1. Introduction

Laser filamentation is a nonlinear effect resulting from the propagation of ultrashort femtosecond (fs) laser pulses in transparent media [1,2]. When the peak power of fs laser pulses exceeds a so-called critical power P_cr, a dynamical equilibrium between nonlinear Kerr self-focusing and strong-field-ionization-induced plasma defocusing is created, resulting in a long and thin plasma channel called filament, in which the laser intensity is clamped at ∼50-100 TW·cm⁻² [3,4]. Laser filamentation has showed the versatility for a variety of applications such as harmonic generation [5], atmospheric spectroscopy [6], remote fabrication [7,8], and fuel ignition [9–11]. Among these applications, it was found that the efficiency of laser energy coupled to laser plasma filament plays an essential role. On the one hand, the applications that depend on the high-intensity light bullet to interact with samples, such as filament-induced breakdown spectroscopy and fabrication, would expect less energy to couple into the plasma filament [12–14]; on the other hand, high energy deposition efficiencies are in favor of those applications depending on energy conversion processes, such as harmonic and THz generation [15–17], and filament-induced fluorescence spectroscopy of gases [18,19].

Previously, it was demonstrated that under the external focusing condition using a single lens, the energy deposition efficiency varies largely, depending on the incident laser energies and focal lengths [9,10,20–22]. When a single filament is formed in air, the energy deposition efficiency falls normally in the range of ∼2-35%, which decreases when the focal length increases, but increases when the incident laser energy increases [20,21]. When multiple filaments are formed in air using terawatt laser pulses, more than 60% of the pulse energy can be deposited into the plasma, resulting in a peak lineic deposited energy (i.e., the energy deposition per unit length) of more than 1 J·m⁻¹ [20]. In a methane/air mixture, a similar behavior as that in air was observed [9,10]. Interestingly, for a few mJ incident laser energy, with which the energy deposition efficiency falls in the range of ∼27-35%, the laser filament ignition (LFI) of the methane/air mixture was achieved, providing the opportunity for remote ignition of lean fuels using ultrashort pulses [9,10]. However, the investigations on the energy deposition in laser filament have been so far conducted mostly under the focal condition using a single lens. Although it has been proved that a telescope system is capable of projecting the filament to a far distance, it is unclear yet how the efficiency of the energy coupled to plasma filament varies when the distance between the two lenses in a telescope changes, i.e., the effective focal length changes, and thus it is questioned whether fs-LFI could be achieved in a controllable manner with a telescopic filament.

In this work, we investigate the efficiency of the energy deposited into the plasma filament formed by a two-lenses telescope system, which is one of the simplest types of telescopes, but is representative among a variety of telescopes composing of complex optics components. We show that when the distance between the two lenses in the telescope increases, there are two variation regions: (i) the energy deposition efficiency decreases as the effective focal length increases; (ii) the energy deposition efficiency first increases and then decreases as the effective focal length increases. Moreover, we reveal that there exists an optimal region for the average lineic energy deposition (ALED), which will remain the optimal values for a distance range between the two lenses, forming a plateau ALED region, which differs remarkably from the monotonic change trend in a single-lens focusing system. As a proof of principle, we show that the telescope can efficiently extend the ignition distance of a premixed lean fuel/air mixture when compared with that achieved using a single-lens focusing system under the same incident laser energy condition. Our results provide a deeper understanding of the characteristics of telescopic filaments and are helpful for their potential application to laser ignition of fuels.

2. Experiments

The schematic of the experimental setup is shown in Fig. 1(a). A Ti: sapphire laser system (Spectra Physics, Spitfire ACE) produced a femtosecond laser pulse train with the central wavelength at 800 nm and pulse duration of ∼50 fs. The maximum output energy was about 2.6 mJ, which can be attenuated by a half-wave plate and a polarizer. The repetition rate of the laser pulses was able to change from 4 Hz to 1 kHz, and the laser beam diameter was about 10 mm at 1/e². A telescope system consisting of a 25.4 mm-diameter concave lens (L1, f₁) and a 50.8 mm-diameter convex lens (L2, f₂) was mounted on a one-dimensional translation stage. The effective focal length f_e in the two-lens telescope is given by the back-focal-length (BFL) equation [23,24],

 

Fig. 1. (a) Schematic diagram of the experimental setup for the energy deposition measurement and fs-LFI. HR: high reflective mirror; HWP: half-wave plate; L1: concave lens; L2: convex lens; (b) theoretical simulation of f_e as a function d between two lenses in the telescope.

