Filamentation refers to the phenomenon of the plasma channel generated by the dynamic balance between the optical Kerr self-focusing effect and the plasma defocusing effect caused by the ionization of neutral molecules when an ultrashort pulsed high-intensity laser propagates in a transparent medium such as air^[1–4]. Femtosecond laser filamentation opens up many potential applications involving lightning control^[5,6], atmospheric condensation and precipitation^[7,8], remote sensing^[9–11], THz emission^[12,13], spectral broadening, and pulse compression^[14]. Thus, the diagnosis of the filament, such as plasma channel size and electron density, is of great significance for understanding nonlinear propagation and applications of femtosecond laser pulses. The electron density inside the filament depends on external focusing and pulse energy^[15]. There have been a variety of methods to experimentally characterize the electron density. Some methods, such as plasma electrical conductivity measurements^[16,17], plasma acoustic wave probing^[18], and nitrogen fluorescence detection^[15], are simple and practicable but need to be calibrated. The other group includes interferometry^[19–22], holography^[23–25], and diffractometry^[26–28], which are based on the pump-probe scheme. For these approaches, the wavefront spatial phase of the probe beam is modulated when it passes through the plasma channel due to the difference of the refractive index between the filamentation region and ambient background. The electron density information can be unveiled through the phase change. Thus, time-resolved measurement can be achieved using time delay between the filamenting pulse and probe pulse. Typical interferometry provides more information, especially on axially resolved electron density distribution. In an interferometric scheme, the probe is typically split into two beams to record the interference fringes on the CCD. The spatiotemporal synchronization of the two beams is required. Longitudinal diffractometry is much simpler with the sacrifice of the axially resolved information. The ionized plasmas produced by ultrafast laser pulse recombine to neutrals in a time range of a few nanoseconds with their energy repartition into molecules in translational, rotational, and vibrational energies^[29]. The localized energy deposition subsequently launches an acoustic wave along the radial direction. As a consequence, a low-density air channel is established. After a few microseconds, the air pressure forms a quasi-equilibrium. The temperature decreases radially, and the gas density changes inversely to the temperature. Then, the density depression decay is dominated by thermal diffusion on millisecond timescales^[29–33]. Hence, there exists a low-density air channel generated by the cumulative effect of a high repetition rate pulse train. The phenomenon of femtosecond filaments that generate long-lived underdense channels was observed by Cheng et al.^[29] experimentally using 0.7 mJ pump pulse energy at a repetition rate of kilohertz. Following the study, Point et al.^[32] proved that the low-density channel could last more than 90 ms after filamentation in the case of 5 mJ laser pulse energy under tight focusing. After many pulses pass through, the low air density channel will stabilize when every following pump pulse comes for the laser repetition higher than 10 Hz. Previous studies using longitudinal diffractometry^[15,26,34] considered that the effect of the low-density electrons on the probe only contributes the additional phase shift term of the complex amplitude at the exit of the plasma channel. The defocusing effect on the intensity distribution was assumed to be negligible. But, the intensity of the probe does experience defocusing effects, especially at the electron density in air filament. Moreover, the previous works ignored the existence of the long-lived underdense air channels under the experimental conditions of high repetition rate, which would affect the accuracy of the measurement results.

In this paper, the longitudinal diffractometry based on pump-probe technology is used to record the diffraction imaging patterns of the probe after passing through the plasma channel at different delay times. Here, we establish a segmented diffraction imaging model to precisely simulate the whole diffraction process according to scalar diffraction theory, which is not only suitable for extracting lower electron densities, but also for the case where the defocusing effect in filament needs to be considered at a higher electron density. The temporal evolutions of electron density and plasma filament diameter are obtained from the extraction of diffraction patterns. Furthermore, the influence of long-lived thermal effects under high-repetition laser pulses is also taken into account to precisely measure the electron density and plasma channel size. When the laser pulse is of high repetition rate, the defocusing effect of the low-density air channel on the probe beam needs to be considered, otherwise, the measured electron density and plasma channel size will be larger. The increasing of plasma channel size over decay time is measured, which is contrary to the constant reported by Liu et al.^[26]. Our results are further supported by numerical simulation through solving the rate equations of the charged particles.

