Analysis on the stealth characteristic of two dimensional cylinder plasma envelopes

TL;DR: In this paper, the stealth characteristic of two dimensional cylinder plasma envelopes is studied synthetically, and three cases about plasma refraction efiect, re∞ection characteristic and attenuation by absorbing electromagnetic wave (EMW) are concerned synthetically.

Abstract: Stealth characteristic of two dimensional cylinder plasma envelopes is studied. Three cases about plasma refraction efiect, re∞ection 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 re∞ection stealth and attenuation stealth, re∞ection calculation of multi-layer plasma is presented flrst, and plasma collision frequency as well as corresponding collision absorption is taken into account simultaneously, then EMW re∞ectivity with double-path attenuation is obtained. It is shown that cylinder plasma envelopes considering the three cases above could make distinct stealth.

Summary

1. INTRODUCTION

• The absorption and attenuation characteristics of EMW propagation in unmagnetized plasma were observed experimentally by Hughes Research Laboratories [2] .
• The stealth characteristic of two dimensional cylinder plasma envelopes is studied.
• 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.
• 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.

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] : EQUATION.
• Due to the refraction of plasma cylinder envelopes, EMW rays in plasma will deviate greatly to original direction.
• It is seen from Figures 3 and 4 that when EMW incidence angle is bigger than critical angle θ min but smaller than 90 , the bigger incidence angle is, the smaller refraction angle is, and the bigger m value is, the smaller refraction angle is.
• Therefore, as for refraction stealth, refraction angle affected by EMW incidence angle and plasma density distribution should be considered comprehensively.

2.3. EMW Reflectivity in Non-uniform Un-magnetized Plasma with Cylinder Envelopes

• As mentioned above, plasma density changes gradually along R direction, and thus plasma refraction index function is as formula (8) .
• When plasma collision is considered, there exists attenuation in the processing of EMW propagating trough plasma envelopes.
• It is seen from Figure 5 that, as plasma collision frequency enlarges, EMW reflectivity will decrease when EMW frequency above 5 GHz.
• When plasma collision frequency is constant, as EMW frequency grows, EMW reflectivity will increase.
• Therefore, as for absorption and attenuation stealth, plasma collision frequency should be enlarged.

3. CONCLUSION

• When EMW enters cylinder conductor covered with concentric cylinder plasma envelopes because of EMW refraction, reflection and absorption by plasma, in the case of EMW rays with larger incidence angle, EMW rays will deviate greatly from original direction, and the refraction angle is θ d , so such EMW rays cannot reach inner conductor.
• EMW rays with smaller incidence angle may arrive at inner conductor after refraction and absorption by plasma envelopes.
• The backwave deviates from incidence direction 2θ 0 , with energy attenuates a lot, and the minimal reflectivity is below −40 dB.
• From above, it is concluded that cylinder plasma envelopes can be used to protect objects.

Figures

Figure 5. Reflectivity with different plasma collision frequency.

Figure 6. Reflectivity with different m values.

Figure 2. EMW tracks at different m values with different incidence angle.

Figure 1. Horizontal section sketch of EMW incidence in cylinder plasma envelopes.

Figure 4. Deviation angle varying vs. that of m values.

Figure 3. Relations between deviation angle and incidence angle.

Full text

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

Frequently asked questions

Q1. What are the contributions in "Analysis on the stealth characteristic of two dimensional cylinder plasma envelopes" ?

Stealth characteristic of two dimensional cylinder plasma envelopes is studied. 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. 

Q2. what is the reflection coefficient of i-th to i-th plasma?

When EMW reflects from the interface of air-toplasma, the reflection coefficient is Γ0, and the reflection coefficient about i-th to (i + 1)-th plasma layers is Γi, and the coefficient of n-th plasma layer to inner conductor is Γn. 

Q3. what is the refraction index in plasma circle?

When EMW incidence angle θ0 is 90◦, plasma deviation angle is 180◦, that is, EMW rays propagate straightly along the tangent of plasma cylinder. 

Q4. what is the reflection coefficient of i-th to i-th?

Due to double-path attenuation, the reflection coefficient in i-th layer plasma envelopes should be modified, and it yields∣∣∣Γ̃i ∣∣∣ 2 = |Γi|2 i∏q=1( 1− |Γq−1|2 ) exp −4ki,ql√1− sin2 θ0εp,q (20)where, as p = q = 1, Γq−1 is Γ0; as l = i = n, Γq = −1; ki,q is q-th layer plasma attenuation constant. 

Q5. What is the effect of plasma stealth on the conductor?

The absorption and attenuation characteristics of EMW propagation in unmagnetized plasma were observed experimentally by Hughes Research Laboratories [2]. 

Q6. what is the reflection coefficient of emw?

When EMW enters cylinder conductor covered with concentric cylinder plasma envelopes because of EMW refraction, reflection and absorption by plasma, in the case of EMW rays with larger incidence angle, EMW rays will deviate greatly from original direction, and the refraction angle is θd, so such EMW rays cannot reach inner conductor. 

Q7. how does the law of emw reflect from air to plasma?

By using Snell’s law again, it yields 1 · sin θ0 = √εp1 · sin θt1 = √εp,i · sin θt,i = √εp,i+1 · sin θt,i+1 (18)Substituting formula (18) into formula (17) and arranging, the authors obtain the coefficient expressed by relative permittivity asΓi = √ εp,i cos θt,i+1 −√εp,i+1 cos θt,i√ εp,i+1 cos θt,i + √ εp,i cos θt,i+1=√ εp,i √ 1− sin2 θ0/εp,i+1 −√εp,i+1 √ 1− sin2 θ0/εp,i√ εp,i+1 √ 1− sin2 θ0/εp,i + εp,i √ 1− sin2 θ0/εp,i+1(19)When plasma collision is considered, there exists attenuation in the processing of EMW propagating trough plasma envelopes. 

Q8. What is the refraction angle in plasma?

It is seen from Figures 3 and 4 that when EMW incidence angle is bigger than critical angle θmin but smaller than 90◦, the bigger incidence angle is, the smaller refraction angle is, and the bigger m value is, the smaller refraction angle is. 

Q9. What is the refraction index of a plasma circle?

Supposed non-uniform un-magnetized plasma envelopes are divided into n layers, and the refraction angle of EMW in i-th layer of plasma envelopes is θti. 

Q10. What was the concept of plasma stealth first proposed by Vidmar?

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. 

Q11. What is the refraction index in plasma circle?

As mentioned above, plasma density changes gradually along R direction, and thus plasma refraction index function is as formula (8). 

Q12. What is the equation of EMW in cylinder plasma?

It is seen from Figure 2 that when EMW incidence angle θ0 is 0◦, plasma refraction deviation angle is 0◦ too, and incidence EMWreflects completely along the original path. 

Q13. What is the subject of the paper?

The subject investigated is a model of a cylinder perfect conductor, whose radius is r0, covered with concentric cylinder plasma envelopes, as shown in Figure 1. 

Q14. What is the difference between the two cases?

The other case is that the rays among parallel EMW rays withshorter distance to the circle center, supposed rd < r0, have smaller incidence angle, which may be incidence on the conductor.