1 Introduction

Photoionization is one of the fundamental processes in the interaction of light and matter. Particularly, in an intense laser field, photoionization can be realized by either a perturbative multiphoton or a non-perturbative tunnel process depending on whether the Keldysh parameter is above or below 1, respectively [1]. The strong-field ionization has attracted significant attention over the past three decades thanks to the rapid development of ultrafast laser technology. Today, wavelength-tunable high-peak-power ultrashort-pulsed laser sources have been widely used in the investigations of strong-field phenomena including high-order harmonic generation [2,3,4], above threshold ionization [5, 6] and non-sequential double ionization [7], which not only have enabled the elaborated experimental examination of physics behind strong-field ionization [5,6,7,8], but also provide opportunities for intriguing new discoveries as well as innovative applications [4, 9, 10].

Here we discuss the physics of an unexpected strong-field phenomenon accidentally observed by Yao et al. [11]. The observation shows that the non-perturbative interaction of mid-infrared intense laser fields with nitrogen molecules enables to generate laser-like narrow-bandwidth coherent emissions instantly after the photoionization. The wavelength of each narrow-bandwidth emission is in accordance with one of transitions between the vibrational energy levels of excited \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma_{{\text{u}}}^{+}} \right)\) state and that of ground \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma_{{\text{g}}}^{+}} \right)\) state. Soon afterwards, pump–probe measurements were performed to investigate the dynamics of gain by setting the probe wavelength around 391 nm, which corresponds to the transition between \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma_{{\text{u}}}^{+},~~v^{\prime}=0} \right)\) state and \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma_{{\text{g}}}^{+},~v=0} \right)\) state [12]. The observed gain in the probe pulses clearly indicates that population inversion has been generated in \({\text{N}}_{2}^{+}\) ions, which is difficult to understand in the framework of multiphoton or tunnel ionization [1]. At present, several mechanisms have been proposed for understanding the origin of the population inversion, including field-induced multiple recollisions [13, 14], depletion of ground state via one-photon resonant excitation [15,16,17], and transient inversion induced by the molecular alignment [18, 19].

In the following, we review the recent progress we made toward better understandings of the physical mechanisms behind the lasing actions initiated by the strong-field ionization of nitrogen molecules. Since this phenomenon can be directly observed in atmospheric environment, we inherit the term “air lasing” which has previously been used for describing various types of lasing actions remotely generated in air. Actually, the attempt of realizing laser emissions remotely in atmospheric air dates back to 1988, when molecular nitrogen laser was generated in air using high-power microwave pulses as the pump source [20]. In 2003, Luo et al. observed an exponential growth of backscattering fluorescence with the length of plasma channel generated by femtosecond laser filamentation, and concluded that amplified spontaneous emission (ASE) was responsible for their observation [21]. In 2011, Dogariu et al. successfully generated high-gain ASE from atomic oxygen in air with a picosecond ultraviolet laser, because the intense ultraviolet laser can drive two-photon dissociation of molecular oxygen followed by two-photon resonant excitation of the oxygen atoms [22]. All the observations are related to generation of directional, narrow-linewidth, coherent emissions in air without use of any laser cavity, which can potentially promote the sensitivity in laser-based remote atmospheric sensing [23]. So far, air lasing actions have been generated using air molecules [20, 21, 24,25,26,27,28,29,30,31,32], atoms [22, 23, 33, 34] or ions [11,12,13,14,15,16,17,18, 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58] as the gain media. In addition, lasing actions in the fluorescing fragments of polyatomic molecules have been observed inside an air filament (see, for example, a recent review [59] and references therein). However, because of the significant differences in the underlying physical mechanisms and the schemes of experimental realization of different types of air lasers, we will exclusively focus on the forward lasing actions initiated in strong-field-ionized molecular nitrogen. This is because that this type of air lasing not only provides a bright coherent ultraviolet source for remote sensing application but also challenges our understanding of non-perturbative nonlinear interaction of intense laser fields with molecules. Indeed, various physical effects such as the attosecond-scale tunnel ionization of the molecules, the femtosecond-scale photoexcitation in the molecular ions, and the picosecond-scale amplification of the seed pulses in the aligned molecular ions, can all play significant roles in generation of the lasing actions of \({\text{N}}_{2}^{+}\) ions. Identifying these roles requires systematic pump–probe examination of the dependence of the lasing actions on a large group of parameters (e.g., polarization, wavelength, power, etc.) of both the pump and probe pulses at various combinations.

The remaining part of the review is organized as follows: first, we discuss the physical mechanisms behind the \({\text{N}}_{2}^{+}\) laser-like emission at 391 nm generated in mid-infrared (1.2–2 µm) and 800-nm laser fields, then we reveal that the origin of \({\text{N}}_{2}^{+}~\) laser-like emission at 428 nm is different from that of the \({\text{N}}_{2}^{+}~\) laser-like emission at 391 nm. The review is concluded by a summary on the major results followed with a future perspective.

2 \({\mathbf{N}}_{2}^{+}~\) laser-like emissions generated in mid-infrared laser fields

2.1 Experimental observations

It was reported by Yao et al. that the free-space \({\text{N}}_{2}^{+}\) laser-like emission was observed when 0.5 mJ, ~ 1900 nm mid-infrared femtosecond laser pulses were tightly focused into air [11]. As shown in Fig. 1a, a narrow-bandwidth laser-like emission at ~ 391 nm wavelength appears on top of the fifth harmonic spectrum of the pump laser. For comparison, we also present the spectra measured in the backward and side directions. As shown in Fig. 1b, these emissions show typical characteristics of femtosecond laser-induced fluorescence in air. These fluorescence lines can be assigned to two band systems, i.e., the first negative band system of \({\text{N}}_{2}^{+}\) ions (i.e., \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+} \to {{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) transition) and the second positive band system of neutral \({{\text{N}}_2}\) molecules (i.e., \({{\text{C}}^3}\Pi_ {{\text{u}}} \to {{\text{B}}^3}\Pi_{{\text{g}}}~\) transition) [60]. The fluorescence emissions are much weaker than the forward laser-like emission at ~ 391 nm wavelength.

