1. Introduction

Ultrafast laser filamentation in air [1] is a phenomenon of increasing interest for applications in remote sensing [2–4], remote laser-induced breakdown spectroscopy [5–8], water condensation, discharge guiding [9,10], and air lasing [11,12]. Consequently, there is motivation to accurately characterize the filaments, including their morphology and plasma properties. Ultrafast laser filamentation arises from the balance between the nonlinear self-focusing and the defocusing due to contributions from diffraction and plasma generation. A filament forms when the electric field strength of the laser pulse is sufficient to manipulate the electron clouds and rotate the atoms and molecules, causing an ensemble-averaged dipole moment [13]. This induced dipole nonlinearly increases with the electric field strength, which causes a change in refractive index and leads to self-focusing of the laser pulse. The relative change of refractive index can be determined, for example, by in-line holography and is known to be <10⁻³ [14]. As the beam self-focuses, the laser intensity increases such that multiphoton and tunnel ionization take place, forming a plasma [15]. The laser intensity at this point of balance between the nonlinear focusing and defocusing is referred to as clamped intensity. The laser power threshold for self-focusing depends both on the composition and density of the propagation medium and the laser wavelength, and is referred to as the critical power (P_cr) [1,16]:

Several prior studies investigated the spatially-dependent properties of the filament plasma in the single-filament regime for a driving wavelength of 800 nm from a standard Ti:sapphire-based ultrafast laser system [13, 14, 26–30]. In other studies, various other wavelengths were used, including 248 nm [8], 400 nm [31], and 3.9 µm [32]. Past research provides clear evidence that different laser wavelengths, beam energies, pulse durations, and focusing conditions can have notable effects on the ionization yield, electron density, and plasma temperature of the filament. For example, Théberge et al. [33] have shown that the plasma density and diameter of the filament strongly depend on external focusing conditions. A variety of invasive and noninvasive techniques have been used to measure the electron density in filaments, like terahertz spectroscopy [34], ${\mathrm{N}}_2^{+}$ emission intensity [33], interferometry and diffraction [13,33,35], Stark broadening [27,28], and many others [29,30,36–38]. Reported densities range from 10¹⁵ cm⁻³ [34] to 10¹⁸ cm⁻³ [33]. In several studies, the plasma temperature of single filaments was measured and reported to range from 3900 K [35] to 5800 K [27,28]. Techniques used to determine these temperatures include diffraction and interferometry [35, 37], as well as the ratios of line intensities [27, 28]. In this work, we present an analysis of the multiple filament plasma formed in air by observing the axially resolved nitrogen fluorescence from 400- and 800-nm ultrafast laser pulses at various energies. Mechanisms that lead to the measured fluorescence spectrum are suggested, and we compare our results to previous measurements of single filament plasmas. We estimate the filament plasma temperatures and electron densities via non-invasive optical emission spectroscopy (OES).

2. Experiment

The experimental schematic can be seen in Fig. 1. The laser used is the custom-built Lambda-Cubed (λ³) Ti:sapphire-based chirped-pulse amplification system at the University of Michigan, operating at a central wavelength of ∼800 nm at a 480-Hz repetition rate and with a pulse duration of ∼50 fs. The beam has a 22-mm Gaussian diameter immediately after the compressor. The laser propagates ∼7 m before reaching the silver-coated 1-m focal length (f /40) spherical focusing mirror. For the 400-nm filament, the laser is frequency doubled using a β-Ba(BO₂)₂ crystal, the remaining 800-nm light being removed using a dichroic mirror. The resulting 400-nm pulse duration is calculated to be 42 fs from the pulses at 800 nm measured via frequency-resolved optical gating. Spectroscopy is performed using a 25.4-mm diameter, 30-mm focal length lens coupled to a 400-µm optical fiber, which is connected to a 0.55-m Czerny-Turner spectrograph (Horiba Jobin Yvon, iHR550), and detected by an intensified charge-coupled device (ICCD, Andor, iStar 334T). The spectroscopic system resolution was measured to be ∼40 pm at 632.8 nm. For side imaging of the filament, an electron multiplying-ICCD (EM-ICCD, Princeton, PI-MAX4) is used with an objective lens, offering sub-ns intensifier timing resolution. To isolate ${\mathrm{N}}_2^{+}$ emission, a 50.8-mm diameter, 391.4-nm bandpass filter with a 1.4-nm FWHM-bandwidth (Andover) has been placed in front of the camera objective. The inset of Fig. 1 shows an example radial profile of the multiple filament cores inferred from the damage on a copper target placed at the geometric focus of the spherical mirror from a single 3-mJ, 400-nm pulse. The range of peak powers used in this study is 20–80×P_cr at 800 nm, and a single peak power of 140 ×P_cr for 400 nm is used. The critical powers for self-focusing of 400- and 800-nm are estimated to be 0.5 and 3 GW, respectively [39]. All experiments were conducted in air.

