1. Introduction

The propagation of high-power laser pulses in air leading to ionization of air molecules and plasma formation has been studied since the early years of laser development, motivated by applications. Beginning in the 1970s, the prospect that this interaction could be used to trigger and guide electrical discharges such as lightning was experimentally studied with some success [1–3]. Further investigation using high-energy laser pulses with durations of >100 ns produced high-voltage discharges guided across meter-scale gaps [4–6]. This relatively long pulse duration causes avalanche ionization at the leading edge of the pulse, which produces a dense plasma that is optically opaque to the trailing edge and limits pulse propagation [7]. Long pulses resulted in the formation of a discontinuous chain of conductive plasma balls and, while guiding was possible due to these conductive plasma balls promoting leader formation [5], the limited pulse propagation and high pulse energy required made these systems poorly suited for guiding discharges over long distances. The development of ultrashort pulse, high-intensity lasers enabled by chirped pulse amplification [8] renewed the interest in this area of research due to their high intensity capabilities. For laser pulses with power greater than the critical power for self-focusing, atmospheric propagation becomes heavily affected by nonlinear responses, resulting in a self-sustaining channeling known as filamentation. Discovered by Braun et al. in 1995 [9], filamentation is primarily the result of a dynamic balance between self-focusing via the optical Kerr effect and plasma defocusing. For femtosecond pulses the result is typically a narrow (of the order of 100 µm), weakly ionized plasma channel of uniform density, which is capable of propagating tens or hundreds of meters [10] with plasma lifetimes of the order of several nanoseconds [11,12].

Following this discovery, the continuous plasma channel produced by femtosecond filamentation was seen as an effective way to trigger and guide electrical discharges. As with natural lightning, electrical discharges across long gaps in the laboratory are typically initiated by a highly conductive leader which propagates by the formation of streamers at its head. The first theory of discharge breakdown based on self-propagating streamers was published by Meek in 1940 [13]. For an atmospheric dc discharge at sea level, the critical field for spontaneous breakdown is ∼30 kV/cm. Several experiments were successful in initiating and guiding the discharge process, an effect that was associated with facilitating leader growth between electrodes with the filament-formed plasma [14,15]. However, other experiments have shown that discharges triggered and guided by filamentation can also occur in the absence of streamers [16], including those triggered with laser pulses of > 2 ps duration [17]. Due to the relatively low electron density [18] of the filaments generated by mJ-level femtosecond pulses, the generated plasma is weakly conductive [19] and has a short (ns) lifetime, making it poorly suited for sustaining discharges. The hydrodynamic response of the air following filament formation was identified as a mechanism for triggering and guiding discharges [20]. Using time-resolved interferometry [21,22] and diffractometry [23], a low density channel of similar geometry to the filament was found in the wake of ultrafast pulses. This hydrodynamic response of the air is caused by the deposition of laser energy leading to a pressure wave that expands radially, forming an underdense region [20–24]. This localized depression in air density along the filament path constitutes a channel of lower density that, according to the Paschen curve [25], allows discharges to be initiated by a weaker electric field.

Most laser-guided discharge experiments have been conducted at low repetition rates (f = 10 Hz) with femtosecond lasers [16,26–31]. Systems based on Ti:Sa lasers have been often utilized due to their capability to produce filamentation with low pulse energy [22,32–35] and their widespread commercial availability. Since the localized density depression has been shown to decay by thermal diffusion on a millisecond timescale [22], at such low repetition rates the time between laser pulses allows for complete relaxation of the surrounding air. Results of voltage breakdown for single shot near-infrared laser pulses of up to 10 J energy and durations between 0.7 and 10 ps found the largest breakdown voltage reduction, ∼ 30%, occurring for pulses of 2 ps duration [17]. Pulses of 50 fs duration and 350 mJ energy from a Ti:Sa laser reduced the breakdown field of a 2.5 m gap by up to 55% [30]. A large reduction of up to a factor of 4 in the breakdown electric field was observed in an experiment in which a filament generated by a 15 mJ femtosecond pulse was heated by an auxiliary 350 mJ laser pulse with a duration in the nanosecond range [36]. The more recent development of kHz-level systems capable of generating the high intensity necessary for filamentation has enabled the study of a new regime in which the medium does not completely recover before the arrival of the next discharge-inducing laser pulse. In this regime, the effects from pulses in the pulse train accumulate. A few experiments have been conducted with femtosecond Ti:Sa lasers operating at a 1 kHz repetition rate [32,33,37]. They showed the formation of a long-lived, quasi-permanent density depression [37] of greater magnitude than that observed at low repetition rates [32]. This cumulative effect was recently investigated in a study that also reports a reduction in breakdown voltage compared with low repetition rate discharges when voltage is rapidly applied by triggering a spark gap before the arrival of a laser pulse [32]. The application of a low energy pulsed electric field from a Tesla coil to the plasma filaments generated by Ti:Sa ultrashort laser pulses allowed the generation of a very conductive channel along the laser path, guiding a subsequent 30-kV DC discharge up to an atmospheric gap of 200 cm [38].

