Abstract
Multiphoton ionization (MPI) is a fundamental first step in highenergy lasermatter interaction and is important for understanding the mechanism of plasma formation. With the discovery of MPI more than 50 years ago, there were numerous attempts to determine the basic physical constants of this process in direct experiments, namely photoionization rates and crosssections of the MPI; however, no reliable data was available until now, and the spread in the literature values often reaches 2–3 orders of magnitude. This is due to the inability to conduct absolute measurements of plasma electron numbers generated by MPI, which leads to uncertainties and, sometimes, contradictions between MPI crosssection values utilized by different researchers across the field. Here, we report the first direct measurement of absolute plasma electron numbers generated at MPI of air, and subsequently we precisely determine the ionization rate and crosssection of eightphoton ionization of oxygen molecule by 800 nm photons σ_{8} = (3.3 ± 0.3)×10^{−130} W^{−8}m^{16}s^{−1}. The method, based on the absolute measurement of the electron number created by MPI using elastic scattering of microwaves off the plasma volume in Rayleigh regime, establishes a general approach to directly measure and tabulate basic constants of the MPI process for various gases and photon energies.
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Introduction
Since the mid1960s^{1,2,3} laserinduced plasmas have found numerous applications in the laboratory ranging from fundamental studies of nonequilibrium plasmas^{4,5}, soft ionization in mass spectroscopy^{6}, development of compact particle accelerators^{7,8}, and Xray and deep UV radiation sources^{9,10} to diagnostic techniques such as laserinduced breakdown spectroscopy and laser electronic excitation tagging^{11,12,13}. In addition, the laserinduced plasma is related to studies of various nonlinear effects at beam propagation, such as laser pulse filamentation, laser beam collapse, selftrapping, dispersion, modulation instability, pulse splitting, etc.^{5,11,12,14,15}.
Multiphoton ionization (MPI) is a key first step in all laserinduced plasma. However, basic physical constants of the MPI process, namely photoionization rates and crosssections, have never been precisely determined in direct experiments. This refers to the fact that there are no diagnostic tools available today to provide absolute measurements of the total number of electrons in plasma volume N_{ e } or local plasma density distribution n_{ e }(r) generated by femtosecond laser pulse in a relatively low intensity linear regime. For MPI of atmospheric air, plasma density has to be below n_{ e } ≤ 10^{15}–10^{16} cm^{−3} to ensure that the contribution of plasma nonlinearities to the refraction index is negligible^{5}. At the same time, the sensitivity of laser interferometry is limited to n_{ e } ≥ 10^{16}–10^{17} cm^{−3} due to the minimal measurable shifts of the interference fringes^{16,17,18}. A number of semiempirical methods for relative measurements of plasma density were proposed, but all require absolute calibration based upon theoretically predicted values of plasma number density. Timeofflight (TOF) mass spectrometer measurements of ion currents generated by laserinduced plasma filament have been conducted to measure photoionization rates^{14,19,20,21}. The fundamental limitation of this technique is the inability to conduct absolute calibration of the system since the reference object against which the calibration can be completed is not readily available. Therefore, the TOF mass spectrometer measurements relied on theoretical estimation of total number of electrons in the focal zone in order to conduct absolute calibration of the system. Very recently, scattering of THz radiation from laserinduced plasmas was proposed for spatially unresolved relative measurements of n_{ e }^{16,17}. Other recently proposed measurement techniques were based on measurements of capacitive response times of the system, including a capacitor coupled with laserinduced plasma loaded inside^{22,23,24}. These attempts to measure n_{ e } in laserinduced plasmas were characterized by varying degrees of success and reliability of the obtained data, but none of them provided an ultimate solution for absolute plasma density measurements.