 

Fig. 1 Schematic of the experimental setup. The inset image shows an example radial profile of the multiple filament cores (bright spots) observed from the damage on a copper target (3 mJ, 400 nm). The copper target is placed near the geometric focus of the spherical mirror. The damage pattern is from a single laser shot.

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3. Results & discussion

3.1. Spatially and temporally resolved emission from the filament plasma

The optical emission from filament-air plasmas contain two dominant nitrogen fluorescence systems, as seen in Fig. 2. The first of them is the N₂ second positive system (C³Π_u → B³Π_g), where the upper excited levels of N₂ are populated indirectly by recombination of electrons with ${\mathrm{N}}_2^{+}$ [40,41]. The second is the ${\mathrm{N}}_2^{+}$ first negative system $({\mathrm{B}}^2{\displaystyle {\sum }_{\mathrm{u}}^{+}\to }{\mathrm{X}}^2{\displaystyle {\sum }_{\mathrm{g}}^{+}})$, where the inner-shell ionization of N₂ by multiphoton and tunnel ionization populates the excited states of ${\mathrm{N}}_2^{+}$ [40]. The (x_u, x_ℓ) notation in Fig. 2 refers to a transition from a vibrational state in the upper level (excited) electronic state (x_u) to a vibrational state in the lower level electronic state (x_ℓ); for example, (0,0) represents a direct transition from the ground vibrational state in the upper electronic level to the ground vibrational state in the lower electronic energy level. According to Fig. 2, the peak of the 337-nm (0,0) band-head from the N₂ second positive system is 2.6× more intense for the 800-nm filament than for the 400-nm filament, both driven by 3-mJ pulses. On the other hand, the peak intensity of the 391-nm band-head arising from the ${\mathrm{N}}_2^{+}$ first negative system (0,0) transition is 4.4× greater for the 400-nm filament than for the 800-nm filament. It is well understood that multiphoton ionization is increasingly probable for decreasing laser wavelength, where the individual photon energy increases. Daigle et al. performed a similar experiment in the single filament regime with 800- and 400-nm input laser energies of 2.3 mJ focused using a 15-cm focal length lens, and reported a 6.4× higher intensity for the 391-nm peak when using 400-nm compared to 800-nm driving wavelength; however, the N₂ emission was observed to be constant for both wavelengths. The equal N₂ emission between the two wavelengths is attributed to similar total ionization yields at 400 and 800 nm [31]. Identical N₂ emission between the two wavelengths is not observed in our case using looser focusing conditions and entering the multiple filament regime for laser peak powers 20×P_cr and 140×P_cr at 800 nm and 400 nm, respectively. The increased ${\mathrm{N}}_2^{+}$ emission may be explained by higher efficiency for inner-shell ionization of N₂ in the 400-nm case, leaving a larger fraction of ${\mathrm{N}}_2^{+}$ in the excited state [31,40,41], which may also contribute to the lower observed N₂ emission.