Recently, diode-pumped Yb:YAG-based IR lasers producing high-power picosecond pulses at kHz repetition rates have become available [39–41] for applications that include filamentation and discharge guiding research [42–44]. The time between shots at 1 kHz repetition rate is 1 ms, which allows a given pulse to interact with the air perturbation caused by the preceding pulses [37,42–44]. Filament lengths up to 70 m have been recently reported with 720 mJ pulses of 0.92 ps duration at 1 kHz [45]. Recently, one such laser system has successfully guided a lightning discharge over a distance of 50 m [46]. In an experiment conducted with 1030 nm 0.9 ps laser pulses of 150 mJ energy at a 1kHz repetition rate, a reduction in breakdown voltage by a factor of more than 2X was reported [42]. A study with 1030 nm, 1.5 ps pulses of 100 mJ energy found a factor of 3X reduction in breakdown voltage when the repetition rate was increased from 10 Hz to 1 kHz [43]. This reduction in breakdown voltage was attributed to the cumulative effect of the hydrodynamic response of the air around the filament at high repetition rates.

Here we examine discharges guided by the filamentation of high-energy picosecond pulses of 7 ps duration and up to 250 mJ energy from a λ = 1030 nm wavelength laser operating at repetition rates of up to 1 kHz. We demonstrate the repetitive generation of stable discharge plasma columns at this repetition rate. A record breakdown voltage reduction of ≥ 4.7 X for gaps of up to 10 cm between circular electrode plates with axial holes is observed to result predominantly from the perturbation caused by a single laser pulse, leading to measured density depressions reaching >80%. The breakdown voltage reduction is observed to be similar at 100 Hz and 1 kHz repetition rate. A current proportional to the laser pulse energy arises as soon as the laser pulse arrives, initiating a high-impedance discharge. Full breakdown, characterized by impedance collapse and the onset of high current conduction, occurs 100s of ns to a few µs later, following the formation of concentrated conducting channels in the radial gaps between the filamentary plasma channel and the electrodes. The experimental evidence shows that the role of these gaps, often neglected, needs to be considered in analyzing the breakdown dynamics of short-distance, laser filament-guided discharges.

2. Experimental setup and methodology

The experiments were performed using λ = 1030 nm wavelength laser pulses of 7 ps duration from a cryogenically cooled, diode-pumped chirped pulse amplification Yb:YAG laser system developed at Colorado State University that operates at repetition rates up to 1 kHz [39]. A schematic of the experimental setup is shown in Fig. 1. After compression, a 3-cm-diameter S-polarized beam is focused to a ∼70 µm (FWHM) diameter spot using a 2.2 m focal length lens, (f/73). The repetition rate (f) of the laser pulses was varied from f = 0.01 to 1 kHz, and the laser pulse energy (E_L) was varied in the range E_L = 10–250 mJ by selecting the energy of the seed pulse injected into the cryo-cooled power laser amplifiers. The pump pulse energy to the power amplifiers was maintained constant to avoid significant variation in the thermal lens. The electrical discharge electrodes are two circular parallel brass plates with a face diameter of 13 cm, which are curved around the edges to increase field uniformity. Holes of 3.8 mm diameter were drilled on axis to allow the laser beam to pass through. The separation of the two plates was varied from d = 2 cm to 10 cm. A 1.7 nF ceramic capacitor was directly connected to the anode electrode and the cathode was grounded. The capacitor was charged using either a current-limited voltage supply (Glassman PS/LT015P132) or a voltage supply (Hipotronics 8100-10), operating at up to 15 kV or 50 kV, respectively. The output of the current-limited voltage supply is more highly filtered, offering a more stable source for repetition-rated discharges, while the voltage supply, which can provide a higher output voltage, was used for larger electrode separation measurements. The charging resistor was selected to be either R_c = 280 kOhm or R_c = 175 MOhm. The lower value was used for faster charging of the capacitor in the high repetition rate measurements, such that the capacitor is nearly fully charged when the next laser pulse arrives. The higher resistor value was used for breakdown measurements at lower repetition rates, for which the train of laser pulses samples the gap for breakdown as the voltage slowly increases. Figure 1(a) shows a photograph of a typical laser filament, and Fig. 1(b) shows a 10-cm-long straight laser-guided discharge channel integrated over 17 discharge events at a 1 kHz repetition rate.

