1. Introduction

Tunnel ionization is one of the most fundamental processes for molecules in intense laser fields. Depending on the molecular orbital which the electron is ionized from, the generated molecular ion can be in the ground or excited electronic state. According to molecular Ammosov-Delone-Krainov (MO-ADK) model [1], the ionization rate decreases exponentially with the electron binding energy. It is therefore assumed that the electron in the highest occupied molecular orbital (HOMO) is firstly removed by the intense laser field and the molecular ion is dominated by the ground electronic state. However, optical gain is observed around 391 nm and 428 nm from the ionized nitrogen molecules generated by 800-nm femtosecond laser field [2,3], known as air lasing, which correspond to the transitions between N₂⁺ (B²Σ_u⁺, ν’ = 0) and N₂⁺ (X²Σ_g⁺, ν = 0, 1) states, respectively. These observations imply that the population inversion can be established between the ground and the excited states of the tunnel ionized nitrogen molecules, which cannot be explained by conventional strong field ionization theories. Because of the potential application of air lasing in remote atmospheric sensing, the phenomenon has attracted great attention, and the interaction mechanism between nitrogen molecules and intense femtosecond laser has been extensively investigated [4–18]. In two pioneering works [6,7], it was proposed that the population inversion between N₂⁺ (B²Σ_u⁺, ν’ = 0) and N₂⁺ (X²Σ_g⁺, ν = 0) states is achieved through the post-ionization population redistribution mechanism. In the case of 800-nm laser field, N₂⁺ is dominantly populated in the ground vibrational and electronic state at the moment of tunneling ionization. However, the laser field simultaneously causes strong coupling and population transfer among the three lowest energy electronic states of N₂⁺ ions, i.e., ${\textrm{X}^\textrm{2}}\mathrm{\Sigma }_\textrm{g}^\textrm{ + }$, ${\textrm{A}^\textrm{2}}{\mathrm{\Pi }_\textrm{u}}$ and ${\textrm{B}^\textrm{2}}\mathrm{\Sigma }_\textrm{u}^\textrm{ + }$ (abbreviated as X, A and B). The efficient population transfer between N₂⁺ (X) and N₂⁺ (A) occurs through one-photon near-resonant process. Most ions in the N₂⁺ (X) state are evacuated with mediation of the intermediate N₂⁺ (A) state, leading to the establishment of population inversion between N₂⁺ (B, ν’ = 0) and N₂⁺ (X, ν = 0). More and more evidences support the post-ionization population redistribution mechanism and the involvement of N₂⁺ (A) state is demonstrated to be indispensable to the optical gain [19–22].

The optical gain depends on the degree of population inversion, i.e., the population difference between the upper level (i.e., B state) and the lower level (i.e., X state). The population of the related states is determined by both tunnel ionization and post-ionization population redistribution processes. The dipole moment between N₂⁺ (B) and N₂⁺ (X) states is parallel to the molecular axis, while the dipole moment between N₂⁺ (A) and N₂⁺ (X) states is vertical to the molecular axis. As a result, both the ionization and the post-ionization population redistribution strongly depends on the polarization of the pump laser. It has been shown that the optical gain around 391 nm is greatly suppressed when the laser polarization is changed from linear polarization to circular polarization [23–25]. In order to further enhance the optical gain, various polarization-modulated methods have been designed to optimize both ionization and post ionization population redistribution processes [26–29]. Based on the polarization gating (PG) technique or ellipticity gating (EG) technique, a polarization-modulated laser field is generated, in which the laser polarization changes in time. It is demonstrated that the optical gain around 391 nm achieved by PG laser field is larger than that achieved by linearly polarized pulse by two order of magnitude [26]. Further, the optical gain generated with the EG technique is ∼2 times stronger than that with the PG technique [27]. Combined with theoretical simulations, it is concluded that these polarization-modulated laser fields can greatly deplete the population in the lower state by transfer the population from X to A state. Very recently, Li et al. demonstrated that the population of the N₂⁺ (X, ν = 0) state can be depleted almost completely by using a synthesized laser field composed of a PG laser field and an infrared 1.6-µm field [29]. The excessive population inversion between N₂⁺ (B, ν’ = 0) and N₂⁺ (X, ν = 0) states is thus achieved and leads to giant enhancement of lasing around 391 nm from nitrogen molecular ion.