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The energy deposition efficiency was measured by a power meter (Spectra Physics, 1919-R) by recording the powers of the incident and transmitted laser pulse before and after the telescope-induced laser filament at the positions where the whole area of the beam occupies ∼40% of the detection area of the power meter. The filament image was recorded by a CCD camera (Olympus, TG-5) with the exposure time of 0.4 s, corresponding to 400 laser shots to be accumulated. The ALED was calculated as ALED = E/L_f, where E is the deposited energy (E) measured by the power meter and L_f the filament length that is determined from the N₂⁺ emission intensity distribution along filament measured by a grating spectrometer (Andor Shamrock SR-500i) coupled with a gated-intensified charge-coupled device (ICCD, Andor iStar). The emission spectrum of the laser filament was also measured by the same spectrometer. In this measurement, the filament was first aligned to be parallel to the entrance slit of the spectrometer, and then imaged by a fused silica lens (50.8 mm in diameter, f = 6 cm) onto the entrance slit with a 3.5:1 imaging scheme. For the spectral measurement, the slit width of the spectrometer was set at 300 µm, and the signal was dispersed by a 600 grooves/mm grating with the blazed wavelength at 500 nm. The ICCD gate width and delay were set at Δt = 10 ns and t = -5 ns, respectively (note that t = 0 means the timing when the laser pulse arrives at the interaction zone). The data for each measurement were accumulated over 2000 laser shots to increase the signal-to-noise ratio, and the spectra shown in the following were the averaged results of three measurements, with which the uncertainties are determined to be less than 4.8%.

For the fs-LFI measurement, the filament was located at 1 cm above a modified McKenna burner that released a premixed CH₄/air mixture with an equivalence ratio of φ = 0.87 and a flow velocity of 1.04 m/s (Reynolds Number: ∼670). The flame ignited by the filament was imaged by the above spectrometer, but the slit width and the grating mode were changed to be 15 mm and the zero-order mode, respectively. In this case, the gate width and delay of ICCD were set at Δt = 20 ms and t = -5 ns, in order to obtain high-resolution flame images based on the previous measurements on flame kernel evolution [10].

3. Result and discussion

We first performed the measurement of the energy deposition using a single-lens (SL) focusing system with the results shown in Fig. 2. In this measurement, the incident laser energy was fixed at 2.0 mJ, which is about 5 times higher than the critical power P_cr for self-focusing of the laser pulses in air [4]. This energy is far below the laser power (20 P_cr ∼50 P_cr) required for the formation of multifilaments [25]. Shown in Fig. 2(a) are the images of filaments under different focal lengths, from which it can be intuitively seen that the filament length increases as the focal length increases. To preciously characterize the plasma filament, we measured the emission spectrum of the filament (see the inset of Fig. 2(b)) and focused our attention on the emission band of the first negative system of N₂⁺ at ∼391 nm because its intensity is proportional to the area-integrated electron density [20]. We thus obtained the emission intensity distribution of N₂⁺ along the filament by integrating the P-branch spectral signals of the B²Σ_u ⁺-X²Σ_g⁺ transition in the range of from 390.85 nm to 391.97 nm, as shown in Fig. 2(b), where the signal peak representing the plasma density shows a sharp decrease as the focal length increases. This is consistent with earlier studies, in which a huge discrepancy of the plasma density from 10¹³-10¹⁸cm⁻³ was revealed to be strongly dependent on the external focusing conditions [26].

 

Fig. 2. (a) The filament images, (b) the 391-nm intensity distributions along the filament obtained from the emission spectrum (inset), (c) the filament length L_f (red dot) and the energy deposition efficiency (blue square), and (d) ALED obtained under different focal conditions.

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We then determined the filament length (L_f) by adopting the region with the emission intensity exceeding 10% of the peak intensity at 391 nm [27]. It should be pointed out that the adopted definition of this filament length aims at the easy measurement, and that the filament normally extends beyond this visible emission region, because the term “filamentation” more generally refers to the entire region/phenomenon corresponding to the balance of self-focusing, plasma formation, subsequent defocusing [1,2]. Figure 2(c) presents the filament lengths L_f (red dots) obtained with different focal lengths in the SL system. As a result, it can be seen in Fig. 2(c) that L_f increases monotonously as a function of the focal length f. In addition, we plotted in Fig. 2(c) the energy deposition efficiencies (blue square) measured with different focal lengths f. It is known that the efficiency of the energy deposition is related to many dynamical processes during filamentation, such as optical field ionization of atoms/molecules [28], heating of free electrons [29], plasma absorption, and non-resonant rotational Raman absorption [28], all of which are related to the plasma generation [20,21,28]. Therefore, it is reasonable to link the energy deposition results shown in Fig. 2(c) with the N₂⁺ emission intensities shown in Fig. 2(b). That is, as f increases, the decrease in the energy deposition efficiency in Fig. 2(c) can be ascribed to the reduction in the plasma density even if the filament length increases. This can be seen more clearly from the ALED obtained by dividing the total deposited energy by L_f, as shown in Fig. 2(d), from which it can be seen that the ALED also shows a decrease behavior as f increases in the SL focusing system.

