The propagation of powerful femtosecond laser pulses in air is a very complicated phenomenon that involves a number of nonlinear processes, including four- and six-wave mixing^[1,2], odd harmonics^[3,4], self-focusing and filamentation^[5], self-phase modulation^[6], supercontinuum generation^[7,8], and so on. The self-focusing phenomenon during the nonlinear propagation leads to the ionization of air molecules. The dynamic balance between self-focusing, owing to the Kerr effect, and plasma defocusing results in the formation of laser plasma filament, which stabilizes laser pulse parameters (beam diameter and pulse duration) in the filament^[9]. Due to its unique properties, femtosecond laser filamentation finds applications in remote spectroscopy^[10] and fabrication^[11], air lasing^[12], fuel ignition^[13], and combustion diagnostics^[14]. Furthermore, the plasmas ionized by the laser pulse have a significant impact on all of the nonlinear processes mentioned above as well as for the related applications in practice^[10,15]. Therefore, from both scientific and practical perspectives, it is crucial to have the femtosecond laser pulses produce the plasma properties in the filaments. So far, the plasma density and electron temperature inside the filaments in air have been measured and analyzed mostly in the case of air excitation by the femtosecond laser pulses under a single repetition rate^[16–18]. Recently, laser repetition-related THz production^[11], wake dynamics^[19], fluorescence^[20], clamping intensity^[21], and the cumulative effects^[22] from air filaments have been reported as high-energy, ultrashort, high-repetition-rate laser systems become commercially accessible. Through pump-probe experiments^[23], we recently found that the low-density region, due to the accumulative effect by 1 kHz repetition filaments, results in higher plasma density obtained in the measurements. However, the pulse repetition effect on the laser ionized plasmas inside femtosecond air filaments is not fully explored yet.

In this Letter, the pulse repetition rate effect on the plasma temperature and density was studied using spectroscopic analysis of the side fluorescence from a femtosecond laser plasma filament in air. Laser exciting oxygen atomic fluorescence lines at 777 nm and 844 nm were employed to look into the plasma density and temperature using the Stark broadening effect and Boltzmann plot^[18]. The plasma density decreases, and the plasma temperature increases as the laser repetition rate increases from 100 Hz to 1 kHz. Numerical simulation based on nonlinear Schrödinger equations was performed to support the experimental observations.

Figure 1 shows an experimental setup designed particularly to monitor the side fluorescence produced by oxygen from a laser-created air plasma filament. In the experiments, the ultrafast laser pulses (duration 35 fs, pulse energy 4.5 mJ, repetition rate up to 1 kHz) produced by a Ti:sapphire laser amplifier were focused in the air by an f=30cm plano-convex lens (L1) to generate laser filaments (Fig. 1). The radius of the laser beam waist was about 7 mm (1/e). The oxygen atomic fluorescence in the near-infrared band emitted by the filament was imaged from the side into the slit of a spectrometer. The imaging system consisted of a pair of lenses, L2 and L3 (30 cm and 10 cm focal lengths, respectively), and a pair of orthogonally placed aluminum mirrors. The height of the fluorescence propagation plane of the filament was lowered and the filament fluorescence was rotated vertically to be parallel to the slit of the spectrometer by the pair of aluminum mirrors. To remove the secondary spectrum of the spectrometer, a long pass filter cut at 550 nm was placed in front of the spectrometer slit. An intensified charge-coupled device (ICCD) mounted on the spectrometer trigged by the laser system was used to capture the fluorescence spectrum generated by the filament. The gate width of the ICCD was set as 10 ns. The delay of the ICCD gate with respect to the laser arrival was 0, which ensures all the fluorescence was collected. 15,000 laser shots were accumulated and averaged for the spectral analysis. A 1200 grooves/mm grating of the spectrometer was chosen for measuring the Stark broadening data of the O I (777.4 nm and 844.3 nm) lines. To precisely monitor the laser energy at different repetition rates, an energy meter (Ophir PE50-DIF-C) was inserted after the lens L1.

The typical spectra in the near-infrared band emitted by the filament are shown in Fig. 2. The principles of laser induced breakdown spectroscopy were used to analyze the side-emitted fluorescence line width of the filament. The instrumental broadening was equivalent to 50 pm determined by the full-width at half-maximum (FWHM) of the spectral lines Ar I 750.387 nm and Ar I 772.376 nm produced from a mercury argon lamp. Compared with the measured line width, natural broadening was assumed to be negligible. A multi-Voigt fit considering Stark broadening by Lorentzian profiles and instrumental broadening (0.05 nm, fixed) by Gaussian profiles was performed, as shown in Fig. 2. The spectral bandwidth of the O I (777.19 nm) line was utilized to determine the electron density. The plasma density was estimated by using the FWHM of the spectral line according to the empirical formula^[24], Ne=Δλ2ω×1016cm−3,(1)where Ne is the density of plasma, Δλ is the Stark broadening obtained from the O I (777.19 nm) line, and ω=0.0166nm is the electron broadening parameter for the O I (777.19 nm) line^[16]. The measured plasma densities of the filaments generated by different repetition rates laser pulses are shown in Fig. 3(b). The measured average plasma density of the 100 Hz filament is 1.54×1017cm−3. The plasma density of the filament decreases with the increase of the filament repetition rate. The measured average plasma density of the 1000 Hz filament decreases by ∼10%, down to 1.43×1017cm−3. The measured plasma density is in agreement with Refs. [17,18] under similar conditions.