 

Fig. 1. (a) Laser filament generated at 1 kHz. (b) Photograph integrated over 17 discharge events, taken with a 60 frames per second camera for a laser pulse energy of 200 mJ, repetition rate 1 kHz, and plate separation of 10 cm. (c) Schematic of the experiment set-up. The probe beam is shown in the longitudinal configuration. The charging resistance (R_c) varies in the range R_c= 280 kOhm – 175 MOhm. The power supply (HV) is the voltage supply, as detailed in the text. The current pulse is monitored with a current transformer placed in the ground loop, or by measuring the voltage drop in a shunt resistor.

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A combination of optical and electrical diagnostics was implemented to characterize the discharges over a wide range of conditions. The electrical diagnostics consisted of a high voltage probe to measure the breakdown voltage, and a current transformer to measure the current pulse. The voltage drop across a resistor placed in the ground electrode connection was measured with a differential probe to provide an additional measurement of the current. Photodiodes were used to measure the time of arrival of both the filament-creating laser pulse and an optical probe pulse with respect to the discharge onset. These signals were collected by a deep-memory digital oscilloscope with 3 GHz analog bandwidth (LeCroy Wavepro 7300A), capable of acquiring traces with durations of seconds and a 10 ns temporal resolution. Several optical diagnostics were fielded. An imaging system measured the light generated by the discharge, recording either time-integrated or time-resolved image sequences. The latter made use of a fast microchannel plate intensified framing camera (Invisible Vision Ultra UHSi-12) capable of recording up to 12 frames of variable temporal separation with a minimum exposure window of 5 ns. Its sequential gating capability allowed us to monitor the early formation of the discharge by recording sequences of images of the discharge channel fluorescence with high time resolution. Mach Zehnder interferometers using a second harmonic (515 nm wavelength) probe beam were deployed to probe the discharge channel evolution in three different angular configurations: longitudinally along the discharge axis (case illustrated in Fig. 1(c), transversely at a grazing incidence angle of 11.5 degrees, or at normal incidence with respect to the filament axis. Time-resolved interferograms were recorded using either the gating provided by the second harmonic probe laser pulse (∼ 5 ns resolution) and a standard CCD camera, or the high-speed gating camera (also 5 ns image resolution) that allows for a sequence of 5 ns images to be acquired for a single discharge event.

3. Results and discussion

3.1 Discharges at 1 kHz

Figure 2 shows an example of electrical data for a train of laser filament-guided discharges at f = 1 kHz repetition rate. Displayed are the laser pulse time-of-arrival measured with a photodiode (a), the current pulse (b), and the voltage drop across the gap (c) for approximately 100 consecutive discharges. The electrode gap was d = 2 cm, and the laser pulse energy was E_L = 150 mJ. The voltage of the current-limited voltage supply was set to 15 kV to charge the capacitor through a R_c = 280 kOhm resistor, corresponding to a charging time constant of τ = 0.5 ms. The time interval between laser pulses is 1 ms, meaning that at the time the subsequent laser pulse arrives the capacitor is charged to about 86% of its maximum voltage, sufficient to consistently initiate a discharge event upon the arrival of each laser pulse. Notice in Fig. 2(b) that the current of the initial discharge of the sequence, which corresponds to having the capacitor fully charged and suddenly unblocking the laser beam, is larger due to the larger initial overvoltage with respect to the reduced breakdown voltage induced by the laser pulses. Figures 2(b) and 2(c) show that each subsequent laser pulse triggers a breakdown event with highly reproducible current pulses. The average power dumped into the discharge at the conditions of Fig. 2 is ∼ 150 W. Figure 2(d) displays a zoomed-in image of a typical set of current and voltage measurements for a single discharge event. The current is underdamped [Fig. 2(d)]. The time of laser pulse arrival is indicated.

