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Abstract
The temporal evolution of electron density of air plasma filament induced by femtosecond laser pulse has been studied experimentally by a single shot electromagnetic induction method. Based on the detection of transient magnetic field around the electrified filament by an induction coil, the current in filament as well as electron density is estimated from the time dependent electromotive force signal. The experimental results indicate that our method significantly reduced the self-oscillation and interference signal, which are difficult to avoid in previous standard electrical diagnostic of filament, meanwhile the reliable temporal evolution of the electron density of filament is obtained.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When intense femtosecond laser pulse propagates in air with peak power higher than the critical value of self-focusing, the filamentation process will takes place due to dynamic balance between optical Kerr effect and plasma defocusing [1–4]. Femtosecond laser filamentation has attracted intense interests for a wide range of potential applications, like lightning and electric discharge control [3–8], remote sensing [2,4,9,10], microwave guiding [11], terahertz generation [12–15] and even artificial precipitation [16,17]. The electron density of femtosecond laser filamentation and its temporal evolution are critical characteristics for conductivity based applications. Much effort has been paid on diagnostics of the electron density of filament using electrical conductivity measurement [18–25], transverse or longitudinal interferometry and diffractometry [26–31], quantitative fluorescence emission measurement [7,32,33], terahertz scattering [28,34] and spectroscopic analysis [35]. The temporal evolution of electron density of filament has been studied by pump-probe scheme [28,29,34,36], time-resolved measurement using ultra-fast camera (ICCD) [7,24] and microwave diagnostics [37,38] etc.
Among these methods, electrical conductivity (EC) measurement is both easy and sensitive. In standard EC measurement, the filament plays as a resistor in a simple volt-ampere circuit, as shown in Fig. 1(a). The electron density of filament can be estimated from the resistance of filament. In the ideal conditions where the circuit has no response delay, the single-shot time-dependent electric signal can be considered as temporal evolution of electron density of filament. However, each circuit has its own response time, relaxation and self-oscillation characteristics, which are dependent on the parameters of elements in circuit. For example, due to self-oscillation, negative current signal often appears in EC measurement [25], which definitely cannot reflect the electron density. Moreover, the electric signal could extend to tens of nanosecond due to long relaxation time of circuit, which are much longer than the typical life-time of filaments [25].
Fig. 1 (a) Typical setup of standard EC measurement. The orderly arrayed voltage sources are DC power supply, chemical battery pack and charged capacitor. (b) Setup of EMI method. (c) The relative position of coil and filament.
In this work, an electromagnetic induction (EMI) method has been demonstrated to investigate temporal evolution of electron density inside the filament. In EMI method, the electron density of filament is measured by non-contact detection of transient magnetic field around the current in filament using an induction coil. Due to the successful decoupling of self-oscillation in circuit from the coil, a clean and reliable temporal evolution of the electron density inside filament is obtained.
2. Experimental setup
The temporal evolution of electron density of filament has been investigated by both EC and EMI method. The experiment is conducted using a Ti:sapphire laser system that delivers 35 fs infrared pulse with an initial beam diameter of 8 mm. A 16 mJ pulse is focused by a lens with focal length of 4 m to generate 40 cm long single laser filament. Figure 1(a) shows the typical setup of the standard EC measurement. Three types of voltage source, i.e. a commercial DC power supply, a chemical battery pack and a charged 1 μF capacitor are used to show the different self-oscillations of the circuit. Figure 1(b) shows the setup of EMI method. Two copper plane electrodes are fixed on the pins of a 1 μF capacitor, the gap between two electrodes is 30 mm. Instead of directly measuring the current in the electrical circuit connected by the filament, we detect the transient magnetic field induced by the electrified filament using a 10mm × 10mm square single-turn coil. The coil is made of 0.5 mm insulated coating Cu wires and is located 2.0 mm away from the laser filament and in the same plane with it, as shown in Fig. 1(c). It has been used as a B-dot in many strong transient magnetic field measurements [39–43] and THz field measurement [44]. Even though the current in the filament and the surrounded transient magnetic field are rather weak, considerable electromotive force (EMF) can still be induced due to sharply change of the current. The EMF signal detected by the coil is recorded by a 25 GS/s high resolution oscilloscope (Tektronix DPO70804). The voltage loaded in the circuits in the experiment is fixed at 100 V. For the circuits using capacitor as voltage source, the charge released by filament bridging is negligible small and could not induce observable reduction of the voltage on capacitor. The filament passes through two electrodes by burning holes on themselves to reduce contact resistance between the filament and the electrodes. In order to minimize the additional transient process caused by laser irradiance on the electrodes, the measurement is started when the holes become steady and no visible plasma is generated on the edge of holes.
3. Results and discussion
Figure 2 presents typical electric signal by standard EC method using commercial DC power supply, battery pack and capacitor as voltage source. The results show that different voltage source lead to different self-oscillation characteristics of electric signal, but the peak values of electric signal in these three cases have no significant difference. The self-oscillation of the circuit using DC power supply is stronger and lasts for hundred nanoseconds due to the longer connect circuit and the interference from complex inner electronic components of power supply. The circuits with battery pack or capacitor are much more compact and faster responsive. Therefore, the self-oscillation has been greatly shortened. Even so, the detecting signal still lasts for tens of nanoseconds, which is still much longer than typical lifetime of filament.