The schematic diagram of the pump-probe experimental setup is shown in Fig. 1. A Ti:sapphire chirped pulse amplification (CPA) laser system was employed to generate 32 fs laser pulses at a central wavelength of 800 nm. The repetition rate of the output pulse from the CPA system can be adjusted from 10 Hz to 1 kHz. Here, we use 1 kHz in our experiments in order to look into the high-repetition-related low-density effect. The output laser pulse is split into two beams. One is focused by a lens (L1) with 30 cm focal length to generate a filament in air. The energy of the filamenting pulse is 4.5 mJ detected by an energy meter behind the dichroic mirror (M4). The B-integral after passing through the bulk materials and air is estimated to be 1.55. The length of the filament ($L_{\mathrm{fil}}$) obtained by the fluorescence is $\sim 1\mathrm{cm}$. The other beam is frequency doubled to 400 nm as a synchronous probe pulse. The probe collinearly propagates with the filamenting pulse after passing by the delay unit (mirrors M2 and M3) and the dichroic mirror (M4). Then, it is recorded on a 16-bit CCD camera (pixel size: $6.45\mathrm{\mu m}\times 6.45\mathrm{\mu m}$) after passing through a UV imaging lens (L2 with focal length of 20 cm). The distances from L2 to L1 and CCD are 71 cm and 45 cm, respectively. The other dichroic mirror (M5) behind the filament is used to cut off the 800 nm pump laser. A neutral density filter (F1) together with a UG11 band-pass filter (F2) is placed before L2. Another band-pass filter (F3) with central wavelength of 400 nm and FWHM of 10 nm is inserted behind L2. Theses filters (F1, F2, and F3) are used to filter the plasma luminescence, third harmonic, and the residual pump pulse out of the CCD as well as avoiding its saturation. The polarization of the laser pulses is linearly polarized in the same direction. Previous works proved that the plasma density decreases, and the clamping intensity increases when the polarization of the filamentation laser is tuned from linear to circular^[35,36]. A series of diffraction patterns of probe pulses recorded by CCD for various delays are shown in Fig. 2. The image of the probe beam alone is also recorded by blocking the pump beam to work as the reference. The median time of the two delay time slots, when the central bright spot firstly appears and finally disappears, is taken as the zero delay. For each pump-probe delay, 30 images are recorded.

The diffraction process of the probe pulse can be depicted by scalar diffraction theory, which is often used to describe free space optical propagation. A reasonable approximation of the propagation effect can be obtained. The Fresnel diffraction integral is widely used in the bidimensional free space propagation along $z$, which can be written as^[37]$$U_2(x,y)=\frac{e^{jkz}}{j\lambda z}\iint U_1(\xi ,\eta )\exp \{j\frac{k}{2z}[{(x-\xi )}^2+{(y-\eta )}^2]\}\mathrm{d}\xi \mathrm{d}\eta .$$(1)

The equation above connects the complex amplitude distribution ($U_1$ and $U_2$) at two positions on the $z$ axis through the impulse response. The refractive index $n$ in the plasma channel can be expressed as^[22]$$n={(1-\frac{{\omega }_p^2}{{\omega }^2})}^{\frac{1}{2}}={(1-\frac{n_e}{n_c})}^{\frac{1}{2}}\approx 1-\frac{n_e}{2n_c},$$(2)where ${\omega }_p$ is the plasma frequency, $\omega$ is the probe pulse frequency, and $n_e$ is the electron density. The critical density $n_c$ is given by $$n_c={\omega }^2{m}_e{\epsilon }_0/e^2.$$(3)

