1. Introduction

Air lasing, referring as remote no-cavity/mirrorless optical amplification in air with air molecules as the gain media [1–11], is promising for a variety of atmospheric applications, such as Raman scattering of greenhouse gas and standoff spectroscopy [12,13]. With pumping of intense laser pulses, it has been proved that molecular oxygen and nitrogen in air can be prepared to be population-inverted for generation of forward and backward air lasing emissions [2–8,11]. Especially, strong-field-induced population inversion in N₂⁺ between the B²Σ_u⁺ and X²Σ_g⁺ states, which leads to forward narrow-bandwidth coherent emissions at 391 and 428 nm, has received considerable attention [4,14–21]. Several scenarios such as field-induced electron recollision [22], post-ionization multi-state coupling [23,24], alignment-induced transient inversion [25] have been proposed to interpret the mechanism of population-inverted N₂⁺, since it is contrast to the common understanding that most of nitrogen molecule ions prepared by multiple/tunnel ionization by intense laser fields are lying on their ground X²Σ_g⁺ state [26,27].

One impediment to settle down the controversy over the inversion mechanism in N₂⁺ is that the consensus on the mechanistic explanation of the ellipticity dependence of N₂⁺ lasing has not been totally reached yet. Earlier, it was demonstrated that under certain conditions the influences of the ellipticity of the pump laser on the two lasing lines at 391 and 428 nm are remarkably different, where the maximum intensities of the 391 and 428 nm lasing emissions appear at the ellipticities of ε ∼ 0.3 and ε ∼0, respectively [28]. Subsequently, by comparing the ellipticity effect of the pump laser on N₂⁺ lasing with that on high harmonic generation (HHG), it was suggested that field-induced inelastic electron recollision, a unique process in strong field ionization [29], could play a key role in promoting the population transfer from X²Σ_g⁺ to B²Σ_u⁺ leading to the N₂⁺ inversion [22]. However, the contribution of the recollision to the N₂⁺ inversion was recently reexamined by comparing the ellipticity effect of the pump pulses on the N₂⁺ gain with that on the HHG produced in room air and a thin gas jet [30], and also by observing the lasing emission variations in co-rotating and counter-rotating bi-circular two-color laser fields [31], both of which suggested that the electron recollision plays a minor contribution on the N₂⁺ lasing.

On the other hand, it was demonstrated that by modulating in time the polarization state of the pump pulse, the 391-nm lasing emission of N₂⁺ can be significantly enhanced by about two orders of magnitude [32], which was well interpreted based on the model of post-ionization multi-state coupling [23]. Furthermore, it was observed in a pump-probe scheme that, as the time delay between the 800-nm pump and probe pulses changes, the intensity of the 391-nm lasing oscillates with the period corresponding to the A²Π_u-X²Σ_g⁺ transition [20, 33], and that the A²Π_u-X²Σ_g⁺ transition is polarization-sensitive [34], showing that the light-induced coupling in N₂⁺ may play an important role in building up the population-inverted N₂⁺. Importantly, it was recently revealed that the different ellipticity dependences of the lasing emissions at 391 and 428 nm can be explained on the basis of the coupling model, in which the elliptical laser pulse induces different efficiencies of population transfer from the X²Σ_g⁺ (v”=0, 1) to A²Π_u states [35, 36].

In the present study, we observe an unexpected ellipticity dependence of the N₂⁺ lasing at 391 nm in an intense laser field with the ellipticity modulated by a 7-order quarter-wave plate (7-QWP), and show that the maximum of the 391-nm lasing intensity appears at ε ∼ 0.3, but with an asymmetric enhancement feature as we change the polarization of the pump laser pulse from linear to circular, and then to linear by continually rotating the 7-QWP from α=0 to 90°, where α is the angle between the polarization direction of the input laser field and the slow axis of the 7-QWP. Contrary to the case of the 7-QWP, a symmetric enhancement feature is observed as the laser polarization state is modified by a zero-order quarter-wave plate (0-QWP). We also find that for both the 7- and 0-QWP cases, the fluorescence from the B²Σ_u⁺ (v = 0) state varies with the same trend. We interpret the above observations based on the post-ionization multiple-state coupling model, in which the varying relative amplitudes of the ordinary (E_o) and extraordinary (E_e) electric components induced by the birefringence of the 7-QWP give rise to different efficiencies of population transfer from the X²Σ_g⁺ (v ”=0) to A²Π_u (v’=2) states.