 

Fig. 2. (a-c) Electrical measurements (labeled) of a laser-guided discharge operating at a repetition rate f = 1.0 kHz, with laser energy E_L = 150 mJ and electrode separation d = 2 cm. A sequence of approximately 100 laser-triggered discharges is shown. (d) Example of voltage and current evolution in a single discharge event.

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In general terms the action of the laser pulse is twofold: it reduces the breakdown voltage V_b required to initiate a discharge event, and it provides a straight channel for the current to flow. To accurately determine V_b(E_L, d, f), the capacitor was charged slowly with R_c = 175 MOhm (corresponding to τ = 300 ms), and the laser was left unblocked while operating at 1 kHz repetition rate, or an alternatively chosen repetition rate. In this way, the high repetition rate pulse train samples the constantly increasing voltage until breakdown is achieved. In the absence of a laser pulse, the breakdown voltage is expected to follow the typical behavior of a high-voltage air gap. The breakdown voltage dependence on gas density and gap distance has been thoroughly studied for many years for different gases, in particular in connection with spark gaps and pulsed-power switching [47–50]. The relationship between air density and breakdown voltage V_b for gaps of sufficient length is described by the well-proven relationship [51]:

Figures 3(a)-(d) display V_b as a function of laser pulse energy E_L for four different discharge gap distances d = 2, 4, 8, and 10 cm and two different repetition rates, f = 100 Hz and 1000 Hz.

 

Fig. 3. (a-d) Breakdown voltage (V_b) as a function of laser pulse energy (E_L), for two laser pulse repetition rates (0.1 and 1 kHz) and four electrode separations (a) d = 2 cm; (b) d = 4 cm; (c) d = 8 cm and (d) d = 10 cm. The breakdown voltage without a laser pulse (E_L = 0 mJ) indicated for the longer discharges is calculated using Eq. (1). (e-f) V_b as a function of d, for E_L = 200 mJ alongside the no-laser case. (e) Zoomed-in voltage axis; (f) full voltage axis. For the no-laser data, the dashed line corresponds to Eq. (1). For the laser-initiated discharges the dashed line corresponds to a linear fit to the measured data.

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To be able to apply a sufficiently high voltage to achieve breakdown in the longest gaps, the current-limited voltage supply (limited to 15 kV) was substituted by the voltage supply with higher output voltage capability. The measured (or calculated) breakdown voltage in the absence of a laser pulse is indicated in black. For d = (0.5, 1, 1.5, and 2 cm), the breakdown voltage V_b (E_L= 0 mJ) was measured to agree well with the expected dependence as a function of inter-electrode distance given by Eq. (1) [Fig. 3(e),(f)].

Figures 4(a) and (b) show the breakdown delay, defined as the time between the arrival of the laser pulse and the breakdown, as a function of applied voltage and laser pulse energy. The data here corresponds to a 2-cm-long discharge at a 5 Hz repetition rate. The breakdown delay time and the jitter both decrease as either the applied voltage or the laser pulse energy is increased. However, for voltages larger than 20 kV no significant further decrease in breakdown delay is observed when the laser pulse energy is increased above 100 mJ. With 100 mJ applied the breakdown delay decreases from 1 µs when 15 kV is applied to 30 ns for 40 kV.

 

Fig. 4. Discharge breakdown delay as a function of applied voltage for discharges initiated by laser pulses with three different energies. The electrode gap was 2 cm. A zoom of the data in a) for delays less than 500 ns is illustrated in b) for better visualization. For voltages above 25 kV, no further reduction in breakdown delay is observed for laser pulse energies greater than 100 mJ.

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3.2 Breakdown voltage reduction

The laser pulse energy has a clear effect in reducing V_b. For d = 2 cm, Fig. 3(a) shows a sharp drop in V_b as the laser pulse energy E_L is increased from E_L = 0 mJ to 50 mJ. For E_L> 50 mJ, the effect of increasing E_L becomes less pronounced, with V_b finally reaching a plateau. For 200 mJ the measured voltage reduction is 4.7 X. Figures 3(e) and (f) show that for this pulse energy the breakdown voltage reduction computed as the ratio of the $V_b\left(E_L=0 \mathrm{~mJ}\right)$ calculated using relation (1) and the measured values is between 4.7 X and 5.1 X over the entire range of interelectrode distances investigated.