Theoretical calculation indicates that ionization itself mainly generates the ground vibrational and electronic state of N₂⁺, i.e., X (ν = 0) state. The post-ionization population redistribution greatly reduces the population on the state of X (ν = 0). Instead, the population is improved for high vibrational states of N₂⁺(X) [7,22]. Previous works about nitrogen ion lasing mainly concentrate on the two typical laser lines around 391 nm and 428 nm, which are related with low vibrational states of N₂⁺ (B) and N₂⁺ (X). The study is scarce for the laser emission lines related with high vibrational states of N₂⁺ (B) and N₂⁺ (X) [30–32]. In this work, we explored the optical gain related with high vibrational states using typical pump-seed scheme. Two emission lines around 353.3 nm and 353.8 nm are observed to be greatly amplified no matter the 800-nm pump laser is circularly or linearly polarized. The two spectral lines correspond to the transition of N₂⁺ (B, ν’ = 5 → X, ν = 4) and N₂⁺ (B, ν’ = 4 → X, ν = 3), respectively. The measurements imply that the population inversion is also built between these high vibrational states of N₂⁺ (B) and N₂⁺ (X). But the degree of population inversion exhibits different polarization dependences of the pump laser in comparison with the spectral lines related with low vibrational states. The new findings can be attributed to the population variation of high vibrational states caused by the pump laser with different polarizations.

2. Experimental setup

Figure 1 shows the schematic diagram of experimental setup. The experiments were carried out by using a commercial Ti:sapphire laser amplifier, which delivers linearly polarized femtosecond laser pulses with a central wavelength of 800 nm, a repetition rate of 1 kHz, a pulse duration of 35 fs, and a maximum pulse energy of 6 mJ. The 800 nm laser beam was split into two beams by a 50:50 beam splitters. One beam with a single pulse energy of about 3 mJ was severed as the pump laser to ionize neutral nitrogen molecules. A quarter wave plate (QWP) was placed in the pump laser beam routine to change its polarization. The other beam with a single pulse energy of about 3 mJ was steered into an optical parametric amplifier to generate a tunable mid-infrared laser source. Then the tunable mid-infrared laser was directed into two BBO crystals to generate its fourth harmonics and utilized as an external seed. The pump pulses and the seed pulses were recombined collinearly by a dichroic mirror (DM) with high reflectivity around 320–470 nm and high transmission around 680–900 nm. Then the two pulses were focused by an f = 30 cm convex lens into a gas chamber filled with pure nitrogen gas. After filtering the residual 800 nm pump pulses by another pair of dichroic mirrors, the forward emission was focused into a fiber spectrometer and recorded by the computer. The resolution of the spectrometer is about 0.1 nm. The intensity of the forward emission was measured as a function of the time delay between the 800 nm pump pulses and the external 355 nm seed pulses by scanning the program controlled translation stage (Newport, XMS100).

 

Fig. 1. Schematic diagram of experimental setup.

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3. Results and discussion

Figure 2 shows typical forward emission spectra from ionized nitrogen molecules generated by intense 800-nm pump pulses and a delayed 355-nm seed pulses. The gas pressure is 50 mbar, and the energy of 800-nm pump pulses is 1.7 mJ. It can be seen that the forward emission around 352-360 nm can be neglected when only the pump laser was irradiated. When the seed laser with wavelength covering 352-360 nm is injected, two emission lines around 353.3 nm and 353.8 nm are greatly amplified. The wavelength of the seed laser covers all transitions of N₂⁺ (B, ν’ → X, ν) with Δν = ’ - ν = 1. However, only the transition of N₂⁺ (B, ν’ = 5 → X, ν = 4) and N₂⁺ (B, ν’ = 4 → X, ν = 3) are greatly amplified, whose wavelengths lie around 353.3 nm and 353.8 nm, respectively. The magnitude of amplification also depends on whether the pump laser is circularly polarized or linearly polarized. Both the emission lines in the case of circularly polarized pump laser are much stronger than those in the case of linearly polarized pump laser. In addition, the emission intensity around 353.8 nm is higher than that around 353.3 nm in linearly polarized laser pulse. However, the situation reverses when the laser polarization is changed. The emission intensity around 353.3 nm is higher than that around 353.8 nm in circularly polarized laser pulse.

