## Abstract

We investigate experimentally spatiotemporal characteristics of fluorescence emission from fs-laser-induced filaments in air. Emissions accompanying the transitions of N_{2} (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}-{\mathrm{B}}^{3}{\mathrm{\Pi}}_{\mathrm{g}}$) and N${}_{2}^{+}$ (${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}-{\mathrm{X}}^{2}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$) are dominant. The decay dynamics of fluorescence from different radial positions and longitudinal sections of a filament column are obtained along with high resolution spectra. A decay curve contains two exponential components: a fast one (with a decay time constant ∼10s ps), and a slow one (∼sub-ns). The lifetime of the N${}_{2}$ fluorescence is about three orders shorter than its spontaneous emission lifetime, indicating that most of the N${}_{2}$ molecules in the excited state (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) are de-excited through collision. Different de-excitation mechanisms of N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules contributing to fluorescence decay constants, e.g., the e${}^{-}-{\mathrm{N}}_{2}$, N${}_{2}-{\mathrm{N}}_{2}$, and O${}_{2}-{\mathrm{N}}_{2}$ collisions, are elucidated. We analyze the variations of decay constants together with corresponding fluorescence intensities, and obtain temperature distributions by fitting band spectra of N${}_{2}$ molecules and N${}_{2}^{+}$ ions with a synthetic spectral model. Our results suggest that the fast and slow decay processes originate from the e${}^{-}-{\mathrm{N}}_{2}$ and O${}_{2}-{\mathrm{N}}_{2}$ collisions, respectively.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Intense ultrashort laser pulses propagating in air can create filaments, as a result of dynamic balance between non-linear Kerr self-focusing and defocusing induced by air ionization. Fs-laser filamentation has been of interest because it involves complex nonlinear processes such as nonsequential double ionization of molecules, high order harmonic generation [1], THz generation [2], and superradiance [3, 4]. A variety of potential applications of laser filamentation have been proposed, such as detection of explosives and isotopes at standoff distances [5], remote sensing [6, 7] and remote material processing [8]. Filament luminescence features clear molecular band spectra from excited N${}_{2}$ molecules and N${}_{2}^{+}$ ions [9, 10]. Several recent studies showed that population inversion of N${}_{2}$ and N${}_{2}^{+}$ can be achieved in filaments, and optical gain can be obtained [11, 12]. Air lasing was demonstrated with filaments by injection of additional seeding fs-laser beam with appropriate wavelengths [11, 13]. Backward lasing was also realized, which advances the potential applications of filaments in remote atmospheric sensing [7, 14, 15]. Lasing duration is related to optical gain, and the duration of optical gain is pressure-dependent [16, 17]. In air, the duration of population inversion is quite short [11]. De-excitation of molecules in the inversely populated energy levels destroys optical gain and lasing becomes unsupported. The de-excitation mechanism of excited molecules and ions in filaments is unclear. However, the population of molecules or ions in excited state can be investigated via time-resolved fluorescent emission [18].

The dynamics of N${}_{2}$ and N${}_{2}^{+}$ fluorescence from fs-laser induced plasma in air was studied initially by Martin $et\text{}al$. [19]. A streak camera was used to record the plasma luminescence without spectral resolution. Plasma emission is initially dominated by molecular lines for tens of ps (∼60 ps), followed by continuum emission and atomic lines. Ivanov $et\text{}al$. [20] measured the lifetime of N${}_{2}^{+}$ and N${}_{2}$ fluorescence of filaments in air by focusing a fs-laser beam with different numerical apertures (NA), and found that molecular lines only appear at NA below 0.03. For NA < 0.01, the characteristic luminescence decay time is 18 ps for N${}_{2}^{+}$, and 27 ps for N${}_{2}$. The decay time of ions and molecules can be extended to sub-ns when gas pressure is several millibars [18]. Wang $et\text{}al$. [21] measured the fluorescence lifetime of N${}_{2}^{+}$ at different gas pressures by using a photoelectric multiplier with a temporal resolution of 10 ns, and revealed that the fluorescence decay is slower as pressure decreases. At $1.9\times {10}^{-5}$ mbar, the fluorescence lifetime of N${}_{2}^{+}$ is close to its spontaneous emission lifetime (∼60 ns). Lei $et\text{}al$.[18] proposed that fast fluorescent quenching of N${}_{2}^{+}$ originates from the collision between free electrons and excited N${}_{2}^{+}$ molecules in filaments created in a low pressure pure nitrogen gas, which leads to population redistribution in electronic states (${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}$). In air, the mechanisms governing the fluorescence dynamics in fs-laser filamentation are more complicated given its higher molecule density and multi-chemical-components involved.