Assuming that the electron density has a Gaussian distribution across the plasma channel diameter and is cylindrically symmetric, the radial distribution of the electric density, $n_e(r)$, can be written as $$n_e(r)=n_{e0}\exp (-r^2/w^2),$$(4)where $n_{e0}$ is the peak electron density, and $w$ is the radius at which the electron density drops to $1/e$. Hence, the spatial phase shift of the probe beam caused by the filament has the same Gaussian profile and can be written as $${\phi }_{\mathrm{fil}}=-\frac{n_e(r)}{2n_c}k\mathrm{\Delta }d,$$(5)where $\mathrm{\Delta }d$ is the distance of the probe beam passing through the plasma channel. A larger phase change $|{\phi }_{\text{fil}}|$, in other words, a higher electron density, means an intense diffraction effect on the probe beam. What is mentioned above is exactly embodied in experimental diffraction patterns at different delays (Fig. 2). The ring number and the radius of the first-order bright rings increase statistically when reducing the pump-probe time delay [see from Figs. 2(f) to 2(c)]. However, the feature near zero delay, as shown in Fig. 2(b), no longer conforms to this variation trend, and there exists a distinctive bright spot in the center. The appearance of the central bright spot and the abnormal contraction of the inner ring could be attributed to the optical Kerr effect produced by cross phase modulation (XPM) on the probe beam^[4]. Suppose the pump laser intensity in the filament region is a Gaussian profile with a $1/e$ radius of $r_p$, so the phase shift of the probe beam caused by the optical Kerr effect is given by $${\phi }_{\text{Kerr}}=2n_{2}k\mathrm{\Delta }d\times I_0\exp (-r^2/r_p^2),$$(6)where $n_2$ is the nonlinear index due to the Kerr effect, and $I_0$ is the clamp intensity, which is assumed to be ${10}^{14}\mathrm{W}·{\mathrm{cm}}^{-2}$ in our experiment^[36]. Equations (5) and (6) show the opposite effect on the wavefront of the probe beam, which means that Kerr phase shift will weaken the plasma filament phase shift. The bright spot appears when the center is a positive phase shift relative to the outside, which indicates that the center is focused with respect to the outside. Liu et al.^[26] assumed that in the case of lower electron density ($<{10}^{17}{\mathrm{cm}}^{-3}$), the complex amplitude of the probe at the end of the plasma channel can be expressed as $U_{\mathrm{out}}=\exp (-a{r}^2)\exp [j\varphi (r)]$, where $\varphi (r)$ is the phase shift of the probe beam caused by filament at the exit of the plasma channel. This expression assumes that the intensity distribution of the probe beam after passing through the filament does not change relative to initial distribution, which means both are the Gaussian profile $\exp (-2a{r}^2)$. Considering the negative phase change in the filament region, the diffraction effect can also be considered as a defocusing effect, leading to the change of the amplitude distribution at the exit of the plasma channel. Here, we utilize the following procedure to mimic the defocusing effect of the probe in filament. As is shown in Fig. 3, the plasma channel is evenly divided into a set of sections ($N$) with a length of $\mathrm{\Delta }d=L_{\text{fil}}/N$. The influence of the filament on the probe in each section can be treated as phase change only. The initial scalar field of the probe at the entrance of the filament ($z=0$) can be written as $U_{\text{in}}=\exp (-a{r}^2)$. Then, the diffraction process over distance $\mathrm{\Delta }d$ is calculated using Eq. (1). Subsequently, the complex amplitude at position $z=\mathrm{\Delta }d$ is multiplied by a phase shift term $\exp (j{\phi }_0)$ as the result of this section, where ${\phi }_0={\phi }_{\text{fil}}+{\phi }_{\text{Kerr}}$ for the delay near zero, and ${\phi }_0={\phi }_{\text{fil}}$ for others. After that, the diffraction process over another distance $\mathrm{\Delta }d$ is calculated using the result of the previous section as the initial condition, and then the phase shift term is multiplied once more. We repeat this iterative process to get the final field distribution $U_{\mathrm{out}}$ at the exit of the filament. For a filament in our experiment with a length of 1 cm, when the section number is 25 and above, the intensity distribution at the exit will tend to be stable, which has insignificant impact on subsequent calculations till the CCD imaging plane. The field $U_{l\prime }$ in front of the lens L2 is obtained by Eq. (1) with $U_{\mathrm{out}}$. The complex amplitude relationship between the two sides of the lens L2 can be written as^[37]$$U_l(r)=U_{l\prime }(r)\exp (-j\frac{k}{2f}{r}^2),$$(7)where the exponential term is the transmittance function for an ideal lens, and $f$ is the focal length of the lens L2. Finally, we have the field at the CCD imaging plane through free space propagation using $U_l$. To summarize, the calculated intensity of the diffraction pattern is determined by the peak electron density $n_{e0}$ and the filament size 2w. These two parameters can be extracted via the comparison between simulation and normalized experimental results. In our calculation, the spacing between points on the field plane is set to 6.45 µm, which is consistent with the pixel size of the CCD camera in order to facilitate subsequent fittings. With various combinations of these two variables $n_{e0}$ and $w$, a group of calculated diffraction patterns are obtained, which contain the most consistent results with the experimental patterns. Each of the simulation results is used to compute the sum squared error (SSE) compared with the experimental result. When the SSE is minimum, the obtained peak electron density and filament size are determined, respectively. The extraction procedure is used for each diffraction pattern. Hence, the temporal evolution of the transverse profile of the plasma channel is plotted.

