When a high-power femtosecond laser pulse propagates through a transparent medium such as air or glass, it can overcome natural diffraction effects by maintaining a dynamic balance between plasma defocusing and Kerr self-focusing. This enables the laser to propagate over long distances, reaching up to 10 km [1–3]. Since Braun and his colleagues first reported the filamentation of femtosecond laser pulses in the air in 1995, it has garnered significant attention due to its diverse applications [4,5], including intense terahertz emission [6], high-order harmonic generation [7], artificial rainfall [8,9], remote sensing of atmospheric composition [4,10,11], air lasing [12], and others [13–15]. Within the filament, the light intensity can reach up to $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Part of the molecules and atoms in the air are ionized [16], and part of the small molecular fragments enter an excited state, both of which can emit clean fluorescence [17]. Fluorescence measurements along the filament propagation path provide valuable information, including plasma density and electron temperature, while offering insights into the excitation and ionization processes [18–20]. Moreover, remote sensing of atmospheric pollutants can be achieved by detecting the backward-propagating fluorescence produced by plasma generated during filamentation.

However, femtosecond laser filamentation is a highly nonlinear process, significantly influenced by the polarization of the laser. For example, experimental evidence shows that the multiphoton ionization rate and the conversion efficiency of supercontinuum generation during the filamentation process depend on the laser polarization [21]. Self-focusing and pulse splitting can be controlled by altering the polarization. Adjusting the polarization state can also suppress the generation of multiple filaments [22]. Observing the polarization-dependent characteristics of ultraviolet (UV) fluorescence emitted by nitrogen molecules provides crucial insights into the underlying fluorescence mechanism.

There have been numerous studies discussing the fluorescence mechanism of neutral nitrogen molecules excited by laser filamentation at short distances. Xu et al. proposed a dissociation-recombination model [23], in which the fluorescence intensity of nitrogen molecules was found to be proportional to the square of the pressure in filamentation experiments with a focal length of 1 m. In contrast, Arnold et al. argued that the contribution of the dissociation mechanism is negligible [24], with intersystem crossing being the primary pathway for forming the ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ state. This explanation is based on varying the gas mixture ratio and laser power in filamentation experiments with a 0.5 m focal length. The electron collision model proposed by Liu et al. is currently the most widely accepted explanation [25]. They studied the effect of laser polarization on the excitation of neutral nitrogen molecule fluorescence generated by filamentation at a distance of 1 m. Below the filamentation threshold (250 μJ), linearly polarized pulses generate higher fluorescence. When using a circularly polarized laser, the fluorescence intensity matches that generated by linearly polarized pulses with an energy of $E_{\mathrm{in}}=600\mathrm{\mu }\mathrm{J}$ and becomes dominant at even higher laser energies, ranging $E_{\mathrm{in}}=2.7–10\mathrm{mJ}$. They attributed this phenomenon to the onset of new impact excitation channels from energetic electrons produced by circularly polarized laser pulses above a threshold intensity. Further analysis revealed that free electrons excited by linearly polarized laser pulses undergo alternating acceleration and deceleration during each optical cycle, resulting in relatively low kinetic energy (below 1 eV). In contrast, circularly polarized laser pulses consistently accelerate electrons away from the molecular ion, generating electrons with higher kinetic energy (around 14.6 eV). Subsequently, using a laser-induced fluorescence depletion technique, they measured the formation dynamics of these excited neutral nitrogen molecules with femtosecond time resolution. Their findings ultimately refuted the previously proposed dissociation-recombination model.

Jin and his team reported similar results in their short-distance filamentation experiments [26], showing that the effect of laser polarization on the fluorescence spectra of nitrogen was nearly uniform across different wavelengths. Furthermore, when using a lens with a shorter focal length ($f=40\mathrm{cm}$) for filamentation, the energy threshold is much lower ($E_{\mathrm{in}}=0.8\mathrm{mJ}$). However, after refuting the dissociation-recombination model, it remains unexplained why linearly polarized laser pulses with lower energy produce stronger UV fluorescence from neutral nitrogen molecules. More importantly, to the best of our knowledge, no experimental studies or analyses have been reported on the fluorescence mechanism of neutral nitrogen molecules during long-distance filamentation.

