II. Electromagnetic Transients in Time-Varying Waveguides and Resonators
Tải bản đầy đủ - 0trang
This page intentionally left blank
July 25, 2012 13:29
PSP Book - 9in x 6in
Chapter 5
An Electromagnetic Field in a Metallic
Waveguide with a Moving Medium
The ﬁrst part of the book was devoted to the basic electromagnetic
transient phenomena caused by time-varying properties of a
homogeneous medium or a medium with very simple discontinuity
in the form of a plane boundary. The time variation can be
caused by changing time medium properties or by moving medium
boundaries. In the second part of the book the transients in
waveguide and resonator structures are investigated. Waveguides
with perfectly conducting walls, the dielectric waveguides and the
dielectric resonators will be considered.
The inﬂuence of a moving medium on a guided wave in a
rectangular waveguide with perfectly conducting walls, as well as
the wave evolution, is considered in this chapter. The transformation
of a guiding wave in the waveguide ﬁlled by a dielectric medium
which uniformly moves with relativistic velocity is examined. Based
on this investigation it is shown that it is possible to model various
phenomena concerned with the interaction of electromagnetic
waves with the boundary of a relativistic moving medium in the
presence of waveguide dispersion. Speciﬁcally, the interaction of the
Non-Stationary Electromagnetics
Alexander Nerukh, Nataliya Sakhnenko, Trevor Benson, and Phillip Sewell
Copyright c 2013 Pan Stanford Publishing Pte. Ltd.
ISBN 978-981-4316-44-6 (Hardcover), 978-981-4364-24-9 (eBook)
www.panstanford.com
05-Alexander-c05
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
348 An Electromagnetic Field in a Metallic Waveguide with a Moving Medium
electromagnetic ﬁeld with a plasma “ﬂashed” at zero moment of
time in the waveguide and then uniformly expanding is considered.
Investigations are made by using the evolution approach
developed in this book. A chain of evolution equations for a nonstationary electromagnetic ﬁeld in a waveguide is derived and
solutions to these equations are obtained by virtue of the resolvent
method.
5.1 Expansion of an Electromagnetic Field by the
Non-Stationary Eigen-Functions of a Waveguide
The general integral equation (Eq. 1.3.3)
E = E0 −
1 ∂2G 0
1
∗χ
(P1 − Pex )
2
2
v ∂t
ε0 ε
∂G 0
∗ {curlχ μ0 μ(M1 − Mex ) + μ0 μ(χ j1 + iδ(S))} (5.1.1)
∂t
must be concretised for the case of a ﬁeld in a rectangular metallic
waveguide, as the Green function in this case has the dyadic form
(Eq. 1.2.18)
−
Gˆ 0
ij
∂2
1 ∂2
− 2 2 δi k
∂ xi ∂ xk
v ∂t
−v 2
4π
= G ij =
˜f E
kj
(5.1.2)
The elements of this tensor are deﬁned by the functions (1.2.23) for
the rectangular waveguide whose symmetry axis is directed along
the x axis and whose transverse dimensions are equal to a along the
y axis and b along the z axis:
˜f E = 8π v δ
ij
ij
ab
˜ f E,j f
mn mn
E,j
mn
(5.1.3)
m,n
where the coeﬃcients
τ
t
˜ =
mn
dτ
−∞
dτ
mn
τ −t ,x −x
(5.1.4)
−∞
are determined by virtue of the functions
mn
(t, x) = J 0
ωmn
t2 −
x2
v2
θ
t−
|x|
v
(5.1.5)
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
Expansion of an Electromagnetic Field by the Non-Stationary
Here ωmn = vπ (m/a)2 + (n/b)2 is the eigen-frequency of the mn
mode.
Substitution of Eq. 5.1.2 into Eq. 5.1.1 gives the speciﬁc form of
the integral equation for waveguide
E i = E 0i +
1
ε
dt
−∞
∂
χ
∂t 2
2
×
+
∂
∂t
∞
∞
a
dx
−∞
Pj −
b
dy
0
dz
0
ε−1
Ej
4π
+ ce j kl
∂2
1 ∂2
− 2 2 δi j
∂ xi ∂ x j
v ∂t
∂2
∂ xk ∂t
Pl −
˜f E
ij
μ−1
Bl
4π μ
χ j j + i j δ (S)
(5.1.6)
where e j kl is the completely anti-symmetric tensor of rank 3.