In this work, we experimentally investigate fluorescence decay at different radial and longitudinal sections of fs-laser induced filaments with high temporal resolutions. Band (0,0) spectra of the N${}_{2}{\text{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}-{\mathrm{B}}^{3}{\mathrm{\Pi}}_{\mathrm{g}}$ and N${}_{2}^{+}{\text{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}-{\mathrm{X}}^{2}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$ systems are recorded, and rotational temperature is calculated via fitting to the molecular spectra with a synthetic spectral model. Through an analysis of fluorescence intensity distribution, fluorescence quenching constants across the longitudinal cross-section of the filaments, the temperature distribution, and the dominant quenching mechanism governing the fluorescence decay, are obtained.

## 2. Experimental setup

The experimental system is shown in Fig. 1. Experiments are performed with a Ti:sapphire laser amplifier system (Coherent) which delivers linearly polarized laser pulses with a central wavelength of 800 nm and 35 fs pulse duration at 1 kHz and 2.5 mJ/pulse as shown in Fig. 1. A short filament is generated by focusing the fs-laser with a plano-convex lens with a focal length of 25 cm. The fluorescence of excited N${}_{2}$ and N${}_{2}^{+}$ molecules is measured with a noncommercial streak camera. The temporal resolution is about 10 ps, as measured via accumulation of ${10}^{4}$ laser pulses.

The filament image is magnified by a 10× UV objective lens with a focal depth of 3.5 μm. A fiber with 100 μm diameter is placed at the image plane for fluorescence signal collection, and the collected light is coupled into a spectrometer (Princeton Instruments) and then the temporally resolved spectrum is recorded by a streak camera. The streak camera is triggered by scattered laser light detected by a photo diode placed at the upstream of the laser beam. Laser pulses are allowed to propagate about 10 m before being focused, in order to compensate the inherent delay of the streak camera. Radially and longitudinally resolved measurements of the filament are achieved by varying the position of fiber on the image plane and the focusing lens along the laser beam propagation direction, respectively. The resolution of spectra detected by the streak camera only is about 0.2 nm. A spectrometer (Princeton Instruments) with a grating of 3600 g/mm is added to achieve higher spectral resolution (∼0.05 nm), which enables the identification of the rotational structure of emission spectra from N${}_{2}$ and N${}_{2}^{+}$ molecules/ions. The spectra are recorded by an intensified-CCD (Princeton Instruments).

## 3. Results and discussion

#### 3.1. Intensity distribution of fluorescence and formation mechanism analysis

A typical temporally and spectrally resolved image recorded by a streak camera is shown in Fig. 2(a). Molecular emissions from N${}_{2}$ molecules and N${}_{2}^{+}$ ions dominate the spectrum, and the spectrum background is close to zero, indicating that the Bremsstrahlung emission is negligible. The emission lines located at 337, 357, 375, 380 and 391 nm are assigned to bands (0,0), (0,1), (1,3) and (0,2) of the N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}-{\mathrm{B}}^{3}{\mathrm{\Pi}}_{\mathrm{g}}$) system, and band (0,0) of the N${}_{2}^{+}$ (${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}-{\mathrm{X}}^{2}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$) system, respectively.