 

Fig. 2 Measured emission spectrum of the N₂ second positive system and the ${\mathrm{N}}_2^{+}$ first negative system for 3-mJ pulses at 400 nm (blue) and 800 nm (red) 20 mm measured before the geometric focus in the direction of beam propagation. The ICCD delay and gate width were 0 ns and 10 ns, respectively, measured with respect to arrival the incident laser pulse at the filament region. The shaded regions highlight the two molecular band-heads primarily discussed.

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We next explore the spatiotemporal dependence of the filament plasma emission with gated side-on imaging in order to gain more insight into the distribution of ionic species and the axial distribution of plasma properties. The filament lifetime is measured by the side-view optical emission intensity distribution at 1-ns intervals, as shown in Fig. 3. The total emission dissipates over a period of ∼5 ns, in agreement with previous measurements of the filament lifetime [37] as well with the more recent measurements of the surrounding magnetic field, which suggest a 5–6 ns lifetime [36]. The decay of total emission intensity in Fig. 3 can be used to infer the temporal evolution of plasma temperature and density. Moreover, Fig. 3 provides information on the spatial distribution of the plasma emission. The rapid decay and spatial distribution of the total optical emission motivate subsequent exploration of the behavior and mechanisms governing air ionization in the filamentary regime. In this way, we can compare the plasma characteristics determined under various controllable laser parameters under the conditions when multiple filaments form with those reported in previous studies, where single filaments were studied.

 

Fig. 3 Time evolution of the filament plasma measured by side imaging for 3-mJ, 800-nm pulses. The gate widths for each time step were 1 ns with 100 accumulations. The white dashed line represents the geometric focus of the spherical mirror.

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We further examine the spatial distribution of the 391-nm ${\mathrm{N}}_2^{+}$ from the filament plasma in Fig. 4. The brightest emission occurs before the geometric focus (represented by the dashed line in Fig. 4), demonstrating the contribution of self-focusing. The 391-nm emission shown in Fig. 4(c) and (f) displays a similar axial profile to the total emission, as can been seen in Fig. 4(a) and (d), implying similar emission trends for N and ${\mathrm{N}}_2^{+}$. The emission diameter from the filament plasma increases from ∼600 µm with 3 mJ to ~800 µm with 12 mJ driving laser energy, respectively. The plasma channel length also increases from ~57 mm to ~85 mm. The large diameter of emission compared to the previously reported filament core diameters of ≤100 µm [21,33] is attributed to the formation of several filament cores; the increased diameter with greater input laser energy implies the formation of more cores with increased laser peak power.

 

Fig. 4 Side imaging of the total and ${\mathrm{N}}_2^{+}$ emission from the 800-nm filament with 3 mJ (a)–(c) and 12 mJ (d)–(f) laser energies. Profiles (a) and (d) show the vertically-integrated emission. Images (b) and (e) show the total emission. Images (c) and (f) show 391 nm emission with a FWHM of 1.4 nm band pass (BP) filter. The white dashed line represents the geometric focus of the spherical mirror. Images were recorded at a delay of 0 ns with respect to the incident laser pulse, gate width of 10 ns, and accumulated for 100 laser shots.

