7.3 Surface waves - the plasma-filled waveguide 215plasma to vacuum – in other words, the plasma has an edge or surface. A qualitatively new
mode, called a surface wave, appears in this circumstance. The physical basis of surface
waves is closely related to the mechanism by which light waves propagate in an optical
fiber.
Using the same analysis that led to Eq.(7.3), Maxwell’s equations in an unmagnetized
plasma may be expressed as
∇×B=−iω
c^2PE, ∇×E=iωB (7.19)wherePis the unmagnetized plasma dielectric function
P=1−
ω^2 pe
ω^2. (7.20)
We consider a plasma which is uniform in thezdirection but non-uniform in the direc-
tions perpendicular toz.The electromagnetic fields and gradient operator can be separated
into axial components (i.e.zdirection) and transverse components (i.e., perpendicular to
z)as follows:
B=Bt+Bzˆz, E=Et+Ezz,ˆ ∇=∇t+ˆz∂
∂z. (7.21)
Using these definitions Eqs.(7.19) become
(
∇t+ˆz
∂
∂z)
×(Bt+Bzˆz)=−iω
c^2P(Et+Ezzˆ),
(
∇t+ˆz∂
∂z)
×(Et+Ezzˆ)=iω(Bt+Bzˆz).(7.22)
Since the curl of a transverse vector is in thezdirection, these equations can be separated
into axial and transverse components,
zˆ·∇t×Bt=−
iω
c^2PEz, (7.23)ˆz·∇t×Et=iωBz, (7.24)ˆz×∂Bt
∂z+∇tBz×zˆ=−iω
c^2PEt, (7.25)ˆz×
∂Et
∂z+∇tEz×zˆ=iωBt. (7.26)The transverse electric field on the left hand side of Eq.(7.26) can be replaced using Eq.(7.25)
to give
iωBt=ˆz×∂
∂z
zˆ×∂Bt
∂z+∇tBz×zˆ−iω
c^2P
+∇tEz׈z. (7.27)It is now assumed that all quantities have axial dependence∼exp(ikz)so that Eq.(7.27)
can be solved to giveBtsolely in terms ofEzandBz,i.e.,
Bt=(
ω^2
c^2P−k^2)− 1 [
∇t∂Bz
∂z−
iω
c^2P∇tEz׈z