Figures 3(a) and (b) also show that the breakdown voltage is practically the same at f = 1 kHz and f = 100 Hz. This indicates that the repetition rate plays no major role in the value of V_b, implying that the large measured reduction in V_b is in our case mostly a single-pulse phenomenon instead of the result of the cumulative pulse effects that occur at high repetition rates. Previous studies of laser filaments generated at high repetition rates have reported the creation of a permanent density depression resulting from the cumulative effect of multiple shots [22,32]. In a previous experiment conducted with the same laser, a density depression of ∼ 20% induced by the previous pulses was measured 10 µs before the arrival of a laser pulse in a 1 kHz pulse train [44]. It has been suggested that such lower density channel can reduce the breakdown voltage and guide electrical discharges by producing a preferential channel for the current to flow [22,32,43]. At the conditions of our experiment (up to 200 mJ, 7 ps, f/73), it was observed that the reduction in breakdown voltage far exceeds the value that can be attributed to the existence of the measured 20% pre-formed air density depression. Equation (1) indicates that such a 20% density reduction for d = 2 cm and d = 4 cm should lead to V_b reductions of 18.4 and 18.8 percent, respectively. In contrast, the V_b reduction we measured reaches ∼ 79%. The same expression predicts that the relative density required for such voltage reduction should be $\rho $ ∼0.13, corresponding to a density reduction of 84% with respect to the atmospheric pressure during the experiment. In agreement with this expectation, the density depression measured with grazing incidence interferometry shortly after the arrival of a laser pulse approaches this value. Figures 5(a) and (b) show an interferogram corresponding to the channel formed 300 ns after the arrival of a 200 mJ laser pulse and the corresponding phase-shift map. Figure 5(c) shows the corresponding density profile obtained assuming cylindrical symmetry, which indicates that at this time the density depression reaches ∼ 75%.

 

Fig. 5. Grazing incidence interferograms acquired 300 ns after the arrival of a 200 mJ laser pulse (no external electric field applied). The phase shift (b), and density as a function of radial distance (c) are shown. The air density depression on axis reaches 75%.

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The evolution of the laser-generated channel was modeled with the hydrodynamic code RADEX [52] using an initial electron temperature of 2.5 eV estimated with a particle-in-cell (PIC) simulation and an initial column diameter of 60 µm. The electron density and temperature are initially calculated simultaneously by the PIC simulation which ionization model includes optical field ionization and collisional ionization. The hydrodynamic simulation results are found to be relatively insensitive to initial electron temperatures between 2 and 5 eV. Electron thermalization occurs in picoseconds due to collisions with neutral and ionized species. Figure 6 shows the computed radial density distribution as a function of time referenced to the arrival of the laser pulse. The air density depression reaches 82% at ∼100 ns after the arrival of the laser pulse and remains at practically the same value for at least several µs. This is in good agreement with the interferometrically measured density values at three different times of the channel evolution illustrated in Fig. 7. Both show similar channel depth and widths, and speed of propagation of the shockwave. The profile of the latter is, however, smoother in the experimental plot. This can be expected as it is an average of different regions of the channel which might not be perfectly uniform.

 

Fig. 6. Computed density profile of channel generated by a 150 mJ laser pulse. A density depression of ∼82% develops ∼ 100 ns after the arrival of the laser pulse.

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Fig. 7. Interferometrically measured molecular air density profile at three times of the channel evolution: 300 ns (orange line), 800 ns (green line), and 1800 ns (blue line).

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In conclusion, the above measurements and simulations indicate that at the condition of this experiment, the large density depression (75-80%) that follows a single laser pulse, possibly combined with the mildly ionized conductive channel that is created with the aid of the detected pre-breakdown current discussed below, is the dominant factor in decreasing the breakdown voltage. Supporting this conclusion is the fact that V_b for 100 Hz and 1000 Hz laser filament-induced discharges was measured to be practically the same, as illustrated by the measurements in Fig. 3(a) and (b). This independence of the breakdown voltage with repetition rate differs from the results of an experiment conducted with 100 mJ pulses of 1.5 ps duration that reported a reduction of the breakdown voltage at increased repetition rates [43]. It should be noted that this experiment used a softer f/238 focus that is expected to produce a shallower but broader (and hence longer lasting) density depression, reducing the contribution of a single filament with respect to the cumulative depression at high repetition rate. On the other hand, in another experiment conducted with a similar setup, using 35 fs pulses of 0.8 mJ and f/100 [32], similar breakdown voltages were observed at 10 Hz and 1 kHz when the voltage was applied shortly after the laser pulse time using a triggered spark gap (t = 0 in Fig. 5 of Ref. 32), consistent with the measurements presented in this study. In any case it should be noted that if voltage were to be suddenly applied long after the arrival of a laser pulse, the deeper cumulative air depression present at high repetition rate (e.g., 1 kHz) would result in a lower breakdown voltage, as reported in the literature [32]. Hence, the role of repetition rate on reducing breakdown voltage should be separated into two scenarios: soon after the laser pulse’s arrival, when the increased depression caused by a single pulse dominates the breakdown; or long after, when it becomes smaller than the cumulative depression caused by all previous pulses.