As the integrals with respect to t and x represent convolutions,
the diﬀerentiation by these variables can be brought from the
ﬁeld functions to Green’s function. After diﬀerentiation and some
manipulations we express Eq. 5.1.6 in the form
E = E0 +
8π v
ab
dx
m,n
1
ε
2
E
E 1 ∂
κ∗)κ − ˆfmn
( ˆfmn
v 2 ∂t2
ε−1
E + ˜ mn (χ ji n + iδ (S))
4π
μ−1
E
E μ ∂
B
− ˆfmn
κ × mn χ fˆ mn Mi n −
c ∂t
4π μ
×
mn χ
fˆ
E
mn
Pi n −
(5.1.7)
E,j
E
E ,i j
E,j
where ˆfmn
is the tensor with the elements fmn
= δi j mn fmn
f mn
,
κ = (∂/∂ x1 , −κm , −κn ), κ∗ = (∂/∂ x1 , κm , κn ), κm = mπ/a, κn =
nπ/b and the product ˆf A is a product of a diagonal tensor ˆf with a
vector A.
The kernel in Eq. 5.1.7 is a series of waveguide eigen-functions. It
is natural to represent all ﬁeld values in the form of an expansion
in terms of waveguide eigen-functions which have to be vectorial
ones. It is common to derive these functions with the assumption
that ﬁelds can be represented in the form of the expansion on
modes which are monochromatic plane inhomogeneous waves with
a certain frequency and a certain wave number. In the general case
of non-monochromatic ﬁelds with inseparable dependence on time
and spatial coordinates the expressions for vector eigen-functions
349
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
350 An Electromagnetic Field in a Metallic Waveguide with a Moving Medium
have to be corrected. The derivation of these eigen-functions is given
in this sub-section following the scheme for the investigation of
ﬁelds in waveguides, as in Ref. 1.
It is well known that in a regular waveguide all waves are divided
into two classes: E- or TM-waves (transverse magnetic waves)
and H- or TE-waves (transverse electric waves). For the TM-waves
the longitudinal components of the electric ﬁeld satisfy the ﬁrst
boundary value problem which consists of the equation and the
boundary condition
∂2 E
∂2 E
+
+ κ2 E = 0
∂ y2
∂z2
(5.1.8)
E |L = 0
where L is the rectangular contour of the waveguide cross-section.
The eigen-functions of this problem are
E x = sin κm y sin κn z κm = mπ/a
κn = nπ/b
(5.1.9)
Expansion for the transverse electric components is given by the
vector eigen-functions emn⊥ which are expressed through the scalar
function (Eq. 5.1.9) by the following
E ,22
E ,33
e y + κn fmn
emn⊥ = ∇⊥ E x = κm fmn
ez = κm cos κm y sin κn ze y + κn sin κm y cos κn zf3E ez (5.1.10)
Here the scalar functions are given by (Eq. 1.2.20)
⎫
E ,11
fmn
= sin κm y sin κn z ⎬
E ,22
= cos κm y sin κn z
fmn
⎭
E ,33
= sin κm y cos κn z
fmn
(5.1.11)
The axes are directed as shown in Fig. 5.1 where the coordinates
x, y,and z will be numbered as 1, 2, and 3, respectively. It is easy
to check that these functions satisfy the ﬁrst vector boundary value
problem
2
emn⊥ = 0
∇⊥2 emn⊥ + κmn
2
κmn
κm2
emnt | L = 0
div⊥ emnt | L = 0
κn2
where
=
+ are the eigen-values.
The vectors (Eq. 5.1.10) give the system of the vector functions
for a magnetic ﬁeld
E ,33
E ,22
eˆ y + κm fmn
eˆ z
e∗mn = [ex , ∇⊥ E x ] = −κn fmn
(5.1.12)
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
Expansion of an Electromagnetic Field by the Non-Stationary
Figure 5.1. Geometry of the problem.
The system of vector functions for an electric ﬁeld is given by
t
the Maxwell’s equation E = curl Bdt. Omitting integration by time
(which is not essential in this case) we obtain
∂
∂
2
E ,11
E ,22
E ,33
eˆ x + κm fmn
eˆ y
eˆ z .
