Interaction of Microwave Radiation Undergoing Stochastic Phase Jumps with Plasmas or Gases

Karas` V.I., Alisov A.F., Artamoshkin A.M., Gavrilenko I.V., Zagorodny A.G*.,
Zagrebel`ny I.A., Lontano M**., Mirny V.I., Potapenko I.F***., Us V.S.
NSC “Kharkov Institute of Physics &Technology” of NAS of Ukraine,61108, Kharkov,1Akademicheskaya Str.
*Bogolyubov Institute of Theoretical Physics of NAS of Ukraine,03143, Kiyv,14b Metrologichna Str.
** Istituto di Fisica del Plasma, Associazione Euroatom-ENEA, Italy,20125,Milan, 53 Roberto Cozzi Str.
***Keldysh Institute of Applied Mathematics of RAS, 125047, Moscow, Miusskaya sq.4
Abstract. The objective of the paper is to discuss the results of theoretical and experimental studies and
numerical simulations of the normal and oblique incidence of linearly polarized electromagnetic waves on an
interface between a vacuum and an overcritical plasma. The anomalously large transmission coefficient for
microwaves, the anomalous character of the breakdown conditions, the anomalous behavior of microwave gas
discharges, and the anomalous nature of collisionless electron heating, are attributed to stochastic jumps in the
phase of microwave radiation.
Keywords: intense stochastic microwave radiation, hybrid plasma waveguide, overcritical plasma, microwaves
with a stochastically jumping phase, transmission coefficient, gas breakdown, microwave gas discharge,
collisionless electron heating.
PACS numbers: 52.75 Di
Fainberg et al.[1] showed that stochastic electric fields with a finite phase correlation time can efficiently heat
particles in a collisionless plasma, so physically the inverse correlation time in the interaction between a particle and an
electromagnetic wave has in fact the meaning of an effective collision frequency [1]. The objective of the present paper
is to determine the conditions for the effective penetration of microwaves with a stochastically jumping phase into a high
density plasma, to study the collisionless electron heating by it, and to utilize microwaves to initiate microwave
discharges in light sources.
In this section we discuss the results of theoretical investigations of the normal and oblique incidence of linearly
polarized electromagnetic waves on an interface between a vacuum and an overcritical plasma. The electron dynamics is
described by the relativistic Vlasov equation for the electron distribution function together with Maxwell’s equations for
self-consistent electromagnetic fields under the assumption that the ions are immobile. We use the method [2,3] that not
only provides a complete nonlinear kinetic description of the electron dynamics for a plasma of arbitrary density and for
electromagnetic waves of arbitrary intensity but also makes it possible to carry out numerical simulations by taking short
time steps in comparison to the electron plasma period. We consider how an overcritical plasma (ω <ω p , where ω is
the wave frequency and ω p is electron Longmuir frequency) is penetrated by three types of waves, namely, a
microwave with a stochastically jumping phase (MWSJP), a broadband regular wave with the same spectral density
(BBRWSSD), monochromatic wave. The possible mechanisms whereby electromagnetic waves penetrate through a
wave barrier in a plasma considered in detail [3].
Numerical simulations of a normally incident electromagnetic wave were carried out for the following parameter
values: eE / mωVT = 3, ω0 = 0.5ω p , τ = 40 /ω p , and X R − X L ≥ 2000λ D (where E is the wave electric field
intensity, VT is the electron thermal velocity, τ is the time between wave phase jumps, X R(L) is the right (left)
boundary coordinate of a numerical region respectivelly, and λ D is the Debye length). The total run time of the code is
T = 5000/ω p . The transmission coefficient for a wave is defined as the ratio of the electromagnetic wave energy at the
point x = X R (i.e., the energy that has passed through the plasma) to the energy of the incident wave at the point x =
X L (with allowance for the corresponding time shift). A monochromatic wave is reflected from the plasma almost
totally (except for its front). MWSJP is reflected to a lesser extent due mainly to the penetration of the wave pulses
associated with stochastic jumps in the wave phase. The transmission coefficient for a BBRWSSD as that of MWSJP is
one order of magnitude less because, in this case, the plasma slab simply acts as a filter that transmits waves with the
frequencies ω> ωp and reflects others. The longitudinal fields in the plasma are weak (two to four orders of magnitude
weaker than the transverse fields). During the run time of the code (5000/ω p ), the longitudinal energy of the electrons
(as well as their temperature) changes by no more than 1%. The electron distribution function remains nearly
Maxwellian, but a small fraction of the electrons (about 0,0001) are accelerated in both directions from the plasma
The oblique incidence of an electromagnetic wave was simulated for the same parameter values as those for the
normal incidence, the only difference being in the length of the time interval, T = 2500/ω p . In this case, the
electromagnetic wave incident on the plasma has a strong impact on the electron dynamics, especially at large angles of
incidence. The longitudinal electric fields in the plasma are close in strength to the transverse fields. The longitudinal
energy of the electrons and their temperature increase severalfold. The electron distribution function becomes non-
Maxwellian: it has a tail of accelerated electrons. The energy of the incident transverse MWSJP is partially converted
into the energy of the longitudinal wave and partially into the electron energy [3].