In this work, we investigate the fluorescence mechanism in long-distance filamentation (30 m) by analyzing the polarization-dependent fluorescence of neutral nitrogen molecules. Our experimental results showed that the total path-integrated fluorescence intensity for linear polarization is about 7 times that excited by circular polarization. Based on these findings, we conclude that during long-distance filamentation, the dissociation-recombination mechanism is the most likely pathway for the formation of ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ excited states. In addition, linearly polarized laser pulses have a larger nonlinear coefficient (1.5 times) compared to circularly polarized pulses, resulting in a lower self-focusing power threshold during filamentation. This study distinguishes the dissociation mechanisms in short- and long-focal-length filamentation. Short-focal-length filamentation, with higher local intensities and stronger ionization effects, supports electron collision models, while long-focal-length filamentation, limited by the clamping effect, favors photoionization followed by dissociation-recombination. Our results highlight the relevance of the dissociation-recombination model in long-focal-length filamentation.

The experiments were performed using a Ti:sapphire femtosecond laser system (Spectra Physics), as illustrated in Fig. 1. This system produces 50 fs pulses at a central wavelength of 800 nm, with a maximum single pulse energy of 7 mJ and a repetition frequency of 1000 Hz. The beam diameter is approximately 13 mm at the $1/{\mathrm{e}}^2$ level of intensity. Femtosecond laser filamentation occurs at a distance of 30 m through an off-axis reflection system, with the filament length reaching approximately 26–48 cm. The initiation of filamentation can be precisely controlled by adjusting the relative distance between the concave lens ($f=-150\mathrm{mm}$) and the concave mirror ($f=2000\mathrm{mm}$). The polarization state of the laser pulse is changed by rotating a quarter-wave plate (QWP), as illustrated in Fig. 1.

The fluorescence emission at 337 nm, produced by filament-excited neutral nitrogen molecules, was initially collected by a laterally placed 2-inch convex lens ($f=50\mathrm{mm}$), focused, and then channeled into a monochromator (WGD-100, Gang Dong Sci. & Tech. Co., Ltd.) and a photomultiplier tube (PMT, Hamamatsu, H11902). To enhance fluorescence collection efficiency, the entrance of the monochromator was aligned parallel to the filament. The fluorescence collection apparatus can be moved parallel to the filament path using a rail system, allowing the fluorescence intensity at specific points and the cumulative fluorescence intensity along the entire filament path to be measured. Furthermore, to improve the signal-to-noise ratio, each data point in this study represents the average of 10 sets of 128 laser shots.

In the experiment, filament length was precisely characterized by observing nitrogen fluorescence signals using imaging techniques such as a CCD camera. Typically, the beginning of the filament is defined as the position where the fluorescence intensity first exceeds $3\sigma$ of the background signal and begins to rise; the ending of the filament is defined as the position where the fluorescence intensity decreases to approximately $3\sigma$ of the background signal [27]. The distance between the starting and ending points defines the filament length. Since nitrogen fluorescence intensity is closely related to the local intensity of the filament and the ionization rate, this method better reflects the ionization rate changes induced by the filament. Similarly, ultrasonic signals generated by inelastic collisions between the filament and the surrounding medium can also precisely characterize filament length [28], with accuracy comparable to fluorescence-based measurements. The intensity of the acoustic signal is proportional to the laser energy absorbed by the filament, and ultrasonic measurements are faster and more convenient. Therefore, the experimental section exploring filament length variations uses acoustic signals for characterization.

To minimize the potential influence of polarization sensitivity from reflective and transmissive elements in the optical path, we measured the variation in laser power with polarization before filamentation. The relative standard deviation (RSD) of the measured power was 1.456%, indicating that its impact on the subsequent experimental results is negligible.