On the other hand, the temporal evolution of the plasma channel can be numerically estimated by solving the rate equations of the charged particles^[38]: $$\begin{gathered} \frac{\partial n_e}{\partial t}=\alpha n_e-\eta n_e+{\beta }_{ep}{n}_e{n}_p, \\ \frac{\partial n_p}{\partial t}=\alpha n_e-{\beta }_{ep}{n}_e{n}_p-{\beta }_{np}{n}_n{n}_p, \\ \frac{\partial n_n}{\partial t}=\eta n_e-{\beta }_{np}{n}_n{n}_p, \end{gathered}$$(8)where $\alpha$ is the collisional ionization rate, $\eta$ is the electron attachment rate, and ${\beta }_{ep}$ and ${\beta }_{np}$ are the electron-ion recombination coefficient and ion-ion recombination coefficient, respectively. These two recombination coefficients can be taken to be equal since they are in the same order of magnitude^[38]. ${\beta }_{ep}$ is given by^[38]$$\begin{gathered} {\beta }_1({\mathrm{m}}^3/\mathrm{s})=2.035\times {10}^{-12}{T}_e^{-0.39}({\mathrm{e}}^{-}-{\mathrm{N}}_2^{+}), \\ {\beta }_2({\mathrm{m}}^3/\mathrm{s})=1.138\times {10}^{-11}{T}_e^{-0.70}({\mathrm{e}}^{-}-{\mathrm{O}}_2^{+}), \\ {\beta }_{ep}=0.79{\beta }_1+0.21{\beta }_2, \end{gathered}$$(9)where $T_e$ is the electron temperature.

As previously mentioned, the low-density air channel is an important factor to be considered in the propagation of the laser beam. The negative refractive index variation due to the low-density channel will defocus the next filamenting pulse as well as the probe pulse propagating in the same direction because the time interval between two consecutive laser pulses is too short for the low density channel to completely relax. The intensity distribution of the probe owing to the low-density channel can be recorded when the plasma is gone. It is the case that the probe beam is ahead of the pump beam in time, which is shown in Fig. 2(a). We calculate the average intensity distribution along the radial direction from Fig. 2(a), which shows the influence of the low-density region on probe intensity. The result is shown in Fig. 4(a). A laser filament will be formed in the central area when the next filamenting pulse comes. In order to estimate the refractive index variation caused by the low air density channel, the above method for determining the lateral distribution of electron density is applied to extract the refractive index profile in a similar manner. Here, the refractive index change is also assumed to have a Gaussian distribution radially, $\mathrm{\Delta }{n}_{\mathrm{air}}=\mathrm{\Delta }{n}_0\exp (-r^2/r_{\mathrm{air}}^2)$. The phase shift can be written as ${\phi }_{\mathrm{air}}=\mathrm{\Delta }{n}_0\exp (-r^2/r_{\mathrm{air}}^2)\times k\mathrm{\Delta }d$. The parameters $\mathrm{\Delta }{n}_0$ and $r_{\mathrm{air}}$ are also determined by the minimum SSE value. The transverse refractive index variation is shown in Fig. 4(b), where $\mathrm{\Delta }{n}_0=-2\times {10}^{-5}$ and $r_{\mathrm{air}}=600\mathrm{\mu m}$ indicate the density and radius of the low-density channel. Since the thermal diffusion is a slow relaxation process (on the order of tens of milliseconds), we can assume that $\mathrm{\Delta }{n}_{\mathrm{air}}$ is time independent in the following discussion of the temporal evolution of the electron density distribution.