The fluorescence intensities of N${}_{2}$ and N${}_{2}^{+}$ exhibit different distributions across the filament column, as shown in Fig. 2(b). The emission zone of N${}_{2}^{+}$ is mainly located at the central part of the filament column (about 4 mm in length), whereas the intensity distribution of N${}_{2}$ emission is more spread out. The reason for the distribution difference of N${}_{2}$ and N${}_{2}^{+}$ fluorescence can be attributed to different formation mechanisms of the N${}_{2}^{+}{\text{B}}^{2}{\displaystyle \sum}_{\mathrm{u}}^{+}$ and N${}_{2}\text{}\overline{m}{C}^{3}{\mathrm{\Pi}}_{u}$ states in the filament. The N${}_{2}^{+}{\text{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}$ state is mainly achieved through tunneling ionization and photoelectron recollision excitation. The ground state electronic configuration for N${}_{2}$ molecule is KK${({\sigma}_{\mathrm{g}}2\mathrm{s})}^{2}{({\sigma}_{\mathrm{u}}2\mathrm{s})}^{2}{({\pi}_{\mathrm{u}}2\mathrm{p})}^{4}{({\sigma}_{\mathrm{g}}2\mathrm{p})}^{2}$. In a laser field, electrons in the highest occupied molecular orbital (HOMO) is first removed because ionization rate decays exponentially over the electron binding energy. The bonding energies are 15.58, 17.07 and 18.75 eV for HOMO (*σ*_{g}), HOMO-1 (*π*_{u}) and HOMO-2(*σ*_{u}), respectively [21]. The removal of one electron from HOMO, HOMO-1 and HOMO-2 of N${}_{2}$ molecule leaves N${}_{2}^{+}$ in the electron state of ${\mathrm{X}}^{2}{\displaystyle \sum}_{\mathrm{g}}^{+}$, ${\mathrm{A}}^{2}{\mathrm{\Pi}}_{\mathrm{u}}$, and ${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}$, respectively [22]. N${}_{2}^{+}$ fluorescence at 391 nm, emitted from N${}_{2}^{+}$ ions in the ${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}$ state, only occurs for strong laser fields. In the photoelectron recollision excitation scheme, the correlated photoelectrons collide inelastically with parent ions, and N${}_{2}^{+}$ is excited from the ${\mathrm{X}}^{2}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$ state to the ${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}$ state.

Excited states of N${}_{2}$ molecules cannot be achieved through multiphoton excitation because the transition from ground state ${\mathrm{X}}^{1}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$ to ${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$ is forbidden. The formation possibility through collision between N${}_{2}$ molecules in groundstate and free electrons [N${}_{2}\left({\mathrm{X}}^{1}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}\right)+{\mathrm{e}}^{-}\to {\mathrm{N}}_{2}\left({\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}\right)+{\mathrm{e}}^{-}$] in the filament column is very low, because electrons with very high threshold energy (11 eV) are required for the collision-induced transition of N${}_{2}$ molecules from ground state ${\mathrm{X}}^{1}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$ to the ${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$ state [23]. In addition, the average kinetic energy of electrons produced in a linearly polarized laser field is much lower than the electron energy required, according to numerical simulations [24]. Thus, the primary reactions, ${\mathrm{N}}_{2}^{+}+{\mathrm{N}}_{2}\to {\mathrm{N}}_{4}^{+}$, and ${\mathrm{N}}_{4}^{+}+{\mathrm{e}}^{-}\to {\mathrm{N}}_{2}\left({\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}\right)+{\mathrm{N}}_{2}$, as reported by Xu $et\text{}al$. [25], are almost fully responsible for populating the excited electronic state ${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$ of N${}_{2}$ molecules. N${}_{2}^{+}$ ions act as precursors to N${}_{2}$ molecules in the ${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$ state. However, Fig. 2(b) shows that the N${}_{2}$ fluorescence appears first along the propagation direction and its intensity reaches maximum at the filament position ($L=-2.5$ mm) with weak N${}_{2}^{+}$ emission, implying that N${}_{2}^{+}$ ions in ground state act as precursors to N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$). N${}_{2}^{+}$ (${\mathrm{X}}^{1}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$) ions are more easily to form according to the HOMO analysis, and plenty of them exist in the filament column, according to the laser induced fluorescence measurements result reported by Wang $et\text{}al$. [21].