 

Fig. 2. Typical forward emission spectra from ionized nitrogen molecules generated by 800-nm pump laser and 355-nm seed laser with an optimized time delay. The laser polarization is (a) linear and (b) circular for the 800-nm pump laser.

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Furthermore, we measured the polarization dependence of amplified emission by placing a Glan-Taylor prism in front of the spectrometer. As shown in Fig. 3, no matter whether the 800 nm pump pulses is linear or circular, the amplified emission has a nearly perfect linear polarization as that of the seed pulses. The coherent emission lines around 353.3 nm and 353.8 nm have been studied by circularly polarized 800 nm laser pulses [31,32]. With the injection of an external seed pulses, strong optical amplification has been exhibited. The optical gain depends on the wavelength of the seed pulse as well as the gas pressure. Very recently, Zhou et al. experimentally measured the temporal profile of the coherent emission around 353.8 nm generated by circularly polarized 800-nm pump laser and a delayed seed laser [32]. The results show that there is a delay of few picoseconds between the coherent emission and the seed laser. Meantime, the coherent emission lasts several tens of picoseconds. The temporal profile is very similar to the coherent emission around 391 nm generated by linearly polarized 800-nm pump laser and a delayed seed laser [24]. Based on these characteristics, the authors attributed the 353.8 nm coherent emission to the seed-triggered superradiance [32]. The present measurement demonstrates that the polarization of the amplified emission is determined by the seed laser polarization, no matter whether the 800-nm laser is circularly or linearly polarized. This observation is consistent with the prediction of superradiance.

 

Fig. 3. The polarization of spectral line around (a) 353.8 nm and (b) 353.3 nm generated by circularly and linearly polarized pump pulses.

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To investigate the ultrafast dynamics of these coherent emissions, we measured the optical gain as a function of the time delay between the pump pulses and the seed pulses at different gas pressures. Figure 4 shows the optical gain of 353.8 nm in linearly and circularly polarized pump laser at the gas pressures of 15, 50, and 110 mbar. It can be seen that the gain exhibits similar time dependence in linearly and circularly polarized pump laser. It increases rapidly on a time scale of hundreds of femtoseconds, and then shows a slow decay with a time scale of several picoseconds. In addition, there exist some periodic modulations. The interval is ∼4 ps, corresponding to a half of fundamental rotational period of the rotational wave packet of N₂⁺ (B²Σ_u⁺) [3]. It should be emphasized that the formation time of the optical gain has no relation to the gas pressure, but is close to the duration of the seed laser. The decay process depends on the gas pressure, which becomes faster with increasing gas pressure. In previous reports [30,31], the optical amplification around 353.3 nm and 353.8 nm was observed only when the 800 nm pump laser is circularly polarized. The authors proposed that the optical gain is related with electron impact excitation. The high vibrational energy levels of N₂⁺ (B) are generated by hot electron impact excitation, i.e., N₂⁺ (X, ν = 0) + e → N₂⁺ (B, ν’ = 4, 5). Because of the threshold for electron impact excitation, higher kinetic energy is required for the electron, which can be achieved in circularly polarized laser field. In the present experiment, the optical gain is exhibited in both circularly and linearly polarized 800-nm pump laser. According to our best knowledge, this is the first time that the optical gain is observed for the emission lines around 353.3 nm and 353.8 nm generated by linearly polarized 800 nm laser pulses. In addition, the formation time of the optical gain is independent of the gas pressure. These observations are contradicted with the predictions of electron impact excitation mechanism. The time dependence of the optical gain shown in Fig. 4 is very similar to that of coherent emission around 391 nm generated by 800 nm linearly polarized pump laser and a delayed seed laser [3]. Even though the mechanism for optical gain around 391 nm is still under hot debate, experimental evidences show that population inversion is achieved between the two levels related to the transition. The degree of population inversion is determined by the population of the upper level as well as the lower level, which depends on both the ionization and the post ionization population redistribution processes. When only the ionization itself is taken into account, the ionized nitrogen molecules are dominated by the ground vibrational and electronic state, i.e., N₂⁺ (X, ν = 0). After including the coupling, most ions in the N₂⁺ (X, ν = 0) state are transferred to the N₂⁺ (A) state. The population redistribution leads to the establishment of population inversion between N₂⁺ (B, ν’ = 0) and N₂⁺ (X, v = 0). It should be emphasized that the optical gain also strongly depends on the laser intensity. Chen et al. [20] reported that the amplification occurs when the 800-nm pump laser intensity is 1.2 × 10¹⁴ W/cm². Instead, the absorption occurs when the 800-nm pump laser intensity is 3 × 10¹³ W/cm². According to these theoretical calculations, we know that it is very difficult to accurately determine the population due to the simultaneous occurrence of the ionization and population redistribution. Here it should be pointed out that the population of high vibrational states of N₂⁺ (X) is also increased due to the postionization population redistribution [7,22]. In the present measurement, we found that the optical amplification in circularly polarized pump laser is much stronger than that in linearly polarized pump laser for both the transitions of N₂⁺ (B, ν’ = 5 → X, ν = 4) and N₂⁺ (B, ν’ = 4 → X, ν = 3). In addition, the relative strength between them reverses when the laser polarization is changed from circular to linear. In the case of circularly polarized pump laser, the intensity from the transition of N₂⁺ (B, ν’ = 5 → X, ν = 4) is higher than that of N₂⁺ (B, ν’ = 4 → X, ν = 3). In the case of linearly polarized pump laser, the intensity of N₂⁺ (B, ν’ = 4 → X, ν = 3) is higher than that of N₂⁺ (B, ν’ = 5 → X, ν = 4). We attributed the change of the optical amplification to the population variation caused by the pump laser with different polarizations.

