Progress In Electromagnetics Research Letters, Vol. 13, 83–92, 2010
ANALYSIS ON THE STEALTH CHARACTERISTIC OF
TWO DIMENSIONAL CYLINDER PLASMA ENVELOPES
L.-X. Ma and H. Zhang
Radar Engineering Department
Missile Institute of Air Force Engineering University
Sanyuan, Shaanxi 713800, China
Z. Li
Missile Department
Air Defence Forces Command College
Zhengzhou, Henan 450052, China
C.-X. Zhang
Radar Engineering Department
Missile Institute of Air Force Engineering University
Sanyuan, Shaanxi 713800, China
Abstract—Stealth characteristic of two dimensional cylinder plasma
envelopes is studied. Three cases about plasma refraction effect,
reflection characteristic and attenuation by absorbing electromagnetic
wave (EMW) are concerned synthetically. As for plasma refraction
stealth, EMW traces equation in cylinder plasma is deduced; a
novel concept of plasma refraction deviation angle is presented; the
relation between refraction deviation angle and incidence angle of
EMW is yielded; the relation between refraction deviation angle and
plasma density distribution is made out. As for reflection stealth
and attenuation stealth, reflection calculation of multi-layer plasma is
presented first, and plasma collision frequency as well as corresponding
collision absorption is taken into account simultaneously, then EMW
reflectivity with double-path attenuation is obtained. It is shown that
cylinder plasma envelopes considering the three cases above could make
distinct stealth.
Corresponding author: L.-X. Ma (mars982133@163.com).
84 Ma et al.
1. INTRODUCTION
Since 1990s, academic institutions and researchers, overseas and
domestic, have been studying plasma stealth technology [1–15].
The concept of plasma stealth was proposed first by Vidmar [1],
who theoretically studied the reflection, transmission and absorption
characteristic of EMW propagation in un-magnetized plasma. The
absorption and attenuation characteristics of EMW propagation in un-
magnetized plasma were observed experimentally by Hughes Research
Laboratories [2]. Laroussi [3] and Hu [4] carried out the studies of
the reflection, transmission and absorption characteristics of EMW
propagation in magnetized plasma, by using numerical method and
scattering matrix method respectively. Pertrin [5–7] studied the
transmission of microwave through un-magnetized plasma layer and
magnetoactive plasma layer. Liu et al. analyzed the stealth mechanism
of sphere plasma [8, 9]. From above references, it can be seen that,
when EMW propagated in plasma, the plasma stealth effect caused by
plasma refraction, reflection and absorption had not been considered
simultaneously in the same document.
In this paper, the stealth characteristic of two dimensional cylinder
plasma envelopes is studied. The subject investigated is a model of a
cylinder perfect conductor, whose radius is r
0
, covered with concentric
cylinder plasma envelopes, as shown in Figure 1. When plane EMW
propagates in plasma envelopes, two cased are concerned. One case is
that the rays among parallel EMW rays with longer distance to the
circle center, supposed r
d
> r
0
, have bigger incidence angle, which
will be refracted by plasma before they arrive at conductor cylinder.
The other case is that the rays among parallel EMW rays with
Figure 1. Horizontal section sketch of EMW incidence in cylinder
plasma envelopes.
Progress In Electromagnetics Research Letters, Vol. 13, 2010 85
shorter distance to the circle center, supposed r
d
< r
0
, have smaller
incidence angle, which may be incidence on the conductor. However,
EMW energy will be partially attenuated because of absorption in
the processing of propagating in plasma. Thus, efficiency reflection
energy only takes a little part of the whole EMW energy. As for the
first case, refraction is dominant, and it may be taken as refraction
stealth, then EMW tracks equation in cylinder plasma is deduced,
and refraction deviation angle is presented. Sequentially, the relation
between refraction deviation angle and incidence angle is obtained that
the bigger incidence angle is, the bigger refraction deviation angle
and the better stealth effect are. In addition, the relation between
refraction deviation angle and plasma density distribution is obtained
as well, which is that the stronger plasma numerical density is (i.e., the
bigger m value is), the smaller refraction deviation angle is. As for the
second case, reflection is dominant, when double-path attenuation is
concerned, and it can be considered as reflection stealth and absorption
stealth. Afterwards, the calculation formula about reflection coefficient
of EMW incidence in multi-layer plasma is discussed. Furthermore,
the reflectivity with double-path attenuation is worked out as plasma
collision is concerned. Moreover, the principle is that the stronger
plasma numerical density is, the less reflectivity is. As the cases above,
all considered, the stealth function of plasma cylinder will be reached.
In Figure 1, r
0
is the radius of conductor cylinder; R
0
is the
radius of plasma cylinder; r
d
is the distance between EMW rays to
the circle center. Solid arrow denotes incidence EMW; hollow arrow
denotes reflection EMW; θ
10
is EMW incidence angle, θ
0
; θ
20
is EMW
emergence angle.