2. Experimental setup

The experiments were conducted with a Ti:sapphire laser system (Spectra Physics, Spitfire ACE), operating at a central wavelength at 800 nm, a 40-fs pulse duration, an energy of 4 mJ and a repetition rate of 500 Hz. The laser beam was divided into two by a 90/10 beam splitter. As shown in Fig. 1, the stronger beam served as the pump to generate a plasma channel in the pure N₂ gas, and the weaker was frequency-doubled by a β-barium borate (β-BBO) crystal to generate the second harmonic light at 400 nm as the probe. The pump laser energy was controlled by an 800-nm half-wave plate (HWP) and a polarizer (P). The pump beam was then focused by a fused silica lens (L1: 40 cm focal length) into a stainless-steel chamber filled with pure N₂ gas at 8 mbar. It should be emphasized that in such focusing geometry and pressure condition, the self-focusing effects can be efficiently reduced, so that the beam focusing is mainly determined by the external focal lens. For the pump-probe experiment, the probe beam was combined with the pump beam by a dichroic mirror (DM2) with high reflectivity at 800 nm and high transmission at 400 nm. The temporal delay between the pump and the seed pulses was finely controlled by a motorized delay line with a temporal resolution of 10 fs. The polarization state of the pump pulse was modified by a 7- or 0-QWP inserted in the pump beam path, in which the relative amplitudes of the E_o and E_e components can be modified by rotating the angle α. The temporal delay Δt between the E_o and E_e components is calculated to be 19.3 and 0.67 fs respectively for 7-QWP and 0-QWP.

 

Fig. 1. Schematic of the experimental setup. HWP: half-wave plate; P: polarizer; DM: dichromic mirror; 0/7-QWP: 0 or 7-order quarter-wave plate; L1: fused silica lens with f=40 cm; L2: fused silica lens with f=30 cm; L3 and L4: fused silica lens with f=6 cm; HR1 and HR2: reflection mirror with high reflectivity at 400 nm.

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After passing through the chamber, the forwardly generated 391 nm light was collimated by a fused silica lens (L2: 30 cm focal length) and then separated from the pump and probe light by another dichroic mirror (DM3). The generated lasing signal at 391 nm was then focused by a fused silica lens (L3: 5.08 cm diameter, 6 cm focal length) on the slit of an Andor spectrometer (Shamrock 303i) coupled with an intensified charge coupled device (ICCD, iStar, Andor). The lasing signal was dispersed by a 1200 grooves/mm grating and recorded by the ICCD. On the other hand, the pump-laser-induced fluorescence of N₂ gas was collected by a focal lens (L4: 5.08 cm diameter, 6 cm focal length) from the side of the laser propagation direction, and then focused on the slit of the spectrometer with a 2f –2f imaging scheme. In all the experiments, the ICCD gate was opened for a period of 100 ns with the opening time of 5 ns before the laser pulse arrived at the interaction zone. The data for each measurement were accumulated over 2,500 laser shots.

3. Results and discussion

3.1 Measurement of self-seed 391-nm N₂⁺ lasing

We first performed the measurement of forwardly-propagating N₂⁺ lasing spectra induced only with the elliptically-modulated pump laser pulse. In this measurement, the pump laser energy was fixed at 1.5 mJ. The ellipticity of the pump laser is modulated by changing the angle α between the polarization direction of the input laser field and the slow axis of the 7-QWP, where α = 0 and 90° represent linear polarization. Figure 2(a) shows the lasing spectra in the spectral range of 388-393 nm obtained respectively at α = 18° (navy short dash dot) and α = 70° (magenta solid). It can be seen from Fig. 2(a) that the lasing lines at around 389.8 nm and 391.4 nm appear, which correspond respectively to the R- and P-branches of the B²Σ_u⁺ (v = 0) - X²Σ_g⁺ (v”=0) transition of N₂⁺. Interestingly, it can be noted that the lasing intensity obtained at α = 18° is much weaker than that obtained at α = 70°, although the pump laser pulses in the two angle cases have the nearly same ellipticity of ε ∼ 0.3.

