1. Introduction

Femtosecond laser filamentation in air is an interesting phenomenon [1–3], and has intensive applications including few-cycle pulse generation, THz radiation source, remote-sensing of atmospheric pollution, lighting and discharge triggering, to name a few. During the femtosecond laser filamentation, the laser beam maintains, in general, a very high intensity over long distance, which is much larger than the optical Rayleigh length, leaving a weakly ionized plasma channel named as filament in its wakefield. Filament appears as a result of dynamic competition between laser beam focusing and plasma defocusing when the peak power exceeds the critical value $P_{cr}=3.77{\lambda }_0^2/8\pi n_0{n}_2$, where λ₀ is the laser central wavelength in vacuum, n₀ and n₂ are the linear refractive index and the nonlinear refractive index coefficient respectively. For applications of femtosecond laser filamentation, it is of great concern to control the propagation of incident ultra-intense and -short laser pulses and optimize the characteristics of filament such as length, diameter in the perpendicular plane, clamping intensity. One example is the formation of plasma grating by few filaments interaction, which introduces a periodically modulated change in the local refractive index of the medium. A stationary plasma grating can be used as an interceptor breaking up the propagation path of filament-forming pulses to enhance the efficiency of third-order harmonic generation [4, 5]. It can also diffract the laser pulses with different frequencies into directions deviating from its propagation direction [6].

As another promising nonlinear process for controlling the propagation of intensive ultra-short laser, energy exchange between few filament-forming pulses has been demonstrated in several experiments involving femtosecond laser filamentation in air and liquid [7–9]. Energy transfer occurs when linearly polarized laser beams are focused and spatiotemporally overlapped. It has been found that the magnitude and direction of energy transfer depend on the relative time delay, initial chirp, laser intensities, interacting location, intersecting angle and relative polarization. It is noticeable that in the first work [7] about energy exchange between laser pulses in the regime of femtosecond laser filamentation, the plasma generation or filament formation is an insignificant contribution desipite its presence. The possible reason is that the volume of filament under this condition was too samll compared to the area of the overall beam and the most of air molecules participate in the molecular rotational excitation rather than being ionized. Its results are well consistent with the classical two-beam coupling theory and attributed to impulsive stimulated Raman scattering (SRS). With a shorter focal length and higher incident laser power, the direction of energy transfer from the lower frequency pulse to the higher one was observed [9]. This result is mediated by a moving plasma grating [10], which is formed at the intersection region of the filaments. It is clearly different from that in a traditional Kerr nonlinearity mediated grating [7]. Efficient energy exchange was also reported and attributed to plasma-mediated forward SRS, facilitated by supercontinuum generation [8]. In plasma-mediated SRS, the electromagnetic fields of two lasers are coupled by an electrostatic plasma wave. In the nonlinear regime involving filamentation, photons are transferred from one beam to another due to blueshift and supercontinuum generation in the plasma, resulting an energy difference between them. By using a motorized translation-stage setup with higher resolution, a fine spectral structure of energy exchange was also observed at small intersecting angles [11,12], in which the direction of energy transfer varies with the spectral components of incident laser pulses.

However, the effect of plasma fromation has not been completely investigated. One effect affecting the energy exchange efficiency is the nonlinear absorption. Accompanying plasma formation in the intersecting region, the energy loss of lasers takes up, which should lead to an influence on the energy exchange process. In this work, we present experimental results on the energy exchange between two filament-forming pulses in air. Spectra of output laser pulses were measured in order to further understand energy exchange process. Different from reported literatures [7–9], the energy exchange efficiency is redefined concerning the inbalance between energy exchange at the negative delays and that at the positive delays. Its dependence on the incident laser power was also investigated.

2. Experimental setup

The schematic diagram of experimental setup is illustrated in Fig. 1(a). The laser system used is a Ti:sapphire pulse amplifier system which is capable of producing 43 fs pulses centered at 790 nm with energy of 3 mJ, at a repetition rate of 1 kHz. The chirp of laser pulse can be neglected, which has been confirmed by FROG measurement. The laser beam was split into two arms by a beamsplitter (splitting ratio 55:45). One arm is defined as the delay-tuned beam (Beam-1) and another arm defined the delay-fixed beam (Beam-2). Both beams were focused by two lens with focusing length of 400 mm to generate two filaments. A sensitive CCD was installed on the top of the filament intersecting region to record the fluorescence image of the plasma, which was collected by using an Al-coated concave mirror installed under the bottom of the filament intersecting region. Filament diameters were measured by CCD flurescent imaging technique to be approximately 100 μm. If the filaments interact without temporal overlap, they simply pass through each other as shown in Fig. 1(b). By tuning the delay-tuned beam passing through a motorized translation-stage setup (precision of 2.5 μm), the two beams can be spatiotemporally overlapped to generate a bright sparking spot accompanying blast sound as shown in Fig. 1(c). The minimal translation setup corresponds to a temporal resolution of 0.495 fs. By concerning the influence of effective interaction length, the crossing angle between the two beams marked as θ was fixed at 17 degree by referring to [8]. The spectra of output filament-forming pulses were measured by a spectroscopy (USB4000). The energy of each beam is controlled by passing through several neutral density filters and measured by an energy meter.

