly demonstrated that the observed behavior is attributed to the interaction of the Signal filament with plasma produced by the Pump pulse. Intensity related effects on the TH generation process were insignificant and limited to the delay times for which the two pulses partially overlapped. Finally, possible mechanisms behind the observed phenomenon have been discussed.