 

Fig. 4. The optical gain evolution of the 353.8 nm spectral line in (a) linearly and (b) circularly pump pulses with the time delay between the pump and seed pulses at the gas pressure of 15, 50 and 110 mbar, respectively.

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Irradiated by the 800-nm pump laser, the nitrogen molecules are ionized firstly and dominantly populated in N₂⁺ (X, ν = 0). When the post-ionization population redistribution is included, the population of N₂⁺ (B, ν’ = 5 and 4) is increased via three-photon near-resonant excitation from N₂⁺ (X, ν = 0) with the assistance of laser-induced dynamic Stark shift. At the same time, the population of N₂⁺ (X, ν = 4 and 3) is also increased due to the coupling between N₂⁺ (A) and N₂⁺ (X). The wavelength for the transitions of N₂⁺ (B, ν’ = 4 ← X, ν = 0) and N₂⁺ (B, ν’ = 5 ← X, ν = 0) lies around 288.3 nm and 271.7 nm, respectively. The energy is close to the three photon energy of 800-nm pump laser, which is equivalent to 266.7 nm. In general, the diagonal dynamic Stark level shifts could be time and position dependent. In the lowest order approximation, we take them to be position independent. We take an approximate polarizability α = 2 ų, the diagonal energy shift is △E = αE²/4 for the N₂⁺ (B) state with E being the laser electric field. When the laser intensity is 1.0 × 10¹⁴ W/cm², the maximum Stark shift is 0.27 eV in linearly polarized laser field and the excitation of N₂⁺ (B, ν’ = 4 ← X, ν = 0) is closer to the resonance condition and becomes more favorable. Instead, the maximum Stark shift is 0.18 eV in circularly polarized laser field and the excitation of N₂⁺ (B, ν’ = 5 ← X, ν = 0) is more favorable. As a result, in the case of linearly polarized 800-nm pump laser, the population inversion between B (v’ = 4) and X (v = 3) is higher than that between B (ν’ = 5) and X (ν = 4). In the case of circularly polarized 800-nm pump laser, the population inversion between B (v’ = 5) and X (v = 4) is higher than that between B (ν’ = 4) and X (ν = 3). The change can explain our observation that the amplified emission around 353.8 nm is stronger than that around 353.3 nm in the former case and the amplified emission around 353.3 nm is stronger than that around 353.8 nm in the latter case. In addition, when the laser polarization becomes from linear to circular, the population transfer becomes less efficient between N₂⁺ (X) and N₂⁺ (A). Accordingly, the population transfer becomes smaller from N₂⁺ (A) to high vibrational states of N₂⁺ (X). Therefore, the population inversion between B (ν’ = 5,4) and X (ν = 4,3) in circularly polarized pump laser are higher than those in linearly polarized pump laser. The stronger population inversion can explain the experimental observation shown in Fig. 2 that both the emission lines around 353.3 nm and 353.8 nm in circularly polarized pump laser are much stronger than those in linearly polarized pump laser.