 

Fig. 1 Schematic diagram of experimental setup (a) and images of the filament produced by the intersecting filament-forming laser pulses at the crossing angle of 17 degree, with (b) 0.15 ps and (c) zero time delay between the pulses. Apparatus components include two focusing lenses (L1, L2), a beam splitters (BS1), a delay line (DL), a neutral density filter (ND), a spectrograph (SM) and an energy monitor.

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3. Results and discussion

Firstly we focus on the transmission of Beam-1 when the relative time delay between these two lasers is adjusted around their overlap. It can be simply decribed as E₁(τ)/E₁₀ where E₁(τ) and E₁₀ represent the energy at time delay of τ and the initial energy of Beam-1, respectively. The positive delay corresponds to Beam-1 delayed with respect to Beam-2. Figure 2 shows results obtained with different laser powers. With an incident laser power of 0.7P_cr, the experiment result didn’t exhibit obvious energy exchange between two laser pulses, as shown in Fig. 2(a). In this case, laser filamentation was not observed by CCD. With a higher incident laser power as shown in Fig. 2(b), two distinct regimes appear with a FWHM of 300 fs. This width is related to the overlap time of two filament-forming pulses, which corresponds to a coupling distance of 90 μm. Beam-2 transfers energy to Beam-1 during the negative delay, while Beam-2 gains energy from Beam-1 as the relative time delay is positive. It is similar to the observation by Y. Liu et al in [8] and clearly different from that by Bernstein et al in [6], in which the trailing pulse obtains energy from the leading pulse. Besides, there is an inbalance of energy exchange, e.g., the maximum energy gain of Beam-1 at the negative delays is smaller than its maximum energy loss at the positive delays. When the incident laser power was 3.0P_cr (Fig. 2(b)), the transmission addition is about 0.18 (−0.30) at the negative (positive) time delays, leading an absolute difference of 0.12. With the increase of incident laser power, this inbalance correspondingly grows up. When the incident laser power is increased to 6.6P_cr as shown in Fig. 2(c), this absolute difference increases to 0.20.

 

Fig. 2 (a) The energy of two laser pulses versus the relative time delay with initial laser power of 0.7P_cr. (b)–(c) The transmission of Beam-1 as a function of the relative time delay when the initial laser powers are 3.0P_cr (b) and 6.6P_cr (c) respectively. The red solid (green dashed, blue dash-doted) line represents the best fit by using the formula T(τ) (T₁(τ), T₂(τ) + T₃(τ)).

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We next turn to the spectral characteristics of energy exchange, which is an important way to understand its underlying physical mechanism. The spectra of two pulses as a function of the relative time delay are shown in Fig. 3. It can be seen that the spectrum of Beam-1 exhibits a large degree of blueshift and enhancement of intensity before the zero time delay, indicating that Beam-1 transfers energy to Beam-2 at the negative delays. On the contrary, the spectrum of Beam-2 beam shows an exactly opposite behavior. For both beams, the laser intensity is dramatically reduced near the zero time delay. With a higher incident laser energy of 1.04 mJ, the Beam-1 spectrum is largely broadened and blueshifted to 700 nm at zero time delay, as shown in Fig. 4. The Beam-1 spectra at the time delays of −74 fs and 99 fs, which are far away from the zero time delay, are presented for comparison. This broadening of the supercontinuum emission (> 60 nm) mainly goes toward to the blue side from the fundamental frequency and almost does not extend to the red side. When the femtosecond laser pulse experiences filamentation and interacts another filament, the main reason for spectral broadening is regarded as the cross-phase modulation effect (XPM), not the self-phase modulation effect (SPM). The ionization effect introduces a negative refractive index change, which only shifts the frequency to the blue side. In our experiment, the external focusing geometry (focal length ∼ 400mm) and filament interaction lead to strong ionization of air. If the contribution of ionization effect exceeds that of the laser intensity variation, the spectral broadening to the red side should be suppressed. Note that the laser intensity at positive time delay of 49.5 fs is much less than that at negative time delay of −49.5 fs. One reason for this is that third harmonic generation at the positive time delays is more effecient than that at their corresponding negative time delays [5].

 

Fig. 3 The spectra of the delay-tuned Beam-1 (a) and delay-fixed Beam-2 (b) verse the relative time delay when the initial total energy was 0.52 mJ.

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Fig. 4 The spectra of the delay-tuned Beam-1 verse the relative time delay when the initial total energy was 1.04 mJ.

