1. Introduction

Femtosecond laser filamentation, which results from extreme nonlinear propagation of intense femtosecond laser pulses in transparent media, has been one of the research frontiers in modern nonlinear optics [1–3]. Due to the strong confinement of intense light field over an extended propagation distance [4,5] and self-maintained phase matching [6–8], efficient wavelength conversion can be achieved in a filament induced by femtosecond laser pulses [8,9]. Nevertheless, so far most of the investigations of nonlinear wavelength conversion in femtosecond laser filamentation are focused on neutral atomic/molecular systems. Recently, we have observed generation of forward coherent emissions at wavelengths corresponding to transitions between different vibrational levels of the electronic B²Σ_u⁺ and X²Σ_g⁺ states of molecular nitrogen ions induced by femtosecond laser filamentation in air [10]. Although remote lasing in air based on amplified spontaneous emission (ASE) has been under intensive investigation in the past ten years owing to its potential application in remote sensing [11–14], the new coherent emissions observed in our experiment show some fundamentally different characteristics. First of all, the latter requires an injection of seed, which can be either the third or the fifth harmonic of the pump laser pulse generated during filamentation of the pump in air; thus the new coherent emission has the same polarization direction as those of the pump and the harmonics (whereas ASE-based remote laser has a random polarization). Secondly, and probably more important, because both the pump laser pulses and the harmonic seed pulses are of short pulse durations of a few tens or hundreds of femtoseconds, our observation clearly suggests an ‘instantaneous’ amplification of the seed pulse. Based on these facts, a tentative scheme in which ultrafast generation of a population inversion in N₂⁺ molecular ions is achieved in a time scale comparable to the pump pulse duration was proposed [10,15–17]. However, based on the previous measurements, other nonlinear amplification processes such as four-wave mixing and resonant Raman scattering that could also enable similar observations in Refs [10,15–17] cannot be simply excluded. Thus, it is necessary to carry out additional experiments to identify the underlying mechanism for the generation of coherent N₂⁺ emissions in air by femtosecond laser excitation.

For the above-mentioned purpose, in the present work, we perform systematic experimental investigations based on a pump-probe scheme. As shown in Fig. 1(a) and 1(b), in this scheme, an intense femtosecond laser pulse at 800 nm wavelength serves as the pump for excitation of the molecular system. In such a case, due to the inversion symmetry of air, second harmonic generation of the pump pulses cannot occur in air. Consequently, amplification of self-generated harmonic seed pulses at wavelengths near ~400 nm is impossible. We thus introduce a weak second harmonic pulse generated by a nonlinear crystal to serve as the probe, which can be temporally arranged with an arbitrary delay time behind the 800 nm pump pulse. The amplification of the seed critically depends not only on the temporal delay between the pump and probe pulses [18], but also on the intensity, spectrum, and the incident direction of the probe pulse as we will show in this study. Below, we present three pump-probe experimental investigations by manipulating the (1) incident direction, (2) spectral property, and (3) intensity of the probe pulses. These investigations provide rich information on the physics behind the coherent emissions generated in air by femtosecond laser excitation.

 

Fig. 1 (a) Typical schematic of the interaction of nitrogen molecules with two color pulses. Inset: the strong and coherent emission with ~0.3 nm bandwidth (FWHM) appearing on the spectrum of the probe pulse. (b) Energy-level diagram of ionized and neutral nitrogen molecules in which the vibrational transition between B²Σ_u⁺ (v = 0) and X²Σ_g⁺ (v = 0) states corresponding to 391.4 nm wavelength [10] is indicated.

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2. Experiments and results

2.1. Seed amplification in the counter-propagating case of the probe and the pump pulses

In our previous studies on the remote lasing excited by mid-infrared femtosecond laser pulses [10,15–17], the third or fifth harmonic pulse is spontaneously generated in the process of filamentation and copropagates with the pump pulse. In the present experiment, we attempt to examine the amplification of the seed with the probe pulse propagating in the opposite direction of the pump pulse. This would provide information on whether the amplification of the seed pulses requires a phase matching condition. The experimental setup is schematically illustrated in Fig. 2(a) . A commercial Ti: sapphire laser system (Legend Elite-Duo, Coherent, Inc.) produces intense 800 nm pulses with a pulse duration of ~40 fs (FWHM) and a single pulse energy up to ~6 mJ at a repetition rate of 1 kHz. The laser beam is divided into two using a beam splitter (1:1). One beam serves as the 800 nm pump pulse. The other is frequency doubled by a beta-barium borate (BBO) crystal (200 μm thick, type I), producing the probe pulse whose spectrum covered the wavelength 391.4 nm. The 800 nm pump pulse with a pulse energy of ~371 μJ is focused by a fused-silica lens (F1) with the focal length of 10 cm, forming a laser spark with ~1 mm length in atmospheric air. The probe laser pulse (~1 μJ) is focused from the opposite direction into the laser spark by another fused-silica lens (F2) with a focal length of 10 cm. After being collimated, the probe pulse is separated from the pump pulse by a dichroic mirror, and is recorded by a spectrometer (Andor Shamrock 303i). The polarization of the probe in this case was orthogonal to that of the pump laser field and the temporal overlapping of the probe pulse and pump pulse is adjusted to achieve a maximum amplified signal at 391.4 nm.