Next, we performed the measurement of the energy deposition in the fs plasma filament formed with the telescope T1. The incident laser energy remained at 2.0 mJ. Figures 3(a) and (b) show the measured images of the plasma filaments and the corresponding intensity distributions of the emission at 391 nm along the filament for several different distances d between the two lenses, from which it can be seen that as d increases, both the intensity and length of the filament change. In this case, we experimentally measured the effective focal length f_e as a function of d as shown in Fig. 3(c), and found that the experimental results (black triangle) can be well fitted by the BFL equation (magenta solid line) with the uncertainties of less than 5%. It can be noted that the increase of f_e in T1 can be divided into two regions: (i) when d > 8 cm, f_e increases as d becomes larger, and (ii) when d < 8 cm, f_e increases as d becomes smaller. We also characterize the filament length L_f (yellow dot) as a function of d in Fig. 3(c), from which it can be seen that, as d increases, L_f first decreases sharply and then decreases slowly. Moreover, as shown in Fig. 3(f), we measured the energy deposition efficiency (black square) as a function of d at two different incident energies, i.e., 1.6 mJ (black open triangle) and 2.0 mJ (black open rectangle). For both the incident energies, the variation in the energy deposition efficiency shows the same trend, that is, first increases and then decreases as d increases, but the energy deposition efficiencies are smaller for the lower incident energy. The linear dependence of the energy deposition efficiency on the incident laser energy is consistent with those obtained with single lens focusing condition [9,10], and can be attributed to enlargement of plasma volume in the filament with a higher incident laser energy [26,30].

 

Fig. 3. (a) The measured filament images and (b) the intensity distributions at 391 nm obtained with T1; (c-e) the measured (black triangle) and simulated (magenta line) effective focal lengths, as well as the measured (yellow dotted line) filament lengths, as a function of d for T1 (c), T2 (d) and T3 (e); (f-h) the measured energy deposition efficiencies (black squares or triangles) and ALEDs (red squares or dots) as a function of d for T1 (f), T2 (g) and T3 (h).

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Surprisingly, it can be seen from Fig. 3(c) and (f) that the maximum of the energy deposition efficiency does not appear at the smallest L_f. By examining the results shown in Figs. 3(c) and (f), we find that as f_e increases in T1, the variations in the energy deposition efficiency can be categorized into two regions: (i) the energy deposition efficiency decreases in the range of d > 8 cm; (ii) the energy deposition efficiency first increases and then decreases in the range of d < 8 cm. Furthermore, by dividing the total deposited energy by L_f, the ALEDs of the telescope T1 for the two incident energies are obtained, as shown in Fig. 3(f). Interestingly, the ALEDs show a plateau maximum region, which shows a remarkable different feature from the monotonic change shown in Fig. 2(d) in the SL focusing system. The plateau area is located in the range where f_e and L_f are relatively small, but not at the maximum of the energy deposition efficiency. Beyond the plateau region (d < 8 cm), the rapid increase in both f_e and L_f results in a sharp decrease in the ALEDs.

To verify the generality, we also conducted the same experiment using the telescopes T2 and T3 with the results shown respectively in Figs. 3(d) and (g) for T2, and Fig. 3(e) and (h) for T3. In the measurements, the incident laser energy was kept at 2.0 mJ. Similar to the result in Fig. 3(c), in both T2 and T3 cases, it can be seen that the experimentally determined effective focal length f_e can be well fitted to the BFL equation, and that the filament lengths, energy deposition efficiencies and ALEDs show the same trend as those in T1. It should be pointed out that for ALEDs, the entire plateau region moves towards the right side with a larger d.

In order to examine the applicability of telescopic systems in extending the operation distance of fuel ignition, we carried out the fs-LFI of the lean methane/air mixture (φ = 0.87) by using T1, T2 and T3, respectively. For comparison, the fuel ignition was also conducted with a SL focusing system. Figure 4(a) shows the side-view images of the laminar premixed methane/air flow under different focusing conditions of SL, T1, T2 and T3, where the parameter d_i is defined as the ignition distance (note for the telescope system d_i = f_e - d, and for the SL system d_i = f). In this measurement, the incident laser energy was fixed at 2.6 mJ and the laser repetition rate was set at 4 Hz for all the cases. The bright flame image means that the ignition is successfully achieved, while the dark image reflects that the ignition fails. Clearly, it can be seen from the images that the distance ranges for the successful ignition are different in the SL, T1, T2 and T3 systems. To see the differences more clearly, we plot in Figs. 4(b-e) the successful ignition (dot) and misfire (cross) events at different d_i positions which vary as functions of f in SL (b) and d in T1 (c), T2 (d), and T3 (e).