Analysis of various theoretical and semiempirical approaches undertaken previously led to a large variability of photoionization process constants available in the literature, some of which were controversial. For example, photoionization rates for O_{2} reported by Mishima in ref.^{25} are approximately 2 orders of magnitude higher than that reported by Talebpour in ref.^{20}. In addition, comparison of photoionization rates for N_{2} and O_{2} yields to 3 orders of magnitude higher photoionization rates for O_{2} compared to that of N_{2} due to difference in ionization potentials (15.576 eV and 12.063 eV respectively)^{24}, while experimentally determined photoionization rates reported by Talebpour for N_{2} and O_{2} yields doubtful proximity: 1.5 × 10^{9} s^{−1} for N_{2} and 3 × 10^{9} s^{−1} for O_{2} for the laser intensity 3 × 10^{13} W/cm^{2} ^{20}.
Therefore, photoionization rates and crosssections of the MPI process still remain unknown 50 years after the discovery. Large discrepancies in photoionization rates available in the literature cause contradictory conclusions and are generally disadvantageous for theoretical modeling of a wide class of problems involving laserinduced plasmas. In this work, we propose a direct experimental approach to measure the total number of electrons created at the MPI of air and directly determine the ionization rate and crosssection of MPI for the oxygen molecule. The proposed approach has tremendous fundamental significance and great potential for applications, since it paves the way to directly measure and tabulate basic constants of the MPI process for various gases and photon energies.
Methodology of MPI crosssection determination
Ionization of gas in laserinduced plasma is associated with multiphoton (MPI) and tunneling processes, two limiting cases of essentially the same physical process of nonlinear photoionization. The choice of the governing mechanism is dictated by Keldysh parameter γ, defined as a ratio of laser frequency ω to tunneling frequency ω_{ t } characterizing time of electron tunneling through the potential barrier: \(\gamma =\frac{\omega }{{\omega }_{t}}=\frac{\omega \sqrt{2m{ {\mathcal E} }_{i}}}{eE}\), where E amplitude of incident electric field,\(\,{ {\mathcal E} }_{i}\) ionization potential, and m and e are electron mass and charge, respectively. In the case of low frequency limit (and/or large laser intensity) ω < ω_{ t }, the electron has sufficient time to tunnel through the barrier and ionization is driven by the tunnel effect, while for highfrequency limit (and/or low laser intensity) ω > ω_{ t }, the electric field varies faster than the time required for tunneling and ionization is governed by the MPI process.
In our method, the crosssection of the MPI is determined experimentally based on absolute measurement of total electron numbers (N_{ e }) generated by a femtosecond laser pulse and precise measurements of the laser pulse characteristics. The experiments have been conducted at low laser intensities (<2.7 ×10^{13} W/cm^{2}, as detailed below) in order to ensure a pure linear operation regime when nonlinearities associated with plasma creation and optical Kerr effect are negligible (see below for details). In this case, plasma formation due to MPI by the femtosecond laser is described by the simple differential equation \(\frac{\partial {n}_{e}}{\partial t}=\nu \cdot ({n}_{0}{n}_{e})\), where n_{ e }  plasma density, ν = σ_{ m }I^{m} – ionization rate, σ_{ m } – crosssection of mphoton ionization process with \(m=\,{\rm{Int}}(\frac{{ {\mathcal E} }_{i}}{\hslash \omega })+1\), I– local instantaneous value of laser field intensity, and n_{0} – background gas density, while other physical processes can be neglected on the extremely fast time scale of the laser pulse^{1,26}. This equation can be easily integrated and plasma density n_{ e } created as result of action of femtosecond laser pulse can be found: \({n}_{e}={n}_{0}(1{e}^{\int vdt})\). For the case of low ionization degree n_{ e } ≪ n_{0}, plasma density distribution immediately after the laser pulse can be written in the form:
where time integration is taken over the duration of the laser pulse at particular location r. Total electron number N_{ e } generated by the laser pulse can be expressed by integrating Equation (1) over the entire plasma volume:
Equation (2) provides a general expression that can be used to determine the MPI crosssection as follows. Total electron numbers (N_{ e }) generated by the femtosecond laser pulse are measured using the Rayleigh Microwave Scattering (RMS) technique (see Methods for details). Spatial and temporal intensity distribution I(r, t) are determined in precise measurements of the laser beam and the integral in the righthand side is calculated. Then, one can determine σ_{ m } from Equation (2) for the known background gas density n_{0}.