3.3 Initial current flow and breakdown

We now turn our attention to the process leading to full breakdown. In our experiments, a static electric field is established between the electrodes before the arrival of a laser pulse. The laser pulse generates a filament that extends beyond the electrodes. Initially, optical field ionization strips electrons from the molecules creating a heated plasma channel that generates a radially expanding shockwave and an air density depression. In the case of the 7-ps long laser pulses used in these experiments, avalanche ionization is the dominant ionization mechanism [44]. A current flow of typically several tens of mA (but as high as 400mA) is observed to arise immediately after the arrival of the laser pulse. This initial current flows during an initial high-impedance phase of the discharge that precedes the time of full breakdown characterized by the collapse of the impedance. In a previous experiment involving a short spark gap driven by a laser filament, the formation of a field-driven current (< 0.15 µA) was identified. This current was reported to contribute to both the deepening of the channel and an increase in its conductivity through resistive heating, ultimately leading to breakdown [22]. In our case, the amplitude of the pre-breakdown current is much larger but is still typically three to four orders of magnitude smaller than the current after breakdown. The onset of this pre-breakdown current is characterized by a rapid rise that is likely to be associated with a displacement current caused by the re-arrangement of charges generated by the laser pulse in the applied electric field. This initial current rise is followed by a sustained current whose magnitude and temporal evolution depend on the laser pulse energy and the applied electric field. Figure 8(a) illustrates the pre-current pulse for the case of a 2-cm-long discharge with an applied voltage of 15 kV initiated by a 150 mJ pulse. Figure 8(b) illustrates how the pre-current magnitude increases as a function of laser pulse energy from 15 mJ to 200 mJ, in the case of a 2-cm gap biased at 20 kV. The pre-current magnitude is similar at 100 Hz and 1000 Hz, and for discharges of 2 cm and 4 cm length when the same electric field is applied across the electrodes. Following its initial fast rise, the evolution of the pre-current can vary, stabilizing or even decreasing in magnitude before the breakdown occurs. These scenarios have been previously observed in spark discharges in air and other gases at atmospheric pressure [53]. This includes cases in which the initial current spike is followed by a transition phase during which electron attachment to oxygen temporarily prevents the full breakdown [54,55].

 

Fig. 8. a) Current evolution illustrating the pre-breakdown current initiated with a 150 mJ laser pulse in a 2 cm discharge gap biased at 15 kV. b) pre-breakdown current as a function of laser pulse energy for discharges at 100 Hz and 1 kHz repetition rate.

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The typical value of the impedance in this pre-breakdown phase ranges from 0.2 MOhm to a few MOhm, depending on laser pulse energy and the voltage applied. The discharge impedance is the sum of the resistance of the axial plasma column between the electrodes, and the resistance of the radial gaps between the column and the electrodes (radial gaps between the filament channel and the electrodes are <1.9 mm). The non-negligible role of these gaps becomes evident by conducting measurements in discharges with similar initial electric field/molecular density ratio (E/N), a well-known scaling parameter in electrical discharges. For this purpose, we performed measurements comparing discharges across 2- and 4-cm electrode gaps applying twice the voltage in the 4-cm case to have the same initial E/N in both cases. Both measurements were performed with 1 kHz laser beams while the capacitor was charged through a 20 MOhm resistor, corresponding to charging time constant of about 40 ms. The measured breakdown voltage was found to nearly double in the case of the longer channel, as shown in Fig. 9(a). It should be noted that the value of E/N at the time of breakdown can be expected to change as a function of laser pulse energy since the density depression varies with laser pulse energy. The time delay between the arrival of the laser pulse and the breakdown was measured to significantly decrease in the case of the 4 cm discharge [Fig. 9(b)]. This can be ascribed to the fact that in this case the gaps between the filament-initiated plasma column and the electrodes are subjected to twice the electric field. This indicates that the role of these small gaps in the breakdown cannot be neglected. The delay is also observed to increase with laser pulse energy because a larger energy lowers the breakdown voltage (Fig. 9(b)). These delays are also much larger than those shown in Fig. 4. This is because the 1 kHz pulse train effectively samples the constantly increasing voltage of the capacitor every 1 ms, leading to breakdown with very small overvoltages as compared with the case of Fig. 4 where the low repetition rate of 5 Hz results in very large overvoltages.