Fig. 1
figure 1

a A typical spectrum of laser-like emission at 391 nm generated by focusing ~ 1900 nm, 0.5 mJ femtosecond laser pulses into air (logarithmic scale). b Fluorescence spectra recorded in the backward (black solid line), and side (red dashed line) directions in the same conditions. For clarity, the fluorescence spectrum measured in the side direction was vertically shifted. The numbers in parentheses denote the vibrational quantum numbers of the upper and lower energy states of corresponding transitions. c The intensity of ~ 391-nm forward laser as functions of the angle of polarizer placed before the spectrometer. d The intensity of the ~ 391-nm lasing signal measured at different lengths of plasma channel [11]

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The laser-like emission shows clear signatures of a coherent light source, i.e., the characteristics of stimulated emission or nonlinear frequency conversion. First, it is perfectly linearly polarized and its polarization direction is parallel to that of the pump laser and that of the fifth harmonic as well, as shown in Fig. 1c. Second, the laser-like emission shows an exponential growth along the plasma channel, as shown in Fig. 1d. The gain curve is obtained by truncating the plasma channel using a pair of uncoated fused silica plates and measuring the spectral intensity of 391-nm laser at each truncation location. The measurement result can be well fitted by the function \(y=a{e^{{\text{gx}}}}\) with a gain coefficient g ≈ 5 cm−1. Third, the laser-like emission has a good directionality. Its divergence angle is comparable to that of the fifth harmonic generated in the pump laser field. These properties can hardly be attributed to any spontaneous emissions.

In addition, the \({\text{N}}_{2}^{+}\) laser-like emission can be generated at various wavelengths by tuning the central wavelength of the pump laser. As shown in Fig. 2a, the \({\text{N}}_{2}^{+}\) laser-like emissions can operate at multiple wavelengths which correspond to the transitions between the different vibrational levels of \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) states of \({\text{N}}_{2}^{+}\) ions. Figure 2b indicates the energy-level diagram of transitions from \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) state to \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state and the corresponding lasing wavelengths. The relation between the wavelengths of the \({\text{N}}_{2}^{+}\) laser-like emission and the pump laser reveals that in the atmospheric environment the \({\text{N}}_{2}^{+}\) laser-like emission is generated with a seed amplification scheme which requires the spectral overlap between the harmonic of the pump laser and the laser-like emission.

Fig. 2
figure 2

a \({\text{N}}_{2}^{+}\) lasers excited by the pump laser with different wavelengths [11]. From left to right, the central wavelength of the pump laser is 1682, 1760, 1920, 2050, 1415 nm. The numbers in parentheses denote the vibrational quantum numbers of the upper and lower energy states of corresponding transitions. b Schematic diagram of the transitions between different vibrational levels of \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) states of \({\text{N}}_{2}^{+}\) ions

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Interestingly, when the similar experiments were performed in nitrogen molecules at low gas pressures, strong laser lines could be observed even if there was no overlap between the harmonic spectrum and the laser lines [61]. Figure 3 shows the spectra of forward emissions excited by 1580 nm, 1 mJ laser pulses as a function of the gas pressure. Two strong laser emissions were observed at 391 and 358 nm wavelengths, which were not covered by either the third or the fifth harmonics of the pump laser. The observed laser-like emissions also show the properties of coherent emission, i.e., linear polarization, good directionality, and high spatial coherence [61]. Surprisingly, at the low gas pressures between 10 and 30 mbar, the spectrum of laser-like emission is significantly broadened to form a supercontinuum-like spectrum with a bandwidth of ~ 80 nm. For the gas pressures above 40 mbar, the supercontinuum disappears accompanied with a rapid decay of laser-like emissions at 391 and 358 nm wavelengths.

Fig. 3
figure 3

a The forward spectra measured at different pressures of nitrogen gas (logarithmic color scale). b The forward spectra obtained in nitrogen and argon gases at a gas pressure of 20 mbar. To facilitate a quantitative comparison, the harmonic spectrum of argon is multiplied by a factor of 10 [61]

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Figure 3b shows a quantitative comparison of the fifth harmonic spectra measured in nitrogen and argon gases at a gas pressure of 20 mbar. Notably, nitrogen molecules and argon atoms have similar ionization energies and nonlinear coefficients [62], thus it is expected that similar nonlinear optical behaviors should be observed. In contrast, in comparison with the result in argon, the harmonic signal generated in nitrogen shows two striking differences. First, the fifth harmonic generated in nitrogen covers an extremely broad spectral range with strong \({\text{N}}_{2}^{+}\) laser-like emissions superimposed on the supercontinuum spectrum. Second, the fifth harmonic produced in nitrogen shows pronounced red shift in its spectrum. Also, the signal of the fifth harmonic is enhanced by approximately one order of magnitude. These differences can be attributed to the different origins of harmonic generation processes, namely, the fifth harmonic generated in argon is mainly from the neutral atoms, whereas the fifth harmonic generated in nitrogen molecules is mainly from the molecular ions.

Furthermore, we performed the pump–probe measurements on the laser-like emissions generated with the mid-infrared pump laser at 1580 nm wavelength [63]. The schematic diagram of experimental setup is shown in Fig. 4a. The probe pulse was generated through frequency doubling of the 800-nm laser. Figure 4b presents the evolution of probe spectrum with the time delay. When the pump lagged behind the probe (i.e., negative delay), the spectrum of probe pulse was hardly affected by the pump beam. When the pump was ahead of the probe (i.e., positive delay), a series of fine absorption dips appeared at the wavelengths of R-branch rotational transitions from \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=0} \right)\) state to \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) state of \({\text{N}}_{2}^{+}\) ions. Meanwhile, there was a prominent absorption band close to 391 nm which corresponds to the P branch of rotational transitions. The strong absorption of probe pulses shows that no population inversion is established in the mid-infrared laser field. Interestingly, the R-branch absorption spectra show pronounced fast oscillations with the increase of pump–probe delay. The oscillation frequency increases with the rotational quantum number J, and thus some arc-shaped structures form in the time-resolved absorption spectrum. Fourier analyses show that these oscillations result from quantum beats between J and J − 2, and that between J and J + 2 of \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}(v=0)\) state, which is out of the scope of this review.