By using EMI method, the induced EMF signal from the transient magnetic field around the filament has been measured at three different positions along the filament, as presented in Figs. 3(a)-3(c), where x = 0 represents the strongest ionized point. Figure 3(d)-3(f) show the integrated curves of the EMF signal along time, i.e. transient magnetic flux (), which are proportional to the current and the electron density of filament. The electron density of filament drops down rapidly in the first nanoseconds and then approaches to a low value. It can be seen that the integrated curves are not accurately turned to zero. In our opinion, the reason is that the long weak tail part of EMF signal in the decay period of filament was insufficiently detected by oscilloscope due to the signal attenuation in transmission line. In case of strong EMF signal with higher signal-noise-ratio, the return-to-zero process gets much improved as presented in Fig. 3(b) and 3(e). The overall evolution trend of electron density is in good agreement with previous results using microwave diagnostics [37, 38]. The small return-to-zero mismatch is not significantly impact the validity of results. The lifetime of filament can be estimated by the width of the integral signal. When the integral signal drops down to almost unchanged value (at 5~6 ns in our experiment), the lifetime of filament is considered to be basically terminated.
Fig. 3 (a)-(c) EMF signal of transient magnetic field around the filament; (d)-(f) The corresponding integral signal. Where x = 0 represents the strongest ionized point.
The value of current in the filament can be calculated by Faraday electromagnetic induction law and Biot-Savart Law. The magnetic flux through the coil is mainly induced by the current in filament, and it can be expressed as follows.
whereis the magnetic induction; is the area of coil; is the magnetic permeability of vacuum. The coefficient is decided by the area of coil and its relative position to the filament,it can be calculated as follow in unit of mm:The variables in Eq. (2) are described in Fig. 1(c). The value of is 32 mm in our experimental condition. As a result, the current flow in the filament can be obtained by
Then the electrical conductivity of the filament can be calculated from its resistance:
whereis the resistance of filament between the electrodes;is the radius of filament;,,are respectively electrodes gap length, current and voltage. On the other side, the conductivity can also be calculated by electron collision theory as follows ref [18].where is the elementary charge; is the electron density; is the electron mass; is the collision frequency for electrons in laboratory conditions; Combine the Eqs. (3) and (4), the electron density of filament can be obtained asThe plasma current and corresponding electron density of the strongest ionization point of filament are calculated using Eqs. (3) and (6). In Figs. 3(b) and 3(e), the peak value ofis about 300 mV·ns, corresponding to the current in filament 94 mA. Then resistance of filament between electrodes is obtained to be 1.05 kΩ. Typical radius of the filament under similar experimental conditions is about 50 μm [3,9,18]. Then the electron density is estimated to be about 1.2 × 1017 cm−3. The whole temporal evolution of electron density on different positions are calibrated by right vertical axis in Figs. 3(d)-3(f). The electron density of filament at strongest ionized position is also measured by standard EC method to be about 2.0 × 1016 cm−3, which is less than that measured by EMI method. In our opinion, the electron density obtained by EMI method is closer to the true value because the EMF signal is directly from the filament. Furthermore, the main advantage of EMI method is the reliable inversion of time dependent electron density inside the filament due to successful decoupling of the self-oscillation and interference in circuit.To verify that the signal of coil is truly induced by the transient magnetic field around the filament, the following comparative experiments have been performed. The idea of the test is that if the plane of coil is perpendicular to the filament, or coil is parallel to filament but no voltage applied, as shown in Fig. 4(a), then theoretically no magnetic flux passes through the coil, and no EMF signal will be detected. Figure 4(b) presents an example of the EMF signal (black line), the signal when the coil is perpendicular to filament (green line), and the signal when the coil is parallel to the filament but without external voltage (red line). It can be seen that the EMF signal is about 4 times higher than the last two cases, which demonstrates that the electromagnetic induction is the dominant process in the electron density measurement. This point is well supported by the integrated curves of oscilloscope signals in these three cases, as presented in Fig. 4(c). The weak signals (green and red line) detected in additional tests are mainly induced by the electromagnetic pulse radiated from free filament due to the ponderomotive force of laser pulse on the free electrons [38, 39], and also possibly induced by the laser irradiance of electrodes. The static electric field built by 100 V voltage is rather weak and could not significantly affect the radiation characteristics, therefore the radiation signals received by coil for these two cases in Fig. 4(a) are similar and much lower than EMF signal.
Fig. 4 (a) The attitudes of coil in additional test experiment; (b) Signals from coil in conditions of different external voltage and attitudes of the coil; (c) The integral signals of (b).
4. Conclusion
In conclusion, the temporal evolution of electron density inside femtosecond laser filament has been investigated experimentally using electromagnetic induction method, which is based on detection of the transient magnetic field around electrified filament by a single turn coil. The time dependent electron density of filament is estimated from the electromotive force signal of coil. The comparative study shows that the standard electrical conductivity method gives a lower initial electron density than the true value within an order of magnitude and strongly disturbed temporal characteristics of filament in most of cases. As an alternative, the electromagnetic induction method can obtain reliable temporal evolution of electron density of filaments easily by single shot operation.
Funding
National Natural Science Foundation of China (grant nos. 11574387, 11334013, 11375262, 11404335); National Basic Research Program of China (grant nos. 2013CBA01501); Science Challenge Project (No. TZ2016005); Strategic Priority Research Program of the Chinese Academy of Sciences (grant nos. XDB16010200, XDB07030300, XDB17030400).
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