ABSTRACT
F( , ) ( ) exp( ).r F rt t= jω (10.2)
In the above, F(r) stands for any of E, H, or J phasors. The basic £eld equations are then given by
∇ × = −E Hjω μ0 , (10.3)
∇ × = +H E Jjω ε0 , (10.4)
j pω ε ωJ r E= 0 2 ( ) . (10.5)
Combining Equations 10.4 and 10.5, we can write
∇ × =H r Ej pωε ε ω0 ( , ) , (10.6)
where εp is the dielectric constant of the isotropic plasma and is given by
ε ω
ω
ω p
p( , ) ( )
.r r
= −1 2
(10.7)
The vector wave equations are
∇ + = ∇ ∇ ⋅2
2E r E E ω
ε c
p( ) ( ),
(10.8)
∇ + = − ∇ ×
= − ∇ ×
H r H r E
r E
ω ε
ε
ω ω
ωε ε
j
j
( ) ( )
( ) . (10.9)
The scalar one-dimensional equations take the form
∂ ∂ ωμ E z
H= − j 0 ,
(10.10)
− = +
∂ ∂ ωε H z
E Jj 0 ,
(10.11)
j pω ε ωJ z E= 0 2 ( ) . (10.12)
Combining Equations 10.11 and 10.12, or from Equations 10.6, we have
− =
∂ ∂ ωε ε ω H z
z Ej p0 ( , ) ,
(10.13)
where
ε ω
ω
ω p
p( , ) ( )
.z z
= −1 2
(10.14)
Figure 10.1 shows the variation of εp with ω . εp is negative for ω < ωp, 0 for ω = ωp, and positive but less than 1 for ω > ωp. The scalar one-dimensional wave equations, in this case, reduce to Equations 10.15 and 10.16:
d d p
2 0 E z c
z E+ = ω
ε ω( , ) ,
(10.15)
d d
d d
d dp p
1H z c
z H z
z z
H z
+ = ω
ε ω ε ω
ε ω ( , )
( , ) ( , )
.