General expression (2) can be simplified if additional assumptions are made. Firstly, we will consider in this work the most practical case of Gaussian beam. In this case, spatial and temporal intensity dependences can be expressed as follows:
where I_{0} – intensity in the beam center, \({w}_{0}1/{e}^{2}\) waist radius (at z = 0), \(w(z)={w}_{0}\sqrt{1+{(\frac{z}{{z}_{R}})}^{2}}1/{e}^{2}\) beam radius at location z along the beam, z_{ R }  Rayleigh length and τ – characteristic temporal width of the beam. This approximation uses the standard for the nondispersing medium Gaussian beam optics spatial distribution \(({I}_{{\bf{r}}}(r,z)={I}_{0}\frac{{w}_{0}^{2}}{w{(z)}^{2}}{e}^{\frac{2{r}^{2}}{w{(z)}^{2}}})\) with Gaussian temporal shape \({e}^{\frac{{(t{t}^{\ast }(r,z))}^{2}}{{\tau }^{2}}}\), where t^{*} = t^{*}(r, z) indicates moment of time when the beam reaches particular (r, z)location (it is taken as a given that beam peak reaches the waist at z = 0 at time t = 0, so that t^{*}(r, 0) = 0 and t^{*}(0, z) = z/c)^{27,28}. All parameters of the beam in Equation (3), namely I_{0}, w_{0}, z_{ R }, and τ, are determined experimentally.
Secondly, we will consider in this work the case of atmospheric air and 800 nm laser. In this case, MPI of oxygen molecules is the dominant process since the O_{2} photoionization rate is 2–3 orders larger than that for N_{2} due to its lower ionization potential^{5,24}. Thus, by using ionization energy of oxygen molecule \({ {\mathcal E} }_{i}\) = 12.2 eV and energy of 800 nm ionizing photons of ħω = 1.55 eV, it is clear that the eightphoton photoionization process should be considered, namely m = 8.
A simplified form of the expression (2) for MPI of air with femtosecond laser pulse of Gaussian shape in temporal and spatial domains can be deduced by analytical integration of the intensity in form (3), namely: \({\int }^{}{\int }^{}I{(r,z,t)}^{8}dVdt=\frac{231\pi }{1024\cdot 16}\sqrt{\frac{\pi }{8}}{{I}_{0}}^{8}\pi {w}_{0}^{2}{z}_{R}\tau \) (see Supplementary Materials), and plugging it into the righthand side of Equation (2). Finally, Equation (2) can be reduced to the form:
The determination of MPI crosssection of oxygen is conducted using Equation (4) as follows. N_{ e } in the lefthand side of the equation is measured using the Rayleigh Microwave Scattering (RMS) technique (see Methods). Spatial and temporal characteristics of the laser beam (I_{0}, w_{0}, z_{ R }, and τ) in the righthand side of Equation (4) are measured directly. Then, the experimental dependence of N_{ e } vs. laser intensities I_{0} is plotted, and σ_{8} is determined by obtaining the best fit of that dependence using Equation (4).
