Electron Beam Sustained Plasma as a Medium for Amplification of
Electromagnetic Radiation in Subterahertz Frequency Band
A. V. Bogatskaya
1,2,3
and A. M. Popov
1,2,3
1
Department of Physics, Moscow State University, Moscow, 119991, Russia
2
D. V. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991, Russia
3
P. N. Lebedev Physical Institute, RAS, Moscow, 119991, Russia
Keywords: Amplification of an Electromagnetic Radiation, Electron Energy Distribution Function, Kinetic Boltzmann
Equation, Electronegative Gases.
Abstract: It is demonstrated that low-temperature weakly-ionized nonequilibrium plasma created by a high-energy
electron beam in gases (gas mixtures) with Ramsauer minimum in a transport cross section and effective
attachment of slow electrons can be used as a medium for amplification and generation of electromagnetic
radiation in subterahertz frequency band. Analysis of the electron energy distribution function (EEDF) in Xe
– F
2
mixture is performed. Energy interval with growing EEDF is found to exist in such a mixture. Such an
interval provides the existence of population inversion of the electron spectrum in continuum and is
responsible for positive value of a gain factor of microwave radiation. A gain factor depending on an
amplified radiation frequency and plasma parameters is analysed.
1 INTRODUCTION
The sources of terahertz (THz) and subterahertz
radiation are of significant interest in a number of
practical applications, in particular, in chemistry
(Skinner, 2010), molecular biology (Meister et al.,
2013), medicine (Titova et al., 2013), materials
science (Grady et al., 2013). The interest in terahertz
and subterahertz radiation and its possible
applications is caused by its ability to penetrate
through a lot of materials (Jepsen et al., 2011),
which are usually opaque in the infrared and visible
ranges. Also, a number of physical and chemical
processes can be controlled and governed by such a
low frequency radiation.
Recently (Bogatskaya and Popov 2013;
Bogatskaya et al., 2014; Bogatskaya et al., 2015)
propose to use plasma channels created in gas by
powerful femtosecond UV laser pulse for
amplification of the electromagnetic radiation in
subterahertz frequency band. To obtain the effect of
amplification one needs to have a population
inversion in the system. Such a population inversion
obviously arises in the process of multiphoton
ionization of gaseous media by a powerful laser
pulse of femtosecond duration. Really, in this case
the pulse duration is less than the time interval
between the elastic electron – atomic collisions and,
hence, photoelectron spectrum consists of a number
of peaks corresponding to the absorption of a certain
number of photons. Such a spectrum has energy
ranges characterized by population inversion. It was
demonstrated that the regime of amplification of
microwave radiation can be achieved only in a gas
with the energy interval of growing with energy
transport cross section (Bekefi et al., 1961; Bunkin
et al., 1973). In this case positive value of a gain
factor appears to exist if the position of energy peak
in the photoelectron spectrum is located within the
energy range with increasing transport cross section
(Bekefi et al., 1961; Bunkin et al., 1973) at time
intervals corresponding to the relaxation of the of
electron energy distribution function (EEDF). It was
found that among different atomic and molecular
gases that can be used for amplification xenon has
some advantages due to the large value of transport
cross section with a pronounced Ramsauer
minimum, large atomic mass and also absence of
excited levels in the energy interval where the
photoelectron peak is formed. For the microwave
frequency
11
105
s
-1
a gain factor 01.0
k
cm
-1
can be reached for the durations of tens of
Bogatskaya, A. and Popov, A.
Electron Beam Sustained Plasma as a Medium for Amplification of Electromagnetic Radiation in Subterahertz Frequency Band.
DOI: 10.5220/0005646702810288
In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 283-290
ISBN: 978-989-758-174-8
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
283
nanoseconds (Bogatskaya et al., 2015). The
possibility to amplify radio-frequency (RF) radiation
in nonequilibrium plasma was firstly experimentally
proved in (Okada and Sugawara, 2002).
It was mentioned by (Bogatskaya and Popov,
2013) that the effect of amplification of
electromagnetic radiation in a plasma channel
created by an intense femtosecond laser pulse is
close from physical point of view to the effect of
negative absolute conductivity in a gas discharge
plasma predicted by (Rokhlenko, 1978; Shizgal and
McMahon, 1985), experimentally detected by
(Warman et al., 1985), and discussed in detail in
reviews (Aleksandrov and Napartovich, 1993;
Dyatko, 2007). In (Dyatko, 2007) different physical
situations leading to the possibility of absolute
negative conductivity to appear are analyzed.