Figures 2(a) and 2(b) illustrate the evolution of the 337 nm fluorescence line intensity along the propagation axis, as the neutral nitrogen molecules are excited by circularly, elliptically, and linearly polarized laser pulses with the same energy. In Fig. 2(a), the fluorescence intensity along the filament propagation axis initially increases and then decreases, a pattern consistent across all polarization states. The polarization of the laser pulse is gradually changed by rotating the QWP, transitioning from horizontal linear polarization to left-handed circular polarization, and finally to vertical linear polarization. The ratio of the fluorescence peak intensities between the linear and circular polarizations is approximately 3.6.

In Fig. 2(b), the laser polarization transitions from vertical linear polarization to right-handed circular polarization and finally returns to horizontal linear polarization. The fluorescence peak intensity under horizontal linear polarization is about 3.5 times that under right-handed circular polarization. It can be observed that the polarization state of the laser does affect the fluorescence intensity of nitrogen, with linear polarization resulting in significantly stronger fluorescence than circular polarization. According to physical principles, nitrogen molecules are symmetric, achiral, and isotropic [29]; thus, the fluorescence intensity under left- and right-handed circular polarization should theoretically show no significant difference. However, in Figs. 2(a) and 2(b), the observed fluorescence intensity differences at 45° and 135° polarization angles arise primarily from the turbulence effect during long-distance filamentation (approximately 11%) [30]. During long-distance filamentation, due to the experimental environment being difficult to completely enclose, the turbulence effect leads to asymmetric distribution of the filament, which in turn affects the spatial distribution and intensity stability of the fluorescence. Based on the fluctuation range represented by the error bars from multiple actual fluorescence signal measurements in the experiment shown in Fig. 2, we set an error threshold of 0.01. When the fluorescence intensity differences are below this threshold, their impact on the analysis of the fluorescence mechanism can be ignored.

Filament length refers to the actual distance over which the femtosecond laser pulse propagates through the medium and forms a stable plasma channel. Although the maximum filament length measured in the experiment was 48 cm, for the purpose of subsequent comparative analysis of different models and to minimize the influence of signal fluctuations, we selected a 20 cm segment from the central region of the filament for analysis. Within this region, points “I,” “II,” and “III” are defined as the positions where the fluorescence intensity begins to rise significantly (approximately 30% of the maximum intensity), reaches its peak, and decreases significantly, respectively. Next, we analyzed the fluorescence intensity and polarization state at different positions along the filament. Points “I,” “II,” and “III” correspond to positions at 30 m, 30.1 m, and 30.2 m, respectively. Figures 2(c)–2(e) show the variation in nitrogen fluorescence intensity with the polarization state at these positions. At points “I,” “II,” and “III,” the fluorescence intensity under linear polarization is approximately 2.7, 3.6, and 2.4 times that under circular polarization, respectively.

There is currently some debate on the fluorescence mechanism of neutral nitrogen molecules excited by laser filamentation at the 337 nm spectral line [31]. The primary models proposed to explain this mechanism include the electron collision model, the dissociation-recombination model, and the collision-assisted intersystem crossing model [32].

The collision-assisted intersystem crossing model relies on a high concentration of heavy atoms with resonant energy levels to facilitate energy exchange [24]. However, such conditions are absent in the air, rendering this model inapplicable to our experimental scenario.

According to the electron collision model [25,33], circularly polarized laser pulses are expected to produce stronger fluorescence from neutral nitrogen molecules. This is attributed to the higher-energy electrons produced by circular polarization, which can activate new impact excitation channels at laser intensities above the threshold. However, this prediction is inconsistent with our experimental observations, where linearly polarized pulses resulted in significantly stronger fluorescence from neutral nitrogen molecules.

In the dissociation-recombination model [23], the excitation of ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ depends on the concentrations of the intermediate products ${\mathrm{N}}_4^{+}$ and ${\mathrm{N}}_2^{+}$. Since the ionization rates of nitrogen molecules vary between linearly and circularly polarized laser pulses, with linear polarization being more effective, this aligns with the experimental observations from long-distance filamentation in our study.

