April 5, 2000

Notes for Lecture #30

Electromagnetic wave guides

In order to understand the op eration of a wave guide, wemust rst learn how electromagnetic

waves behave in a dissipative medium. Aplanewave solution to Maxwell's equations of the

form:

E

=

E

0

e

ik

^

k



r



i!t

and

B

=

k

!

^

k



E

0

e

ik

^

k



r



i!t

(1)

for the electric and magnetic elds, with the wave vector

k

satisfying the relation:

k

2

=

!

2

"

R

+

i

I

:

(2)

We can determine the complex wavevector

k

r

+

ik

i

according to

k

r

=

p

R

2

+

I

2

+

R

2

!

1

=

2

and

k

i

=

p

R

2

+

I

2

R

2

!

1

=

2

(3)

The form of the frequency dependent constants

R

and

I

depend on the materials. For the

Drude model at low frequency (Eq. 7.56),

R

=

!

2

"

b

and

I

=

!

,for example. The

value of

k

i

determines the rate of decay of the eld amplitudes in the vicinity of the surface,

with the skin depth given by

Æ



1

=k

i

. In the limit that

I  R

,as in the case of a good

conductor at low frequency,

Æ



(2

=

(

!

))

1

=

2

.

For an "ideal" conductor

I!1

, so that the elds are conned to the surface. Because of

the eld continuity conditions at the surface of the conductor, this means that,

B

tangential

6

=0

(because there can be a surface current),

E

normal

6

= 0 (because there can be a surface charge),

and

B

normal

=0.

Suppose we construct a wave guide from an "ideal" conductor, designating

^z

as the propa-

gation direction. We will assume that the elds take the form:

E

=

E

(

x; y

)e

ikz



i!t

and

B

=

B

(

x; y

)e

ikz



i!t

(4)

inside the pipe, where now

k

and

"

are assumed to be real. Assuming that there are no sources

inside the pipe, the elds there must satisfy Maxwell's equations (8.16) which expand to the

following :

@B

x

@x

+

@B

y

@y

+

ikB

z

=0

:

(5)

@E

x

@x

+

@E

y

@y

+

ikE

z

=0

:

(6)

@E

z

@y



ikE

y

=

i!B

x

:

(7)