Among them is the non-self-sustained steady-state
discharge supported by high energy electronic beam
in a gas mixture with an efficient attachment of slow
electrons and pronounced Ramsauer minimum in
transport cross section. Such a situation can be
realized, for example, in Ar:CCl
4
(Dyatko et al.,
1987), Ar:F
2
(Rozenberg et al., 1988) or Xe:F
2
(Golivinskii and Shchedrin, 1989) mixtures.
In this paper we apply the analysis based on the
Boltzmann kinetic equation in two-term
approximation to study EEDF in a plasma in Xe:F
2
sustained by a high-energy e-beam in the presence
of a microwave field. The possibility to use such
plasma as a medium for RF field amplification is
studied in dependence on mixture parameters
(partial concentrations of mixture species, electronic
density) as well as frequency and intensity of RF
radiation.
2 BOLTZMANN EQUATION FOR
THE EEDF IN A PLASMA
SUSTAINED BY HIGH-ENERGY
E-BEAM
The energy spectrum of electrons in a plasma
),( tf
, normalized according to the condition
)(),( tNdtf
e
(1)
(
e
N is the electron density) was analyzed using the
kinetic Boltzmann equation in the two-term
approximation (Ginzburg and Gurevich, 1960):
).()(
)(
2
)(3
)(
),(
*
23)(
23
22
2
0
2
b
i
g
i
tr
i
tr
tr
QfQ
f
Tf
M
m
f
m
Ee
t
tf
(2)
Equation (2) has the form of the diffusion equation
in an energy space. Here,
0
E and
are the
amplitude and frequency of the amplified RF field,
T
g
is the gas temperature (below, we take
03.0
g
T
eV),
m
is the mass of the electron,
i
M (
2,1i
) are
the masses of the xenon atom and fluorine molecule
respectively,
mN
i
tri
i
tr
2)(
)()(
is the partial
transport frequency, where
)(
)(
i
tr
is the transport
scattering cross section for Xe (i=1) and F
2
(i=2)
molecule,
)(i
trtr
is the total transport
frequency,
1
N and
2
N are concentrations of Xe and
F
2
in a gas mixture respectively, )(
b
Q is the rate of
slow electron production by a high-energy electron
beam,
)(
*
fQ
is the integral of inelastic collisions.
These integrals are described in detail in the review
(Ginzburg and Gurevich, 1960). Further we will
assume that excitation of vibrational and electronic
levels of fluorine molecules do not contribute to the
EEDF evolution. This is possible if relative
concentration of fluorine molecules satisfies to the
inequality
**
)1(
112
)2(
I
MmNN
tr
(3)
024681012
0
10
20
30
tr
*10
-16
, cm
2
eV
Figure 1: Transport cross section of xenon atoms.
Here
*
is the cross section of excitation of a F
2
vibrational level with excitation energy
1.0
*
I
eV,
is the averaged over EEDF electronic
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
284
energy. If we assume that
1
eV and
10
*)1(
tr
we obtain that (3) is fulfilled for the
partial fraction of F
2
molecules in xenon being rather
small,
3
10
. Total transport cross section in
such a mixture is very close to the partial transport
cross section for Xe atoms (see fig. 1). This cross
section was taken from (Hayashi, 1983). The most
important thing for our consideration is the existence
of energy interval with the positive derivative
0
dd
tr
in the range of 0.56.0 eV.
The term
b
Q is the source of slow electrons
produced by fast electron beam in the ionization
process. According to (Dyatko, 2007) it is chosen in
a form
XeXe
b
II
qQ
1
2
)(
(4)
in the energy range of
Xe
I
0 (
Xe
I is the
ionization potential of xenon atom) and
0)(
b
Q
outside this interval.
123
0,01
0,1
1
10
a
*10
-16
, cm
2
eV
Figure 2: Cross section of electron attachment of F
2
molecule.
The energy spectrum of produced electrons is
normalized according to
qdQ
b
)(
(5)
So, q is the production rate, reflecting ionization of
atoms by fast electrons. As the electron – electron
collisions provide the tendency for maxwellization
of the EEDF, the production rate q was chosen in a
way to exclude contribution of ee-collisions to the
formation of an electron energy spectrum. For Xe
plasma it is possible only for ionization degree
7
10
NN
e
(Bogatskaya et al., 2013).