Consequently, considering the defocusing effect due to the low-density channel, the phase shift settings for extracting the electron density and filament size need to be corrected. The phase term added at the end of each section should be modified to $\exp (j\phi )$ for calculating the diffraction patterns, where $\phi ={\phi }_0+{\phi }_{\mathrm{air}}$. Figure 5 shows simulated results at a delay time of 0.98 ps. Figure 5(a) is the corresponding radial intensity distribution at the exit of the filament, which shows the strong defocusing effect of the high-density plasma channel. Figure 5(b) shows the radial fitting and experimental curve of the diffraction patterns imaged on the CCD camera. The CCD records the diffraction image of the modulated probe beam after free space propagation. The maximum electron density of the transverse profile and the filament diameter are $n_{e0}=9.4\times {10}^{17}{\mathrm{cm}}^{-3}$ and $2w=90\mathrm{\mu m}$, respectively. The temporal evolution of the electron density profile is shown in Fig. 6. As can be seen from Figs. 6(a) and 6(b), there is a clear difference between the results with and without considering the low-density effect. When the influence of the low-density channel is not taken into account, larger electron density and filament diameter will be obtained, which is due to the negative phase shift from the low-density channel. The initial peak electron density as shown in the inset is $\sim {10}^{18}{\mathrm{cm}}^{-3}$ in our experimental conditions, which is in agreement with other measurements^[15,23] and decays rapidly by nearly two orders of magnitude within 200 ps. Moreover, the plasma channel size rises from $\sim 90\mathrm{\mu m}$ to $\sim 120\mathrm{\mu m}$ as the delay time increases. The electron density and filament size cannot be extracted effectively after 200 ps since the diffraction patterns are rather weak to distinguish from intensity noise.

The temporal evolution of electron density can be obtained by solving Eq. (8) with the following initial conditions, $n_e(t=0)=n_p(t=0)=1.4\times {10}^{18}{\mathrm{cm}}^{-3}$, $n_n(t=0)=0$, and $T_e=3000\mathrm{K}$ is assumed, so ${\beta }_{ep}$ is $0.8\times {10}^{-13}{\mathrm{m}}^3/\mathrm{s}$. This assumption is reasonable considering that electrons are heated. A similar assumption was also made by Tzortzakis et al.^[28], where ${\beta }_{ep}=1.2\times {10}^{-13}{\mathrm{m}}^3/\mathrm{s}$. The numerical result in Fig. 6(a) shows that the peak electron density decays rapidly in the early stage and then relaxes slowly to ${10}^{16}{\mathrm{cm}}^{-3}$, which agrees with the results obtained in the experiment. It is noted that the time dependence of the refractive index is ignored in our simulations. At the early decay stage when the probe pulse width is on the order of a few picoseconds, the time dependence has to be considered since the electron density drops too fast (by more than 30% in the first few picoseconds). In our experiment, the probe pulse duration is in several tens of femtoseconds. Therefore, the time dependence in the model can be safely ignored. The approximation is valid for the probe pulse duration shorter than 1 ps if the change of the electron density is within 10%. Additionally, assuming initial condition ($t=0$) of a Gaussian radial profile with a peak of $1.4\times {10}^{18}{\mathrm{cm}}^{-3}$ and a $1/e$ radius of 34 µm, the radial evolution of electron density at each delay time is obtained. The calculated temporal evolution of plasma channel size is shown in Fig. 6(b), which is qualitatively in agreement with the experimental observation. The electron density decays more rapidly near the center, resulting in an increase in filament size. The expansion of the plasma channel size as the electron density decays was not observed in the paper by Liu et al.^[26]. The extracted transverse distributions of electron density after correcting the low-density effect are shown in Fig. 6(c) for different decay times.

In conclusion, we developed a time-resolved longitudinal diffraction imaging method to characterize the filament generated by 1 kHz femtosecond laser pulses in air. Filament induced diffraction fringes of a 400 nm probe pulse were recorded by a CCD camera at different delay times, which are used to extract the temporal evolution of electron density and diameter of the plasma filament. Since the probe has not only phase disturbance, but also intensity changes when passing through the plasma channel, a segmented process was proposed to deal with the probe propagation on the basis of scalar diffraction theory. The low air density effects from hydrodynamic progress of laser filamentation on the measurement results were quantitatively analyzed. The low air density generated at the 1 kHz filament contributes to the negative phase shift of the probe, resulting in a larger electron density and filament diameter. The measured initial peak electron density is $\sim {10}^{18}{\mathrm{cm}}^{-3}$ in our experimental conditions, which then decays rapidly by nearly two orders of magnitude within 200 ps. Moreover, the measured initial plasma channel diameter is $\sim 90\mathrm{\mu m}$, which then expands to 120 µm as the delay time increases to 200 ps. Numerical simulation by solving the rate equations of the charged particles is performed. The simulated results are in agreement with experimental results. The precise characterization of filaments in this work not only contributes to high repetition or short distance filamentation applications, such as pulse compression, terahertz (THz), and harmonic radiation sources and laser fabrication, but also benefits the understanding of filamentation.