Fig. 4
figure 4

a Schematic of the experimental setup for pump–probe measurements. The energy of the 1580 nm pump laser is 0.92 mJ, while the probe pulse energy is 20 nJ. b Spectra of the probe pulse recorded after the gas chamber as a function of time delay between the pump and the probe pulses. c The spectra of probe pulses measured at the time delays \({t_0}\) (gray dot line), \({t_1}\) (red dot-dash line) and \({t_2}\) (blue solid line). Inset: the enlarged spectrum in the spectral range 390.9–392.5 nm at the time delay \({t_2}\) [63]

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One can easily notice that both the P- and R-branch absorption spectra show a transient change at the time delays near every half and full revival period of \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}(v=0)\) state. Thus, we focus on the spectral structures near the first revival period. For comparison, we chose three specific time delays t0 = − 0.3363 ps, t1 = 8.6000 ps and t2 = 8.7670 ps as indicated by the white arrows in Fig. 4b. The absorption efficiencies of all the R-branch rotational transitions reach either minimum (at time delay \({t_1}\)) or maximum (at time delay \({t_2}\)) regardless of the quantum number J, as shown in Fig. 4b. The spectrum of the probe laser at time delay \({t_0}\) is almost unchanged compared with its original spectrum measured before the gas chamber, and thus can be used as a reference. Figure 4c shows the typical spectra measured at the three time delays. To facilitate our analysis, the spectra are divided into three regions, i.e., the R branch (region I) and P branch of rotational transitions (region II), and the region on the red side of the P-branch bandhead (region III). It can be observed that at time delay \({t_2}\), a gross loss appears in the spectral range from 386 to ~ 391 nm in region I compared to the reference spectrum (gray dotted line), while the probe pulse has been significantly enhanced at some specific wavelengths of regions II and III. A completely opposite behavior has been observed in the spectrum recorded at time delay \({t_1}\) as shown by the red dash dot curve in Fig. 4c. If we zoom into the enhanced emission in region III, we observed a multiple-plateau structure, as illustrated in the inset of Fig. 4c.

Below, we will provide a unified physical picture on all the experimental observations mentioned above. This picture indicates that all of these lasing behaviors observed with mid-infrared pump lasers root from one origin—the resonantly enhanced nonlinear interaction in the quantum wavepackets of \({\text{N}}_{2}^{+}\) ions coherently prepared by the strong-field-induced tunnel ionization.

2.2 The underlying physical mechanisms

We begin with discussion on the origin of the supercontinuum emission generated in the 1580-nm pump laser field [61]. The proposed physical picture is illustrated in Fig. 5. First, the fifth harmonic generation can be greatly enhanced by resonant four-wave mixing (i.e., \(~5\omega =3\omega +\omega +\omega\)), as shown in Fig. 5a. Here, the \(3\omega\) photons are mainly produced by the third harmonic generation in the neutral nitrogen molecules; therefore, the conversion efficiency of the third harmonic is high. The efficient third harmonic generation facilitates the subsequent four-wave mixing in \({\text{N}}_{2}^{+}\) ions. As a result, the fifth harmonic in nitrogen is about one order of magnitude stronger than that in argon, and the resonant process shifts the central wavelength of the fifth harmonic to near the wavelength of 331 nm (i.e., corresponding to transition between \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=2} \right)\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}(v=0)\) states), as observed in Fig. 3b. Subsequently, the \(5\omega\) photons at 331 nm can initiate the stimulated Raman scattering (SRS) in the excited \({\text{N}}_{2}^{+}\) ions, leading to the generation of photons at ~ 358 nm. The coherent emission at ~ 358 nm triggers further SRS to produce the photons at ~ 391 nm. In general, the Raman processes as illustrated in Fig. 5b, c require an initial population in the \({{\text{B}}^2}\Sigma _{{\text{u}}}^{+}\) state of \({\text{N}}_{2}^{+}\) ions. In our experimental conditions (i.e., tight focusing of intense laser fields into low-pressure gases), a decent amount of ions can be populated to \({{\text{B}}^2}\Sigma _{{\text{u}}}^{+}\) state immediately after tunnel ionization. The population on \({{\text{B}}^2}\Sigma _{{\text{u}}}^{+}\) state can be further promoted with the resonant single-photon absorption of fifth harmonic photons.

Fig. 5
figure 5

Schematic of the physical mechanism of generation of \({\text{N}}_{2}^{+}\) laser and supercontinuum emission [61]. a The signal around 331 nm was generated with a resonant four-wave mixing. b The emission at ~ 358 nm was generated through stimulated resonance Raman scattering. c The Raman lasing line at ~ 358 nm triggered further stimulated resonance Raman scattering to produce the photons at ~ 391 nm. d The emission lines at ~ 358 and ~ 391 nm were spectrally broadened by a resonant cross-phase modulation to form an unusual supercontinuum

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One concern is that the fifth harmonic generated in \({\text{N}}_{2}^{+}\) ions might be too weak to initiate SRS processes. To understand this, it should be noticed that the resonance of the Stokes photons generated by the Raman process in Fig. 5b, c with the electronic states of \({\text{N}}_{2}^{+}\) ions will efficiently promote the gain and meanwhile reduce the threshold of SRS [64]. Moreover, the Raman process can be more efficient for the vibrational ground state because of its larger population [65], smaller damping rate [66], and larger dipole moment [60]. The molecular oscillation with a frequency of \(\Delta v^{\prime}=1\) has been coherently excited by SRS from 331 to 358 nm, which further reinforces the Raman process from 358 to 391 nm. Through the cascaded resonant SRS processes, the 331-nm photons are efficiently converted to 358 nm and then 391-nm photons. This is why the \({\text{N}}_{2}^{+}\) emission line at 391 nm is the strongest even though the harmonic spectrum cannot cover the lasing line.