 

Fig. 9. Breakdown voltage (a), and breakdown delay with respect to the arrival of the laser pulse as a function of laser pulse energy (b) for 2 cm and 4 cm electrode gaps. The voltage applied was adjusted to have initially equal E/N values.

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On-axis interferograms acquired at different times of the plasma evolution shown in Fig. 10 give additional information on the transition from the high impedance discharge phase into the high current phase following breakdown. Interferograms taken early during the pre-breakdown current-conducting phase do not identify a concentrated current path across the radial gaps between the filament and the electrodes [Fig. 10(a)]. It is possible that during this high-impedance discharge phase, the current is conducted across the gap by a spatially distributed, high-impedance glow-type discharge. Glow discharges are known to take place in air at atmospheric pressure and have been the subject of several studies [56,57]. The regions near the electrodes in glow discharges are locations with high electric fields and therefore the cradle of instabilities that lead to constriction and arc formation. These discharges are known to undergo glow-to-arc transitions [57] in which the high-impedance glow plasma collapses into a low-impedance arc. Figures 10(b)-(d) reveal the formation of concentrated discharge channels in the radial gap between the filament-initiated discharge column and the electrodes. This occurs 100 s of ns to a few µs after the arrival of the laser pulse. Two radial arcs are observed to form, each linking the filamentary column to one of the electrodes. Once these concentrated conductive channels are formed across the radial gaps, the collapse of the impedance leads to the sudden increase of the current which rapidly heats the plasma in these channels, causing them to expand at a higher rate [Fig. 10(c), (d)]. At this stage the capacitor, whose voltage drops only very slightly during the pre-breakdown phase, is rapidly discharged.

 

Fig. 10. a-d) On-axis interferograms showing the development of plasma channels linking the laser filament-initiated plasma column to the hollowed electrodes. The region with fringes corresponds to the electrode hole diameter of 3.8 mm. e-g). Current pulses indicating the time at which the data corresponding to each frame was taken (arrow). The data corresponds to four individual shots, and an electrode distance of 2 cm and laser pulse energy of 136 mJ for a pulse train at 100 Hz.

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3.4 Onset and evolution of the high current discharge channel

The evolution of the plasma column immediately before and after the time of breakdown was observed using fast sequential imaging with a temporal resolution of 5 ns. Figure 11 shows the first five frames of a sequence of 12 contiguous 5-ns resolution images mapping the evolution of the discharge during the first 60 ns after breakdown for a 4-cm gap. Before breakdown, no light is detected by the MCP-intensified camera. When plasma light emission is first detected in the first 5 ns frame after breakdown, it is observed to be uniform across the entire gap. It is observed to grow in intensity on every frame thereafter as the current increases. No evidence of streamers propagating across the electrode gap is detected. At a velocity of 1 × 10⁸ cm/s, on the upper end of the typical velocity for streamers [58], it would take 40 ns for a streamer to bridge the electrode gap, a sufficiently long time for the framing camera to detect it.

 

Fig. 11. Sequence of time resolved images of the plasma column near the time of breakdown taken 5 ns apart (right). The time of each image with respect to the current pulse is indicated (left).

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Interferograms acquired at different times during the low current phase that precedes the breakdown also show a uniform channel bridging the electrode gap, providing additional evidence suggesting that the filamentary charge created by the laser pulse starts to flow uniformly across the gap, without evidence of streamers. As mentioned above, other laser filament-initiated discharges free of streamers have been previously reported [16,17].

On-axis interferograms/shadowgraphs were used to quantify the propagation of the shockwave and channel expansion. Figure 12 shows a sequence of on-axis interferograms corresponding to 100 Hz and 1 kHz. A shockwave is observed to propagate outwards at a velocity of ∼ 350 m/s, in agreement with the simulation in Fig. 6. Absorption of the probe beam along the column creates a darkened region that grows with time without fringes. This dark area associated with the channel has a complex nature that depends not only on density but also on temperature, charge, and propagation of light in the channel. Figure 13 illustrates the growth of this region as a function of time for the case of a filament generated by a 150 mJ laser pulse. When the discharge is present, the current that flows during the high impedance phase of the discharge contributes to increasing the expansion [red markers in Fig. 13(b)], as reported in Ref. 22. After the breakdown occurs, the rapid current increase results in greatly enhanced ohmic heating that increases the channel expansion velocity to about 2400 m/s. In the example of Fig. 13 the channel is observed to expand more than 0.3 mm in 100 ns.