Finally, we investigated the optical amplification of these two emission lines as a function of gas pressure. As shown in Fig. 5, the two emissions have similar dependence on the gas pressure, whether the laser polarization is circular or linear. With the increase of the gas pressure, the emission intensity firstly increases and then decreases. The pressure dependence of the emission intensity can be attributed to the competition between the generation and the decay of the population inversion between related states. When the pressure is low, both the population of N₂⁺ (B, ν’ = 5 and 4) and N₂⁺ (X, ν = 4 and 3) are increased with increasing the gas pressure. As a result, the population inversions between B (ν’ = 5) and X (ν = 4) as well as between B (ν’ = 4) and X (ν = 3) are increased with increasing gas pressure. When the gas pressure is further increased, the collision between N₂⁺ (B) and free electron plays an important role and leads to the fast decay of N₂⁺ (B), which has been confirmed by monitoring the population evolution of N₂⁺ (B) as a function of free electron density [33,34]. As a result, the population inversions between B (ν’ = 5) and X (ν = 4) as well as between B (ν’ = 4) and X (ν = 3) are decreased with further increasing gas pressure. Therefore there was an optimal pressure at which the maximum population inversion is achieved. The pressure-dependent variation of population inversion leads to the optical gain shown in Fig. 5. It should be mentioned that the optical gain is also observed for the spectral line around 354.9 nm, which corresponds to the transition of N₂⁺ (B, ν’ = 3 → X, ν = 2). In comparison with the transition of N₂⁺ (B, ν’ = 5 → X, ν = 4) and N₂⁺ (B, ν’ = 4 → X, ν = 3), the optical gain for the transition of N₂⁺ (B, ν’ = 3 → X, ν = 2) only exists in a relatively low pressure and a much narrowed pressure region whether the laser polarization is circular or linear. The different pressure dependences imply that gas pressure is an important parameter to optimize the relative intensity of spectral line.

 

Fig. 5. The intensities of amplified emissions in linearly and circularly pump pulses as a function of gas pressure.

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4. Conclusions

In summary, we experimentally investigated the dynamics of nitrogen molecules irradiated by an 800-nm pump laser and a delayed 355-nm seed laser. The spectral lines around 353.3 nm and 353.8 nm are observed to be greatly amplified whether the pump laser is circularly or linearly polarized. But the optical amplification in circularly polarized pump laser is much stronger than that in linearly polarized pump laser. In addition, the relative intensity reverses for these two spectral lines when the laser polarization is changed from circular to linear. The intensity around 353.3 nm is higher than that around 353.8 nm in circularly polarized pump laser field. Instead, the intensity around 353.8 nm is higher than that around 353.3 nm in linearly polarized pump laser field. The observed change of the optical amplification can be explained by the population variation caused by the pump laser with different polarizations. When nitrogen molecules are irradiated by intense 800-nm pump laser, the generated nitrogen molecular ions are populated in various vibrational and electronic states through the processes of tunneling ionization and postionization population redistribution. The maximum electric field decreases when the laser polarization is changed from linear to circular. Accordingly, the population transfer between N₂⁺ (X) state and N₂⁺ (A) state becomes less efficient and the dynamic Stark shift becomes smaller. As a result, the population inversion in circularly polarized pump laser is higher than those in linearly polarized pump laser. Meantime, due to the dynamic Stark shift, the three-photon near-resonant excitation is more favorable for the transition of N₂⁺ (B, ν’ = 4 ← X, ν = 0) in linearly polarized laser field and N₂⁺ (B, ν’ = 5 ← X, ν = 0) in circularly polarized laser field. The polarization-dependent population variation leads to the distinctive change of the optical amplification for the spectral lines related with high vibrational states of ionized nitrogen molecules.