Experimental Details
The experimental layout including femtosecond laser and RMS system are shown schematically in Fig. 1. Photoionization of air (relative humidity 30%, temperature about 300 K) was achieved by focusing laser pulses having Gaussian temporal and spatial shape from a 800 nm Ti:Sapphire laser of 164 fs FWHM having repetition rate of 100 Hz using a 1000 mm planoconvex lens. The laser repetition rate was decreased from nominal f_{ rep } = 1 kHz to ensure no “memory effect” in the interrogated volume. To this end, the number of electrons generated by the laser pulse was measured as a function of the laser repetition rate (using Rayleigh Microwave Scattering as detailed below). The number of electrons being generated was independent on the repetition rates for f_{ rep } ≤ 100 Hz indicating absence of “memory effects,” while an increase was observed at higher repetition rates. The diameter of the incident beam on the lens was 7 mm. The pulse energy was varied using a linear polarizer and measured using a laser power meter (GentecEO XLP123SH2DO). Images of the plasma were taken using a 1024i Pi Max 4 ICCD camera. Coordinate z = 0 was chosen at the beam focal plane of the strongly attenuated laser beam (no plasma present). Several experiments were also conducted using 400 mm lens. The results similar to that with 1000 mm lens we obtained; however, focal length increase to 1000 mm led to increase of the total number of electrons generated by the laser due to increase of Rayleigh length and beam waist. This increased the accuracy of the measurements with Rayleigh Microwave Scattering system, and therefore, 1000 mm lens was used in the experiments presented below.
Measurements of the total number of electrons in the plasma volume were conducted using Rayleigh Microwave Scattering (RMS) diagnostics in which absolute calibration was completed using dielectric scatterers with known physical properties (see Methods). A homodynetype RMS system operating at the microwave frequency 10.8 GHz was used, as shown schematically in Fig. 1. The microwave signal from the source was split in two arms. One arm sent microwaves to the plasma using the radiating horn, while the second arm delivered the signal directly to the LOinput of the I/Q mixer. Microwave radiation was linearly polarized along the plasma channel orientation. Radiating and detecting horns were mounted at a distance of 6 cm from the plasma. The signal scattered from the plasma was received by the detecting horn, amplified, and sent to the RFinput of the I/Q mixer. The two outputs of the I/Q mixer were again amplified and captured on the oscilloscope. All components of the microwave system operated in the linear range of powers to ensure a measured response proportional to the amplitude of signal scattered from the plasma volume. The overall time response of the system was measured to be about 250 ps.
Measurements of MPI crosssection of oxygen
The spatial distribution of laser beam intensity was determined using beam profiler measurements conducted with a strongly attenuated laser beam. To this end, a set of attenuator plates was used to reduce the beam intensity manifold (about 2–3 orders of magnitude) to completely eliminate plasma creation. The \(1/{e}^{2}\) radius of the beam, measured using the beam profiler at various zlocations, is shown in Fig. 2(a,b). Location of beam waist refers to coordinate z = 0. Waist radius in x  and y directions were R_{ x } = 92.17 μm and R_{ y } = 94.41 μm, respectively. The average radius of the beam (w) was chosen to satisfy condition πw^{2} = πR_{ x }R_{ y } at each measured zlocation and was approximated by analytical function \(w(z)={w}_{0}\sqrt{1+{(\frac{z}{{z}_{R}})}^{2}}\) with waist radius w_{0} = 93.28 μm and Rayleigh length z_{ R } = 26.98 mm to achieve the best fit of the experimental data. Based on the data fit the beam quality factor M^{2} = 1.3, which is slightly worse than expected M^{2} = 1.1 from the laser and is attributed to the laser alignment and use of spherical lenses for beam size control.
The temporal shape of the laser pulse was determined using measurements of intensity autocorrelation function by means of a second harmonic generation crystal. The autocorrelation function had a nearly Gaussian shape with full width at half maximum (FWHM) equal to FWHM_{ τ } = 232fs, as shown in Fig. 2(c). Thus, it may be concluded that laser intensity in the time domain was also Gaussian, with \(FWH{M}_{t}=\frac{232\,{\rm{fs}}}{\sqrt{2}}=164\,{\rm{fs}}\). Finally, temporal dependence of the laser intensity was approximated by Gaussian distribution \(I\propto {e}^{\frac{{t}^{2}}{{\tau }^{2}}}\) with \(\tau =\frac{FWH{M}_{t}}{2\sqrt{\mathrm{ln}\,2}}=98.6\,{\rm{fs}}.\)
Measured temporal and spatial parameters of the femtosecond laser pulse utilized in this work are summarized in Table 1. Mean values averaged over the multiple measurements of the corresponding quantities and their standard errors are shown in the Table 1.