A comparative discussion was conducted to assess whether the electron collision model or the dissociation-recombination model provides a better explanation for the fluorescence mechanism observed in long-distance filamentation. According to the electron collisional excitation model, the excited state of ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ is produced when a neutral nitrogen molecule is impacted by a high-energy electron. This process can be expressed as follows: $${\mathrm{e}}^{-}+{\mathrm{N}}_2({\mathrm{X}}^1{\mathrm{\Sigma }}_{\mathrm{g}}^{+})\to {\mathrm{e}}^{-}+{\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}}),$$(1a)$${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})\to {\mathrm{N}}_2{(\mathrm{B}}^3{\mathrm{\Pi }}_{\mathrm{g}})+h\nu .$$(1b)

From Eqs. (1a) and (1b), it is evident that the intensity of nitrogen fluorescence excited by filamentation is proportional to the electron collision excitation rate. A significant difference between linear and circular laser polarization in gas plasma generation lies in the kinetic energy of free electrons remaining after the intense laser field passes. According to the literature [34], for linearly polarized laser pulses, most electrons retain energies below 1 eV. By contrast, under circular polarization, the electron kinetic energy is approximately 14.6 eV, exhibiting a nearly monoenergetic distribution. Based on the electron collision model, the fluorescence intensity of neutral nitrogen molecules excited by linearly polarized laser pulses should be much lower than that excited by circularly polarized laser pulses. However, this prediction is clearly inconsistent with our experimental results.

Unlike tight focusing ($I>1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) or multiple filaments, which are easily achieved in short-range filamentation (less than 1 m), long-distance filamentation is limited by the clamping intensity ($I=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$). Under intensity clamping, the ponderomotive energy ($U_p$) in the core of the filament formed by linearly polarized laser pulses is approximately 3 eV [23]. Even when considering that, in linear polarization, electrons may be accelerated by the laser field and return to their initial position for a secondary collision with the parent ions, the maximum kinetic energy gained, around $3.2U_p$ (9.6 eV) [23], is still insufficient to reach the threshold for collisional ionization. In the core of the filament formed by circularly polarized laser pulses, the ponderomotive energy ($U_p$) is approximately 6 eV [35]. However, since circularly polarized laser pulses continuously accelerate electrons, secondary collisions are unlikely to occur, and the energy gained remains insufficient to reach the threshold for collisional ionization. The threshold electron energy required for effective collision excitation is approximately 11 eV [34]. Therefore, the electron collision model is not suitable for explaining our experimental observations.

Next, we consider the formation process of the ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ excited state within the framework of the dissociation-recombination model. Xu et al. pointed out that the ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ molecules might be generated through the following steps [23]: $${\mathrm{N}}_2+{\mathrm{N}}_2^{+}\to {\mathrm{N}}_4^{+},$$(2a)$${\mathrm{N}}_4^{+}+{\mathrm{e}}^{-}\to {\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})+{\mathrm{N}}_2.$$(2b)

In a strong laser field, such as during filamentation, nitrogen molecules (${\mathrm{N}}_2$) undergo ionization to form positively charged (${\mathrm{N}}_2^{+}$) and then collide with nitrogen molecules to form an intermediate state, the tetranitrogen ion (${\mathrm{N}}_4^{+}$). When ${\mathrm{N}}_4^{+}$ recombines with electrons, it produces excited molecules ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$, which emit fluorescence. In the dissociation-recombination model, the final density of ${\mathrm{N}}_2({\mathrm{C}}^3{\mathrm{\Pi }}_{\mathrm{u}})$ molecules depends on the density of ${\mathrm{N}}_2^{+}$. Since the tunneling ionization rate is exponentially related to the instantaneous electric field of the laser pulse, reducing the maximum electric field greatly suppresses ionization as the laser’s polarization state changes from linear to circular [36,37]. To further investigate the differences in ionization rates between the two polarization states in detail, we calculated the single ionization rate of nitrogen molecules using the Ammosov–Delone–Krainov (ADK) tunneling model [38,39], with the expression as follows: $$P_{\mathrm{ADK}}=\frac{2^{2n_l}}{\omega }{\left(\frac{I}{U_p}\right)}^{n_l+1/2}\exp [-\frac{2}{3}\frac{{(2I_p)}^{3/2}}{U_p}],$$(3)where $P_{\mathrm{ADK}}$ is the probability of electrons tunneling away from the molecule, $I$ represents the laser intensity, $I_p$ denotes the ionization potential, $n_l$ is the principal quantum number, $\omega$ denotes the frequency of the laser, and $U_p$ is the ponderomotive potential of the electron in a laser field.