For low partial fraction of F
2
in a gas mixture
only the following processes were taken into
account: the elastic scattering of electrons on Xe
atoms and dissociative attachment of electrons to F
2
molecules. Cross section for electron attachment to
F
2
molecules was taken from (Morgan, 1992). This
cross section is presented at fig. 2. Integration of the
equation (2) over energy allows to obtain the
equation for the electron density in the plasma:
ea
e
Ntq
d
t
dN
)(
(6)
where
dtfNmtv
aFa
),()(2)(
2
(7)
Here
2
F
N
is the concentration of F
2
molecules and
)(
a
is the cross section for electron attachment.
For numerical solution the EEDF was presented
in a form
),()(),( tntNtf
e
, where the function
),( tn
is normalized to unity:
1),(
dtn
.
The Boltzmann equation for
),( tn
and eq. (6) for
electronic density were solved self-consistently
using an explicit scheme in the energy range
5.120
eV. Steps of integration in space and
time domain were the same as in (Bogatskaya and
Popov, 2013). Steady-state EEDF was obtained as a
result of temporal evolution of the initial
Maxwellian function with given temperature
0
T and
electronic density
)0(
tN
e
. Calculations were
performed for the gas mixture Xe:F
2
=
45
1010
,
gas pressure 4 atm (
20
10N cm
-3
) and
17
1082 q
cm
-3
s
-1
. Initial electron density was
obtained from the stationary solution of eq. (6) with
initial Maxwellian EEDF (with temperature T
0
=1.0
eV) used to calculate the rate of attachment (6).
Under considered conditions given q value is
provided by the electron beam with the following
characteristics (Cason et al., 1977): current density ~
100 mA/cm
2
and energy of the beam ~ 300 keV.
Results of simulation of the EEDF in the Xe:F
2
mixture for the total concentration
20
10N cm
-3
and the concentration of F
2
at a level of
15
104 cm
-3
and electron production rate
17
108q
cm
-3
s
-1
are
presented at fig.3 for different time intervals from
the initial instant of time corresponding to the
Maxwellian distribution. First we note that for given
conditions the relaxation time of the final steady-
state distribution is about 200 ns. The obtained
distribution has the energy range (~0.4 – 0.8 eV)
with population inversion that appears as a result of
Electron Beam Sustained Plasma as a Medium for Amplification of Electromagnetic Radiation in Subterahertz Frequency Band
285
slow electron losses in the attachment process, while
the electron production mainly takes place in the
energy range above 1 eV. The electron density
corresponding to the given conditions is
10
1011.7
e
N
cm
-3
.
Figure 3: EEDF in Xe:F
2
plasma for different instants of
time (ns): t=0 (1), 25 (2), 50 (3), 200 (4), 400 (5); Gas
density is
20
10N cm
-3
, fluorine concentration is N
F2
=
15
104 cm
-3
, production rate is
17
108q
cm
-3
s
-1
.
The set of steady-state EEDFs and plasma
parameters obtained for different F
2
molecule
concentrations and different electron production
rates
q are presented at fig.4 and in the table 1. It
can be seen that the increment of the F
2
concentration for a given
q and vice versa the
decrement of
q for a given F
2
concentration results
in electron density reduction. As about the EEDF it
was found that it is the same one for any value of
electron production rate
q used for the modeling.
Definitely, the reduction of the electron density will
result in the decrement of a plasma gain factor. On
the other hand, the less the plasma electron density,
the higher is the position of the peak in energy
spectrum. Later we will see that this fact allows to
increase the frequency band of amplified RF
radiation.
Table 1: Electron densities and EEDF peak position in the
e-beam sustained discharge.
q
, cm
-3
c
-1
F
2
concentration,
cm
-3
electron
density, cm
-3
electron peak
position, eV
17
108
15
102
11
1010.1
0.68
17
108
15
104
10
1011.7
0.78
17
108
15
108
10
1089.4
0.88
17
108
16
106.1
10
1044.3
1.00
17
104
15
108
10
1045.2
0.88
17
102
15
108
10
1022.1
0.88
Figure 4: Steady-state EEDF in Xe:F
2
plasma for any
value of the production rate and concentrations of F
2
molecules (cm
-3
)
15
104 (1),
15
108 (2) and
16
106.1
(3).