Optical images of the laserinduced plasma created by MPI of air were analyzed to demonstrate that nonlinear effects in the nonattenuated laser beam were small and, thus, intensity approximation used in Equation (3) still applies when the plasma was on. A typical photograph of the laserinduced plasma taken by ICCD camera (exposure time t = 0–10 ns) is shown in Fig. 2(e) (energy in pulse 620 μJ, intensity at the beam center I_{0} = 2.68 × 10^{13}W/cm^{2}). Figure 2(f) shows the corresponding distribution of plasma radiation intensity (S) along the zaxis for two laser intensities I_{0} = 2.68 × 10^{13}W/cm^{2} and 3.01 × 10^{13}W/cm^{2}. It was observed that the focal plane of the beam coincided with coordinate z = 0 for I_{0} ≤ 2.68 × 10^{13}W/cm^{2}. A shift of the focal plane toward the direction of the laser was observed for higher intensities, which can be explained by action of focusing Kerr nonlinearity. Based on that experimental evidence, we have concluded that nonlinear effects (Kerr and plasma nonlinearities) were negligible for I_{0} ≤ 2.68 × 10^{13}W/cm^{2}.
The electron number generated by fslaser laser pulse was measured using the RMS system shown in Fig. 1. Figure 3(a) presents a typical temporal evolution of the number of electrons and amplitude of scattered microwave signal for two values of intensity: I_{0} = 2.68 × 10^{13} W/cm^{2} and 2.93 × 10^{13}W/cm^{2}. Right and left vertical axes indicate the signal directly measured by the RMS and the total number of electrons in the plasma volume N_{ e } determined using the approach described in Methods. It was observed that plasma decayed faster for larger laser intensities; specifically, twofold decay occurs on characteristic times 2.5 ns and 2 ns for I_{0} = 2.68 × 10^{13} W/cm^{2} and 2.93 × 10^{13}W/cm^{2}, respectively. Several experiments were also performed in conditions of higher humidity (relative humidity 65%). Higher humidity had no effect on the yield of electrons, but resulted in faster plasma decay.
Two distinct physical processes occurring on significantly different time scales can be traced on Fig. 3(a). The first process is the fast rise at the moment of plasma creation (around t = 0) associated with the laser pulse passing the waist region and reaching the peak value of N_{ e }(t). The characteristic time of plasma creation is about the time required for light to pass the Raleigh length around the beam waist, namely \(\frac{{z}_{R}}{c} \sim 0.1\,{\rm{ns}}\). The second process is the decay of the plasma remaining after the laser pulse, which occurs on characteristics times of about several nanoseconds, according to Fig. 3(a) (see section below). The RMS diagnostic was unable to temporally resolve the precise details of plasma creation due comparable response times of the system used (about 0.25 ns). However, the RMS system precisely measured N_{ e }peak value and the following plasma decay N_{ e }(t) since plasma recombination is occurring on a time scale several nanoseconds slower and, therefore, the difference between the true peak value and the measured value is negligible. Note that the maximal electron number occurring immediately after the plasma creation is denoted as N_{ e } throughout the manuscript, while decay of the plasma refers to the dependence N_{ e }(t). Figure 3(b) presents the experimentally measured dependence of N_{ e } immediately after the plasma creation versus intensity at the beam center I_{0}. This peak value N_{ e }is used for the purpose of determination of MPI crosssection σ_{8}.