fmn
emn = κmn
+ κn fmn
(5.1.13)
∂x
∂x
Thereby the electric and magnetic ﬁelds of the TM-waves are
represented by series of the systems of the vector eigen-functions
(Eqs. 5.1.12 and 5.1.13)
E=
emn E mn (t, x)
(5.1.14)
e∗mn Bmn (t, x)
(5.1.15)
m,n
B=
m,n
The vector eigen-functions for TE-waves generated by the scalar
functions
H x = cos κm y cos κn z
(5.1.16)
are equal to
B,22
B,33
eˆ y − κn fmn
bmn⊥ = ∇⊥ H x = −κm fmn
eˆ z = −κm sin κm y cos κn zeˆ y − κn cos κm y sin κn zeˆ z (5.1.17)
where
B,11
= cos κm y cos κn z
fmn
B,22
E ,33
fmn
= fmn
B,33
E ,22
fmn
= fmn
(5.1.18)
351
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
352 An Electromagnetic Field in a Metallic Waveguide with a Moving Medium
This system of functions satisﬁes the second vector boundary
value problem
2
∇⊥2 bmn⊥ + κmn
bmn⊥ = 0
(bmn⊥ , n)| L = 0
(curl⊥ , bmn )⊥
L
=0
The function (Eq. 5.1.16) gives the system of the functions for an
electric ﬁeld
B,33
B,22
b∗mn = [ex , ∇⊥ H x ] = κn fmn
eˆ y − κm fmn
eˆ z
and the Maxwell’s equation B = −curl
functions for a magnetic ﬁeld
2
B,11
B,22
eˆ x − κm fmn
eˆ y
fmn
bmn = κmn
(5.1.19)
t
Edt gives the system of
∂
∂
B,33
− κn fmn
eˆ z
∂x
∂x
(5.1.20)
Thereby the ﬁelds for the TE-waves are given by the series
E=
b∗mn E mn (t, x)
(5.1.21)
bmn Bmn (t, x)
(5.1.22)
m,n
B=
m,n
The series for the ﬁelds (Eqs. 5.1.15, 5.1.21 and 5.1.22) allow one
to reduce the vector integral Eq. 5.1.17 to the system of the scalar
integral equations with respect to the expansion coeﬃcients. If a
material object in the waveguide has an arbitrary shape then the
ﬁeld contains the waves of both classes and the system of integral
equations is very intricate. As the main interest in this book is
non-stationary behaviour of electrodynamics processes we conﬁne
ourselves further to the case where the material object is a planeparallel insertion ﬁlling the whole waveguide cross-section. In this
case the object characteristic function χ = χ (t, x) depends on one
spatial coordinate only.
Let us consider the case when the initial ﬁeld consists of H- or
TE-waves. It is evident that the geometric properties of the object
eliminate the appearance of the waves of the class other than the
initial one. So, the vectors E, P and j can be represented in the form
of Eq. 5.1.21 and the vectors B and M in the form of Eq. 5.1.22. These
expansions allow integration over the transverse cross-section of
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
Expansion of an Electromagnetic Field by the Non-Stationary
the waveguide in the integral Eq. 5.1.7
∞
∞
b∗mn
E = E0 − 2π v
m,n
dt
−∞
1 ∂2
εv 2 ∂t2
dx
−∞
ε−1
E mn + ˜ mn (χ jmn + i mn δ (S))
Pmn −
mn χ
4π
μ−1
μ ∂ ∂ mn ∂
2
− κmn
+
M mn −
Bmn (5.1.23)
mn χ
c ∂t
∂x ∂x
4π μ
Equating coeﬃcients of the terms with the same vector-functions
reduces Eq. 5.1.23 to the temporal and 1D spatial integral scalar
equation for an arbitrary mode of the electric ﬁeld whose index is
omitted
2π
E = E0 −
εv
∞
∞
dt
−∞
dx
−∞
+ ˜ k (χ ji n + i δ (s)) + c
Mi n −
×
where
k
⎛
= J 0 ⎝ωk
∂2
∂t2
∂
∂t
kχ
Pi n −
ε−1
E
4π
∂ k ∂
− κk2
∂x ∂x
μ
B
4π μ
k
χ
(5.1.24)
⎞
2
(x
)
−
x
⎠θ
(t − t )2 −
v2
t−t −
x−x
v
(5.1.25)
t
˜k =
dτ
k
τ −t ,x −x
ωk = vκk
(5.1.26)
−∞
and the index k designates the oscillation mode considered, κk =
κmn . The diﬀerentiation operations in this equation can be brought
out as the integrals as the latter are convolutions.
If the background medium is not magnetic (μ = 1) and the
object consists of such a medium that the magnetisation vector M
has transverse components only then the integral equation can be
simpliﬁed. In this case the magnetic ﬁeld B does not enter directly in
the equation and the magnetisation M can be expanded in a series of
the transverse eigen-vectors
B,22
B,33
eˆ y − κn fmn
eˆ z
bmn⊥ = −κm fmn
(5.1.27)
353
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
354 An Electromagnetic Field in a Metallic Waveguide with a Moving Medium
in which, in comparison with vectors (Eq. 5.1.20), the derivative
∂/∂ x is brought into the coeﬃcients M mn . In this case the coeﬃcients
¯ mn (t, x) are connected with the
bmn⊥ M
of the new expansion M =
m,n
¯ mn = ∂ M mn . One has to take into account
old ones by the relation M
∂x
also that the equality κmn = 0 would be stated in Eq. 5.1.20. After
manipulations, the Eq. 5.1.24 takes the form
∞
∞
E = E 0 − 2π
dt
−∞
dx
−∞
1 ∂2
εv ∂t2
1 ∂2
+ ˜ k (χ ji n + i δ (s)) + √
ε ∂t∂ x
k
ε−1
E
4π
Pi n −
k χ Mi n
¯
.