 

Fig. 12. Longitudinal interferograms for E_L = 150mJ at f = 100 Hz (a-e) and f = 1 kHz (f-j) showing channel and shockwave formation. No external field was present. Times are indicated with respect to the arrival of the laser pulse. A noticeable air perturbation is present at 1 kHz (f) and persists at 10 µs before the arrival of the laser pulse. No such perturbation exists at this time for 100 Hz (a-e). The scale bar is 1 mm.

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Fig. 13. (a) Interferograms obtained for the filament with (top row, red) and without (bottom row, blue) voltage applied, for a discharge initiated by 1 kHz, 150 mJ laser pulses. The labels indicate the delays. The scale bar is 500 µm. (b) Evolution of the dark region radius before and after breakdown for the case of a laser filament alone (no voltage applied) and a filament with voltage applied. A rapid increase is observed to occur after breakdown. The current pulse is shown as a reference (gray line).

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4. Conclusions

We have used laser filaments generated by λ = 1030 nm, 7-ps long pulses of up to 250 mJ energy and f/73 optics to initiate and guide stable atmospheric electrical discharges at repetition rates of up to 1 kHz. The laser-created filament is shown to reduce the discharge breakdown voltage by a factor of 4.7 X, which is to our knowledge the largest reported reduction for a discharge between dc-biased electrodes. The laser pulse arrival is observed to immediately induce the onset of a current flow between the biased electrodes whose magnitude, typically several tens of milliamperes, is directly proportional to the laser pulse energy. This phase of the discharge evolution precedes the full breakdown and is characterized by an impedance in the range of 0.2 MOhm to a few MOhm. The gaps between the filament and the discharge electrodes are observed to play a significant role in the breakdown dynamics. Axial interferograms taken during this pre-breakdown phase show the absence of concentrated current flow between the filament and the electrodes. This suggests that during this initial phase, the current across these gaps is conducted by a diffused, high-impedance glow. Full breakdown, characterized by the collapse of the impedance and a rapid increase of 3-4 orders of magnitude in current, is observed to follow the formation of localized radial current channels linking the filament to the electrodes. Breakdown occurs 100s of ns to a few µs after the arrival of the laser pulse, depending on the laser pulse energy and voltage applied. After breakdown, the increased discharge heating causes the rate of expansion of these arcs and of the filament-initiated discharge channel itself to rapidly increase.

The breakdown voltages for discharges initiated by laser filaments generated at 100 Hz and 1 kHz were measured to be practically the same. This is consistent with interferometry and hydrodynamic modeling results which show that at our conditions the density depression that develops after the arrival of a single laser pulse can reach 80%, overwhelming the cumulative density depression measured at 1 kHz. The measured magnitude of the density depression caused by a single laser pulse is consistent with that required to obtain the observed 4.7 X reduction in breakdown voltage. This implies that for our experimental conditions the measured large reduction in Vb is dominantly a single-pulse phenomenon resulting from a large density depression, possibly aided by the increased conductivity of the mildly ionized filamentary channel heated by the pre-breakdown current. However, the influence of the cumulative density depression in aiding breakdown cannot be disregarded. In the case of experiments with weaker focusing it can be expected that a shallower but broader, and hence longer lasting density depression would be formed, decreasing the contribution of a single pulse relative to the cumulative density depression at high repetition rates. Also, if voltage were to be applied abruptly at any point during a laser pulse train, except for the period shortly after a laser pulse when the single pulse effect can prevail, the observed cumulative air depression would reduce the breakdown voltage, as reported in the literature [32]. In all cases, for the cumulative effect to be a major contributor to the reduction in breakdown voltage, the pre-formed density depression at the time of the arrival of the laser pulse has to be significant with respect to the depression caused by the single pulse itself. The effect of the repetition rate is always present at 1 kHz, but at our conditions it becomes a minor contributor compared to the effect of the single filament.

From the point of view of applications, the results obtained demonstrate the generation of stable discharge channels in the atmospheric air at a 1 kHz repetition rate, with a record reduction in breakdown voltage. The increased understanding of the physics of the formation of laser filament-guided discharges in air at high repetition rates can be expected to benefit applications.