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Assessing the axially resolved and energy-dependent optical emission spectrum helps to identify the excitation pathways that produce this distribution of species. The axial position is measured with respect to the geometric focus of the spherical mirror (0 mm). The maximum intensities for the 391-nm and 337-nm band-heads in the spectra occur 10 mm before geometric focus for both 400- and 800-nm beams, as seen in Fig. 5. The 337-nm N₂ emission dominates for 800-nm, while the 391-nm ${\mathrm{N}}_2^{+}$ emission dominates for the 400-nm filament, which is consistent with the result of the spectra shown in Fig. 2. Figures 5(a) and (b) show that both N₂ and ${\mathrm{N}}_2^{+}$ emission bands follow similar trends along the filament axis. While the nitrogen molecular emissions peak near the same location, O I peaks at the geometric focus for 400 nm and 5 mm behind the geometric focus for 800 nm. The different axial distribution of O I emission as distinguished from that of N₂ and ${\mathrm{N}}_2^{+}$ implies a different origin for O I excited species. Consequently, plasma processes (e.g., collisional or thermal dissociation and excitation) may cause the O I emission to peak at an axial position beyond the region of maximum filament intensity along the beam propagation direction. Results from previous work on temporal plasma emission characteristics illustrated that atomic lines of N and O in filament plasmas appear after both N₂ and ${\mathrm{N}}_2^{+}$ emissions and after the decay of continuum emission [42], suggesting further that collisional interactions in the plasma result in atomic O-I emission. The axial gradient (dI/dz) is highest for the 337-nm band-head for both filaments at the beginning of the filament formation, as seen in Fig. 5(c) and (d). Previous work indicates that the N₂ 337-nm emission band-head has a longer persistence (∼1 ns) than the ${\mathrm{N}}_2^{+}$ 391-nm emission band-head (∼400 ps) [42]. This is primarily due to the means by which each upper level is pumped: photoionization and collisional processes for the ${\mathrm{N}}_2^{+}$, and solely collisional processes for the N₂. We observe also that the axial gradient in the excited N₂ population is greater for the 800-nm filament than for the 400-nm case. The different axial gradients for each wavelength may imply different excitation and ionization paths for the two driving laser wavelengths.

 

Fig. 5 Measured peak intensities for the N₂(337 nm) and ${\mathrm{N}}_2^{+}$ (391 nm) band-heads, and O I (777.19 nm) line from the (a) 800-nm and (b) 400-nm filament spectra and axial gradients (dI/dz) for the measured intensities of each band and line for the (c) 800-nm and (d) 400-nm filaments. Driving laser pulse energy was 3 mJ for both wavelengths. Geometric focus of the spherical mirror is denoted as 0 mm. Data is fit with a spline curve to help guide the eye.

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3.2. Axially-resolved, driving laser energy- and wavelength-dependent filament plasma diagnostics

We perform diagnostics of the optical emission in order to determine the axially-resolved plasma properties and how they vary with incident laser parameters. We fit the experimental nitrogen spectra using SPECAIR [43] in order to determine plasma temperatures, assuming Boltzmann velocity distributions. We iteratively vary individual temperatures to approximate the assumed case of incomplete or partial local thermodynamic equilibrium (LTE). Temperatures are derived from a “best fit” by minimizing the residuals between the experimental and simulated spectra. Figure 6 shows an example of experimental and simulated spectra used to estimate rotational (T_r), vibrational (T_v), and electronic temperatures (T_e).

 

Fig. 6 Spectrum measured with 3-mJ, 800-nm pulses and simulated by SPECAIR for the (1,3) and (0,2) N₂ emission bands.

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As stated previously, a variety of methods, both invasive and non-invasive, have been used to date to estimate and measure the electron density in transient filament plasmas. Here we measure the degree of Stark broadening for the O I 777.19-nm line as a non-invasive method for evaluating electron density. Figure 7 shows a sample multi-Voigt fit that combines the natural and Stark broadening (Lorentz shape) and instrumental and Doppler broadening (Gaussian shape) that determine the measured line shape and line width. Natural broadening is assumed negligible in comparison to the measured line width. Instrumental broadening for the spectroscopic system is measured using a Hg-Ar source, and Doppler broadening is calculated using the electron temperatures from SPECAIR ($w_{\text{Doppler}}=7.2\times {10}^{-7}{\lambda }_0\sqrt{T_e/M}$, where w_Doppler is the Doppler half-width, λ₀ is the line center, and M is the mass of the emitter). These contributions are subtracted from the full width at half maximum (FWHM) from the Voigt fit of the selected spectral lines, isolating the Stark broadening contribution. Assuming negligible broadening from ions in the plasma, the electron density can then be estimated from [44]

 

Fig. 7 Sample multi-Voigt fit for the O I 777.19-, 777.42-, and 777.54-nm lines used to determine the degree of Stark broadening. The spectrum shown is taken for 800-nm with driving laser energy of 12 mJ.

