# Experiments Designed to Study the Non-Linear Transition of High-Power Microwaves through Plasmas and Gases

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{16}W/cm

^{2}) ultra-short pulses (~30 fs), is directed into a high density plasma (~10

^{18}cm

^{−3}), the extremely high laser electric fields act as a ponderomotive force generating a charge separation in the plasma, a wake. The latter is accompanied by strong electric fields, a wakefield (>100 GV/m) which can be used instead of a vacuum accelerator cell as an accelerating gradient for charged particles [1,2,3,4]. The characteristic times and lengths of the plasma in such wakes are tens of fs and tens of µm respectively which make diagnostics rather difficult. There are several sophisticated experimental studies using a unique longitudinal interferometric technique [5] and time-resolved polarimetry with plasma shadowgraphy [6] to measure plasma density gradients and the parameters of the electron bunches during the formation and evolution of a wake. Most of the diagnostics was focused on the accelerated particles or using the driving laser to diagnose the plasma density itself [7,8,9,10]. An experimental system to study this problem is affordable only to a few laboratories because of the unique features of such laser system and the expenses involved.

^{11}cm

^{−3}density. The plasma density modulation up to 40% was obtained when the microwave pulse duration is near the ion oscillation period. Other experimental studies of plasma wakefield generation, ponderomotive force effects, and self-focusing of microwaves (with frequencies in the range of 3–10 GHz) in plasma were also carried out [16,17,18]. All these experiments were characterized by a relatively low power microwave sources (10–250 kW) with much longer pulses (60 ns–1 µs) than the periods of the plasma electron and ion oscillation.

^{7}W/cm

^{2}HPM, a longitudinal electric field of ~10 MV/m can be generated [24].

_{mw}are the electron charge, mass, the microwave angular frequency and the amplitude of the microwave electric field respectively. For a cylindrical microwave beam with a radial Gaussian profile,$\text{}{E}_{mw}=\text{}{E}_{0}{e}^{(-{r}^{2}/{\rho}_{0}^{2})}$, focused to ${\rho}_{0}\approx 2\text{}\mathrm{cm}$, one can estimate the density modulation by equating F

_{P}with the space charge force, ${F}_{SC}=-e{E}_{SC}=\text{}-{e}^{2}\delta n\text{}r/{\epsilon}_{0}$, assuming that the latter it is directed mainly radially. Then the density modulation is given as $\frac{\delta n}{{n}_{0}}\text{}=2(\frac{e{E}_{mW}}{{\rho}_{0}m\mathit{\omega}{\mathit{\omega}}_{\mathrm{P}}})$, where n

_{0}is the initial uniform plasma density and ${\omega}_{P}={(\frac{{n}_{0}{e}^{2}}{{\epsilon}_{0}m})}^{1/2}$ is the plasma electron frequency. If we assume consider a microwave pulse of peak power P ≥ 200 MW with an electric field component of E

_{mw}≥ 20 MV/m, a density modulation of at least $\text{}\frac{\delta n}{{n}_{0}}\text{}\ge 0.4\text{}$can be obtained for an initial electron density of n

_{0}~ 10

^{10}cm

^{−3}below the critical density. The density modulation period is of the order of 1 ns over a range of ~10 cm, a parameter range much more tractable than the laser/plasma experiments though realized with much lower ponderomotive forces. Early studies of the interaction of microwaves with plasmas were performed at much lower powers (<250 kW and pulse lengths of 50 ns–3 µs) [11,15,16,17,18].

## 2. The Experiment

_{01}mode electromagnetic wave but for the X-SR-BWO we used a mode convertor to convert this mode to a TE

_{11}mode, more appropriate for focusing.

_{01}mode electromagnetic wave produced by the X-SR-BWO is transmitted through a mode-converter which transforms it to TE

_{11}, followed by an appropriate antenna which radiates the power towards a hyperbolic dielectric lens (34 cm diameter) which focuses the power ~8 cm from its tip inside a Pyrex tube (see Figure 1). This tube may be filled with gas to controllable pressures, with RF plasma produced by a set of quadrupole antennas placed near its walls [41], or pumped to near vacuum. A set of diagnostic tools, receiving antennas to measure microwave transmission, a fast frame camera to visualize the plasma in time and space, a spectrometer to study the plasma parameters and scintillators attached to photomultipliers to measure electron energies, are placed around this experimental chamber as seen in Figure 1. A matrix of Neon lamps placed on plane perpendicular to the microwave beam axis inside or outside the chamber, is also used to produce a two-dimensional spatial integrated distribution of the electromagnetic beam power. All our experiments so far have been performed in this chamber [35,42,43,44].

