Fundamentals of Plasma Physics

7.1 Wave propagation in inhomogeneous plasmas 211

Substituting Ampere’s law into the curl of Faraday’s law gives

∇×∇×E 1 =

ω^2

c^2

(

1 −

ω^2 pe(x)

ω^2

)

E 1. (7.4)

Attention is now restricted to waves for which∇·E 1 = 0;this is a generalization of
the assumption that the waves are transverse (i.e., haveik·E= 0) or equivalently are
electromagnetic, and so involve no density perturbation. In this case, expansion of the left
hand side of Eq.(7.4) yields
(
∂^2
∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

E 1 +

ω^2

c^2

(

1 −

ω^2 pe(x)

ω^2

)

E 1 =0. (7.5)

It should be recalled that Fourier analysis is restricted to equationswith constant coeffi-
cients, so Eq.(7.5) can only be Fourier transformed in theyandzdirections. It cannot be
Fourier transformed in thexdirection because the coefficientω^2 pe(x)depends onx.Thus,
after performing only the allowed Fourier transforms, the wave equationbecomes
(
∂^2
∂x^2

−k^2 y−k^2 z

)

̃E 1 (x,ky,kz)+ω

2

c^2

(

1 −

ω^2 pe(x)

ω^2

)

E ̃ 1 (x,ky,kz)=0 (7.6)

whereE ̃ 1 (x,ky,kz)is the Fourier transform in theyandzdirections. This may be rewrit-
ten as (
∂^2
∂x^2

+κ^2 (x)

)

E ̃ 1 (x,ky,kz)=0 (7.7)

where

κ^2 (x)=

ω^2

c^2

(

1 −

ω^2 pe(x)

ω^2

)

−k^2 y−k^2 z. (7.8)

We now realize that Eq. (7.7) is just the spatial analog of the WKB equation for a pendulum
with slowly varying frequency, namely Eq.(3.17);the only difference is that the indepen-
dent variablethas been replaced by the independent variablex.Since changing the name
of the independent variable is of no consequence, the solution here is formally the same as
the previously derived WKB solution, Eq.(3.24). Thus the approximate solution to Eq.(7.8)
is

E ̃ 1 (x,ky,kz)∼√^1

κ(x)

exp(i

∫x

κ(x′)dx′). (7.9)

Equation (7.9) shows that both the the wave amplitude and effective wavenumber change
as the wave propagates in thexdirection, i.e. in the direction of the inhomogeneity. It
is clear that if the inhomogeneity is in thexdirection, the wavenumberskyandkzdo not
change as the wave propagates. This is because, unlike for thexdirection, it was possible
to Fourier transform in theyandzdirections and sokyandkzare just coordinates in
Fourier space. The effective wavenumber in the direction of the inhomogeneity, i.e.,κ(x),
is not a coordinate in Fourier space because Fourier transformation was not allowed in the
xdirection. The spatial dependence of the effective wavenumberκ(x)defined by Eq.(7.8)
and the spatial dependence of the WKB amplitude together provide the means by which
the system accommodates the spatial inhomogeneity. The invarianceof the wavenumbers
in the homogeneous directions is called Snell’s law. An elementary example ofSnell’s
law is the situation where light crosses an interface between two media having different
dielectric constants and the refractive index parallel to the interface remains invariant.