#### 3.2. Radially resolved decay dynamics of N_{2} fluorescence

Figure 3(a) show the temporally and radially resolved fluorescence emission around 337 nm recorded at a longitudinal position *L* = −1 mm. The fluorescence decays fast at the core position of the filament column, and the fluorescence emission persists longer time at larger radial positions. The fluorescence decay curves can be well fitted with two distinct exponential decays,

Here, $I\left(t\right)$ is fluorescence intensity, IRF represents the instrumental response with a Gaussian function, which represents the probability distribution of instantaneous electrons striking at the phosphor screen of the streak camera system. *A*_{1} and *A*_{2} are the amplitudes of the two decay processes. *τ*_{f} and *τ*_{s} denote fast and slow characteristic decay time constants, respectively. At radial position *R* = 0 m, ${\tau}_{\mathrm{f}}=26$ ps and ${\tau}_{\mathrm{s}}=240$ ps, as seen in Fig. 3(b); the latter is about order of magnitude larger than the former. However, ${\tau}_{\mathrm{f}}=73$ ps and ${\tau}_{\mathrm{s}}=530$ ps at $R=-75$ m. Both *τ*_{f} and *τ*_{s} vary with locations in the filament, and their radial and longitudinal variations will be discussed below. The radiative lifetimes (${\tau}_{\text{rad}}$) of the N${}_{2}^{+}$ (${\mathrm{B}}^{2}{\mathrm{\Sigma}}_{\mathrm{u}}^{+}-{\mathrm{X}}^{2}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}$) and N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}-{\mathrm{B}}^{3}{\mathrm{\Pi}}_{\mathrm{g}}$) transitions are about 60 ns and 40 ns, respectively [21, 26], while the measured fluorescence decay times are very short. For N${}_{2}$, the fluorescence decays within 1 ns. The decay dynamics of N${}_{2}^{+}$ fluorescence cannot be resolved and its lifetime is less than the temporal resolution of the streak camera (10 ps). The excited N${}_{2}$ and N${}_{2}^{+}$ molecules/ions are most likely to be de-excited through non-radiative process, and collision is the most possible way. The filament column contains electrons, molecules and ions. For this study, the main collision partners of excited N${}_{2}$ molecules are electrons, oxygen molecules and nitrogen molecules. Nitrogen ions are negligible, given the low degree of ionization in the filament.