2. THEORY ANALYSIS AND RESULT DISCUSSION
2.1. The Dispersion of EMW in Un-magnetized Collision
Plasma
As EMW interacts with collision plasma, the equivalent permittivity
is a complex [1–4, 15].
ε
pr
= 1 −
ω
2
p
ω(ω − jv
c
)
(1)
where ω
p
is plasma angle frequency; ω is incidence EMW angle
frequency; v
c
is plasma effective collision frequency.
Consequently, the propagation constant is also a complex [15].
k = k
0
√
ε
pr
= k
r
+ i ·k
i
(2)
86 Ma et al.
where k
0
=ω/c is the free-space wave number, and k
r
and k
i
are the
real and imaginary parts of k respectively, corresponding to the phase
shift constant and attenuation constant. Combining Equations (1) and
(2), there yields
k
r
= k
0
p cos(θ/2) (3)
k
i
= k
0
p sin(θ/2) (4)
where p and θ are defined as
p =
"
1 −
ω
2
p
ω
2
+ v
2
c
Ã
2 −
ω
2
p
ω
2
!#
1/4
(5)
θ =
(
θ
c
= tan
−1
h
−
v
c
ω
2
p
ω(ω
2
+v
2
c
−ω
2
p
)
i
: Re(ε
r
) > 0
θ
c
+ π : Re(ε
r
) < 0
(6)
2.2. Non-uniform Un-magnetized Plasma Cylinder EMW
Tracks and Refraction Deviation Angle
EMW tracks in non-uniform un-magnetized plasma depend on plasma
refraction index. When plasma collision is omitted, its refraction index
is expressed approximately as [8, 9, 11]:
n
2
p
= ε
pr
= 1 −
ω
2
p
ω
2
(7)
Supposed plasma density changes gradually along R direction,
that is, plasma density is just a function of radius R, and R changes
from r
0
to R
0
and the same bellow. As for the ideal case, plasma
density is zero at the position of outer radius of plasma circle, that is
n(R
0
) = 1. The refraction index in plasma circle is defined as [8, 9]:
n(R) = (R)
m
/R
m
0
(8)
According to Fermat principle and variation method and
combining with formula (8), the equation in polar coordinates is
d
dθ
∂
∂
˙
R
³
R
m
p
R
2
+
˙
R
2
´
−
∂
∂R
³
R
m
p
R
2
+
˙
R
2
´
= 0 (9)
where
˙
R = dR/dθ, let
¨
R = d
2
R
±
dθ
2
, then combining formula (8) we
get
R
¨
R − (m + 2)
˙
R
2
− (m + 1)R
2
= 0 (10)
Progress In Electromagnetics Research Letters, Vol. 13, 2010 87
Make tracks parameters of EMW (R , θ) and solve Equation (10),
then four solutions are respectively
θ
1
= −arcsec
³
p
C
1
R
m+1
´
/ (m + 1) + C
2
(11a)
θ
2
= arcsec
³
p
C
1
R
m+1
´
/ (m + 1) + C
2
(11b)
θ
3
= −arcsec
³
p
C
1
R
m+1
´
/ (m + 1) − C
2
(11c)
θ
4
= arcsec
³
p
C
1
R
m+1
´
/ (m + 1) − C
2
(11d)
Choosing the coordinate of incidence EMW (R
0
, θ
0
), where θ
0
is EMW
incidence angle, which is an angle between EMW incidence ray and
plasma circle normal, integral constants are expressed as
C
1
=
cot
2
θ
0
+ 1
R
2m+2
0
(12)
C
2
= arcsec
³
p
C
1
R
m+1
´
/(m + 1) + θ
0
(13)
From formulae (11), we can obtain EMW tracks in cylinder plasma
as shown in Figure 2.
Based on Snell’s law and combined with Figure 1, we get
R
m
0
sin θ
0
= R
m
sin θ = r
m
d
sin(π/2) = r
m
d
(14)
Suppose r
d
= r
0
, and we will get the minimal EMW rays incidence
angle
θ
min
= arcsin(r
0
/R
0
)
m
(15)
According to formula (15), we can obtain θ
min
in the case of r
0
.
R
0
and m are sure.
Due to the refraction of plasma cylinder envelopes, EMW rays in
plasma will deviate greatly to original direction. Combing Figures 1, 2
and formulae (11), we define EMW refraction deviation angle, which is
the difference between incidence angle and emergence angle that return
to air, as
θ
d
= θ
20
− θ
10
= 2arcsec
³
p
C
1
R
m+1
0
´
(16)
where θ
10
is EMW incidence angle, θ
0
, and θ
20
is EMW emergence
angle.
It is seen from Figure 1 that the angle between initial reflecting
wave and incident wave is 2θ
0
.
It is seen from Figure 2 that when EMW incidence angle θ
0
is
0
◦
, plasma refraction deviation angle is 0
◦
too, and incidence EMW
