Symmetric Surface Wave at Coaxial Structure with Azimuth External Magnetic Field Filled
N.A. Azarenkov, V.P. Olefir, A. E. Sporov
Karazin National University, Institute of High Technologies, Department of Physics and Technology
Kurchatov av. 31, 61108 Kharkiv, Ukraine
Abstract. This report is devoted to the investigation of dispersion properties, attenuation coefficient and radial wave field
structure of axially-symmetric high-frequency electromagnetic E–wave. The wave considered propagates in coaxial magnetized
waveguide structure, partially filled by radially non–uniform dissipative plasma with radially non–uniform azimuth
The influence of the geometric parameters of waveguide structure, plasma non–uniformity, effective collision frequency,
direction and value of the direct current on phase characteristics, attenuation coefficient and radial wave field structure of the
wave considered is studied.
Keywords: coaxial waveguide, non–uniform plasma, electromagnetic wave
PACS: 52.35g, 52.50.Dg
The study of eigen electromagnetic waves of plasma filled metal waveguide structures is of a great importance for
plasma electronics, various plasma technologies, etc. Among different types of plasma filled waveguide structures it is
possible to separate the cylindrical devices with coaxial elements. At present such cylindrical plasma waveguide structures
are the subject of intensive experimental research [1, 2]. Both, spatial wave field distribution and electromagnetic
wave phase characteristics significantly determined by the radial structure of coaxial waveguide. The dielectric rod or
conductive cylinder can be placed inside coaxial waveguide along the axis of the system.
The realized experimental study of coaxial waveguide structures with central metallic rod have shown that properties
of electromagnetic waves and gas discharge plasma maintained by these waves, differ considerably from the
corresponding properties of cylindrical plasma – metal waveguide structures without central conductor . Properties
of plasma maintained in coaxial structure with dielectric rod inside have been study in paper .
It is necessary to mention that the possibility of effective control of the eigen wave properties is the characteristic
feature of coaxial structures with metallic rod at the axis of waveguide system. Plasma properties and electromagnetic
wave characteristics essentially depend on the external magnetic field value and direction. So, with the help of varying
the value of the direct current that flows along the inner conductor and changing its direction one can effectively create
the azimuth external magnetic field and control its value and, therefore, the characteristics of electromagnetic waves
and properties of plasma .
It is necessary to note that in spite of perfect plasma parameters obtained in experimental devices with coaxial
structures, theoretical study of eigen wave properties of coaxial waveguide structures and efficiency of such structure
use in various applications is insufficient. This especially concerns the theoretical study of plasma density radial
non–uniformity and electron collision frequency influence on the phase characteristics and spatial damping of electromagnetic
eigen waves of coaxial structure with central metal conductor. These circumstances greatly determine the
urgency of theoretical study of eigen wave properties of coaxial structures.
Let consider the axially–symmetric (azimuthal wavenumber m = 0) high–frequency electromagnetic wave that propagates
in cylindrical coaxial magnetized waveguide structure, partially filled by radially non–uniform dissipative
plasma. The waveguide structure consists of metal rod of radius R1, which is placed at the axis of plasma column.
The direct current Jz flows along this rod, creating radially non-uniform azimuth magnetic field H0(r). This rod is enclosed
by the cylindrical plasma layer of radius R2. The vacuum region (R2 < r < R3) separates the cylindrical plasma
layer from waveguide metal wall with radius R3. It was supposed, that plasma density is radially non–uniform and
possesses the bell–shaped profile of the following form:
Here, rmax — is the radius value where plasma density culminates its maximum, parameters rd and μ describes
the width and the gradient of the bell–shaped profile, respectively. The proposed method of investigations gives the
possibility to model different radial profiles, that can be occurred in experiments. In this study the parameter rmax was
chosen to be the center of plasma layer and rd was chosen to be approximately 10% of the width of plasma layer. The
variation of parameter μ from μ = 0 (radially uniform plasma) up to μ = 1 (strong radial non–uniformity) gives the
possibility to study the influence of plasma density radial profile on the properties of wave considered.
