Download Electromagnetic Wave Propagating in Gyroelectric Slab in the

PIERS Proceedings, Marrakesh, MOROCCO, March 20–23, 2011
152
Electromagnetic Wave Propagating in Gyroelectric Slab in the
Perpendicular Configuration
Hui Huang1, 2 , Bo Yi2 , and Bo Huang3
1
School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China
2
State Key Laboratory of Millimeter Waves, Nanjing 210096, China
3
School of Electronics Engineering and Computer Science, Peking University, 100871, China
Abstract— This paper present the characteristics of electromagnetic wave propagating in gyroelectric slab with an external magnetic field perpendicular to the interface between gyroelectric
medium and a perfect conductor. First, using KDB coordinate system, we decomposed an electromagnetic wave in infinite gyroelectric medium into 2 types and got the dispersion relation
respectively. Second, we discuss the reflection from the interface between gyroelectric medium
and a perfect conductor for inclined incidence case. The conclusion that the reflected wave has
the same ellipticity but the opposite rotate direction to the incident wave is proofed theoretically. Finally, the characteristics of metallic waveguide in Perpendicular Configuration have been
discussed, and the guidance condition has been derived. And we found the main mode is zero
mode, which is similar to the ordinary wave in infinite gyrotropic medium.
1. INTRODUCTION
Applied a magnetic field on an electron plasma, it becomes gyroelectric medium, an anisotropic
medium. The characteristics of electromagnetic waves propagation in gyrotropic plasmas have been
theoretically investigated for years. Kushwaha and Halevi have been studied the magnetoplasma
modes in Voigt, perpendicular, and Faraday configurations [1–3], Gillies and Hlawiczka have done
some researches on gyrotropic waveguide [4–8], and Eroglu, as well as Li, have investigated dyadic
Green’s functions for gyrotropic medium [9–11]. Furthermore, there are some studies focusing
upon the effects of magnetic field on semiconducting plasma slab and negatively refracting surfaces [12, 13]. Moreover, propagation and scattering characteristics in gyrotropic systems [9, 14–18]
and surface modes at the interface of a special gyrotropic medium [19] have been investigated
extensively.
In this paper, we focus on the characteristics of electromagnetic wave propagating in gyroelectric
slab with an external magnetic field perpendicular to the interface between gyroelectric medium
and a perfect conductor.
2. DISPERSION RELATIONS
consider the gyroelectric medium with permeability µ2 and permittivity ε¯2 , which is a tensor and
takes the following form:
"
#
εxx
iεg 0
ε̄¯2 = −iεg εyy 0
,
(1)
0
0
εzz
where elements are given by
εxx = εyy = ε∞ (1 −
ωp2
),
ω 2 − ωc2
εzz = ε∞ (1 −
ωp2
),
ω2
εg = ε∞ [−
ωp2 ωc
].
ω(ω 2 − ωc2 )
(2)
q
±
±
Here, ωp = N qe2 meff ε∞ and ω̄c = qe B̄0 meff are the plasma and cyclotron frequencies respectively, ε∞ is the background permittivity, N is the electron density, meff is the effective mass, and
qe is the electron charge.
Using KDB coordinate system, we can get the dispersion relations
·
¸
q
¡ 2
¢
ν
2 4
2
2
2
2
2
2
ω =
κ k + kz + κz ks ± (κ − κz ) ks + 4κg kz k
(3)
2
where kz is z component of k̄, ks is its component in xy plane, and
κ=
ε2
ε
,
− ε2g
κg =
−εg
,
− ε2g
ε2
κz =
1
.
εz
Progress In Electromagnetics Research Symposium Proceedings, Marrakesh, Morocco, Mar. 20–23, 2011 153
z
Perfect
conductor
θ'
y
k
Gyroelectric
B0
θ
k'
Figure 2: The incidence of elliptical polarization
wave to the interface between gyroelectric medium
and perfect conductor.
Figure 1: The KDB system.
We call the characteristic wave with the plus sign in Eq. (3) Type I wave, while the wave with the
minus sign Type II wave.
3. REFLECTION ON THE SURFACE BETWEEN GYROELECTRIC MEDIUM AND
PERFECT CONDUCTOR
We consider the configuration in Fig. 2, where a plane wave (suppose it is either Type I or Type II)
is incident from an semi-infinite gyroelectric medium into a perfect conductor at an oblique angle
θi with respect to the normal of the interface.