 

Fig. 4. (a) Side-view images of the CH₄/air mixture flow with the 2.6 mJ fs laser irradiation at different ignition distances under the SL, T1, T2 and T3 focusing conditions; (b-e) successful ignition (dot) and misfire (cross) events at different ignition distance d_i for SL (b), T1 (c), T2 (d) and T3 (e); (f) the MPEs measured as a function of d_i under the SL (rectangle), T1 (dot), T2 (upward triangle) and T3 (downward triangle) focusing conditions.

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On the one hand, it can be seen in Fig. 4(b) that the ignition distance is less than 40 cm in the SL case. This has been interpreted based on the crucial balance between the plasma density and the plasma column length, which plays the key role in successful fs-LFI [10]. As shown by the results in Fig. 2(c) and (d), as f (d_i) ≥ 40 cm, both the ALED and the deposition efficiency are very low. In this case, even the filament length increases, the reduced ALED is insufficient to trigger effective ignition, leading to the failure of the fs-LFI.

On the other hand, under the T1-T3 focusing conditions, the effects of both the ALED and filament length on the ignition can also be clearly seen. For T1, the distance d_i with 100% ignition success ratio occurs in the range of 16.3-27.5 cm for d = 6-18 cm. As shown in Fig. 3(c) and (f), when d becomes smaller than 6 cm, the ALED decreases sharply even the filament length becomes larger; while when d becomes larger than 18 cm, despite of a relatively high-level ALED, the filament length becomes very short. Therefore, it is concluded that a balance between the ALED and filament length should be satisfied to realize the fs-LFI in the telescope. For T2 and T3, the successful ignitions occur in the range of d = 15-40 cm (L_f = 13.2-6.5 mm) and d = 23-40 cm (L_f = 11.8-7.9 mm), respectively. It should be pointed out that the measurement of d > 40 cm was not performed because the diameter of divergent beam on the convex lens L2 induced by the concave lens L1 is larger than the diameter of L2. In this case, the maximum ignition distances reach to ∼48 cm (d = 15 cm) and ∼62 cm (d = 23 cm) for T2 and T3, respectively. Based on the variation trends of the ALEDs shown in Figs. 3(g) and (h), when d becomes smaller than 15 cm (T2) and 23 cm (T3), the ALEDs decrease sharply, which leads to the failure of the fs-LFI. Interestingly, it is found that by using the T3 system, the operation distance of fs-LFI can be extended by a factor of 2 when compared with that obtained in the SL system under the same incident laser energy, indicating the ability of this telescope approach for improving the operation distance of fs-LFI.

Moreover, we measured the minimum pulse energies (MPEs) at different ignition distances d_i under the SL and T1-T3 conditions, as shown in Fig. 4(f). Clearly, it can be seen in Fig. 4(f) that as d_i increases the MPE first decreases and then increases for each case, i.e., there exists an optimal MPE. The optimal MPE in the SL case is about 1.4 mJ when d_i (f) = 20 cm, and it rises sharply to 2.4 mJ when the ignition distance is extended to d_i =30 cm. For the T1-T3 cases, the optimal MPEs are 2.0 mJ at d_i = 20 cm for T1, 2.1 mJ at d_i = 41.5 cm for T2, and 2.2 mJ at d_i = 52 cm for T3, respectively. The optimal MPEs in the telescope system are slightly larger than that in the SL system, but different pairs of lenses in the telescope only give a small variation in the optimal MPE, which increases slowly when the ignition distance induced by different pairs of lenses increases. This result indicates that the employment of a telescope in fs-LFI enables the extension of the ignition distance with a reasonable MPE.

4. Summary

To summarize, we have investigated the energy deposition in a telescopic laser filament, and found that the change in the distance between the two lenses in a telescope can efficiently modulate the energy deposition efficiency. By measuring the filament length, the intensity distribution of the N₂⁺ fluorescence, and the energy deposition efficiency, we have revealed that there exists an optimal ALED plateau region in the telescope, which is remarkably different from the monotonic change in a single-lens focusing system. Based on the energy deposition characteristics in the telescopic filament, we have successfully extended the operation distance of fs-LFI of a lean CH₄/air mixture flow by a factor of 2 when compared with that in a single lens focusing system under the same incident laser energy condition. This proof-of-principle demonstration of fs-LFI by the telescopic filament breaks through the so-called Goldilocks focal zone fs-LFI limitation, and provides a viable strategy for manipulating the distance of fs-LFI of lean fuels with a relatively low incident laser pulse energy. In addition, our results may offer a deeper understanding of the energy deposition characteristics of telescopic filaments formed with various telescopes composing of complex optics components at extended distances. Although it has long been known that the generation of telescopic filaments over a long distance in the range from hundreds of meters to kilometers are practically limited by many factors such as the diameter of the convex lens and the laser energy, our findings may be useful for promising applications of the telescopic filament in areas such as remote sensing, laser fabrication and discharge guiding.