Numerical simulations on the laser filamentation in air under different laser repetition rates were performed by solving the nonlinear Schrödinger equations corrected by the “low density hole”^[21]. The equations read as $$\begin{gathered} ∂E∂z=i12k0Δ⊥E−ik′′2∂2E∂t2+iω0cn2E−βK2IK−1E \\ −σ2(1+iω0τ)ρE+ik0ΔnE, \end{gathered}$$(2)∂ρ∂t=βKKℏω0(1−ρρair)IK.(3)

The values of the parameters in Eqs. (2) and (3) vary with the air density, $$\begin{gathered} τ=τ0/ρindex,σ=σ0×ρindex(1+ω02τ02)ρindex2+ω02τ02, \\ βK=β0K×ρindex,n2=n2,0×ρindex, \\ k′′=k0′′×ρindex, \end{gathered}$$(4)where ρindex=ρair/ρat denotes the relative air density^[21]. When the ambient air is under 1 atm, the nonlinear coefficient of the Kerr effect is n2,0=3.2×10−23m2/W, the momentum transfer collision time is τ0=350fs, the cross section for the inverse bremsstrahlung is σ0=2×10−24m2, the coefficient related to the multiphoton ionization is β0K=1.27×10−160m17/W9, and the coefficient of group velocity dispersion is k0′′=2×10−29s2/m.

The process of heating caused by filamentation can be regarded as an isochoric (constant volume) process^[25]. The peak temperature variation ΔTpeak through plasma recombination in air is calculated as^[21]ΔTpeak(z)=Uρplasma(r=0,z)cvρat,(5)where the ionization potential energy for the air molecules is U=14.6eV. ρplasma(z) is the longitudinal distribution of the plasma density in the single-pulse case, and cv is the gas isochoric heat capacity per molecule. The ambient air under 1 atm can be approximately regarded as an ideal diatomic molecular gas, and cv=5kB/2. kB=1.38×10−23J/K is the Boltzmann constant. ρat is the neutral molecule density in the air under 1 atm.

The air density of the density hole induced by the energy deposition in the air is described as^[21]ρair(z)=ρat−Δρairpeak(z)exp(−r2R(z)2),(6)Δρairpeak(z)=ρatΔTpeak(z)R02(z)(ΔTpeak(z)+Ta)R2(z),(7)where Ta=300K is the ambient air temperature, and R(z)=(R02(z)+4αΔt)1/2 is the radius of the density hole. The initial radius of the density hole R0(z) is the radial distribution of air density in FWHM at the z position. Then, the “low density hole” will evolve with the thermal diffusivity α=0.19cm2/s^[26,27]. Δt is the pulse temporal spacing of the subsequent pulse.

The refractive index change contributed by the density hole is written as^[27]Δn(z)=−Δnm(z)exp(−r2R2(z)),(8)Δnm(z)=(n0−1)ΔTpeak(z)TaR02(z)R2(z),(9)where n0=1.000275 is the refractive index of the ambient air, and Δnm is the maximal refractive index change^[27].

The initial laser beam is assumed to be Gaussian type as described by E=E0exp(−r2/w02)exp(−t2/τp2)exp(−ik0r2/2f).(10)

In our experiment, the waist of the initial beam is w0=7mm, the pulse duration is τp=35fs, and the focal length is f=30cm.

Both experimental and theoretical results [Fig. 3(b)] show that the average plasma density of the filament zone decreases with the increase of the laser repetition rate, which confirms that the repetition-rate dependent low-density hole plays a significant role in the ionization of air molecules during the process of laser filamentation in air. The low-density hole will cause a smaller nonlinear refractive index coefficient and lower ionization rate. The effect caused by the low-density hole is enhanced with the increase of the laser repetition rate, which is due to the shorter pulse interval time.

The two key parameters in comprehending the intricate phenomena occurring in plasma are electron temperature and its density. When the plasma is in the local thermal equilibrium (LTE) condition, these parameters can be measured. The LTE condition of plasma was justified by the fulfillment of the McWhirter criterion, which describes the minimum plasma density required for the plasma to be in the LTE state. The McWhirter criterion reads^[28]Ne≥Ncr=1.6×1012T(ΔE)3,(11)where T (in K) is the temperature of the plasma, ΔE (in eV) is the largest gap between adjacent energy levels, and Ncr is the critical density for the LTE. In the case of the O I (777.4 nm) line, the energy gap is about 1.59 eV, and the temperature of the plasma in the filament is about 5000 K. The minimum plasma density required to satisfy the McWhirter criterion is about 4.5×1014cm−3. The average plasma density of the filament zone is above 1.4×1017cm−3, which satisfies the McWhirter criterion [Eq. (11)].