 

Fig. 2 (a) Schematic of the experimental setup for the counterpropagating pump and probe pulses configuration. (b) Measured spectra of the probe pulses before and after the laser spark. (c) Measured Intensities of the coherent emission line at ~391 nm at different delay times.

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Figure 2(b) shows the spectrum of the probe pulse recorded. In the absence of the pump pulses, the spectral profile of the probe pulses appears smooth and does not exhibit any emission lines at wavelength ~391 nm. In contrast, when the pump pulse is sent into the molecular system ahead of the probe pulse, a strong, narrow-bandwidth emission located at 391.4 nm appears on the top of the probe spectrum, indicating amplification of the probe pulse at 391.4 nm even when it counterpropagates with respect to the pump pulse. The amplified signal intensity is measured with varied delay times as depicted in Fig. 2(c). The zero delay is defined as the occurrence of the amplification. Positive delay time suggests the pump pulse interact with medium at an earlier time than the probe pulse.

2.2. Dependence of the coherent emission on the spectral property of the probe pulse

In our previous studies [10,15–17], the spectrum of the harmonic seed has a broad bandwidth, leading to missing information on how the coherent emission depends on the spectrum of the seed pulse. In the second experiment of this work, we attempt to study the influence of the spectral property of the probe pulse on the coherent emission by spectrally modulating the probe pulse. The experimental setup is depicted in Fig. 3(a) . The pump pulse is focused by a focusing lens with 20 cm focal length, forming a laser spark of ~4 mm length. The probe pulse is generated by a BBO crystal (thickness: 2 mm) and then arranged to co-propagate with the pump pulse and passed through the focal lens. By varying the incident angle on the BBO crystal, the spectrum of the seed pulses can be changed. In the present experiment, the probe pulse (pulse energy: 0.1 μJ) co-propagates with the pump pulse (pulse energy: 1.1 mJ) as shown in Fig. 3(a). The spectrum of the probe pulse after passing through the laser spark is recorded by a spectrometer after being reflected by a dichroic mirror. From the spectra shown by the dash lines in Figs. 3(b) and 3(c), spectral modulation in the second harmonic spectrum can be clearly observed. Such periodic modulation could be the result of interference between time-separated harmonic pulses due to pulse splitting in the fundamental pulse caused by significant self-focusing and self-phase modulation in thick BBO crystal, as explained by ref [19]. By slightly rotating the BBO crystal, the spectral modulation can be changed. Implementation of such modulation enables us to probe the amplification of the seed with a narrow bandwidth spectral gate, and thus the dependence of the coherent emission on the frequency of the probe pulse.

 

Fig. 3 (a) Schematic of the experimental setup. (b) Measured spectra of the probe pulses with one of the dip positions of the modulated harmonic spectrum located at 391.4 nm. (c) Measured spectra of the probe pulses with one of the peak positions of the modulated harmonic spectrum located at 391.4 nm.

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It can be seen in Fig. 3(b) that when the 391.4 nm wavelength, which corresponds to the transition between the B²Σ_u⁺ (v = 0) and X²Σ_g⁺ (v = 0) states coincides with the dip of the modulated harmonic spectrum, there is no amplification of the probe pulse. Since the probe pulse was severely diffracted by the plasma formed by the pump pulse, the spectral amplitude of the probe pulse in the presence of the pump pulse (solid black curve) is weaker than its original spectrum (dash red curve). Apart from amplitude attenuation due to diffraction, the spectrum of the probe pulse after the plasma appears to be slightly blue shifted by comparing the positions of peaks in the black and red curves. We currently speculate that the blue shift in the spectrum is attributed to the passage of the probe pulses through the weak plasma region, which has been explained in Ref [20]. In contrast of the failure of amplification in Fig. 3(b), in Fig. 3(c), when the 391.4 nm wavelength coincides with one of the peaks of the spectrum of the seed pulses, a strong signal with a bandwidth (FWHM) of ~0.3 nm centered at the wavelength of 391.4 nm is superposed on top of the spectrum of the probe pulse. This observation indicates that the amplified emission is strongly dependent on the spectral intensity at 391.4 nm of the probe pulse. To achieve distinct amplification, the spectrum of the probe pulse must cover the frequency that corresponds to the transition of molecular ions.

