1. Introduction

In the past few years, optically pumped air lasers remotely generated in free space have attracted much attention due to their great promise for remote chemical sensing and the involved fruitful physics [1–16]. Among all candidates of air lasing, the ${\text{N}}_2^{+}$ lasing has aroused much interest owing to the mysterious mechanism underlying the gain in the strong field ionized molecules [9–16]. Several scenarios have been proposed for revealing the origin of ${\text{N}}_2^{+}$ lasing, including population inversion induced by multiple-states couplings [17–20], superradiance induced by electron recollision [21,22], alignment-induced transient inversion [23,24], etc., whereas none of them has been completely accepted. It should be stressed that only recently strong nonlinear optics effects were observed in ${\text{N}}_2^{+}$ ions during the generation of ${\text{N}}_2^{+}$ lasing [25–27]. The physical origin is that the resonant interaction of the laser field with ${\text{N}}_2^{+}$ ions significantly enhances the nonlinear response of ions, and creates coherence among different electronic, vibrational and rotational states. The findings motivate us to further examine the contribution from the coherently and resonantly enhanced optical nonlinearity on the generation of ${\text{N}}_2^{+}$ lasing.

Here, we will explain the different behaviors observed in ${\text{N}}_2^{+}$ laser-like radiations at various wavelengths such as 358 nm, 391 nm and 428 nm based on the understandings built upon the latest investigations. Our study is carried out using a traditional pump-probe scheme with the wavelength-tunable probe pulse, and we quantitatively examine the dependence of these ${\text{N}}_2^{+}$ radiations on the key parameters such as the probe laser power and gas pressure.

2. Experimental setup

The experiment was carried out with a pump-probe scheme, which is shown schematically in Fig. 1(a). A commercial Ti:sapphire laser system (Legend Elite-Duo, Coherent, Inc.) delivers 800 nm, 1 kHz, 6 mJ, 40 fs laser pulses. The laser pulse was split into two beams. One beam with an energy of 2.1 mJ was used as the pump to ionize nitrogen molecules and excite vibrational coherence of ${\text{N}}_2^{+}$ ions. Another beam was used to generate the wavelength-tunable probe pulse. Similar to previous method [28], the probe pulse was obtained by frequency doubling of the spectrally-broadened 800 nm laser beam. First, the 800 nm pulse in the probe beam was focused by an f = 20 cm lens and collimated by an f = 10 cm lens. Then, a 4-mm-thickness sapphire plate was inserted before focus to broaden the spectrum, whose frequency was doubled by a 0.4 mm β-barium borate (BBO) crystal. The residual fundamental wave was removed by filters. The probe pulses centered at 391 nm, 396 nm and 401 nm were generated by changing the phase-matched angle of BBO. At all the wavelengths, the energy of the probe laser was remained at 7 nJ by adjusting the pulse energy into BBO. The polarization of the probe was linear and parallel to that of the pump by using a polarizer and a half wave plate. The pump and probe beams were combined by a dichroic mirror and focused collinearly by an f = 30 cm lens into a gas chamber filled with nitrogen gas. A 10-mm-long plasma channel was formed near focus of the pump laser. The time delay between the two beams was controlled by a motorized translation stage. The zero delay was determined by the maximum spectral broadening of the probe pulse caused by the cross-phase modulation.

 

Fig. 1 (a) Schematic diagram of experimental setup. L1: f = 20 cm lens; L2: f = 10 cm lens; L3: f = 30 cm lens; L4: f = 20 cm lens; SP: sapphire plate; BBO: β-barium borate; P: polarizer; HWP: half wave plate; DM: dichroic mirror with high reflectivity around 400 nm and high transmission around 800 nm. Energy diagram of coherent (b) Stokes and (c) anti-Stokes Raman scattering in${\text{N}}_2^{+}$ ions. (d) The original (black dashed line) and broadened (blue solid line) spectra of the pump pulses measured before and after the gas chamber, respectively.

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After being collimated with a lens, the forward radiations were completely collected into a grating spectrometer (Shamrock 500i, Andor) by a lens for spectral analysis. Some filters were used in the measurements to enhance the signal to noise ratio. Figures 1(b)–1(d) illustrate the energy level diagrams of Raman scattering in ${\text{N}}_2^{+}$ ions and the spectra of the pump laser measured before and after the gas chamber, which will be further discussed later.