In the LTE condition, the plasma temperature is the electron temperature^[29], not the thermal temperature. The plasma temperature can be determined by applying the formula to the intensity of the spectral lines for a transition from an upper level k to lower level i^[30], ln(IkiλkigkAki)=−EkkBT+C,(12)where λki is the emission wavelength, Iki is the integral intensity of the emission line, gk is the degeneracy of the upper-level energy, Aki is the transition probability, Ek is the energy of the upper-level k, kB=1.38×10−23J/K is the Boltzmann constant, and C is a constant. In order to determine the temperature, neutral atomic O I lines were used. The spectroscopic information for Aki, Ek, and gk of the O I spectral lines is indexed from the National Institute of Standards and Technology (NIST) database^[31]. The Boltzmann plot was drawn according to Eq. (12) based on the O I triplet centered at 777.4 nm and 844.6 nm, as shown in Fig. 4(a). The plasma temperature was obtained by fitting the line in the Boltzmann plot, as shown in Fig. 4. The plasma temperature is obtained through the slope (−1kBT). The slope (−1kBT) from the 100 Hz filament is more inclined than that of the 1 kHz filament in the Boltzmann plot, which indicates that the plasma temperature of the 1 kHz filament is higher than that of the 100 Hz. The measured plasma temperature of the 100 Hz filament is ∼5100K. The plasma temperature increases with the increase of the laser repetition rate, as shown in Fig. 4(b). The measured plasma temperature of the 1000 Hz filament increases by 22% up to ∼6230K, which is beneficial for plasma filament-related applications, such as air lasing^[12], three harmonic generation^[23], and supercontinuum generation^[8]. The measured plasma temperature is in agreement with Ref. [18] under the similar conditions. The numerically obtained average intensities of the filament zone under different laser repetition rates are shown in Fig. 4(b). The variation trend of the average intensities of different repetition rate filaments agrees with that of the plasma temperature. After the ionization of air molecules, free electrons are accelerated in the laser field. The free electrons gain more energy under a high-repetition rate due to the higher intensity inside the filament^[21], which results in a higher plasma temperature.

The effect of the pulse repetition rate on the plasma inside the filament can be understood as the following. The laser energy is deposited into air molecules through the ionization and the fast recombination of the plasmas (∼10ns) inside the filament zone. A high temperature and pressure zone is formed after ionization. Subsequently, the shock wave is formed (∼1μs). The pressure recovers to the environment level at millisecond time scales. According to the ideal gas equation, the gas density distribution is opposite the temperature distribution. A low-density zone is formed and weakened over time. For a higher repetition-rate laser with a pulse interval time that is shorter than the thermal diffusion time, the succeeding pulse propagating in the zone experiences the lower air density zone as well as the lower ionization rate. As a result, fewer air molecules are ionized, as shown in Fig. 3, and less laser energy is spent on the ionization in the filament. The intensity inside the filament is higher. The free electrons from the ionization of the air molecules gain more energy under the high repetition rate due to the higher intensity. The plasma temperature is increased under the high pulse repetition rate, as observed in Fig. 4.

Note that when keeping the laser pulse energy constant and increasing the laser repetition rate, the laser power will increase. As a consequence, a decrease in plasma density and an increase in plasma temperature were observed in this work. When keeping the laser repetition rate constant and increasing the laser pulse energy, the laser power will increase too. Then, an increase in both plasma density and plasma temperature was observed in the literature^[19].

In summary, we have carried out a spectroscopic analysis of the side fluorescence produced by oxygen from a laser-ionized air plasma filament under different pulse repetition rates. The plasma temperature and its density were estimated through the atomic fluorescence produced by the oxygen. The measured plasma density decreases by 10% from 1.54×1017cm−3 to 1.43×1017cm−3, and the temperature of the plasma increases by 22% from 5100 K to 6230 K when the filament pulse repetition rate increases from 100 Hz to 1000 Hz. Nonlinear Schrödinger equations based on numerical simulation results indicate the low-density hole, resulting from the pulse cumulative effect of the high repetition rate filament, which leads to a lower ionization rate. Under a higher pulse repetition rate, the effect caused by the low-density hole is enhanced due to the shorter pulse interval time. Consequently, the plasma density of the laser filament is lower, and the intensity inside the filament is higher for the higher repetition rate filament. The temperature of the plasma is higher due to the free electrons that are accelerated in a higher laser field. We believe that the results not only improve the understanding of the laser repetition effect on the laser filamentation in air but also provide scientific guidance for high repetition rate filament applications.