In order to illustrate the practical importance of the situation under examination, we present characteristic waveforms of
stochastic signals in new types of beam–plasma devices generating intense stochastic microwave radiation in the
interaction of electron beams with hybrid plasma waveguides that were developed and put into operation at the National
Science Center Kharkov Institute of Physics and Technology (Ukraine) [4-6]. From Fig. 1 we can see that

The passage of stochastic electromagnetic radiation from a broadband generator through a plasma was investigated
experimentally on the device [7]. The plasma in cavity was created by M-571 magnetron with a controlled output power
(W ≤2.5 kW), operating at a frequency of f = 2.475 GHz. The working gas (deuterium) was puffed into and pumped out
of cavity through pipes. The mirror magnetic field was produced by solenoids, positioned at the ends of cavity. It
should be noted that, although the experimental investigations were carried out under conditions different from those
analyzed theoretically in section 1 (the experiments were performed with a short plasma cavity, rather than with a semiinfinite
plasma, and with a nonzero external magnetic field), an important point, as will be clear later, is that they
justified the main conclusion of section 1 that microwaves with a stochastically jumping phase can penetrate far deeper
into the plasma than a broadband wave with the same spectral density. Preliminary experimental results were reported in
[7,8]. Our experimental investigations of the excitation of regular and stochastic electromagnetic waves in plasmas of
different densities and their passage through a cavity allow us to draw the following conclusions:
(i) A regular wave excites a cavity less efficiently than does a wave with a stochastically jumping phase (in order for the
transmitted signals from an incident regular wave and from an incident wave with a stochastically jumping phase to have
the same amplitude, the amplitude of the former should be one to two orders of magnitude larger than that of the latter).
(ii) As a regular monochromatic signal excites a cavity and passes through it, selectivity between eigenmodes and
unnatural waves is lacking.
The results of our experimental investigations are in satisfactory qualitative agreement with the theoretical predictions.
In 1992, specialists from the Fusion System Corporation (Maryland) designed a highly efficient light source operating
in the quasi-solar spectral region and based on an electrodeless microwave gas discharge in a sulfur-containing tube [9].
The continuous (molecular) spectrum of high-power optical radiation from a sulfur-containing lamp resembles that of
the Sun, but with depressed levels of IR and UV radiation.
The main problems associated with microwave pumping are as follows (see, e.g., [10]):
(i) To choose the power of a microwave signal and its shape (continuous or amplitude-modulated).
(ii) To design a microwave transmission line from a microwave source (generator) to a load (electrodeless lamp), to
construct a transmitter (whose operating regime should depend on the mode of microwave radiation), and to provide an
appropriate topography of the microwave field in the region where it interacts with the working substance of the lamp
(just after the generator is switched on and in the plasma operating mode).
(iii) To maintain the stable operation of the microwave generator loaded by the lamp, whose parameters change
substantially during the development of a microwave discharge (from the switching on of the generator up to the
beginning of the steady-state plasma operating mode).