Plasma was considered in the hydrodynamic approach as cold medium with collisions. The collisions were characterized
by the effective collisional frequency n that is constant in the whole volume of cylindrical
To obtain the full set of equations that describes propagation of the wave considered one must determine the initial
conditions for the system of ordinary differential equations (4). These initial conditions for Ez(r) and Hj(r) at the
inner conductive rod of radius R1 can be obtained from the boundary conditions for Ez (Ez(R1) = 0) and from the
normalizing conditions (all wave field components will be normalized on the Hj(R1)).
The obtained dispersion equation (7) is solved in complex algebra. For this purpose the system of ordinary differential
equations was numerically solved with the help of Badler and Deuflhard version of semi-implicit extrapolation
method . This method give the possibility to obtain accurately numerical solution even in the region where the
conditions of upper hybrid resonance take place. The dispersion equation (7) was solved with the help of Muller
It is necessary to mention that, in the case when external current flows along the propagation direction of the wave
considered the dispersion equation (7) possesses two solutions with different values of frequency for the fixed value
of dimensionless wavenumber Re(k3)R1. One of them with comparatively more high frequency will be called further
high frequency (HF) wave, and other—low frequency (LF) wave. Properties of these waves substantially determined
by the direct current value and its direction. Thus, in the limiting case, when the azimuth magnetic field H0(r) trends
to zero the LF wave vanishes. The increase of the direct current leads to the decrease of the HF wave frequency and to
the increase of the LF wave frequency. So, for rather high dimensionless direct current value ( j = eJz/(2mc3) 2.0)
the frequencies of HF and LF waves for rather high Re(k3)R1 values are close.
The influence of parameter n/w value on the dispersion and attenuation properties of HF and LW waves was study
for the case of radially uniform plasma (μ =0). For rather small values of effective collisional frequency (n/w <1) the
increase of n/w parameter leads to the increase of the LF dimensionless frequency w/wp and attenuation coefficient
Im(k3)R1. Dispersion and attenuation properties of HF wave depend on the n/w parameter much weaker. It is
necessary to note that the attenuation coefficient value of LF wave is approximately of one order greater than the
value of attenuation coefficient of LF wave.
The influence of plasma density radial profile (non-uniformity parameter μ value) on the dimensionless frequency
and attenuation coefficient for HF and LF wave at fixed point of dispersion curve (for Re(k3)R1 = 0.01)
is shown on fig. 1. Other external parameters were equal to r1 = R1wp(rmax)/c = 4.0, r2 = R2wp(rmax)/c = 5.0,
r3 = R3wp(rmax)/c = 6.0, n/w = 0.001, rmax = 4.5, rd = 0.1, j = eJz/(2mc3) = 2.0. One can see that the increase of
non-uniformity parameter μ leads to the increase of the dimensionless wave frequency of HF wave and to the decrease
of the dimensionless wave frequency of LF wave. The wave attenuation coefficient shows more complicated behaviour.
The calculations carried out have shown that the spatial attenuation coefficient for HF wave possesses the maximum
in the range of rather small μ values (μ 0.2). In contrast, the attenuation coefficient for LF possesses the minimum
for approximately the same μ values. Such complicated behaviour may be very important
The influence of plasma density radial non–uniformity, electron effective collision frequency, direction and value of the
direct current on phase characteristics, attenuation coefficient and radial wave field structure of the wave considered
was studied. It was shown that it is possible to control effectively the dispersion properties of E-wave by varying
the value and direction of direct current. The influence of dimensionless collision frequency on the dispersion and
attenuation properties of HF and LF waves was study as well. It was shown that LF wave attenuates more effectively
than HF wave. It was shown also that in the case of bell–shaped plasma density profile the increase of plasma density
radial non–uniformity results in the growth of wave frequency of HF waves and in the decrease of wave frequency of
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