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Fig. 8 Axially-resolved (a) rotational temperatures (T_r), (b) vibrational temperatures (T_v), (c) electronic temperatures (T_e), and (d) electron densities (n_e) along the filament length. Geometric focus of the spherical mirror is at 0 mm. Results shown are for 3 mJ energies with both 400- and 800-nm beam wavelengths.

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Table 1. Driving laser energy- (E_λ) and wavelength- (λ) dependence of filament optical emission and plasma properties. Energy is given in mJ. Intensity ratios $(I_{{\mathrm{N}}_2}/I_{{\mathrm{N}}_2^{+}})$ and peak emission (I_peak) compare the emissions from N₂(337 nm) and ${\mathrm{N}}_2^{+}$ (391 nm) (0,0) band-heads at the axial peak of the optical emission. Plasma temperatures are also given at the axial peak of the optical emission in units of K. Electron densities are calculated from Stark width parameters measured for filaments [27] and given in units of cm⁻³.

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Axially resolved T_r, T_v, and T_e are shown in Fig. 8; temperatures vary less than 10% along the axial distance. The temperatures are found to be unequal, implying the filament plasma is not in complete LTE, which may contribute to the calculated errors in the fits seen in Fig. 8. The temperatures, however, are comparable to those previously reported for single filaments where LTE is assumed [27,28,45]. While the electronic temperature is most commonly used for characterizing plasmas, it is interesting to note that T_r and T_v primarily determine the appearance of the (2,5) N₂ vibrational band. The average T_r and T_v are ∼900 K and ~3300 K for the 800-nm filament, and ∼1150 K and ∼2300 K for the 400-nm filament. The lower T_v in the 400-nm plasma indicates that the higher lying (2,5) vibrational transition is less probable than transitions involving lower lying vibrational levels in the upper electronic state. We calculate the vibrational partition functions in order to determine how many accessible vibrational states there are at each respective temperature. Vibrational energy levels are described by the harmonic oscillator model, hence T_v can be determined as follows:

Figure 8(d) shows the axially-resolved electron densities, where the maximum is near the geometric focus. The electron density distribution provides further evidence that plasma processes may be the dominant cause for the dissociation and excitation of oxygen. The dissociation energy for O₂ is ∼5.1 eV at room temperature from the ground vibrational state. Dissociation of O₂ at 5000 K is possible from O₂-O₂ and e⁻-O₂ collisions, and is especially likely when O₂ is in an excited vibrational state [47]. The photoabsorption cross section which dictates the dissociation of O₂ significantly decreases with increasing wavelength, and most photo-dissociative processes occur below 400 nm [48]. Therefore, it is possible that multiphoton dissociation of O₂ by 400- and 800-nm pulses is not the main process for production of O I. We see that the optical emission in Fig. 5 peaks in the region before the maximum electron density is reached, while the temperatures remain fairly constant along the length of the filament. Similar observations about electron temperature were reported by Liu et al. [28] in the single-filament regime, where the average and maximum temperature were similar (within ∼6%). Further work is necessary to understand more details in the process of filament interaction with air, which causes the maximum electron density to occur at a different position along the filament axis than the maximum molecular optical emission intensity. The processes that may play a role in explaining this difference include the spatial evolution of the filament profile (including the filament core and reservoir), compression of the laser pulse while propagating along the filament, and the gradual decrease of laser pulse energy along the filament.

We evaluate the McWhirter criterion [49], which states $n_e\ge 1.6\times {10}^{12}{T}_e^{1/2}{(\mathrm{\Delta }E)}^3$, where E∆ is the largest gap between adjacent energy levels. For an energy gap of 1.78 eV [28] and electronic temperature of 5000 K, the minimum electron density required to satisfy the criterion is 6.4×10¹⁴ cm⁻³. The measured densities, on the order of 10¹⁶ cm⁻³, satisfy the McWhirter criterion, implying the density is sufficiently high for the electrons to be in LTE. However, the different measured rotational, vibrational, and electronic temperatures suggest that the LTE conditions may differ between subsets of particles (e.g. electrons vs. neutral or ionic molecules in the excited state). For transient and inhomogeneous plasmas, such as laser-produced [50] or filament plasmas, further studies are required to verify the existence of equilibrium conditions.