For N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules at room temperature, the deactivation rate constants with different deactivators, N${}_{2}$, O${}_{2}$ and electrons are $1.3\times {10}^{-11},3\times {10}^{-10}$ and $4\times {10}^{-7}\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{3}{\mathrm{s}}^{-1}$ [27], respectively. Given that the initial electron density at the filament core is ${10}^{18}\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{-3}$ [20], the collision frequency is on the order of ${10}^{11}{\text{s}}^{-1}$, giving rise to a characteristic fluorescence decay time of tens of ps. The lifetime associated with deactivators, N${}_{2}$ and O${}_{2}$, can be evaluated with ${\tau}^{-1}={k}^{{\mathrm{O}}_{2}}\left[{\mathrm{O}}_{2}\right]+{k}^{{\mathrm{N}}_{2}}\left[{\mathrm{N}}_{2}\right]$. Herein $\left[{\mathrm{O}}_{2}\right]$ and $\left[{\mathrm{N}}_{2}\right]$ donate the densities of the deactivators, and *k* denotes deactivation rate constants. The corresponding decay constant is calculated as $\tau \approx 430$ ps. Here the *k*-constants used in the calculation are for room temperature. However, laser irradiation heats the deactivators, increases collision probability at elevated temperatures, and hence results in a decrease in *τ*. Laser irradiation also induces a slight decrease in deactivator density due to minor ionization, and thus a slight increase in lifetime. Actually, the fluorescence decay resulted from deexcitation of N${}_{2}$ in the N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) state through collision with N${}_{2}$ and O${}_{2}$ is dominated by O${}_{2}$, since the decay constants of O${}_{2}$ (${\tau}_{{\mathrm{O}}_{2}}$) and N${}_{2}$ (${\tau}_{{\mathrm{N}}_{2}}$) are 0.5 and 3.3 ns, respectively. This is consistent with the experimental finding that *τ*_{s} in an air filament is much smaller than that formed in pure nitrogen, as shown in Fig. 4. Yao $et\text{}al$. [28] also reported that oxygen molecules have significant quenching effect. The underlying physical mechanism of collisional deactivation of N${}_{2}$-O${}_{2}$ is a collision reaction: ${\mathrm{N}}_{2}\left({\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}\right)+{\mathrm{O}}_{2}\to {\mathrm{N}}_{2}\left({\mathrm{X}}^{1}{\mathrm{\Sigma}}_{\mathrm{g}}^{+}\right)+\mathrm{O}+\mathrm{O}$, which has been considered as a de-excitation mechanism of N${}_{2}$ molecules in conventional discharge-pumped nitrogen ion laser [26]. The measured fast decay constants (*τ*_{f}) and slow decay constants (*τ*_{s}) are in the range of 20–100 ps and 200–600 ps, respectively (c.f. Fig. 3). Thus, we suggest that the fast and slow decay processes are caused by the e${}^{-}-{\mathrm{N}}_{2}$ and O${}_{2}-{\mathrm{N}}_{2}$ collisions, respectively.

Both *τ*_{f} and *τ*_{s} increase radially from the filament center, especially for the longitudinal sections at *L* = 0 and $-1$ mm, as shown in Figs. 5(a) and 5(b). *τ*_{f} has similar values over radial positions for the filament section at *L* = −2 mm; the reason is that the fluorescence intensity dominated by the fast decay process is very weak, and the calculated *τ*_{f} values have relatively large uncertainties. In addition, the integrated intensities of the fast and slow decay processes (${I}_{{\tau}_{\mathrm{f}}}$ and ${I}_{{\tau}_{\mathrm{s}}}$; i.e., the areas under the deconvoluted exponential curves), vary significantly over radial positions and at different longitudinal sections of the filament column. The fluorescence intensity percentages, ${P}_{{\tau}_{\mathrm{f}}}\equiv {I}_{{\tau}_{\mathrm{f}}}/\left({I}_{{\tau}_{\mathrm{f}}}+{I}_{{\tau}_{\mathrm{s}}}\right)$ and ${P}_{{\tau}_{\mathrm{s}}}\equiv {I}_{{\tau}_{\mathrm{s}}}/\left({I}_{{\tau}_{\mathrm{f}}}+{I}_{{\tau}_{\mathrm{s}}}\right)$, are summarized in Figs. 5(c) and 5(d). A higher percentage of N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules is de-excited by electron collision at the core of the filament column, compared to that of the peripheral area, where more N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules tend to be de-excited by collision process with oxygen molecules. This is reasonable because electron density is higher at the filament core. The two de-excitation processes compete with each other. The N${}_{2}-{\mathrm{O}}_{2}$ collision process dominates depopulation of N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules in the peripheral area. At the early-formed sections (c.f. *L* = −2 mm in Fig. 2), electron density is supposed to be very low and only a small percentage ($<10\%$) of the N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules is de-excited via the e${}^{-}-{\mathrm{N}}_{2}$ collision.

One possible reason for the asymmetry in the radial distribution of decay times and percentages of integrated fluorescence is that the intensity distribution of the fs-laser beam is not perfectly radially symmetric (non-perfect Gaussian beam). Moreover, the filament tends to bifurcate during its propagation when too high a laser intensity is confined in the filament core. In addition, slight non-coaxial alignment or tilt of lens with respect to the laser beam may also contribute to the filament asymmetry.