_{01}mode electromagnetic wave can be introduced directly (Figure 3a) [41]. This waveguide is made of parallel wires so that the longitudinal slots between the wires is large enough for the plasma produced by flashboard source [45] placed along the walls of the chamber (Figure 3b), to fill the volume of the waveguide. At the same time, the distance between the wires is small enough so that the electromagnetic power is contained with minimal losses. The flashboard plasma was characterized to be stable in time and space [37]. The chamber has longitudinal observation windows where external diagnostics can be placed.

## 3. Self-Channeling Experiments

^{14}cm

^{−3}. Note, that different well developed optical diagnostic methods based on laser assisted spectroscopy, Stark broadening of spectral lines and the observation of forbidden lines can be applied to measure high-frequency electric fields in the plasma [46,47,48,49,50,51,52,53]. However, the application of these methods in this case is very challenging because the plasma is dilute and the microwave beam interaction with the plasma is over less than a nanosecond resulting in only a limited number of photons emitted by the excited atoms which does not allow one to obtain a reliable spectral line profile.

_{0}≈ 0.14 for an electric field of 150 kV/cm, resulting in an average energy of ~25 keV for the oscillating electrons which is in agreement with the experimental results.

_{0}≤ 10

^{5}cm

^{−3}, then for a microwave beam of Gaussian distribution around r, we obtain an ionization electron density by $\mathrm{ln}\left[{n}_{e}(r,t)/{n}_{0}\right]={n}_{g}{{\displaystyle \int}}_{0}^{t}v(r,{t}^{\prime}){\sigma}_{i}\left[w(r,{t}^{\prime})\right]d{t}^{\prime}$, where n

_{g}is the gas number density, v the electron velocity and w the electron energy. This model predicts that as the microwave power increases during the rise of the pulse, the plasma density shifts from the axis to the beam periphery. It also shows that a threshold field amplitude exists for this process to occur, ${E}_{th}(V/cm)\approx (4-5)\times {10}^{5}/\lambda $, where for λ =3 cm, the wavelength associated to the 9.6 GHz microwave beam E

_{th}= 150 kV/cm. The 1D Particle in Cell (PIC) simulations [43] confirm that high energy electrons are indeed produced during the process (Figure 7).

_{g}= 1.64 × 10

^{17}cm

^{−3}and a 10

^{6}cm

^{−3}background electron density are seen in Figure 8 (for a detailed explanation see Ref. 46).

^{12}cm

^{−3}) walls is seen. In Figure 8. (2nd row) the guiding and reflections of the electric field inside and from the walls of the plasma channel can be depicted whereas in the 3rd row high energy electrons form along the axis. These LSP simulations confirm that the experiments described where an HPM beam interacts with a neutral gas, reveal an ionization induced self-channeling process.

## 4. Wakefield Experiments

^{10}cm

^{−3}density plasma and obtained that a localized wake develops near the focal plane (z = 0, in Figure 9 [35]) after the pulse has left this region.

_{01}mode pulse propagating in a cylindrical waveguide filled with under-dense plasma [57]. The electron motion can be separated into a fast and slow motion. The fast motion, oscillations at the microwave frequency can be averaged out to leave only the slower motion following the time dependence of the amplitude of the pulse envelope. This leaves a set of relatively simple equations of motion, $\frac{d{\overline{v}}_{r}}{dt}={f}_{r}(r){E}_{0}^{2},\text{}\frac{d{\overline{v}}_{z}}{dt}={f}_{z}(r){E}_{0}^{2},$ where f

_{r,z}(r) describe dependencies of the radial and longitudinal forces on the radius r, and E

_{0}(t) is the time dependent pulse envelope for a given waveguide radius. The analysis shows that the radial force f

_{r}(r) is much stronger than the longitudinal force f

_{z}(r) which also changes sign. The radial force is the sum of two opposing forces, the ponderomotive and the Lorentz force. Depending on the waveguide radius, the total radial force can either be in the direction of the Lorentz force, outwards from the axis of the waveguide, or in the opposite direction forcing the electrons towards the waveguide axis. For the radial force to be positive everywhere, that is, all electrons are forced towards the waveguide walls, it is required that the waveguide radius be ${r}_{WG}\le \text{}{r}_{cr}\left[\mathrm{cm}\right]=\frac{19.85}{f\left[GHz\right]}$ [37]. If ${r}_{WG}>\text{}{r}_{cr}$, some electrons move towards the axis and some towards the walls. This model was tested numerically for the Ka-SR-BWO and some of the results are seen in Figure 10.