In the last stage as illustrated in Fig. 5d, cross-phase modulation can occur immediately after the generation of the stimulated Raman lasers in the intense pump laser field. In particular, the Raman lines near 358 and 391 nm wavelengths are in resonances with the transitions between \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma _{{\text{u}}}^{+}} \right)\) and \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma _{{\text{g}}}^{+}} \right)\) states. In such a case, the nonlinear Kerr coefficient will be greatly enhanced, leading to the extraordinary supercontinuum generation which, to the best of our knowledge, has never been observed in any atomic or neutral molecular gases under the similar experiment conditions. In addition, because the cross-phase modulation mainly occurs in the falling edge of the pump pulse after the ions are generated, the spectral blue shift is much more pronounced than red shift [67], as evidenced by our experimental observations in Fig. 3a. Because the three nonlinear processes in Fig. 5 exclusively take place in \({\text{N}}_{2}^{+}\) ions but not neutral molecules, the pump laser must be sufficiently strong to generate a large amount of ions in both the excited and ground states. Therefore, generation of the supercontinuum emission shows a strong dependence on gas pressure and the pump intensity. Furthermore, when a circularly polarized pump laser is used, all the nonlinear signals disappear which is consistent with the theory in Ref. [68].

Besides the vibrational Raman process mentioned above, stimulated rotational Raman scattering can also be excited which is responsible for producing the prominent peak in the region III of Fig. 4c [63]. To facilitate quantitative analyses, we zoom into the spectral range from 390.9 to 392.5 nm, as shown in Fig. 6a. A prominent spectral peak beyond the P-branch bandbead of rotational transitions (i.e., 391.42 nm) can be observed. Clearly, this peak cannot be generated by any resonant transitions in the R and P branches. More interestingly, the spectral peak shows a multiple-plateau structure, and each plateau has a similar width. The experimental result in Fig. 4c shows that the peak in region III becomes most pronounced when the absorption (i.e., the loss) in region I is the strongest. In addition, the peak appears every half alignment period of \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma_{{\text{g}}}^{+},v=0} \right)\) state (i.e., ~ 4.3 ps), as we have observed in Fig. 4b.

Fig. 6
figure 6

a The zoom-in spectra in the range between 390.9 and 392.5 nm obtained from the data in Fig. 4c. For the R-branch photons at the frequency of \(\omega_{{\text{R}}}\left( J \right)\) scattered from J + 1, J + 2, J + 3, J + 4, J + 5 rotational states, the calculated boundary wavelengths of the corresponding near-resonant Raman scattering processes are indicated by the vertical dotted lines [63]. b Schematic diagram of resonant and the near-resonant Raman processes in \({\text{N}}_{2}^{+}\) ions

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The physics underlying the observed spectral peak is schematically depicted in Fig. 6b. The absorption of one R-branch photon at the frequency of \(\omega_{{\text{R}}}\left( J \right)\) and the subsequent emission of one P-branch photon at the frequency of \(\omega_{{\text{P}}}\left( {J+2} \right)\) synergetically give rise to resonant rotational Raman scattering in \({\text{N}}_{2}^{+}\) ions, as illustrated as the process on the left side of Fig. 6b. Likewise, the photon at the frequency of \(\omega_{{\text{R}}}\left( J \right)\) inelastically scattered from J + 1 energy levels of \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=0} \right)\) state will produce the Raman signal at a frequency of \(\omega_{{\text{R}}}\left( J \right) - \Omega _{{{\text{J}}+3}}^{{{\text{J}}+1}}\), as shown by the process in the middle of Fig. 6b, which corresponds to near-resonant Raman scattering. Here, the frequency \(\Omega _{{{\text{J}}+3}}^{{{\text{J}}+1}}\) is determined by the energy difference between J + 3 and J + 1 states of the ground-state ions. We stress that the generation of near-resonance SRS signals can be very efficient as the involved wavelengths are out of the absorption range from \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=0} \right)\) state to \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}(v^{\prime}=0)\) state of nitrogen molecular ions. This off-resonant condition ensures that the gain in region III will not be suppressed by single-photon absorption process. In addition, the intense mid-infrared laser field has initiated coherent rotational wavepackets prior to the arrival of the probe. Subsequent near-resonant interaction of rotational wavepackets with the probe field will give rise to strong SRS.

We would like to point out that the rotational Raman signal generated through the near-resonant process will be spectrally shifted by 4Bc in its frequency with respect to the rotational Raman signal generated through the resonant process, which is determined by \(\Omega _{{{\text{J}}+3}}^{{{\text{J}}+1}}=\Omega _{{{\text{J}}+2}}^{{\text{J}}}+4{\text{Bc}}\). Here, B = 1.9223 cm−1 is the rotational constant of \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=0} \right)\) state [69]. Based on this fact, the near-resonant Raman process can generate Stokes photons at the wavelength up to 391.54 nm as indicated by the vertical dot line in Fig. 6a, which agrees very well with the experimental observation. When the R-branch photons at the frequency of \(\omega_{{\text{R}}}\left( J \right)\) are scattered from J + 2 rotational energy levels, the frequency of the Raman signal will be further shifted by 4Bc, as shown by the process on the right side of Fig. 6b. When the same R-branch photons are scattered from the higher rotational energy levels, Raman signals will be produced at even longer wavelengths, resulting in a multiple-plateau structure in the spectral peak in region III. Remarkably, all the plateaus in the measured spectrum and the calculation results show excellent agreement. With the increase of initial rotational energy levels, the efficiency of near-resonant Raman scattering decreases due to the increased energy detuning. The quantitative evidence unambiguously confirms the occurrence of highly efficient rotational Raman scattering in \({\text{N}}_{2}^{+}\) ions.

