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II. Electromagnetic Transients in Time-Varying Waveguides and Resonators

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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

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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)

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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

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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)

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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)

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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

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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)

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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

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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

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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,

II. Electromagnetic Transients in Time-Varying Waveguides and Resonators