 

Fig. 2. (a) Forward self-seeded N₂⁺ lasing spectra induced by the 7-QWP-modulated laser pulses at α = 18° (navy short dash dot) and α = 70° (magenta solid), respectively. (b) Self-seeded lasing intensities of N₂⁺ at 391 nm pumped by the 7-QWP-modulated laser pulses with different α angles.

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In order to see clearly the intensity dependence of the lasing signals on α, we show in Fig. 2(b) the lasing intensity of the P-branch at 391 nm as a function of α in the range from 0 to 90°. it can be seen from Fig. 2(b) that as α changes from 0° to 45° or from 90° to 45°, that is, the laser polarization state evolves from linear to circular, the 391-nm lasing first increases and then decreases with the optimized values appearing at α ∼15° and α ∼70°, respectively. The optimization of the lasing intensity induced by an elliptically-modulated intense laser field is consistent with previous measurement [36], where the lasing optimization is ascribed to efficient depletion of the population in the ground X²Σ_g⁺ (v’’=0) state through the optical coupling between the X²Σ_g⁺ (v”=0) and A²Π_u (v’=2) states that is sensitive to the relative amplitudes of the two polarization components of the elliptical field [36]. However, it can be noted from Fig. 2(b) that the two optimized values of the 391-nm lasing intensity are remarkably different, showing an asymmetric enhancement feature for the ranges of α ∼ 0°-45° and α ∼ 45°-90°, where the lasing intensity peaked at α = 15° is ∼4-5 times smaller than that at α = 70°.

3.2 Measurement of external seeded 391-nm N₂⁺ lasing

Since in the self-seeded lasing experiment performed only with the pump laser, the polarization and the intensity of the self-generated seed signal would also change as a function of α, we thus performed pump-probe measurements, in which the laser energy of the pump laser was lowered to 0.7 mJ to avoid self-seed generation, and an external 400-nm probe (seed) pulse was used, whose polarization direction is perpendicular to that of the input pump pulse. The laser energy of the probe pulse was measured to be about 100 nJ, and the delay time between the pump and the probe pulses was optimized for the 391-nm lasing generation. Since no self-focusing effect occurs in such a low pressure and pump laser energy, the change in the ellipticity of the pump pulse due to the birefringence of the plasma filament [37, 38] can be negligible. As an example, we show in Fig. 3(a) the forward externally-seeded lasing spectrum in the spectral range of 388.5-393.0 nm. For comparison, we also show in Fig. 3(a) the spectra obtained with only the pump (dot) or the probe (dash) laser pulses. It is obvious that the lasing emissions of N₂⁺ at 391 nm can be observed when both of the pump and seed pulses are adopted, while when only the pump or the probe pulses exist, 391-nm lasing emission cannot be generated.

 

Fig. 3. (a) Forward lasing emission spectrum by the linearly-polarized 800-nm pump and 400-nm seed laser pulses. The spectrum obtained only with the seed pulse (blue dashed) and that only with the pulse (red dotted) are also shown. (b) 391∼nm N₂⁺ lasing intensities by the pump–probe scheme in the 0-(blue square) and 7-(red circle) QWP-modulated laser fields as a function of the angel α, respectively.

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We then examine the intensity variation of the externally seeded 391-nm lasing as the angle α changes from 0° to 90°, as shown by the red circles in Fig. 3(b). Clearly, similar asymmetric ellipticity dependence as that shown in Fig. 2(b) is obtained, but the positions and ratio of the two maximums at around α∼ 10°-20° and α∼ 70°-80° are slightly different from those shown in Fig. 2(b), which can be ascribed to the influence of the self-generated seed that varies as the angle α changes. To explore the physical origin of the asymmetric enhancement of the lasing emissions of N₂⁺ at 391 nm, we also measured the lasing behavior as a function of the ellipticity of the pump laser modulated by 0-QWP. As shown by the blue rectangles in Fig. 3(b), in the case of 0-QWP, the maximum value of the 391-nm lasing intensity at α∼ 14° is almost the same as that at α∼ 76°, showing a symmetric enhancement feature.

