Utilization of self-lasing radiation for characterization of plasma discharge waveguides

Naturally occurring self-lasing of a confined plasma discharge is used as a plasma diagnostic. Together with other readily measurable parameters such as discharge voltage and current, the laser radiation provides the necessary constraints for fitting the parameters of a plasma chemistry model. The model determines the plasma density, electron temperature and excited-state populations as functions of time and space and shows excellent agreement with experiments performed in a nitrogen-filled discharge tube. Plasma self-lasing has been observed in a form of a ring and has a plasma density profile that can be employed for optical guiding.


Introduction
Plasma waveguides have potential applications in multiple areas of science and technology from particle acceleration and non-linear optics to plasma-based electronics, directed-energy and space propulsion [1][2][3][4].The main advantages of a plasma as a waveguide are the extremely high damage threshold [5], the ability to guide a broad range of wavelengths [6] (from soft x-rays to terahertz), and on-demand modification of the guiding parameters [7].The plasma waveguide can be generated by confined electrical discharge [8], thermal expansion of the gas by femtosecond laser filament [9] and other drivers such as microwaves and shockwaves.
Each application of a plasma waveguide requires an appropriate diagnostic tool for monitoring and adjusting of the plasma parameters [10].In many applications only a realtime knowledge of the waveguide parameters can provide * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the necessary control for stable operation.The main parameter of interest is the radial distribution of the neutral gas or free electron number density, which can be measured using appropriate optical diagnostics such as interferometry and optical emission spectroscopy (Stark broadening or Raman shift) whose choice depends strongly on the dynamic range and sensitivity of the diagnostic (see review [10] and references therein).Most of these diagnostics introduce additional elements in the experimental apparatus and require line-of-site capability that is not always available.
The purpose of this paper is to propose and demonstrate a novel approach to diagnosing the plasma characteristics, based on the naturally occurring self-lasing of the plasma, not requiring any additional hardware.Lasing of nitrogen gas, in the form of a confined, cylindrical shell in a capacitively-driven discharge provides the proof-of-concept diagnostic of a plasma channel capable of guiding optical radiation.The diagnostic is capable of measuring a wide range of plasma parameters, including electron density and temperature that are determined using a model incorporating the relevant reactions among the chemical species.Without the self-lasing information, the experiment does not provide enough constraints for the fitting parameters of the model.Moreover, the laser intensity profile gives radial dependence of the plasma parameters in addition to their temporal evolution.
A key characteristic of plasma waveguides is their ability to guide optical beams over a broad range of wavelength.To quantify this statement, consider the case where the plasma has a parabolic radial density profile where n 0 is the plasma density on axis and R ch (z) is the channel 'radius' i.e. the plasma density doubles at that radius.In general, the channel radius and density can vary slowly along the channel axis (i.e.along the coordinate variable z).
The assumption that the plasma density has a quadratic dependence on the radial coordinate is reasonable in many cases, and this form preserves the radial profile of the TEM 00 Gaussian beam in the low power limit.Guided propagation results when the refractive property of the plasma channel balances the diffractive tendency of the laser beam.The matched radius r M (z) of the mode in the channel given by ( 1) is [11] where k p = √ 4π n 0 e 2 /mc 2 is the plasma wavenumber.Here e, m, and c are the electron charge, the electron mass, and speed of light respectively.A significant feature of (2) is that a given channel can guide multiple wavelengths with the same matched radius.In contrast, guiding channels from conventional materials are limited to a narrow range of wavelengths and can guide only specific modes of radiation.It should be noted that the magnitude of the deviation in the refractive index required to guide a laser beam is relatively small.