In KDB system,
D̄ = ê1 D1 + ê2 D2
(4)
We transform it into the Cartesian coordinate system, and in the expression in the matrix form
is
T
D̄ = ( D1 D2 cos θ D2 sin θ )
(5)
Thus we can get the electric field
Ã
!Ã
! Ã
!
κ
iκg 0
D1
κD1 + iκg D2 cos θ
¯ · D̄ =
−iκg κ
0
D2 cos θ
−iκg D1 + κD2 cos θ
Ē = κ̄
=
(6)
0
0
κz
D2 sin θ
κz D2 sin θ
According to the boundary condition of perfect conductor, i.e., continuous tangential component
of electric field, we can get that the x and y components of the reflected wave:
Ex0 = −Ex = − (κD1 + iκg D2 cos θ)
Ey0 = −Ey = iκg D1 − κD2 cos θ
Dx0 = εEx0 + iεg Ey0 = − (εκ + εg κg ) D1 − (εκg + εg κ) iD2
(7)
Dy0 = −iεg Ex0 + εEy0 = (εκg + εg κ) iD1 − (εκ + εg κg ) D2
Simplified them in KDB system, we can get
Dx0 = −D1
Dy0 = −D2 cos θ
(8)
PIERS Proceedings, Marrakesh, MOROCCO, March 20–23, 2011
154
Only on the DB plane can it be electric displacement vector, we can get the electric displacement
vector of the reflected wave in kDB coordinate system
D̄0 = −eˆ1 0 D1 + eˆ2 0 D2
(9)
We can find out from the formula mentioned above that the reflected wave has the same ellipticity
but the opposite rotate direction to the incident wave.
Now we will prove the existence of this kind of reflected wave. Notice that the kDB coordinate
system has changed compared to the incident wave, that is θ0 = π − θ, then,
tan 2ψ 0 =
2κg cos θ0
2κg cos θ
= − tan 2φ
2 0 =
(κ − κz ) sin θ
(κ − κz ) sin2 θ
(10)
so ψ 0 = −ψ, and tan ψ 0 = − tan ψ. Which demonstrate that this type of reflected wave does exist,
and is the same type as the incident one.
Hlawiczka once mentioned this condition as the supposed condition for the study of gyrotropic
slab waveguide. He extended this general condition to gyroelectric medium, supposing that the
reflected wave has the same ellipticity and rotate direction to the incident wave, while the calculation
above shows that the reflected wave has the same ellipticity but the opposite rotate direction to
the incident wave.
4. CHARACTERISTICS OF METALLIC WAVEGUIDE IN PERPENDICULAR
CONFIGURATION
Considering the waveguide mode as vector combination of incident and reflected wave, we derived
the guidance condition. We suppose that incident and reflected wave vectors are
k̄i = ŷky + ẑkz
(11)
k̄r = ŷky − ẑkz
y
Perfect
conductor
Gyroelectric
Perfect
conductor
k
z
O
d
B0
Figure 3: Gyroelectric slab in the perpendicular configuration.
Progress In Electromagnetics Research Symposium Proceedings, Marrakesh, Morocco, Mar. 20–23, 2011 155
And electric displacement vectors are
µ
¶
ky
kz
D̄i = x̂D1 + ŷ D2 − ẑ D2 e−i(ky y+kz z−ωt)
k
k
µ
¶
ky
kz
D̄r = −x̂D1 − ŷ D2 − ẑ D2 e−i(ky y−kz z−ωt)
k
k
kz
D̄ = − x̂2i sin (kz z) D1 e−i(ky y−ωt) − ŷ2i sin (kz z) D2 e−i(ky y−ωt)
k
ky
−i(ky y−ωt)
− ẑ2 cos (kz z) D2 e
k
(12)
According to boundary conditions of perfect conductor, we can get the guidance condition:
kz =
mπ
d
(13)
where m is integer.
If m = 0
ky
D2 e−i(ky y−ωt)
(14)
k
Electric field only has the z component, but magnetic field intensity has the component of x
and y, which equivalents to ordinary wave.
D̄ = −ẑ2
5. CONCLUSION
This paper investigates characteristics of electromagnetic wave propagating in gyroelectric slab
with an external magnetic field perpendicular to the interface between gyroelectric medium and a
perfect conductor. After deriving the dispersion relation of the medium, we discuss the reflection
at the interface between the gyroelectric medium and the perfect conductor. Then we derive the
guidance condition of the slab waveguide.
ACKNOWLEDGMENT
This work is sponsored by State Key Laboratory of Millimeter Waves under Contract K201012.
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