The signature of rotational Raman scattering in the air lasing actions generated with mid-infrared pump laser sources has also been observed in an atmospheric environment [70]. Again, the resonant effect leads to the dependence of \({\text{N}}_{2}^{+}\) laser at 391 nm on the pump wavelengths, as shown in Fig. 7. The comparative measurements in air and argon show that the fifth harmonic signal in air has a strong spectral distortion along with the generation of laser-like emission. The laser-like emission is stronger when the peak of the fifth harmonic spectrum is on the blue side of the wavelength of \({\text{N}}_{2}^{+}\) laser but not on its center. From Fig. 7b, we can also clearly see the loss of the fifth harmonic on the blue side of \({\text{N}}_{2}^{+}\) lasing line. The signature will be more striking if we examine the evolution of the fifth harmonic spectrum with the increase of the pump laser power. As shown in Fig. 8a, the generation of the laser-like emission at ~ 391 nm is always accompanied by the formation of a “hole” on the blue side of the fifth harmonic spectrum. The “hole” locates in the spectral region of R-branch rotational transitions between \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma _{{\text{u}}}^{+}},v^{\prime}=0 \right)\) and \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma _{{\text{g}}}^{+}},v=0 \right)\) states. In contrast, the fifth harmonic generated in argon shows a smooth spectrum whose bandwidth and central wavelength can be well described in the framework of nonlinear harmonic generation, as illustrated in Fig. 8b. With the increase of the pump power, the spectral “hole” in air becomes broader and its peak continuously shifts toward shorter wavelength. The result in air indicates absorption of R-branch photons and emission of P-branch photons, which correspond to the resonant rotational Raman scattering in Fig. 6b. Further measurements at higher spectral resolutions are required to confirm the physical picture.

Fig. 7
figure 7

The spectra of fifth harmonic generated in air (red solid lines) and argon at atmospheric pressure (blue dashed lines) with a 1875 nm, b 1933 nm, c 1955 nm and d 1975 nm pump pulses

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Fig. 8
figure 8

The fifth harmonic spectra generated in a air and b argon as functions of the powers of the pump laser at ~ 1930 nm wavelength [70]

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The experimental observations and theoretical analyses suggest that stimulated Raman processes could be the dominant mechanism of \({\text{N}}_{2}^{+}\) laser generation driven by the mid-infrared laser field. Its pumping mechanism is different from the \({\text{N}}_{2}^{+}\) laser excited by the 800-nm laser field. Furthermore, the highly efficient nonlinear processes in the near ultraviolet region with \({\text{N}}_{2}^{+}\) ions extend the study of nonlinear optics to molecular ions, which provides a means to practice nonlinear optics at short wavelengths.

3 \({\mathbf{N}}_{2}^{+}~\) laser emissions generated in 800-nm laser fields

3.1 Experimental observation

Free-space \({\text{N}}_{2}^{+}\) lasing action in the 800-nm laser field has been systematically investigated using a pump–probe scheme, as shown in Fig. 9a [12]. In this scheme, the pump is an intense 800-nm laser, whereas the probe is a weak 400-nm laser. Figure 9b, c shows two typical spectra measured in the forward propagation direction. It can clearly see that the probe pulse can be amplified at both the wavelengths of ~ 391 and ~ 428 nm, which correspond to the transitions \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right) \to {{\text{X}}^2}\Sigma_{{\text{g}}}^{+}(v=0)\) and \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right) \to {{\text{X}}^2}\Sigma_{{\text{g}}}^{+}(v=1)\) of \({\text{N}}_{2}^{+}\) ions, respectively. The pump–probe delay and the probe spectrum were optimized to maximize the \({\text{N}}_{2}^{+}\) laser signals at 391 and 428 nm. The two \({\text{N}}_{2}^{+}\) laser lines will disappear when the pump beam is blocked, as illustrated in Fig. 9b, c. In addition, the amplified laser signal shows a linear dependence on the intensity of the probe laser [40], and its polarization direction is the same as that of the probe laser [12]. These facts indicate the seeding effect of the probe pulse. If the pump energy is sufficiently high and the gas density is proper, \({\text{N}}_{2}^{+}\) lasers can also be triggered by the 800-nm pump laser alone, which has been independently confirmed by three research groups [37,38,39]. In this case, the seed pulse is provided by the supercontinuum emission during femtosecond filamentation or the weak second harmonic signal in gas.

Fig. 9
figure 9

a Schematic diagram for generating the \({\text{N}}_{2}^{+}\) lasers with a pump–probe scheme. The p-polarized laser pulse at 800 nm wavelength served as the pump, while the s-polarized probe pulse was generated through frequency doubling of the 800-nm laser. Typical spectra of \({\text{N}}_{2}^{+}\) lasers at b 391 nm and c 428 nm generated by the pump–probe scheme [12]. For comparison, the spectra of the probe pulses in the absence of the 800-nm pump laser are indicated by red dotted lines