To provide more insight to the EEDF formation
in Xe:F
2
plasma beam we will perform simple
qualitative analysis of the EEDF in a beam-sustained
plasma. Neglecting the contribution of transported
RF field to eq. (2) as well as the term with non-zero
value of the gas temperature, under above
formulated assumptions the Boltzmann equation for
the steady-state EEDF can be written in a form
.0/)()()(
))()((
2
23
eba
tr
NQn
n
d
d
M
m
(8)
Here
M
and
tr
are the mass of Xe atom, and the
transport frequency for electron scattering on Xe
atoms,
mN
aFa
2)()(
2
is the attachment
frequency. Introducing new function
)()(
2
)(
23)(
n
M
m
F
Xe
tr
(9)
one derives
eb
NQFA
d
dF
/)()()(
)(
(10)
where
/))(2()(
tra
mMA
. As the
attachment process takes place only for slow
electrons with energy
*
, we obtain for
*
*
)(
1
)(
*
dQ
N
FF
b
e
(11)
(here
)(
**
FF
) which results in the parabolic
dependence of
F
on energy. On the other hand, in
low energy range
*
the attachment process is
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
286
of the most importance and solution of eq.(2) can be
approximately expressed in the form
*
)(exp)(
*
dAFF
(12)
In the assumption that constAA
0
)(
in the low
energy limit one obtains from (12)
))(exp()(
*
0
*
AFF
.
(13)
Figure 5: Analytical (solid curve) and numerical (dash
curve) calculations of the EEDF in Xe:F
2
plasma sustained
by an electron beam. See the text for details.
The electron spectrum
23
)()(~)(
tr
Fn
obtained from (11) and (13) for
10
0
A eV
-1
and
85.0
*
eV is presented in a liner scale at fig.5
and is found to be in a good agreement with the
results of numerical simulations also displayed at the
same figure.
3 AMPLIFICATION OF THE RF
RADIATION IN A PLASMA
It is known that an absorption coefficient
(or
gain factor
k ) of electromagnetic radiation
with frequency
in a plasma with the EEDF )(
n
is given by the expression (Bunkin et al., 1973;
Ginzburg and Gurevich, 1960; Raizer, 1977):
0
22
23
2
.
3
2
d
d
dnv
c
tr
tr
p
(14)
Here
mNe
ep
22
4
is the plasma frequency
squared. Typically the EEDF decreases with the
increase of energy, i.e.
ddn is negative and the
integral in (14) is positive. Hence, the absorption
coefficient in plasma is positive,
0
. However,
if energy intervals with positive derivative
ddn
dominate in the integral (14), it is possible to obtain
negative absorption or amplification of
electromagnetic radiation in a plasma. It was
demonstrated in (Bekefi, et al, 1961; Bunkin, et al,
1973) that in order to obtain negative value of the
integral (14) the condition
0
22
23
tr
tr
v
v
d
d
(15)
should be satisfied in the energy range with
population inversion and also this energy range
should mainly contribute to the integral (14). In the
low frequency limit (
tr
) one derives from
(15)
0)(
dd
tr
, i.e. transport cross section
should grow up rapidly than linear dependence.
Such a situation is realized for Xe atoms in the
energy range
47.0
eV.
Results of calculations of the gain factor in a
Xe:F
2
plasma sustained by an electron beam are
presented at fig.6. First we note that for xenon
concentration of
20
10N cm
-3
(
4 atm) and
concentration of F
2
molecules equal to
15
108 cm
-3
the positive value of gain factor is achieved for
microwave radiation with frequencies
11
104
c
-1
(see fig.6a). Electron density is found to be
proportional to the total electron rate production
q
while the position of the peak in the electron energy
spectrum do not depend on
q . Hence the gain factor
(absorption coefficient) is characterized by a linear
dependence on
q . The dependence of plasma
amplification features on the F
2
concentration is
more complicated (see fig.6b). First one notes that in
order to achieve positive value of a gain factor the
concentration of electronegative specie should be
above some critical value
15
103 cm
-3
. A gain
factor reaches its maximum value ~0.0032 cm
-1
for
15
2
106
F
N
cm
-3
and then falls down with
further increment of F
2
concentration due to the
decrement of electron density. Such a
nonmonotonous gain factor dependence on the F
2
concentration results from the sugnificant
reconstruction of the EEDF in dependence on F
2
concentration. In particular, the position of the
electron peak in the EEDF
*
is shifted towards
lower energies with the reduction of concentration of
F
2
specie (see data in the Table 1). The possibility to
Electron Beam Sustained Plasma as a Medium for Amplification of Electromagnetic Radiation in Subterahertz Frequency Band
287
obtain positive gain factor value disappears when
*
is found to be out of the energy interval
47.0
eV suitable for amplification.
Figure 6: Gain factor (absorption coefficient) in Xe:F
2
plasma : (a) concentration of F
2
molecules is
15
108 cm
-3
;
production rates (cm
-3
c
-1
) are
17
102 (1),
17
104 (2) and
17
108 (3) and (b) production rate is
17
108 cm
-3
c
-1
;
concentrations of F
2
molecules (cm
-3
) are
15
102 (1),
15
104 (2),
15
108 (3),
16
106.1 (4).