#### 3.3. Longitudinally resolved decay dynamics of N${}_{2}$ fluorescence

Typical decay curves at different longitudinal positions of the filament column (*R* = 0) are shown in Fig. 6. The fluorescence decays much faster at the filament center, and the fluorescence lifetime increases towards both ends. The fittings for *L* = 0 mm and *L* = −2 mm reveal significant difference in the decay constants (*τ*_{f} and *τ*_{f}). At *L* = 0 mm, the e${}^{-}-{\mathrm{N}}_{2}$ collision dominates the fast decay process with ${\tau}_{\mathrm{f}}=21$ ps, while the O${}_{2}-{\mathrm{N}}_{2}$ collision dominates the slow decay process with ${\tau}_{\mathrm{f}}=202$ ps. Moreover, the fast decay process nearly disappears at *L* = −2 mm, indicating that the e${}^{-}-{\mathrm{N}}_{2}$ collision probability is very small.

The fitted decay constants, *τ*_{f} and *τ*_{s}, together with their corresponding percentages of fluorescence intensity, at different longitudinal positions of the filament column are given in Fig. 7. The ${\tau}_{\mathrm{f}}\left(L\right)$ and ${\tau}_{\mathrm{s}}\left(L\right)$ curves are both concave upward, and approximately symmetric about *L* = 0, where their minima are located, as seen in Fig. 7(a). At the same time, the laser intensity reaches its maximum around the center of the filament and then attenuates along the propagation direction of the filament, due to the focusing effect by lens and the Kerr effect induced self-focusing. The laser beam is defocused after passing the focal point, ${f}_{\text{eff}}={f}_{\text{lens}}{f}_{\text{sf}}/\left({f}_{\text{lens}}+{f}_{\text{sf}}\right)$ [29]. Herein *f*_{lens} and *f*_{sf} are the focal length of the lens and self-focusing, respectively.

The variation of decay constants over the longitudinal positions is negatively correlated to the intensity of N${}_{2}^{+}$ fluorescence, i.e., both decay constants reach their minimum at *L* = 0 mm, where the N${}_{2}^{+}$ fluorescence intensity is the highest, as seen in Fig. 2(b). The reason is that the distribution probability of N${}_{2}^{+}$ ions in excited states is an exponential function of laser intensity. However, higher laser intensity results in higher free electron density, and deposition of more laser energyon air molecules leads to an increased collision probability. As a result, the fluorescence decays faster where the laser intensity is higher. However, similar observations are not found between N${}_{2}$ fluorescence distribution and its decay constants, and a possible reason is that the formation of N${}_{2}$ molecules in the ${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$ state is not directly related to the laser intensity, since two elementary reactions are involved in the formation of N${}_{2}$ molecules, as discussed above. Here we also note that the fluorescence intensity percentage for *τ*_{f} is concave downward, as seen in Fig. 7(b). At $L=\pm 3$ mm, the fluorescence decay is mostly caused by the *τ*_{s} process, which indicates that the free electron density at these positions is close to zero.

#### 3.4. Rotational temperatures of N${}_{2}$ and N${}_{2}^{+}$

The rotational temperatures of N${}_{2}$ and N${}_{2}^{+}$ are determined via fitting to the experimental spectra with a corresponding synthetic spectrum model (Figs. 8(a) and 8(b)). Synthetic spectra of N${}_{2}$ and N${}_{2}^{+}$ are computed within the Herzberg’s framework of diatomic molecules [30]. The molecular constants in Ref. [31] are used for band spectrum simulation. The N${}_{2}$ and N${}_{2}^{+}$ spectral models contain several fitting parameters, e.g., rotational temperature, line broadening, intensity. Detailed spectrum fitting procedure can be found elsewhere [32–35]. Figure 8 depicts representative high resolution spectra of N${}_{2}$ and N${}_{2}^{+}$ and their fittings. The two spectra are obtained at identical experimental conditions but for different wavelengths. The rotational structures of molecular bands and overall intensity are well fitted. The rotation temperature, which represents the distribution of electrons in the excited rotational energy levels of molecules, is obtained through fitting the N${}_{2}^{+}$ spectrum. The rotation temperature from the N${}_{2}^{+}$ spectrum is about twice that from the N${}_{2}$ spectrum. A possible reason is that the N${}_{2}^{+}$ temperature refers to the very early stage of filamentation because of the short lifetime of the N${}_{2}^{+}$ fluorescence (below 10 ps, c.f. Fig. 2), whereas the N${}_{2}$ temperature represents an averaged temperature during the whole lifetime of the filament. The N${}_{2}^{+}$ spectrum acquired in the time-integrated configuration is equivalent to that acquired with a sub-10 ps gate width and zero gate delay because the persistent time of N${}_{2}^{+}$ fluorescence is pretty short.