3. Experimental results

First, we measured the forward spectrum after the weak probe pulse interacts with the plasma produced by the intense pump pulse. The results are shown in Fig. 2. Here, the weak signal generated by the single pump pulse has been subtracted as the background. The spectrum in different regions was measured by choosing suitable filters, and their relative intensities shown in Fig. 2 were calibrated by taking the transmission of filters into account. Here, we choose three probe wavelengths, i.e., 391 nm, 396 nm and 401 nm, while the other experimental parameters are remained the same. In the measurement, the gas pressure is fixed at 42 mbar, and the pump-probe delay was 0.667 ps to ensure no any temporal overlap of two pulses. The intensity of the pump laser in the gas chamber was measured to be about 1 × 10¹⁴ W/cm² using a technique described in [29]. Figure 2(a) shows that the probe pulse centered at 391 nm has prominent absorption near 391 nm and 388 nm wavelengths, which corresponds to the transition $X^2{\Sigma }_g^{+}(v=0)\to B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=1)\to B^2{\Sigma }_u^{+}(v\text{'}=1)$, respectively. At the same time, the narrow-bandwidth emissions at 358 nm and 428 nm wavelengths can be observed, which are ascribed to the transitions $B^2{\Sigma }_u^{+}(v\text{'}=1)\to X^2{\Sigma }_g^{+}(v=0)$ and $B^2{\Sigma }_u^{+}(v\text{'}=0)\to X^2{\Sigma }_g^{+}(v=1)$, respectively. The 428 nm emission is three orders of magnitude stronger than 358 nm emission, and the spectral intensity of the 428 nm signal is even higher than the probe pulse.

 

Fig. 2 The forward spectra obtained by injecting (a) 391 nm, (b) 396 nm and (c) 401 nm probe pulses into the plasma channel induced by 2.1 mJ, 800 nm pump pulse. For comparison, the original probe spectra measured without the pump laser are indicated by red dashed lines. The delay between pump and probe is 0.667 ps. For comparison, the signals on two sides of the probe spectrum are multiplied by a factor, which is indicated on the top of the corresponding spectrum. The inset of Fig. 2(a) is the spatial profile of the strong 428 nm radiation. The insets of Figs. 2(b) and 2(c) are the enlarged spectral regions indicated with blue dashed boxes.

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When the center wavelength of the probe pulse is shifted to 396 nm and 401 nm, both 428 nm and 358 nm radiations are significantly reduced, whereas the probe spectrum almost remains the same as the original one. It is worth noting that some broadband radiations appear on two sides of the probe spectrum. Comparing Fig. 2(b) with Fig. 2(c), we can clearly see that the center wavelengths of these sidebands shift with the probe wavelength. The frequency differences of these sideband radiations with respect to the probe laser are close to the energy separation between $X^2{\Sigma }_g^{+}(v=0)$ and $X^2{\Sigma }_g^{+}(v=1)$ states of ${\text{N}}_2^{+}$ ions. Therefore, we confirm that these frequency sidebands mainly originate from Raman scattering in ${\text{N}}_2^{+}$ ions. Stokes Raman scattering generates the red-shift spectral components, while the anti-Stokes Raman scattering contributes to the blue-shift components. The Raman shifts theoretically calculated are indicated by black arrows in the Fig. 2, which are in good agreement with the experimental results.

Comparing Fig. 2(a) with Figs. 2(b) and 2(c), we can see that Raman scattering will be significantly enhanced when the probe laser is resonant with the electronic transition of ${\text{N}}_2^{+}$ ions. As a result, the probe pulse is efficiently converted as Stokes and anti-Stokes Raman signals, resulting in strong narrow-bandwidth emissions at 428 nm and 358 nm. Previous studies have shown that the population inversion is easily established between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=1)$ states, but it is hardly generated between $B^2{\Sigma }_u^{+}(v\text{'}=1)$ and $X^2{\Sigma }_g^{+}(v=0)$ states owing to the vibrational distribution of population [13,28]. The 428 nm Raman signal can be amplified in the population-inverted ions. Thus, its spectral intensity is even higher than the probe pulse. In contrast, the 358 nm Raman signal is much weaker due to the absence of population inversion. However, with the help of electronic resonance, the 358 nm Raman signal is still two orders of magnitude higher as compared to the case without resonance.