It should be noted that the formation of nitrogen molecules may vary depending on the filamentation conditions. In short-focal-length filamentation, the strong local electric field promotes electron collisions and ablation, with collision ionization explaining the nitrogen molecule formation. In long-focal-length filamentation, where the beam intensity is clamped, photoionization and the dissociation-recombination model are more likely to dominate the fluorescence mechanism. The ionization rate calculation results in Fig. 3(a) show that, under long-focal-length conditions, the polarization-induced differences in ionization rates match the variations in nitrogen fluorescence intensity, supporting the validity of the dissociation-recombination model in these conditions.

Additionally, the variation in filament length caused by laser polarization also affects the fluorescence intensity, a phenomenon that becomes more pronounced at long distances. The polarization state of the laser pulse affects the nonlinear refractive index, $n_2$, thus influencing the critical power for self-focusing in air. For linearly polarized light, the nonlinear refraction coefficient is given by the following expression [40]: $$n_{2,\mathrm{liner}}=\frac{3}{4}\frac{\mathrm{Re}({\chi }_{\mathrm{xxxx}}^{(3)})}{{\epsilon }_{0}c{n}_0^2}.$$(4)

In Eq. (4), $\mathrm{Re}\left({\chi }_{\mathrm{xxxx}}^{(3)}\right)$ represents the real part of the third-order nonlinear susceptibility, ${\epsilon }_0$ is the permittivity of free space, $c$ is the speed of light, and $n_0$ is the linear refractive index of the medium. For circularly polarized light, the nonlinear refraction coefficient is given by $$n_{2,\mathrm{circular}}=\frac{2}{4}\frac{\mathrm{Re}({\chi }_{\mathrm{xxxx}}^{(3)})}{{\epsilon }_{0}c{n}_0^2}.$$(5)

The equation describing external focusing and self-focusing of the femtosecond laser is given as follows [41]: $$1/f^{\prime }=1/f+1/z_f,$$(6)$$z_f=\frac{0.36L_{\mathrm{DF}}}{\sqrt{[(P_{\mathrm{in}}/P_{\mathrm{cr}}{)}^{1/2}-0.852{]}^2-0.0219}},$$(7)$$P_{\mathrm{cr}}=\frac{3.77{\lambda }^2}{8\pi n_2{n}_0}.$$(8)

In these equations, $f^{\prime }$ represents the effective focal length, $f$ represents the focal length of the external focusing lens, $z_f$ represents the self-focusing distance of the collimated Gaussian beam, $L_{\mathrm{DF}}$ represents the Rayleigh length of the beam, $\lambda$ denotes the laser wavelength, $P_{\mathrm{in}}$ represents the input power, and $P_{\mathrm{cr}}$ represents the critical power for filamentation.

When the laser pulse exceeds the critical power and further increases, the focal spot of the laser beam shifts toward the external focusing lens, leading to an increase in the filament length. The calculation results in Fig. 3(b) indicate that, in air, the self-focusing critical power for a linearly polarized laser is approximately 9.6 GW, while that for a circularly polarized laser is around 14.4 GW. The lower self-focusing critical power causes the filamentation onset for linearly polarized lasers to occur earlier at the same energy, resulting in a filament length approximately 2.1 times that of induced circular polarization.