_{01}source traversing a 1.4 cm radius waveguide made up of 24 uniformly distributed 2 mm diameter wires in a 5.25 cm radius tube (Figure 3) filled with a 3 × 10

^{10}cm

^{−3}electron density plasma.

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**General mechanical drawing (not to scale) of the X-band Super Radiance Backward Wave Oscillator (SR-BWO) and interaction chamber.

**Figure 2.**(

**a**) The Slow Wave Structure (SWS) of the X-SR-BWO and the electron beam cross section. (

**b**) The same for the Ka-SR-BWO.

**Figure 3.**(

**a**) The slotted cylindrical waveguide (

**b**) Drawing of the experimental chamber (left), the structure of a flashboard (upper right) and the equivalent circuit of the flashboard breakdown process (lower right) [37].

**Figure 5.**Time integrated side-view of the light emitted along the He filled interaction chamber for various values of the gas pressure [42] (

**left panel**). 1.2 ns long images of the light emission in a region close to the dielectric lens for 4.5 kPa, He pressure from the time light appears at t = 0 [46] (

**right panel**).

**Figure 6.**Time integrated patterns of the microwave beam power distribution measured by the array of Ne lamps at the distance of 10 cm from the focal plane: for He at 20 mPa (

**a**) 4.5 kPa (

**b**) 30 kPa (

**c**) and front view fast framing images of the plasma obtained for He at 4.5 kPa (

**d**) and 30 kPa (

**e**).

**Figure 8.**Results of Large Scale Plasma (LSP) simulations: Time- and space-resolved evolution 1st row: of the density of the plasma; 2nd row: of the electric field |E

_{x}| (electric field ranges in (

**a**) between 0 and 200 kV/cm, whereas in (

**b**,

**c**) 0–150 kV/cm and the contour colors scale accordingly); and 3rd row: plasma electron energy [42].

**Figure 9.**Electron density, ρ, contours (first row), contours of E

_{x}(second row), and contours of E

_{z}(third row) vs. z, for three time points. For electric field contour plots, the arrows represent the direction of the electron motion resulting from the corresponding electric field [35].

**Figure 10.**Radial distributions of the normalized density ${n}_{e}(r)/{n}_{e}^{0}$ for different waveguide radii, for P = 0.5 GW, ${n}_{e}^{0}$ = 2.5 × 10

^{10}cm

^{−3}[37].

**Figure 11.**Contours of the electron density n

_{e}normalized to the initial density, n

_{e}

^{o}= 2.5 × 10

^{10}cm

^{−3}vs. z at 3 ns from the microwave pulse injection start time at z = 0, t = 0, for different waveguide radii 6.2 (

**a**), 6.4 (

**b**), 6.6 (

**c**) and 7.5 mm (

**d**) for a 28 GHz, 500 MW and 0.4 ns Full Width at Half Maximum (FWHM) microwave pulse propagating from z = 0 downstream [37].

**Figure 12.**Relative electron density contours in the [r, z] plane at y = 0 and t = 3.5 ns [57].

**Figure 13.**Electron positions in the [r,z] plane including all electrons within a 1 mm thick slice, at t = 3.5 ns for 1.25 GW, 10 GHz, 0.4 ns FWHM pulse and ne = 3 × 10

^{10}cm

^{−3}[57].

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**MDPI and ACS Style**

Krasik, Y.E.; Leopold, J.G.; Shafir, G.; Cao, Y.; Bliokh, Y.P.; Rostov, V.V.; Godyak, V.; Siman-Tov, M.; Gad, R.; Fisher, A.;
et al. Experiments Designed to Study the Non-Linear Transition of High-Power Microwaves through Plasmas and Gases. *Plasma* **2019**, *2*, 51-64.
https://doi.org/10.3390/plasma2010006

**AMA Style**

Krasik YE, Leopold JG, Shafir G, Cao Y, Bliokh YP, Rostov VV, Godyak V, Siman-Tov M, Gad R, Fisher A,
et al. Experiments Designed to Study the Non-Linear Transition of High-Power Microwaves through Plasmas and Gases. *Plasma*. 2019; 2(1):51-64.
https://doi.org/10.3390/plasma2010006

**Chicago/Turabian Style**

Krasik, Yakov E., John G. Leopold, Guy Shafir, Yang Cao, Yuri P. Bliokh, Vladislav V. Rostov, Valery Godyak, Meytal Siman-Tov, Raanan Gad, Amnon Fisher,
and et al. 2019. "Experiments Designed to Study the Non-Linear Transition of High-Power Microwaves through Plasmas and Gases" *Plasma* 2, no. 1: 51-64.
https://doi.org/10.3390/plasma2010006