We also examined basic properties of the strong 428 nm radiation shown in Fig. 2(a). In the measurements, the 428 nm signal was spectrally filtered using a bandpass filter with a center wavelength of 430 nm and a bandwidth of 10 nm. After the beam was collimated with a lens, its spatial profile was recorded with a CCD camera, as shown in the inset of Fig. 2(a). According to the measured profile, its divergence angle is estimated to be ~22 mrad. Its polarization property was measured by placing a polarizer before the spectrometer. As indicated in Fig. 3(a), the 428 nm emission is nearly perfectly linearly polarized with the polarization direction parallel to that of the probe pulses. We also examined both spatial and temporal coherence of the 428 nm emission using a Young's double-slit interferometer and a Michelson interferometer, respectively. Figure 3(b) illustrates that double-slit interference fringes are clearly visible on a far-field screen, which indicates a good spatial coherence of the radiation. Figure 3(c) shows the fringe visibility as a function of the time delay of two arms of the Michelson interferometer. From the measured data, the temporal coherence is estimated to be tens of picoseconds. All these characteristics show that the 428 nm emission has a good directionality and a high coherence.

 

Fig. 3 (a) The intensity of the strong 428 nm signal as a function of angles of the polarizer (blue circles). The measured data are fitted by the Malus law (blue solid curve). Zero degree indicates the polarization direction of the probe pulses. (b) The measured interference patterns of the 428 nm signal using a Young's double-slit interferometer. (c) The measured fringe visibility as a function of the time delay of two arms of the Michelson interferometer. The solid line is the numerical fit to guide the eyes. The interference patterns measured at the time delay of −10 ps, 0 ps and 20 ps are indicated in the insets.

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We further measured spectra of Stokes Raman scattering with the on-resonance and off-resonance probe pulses as a function of the time delay between pump and probe pulses. In these measurements, a bandpass filter with a center wavelength of 440 nm and a bandwidth of 10 nm was used to enhance the signal to noise ratio. Here, the zero delay represents temporal overlap of two pulses, and the positive delay means that the probe pulse lags behind the pump pulse. As shown in Fig. 4(a), only P- and R-branch rotational components of the transition $B^2{\Sigma }_u^{+}(v\text{'}=0)\to X^2{\Sigma }_g^{+}(v=1)$ are observed with the on-resonance probe laser centered at 391 nm. The wavelength of P-branch bandhead is about 428 nm. The 428 nm signal begins to generate near zero delay and gradually decays in the subsequent several picoseconds. With the off-resonance probe pulse at 401 nm wavelength, the 428 nm signal is superimposed on the supercontinuum spectrum when the pump and probe pulses temporally overlap. With the increase of time delay, we can see that the supercontinuum spectrum is divided into two parts, i.e., the 428 nm narrow-bandwidth signal and the 439 nm broadband Raman signal. As compared to the result at zero delay, the 428 nm signal is four orders of magnitude weaker when two pulses separate temporally, whereas the broadband Raman signal shows a slight change. It is noteworthy that the Raman signal is enhanced at half and full revival periods of ${\text{N}}_2^{+}$ ions, which will be explained in the section of discussion.

 

Fig. 4 Spectra of the nonlinear radiations generated by (a) 391 nm, (b) 401 nm probe pulses and the pump laser as a function of the time delay between two pulses (logarithmic color scale). The same filter was employed in two measurements, thus its transmission was not considered in the spectra. (c) The 439 nm Raman signal (black solid line), the 428 nm emission induced by the 391 nm probe (blue dash dot line) and the 401 nm probe (pink shaded curve) as a function of the time delay. All signals are normalized individually.

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Figure 4(c) indicates that the 428 nm signal obtained by the resonant excitation of 391 nm probe pulses exhibits a slow decay with the lifetime of several picoseconds, which has a similar dynamics with the 439 nm broadband Raman signal. In contrast, when the probe laser is far from resonance, the 428 nm signal mainly appears near zero delay. The difference indicates that the 428 nm signals in two cases originate from different physical processes. In the case of electronic resonance, Raman amplification in the population-inverted system could play a dominant contribution, and thus the 428 nm emission exhibits a similar evolution with the 439 nm Raman signal. Once the probe laser is far from electronic resonance, Raman scattering will become less effective, and the Raman spectrum hardly covers the 428 nm wavelength. In this case, the spectral broadening of the probe pulse induced by cross phase modulation will serve as an initial seed to trigger the amplified 428 nm emission near zero delay, as verified by the experimental result in Fig. 4(b).