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The pump–probe scheme enables us to study the evolution of \({\text{N}}_{2}^{+}\) lasing signals after excitation by the pump laser. As illustrated in Fig. 10, the ~ 391-nm signal first increases rapidly on the time scale of ~ 400 fs, and then slowly decays with a decay constant of ~ 46 ps as fitted by an exponential curve (red dotted line). The fast generation of the coherent emissions suggests an “instantaneous” population inversion compared to the time required for building up the population inversion in the ASE-based air lasing which is on the nanosecond timescale [21, 22, 24, 30]. Particularly, when the pump and the probe pulses are temporally separated, the laser-like emission can still be generated, which not only further confirms the seeding effect of the probe pulse, but also shows that the population inversion can be established with the 800-nm pump pulse alone. In addition, the ~ 391-nm emission is sensitive to the rotational dynamics of the molecular ions, as evidenced by the periodic modulation at each half revival period (\({T_{{\text{rot}}}}\)) in the inset of Fig. 10. The rotational coherence created by the pump pulse can also be faithfully encoded into the amplified lasing signal in the condition of low gas pressures [35, 44, 45, 54]. As an example, Fig. 11a shows the spectrum of \({\text{N}}_{2}^{+}\) laser captured in the 4-mbar nitrogen gas [44]. It can be clearly seen that a strong, narrow-bandwidth emission appears at 391.4 nm wavelength, which corresponds to the P-branch band of \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}(v^{\prime}=0) \to {{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=0} \right)\) transition. Meanwhile, a series of peaks appear on the blue side of the 391.4-nm laser, which corresponds to the R-branch band of the same electronic transition. Each peak corresponds to stimulated emission from a specific rotational energy level, as illustrated in Fig. 11b. Therefore, \({\text{N}}_{2}^{+}\) laser offers an ideal tool for investigating the ultrafast dynamics of molecular ions in a rotational-state-resolvable manner [35, 44, 45, 54].

Fig. 10
figure 10

The coherent emission at ~ 391 nm as a function of time delay of the 800-nm pump and 400-nm probe pulses. The red dotted line shows the exponential fit of the experimental data. Inset: decay dynamic in the range from − 1 to 14 ps [12]

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Fig. 11
figure 11

a The spectrum of \({\text{N}}_{2}^{+}\) lasing emission generated in 4-mbar nitrogen gas with a pump–probe scheme (blue solid line) [44]. For comparison, the spectrum of the probe pulses in the absence of the pump laser was indicated by the red dotted line. The polarization of the probe laser was set to be parallel to that of the 800-nm pump laser. The time delay between the pump and probe pulse was chosen as 1.1 ps. The numbers denote the rotational levels of \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) state in R-branch rotational transitions. b Schematic diagram of P- and R-branch rotational transitions between \({{\text{B}}^2}\Sigma _{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) and \({{\text{X}}^2}\Sigma _{{\text{g}}}^{+}\left( {v=0} \right)\) states

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Lasing action in \({\text{N}}_{2}^{+}\) ions can even take place when the pump and probe pulses are propagating in the opposite directions [40]. The experimental setup is schematically illustrated in Fig. 12a. Figure 12b shows the spectrum of the probe pulse recorded using the counter-propagation scheme. In the absence of the pump pulses, the spectral profile of the probe pulses appears smooth and does not exhibit any emission line at wavelength of ~ 391 nm. In contrast, when the pump pulse is launched into the nitrogen molecules ahead of the probe pulse, a strong, narrow-bandwidth emission located at ~ 391 nm appears in the probe spectrum, indicating amplification of the probe pulse even when it counter propagates with respect to the pump pulse. The gain can last for dozens of picoseconds, as shown in Fig. 12c. Based on these experimental observations, there is no doubt that there is population inversion between the excited and ground states of ionized nitrogen molecules. Below we will discuss the physical mechanism behind the population inversion generated in \({\text{N}}_{2}^{+}\) ions with the 800-nm pump laser.

Fig. 12
figure 12

a Schematic of the experimental setup. b Spectra of the probe pulses in the presence and absence of pump laser. The pump–probe delay was optimized to obtain the strongest coherent emission at ~ 391 nm. c Measured intensities of the amplified signal at ~ 391 nm as a function of the time delay between the pump and the probe pulses [40]

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3.2 The mechanism behind the population inversion

To gain insight into the underlying mechanism of population inversion, we calculated the evolution of the population of three electronic states (i.e., \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\), \({{\text{A}}^2} \Pi_{{\text{u}}}\) and \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\)) in the pump laser field by solving the time-dependent Schrödinger equation:

$$i\frac{\partial }{{\partial t}}\left( {\begin{array}{*{20}c} {\Psi _{{\text{X}}} (R,t)} \\ {\Psi _{{\text{A}}} (R,t)} \\ {\Psi _{{\text{B}}} (R,t)} \\ \end{array} } \right) = \left[ { - \frac{1}{{2\mu }}\frac{{\partial ^{2} }}{{\partial R^{2} }} + V(R,t)} \right]\left( {\begin{array}{*{20}c} {\Psi _{{\text{X}}} (R,t)} \\ {\Psi _{{\text{A}}} (R,t)} \\ {\Psi _{{\text{B}}} (R,t)} \\ \end{array} } \right),$$
(1)

where atomic units are used, µ is the reduced mass and R is the internuclear separation. The potential matrix \(V\left( {R,t} \right)\) for the three-state system can be written as:

$$V(R,t)=\left( {\begin{array}{*{20}{c}} {{V_{\text{X}}}(R)}&{{{\vec {\mu }}_{{\text{XA}}}}(R) \cdot \vec {E}(t)}&{{{\vec {\mu }}_{{\text{XB}}}}(R) \cdot \vec {E}(t)} \\ {{{\vec {\mu }}_{{\text{XA}}}}(R) \cdot \vec {E}(t)}&{{V_{\text{A}}}(R)}&0 \\ {{{\vec {\mu }}_{{\text{XB}}}}(R) \cdot \vec {E}(t)}&0&{{V_{\text{B}}}(R)} \end{array}} \right),$$
(2)

where the diagonal elements denote the potential energy of three electronic states. The off-diagonal elements denote the coupling terms induced by the laser fields under dipole approximation with \({\vec {\mu }_{{\text{XA}}}}\) and \({\vec {\mu }_{{\text{XB}}}}\) being the electronic transition moments between the relevant pairs of electronic states. \(\vec {E}\left( t \right)\) represents the linearly polarized pump laser field.

