Femtosecond laser filamentation is a nonlinear laser propagation effect in transparent media caused by the dynamic interplay among self-focusing, plasma-induced defocusing, and diffraction^[1–3]. This nonlinear phenomenon can generate a high and nearly constant laser intensity of ∼1013–1014W/cm2 in long distances ranging from 1 to 103 m in air^[4–8], providing a versatile platform for atmospheric remote sensing^[9–11], lightning control^[12,13], laser-induced water condensation^[14–16], and terahertz pulse generation^[17–20]. Accurate characterization of the filamentation process is the prerequisite for these applications.

However, the ultra-high intensity inside the filament can damage almost all the detectors and excite various intricate higher-order nonlinear optical effects^[21]. These extreme conditions and complicated phenomena make the diagnosis of filamentation challenging. Fortunately, the abundant photo-thermo-acoustic energy conversion effects during femtosecond laser filamentation open the door to exploring and diagnosing the laser filament. During the past few years, different methods have been proposed to measure the spatial profile of the laser filament. Hosseini et al.^[22] proposed an electromagnetic pulse detection technique in which the length of a filament is obtained by detecting electromagnetic pulses radiated from multipole moments inside the filament with an antenna. The axial profile of the filament can also be obtained by monitoring the longitudinal fluorescence^[23]. Other methods such as plasma-guided corona discharges and electrical conductivity have also been shown to be effective for filament characterization^[24–26]. Yu et al.^[27] used ultrasound methods to determine the length and the spatial profile of the free-electron density of a filament. Compared with other methods, acoustic detection is simpler and more sensitive. However, all the above methods require the detector to move step by step along the filament. As a result, the diagnosing represents the average result of multiple pulses, which cannot give an instantaneous and rapid filament diagnosis.

In this Letter, we introduce a novel filament diagnosing method in which the longitudinal spatial distribution of the filament can be obtained by a single measurement of the acoustic emission generated by the optical filament. By measuring the filament-induced acoustic wave at different detection distances and different detection angles, the analytical dependence of the acoustic signals on the detection distance and angle is summarized. Then the spatial distribution of the filament in air is obtained through Wiener filter deconvolution (WFD) processing based on a single acoustic wave measurement. The spatial distribution of the filament obtained by the proposed method is in good agreement with those of traditional point-by-point acoustic diagnosis methods^[28–31].

The point source model of the optical filament is schematically shown in Fig. 1. In this model, the long plasma channel is divided into a series of short or point-like plasma regions, each of which is assumed to be a point source. In Fig. 1, the filament-induced acoustic wave reaching the microphone at time t can be regarded as the integration of the acoustic waves generated by these point sources. The measured signal G(t) by the microphone can be expressed as the weighted sum of a series of time-shifted acoustic waves f(t) generated by the point sources, G(t)=∑x=1x=Na(x)r(x)g(φ)f[t−r(x)va],(1)where r(x) is the distance between the point source at x and the microphone [r(x)=d2+x2], a(x) is the acoustic amplitude generated by the point source at x and detected at r=1 and φ=0, g(φ) is the amplitude modified function due to the incident angle φ of the acoustic wave on the microphone, and va is the sound velocity. As shown in Fig. 1, the filament is located along the x-axis ranging from x1=0 to xN=L. Therefore, Eq. (1) can be simplified to be G(t)=∫0Lh(x)f(t−x)dx=h(t)∗f(t).(2)

According to Eq. (2), the measured signal function G(t) can be expressed as the convolution of functions h(t) and f(t). Thus, once the form of the point source function f(t) is determined, h(t) can be obtained by deconvolution of Eq. (2). Since h(x)=a(x)g(φ)/r(x), the axial acoustic amplitude distribution a(x) can be obtained after g(φ) and r(x) are determined. Since the acoustic generation can be attributed to purely thermo-elastic effects and the plasma acoustic source is practically massless^[32], in Eq. (2), the interactions between the acoustic point sources and the diffraction of the acoustic wave are not taken into account.