In order to increase the frequency range of the
amplification one also needs to shift the position of
the maximum in the electron spectrum
*
up to
higher energies to satisfy the condition
)(
*
tr
.
This is possible (see data in the Table 1) if one
increases concentration of F
2
molecules in the
mixture, as the balance of electrons production and
attachment will shift to higher energies. For
example, for
17
108q
cm
-3
c
-1
the increment of F
2
concentration from
15
104 cm
-3
to
16
106.1 cm
-3
leads to the change in the energy peak position from
0.78 to 1.00 eV and subsequent expanding of the
frequency band for amplification from 0.25 THz to
0.5 THz (see fig. 6b). On the other hand the
increment of F
2
concentration for a given electron
production rate results in reduction of the electron
density and absolute value of the gain factor. For the
above mentioned parameters the four-time-
increment of the of F
2
concentration results in one
and a half time reduction of the gain factor in low
frequency
)(
*
tr
range.
All data above were obtained for rather low
intensities of RF field that do not contribute to the
EEDF evolution. In the two-term approximation
used for analyzing of the EEDF temporal evolution
external electromagnetic field also results in the
diffusion spreading of initial photoelectron peak.
This diffusion should be taken into account for low-
frequency fields (
tr
) if the condition
g
tr
T
M
m
m
Ee 2
3
2
2
0
2
(16)
Figure 7: Steady-state EEDF in Xe:F
2
plasma for
production rate
17
108 cm
-3
c
-1
and F
2
concentration are
16
106.1 cm
-3
. RF field intensities (W/cm
2
) are 0 (1), 0.1
(2), 1 (3), 10 (4) and 100 (5).
is fulfilled. For example, for xenon plasma with
energy peak position
1
*
eV, gas temperature
03.0
g
T
eV and xenon concentration
20
10N
cm
-3
one derives that the inequality (16) is valid for
RF radiation with the intensity greater than
31
W/cm
2
. The results obtained from the numerical
integration of the Boltzmann equation (see fig. 7) are
in agreement with the above estimations. Above the
intensity of 1 W/cm
2
the reconstruction of the EEDF
by RF field is of importance.
Results of calculations of a gain factor for
different values of radiation intensity in dependence
on RF frequency are presented at fig. 8 and
demonstrate that the absolute value of gain factor
value reduces significantly with the growth of
amplified RF field. The intensity ~100 W/cm
2
destroys the amplification process for any value of
frequency. Also the increment of RF field intensity
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
288
leads to the shift of the upper boundary of the
amplification band.
Figure 8: Gain factor (absorption coefficient) in Xe:F
2
plasma for: (a) concentration of F
2
molecules
16
106.1
cm
-3
, production rate
17
108 cm
-3
c
-1
. RF field intensities
(W/cm
2
) are 0 (1), 1 (2), 10 (3) and 100 (4).
4 CONCLUSIONS
Thus, it was shown that a plasma sustained by an
electron beam in a gas mixture with effective
attachment process and the energy range with
increasing transport cross section can be used as a
media for amplification of RF radiation up to
subterahertz frequency band. For the mixture of
Xe:F
2
for the pressure ~4 atm it was found that the
gain factor
3
1031~
cm
-1
can be achieved in the
stationary regime. The amplification up to intensity
~1-10 W/cm
2
is possible in the examined mixture.
For the discharge chamber with transverse size of
~30 cm it provides the possibility to amplify
subterahertz pulses up to 1-10 kW power. To
achieve terahertz range one needs to increase
electron collisions transport frequency in the energy
range corresponding to the peak position in electron
energy spectrum. The simplest way to do it is to
increase the gas pressure. More complicated but
probably more effective way is to choose optimal
mixture components. For example, if xenon as a gas
that characterized by the energy interval with
increasing transport cross section
47.0 eV is used,
it is desirable to create the peak in the electron
spectrum near the upper limit of this interval as the
transport cross section will be of order of value
larger than that used in our simulations. Hence, one
needs the electronegative component in the mixture
with effective attachment of electrons up to energies
2 - 3 eV.
The absence of spontaneous emission in terahertz
and subterahertz frequency band does not allow to
use an e-beam sustained plasma also as a generator
of subterahertz radiation as it is typically possible
for visible and IR radiation. One still needs the
additional source of RF radiation to be used for
amplification.
ACKNOWLEDGEMENTS
This work was supported by the Russian Foundation
for Basic Research (projects no. 15-02-00373, 16-
32-00123).
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