We will now determine MPI crosssection σ_{8} by fitting the measured dependence of N_{ e } versus I_{0} shown in Fig. 3(b) using analytical expression (4). According to the analytical expression, N_{ e } increases with laser intensity as \({{I}_{0}}^{8}\). RMS data shown in Fig. 3(b) indicates that dependence \({N}_{e}\propto {{I}_{0}}^{8}\) was satisfied with high accuracy at low intensities I_{0} < 2.7×10^{13} W/cm^{2}, which represents a clear manifestation of the pure MPI regime. Deviation from the \({{I}_{0}}^{8}\)law for higher intensities indicates a departure from the pure MPI process at these higher I_{0}, which can be explained by relative proximity of Keldysh parameter γ to 1 [top horizontal axis of the Fig. 3(b)] and action of Kerr nonlinearity. Therefore, N_{ e } was fitted by the \({{I}_{0}}^{8}\)law for intensities I_{0} < 2.7 × 10^{13} W/cm^{2}, and MPI crosssection σ_{8} was determined based on the fit of this initial segment of the dependence as shown by the blue line in Fig. 3(b) using the parameters of the laser system measured above and density of molecular oxygen in the background air n_{0} ≈ 5.13 × 10^{18} cm^{−3}. Finally, the MPI crosssection was determined to be σ_{8} = (3.3 ± 0.3) × 10^{−130} W^{−8}m^{16}s^{−1}.
Photoionization rates, electron and species’ densities
We will now consider oxygen photoionization rates based on the measured data and compare it with data available in the literature. For laser beam center intensities I_{0} < 2.7 × 10^{13} W/cm^{2} (pure MPI regime), dependence of photoionization rate can be readily plotted as ν = σ_{8}I^{8}, shown by the solid blue curve in Fig. 3(c), using the value of σ_{8} obtained above. The comparison with previously available data determined based on theoretical and semiempirical approaches is also shown in Fig. 3(c)^{20,24}. It is clear that semiempirical predictions given in ref.^{20} underestimated the photoionization rates about 2–3 times, while purely theoretical predictions in ref.^{24} seem to slightly overestimate the rates.
The experiments conducted here pave the way to determining plasma density distribution created in the fslaserinduced plasmas. Distribution of the plasma density immediately after the laser pulse for the laser Gaussian intensity distribution used in this work can be written using Equation (1) as: \({n}_{e}(r,z)={\sigma }_{8}{n}_{0}\)\({\int }^{}I{(r,z,t)}^{8}dt={n}_{e}(0,0)\frac{{w}_{0}^{2}}{w{(z)}^{2}}{e}^{\frac{2{r}^{2}}{w{(z)}^{2}}}\). Integrating the left and right side of this expression relates the plasma density at the origin location immediately after the laser pulse n_{ e } (0, 0) with the directly measured quantities of N_{ e }, w_{0}and z_{ R } as follows: \({n}_{e}(0,0)=\frac{{N}_{e}}{\frac{231\pi }{1024\cdot 16}\pi {w}_{0}^{2}{z}_{R}}\). Figure 3(d) shows the dependence of n_{ e }(0, 0) on laser intensity. A 2D distribution of plasma density n_{ e }(r, z) for I_{0} = 2.68 × 10^{13} W/cm^{2} is shown in Fig. 3(e).
We have numerically simulated the plasma decay to validate our experimental measurements. Plasma decay was simulated using a 1D axially symmetric model in radial direction that selfconsistently integrates NavierStokes, electron heat conduction, and electronvibration energy transfer equations^{25}. The model accounts for recombination of molecular ions, attachment processes, formation and decay of complex ions, electron energy losses due to electronic, vibrational excitations, and elastic scattering.
The simulated decay of the densities and temperatures of various plasma species at the point of origin is shown in Fig. 4 for intensity in the beam center I_{0} = 2.7 × 10^{13}W/cm^{2}. An extremely fast (<1 ns) decrease of electron temperature to about 0.4 eV is associated with electron energy loss due to vibrational excitation of molecules, while slower later decay is governed by elastic collisions. Plasma density in the center decayed twice in about 3.2 ns, primarily dominated by dissociative and threebody recombination of molecular ions. Slightly faster plasma decay times observed in the experiments (about 2.5 ns) might be related to the presence of water vapor in the ambient air.