Experimental
To demonstrate the novel diagnostic we designed an experimental system schematically shown in figure 1.A plasma waveguide was generated in a 30 cm long, 1/4 inch (6.35 mm) internal diameter alumina tube attached to metal flanges on each end.The flanges played the role of high voltage electrodes and were enclosed by flat fused silica windows.The ground side flange was connected to a vacuum pump whilst the high voltage side connected to the N 2 gas supply.The flow of nitrogen was sufficiently slow so that no measurable density gradient was created along the length of the tube.Nitrogen in the tube was ionized by a modified 'capacitor transfer' circuit [12] with storage capacitor C1 = 5 nF and peaking capacitor C2 = 2.5 nF, decoupling inductance L = 240 µH, and charging (R1 = 1.3 MΩ) and discharging (R2 = 100 MΩ) resistors.The circuit could be charged up to negative 35 kV DC provided by a high voltage power supply, and the discharge was initiated by a fast solid state switch (S33A Applied Pulsed Power).Current and voltage were monitored by corresponding probes and recorded by an oscilloscope.To avoid heating of the discharge tube the system was operated at repetition rate below 1 Hz.
The optical part of the system included high reflectivity mirror (UV enhanced aluminum mirror) on the high voltage side of the discharge tube.Due to the exceptionally high gain of the nitrogen laser, the laser cavity operated in a double pass mode, without an output coupler.After optimization of the electrical parameters and cavity alignment, a coherent, collimated beam at 337 nm, which corresponds to N 2 (C) → N 2 (B) transition, was emitted from the discharge tube.Part of the beam was diverted by a beamsplitter to a fast photodiode, and the rest was imaged onto a camera by 100 cm focal length lens.A typical image of the beam is shown on figure 1

Plasma chemistry model
In this section, an overview of the plasma chemistry model is given, which is used to simulate the plasma properties (species densities and electron temperature) and lasing, as well as the electrical characteristics of the discharge.The validity of our model extends beyond previous analyses of nitrogen lasing in the atmosphere wherein the range of allowed electron temperature was limited to 0.1 < T e < 2 [13,14].The model has been limited to the minimum set of species and reactions appropriate for the experimental conditions under which lasing has been observed (nitrogen pressures on the order of ∼1 Torr and Table 1.List of reactions and rate coefficients for N 2 .The references for reactions 1-20 refer to the cross section source.The electron temperature is in units of eV, the collisional rate coefficients are in units cm 3 s −1 , and the radiative transition rates are in units s −1   electric pulse duration time of ∼150 ns).The species used in the model are listed in figure 2 and the reactions and rate coefficients are listed in table 1.The excited states, relevant for lasing are the N 2 (B) and N 2 (C) levels, however the N 2 (A) level is included too, because it is a metastable state and is expected to have a high population, feeding the B and C levels via collisional excitation with electrons.The lifetime of this lower level is 40 µs, causing the laser to self-terminate in ≈20 ns on account of the lower-level bottleneck.Since N + 4 is unlikely to form at low pressure on a such short time scale, the only ion under consideration is N + 2 , and we can assume that the N + 2 density is equal to the electron density.The following collisional processes are taken into account: elastic scattering of electrons with ground state N 2 molecules [15], collisional excitation of N 2 vibrational [18] and electronic states [15], ionization from the N 2 ground state [15] and excited states [16], collisional quenching of N 2 excited states [19,20], radiation and dissociative recombination [17].Three-body reactions are deemed slow and have not been included.The cross sections for electron impact excitation and ionization have been integrated over a Maxwellian electron energy distribution function (EEDF) to calculate the collision rate coefficients.The use of a Maxwellian is justified by the very high ionization degree (the ratio of electron-to-neutral density), which is on the order of 10 −2 (see figures 5 and 6).The rate coefficients are then fitted in the Arrhenius form, except for the elastic scattering (reaction 1 in table 1).They are valid in the electron temperature range 0.1 < T e < 20 eV [15].
The species densities, n s , are advanced in time: In the right hand side of the equation, R + s and R − s are the species s gain and loss terms, respectively.In particular, the equation for the electron density reads: dn e /dt = R 4 + R 5 + R 19 + R 20 − R 6 .Electrons are created via ionization (reactions 4, 5, 19 and 20) and destroyed via dissociative recombination (reaction 6).Equation ( 3) is applied at any radial position independently of the others (local approximation).The electron temperature T e is determined from the electron power balance equation, which reads: where k B is the Boltzmann constant.The Joule heating term takes the standard form, P joule = σE 2 , where σ is the plasma conductivity and E is the electric field along the discharge tube.P coll is the rate at which energy density is transferred from electrons to heavy particles.The plasma conductivity is σ = n e e 2 /(mν mom ), with ν mom being the momentum transfer collision frequency ν mom = k 1 N g (k 1 is the rate coefficient for reaction 1).It evolves dynamically in time and space since both n e and T e are functions of time and radial position.One of the assumptions of the model is to use a locally computed values along the tube radius.In reality, each radial zone is coupled to two adjacent ones via diffusion, which is neglected.The diffusion length, L diff ∼ = √ D amb τ , is determined by the electric current pulse duration, τ ≈ 150 ns, and ambipolar diffusion coefficient, 2 ions in N 2 at pressure 1 atm [21].At pressure p = 1 Torr and T e = 10 eV, L diff ∼ = 0.05 cm, much smaller than the tube radius (a = 0.32 cm), which justifies to a large extent the use of local values.
It is further assumed that the plasma is uniform along the tube axis.The gas breakdown occurs as follows.When voltage is applied, a uniform electric field is set up in the entire discharge gap.With the voltage increasing (in time), the reduced electric field E/N g becomes large enough for breakdown to occur in the entire gap simultaneously (from one electrode to the other).Thus, the model describes the gas breakdown process in time and one spatial dimension (radial), assuming that it is uniform along the other (axial).
The electric field in the plasma is determined by the applied voltage V(t) and the tube length L according to E(t) = αV(t)/L.The coefficient α < 1 accounts for the fact that the plasma partially screens the applied voltage.The current I, flowing through the discharge tube at any given time t, is calculated by integrating the local current density, j(r, t) = σ(r, t)E(t)R(r) along the radius r.Here, R(r) is the normalized (and dimensionless) observed radial intensity profile of the laser beam.The profiles are shown in figure 3 and the following argument justifies this approach for j(r, t) calculation.The current density is proportional to the electron density, which grows exponentially with the reduced electric field E/N g .The excitation rate coefficients to the lower laser level, N 2 (B), and upper laser level, N 2 (C), also scale exponentially with E/N g .The excitation rate to these levels and therefore, the laser intensity, have comparable scaling.Thus, the laser intensity and current density grow at roughly the same (exponential) rate.The laser beam profile for 3 Torr gas pressure (figure 3(a)) is narrower than that for 0.6 Torr (figure 3 explained with diffusion processes, whose role decreases with pressure.