The experimental setup is shown in Fig. 2. A commercial Ti:sapphire femtosecond laser amplifier (Legend, Coherent Inc.) was used to deliver 50 fs and 800 nm pulses with an energy of up to 5 mJ at a repetition rate of 500 Hz. The Gaussian femtosecond laser beam (8 mm diameter at the 1/e2 level) was focused by the combination of a plano-concave lens L1 (f=−75mm) and a plano-convex lens L2 (f=500mm). By varying the relative distance between L1 and L2, the position of the filament can be precisely controlled. A sensitive microphone (378C01, PCB Inc.) was used to detect the filament-induced acoustic emission, and the oscilloscope (DPO3034, Tektronix Inc.) was used to monitor and record the acoustic signal. A metal slit with tunable width is placed near the optical filament, and the acoustic signal induced by the filament of different lengths or even point sources can be obtained by varying the slit width. The distance between the filament and the metal slit was 5 mm, and the distance between the filament and the microphone was 15 cm. The microphone together with the metal slit is installed on the translation stage, which moves step by step along the filament during the experiments. The moving step is 10 mm, which is identical to the initial width of the slit.

In order to measure the spatial distribution of the filament via Eq. (2), the effects of the detection distance r and detection angle φ on the temporal profile of the acoustic wave f(t) generated by the point-like plasma must be first investigated. It should be noted that the point-like plasma is generated by focusing the femtosecond laser using a short focal length lens (f=5cm). In this case, the length of the plasma is shorter than 1 mm, which is two orders of magnitude shorter than the optical filament and thus can be considered to be a point source. According to the point acoustic source model, the temporal profile of the acoustic wave generated by a point source is independent of the detection distance and detection angle. The temporal profiles of the acoustic pulses induced by point sources were studied for different detection distances ranging from 10 to 22 cm, and the experimental results are shown in Fig. 3(a). In Fig. 3(a), the incident angle φ is kept at zero. As the distance r between the microphone and the point source increases, the arrival time of the acoustic pulse increases, and a 1/r dependence of the acoustic amplitude is observed, as shown in Fig. 3(b). The relation between the acoustic pulse amplitude A (peak-to-peak value) at distance r and the acoustic pulse amplitude a at the unit distance is A=ar.(3)

Figure 3(c) shows the normalized temporal profiles of the acoustic waves at different detection distances. It is seen that the temporal profiles of the acoustic pulses are nearly constant as the detection distance increases from 10 to 22 cm. The time-domain signal in Fig. 3(a) was Fourier transformed to obtain the acoustic spectra in Fig. 3(d), and the normalized acoustic spectra are presented in Fig. 3(e). At different detection distances, there were no significant changes in the shape of the acoustic spectrum, except that the amplitude and the bandwidth of the acoustic spectrum is kept constant at 300 kHz. The microphone used in the experiment has a flat frequency response curve below 100 kHz, and the responsivity rapidly decreases beyond 100 kHz. Therefore, the upper detection limit of 300 kHz is mainly caused by the attenuation of the ultrasonic waves in air and the bandwidth of the microphone.

In the following, the dependence of the temporal profile of the acoustic pulse generated by the point-like plasma source on the detection angle was studied, and the schematic of the experimental setup is shown in Fig. 4(a). During the measurements, the microphone is fixed on a circle with a diameter of r=1m. The point-like plasma source is located at the center of the circle. The microphone moves along the circle with a step of 10° ranging from 0° to 80°. A larger φ cannot be achieved since the microphone blocks the femtosecond laser in these cases. Note that the normal direction of the sensing area of the microphone is kept perpendicular to the laser propagation direction during the measurements. Figure 4(b) shows the temporal profiles of the acoustic pulses under different detection angles φ. The dependence of the acoustic pulse amplitude on φ is summarized in Fig. 4(c). The acoustic amplitude reaches the maximum when φ=0° and goes to the minimum when φ increases to 80°. The directivity function of the microphone can be written as g(φ)=Sacosφ,(4)where S is the area of the microphone sensing area and a is the acoustic pulse amplitude detected at φ=0° and r=1m. By normalizing the acoustic signal in Fig. 4(b), it is found that the acoustic temporal profile is roughly independent of the detection angle, which is shown in Fig. 4(d). By Fourier transforming the acoustic pulses in Fig. 4(b), the normalized acoustic spectra are shown in Fig. 4(e). It is seen that there is no significant change in the bandwidth and shape of the acoustic spectrum.