Concluding remarks
In this work, we have presented a methodology that is paving the way for precise determination of the physical constants of multiphoton ionization, namely crosssection and ionization rate. The method utilizes precise measurement of the spatial and temporal distributions of the laser beam intensity and absolute measurement of total electron number in the plasma volume by means of elastic scattering of microwaves off the plasma volume and absolute calibration of the microwave system using dielectric scatterers. We have demonstrated the capability of this method on the example of eightphoton ionization of molecular oxygen and determined the corresponding MPI crosssection to be σ_{8} = (3.3 ± 0.3) × 10^{−130} W^{−8}m^{16}s^{−1}. Future studies may focus on the precise tabulation of the crosssections and photoionization rates of the multiphoton ionization for different gases and laser wavelengths using the methodology proposed and validated in this work. This effort would provide critical experimental evidence for the theoretical modeling of laserinduced plasmas.
Methods
Rayleigh Microwave Scattering method description
To more clearly illustrate the method of absolute measurement of electron number in plasma volume, we present the schematics of the Rayleigh Microwave Scattering (RMS) system in Fig. 5 and provide a more detailed description here. In the Rayleigh Microwave Scattering technique, elastic scattering of microwave radiation off the plasma volume is measured and the total number of electrons in the plasma volume is determined. The scattered radiation is created as the result of polarization of the plasma channel in the external microwave field. For the thin plasma channel, when amplitude of the microwave field is nearly uniformly distributed inside the plasma, the radiation in farfield is equivalent to the Hertz dipole radiation. Overall, such a process is analogous to elastic Rayleigh scattering of light, when radiation wavelength significantly exceeds the scatterer size. The signal scattered from the plasma is proportional to the total electron numbers in the plasma volume. Absolute calibration of the RMS system was conducted using dielectric scatterers with known physical properties.
Linearly polarized microwave radiation at frequency of 10.8 GHz was scattered on the collinearly oriented plasma filament and the amplitude of the scattered signal was measured. Microwaves are radiated and detected using a horn (as shown in Fig. 5) mounted at a distance 6 cm from the plasma. A homodynetype detection system was used for the scattered microwave signal measurements by means of an I/Q mixer.
We first demonstrate that the amplitude of the electric field induced inside the scatterer channel as a result of irradiation with microwaves was uniform throughout the channel and equal to the field amplitude E_{0} in the incident wave. Slender prolate plasma channel geometry with length (l) significantly exceeding the diameter (d) was used in this work. For conditions of this experiment, the plasma channel can be considered thin compared to skin depth (so that f[GHz] \(\le \frac{2.5}{\sigma [{\Omega }^{1}c{m}^{1}]\cdot d{[mm]}^{2}}\))^{29}. In this case, the amplitude of the electric field induced inside the scatterer with dielectric permittivity ε and conductivity σ can be written as \(E=\frac{{E}_{i}}{\sqrt{{(1+k(\varepsilon 1))}^{2}+{(k\frac{\sigma }{{\varepsilon }_{0}\omega })}^{2}}}\), where k  depolarization factor governed by the channel geometry, and E_{ i } – amplitude of incident microwave electric field at the channel location^{30,31,32}. For the conditions of the experiments conducted here, the depolarization factor k is small due to large aspect ratio AR = l/d ≫ 1: k ≈ (1)/(AR^{2})ln(AR) ≪ 1^{29,30,31}. Therefore, the amplitude of the electric field inside the channel is close to that in the incident wave E = E_{ i }^{31,33,34}. The flatness of the wave front surface along the scatterer length was ensured by placing the plasma scatterer at distance \(r > \frac{{l}^{2}}{\lambda }\approx \,1\,{\rm{c}}{\rm{m}}\).
Electrons in the plasma volume experience oscillations with an amplitude of \(s=\frac{e{E}_{i}}{m\,\omega }\) due to the incident microwave field (we consider no restoring force due to prolate plasma channel geometries)^{35}. The electron collision frequency in the denominator is governed by those with gas particles (ν_{ eg }) for densities <10^{17} cm^{−3}, since contribution of electronion collisions can be neglected in that range, so that finally \(s=\frac{e{E}_{i}}{{m}_{eg}\omega }\). Electrongas collision frequency ν_{ eg } is independent on plasma density, ensuring that electrons in each location inside the plasma volume experience essentially the same displacement. We used ν_{ eg } = 5.18 × 10^{11} s^{−1} based on electrongas elastic collision crosssection σ_{ eg } = 5 × 10^{−16} cm^{−2} and electron temperature T_{ e } = 0.4 eV.