(b)). It can be
The current I(t) can be calculated where A = 2π ´a 0 r R(r)dr and a is radius of the discharge tube.
The screening parameter α is not a-priori known but the procedure of determining α is very rapid, since the ionization rate is an exponential function of E(t), and so is the electron density growth and plasma conductivity, roughly, I ∼ σ ∼ e α and the desired α can be found with a few iterations by comparing the simulated current (5) with the corresponding experimental measurement.
The model also requires initial conditions.It was established that it is very sensitive to the initial value of the electron density n e0 , which is specified at a location not far from the tube wall.It is surmised that seed electrons are emitted from the wall due to breakdown of the gas by large electrostatic fields caused by bumps and spikes on the wall [22].Initial electron density n e0 and the shielding α were used as fitting parameters of the model.The best fitting was established when simulated current and laser transition population inversion overlap with experimentally measured current and laser pulse while experimental voltage V(t) was fed into (4).

Results and discussion
Detectable laser radiation was observed for nitrogen pressures 0.5-6.5 Torr.Here we discuss the two cases 0.6 and 3 Torr in detail.The shielding coefficient α and initial electron density n e0 changed continually with the pressure change.The fitting parameters for 0.6 Torr were n e0 = 9 × 10 8 cm −3 and α = 0.22.For 3 Torr n e0 = 6 × 10 9 cm −3 and α = 0.61.Simulation results were sensitive to these fitting parameters and even 2% change in n e0 or α produced significant differences with experimental current and laser pulses (both in time and amplitude).
The measured nitrogen laser lifetimes in the experiment (as determined by the width of the green curves in figure 4) is on the order of 10 ns (full width at half maximum), consistent with the low pressure nitrogen laser lifetime of 4-20 ns [23].
In the plasma chemistry model the nitrogen lasing lifetime is determined by the lifetime of the upper laser level, N 2 (C), and collisional quenching with nitrogen molecules and electrons.From the rate coefficients listed in table 1 at electron temperature 10 eV, the lasing lifetime is τ c [ns] ∼ = 39/(1 + 0.016p + 2 × 10 −14 n e ).It is not sensitive to the gas pressure (p ∼ 1 Torr), but can be affected by collisional quenching with electrons, whose density is very large (n e ∼ 10 14 cm −3 ).
Figure 5 shows simulated electron density and temperature for 3 and 0.6 Torr pressures.In the case of 0.6 Torr, ionization starts later and observed over wider area along the radius, proportional to the experimental laser ring profile shown in figure 3.In addition, the electron temperature in 0.6 Torr pressure reaches 12 eV, compare to 6 eV for 3 Torr pressure.The temperatures ratio is proportional to the ratio of the peak currents of 2 and 1 kA correspondingly (figure 4).
In addition to the standard plasma parameters, electron temperature and density, the proposed plasma diagnostic is capable to measure the excited-state populations.The radial function of the population inversion N 2 (C) − N 2 (B) is shown in figure 3(c) for the moment when the this inversion is maximized.Figure 6 shows temporal evolution of the excitedstate populations N 2 (A), N 2 (B), N 2 (C), N 2 (ν) and the electron density n e at the radial position of the highest lasing intensity.
Simulated population inversion matches the experimental time duration of the laser pulse (figure 4), but produces narrower radial extent compared to the experimental laser beam profile (figure 3(c)).This discrepancy can be explained in part by diffusion and radiation that are not included into the simulation.The observed in figure 5 plasma density profiles are similar to step-index optical fiber [24].Diffusion will eventually reshape these density profiles into parabolas [22], capable of guiding Gaussian laser beams described in the introduction.