It is known that the optical gain g can be expressed as g=σ×Δn, where σ denotes the stimulation emission cross section that is constant for a specific transition, and Δn indicates the population difference between the upper and lower states of the transition. Since the 391-nm lasing comes from the transition of B²Σ_u⁺ (v = 0) - X²Σ_g⁺ (v”=0), the dramatic enhancement of the lasing signal at α∼ 70°-80° for the 7-QWP case may thus result from the population increase in the B²Σ_u⁺ (v = 0) state and/or the population decrease in the X²Σ_g⁺ (v”=0) state. In addition, the difference in the two enhancement features of the 391-nm lasing shown in Fig. 3(b) may indicate that the populations on the B²Σ_u⁺ (v = 0) and X²Σ_g⁺ (v”=0) states change differently for the 0- and 7-QWP cases.

3.3 Measurement of 391-nm fluorescence

Since the fluorescence intensity is proportional to the population on the upper level of a transition [29], we examine the population variation in the B²Σ_u⁺ (v = 0) state by measuring the 391-nm fluorescence emissions from the side of the laser propagation direction, as shown in Fig. 1. As a result, we show in Fig. 4 the dependence of the normalized 391-nm fluorescence intensity on the angle α for 0-QWP (blue circle) and 7-QWP (red square), respectively. It can be observed from Fig. 4 that the fluorescence intensities induced by both the QWP-modulated laser fields first decease, and then increase as the angle α changes from 0° to 90° with the minimum at α = 45°. Clearly, the 391∼nm fluorescence intensity induced by 7-QWP modulated laser field at α∼ 10°-20° is comparable to that at α∼ 70°-80°, indicating that there is no obvious difference in the population of the B²Σ_u⁺ (v = 0) state for the two angle ranges. Moreover, in both the QWP cases, the variations of the population of N₂⁺ in the B²Σ_u⁺ (v = 0) state have the same trend as the angle α changes, indicating that the asymmetric enhancement is not due to the population change in the B²Σ_u⁺ (v = 0) state. Therefore, it can be concluded that the asymmetric enhancement mainly results from the different deletion efficiencies of population in the X²Σ_g⁺ (v”=0) state in the two angle ranges of α∼ 10°-20° and α∼ 70°-80°.

 

Fig. 4. Normalized fluorescence intensities of N₂⁺ at 391 nm measured as a function of the angle α with the ellipticity-modulated laser pulse by 0-QWP (blue circle) and 7-QWP (red square).

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3.4 Physical mechanisms

In order to understand how the population depletion of the X²Σ_g⁺ (v”=0) state proceeds differently in the two enhancement angle ranges, we analyze the electric field, E_mod (t), of the ellipticity-modulated laser field, whose expression can be written as

Next, we interpret the physical mechanisms responsible for the asymmetric dependence of the population in the ground X²Σ_g⁺ (v”=0) state of N₂⁺ on the angle α, on a basis of the post-ionization multi-state coupling model [36], as presented in Fig. 5(e). In this model, post-ionization coupling is regarded as playing a key role in establishing the population inversion of N₂⁺ between the B²Σ_u⁺ (v = 0) - X²Σ_g⁺(v”=0), in which the resultant N₂⁺ ions lying on the ground state X²Σ_g⁺(v”=0) can be transferred to the intermediate state A²Π_u. Therefore, it is expected that the intensity of N₂⁺ lasing is sensitive to both the amplitude and the timing of the ionization and the coupling fields. To simplify, it is reasonably assumed that the ionization is induced by the stronger component in the elliptically-modulated laser fields (the pink region). Meanwhile, the weaker component with the polarization direction perpendicular to that of the stronger pulse induce the population transfer from X²Σ_g⁺ (v”=0) to A²Π_u (v’=2) through the perpendicular transition [33,34]. Because the coupling is a post-ionization process, taking place after the ionization, we thus mark the coupling contributions from the weaker component by the green region in Figs. 5(a)-(d), respectively.