Figure 13a shows the evolution of population distribution of three electronic states in the 800-nm laser field [15]. Clearly, after the interaction with the laser field, the population in \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) state is almost the same as its initial value. In contrast, a strong population transfer occurs between \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state and \({{\text{A}}^2}\Pi_{{\text{u}}}\) state, resulting in large growth of population in \({{\text{A}}^2}\Pi_{{\text{u}}}\) state and significant reduction of population in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state. As a result, the final population in \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) state becomes higher than that in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state. In the same condition, if we remove the coupling between \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) and \({{\text{A}}^2}\Pi_{{\text{u}}}\) states, the populations in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state and \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) state almost remain unchanged after the interaction of the molecular ions with the laser field, as shown in Fig. 13b. In this case, no population inversion is established. The simulation results clearly show that the intermediate \({{\text{A}}^2}\Pi_{{\text{u}}}\) state plays a vital role on the population inversion. It acts as a reservoir for evacuating the population in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}~\)state, which helps to establish the population inversion between \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) states.

Fig. 13
figure 13

a The temporal evolution of the population distributions in three electronic states of \({\text{N}}_{2}^{+}\) ions in the 800-nm laser field with an intensity of 2.2 × 1014 W/cm2. b The temporal evolution of the population distributions in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) and \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) states obtained by removing \({{\text{A}}^2}\Pi_{{\text{u}}}\) state [15]

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The efficient population transfer between \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) and \({{\text{A}}^2}\Pi_{{\text{u}}}\) states results from the resonant one-photon process as illustrated in Fig. 14a. However, it is well known that the resonant excitation of atomic or molecular species with coherent light fields will lead to famous Rabi oscillations. In general, the final population distribution after the coherent excitation sensitively depends on the laser parameters (i.e., intensity and pulse duration). This is inconsistent with the fact that the \({\text{N}}_{2}^{+}\) laser has been experimentally confirmed by various research groups with pump laser pulses of different parameters [12,13,14,15,16,17, 35, 37,38,39,40,41, 43,44,45,46,47,48,49,50,51,52,53,54, 56, 57]. Besides, our theoretical simulations also show that the population inversion can survive even when the pulse duration and peak intensity of the pump fields have varied in broad ranges, as indicated in Fig. 14b. To gain more insight, we numerically examined the evolution of nuclear wavepackets in the laser field by calculating the time-dependent nuclear density \({\rho _\alpha }\left( {R,t} \right)={\left| {{\Psi _\alpha }\left( {R,t} \right)} \right|^2}\left( {\alpha =X,~~A,~~B} \right)\), as shown in Fig. 14d–f. In the simulation, the parameters of the pump laser were the same as that used in Fig. 13a but the alignment angle of the molecular ions was fixed at 45°. We found that the wavepacket of \({{\text{A}}^2}\Pi_{{\text{u}}}\) state will start to oscillate immediately after the photoionization due to its larger equilibrium internuclear distance compared to \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state (see Fig. 14a). The fast oscillation of nuclear wavepacket of \({{\text{A}}^2}\Pi_{{\text{u}}}\) state rapidly modulates the coupling efficiency between \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}~\) and \({{\text{A}}^2}\Pi_{{\text{u}}}\) states in a transient decay laser field (see Fig. 14c), and eventually most ions are trapped in \({{\text{A}}^2}\Pi_{{\text{u}}}\) state. Therefore, the nuclear vibration breaks the reversibility of the population transfer between \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) and \({{\text{A}}^2}\Pi_{{\text{u}}}\) states, which ensures the population inversion to be achieved in a broad parameter range.

Fig. 14
figure 14

a Energy-level diagram of \({\text{N}}_{2}^{+}({{\text{X}}^2}\Sigma_{{\text{g}}}^{+})\) state and \({\text{N}}_{2}^{+}\left( {{{\text{A}}^2}\Pi_{{\text{u}}}} \right)\) state obtained by cubic spline interpolation of ab initio data using the MOLPRO package. The ions in \({{\text{X}}^2}\Sigma _{{\text{g}}}^{+}\left( {v=0} \right)\) state can be resonantly excited to \({{\text{A}}^2}{\Pi _{\text{u}}}\left( {v=2} \right)\) state through the 800-nm pump laser. b The calculated population difference \(\Delta P\) of \({{\text{B}}^2}\Sigma _{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) state and \({{\text{X}}^2}\Sigma _{{\text{g}}}^{+}\left( {v=0} \right)\) state as functions of the intensity and the pulse duration of the 800-nm pump laser. The population inversion occurs for the case of \(\Delta P>0\). c The waveform of the pump laser field chosen in the simulation. The evolution of nuclear density of d \({\text{N}}{^{+}_{2}}\left( {{{\text{X}}^2}\Sigma_{{\text{g}}}^{+}} \right)\) state, e \({\text{N}}_{2}^{+}\left( {{{\text{A}}^2}\Pi_{{\text{u}}}} \right)\) state and f \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma_{{\text{u}}}^{+}} \right)\) state in the 800-nm laser field. g Time-dependent population distribution in three electronic states of \({\text{N}}_{2}^{+}\) ions [15]

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Based on the analyses above, Fig. 15 shows a brief physical picture on the establishment of population inversion, which can be divided into three stages [15]. First, in the rising edge of the pump laser field, most nitrogen molecules remain neutral because of the strong dependence of tunnel ionization on the field strength (stage I). When approaching the peak of envelop of the driver laser, the molecules will be rapidly ionized due to the high field strength near the peak (stage II). However, the produced \({\text{N}}_{2}^{+}\) ions are mainly distributed in the ground \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state, which is a result of the exponential decay of tunnel ionization rate with the increase of ionization potential. Subsequently, population is redistributed in three electronic states by the interaction of \({\text{N}}_{2}^{+}\) ion with the remaining pump laser field (stage III). In particular, it is observed that most ions in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state are efficiently evacuated to the intermediate \({{\text{A}}^2}\Pi_{{\text{u}}}\) state through one-photon resonant excitation, giving rise to the population inversion between \({{\text{B}}^2}\Sigma_{{\text{u}}}^{+}\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) states in the 800-nm laser field. Meanwhile, the motion of nuclear wavepackets makes most ions to be trapped in \({{\text{A}}^2}\Pi_{{\text{u}}}\) state, which largely reduces the sensitivity of population inversion on the laser parameters.