Table 1 presents T_e at the axial peak of the filament emission for various laser parameters. Electronic temperatures do not increase significantly with input beam energy, while the peak optical emission intensity (I_peak) does. We note that Table 1 gives the peak emission at a single axial position, which we distinguish from the total emission of the plasma. In Fig. 4, we observe that the diameter and length of the emission increases with energy. A clamped laser intensity implies that the plasma emission also reaches a maximum. The increasing peak emission at this axial position therefore implies the formation of additional plasma cores. This observation is supported by the increased diameter of the emission observed in Fig. 4 between 3 mJ and 12 mJ incident laser energies, from 600 to 800 µm. Becker et al. [51] observed an analogous relationship between total optical emission and beam energy under different experimental conditions for various high pressures. Constant temperature with increasing emission intensity may also be a consequence of multiple filamentation, where multiple cores contain more total emitters, but each may have a similar temperature. However, single-shot measurements are necessary to disambiguate the temperatures of individual cores. Also, the temperatures presented in this study are determined from averaging the emission of 1000 shots, smearing the sporadic behavior typical for multiple filamentation.

The fraction of ions in the excited state also increases with beam energy, as seen by the decrease in intensity ratio between 337 nm and 391 nm $(I_{{\mathrm{N}}_2}/I_{{\mathrm{N}}_2^{+}})$ in Table 1. Approximately 11 photons of 800-nm are needed for MPI of N₂ [31]. The greater laser intensity at higher pulse energies increases the probability of multiphoton inner-shell ionization of N₂, which emit the 391-nm band. The 400-nm pulses produce a larger fraction of ions in the excited state than 800-nm pulses, since fewer photons are required to ionize N₂; the increasing fraction of ions in the excited state correlates to the increasing electron density. The n_e values in Table 1 agree with those calculated in [27], where the values on order of 10¹⁶ cm⁻³ were obtained with 2-mJ, 800-nm pulses with varying focal length lenses. This is also consistent with results from [33], where n_e was deduced from the intensity of the 391-nm peak. In addition, the authors reported that the plasma density in the filament increases slowly with beam energy in the single-filament regime. We observe similar, slowly increasing electron densities with incident beam energy in the multiple-filament regime. We also observe a larger electron density in the 400-nm case when compared to the 800-nm case, which is consistent with the expectation that at 400 nm the probability of MPI is greater [8,31,39].

4. Conclusion

In summary, we examined the optical emission in order to determine the filament plasma properties in the multiple filament regime. The spatiotemporal structure of the filament plasma was evaluated and the visible emission lifetime was determined to be 5 ns, which agrees with previous measurements performed on single filaments. Both the N₂ and ${\mathrm{N}}_2^{+}$ emission bands peak at the same axial location prior to geometric focus and follow similar emission trends along the length of the plasma, suggesting similar relative spatial population distribution of the molecular nitrogen species. The occurrence of O I peak emission near or after geometric focus, however, implies that collisional (plasma) processes may have a more significant contribution to the dissociation and excitation of oxygen than radiative processes directly driven by the laser. We estimate similar electronic temperatures of ∼5000–5200 K as well as uniform axial distributions despite multiple filamentation for both 400- and 800-nm pulses. This negligible change was attributed to increasing total peak emission resulting from the formation of several plasma cores, each with a similar temperature, reported for single filaments. Although more efficient multiphoton inner-shell ionization is observed for the 400-nm filament, the axial gradient for the N₂ emission measured at the beginning of the filament is greater than the ${\mathrm{N}}_2^{+}$ emission for both wavelengths, demonstrating different ionization and excitation paths for these two processes. Electron density has a more significant change with axial position than temperature, and the maximum density appears near geometric focus. These results provide further insights into the characteristics of filament plasma in the multiple filament regime, which dictates the guided laser intensity relevant to applications such as remote sensing. A dedicated modeling study is needed, however, to better understand the fundamental processes leading to these observations.