χ
(5.1.28)
The integrals in this equation represent convolutions, so the
diﬀerentiation operators can be brought out of the integrals
according to properties of generalised functions.
Thereby the integral equation for the electric ﬁeld in the
rectangular metallic waveguide containing a dielectric object ﬁlling
the whole cross-section of the waveguide and restricted by the
plane-parallel boundaries which are normal to the waveguide’s
longitudinal axis is given by Eq. 5.1.28.
Equation 5.1.28 is the starting one for the further investigation
of non-stationary behaviour in the waveguide. It is the Volterra
integral equation and it breaks up into a chain of evolutionary linked
equations which can be solved by the resolvent method as in the case
of an unbounded medium considered in the previous chapters. The
existence of waveguide dispersion gives eﬀects of theoretical and
practical interest in the case of such simple dynamics as uniform
movement of a restricted dielectric, even if the object inside the
waveguide does not change its properties. Such phenomena will be
considered in the next sections.
5.2 Equations for a Field in the Waveguide with a
Non-Stationary Insertion
The inﬂuence of the waveguide dispersion on the electromagnetic
transients is revealed in the stationary movement of the medium or
July 25, 2012 13:29
PSP Book - 9in x 6in
Equations for a Field in the Waveguide with a Non-Stationary Insertion
Figure 5.2. Movement of the insertion in the waveguide.
its boundary [3–6]. The stationary movement is understood as the
movement which does not change its character since the inﬁnitely
remote past. This investigation is interesting because it reveals
peculiarities that are brought about by the waveguide dispersion.
The results can be compared with results for strictly non-stationary
phenomena, for example, when the movement begins at some ﬁnite
moment of time.
We consider a dielectric non-stationary insertion ﬁlling the
whole cross-section of the waveguide and moving along it. This
insertion is restricted by planes x1 (t) and x2 (t) which are perpendicular to the waveguide’s longitudinal axis (Fig. 5.2). The waveguide
itself is ﬁlled by a motionless background non-magnetic, μ = 1,
medium. A guided wave of a given mode E 0 is falling on the layer
from the side x < x1 (t). One must distinguish two cases: (i)
the medium inside the insertion is motionless, and (ii) it moves
together with its boundary. In both cases we consider only the
most practical case when the movement is directed along the
waveguide.
If the medium when at rest is non-conducting and non-magnetic
and has a dielectric polarisability αi j then the general constituent
relations [3] give the expressions for the polarisation vectors for the
medium uniformly moving with the velocity u
05-Alexander-c05
355
July 25, 2012 13:29
PSP Book - 9in x 6in
05-Alexander-c05
356 An Electromagnetic Field in a Metallic Waveguide with a Moving Medium
Pi = γ 2 γi j α j m (γmn E n + emns βn B S ) ,
(5.2.1)
M i = −γ 2 ei j k β j αkn (emns βn Bs + γmn E n ) .
(5.2.2)
Here β = u/c, γ 2 = 1/ 1 − β 2 , γi j = δi j + 1−γ
β β and emns
γβ 2 i j
is the complete anti-symmetric tensor of the third rank. For an
isotropic medium the polarisability is a scalar one, αi j = α δi j ,
so the constituent relations (5.2.1) and (5.2.2) are simpliﬁed as
follows:
P = α γ 2 {E − β (β E) + [βB]}
M = [Pβ] .
(5.2.3)
These relations provide transversality for all the ﬁeld values E, P and
M in the case of the TE-waves as the mode transformation is absent
under the chosen geometry of the insertion.
The stationary nature of the problem allows one to represent all
ﬁeld values in the form of a Fourier transform with respect to the
time variable. In this representation the ﬁelds inside the insertion
consist of two waves
2
∞
dν ˜
E s (ν) ei νt−i ks (ν)x
2π
E =
s=1 −∞
ε−1
E =
P−
4π
2
2
∞
s=1 −∞
∞
¯ =
M
s=1 −∞
(5.2.4)
dν ¯˜
Ps (ν) ei νt−i ks (ν)x
2π
dν ˜¯
M s (ν) ei νt−i ks (ν)x
2π
If the kernel (Eq. 5.1.25) is represented as Fourier transform also
∞
k
=
−∞
dω i ω(t−t )
1
e
e−i k0 (ω)|x−x |
2π
i vk0 (ω)
(5.2.5)
then the integral Eq. 5.1.24 in the previous section, after integration
by x over the interval [x1 (t), x2 (t)] with x belonging to this interval,