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From the two-beam coupling theory, the energy exchange vanishes if there is no difference in frequency, as is the case for unchirped laser pulses with identical carrier frequency. But in the pulsed experiments involving filamentation, the delayed nonlinear response such as the impulsive Raman nonlinear response and plasma formation can effectively chirp the involved laser pulse [8, 13]. In our case, the incident laser pulses have the same carrier frequency ω₀, but due to chirp from the delayed nonlinear responses the instantaneous frequency varies over the pulse duration. The frequency difference Δω = ω₁ − ω₂ between the delay-fixed pulse and the delay-tuned one depends on their relative time delay τ, where ω₁ and ω₂ represent the frequency of delay-fixed pulse and delay-tuned pulse respectively. With this frequency difference, a phase shift occurs between the local optical interference pattern and the refractive index modulation. In the case where the nonlinear refractive index follows the Debye relaxation, the energy exchange between laser beams can be easily derivated from the wave propagation equation [14]. As is well known, the generation of plasma introduces a retarded negative refractive index change by −ρ/2ρ_cr, where ρ_cr is the critical plasma density for 800 nm laser. This could negatively chirp the laser pulses. For pulses with negative chirp, Δω is negative for τ < 0 as the tail of Beam-1 and the front of Beam-2 overlap. Correspondingly, it is positive for τ > 0 as the tail of Beam-2 and the front of Beam-1 overlap. Though the decay process of nonlinear refractive index due to plasma generation is more complicated than a simple exponential law [15], but it doesn’t affect the validity of refractive grating dominating the energy exchange process. Therefore, according to the two-beam coupling theory [13,16,17], one can expect that the delay-tuned pulse gain energy at negative delays and lose energy at positive delays because of plasma formation.

We use the time-domain theory for pump-probe experiments with chirped pulses [18] developed by N. Tang et al. to fit the transmission of Beam-1, as shown in Fig. 2. In our case involving femtosecond laser filamentation in air, the linear absorption can be neglected. Taking the temporal response function R(t) of air as exp(−t/τ_n), where τ_n is the exponential decay constant, the transmission of Beam-1 can be estimated as:

The red solid lines in Fig. 2 are the best fitted curves by using T(τ) with the relevant pulse duration of 43 fs and energies. A good fit of B_xxxx = 9.0 × 10⁻⁶cm³/erg · s was obtained in Fig. 2(b). Also, as a dependent parameter, the temporal width of laser pulse was extracted as τ_p ≈ 60 fs, corresponding a chirp parameter C = 0.97. Then the delayed nonlinear refractive index coefficient n_2,d can be extracted as 3.22 × 10⁻¹⁹cm²/W, which is perfectly consistent with the reported value of ∼ 3.01 × 10⁻¹⁹cm²/W in literatures [19, 20]. The fit in Fig. 2(c) yields n_2,d = 1.70 × 10⁻¹⁹cm²/W and the average of extracted n_2,d for different laser powers is 2.78 × 10⁻¹⁹cm²/W, also close to the reported value. The combined contribution T₂ + T₃ originates from coherent interaction between the laser pulses and the air medium [18], which introduces a deep dip near zero time delay as shown in Fig. 2(b) and 2(c). This contribution was also termed the coherent artifact, coming from the absorptive nonlinearities. Any absorptive nonlinearity has a 90 degree phase shift in its polarization compared to the laser field. It leads to an energy exchange between the laser and the medium. Hence, when the laser intensity increases, the influence of absorptive nonlinearity becomes more important and then one can expect larger inbalance of energy exchange, as shown in Fig. 2(c).

Finally, the energy exchange efficiency was also studied as a function of the incident laser power. Similar to the previous work [7], the energy exchange ratio can be defined as

 

Fig. 5 The energy exchange efficiency (a) and the corresponding time duration (b) as a function of the incident laser power.

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These results can be qualitatively understood as follows: with a higher laser power, a larger volume plasma region and a higher refractive index contrast between the negative contribution due to the plasma formation and the positive contribution due to the delayed rotational Raman effect [9] can be generated, which allows for more efficient energy exchange. Hence, it can be expected that more efficient energy exchange can be achieved at smaller crossing angles because of more effective interaction length. At the same time, the laser energy involving energy exchange process could be reduced by nonlinear absorption. With the competition of these two kinds of contributions, the energy exchange efficiency could reach a maximum value when the plasma density in the interacting region saturates, which also leads to a saturated interaction length. In this case, the laser power corresponding to the maximal efficiency is almost 3.0P_cr. After that, it decreases because of the second contribution, as shown in Fig. 5(a). However, the duration of energy exchange reflects the effective interaction length and has no close relation with strong plasma formation when the interaction length has saturated. Therefore, the duration of energy exchange remains an approximately constant level from P ∼ 3.0P_cr to P ∼ 6.6P_cr (Fig. 5(b)).

4. Summary

In summary, energy exchange between two initially chirp-free femtosecond pulses in air was experimentally studied. The transmission of the delay-tuned beam around the zero time delay were measured. Accompanying energy exchange process, spectral characteristics of output laser pulses shows dramatic blueshift and supercontinuum generation. Because of the inbalance between energy exchange at the negative delays and that at the positive delays caused by nonlinear absorption, the energy exchange efficiency was redefined. When the incident laser power is varied from 0.7P_cr to 3.0P_cr, the energy exchange efficiency increases from 8% to 45%. With higher incident laser power, the exchange efficiency slightly decreases. With a time-domain two-beam coupling model, the experimental results were fitted and the delayed nonlinear coefficient was extracted. These experiments were performed in air, and a higher energy exchange efficiency could be achieved in gaseous media with longer molecule response time. The results may be promising for some applications like filament control and ultra-intense laser propagation.