The comparison of temperatures deduced from N${}_{2}$ and N${}_{2}^{+}$ implies that the filament temperature drops rapidly within hundreds of ps. The temperature distributions at different radial and longitudinal positions are shown in Figs. 8(c) and 8(d). Temperature is higher at the filament core for the *L* = 0 and -1 mm cases, and drops significantly at larger radial positions. The temperature at $R=\pm 75\text{}$m is about $20\%-25\%$ lower than that at $R=0\text{}$m. The radial temperature distributions shown in Figs. 8(c) and 8(d) are negatively correlated to the radial distributions of both decay constants, *τ*_{f} and *τ*_{s}, especially at *L* = 0 and $-1$ mm. Both rotational temperatures deduced from N${}_{2}$ and N${}_{2}^{+}$ peak at the radial position of 0 μm, and decrease with increasing absolute radial distance. However, both decay constants are minimal at *R* = 0 μm, and increase with increasing absolute radial positions, as shown in Figs. 5(a) and 5(b). The faster decay of fluorescence at central position of filament is attributed to that higher collision possibilities of e–N${}_{2}$ and O${}_{2}$-N${}_{2}$ are expected at higher temperature condition. However, the radial variation of temperature at *L* = −2 mm is negligible, likely because the fs-laser is not focused. The significant increase in temperature from *L* = −2 to *L* = 0 mm indicates sharp focusing of the laser beam due to the strong Kerr effect, which is also the main reason for the distinct increase in N${}_{2}^{+}$ fluorescence (Fig. 2). Moreover, higher electron density in the tightly focused zone is expected. The increases in electron density and temperature result in shortened fluorescence decay time, consistent with our experimental observations in Fig. 7.

## 4. Conclusion

We have experimentally investigated the decay dynamics of N${}_{2}$ and N${}_{2}^{+}$ fluorescence in fs-laser filaments formed in air. The fluorescence decay contains two exponential decay processes: a fast one (∼10s ps) and a slow one (100s ps). The radially resolved fluorescence decay curves reveal that both the fast and slow decay processes decay faster at the core of the filament column, and a larger portion of the overall fluorescence intensity is governed by the fast decay process at smaller radial positions. The slow decay process dominates the fluorescence dynamics at peripheral parts of the filament, particularly in the early-formed and vanishing sections of a long filament column, where the fast decay process almost disappears. The decay constants resulted from collisions with different particles are calculated. Through analyzing the characteristic decay processes throughout the whole filament column, we conclude that the fast decay process originates from the collision between free electrons and N${}_{2}$ (${\mathrm{C}}^{3}{\mathrm{\Pi}}_{\mathrm{u}}$) molecules, and the slow decay process is resulted from excited N${}_{2}$ molecules colliding with oxygen and nitrogen molecules, as well as spontaneous radiation, but the N${}_{2}-{\mathrm{O}}_{2}$ collision dominates the lifetime of the slow decay process.

## Funding

National Natural Science Foundation of China (Grant Nos. 61605161 and 11627901) and Fundamental Research Funds for the Central Universities (Grant No. 2682017CX074).

## Acknowledgments

We have benefitted from valuable discussions with Dr. Jin Yu from Shanghai Jiaotong University.

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