(Color online) Experimental setup: L1, L2, lenses; CL, plano-convex cylindrical lens (cylindrical axis is parallel to the plane of the figure); P, prism; IF, interference filter at 266 nm; D, calibrated photodetector. 

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... experiments were conducted using a Ti:sapphire chirped-pulse amplification laser system supplying 35-fs, 800-nm IR pulses with energies up to 30 mJ at a 50-Hz repetition rate. The schematic of the experimental setup is shown in Fig. 1. First, the fundamental laser beam was split into two arms, Pump and Filament, using a beamsplitter. Laser pulses in the Filament arm with 0.5-mJ energy were focused using a 100-cm-focal-length lens L1 creating a light filament in air with a length of about 5 cm. The filament interacted with the plasma string generated by pulses in the ...

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... show that the bulk plasma-enhanced nonlinear susceptibility and not the air-plasma interface effect, is responsible for the observed TH enhancement. Furthermore, we propose a phenomenological model and derive a simple expression for the TH intensity that quantifies the dependence of the TH signal on the plasma density. Using this model, which also takes into account phase matching between the fundamental and the harmonic waves in the plasma volume, we discuss the limitations of the technique in producing energetic pulses at third-harmonic frequency in a low-density plasma. The experiments were conducted using a Ti:sapphire chirped-pulse amplification laser system supplying 35-fs, 800-nm IR pulses with energies up to 30 mJ at a 50-Hz repetition rate. The schematic of the experimental setup is shown in Fig. 1. First, the fundamental laser beam was split into two arms, Pump and Filament, using a beamsplitter. Laser pulses in the Filament arm with 0.5-mJ energy were focused using a 100-cm-focal-length lens L1 creating a light filament in air with a length of about 5 cm. The filament interacted with the plasma string generated by pulses in the Pump arm that were focused perpendicularly to the Filament beam with a combination of a spherical (L2, focal length 10 cm) and a plano-convex cylindrical (CL, focal length ∞ / 10 cm) lens. This astigmatic arrangement leads to the generation of two orthogonally oriented plasma channels in the sagittal and meridional foci. By changing the distance between L2 and CL we were able to vary the length of the horizontally oriented plasma string in the meridional focus, while its spatial overlap with the filament was achieved by moving the two lenses as a whole along the Pump beam path. The delay between the pulses in the Filament and Pump arms was adjusted with an optical delay line. After the filament, a fused silica prism P was used to angularly separate the TH and fundamental waves. Finally, the TH energy was measured using a calibrated photodetector (D) which was preceded by an interference filter at 266 nm. In order to study the nature of the physical mechanism responsible for the harmonic emission enhancement, we first need to distinguish between the two different mechanisms that could, in principle, explain the observed effect, namely the bulk plasma properties and the neutral air-plasma interface. More specifically, the presence of charged species (free electrons and ions) can effectively increase the third-order nonlinear optical susceptibility of a medium [11,12]. Thus, the enhancement of TH generation in this case would be the result of a bulk effect in the plasma volume. On the other hand, since the refractive index of a medium changes in the presence of plasma, then by introducing the Pump plasma string we create an interface between the two media with different refractive indices (under our experimental conditions, this difference in air can be as high as 7 × 10 − 3 [14]). Therefore, at such an interface, the inversion symmetry of the bulk along the Filament beam propagation direction is broken, resulting in a field gradient that could be responsible for the enhancement of TH emission (see for example Ref. [13]). In the case of bulk nature of the effect, one would expect the harmonic emission to grow with the plasma thickness (assuming all other parameters are the same), while this would have no effect in the case of the interface model, as the number of interfaces remains the same. So, in order to clarify the origin of the physical mechanism that drives the enhancement of the harmonic wave, we conducted TH generation experiments with various thicknesses of the Pump plasma string. Using the astigmatic two-lens arrangement, described above, we created three plasma strings in the Pump arm with different thicknesses of 0.4, 0.7, and 1 mm, respectively. To guarantee the same mean plasma density ( ∼ 5 × 10 17 cm − 3 ) inside all three plasma strings, the Pump pulse energy was set at 2.5, 5, and 8 mJ, respectively. The total measured TH energy as a function of the delay τ between the Pump and Filament pulses for all three cases is shown in Fig. 2(a). For better understanding these results, a simple schematic of the relative position of the Pump and Filament pulses at different delays is shown in Fig. 2(b). We define a zero delay point as the delay at which the plasma, produced by the Pump pulse, starts to affect the Filament pulse propagation and, consequently, the energy of the harmonic emission. With further increase of the delay, the Filament pulse will interact with the remaining portion of the Pump plasma channel. Finally, the maximum interaction length will be achieved when the delay (expressed in millimeters of light propagation in air) becomes equal to the Pump plasma string thickness. Therefore, the delays that correspond to the maxima of the TH energy in Fig. 2(a) are equal to the respective plasma string thicknesses, namely 0.4, 0.7, and 1 mm. The results of Fig. 2(a) are a clear indication of the bulk nature of the effect since the maximum TH energy monotonically grows with the plasma thickness. Additionally, the concept of TH enhancement through an interface effect fails to predict our experimental results because in this case the TH energy should depend only on the number of interfaces, which in all three cases is constant (two interfaces). The observed subsequent drop in the harmonic emission in Fig. 2(a) at higher delays is due to the plasma density decay in the Pump channel, as was also observed in Ref. [9]. Since we have proven that the effect of interfaces in our case is not important during TH generation in a plasma channel, we focus on the concept of the enhanced bulk third-order optical susceptibility due to the presence of free-electron–ion plasma. Using this assumption and according to the well-known formula [15,16], the third-harmonic intensity for the case of a Gaussian beam focused in the middle of a plasma medium can be written as I 3 ω ( k, L ) = n 3 ω (3 n ω 3 ω c ) 2 4 ε 0 2 χ pl (3) 2 I ω 3 − L/ L/ 2 2 (1 exp + ( iz/z i kz R ) ) 2 dz 2 , (1) (3) where χ pl is the plasma-enhanced third-order susceptibility, I ω is the fundamental intensity, k = 3 k ω − k 3 ω is the total wave-vector mismatch between the fundamental and TH waves, L is the plasma thickness, z R is the Rayleigh length, and n ω and n 3 ω are the refractive indices at the fundamental and TH frequency, respectively. Equation (1) takes into account the generation of the third-harmonic wave in the bulk of an ionized medium due to effective third-order susceptibility, as well as the effect of the phase mismatch between the fundamental and harmonic waves due to chromatic dispersion. Although Eq. (1) is written for continuous-wave beams, it still holds for ultrashort pulses under our experimental conditions, since the maximum calculated walk-off between the fundamental and TH pulses caused by the group velocity dispersion in a thin plasma layer of ionized air does not exceed 1 fs. In a quasiplanar limit for the fundamental wave phase front ( L z R ), which was also the case in our experiments, Eq. (1) can be simplified ...