According to the research by Luo et al., the relationship between the nitrogen fluorescence signal intensity ($I$) and the laser filament length ($L$) can be expressed as [13] $$I\propto P={\int }_0^L{P}_s{e}^{gl}\mathrm{d}l=\frac{P_s}{g}\times gL=P\times L,$$(9)where $P_s$ denotes the spontaneous emission power per unit length, $L$ represents the length of the filament, and $g$ is the effective gain coefficient. Visible fluorescence emitted during femtosecond laser filamentation in air serves as a key diagnostic tool for identifying the spatial extent of the filament and has been extensively validated through both theoretical and experimental reports [10,42]. Typically, filament length refers to the actual length over which a laser pulse propagates in the medium and generates a stable plasma, while fluorescence length refers to the effective length along the filament propagation axis over which detectable ${\mathrm{N}}_2$ fluorescence is emitted. Since our measurements involve lateral collection, we do not account for the amplification gain from spontaneous emission observed in other experiments [13], such as forward or backward spontaneous radiation gain from the filament. Therefore, in this study, filament length is effectively equivalent to ${\mathrm{N}}_2$ fluorescence length. Consequently, linearly polarized pulses produce a more intense fluorescence signal at 337 nm due to longer filament length. To experimentally verify this relationship, we used ultrasound signals to characterize the filament length [19,43].

The starting point, length, and axial relative intensity distribution of filamentation were recorded in detail by using a manual displacement platform to move the ultrasonic probe along the axis of the filament in 2 cm steps. After simple data processing, we can obtain the spatial information of the complete filamentation. By comparing the axial relative intensity with three times the standard deviation ($3\sigma$) of the blank sample, the starting position and length of filamentation can be determined from Fig. 4(a).

By rotating the QWP in increments of 2.5° from 0° to 180°, we recorded the variations in both the starting point and the length of filamentation with changes in the polarization state, as shown in Figs. 4(b) and 4(c). As the QWP rotation angle increases (resulting in greater ellipticity), the starting point of the filament gradually moves backward, while the filament length decreases correspondingly. In the experiment, with the same pulse energy, the filament length at 30 m was measured to be 48 cm for the linearly polarized laser and 26 cm for the circularly polarized laser, yielding a filament length ratio of 1.85:1.

We systematically analyzed the effects of polarization states on both the ionization rate of nitrogen and the filament length. To comprehensively account for these factors, the total expected fluorescence intensity was calculated by multiplying the ionization rate factor with the corresponding filament length factor for each polarization state. The resulting theoretical values, represented by the blue curve in Fig. 4(d), illustrate the predicted variation of fluorescence intensity with polarization.

Experimentally, the cumulative fluorescence intensity collected along the filament path for different polarization states is shown as red spheres in Fig. 4(d), with longer filament length resulting in greater cumulative fluorescence intensity. It should be noted that due to the size limitation of the lateral collection lens, to obtain the cumulative fluorescence intensity along the entire filament path, the fluorescence collection apparatus was moved parallel to the filament path using a rail system, with the movement range equal to the maximum length of the filament. The comparison reveals that the trends and magnitudes of the theoretical predictions closely match the experimental measurements. This agreement validates the model used to describe the combined effects of the ionization rate and filament length on the fluorescence intensity.

This study investigates the effect of laser polarization on nitrogen fluorescence during femtosecond laser filamentation over a long distance of 30 m and analyzes the relevant theoretical models. The experiment shows that, at long distances, nitrogen under a linearly polarized laser emits fluorescence at 337 nm with an intensity approximately 7 times as strong as that under circular polarization, which contrasts with previous short-distance results where circular polarization produced stronger fluorescence. Short-focal-length filamentation often involves higher local intensities and stronger ionization effects, leading to noticeable ablation, and the electron collision model has been widely used to explain nitrogen excitation in such cases. Under long-focal-length filamentation, the laser intensity is limited by the clamping effect, and the electron energy is generally too low to induce efficient collisional ionization. Instead, fluorescence is more likely generated through photoionization followed by a dissociation-recombination process. The coexistence of different models under different filamentation conditions provides a more comprehensive understanding of the nitrogen fluorescence mechanism and further enriches our understanding of filament dynamics.