From the above experimental results, the temporal profile of the acoustic wave generated by the point-like plasma is independent of the detection distance and detection angle. This implies that once the laser parameters are determined, the point source function f(t) [see Eqs. (1) and (2)] can be measured experimentally. In our experiment, a 5 mJ, 50 fs, and 800 nm femtosecond laser pulse is focused to generate the optical filament. Figure 5(a) shows the temporal profiles of a point-like plasma-induced acoustic pulse by placing a 1 mm metal slit adjacent to the optical filament. Figure 5(b) shows the temporal profile of the acoustic pulse induced by the optical filament. The function h(t) in Eq. (2) can be obtained through the deconvolution of Eq. (2) after the point source function f(t) and the optical filament-induced acoustic pulse G(t) are determined. f(t) and G(t) are, respectively, shown in Figs. 5(a) and 5(b). The Wiener filter is used in the deconvolution since the temporal signal recorded in the experiments has a finite time interval. The Wiener filter is expressed as P(υ)=|G(υ)|2|G(υ)|2+K,(5)where G(υ) is the Fourier transform of G(t) and K is a constant related to the background noise. The frequency estimates of the function h(t) by WFD are H(υ)=G(υ)P(υ)F(υ),(6)where H(υ) and F(υ) are Fourier transforms of h(t) and f(t), respectively. Then, the function h(t) can be easily obtained after the inverse Fourier transformation according to Eq. (6). The calculated function h(t) is shown in Fig. 5(c).

According to the point source model of the optical filament shown in Fig. 1, the spatial coordinate x is expressed as x=r(x)2−d2, where r(x)=va·t, Therefore, the time axis in Fig. 5(c) can be converted to the x-axis along the optical filament, which is shown in Fig. 6(a). In addition, since the amplitude of the acoustic pulse emitted from the optical filament at x can be expressed as a(x)=h(x)r(x)/g(φ), the distribution of the acoustic pulse emitted from different positions of the filament can be obtained, which is shown in Fig. 6(b).

Finally, in order to confirm the accuracy of the above results, the traditional filament characterization method is employed to evaluate the spatial distribution of the optical filament. The experimental setup is shown in Fig. 2. During the measurement, the microphone and the metal slit move along the filament step by step to avoid measurement errors caused by changes in detection distance and detection angle. By measuring the acoustic pulse emitted from the filament and transmitted through the metal slit point by point, the longitudinal spatial distribution of the filament can be obtained.

Figure 7(a) shows the temporal profiles of the acoustic pulses measured at different positions along the filament using the microphone and the metal slit. The laser filament is generated by the lens group with an effective focal length of 5 m and a single pulse energy of 5 mJ. In Fig. 7(a), it is seen that the acoustic pulse duration is roughly 100 µs, and the waveform consists of 2–3 main oscillation cycles followed by a series of small ripples. The temporal profiles in Fig. 7(a) are normalized and shown in Fig. 7(b). It is seen from Fig. 7(b) that the temporal profiles of the acoustic pulses measured at different positions along the filament are identical.

The dependence of the peak-to-peak value of the acoustic pulse on the position along the filament is presented as the red curve in Fig. 8. The blue curve in Fig. 8 is the spatial distribution measured by the proposed method, which is in good agreement with the results obtained by traditional point-by-point methods. However, it should be emphasized that there is a slight discrepancy between these two results based on the different values of the constant K. The proposed method is only possible to characterize the longitudinal spatial profile of the optical filament, not including the transverse profile. Since only a single laser shot-induced acoustic pulse needs to be measured once, the characterization speed (less than one minute) is much faster than that of the traditional point-by-point method.

In this work, a novel single-shot optical filament diagnosing method is proposed. The acoustic wave emitted from the femtosecond laser-induced plasma is systematically studied. It is found that the temporal profile f(t) of the acoustic wave generated by a point-like plasma is independent of the detection distance and the detection angle, which can be regarded as the point source response function in measuring the spatial distribution of the laser filament. After determining the relation among the acoustic emission, the detection distance, and the detection angle experimentally, a single measurement of the acoustic emission generated by a single laser pulse-induced filament can present the spatial distribution of the filament using the WFD algorithm. The results are in good agreement with those obtained by traditional point-by-point acoustic diagnosis methods. These findings provide a new solution for the rapid and remote diagnosis of the optical filament, thereby laying a firm foundation for femtosecond laser filament-based promising applications.