Fig. 15
figure 15

Schematic diagram of the pumping mechanism for establishing population inversion in the 800-nm femtosecond laser field

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4 \({\mathbf{N}}_{2}^{+}\) laser at 428 nm

In the Sect. 3.2, we discussed the population dynamics of multiple electronic states of \({\text{N}}_{2}^{+}\) ions in the 800 nm laser field. In fact, the vibrational population is also critical for the generation of \({\text{N}}_{2}^{+}\) laser. In this section, we will discuss the role of vibrational population of the nitrogen molecular ions in initiating the laser emissions at 391 nm and 428 nm wavelengths. These two lasing lines share the same upper energy level, i.e., \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) but two different vibrational energy levels in the lower \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state, and thus the comparison between the two lasing lines will provide an insight on the role of vibrational dynamics in generating \({\text{N}}_{2}^{+}\)-based air lasers at various wavelengths.

We performed the pump–probe measurements on 391 and 428 nm \({\text{N}}_{2}^{+}\) laser signals generated in 800- and 1500-nm laser fields [71]. The results in Fig. 16 show that a pronounced gain can be observed at either 391 or 428 nm with a pump laser centered at 800 nm, whereas the gain at 391 nm wavelength completely disappears when the wavelength of pump laser is tuned to 1500 nm. In all the measurements, the time delay was chosen to be ~ 1 ps to avoid the influence of molecular alignment as well as the overlap between the pump and probe pulses. The results clearly show that the distribution of vibrational populations plays a key role on the generation of population inversion in \({\text{N}}_{2}^{+}\) ions. The low population in \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=1} \right)\) state facilitates generation of the population inversion between \({{\text{B}}^2} \Sigma_{{\text{u}}}^{+}\left( {v^{\prime}=0} \right)\) and \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\left( {v=1} \right)\); therefore, the coherent population transfer from \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state to \({{\text{A}}^2}\Pi_{{\text{u}}}\) state mandatory for generating the 391-nm \({\text{N}}_{2}^{+}\) laser becomes unnecessary for generating the 428 nm. Note that the coherent population transfer from \({{\text{X}}^2}\Sigma_{{\text{g}}}^{+}\) state to \({{\text{A}}^2}\Pi_{{\text{u}}}\) state can only be realized in 800-nm laser fields, indicating that the \({\text{N}}_{2}^{+}\) laser at 428 nm wavelength can be generated with much flexibility in terms of the choice of the pump laser wavelength. Meanwhile, the above understanding suggests that it will be important to be able to manipulate the vibration population distribution as well as the dynamics of different vibrational levels for promoting the efficiency and power of \({\text{N}}_{2}^{+}\) lasers by tailoring the pump laser field.

Fig. 16
figure 16

The forward spectrum obtained by injecting a ~ 391 nm and b ~ 428 nm seed pulses into the plasma channel generated by an 800-nm pump laser. c, d The same measurements as in a, b, respectively, except that the pump laser is operated at 1500 nm wavelength [71]. For comparison, the spectra of the seed pulses in the absence of the pump laser were indicated by dashed lines

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5 Summary and future perspective

We have discussed the mechanisms behind the strong-field ionization-induced air lasing actions under the different pump laser conditions. Our results show that although the generated laser emissions share some common properties such as high spatial coherence, high brightness, linear polarization and narrow spectral width, they can have different origins. It is found that at the 800-nm pump wavelength, population inversion has been generated in the nitrogen molecular ions which further leads to stimulated emissions triggered by seed pulses at various wavelengths. However, at the mid-infrared pump wavelengths, the laser-like emissions are generated by means of near-resonant nonlinear frequency conversion. This explains the failures in the attempt of establishing a unified physical picture for understanding the nitrogen molecular ion lasers generated under different pump laser conditions.

One of the major findings in the current research is that nitrogen molecular ions generated in strong laser fields are a unique quantum system for demonstrating a series of extreme nonlinear effects. First of all, it should be noted that tunnel ionization always occurs within a fraction of the optical cycle. For the laser fields of wavelengths below 2 µm, the electrons will be ripped off from the nitrogen molecules on the attosecond time scale, leading to highly synchronized excitation of the rotational and vibration wavepackets. The coherent rotational and vibrational wavepackets of the molecular nitrogen ions can dramatically promote the efficiencies of various Raman processes. Second, since the molecular nitrogen ions have an ionization potential as high as ~ 28 eV, they can survive against photoionization at visible and infrared (IR) wavelengths even when the pump laser intensity reaches 8.5 × 1014 W/cm2 (assuming an ionization probability of ~ 1%). In the meantime, the nitrogen molecular ions can also allow the resonant transitions between various vibrational energy levels of \({\text{N}}_{2}^{+}\left( {{{\text{X}}^2}\Sigma_{{\text{g}}}^{+}} \right)\) and \({\text{N}}_{2}^{+}\left( {{{\text{B}}^2}\Sigma_{{\text{u}}}^{+}} \right)\) states to occur at near ultraviolet wavelengths. It is still unclear how the resonance condition will influence the nonlinear interaction of intense laser fields with the nitrogen molecular ions. In other words, strong-field atomic and molecular physics has been focusing on processes in which the resonance conditions only play minor or even negligible roles. Extending such investigations to a quantum system like the \({\text{N}}_{2}^{+}\) ions which allow to involve strong contribution from resonant interaction with intense laser fields is unnoticed and deserves more attention. Together with the opportunity of using the \({\text{N}}_{2}^{+}\)-based air laser as a bright pump source for atmospheric nonlinear spectroscopy, the research will have important implications for both strong field physics and remote atmospheric sensing.