Conclusion
We have demonstrated a novel approach to diagnosing a longitudinal confined, cylindrical plasma discharge, based on the naturally occurring self-lasing of the plasma.Pure nitrogen gas was used as the active medium and the measured voltage and laser profile were used as input parameters for a plasma chemistry model.The electric current amplitude and temporal profile together with laser timing provided a complete set of constraints for fitting the parameters of the model.
The short duration of the electric current pulse, compared to the time for particle diffusion in the radial direction, provided justification for the locally computed values of plasma parameters in the radial direction, thus permitting solution of the relevant equations ( 3) and (4) on a laptop computer in under a minute.Including diffusion in the model, while providing  more realistic lasing profiles, would significantly complicate the modeling effort and require much larger computing resources.
The very strong gain of the nitrogen laser allows operation in a simple, double-pass configuration with only one cavity mirror, or over a single pass, without any mirrors.Single pass beam profiles (not shown) were measured by blocking the high reflectivity mirror and were similar (but significantly weaker) than the double-pass profiles.
As described here, the experiment has significant limitations on the gas pressure at which simple, longitudinal nitrogen lasing is observed.Lasing in nitrogen at higher pressures has been demonstrated in transverse discharge configurations [25][26][27].It may be possible to employ the diagnostic approach discussed here in transverse discharges, but they are not expected to generate waveguide-like plasma density profiles and the initial electron density is likely non-uniform in the lasing direction.
The approach discussed here can readily be extended to lasing action in gasses like argon or carbon dioxide (typically a mix of CO 2 , nitrogen and helium).In longitudinal configuration these lasers can use relatively simple power supplies and operate in the geometry discussed here [28,29].The major difference compared to the nitrogen laser is that the longer duration of the lasing may necessitate the use of a streak camera for recording the instantaneous radial profiles.In addition, diffusion must be included for modeling longer discharges.
The experiment and model presented here represent the proof-of-concept to a new approach for diagnosing plasma parameters.As with any new diagnostic it requires independent verification by comparing with established techniques.In the case of a long, enclosed, low density plasma column such verification presents a challenge.It may be possible to estimate the plasma parameters using time-resolved plasma spectroscopy [30] or pulsed microwave diagnostics [31].Both of these approaches require significant experimental and analytical effort and are beyond the scope of this paper.

Figure 1 .
Figure 1.Optical and electrical schematics of the experimental system.Beamsplitter (BS), photodiode (PD), highly reflective mirror (HR), solid state switch (SS).All other symbols are explained in the text.An example of the image recorded on the camera is shown in the lower left corner.
insert.The following parameters were recorded and used for the plasma characterization: voltage, current and photodiode signals as function of time, laser beam intensity as a function of radius.Examples of these parameters are shown on figures 3 and 4 below.

Figure 2 .
Figure 2. N 2 energy level diagram and list of species used in the model and their energies accounted from the ground state.

Figure 3 .
Figure 3. Laser beam profiles for 3 Torr (a) and 0.6 Torr (b) gas pressures.Solid lines in (c) show laser intensities R(r) along the radius of the discharge tube, dashed lines are normalized, simulated population inversions N 2 (C) − N 2 (B).Dotted lines show location of the wall.

Figure 4 .
Figure 4. Experimental voltage, current, and laser pulse (solid lines) for 3 Torr (a) and 0.6 Torr (b) gas pressures.Dashed lines represent fitting of simulation results.Laser and population inversion are plotted in arbitrary units unrelated to vertical axes.
. DB in the reference column stands for detailed balancing.