The underlying problem is that of choosing the microwave field frequency so as to satisfy the requirement that the
input microwave power be minimum. In order to determine the working microwave frequency, it is necessary to
compare three parameters: the diameter of the shell Λ (Λ≈1–2 cm), the electron mean free path l, and the electron
oscillation amplitude A. Discharges in argon that evaporate sulfur (which is an electronegative element) can be initiated
only when the electrons oscillate within a quartz shell, i.e., when A < Λ/2. The capture of electrons by sulfur molecules
can only be balanced by intense ionization. It is known (see, e.g., [11]) that, for all gases, the dependence of the
threshold field for gas breakdown on the pressure has a minimum that separates two branches.
For regular microwave radiation, the threshold field just obtained is directly proportional to the frequency and is
inversely proportional to the gas density (pressure) and the size of the discharge region, in complete agreement with the
known experimental data (see, e.g., [11]). An important task is to determine the power of a microwave generator that is
required to initiate a discharge in a buffer gas and then to maintain it in a plasma after the evaporation and ionization of
sulfur. Recall that, for microwave discharges in regular electromagnetic fields, the threshold field is minimum when the
collision frequency is equal to the electromagnetic field frequency (see, e.g., [11]). Thus, at a frequency of f ≈3.0 GHz,
the minimum threshold field for breakdown of Ar at a pressure of about 650 Pa is 500 V/cm. Such field strengths can be
achieved in a cavity in which one of the walls is transparent to light.
In the present paper, we propose to initiate microwave discharges in argon containing sulfur vapor by MWSJP. The
advantages of this method are as follows: (i) such microwaves are capable of initiating discharges at lower gas pressures
because the jumping phase slows electron diffusion; (ii) the jumps in the phase ensure that the collisionless electron
heating is not accompanied by energy losses in elastic and inelastic collisions; (iii) a uniform microwave discharge is
easy to initiate because microwaves with a stochastically jumping phase can deeper penetrate into an overcritical plasma.
Let us now consider the conditions for breakdown in argon by microwave radiation from the generator described in
[6]. The working frequency of this generator is 450 MHz, the mean rate of the phase jumps being ν jp = 2⋅108 s-1 . It is
important to keep in mind that, when the electron energy increases from zero to the ionization energy I Ar , the cross
section for elastic collisions of electrons with argon atoms varies greatly (by a factor of about 30), being at its maximum
several times larger than the ionization cross section corresponding to electron energies of 15–20 eV. This makes it
possible to initiate discharges in argon by microwaves with a stochastically jumping phase at pressures as low as 4 Pa. In
this case, the mean rate of phase jumps is equal to the maximum inelastic collision frequency, which corresponds to
electron energies close to the ionization energy. Operation under such conditions is advantageous in that, first, no energy
is lost in elastic collisions, and, second, due to the jumps in the phase, the electron diffusion remains insignificant and the
electromagnetic energy is efficiently transferred to electrons. Our numerical simulations and preliminary experiments
show that, in order to initiate a microwave discharge at a frequency of 450 MHz in argon at a pressure of 4 Pa, the
microwave electric field strength should be about 50 V/cm, whereas sulfur vapor can be excited by an electric field of 25
V/cm, which can easily be microwave cavities. With the use of such chambers, it is possible to substantially reduce the
generator power. The working microwave frequency of this system, 450 ± 50 MHz, is consistent with standards adopted
for industrial, scientific, and medical applications. With the version of the light system proposed by the company, it
becomes possible to design compact low-power SLSs, in addition to the already existing traditional SLSs with output
powers in the kilowatts range [9, 10, 12], which are usually based on 2450 ±50-MHz magnetrons.
The main results of our investigations of the normal and oblique incidence of linearly polarized electromagnetic waves
on an interface between a vacuum and an overcritical plasma are as follows: (i) for the parameter values under
consideration, the transmission coefficient for MWSJP is found to be one order of magnitude greater than that for
BBWSSD; (ii) the electrons are shown to be heated most efficiently by obliquely incident MWSJP and, in addition, the
electron distribution function has a high-energy tail; and (iii) necessary conditions for gas breakdown and for the
maintenance of a microwave discharge in stochastic fields in a light source have been determined.The anomalously large
transmission coefficient for microwaves, the anomalous character of the breakdown conditions, the anomalous behavior
of microwave gas discharges, and the anomalous nature of collisionless electron heating have been attributed to
stochastic jumps in the phase of microwave radiation.
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