Fundamentals of Plasma Physics

216 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations

This result can also be used to solve for the transverse electric fieldby interchanging
−iωP/c^2 ←→iωandE←→Bto obtain

Et=

(

ω^2

c^2

P−k^2

)− 1 [

∇t

∂Ez

∂z

+iω∇tBz×zˆ

]

. (7.29)

Except for the plasma-dependent factorP,these are the standard waveguide equations. An
important feature of these equations is that the transverse fieldsEt,Btare functions of the
axial fieldsEz,Bzonly and so all that is required is to construct wave equations charac-
terizing the axial fields. This is an enormous simplification because, instead of having to
derive and solve six wave equations in the six components ofE,Bas might be expected, it
suffices to derive and solve wave equations for justEzandBz.
The sought-after wave equations are determined by eliminatingEtandBtfrom Eqs.(7.23)
and (7.24) to obtain

zˆ·∇t×

{(

ω^2

c^2

P−k^2

)− 1 [

ik∇tBz−

iω

c^2

P∇tEz×zˆ

]}

=−

iω

c^2

PEz, (7.30)

ˆz·∇t×

{(

ω^2

c^2

P−k^2

)− 1

[ik∇tEz+iω∇tBz׈z]

}

=iωBz. (7.31)

In the special situation where∇tP×∇tBz=∇tP×∇tEz=0,the first terms in the
square brackets of the above equations vanish. This simplification would occurfor example
in an azimuthally symmetric plasma having an azimuthally symmetricperturbation so that
∇tP,∇tEzand∇tBzare all in therdirection. It is now assumed that both the plasma
and the mode have this azimuthal symmetry so that Eqs.(7.30) and (7.31) reduce to

zˆ·∇t×

{(

ω^2

c^2

P−k^2

)− 1

(P∇tEz×zˆ)

}

= PEz, (7.32)

zˆ·∇t×

{(

ω^2

c^2

P−k^2

)− 1

(∇tBz×zˆ)

}

=Bz (7.33)

or equivalently

∇t·

{

P

P−k^2 c^2 /ω^2

∇tEz

}

+

ω^2

c^2

PEz=0, (7.34)

∇t·

{

1

P−k^2 c^2 /ω^2

∇tBz

}

+

ω^2

c^2

Bz=0. (7.35)

The assumption that both the plasma and the modes are azimuthally symmetric has
the important consequence of decoupling theEzandBzmodes so there are two distinct
polarizations. These are (i) a mode whereBzis finite, butEz=0and (ii) the reverse. Case
(i) is called a transverse electric (TE) mode while case (ii) iscalled a transverse magnetic
mode (TM) since in the first case the electric field is purely transverse while in the second
case the magnetic field is purely transverse.