 

Fig. 5. Schematic diagrams of the ellipticity-modulated pump laser pulse in the angle range of (a) 0°<α<45° and (b) 45°<α<90° for 7-QWP, and (c) 0°<α<45°and (d) 45°<α<90° for 0-QWP, respectively. (e) Energy diagram of the N₂⁺ lasing emissions generated by the post-ionization light-induced coupling.

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In the 7-QWP case shown in Figs. 5(a) and 5(b), it can be seen that when 0°<α<45°, the E_e component is stronger, and thus serves as the ionization pulse. The E_o component plays a role in coupling of the X²Σ_g⁺ (v”=0) and A²Π_u (v’=2) states, but only a small portion in the rear part of the E_o component (filled green area) makes a contribution to the coupling because its peak position is 19.3 fs prior to that of the ionization pulse in time. While when 45°<α<90°, the E_e component is weaker, and thus serves as the coupling pulse. The most part of the delayed coupling pulse (filled green area) can contribute to the coupling of the X²Σ_g⁺ (v”=0) and A²Π_u (v’=2) states. As a consequence, the population depletion in the X²Σ_g⁺ (v”=0) state would be more significantly enhanced for 45°<α<90° as compared with that for 0°<α<45°. Therefore, the asymmetric enhancement dependence of the N₂⁺ lasing in the 7-QWP case can be attributed to the difference in the contributions of the coupling pulse to the population coupling between the X²Σ_g⁺ (v”=0) and A²Π_u (v’=2) states in the two angle ranges.

In the 0-QWP case shown in Figs. 5(c) and 5(d), the temporal delay between the E_o and E_e components is about 0.67 fs, which is much smaller than the input pulse duration (40 fs). Therefore, in the two angle ranges of 0°<α<45° and 45°<α<90°, the 0-QWP-modulated laser fields with the same ellipticity provide the almost identical ionization and coupling efficiencies, resulting in the symmetric enhancement of the 391-nm lasing shown in Fig. 3(b).

Furthermore, it can be observed from Fig. 3(b) that when 0°<α<45° the enhancement of the lasing intensity induced by the 7-QWP is smaller than that obtained by 0-QWP, but when 45°<α<90° it is reversed. This could be explained based on the above interpretation. It can be seen in Fig. 4 that when 0°<α<45° the efficient portion of the coupling pulse (filled green area) in Fig. 5(c) (0-QWP) is larger than that in Fig. 5(a) (7-QWP), indicating that more efficient population transfer from X²Σ_g⁺ (v”=0) to A²Π_u (v’=2) proceeds in the case of 0-QWP. Similarly, when 45°<α<90° the more significant enhancement in Fig. 5(b) than that in Fig. 5(d) can also be attributed to the more efficient population depletion of X²Σ_g⁺ (v”=0) by the coupling laser pulse [36].

4. Summary

In summary, we have demonstrated the asymmetric enhancement of the N₂⁺ lasing on the B²Σ_u⁺ (v = 0)-X²Σ_g⁺ (v” = 0) transition induced by an intense and birefringence-modulating elliptical laser fields. We found that the maximums of the 391-nm lasing intensity appear at the ellipticity of ε ∼ 0.3, but the intensity enhancement in the angle ranges of 0°<α<45° is about 4-5 times smaller than that in the angle range of 45°<α<90°. By analyzing the modulated light fields, it was revealed that relative amplitudes of the E_o and E_e electric components induced by the birefringence of the 7-QWP are reverse in the two cases, so that when 0°<α<45° the E_e component induces the ionization of N₂, while when 45°<α<90° the E_o electric components serves as the ionization pulse. Furthermore, the temporal separation between the E_o and E_e components induces different time delay between the ionization and coupling pulses, giving rise to different depletion efficiencies of population in the X²Σ_g⁺ (v” = 0) state through the coupling between the X²Σ_g⁺ (v” = 0) and A²Π_u (v’=2) states.