2.3. Influence of the probe power on the coherent emission

Lastly, we study the dependence of the coherent 391.4 nm signal intensity on the input power of the probe pulse. In this experiment, the probe pulse co-propagates with the pump pulse. The experimental setup is described in detail in section 2.2. In this case, the input pump pulse energy is 1.7 mJ. The focal length of the focusing lens is 30cm. The laser spark generated by the pump pulse is of ~6 mm length. The thickness of the BBO crystal is 400 µm. The polarization of the probe is perpendicular to that of the pump. The power of the probe pulse is varied by rotating a half-wave plate placed before a Glan-Taylor prism. The amplified signal at 391.4 nm intensity is measured by a spectrometer.

The spectrum of the probe pulse after passing through the laser spark is presented by the red line in Fig. 4(a) , while its original spectrum measured before the laser spark is shown by the black line. The amplified signal intensity, S_s, is defined as S_s = S_total-S_harmonic, where S_total represents the spectral peak intensity at 391.4 nm and S_harmonic is the signal intensity of the smooth harmonic spectrum at 391.4 nm, as illustrated in Fig. 4(a). In Fig. 4(b), the signal intensity, S_s, is plotted as a function of the input power of the probe pulse, which demonstrates a perfect linear dependence, as fitted by the solid line.

 

Fig. 4 (a) Measured spectra of the probe pulse in the present and absent of the pump pulse. (b) Measured dependence of the 391.4 nm output signal on the input probe power.

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3. Discussion

Since the first observation of the generation of ultrafast forward coherent emissions in air by the infrared femtosecond filament [10], a consensus about its underlying mechanism has not been reached yet. Currently, it is concluded that three schemes including four-wave parametric amplification [21], stimulated resonant Raman amplification [22], and seed amplification based on the population inversion [10,15–17] are the most likely candidates that could lead to this interesting phenomenon.

However, four-wave parametric amplification and stimulated resonance Raman scattering are all third-order nonlinear optical processes which greatly differ from lasing actions based on population inversion. The first difference between four-wave parametric amplification and lasing action based on population inversion is that the former is a parametric process that strongly depends on phase matching condition and thus on the propagation direction of the probe pulses, while the latter is able to occur for both the co-propagation and counter-propagation cases of the pump and probe pulses even when the pump and probe are temporally separated (no phase matching). As shown in Figs. 2(b) and 2(c), the generation of the coherent emission at 391.4 nm can be achieved in the counter-propagation case of the pump and probe pulses, as well as in the case of temporally separated pump and probe pulses. This excludes the possibility of four-wave mixing as the mechanism to generate the coherent emission at 391.4 nm. Furthermore, the above observation can be regarded as the first evidence for the population inversion mechanism with picosecond population inversion lifetime and stimulated emission cross section $\text{σ}=6.76\times {10}^{-8}{\text{m}}^{\text{2}}{\text{s}}^{-\text{1}}$ [23], leading to the seed amplification at 391.4 nm.

The second difference between a lasing action based on population inversion and nonlinear frequency conversion processes (e. g., four-wave mixing and stimulated Raman scattering) lies in the spectral property of the probe pulse. For population inversion induced amplification, the spectrum of the probe pulse has to cover the transition between the two levels to achieve a high gain. For the two nonlinear frequency conversion processes mentioned above, the spectrum of the probe pulse does not need to overlap exactly with the transition frequency. Therefore, the observation that the amplified emission at 391.4 nm appears in Fig. 3(c), but disappears in Fig. 3(b) indicates that the spectrum of the probe pulse covering the vibrational transition frequency between B²Σ_u⁺ (v = 0) and X²Σ_g⁺ (v = 0) states is a prerequisite for generation of the coherent emission at 391.4 nm. This can be regarded as the second evidence for the population inversion mechanism.

Last but not least, stimulated Raman gain is exponentially proportional to the seed signal intensity, I_seed. On the other hand, a laser signal intensity resulted from the seed amplification in a population inversion system is linearly proportional to I_seed [22]. Therefore, the linear dependence observed in Fig. 4(b) clearly shows that the mechanism due to stimulated resonance Raman scattering can be safely excluded. As a consequence, the observation can serve as the third evidence for the mechanism of the seed amplification based on the population inversion.

4. Summary

To summarize, we have systematically investigated the physical mechanism of generation of coherent emissions induced by femtosecond laser excitation in air using a series of pump–probe experiments. By manipulating the propagating direction, spectral property, and intensity of the probe pulse, it is found that the coherent N₂⁺ emission at 391.4 nm results neither from the parametric process of four-wave mixing nor from stimulated Raman scattering. The results further suggest that the population inversion in nitrogen molecular ions established by femtosecond laser filamentation is the most likely mechanism.