Total dipole moment of the plasma channel (p) can be calculated as follows:
Radiation from the plasma dipole in a plane perpendicular to the dipole orientation was detected by the same horn antenna as incident one (see Fig. 5). The antenna was placed at a distance r = 6 cm from the plasma channel to ensure the dominant contribution of farfield \(( \sim \frac{{k}^{2}p}{r})\), while nearfield \(( \sim \frac{p}{{r}^{3}})\) is negligible (kr > 6)^{32,36}. Thus, the amplitude of the electric field at the location of the detecting horn was:
A homodynetype detection system was used for the scattered microwave signal measurements, which provides output voltage \({U}_{out}\propto {E}_{s}\). The detection was achieved by means of an I/Q Mixer, providing inphase (I) and quadrature (Q) outputs. The total amplitude of the scattered microwave signal is determined as:\(\,{U}_{out}=\sqrt{{I}^{2}+{Q}^{2}}\). The amplifiers and the mixer used in the microwave system operate in a linear mode for the entire range of the scattered signal amplitudes, thereby ensuring that the output signal U_{ out } is proportional to the electric field amplitude of scattered radiation E_{ s } at the detection horn location: \({U}_{out}\propto {E}_{s}\).
It is clear that the RMS system detects the total number of electrons in the plasma volume accurate to the coefficient function of the specific microwave system used. Absolute calibration of the RMS can be conducted using dielectric scatterers with known physical properties. To this end, we now consider RMS system signal generated by the prolate scatterer made of dielectric material with dielectric constant ε and volume V. The only difference from the above consideration for the plasma channel would be that total dipole moment induced in the scatterer is p = ε_{0}(ε − 1)E_{ i }V, and thus:
One particularly convenient form of expression for the measured output of the RMS system is:
where A –proportionality coefficient, which is a property of the specific microwave system (utilized components, geometry, microwave power, etc.) while independent of scatterer properties, and it can be found using scatterers with known properties. The lower part of Equation 9 was used for calibration of the specific microwave system, particularly in order to determine the value of coefficient A. Cylindrical dielectric bullets made of Teflon with diameter 3.175 mm and length 1 cm were shot through the microwave field (along the same axis where plasma was later placed) using a pneumatic gun with velocities below 100 m/s in order to generate a timevarying response on the dielectric bullet passage for separation of signal scattered from the bullet from one DC background caused by reflections from surroundings, elements of microwave circuit, etc.
Calibration of the RMS system with a 1 cm long and 3.175 mm diameter Teflon bullet yielded the response of the RMS system shown in Fig. 6. Coefficient A was found to be A = 2.12 × 10^{5} V·Ω·m^{−2} in this calibration procedure. Thus, the relation between the total electron number in the plasma volume N_{ e } and amplitude of the scattered system measured using the RMS system U_{ out } was: N_{ e } = U_{ out } × 4.81 × 10^{13}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Dr. T. Meyers and Dr. D. Kartashov for useful discussions. This work was partially supported by NSF/DOE Partnership in the Basic Plasma Science and Engineering program (Grant No. 1465061) and by U.S. Department of Energy (Grant No. DESC0018156).
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A.S., X.W. and K.A.R. acquired the data, all authors analyzed the data, M.N.S. conducted numerical simulations, A.S., M.N.S. and A.S. wrote the paper.
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Sharma, A., Slipchenko, M.N., Shneider, M.N. et al. Counting the electrons in a multiphoton ionization by elastic scattering of microwaves. Sci Rep 8, 2874 (2018). https://doi.org/10.1038/s4159801821234y
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DOI: https://doi.org/10.1038/s4159801821234y
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