4. Discussion

The time dependence of the Raman scattering suggests that these signals are coherent Raman scattering (CRS) rather than stimulated Raman scattering (SRS). This is because the dynamics of SRS from molecular ions is mainly determined by the population decay. For molecular ions produced by the femtosecond laser, its population decay depends on the plasma lifetime whose characteristic time is several hundreds of picoseconds. Apparently, it is much longer than the experimental result in Fig. 4(c). However, the pulse duration of pump laser (i.e., 40 fs) seems to be too long to excite the vibrational coherence of ${\text{N}}_2^{+}$ ions. In previous studies, CRS from nitrogen and oxygen molecules has been observed with the pump laser with similar pulse duration [30]. They attributed the result to the formation of spike structures during nonlinear propagation. To confirm the possibility, we also measured the spectrum of the pump pulse exiting from the gas chamber, as shown in Fig. 1(d). It can be seen that the spectrum of the pump laser is significantly broadened after nonlinear propagation in the gas. Nevertheless, the spectral bandwidth is comparable to the energy difference of two lower vibrational energy levels (i.e.,$v\text{=}0,1$) of $X^2{\Sigma }_g^{+}$ state, thus only the weak tails of the broadened spectrum contribute to the creation of vibrational coherence. It is noteworthy that nitrogen molecules are instantaneously turned into molecular ions by tunnel ionization, which is also beneficial to the creation of vibrational coherence in molecular ions. This could also be the main reason that Raman scattering from ${\text{N}}_2^{+}$ ions are much stronger than ${\text{N}}_2$ molecules under our experimental conditions. Therefore, the generation of 428 nm and 358 nm coherent emissions in Fig. 2(a) can be attributed to coherent Stokes and anti-Stokes Raman scattering respectively, as indicated in Figs. 1(b) and 1(c).

Exactly speaking, the pump laser excites the initial coherence of two lower vibrational energy levels (i.e.,$v\text{=}0,1$) of $X^2{\Sigma }_g^{+}$ state due to their high population. Meanwhile, all thermally populated rotational states for the two vibrational states are also excited. The ro-vibrational coupling will cause the dispersion of ro-vibrational wavepackets, and the decay of the vibrational coherence [30,31]. Thus, the temporal dynamics of CRS reflects the decoherence process. In our experimental conditions, the vibrational coherence will be lost on the timescale of several picoseconds, as illustrated in Fig. 4. For resonant Raman scattering at 428 nm, its temporal evolution reflects the decay of the population inversion and the vibrational coherence. For off-resonance Raman scattering at 439 nm, the temporal dynamics mainly originates from the decay of the vibrational coherence. In addition, Q, O and S branches of Raman scattering occur in the experiment and they cannot be resolved spectrally due to the broad bandwidth of the probe laser. At half and full revival periods of molecular orientation, all oscillators corresponding to O- and S-branch Raman transition will be in phase [31]. Their constructive interferences will give rise to the enhancement of the Raman signal, as observed in Fig. 4(b).

To further confirm the picture of CRS, we measured the 391 nm absorption, the 428 nm coherent emission in Fig. 2(a) and the 439 nm Raman signal in Fig. 2(c) as a function of powers of the probe pulses. The intensities of these signals are obtained by integrating the corresponding spectrum. As shown in Fig. 5, both the 391 nm absorption and the 439 nm Raman signal show a linear growth with the increase of the probe power, as predicted by the theory of CRS. However, the 428 nm radiation has a nonlinear dependence on the probe power, which can be well fitted by the formula $y=0.049x^{1.61}$. The discrepancy of the 428 nm coherent emission with the theory of CRS could be due to the existence of population inversion. As the probe power increases, more ${\text{N}}_2^{+}$ ions in $X^2{\Sigma }_g^{+}(v=0)$ state would be excited to $B^2{\Sigma }_u^{+}(v\text{'}=0)$ state by absorbing more 391 nm probe photons, which raises the population inversion between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=1)$ states. As a result, the 428 nm signal grows nonlinearly with the increasing probe power.

 

Fig. 5 The intensity of (a) the 391 nm absorption and (b) the 428 nm emission as a function of the power of the 391 nm probe laser. (c) The intensity of the 439 nm Raman signal as a function of the power of the 401 nm probe laser. All signals are normalized individually. The experimental data are fitted with the functions indicated below the curves.

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Based on these measurements, we believe that these strong emissions at 358 nm and 428 nm in Fig. 2(a) originate from CRS. Thus, they show a small divergence, a linear polarization, and a good coherence, and they can be produced with two temporally separated laser pulses. Generally, phase matching is of vital importance for CRS. However, our experiment was performed in the 42 mbar nitrogen gas, and the plasma density is also very low in the interaction region. In such experimental conditions, the coherence lengths of coherent Stokes and anti-Stokes Raman scattering are about two orders of magnitude longer than the plasma length. Thus, phase mismatch can be ignored in these processes. In the collinear pump-probe configuration, all these Raman signals are produced along the direction of the pump and probe pulses.

For gaining deeper insight into the above-mentioned picture, we examine the intensity of 428 nm emission, 439 nm Raman signal and the intensity change of the probe laser in the vicinity of 391 nm as a function of gas pressures. As shown in Fig. 6(a), at pressures below 20 mbar, the probe laser is amplified near 391 nm (i.e., the intensity change with the respect to its original signal is larger than zero), whereas the 428 nm radiation is extremely weak. As the gas pressure increases, the gain near 391 nm gradually fades out and eventually absorption of the signal at 391 nm is observed above the gas pressure of 25 mbar. Meanwhile, the 428 nm signal begins to grow rapidly around the same gas pressure. We further compare the evolution of the 428 nm signal and the 439 nm Raman signal with the increasing gas pressure. Figure 6(b) clearly illustrates that the 439 nm signal reaches its maximum near 10 mbar, which is one order of magnitude lower than the optimal pressure of the 428 nm radiation. When the gas pressure is above 50 mbar, the 439 nm signal almost remains unchanged. The comparative measurements indicate that the optimal gas pressure of the 428 nm radiation could mainly be determined by the population inversion rather than Raman scattering.

 

Fig. 6 (a) The intensity change of the probe pulse in the vicinity of 391 nm (blue crosses) and the intensity of the 428 nm emission (red dots) as a function of gas pressures. The positive/negative intensity change around 391 nm represents amplification/absorption. (b) The intensity of the 428 nm emission (red dots) and the 439 nm Raman signal (black circles) as a function of gas pressures. All data are fitted to guide the eyes.

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From the result of Fig. 6(a), we can see a strong competition and energy transfer between the emissions at 391 nm and 428 nm wavelengths. This should be attributed to that the two transitions share the same upper energy level. At the low pressures, the pump intensity is high enough to achieve the population inversion between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=0)$ states as well as that between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=1)$ states [17,18,28]. As compared to the transition to generate the 428 nm emission, the transition to generate the 391 nm emission has a larger Franck-Condon factor [32]. In addition, the probe pulse can be used as the seed for generating the amplified 391 nm radiation because its spectrum covers the 391 nm wavelength. For the amplified 428 nm signal, the seed is provided by Raman scattering of the probe laser, which is usually weaker than the 391 nm seed. For two reasons above, the probe pulse is amplified at 391 nm wavelength at low pressures. With the increase of gas pressure, the pump laser intensity decreases due to plasma defocusing, which makes difficult to establish the population inversion between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=0)$ states. Nevertheless, the population inversion between $B^2{\Sigma }_u^{+}(v\text{'}=0)$ and $X^2{\Sigma }_g^{+}(v=1)$ states can be realized at relatively lower intensities due to smaller population in $X^2{\Sigma }_g^{+}(v=1)$ state as compared to $X^2{\Sigma }_g^{+}(v=0)$ state [28]. In this case, the probe laser will be absorbed near 391 nm, while it is amplified at 428 nm. The absorption of 391 nm photons not only resonantly excites Raman scattering at 428 nm but also improves the population in $B^2{\Sigma }_u^{+}(v\text{'}=0)$ state. Therefore, the probe laser is converted as the 428 nm coherent emission more easily at high pressures.

5. Conclusion

To conclude, we report on the generation of coherent vibrational Raman scattering in tunnel-ionized nitrogen molecules. It is found that when the probe laser is resonant with the electronic transitions, Raman signals can be enhanced by at least two orders of magnitude. The 428 nm Raman signal can be effectively amplified in the population-inverted ${\text{N}}_2^{+}$ ions, resulting in the generation of ${\text{N}}_2^{+}$ lasing. Besides, our experiment also demonstrates that vibrational coherence of molecular ions can be excited by the 40 fs pump laser due to pulse shaping and instantaneous response of tunnel ionization to the laser field, which is in good agreement with the previous result reported in [30]. The remarkably high nonlinear optical efficiency provided by the electronically, vibrationally and rotationally coherent system is promising for nonlinear spectroscopic applications such as chemical sensing, environment diagnostic and surface science.