7.2 Geometric optics 2137.2 Geometric optics
The WKB method can be generalized to a plasma that is inhomogeneous in more than
one dimension. In the general case of inhomogeneity in all three dimensions, the three
components of the wavenumber will be functions of position, i.e.,k=k(x).How is the
functional dependence determined? The answer is to write the dispersion relation as
D(k,x)=0. (7.11)Thex-dependence ofDdenotes an explicit spatial dependence of the dispersion relation
due to density or magnetic field gradients. This dispersion relation is now presumed to
be satisfied at some initial pointxand then it is further assumed that all quantities evolve
in such a way to keep the dispersion relation satisfied at other positions. Thus, at some
arbitrary nearby positionx+δx, the dispersion relation is also satisfied so
D(k+δk,x+δx)=0 (7.12)or, on Taylor expanding,
δk·∂D
∂k+δx·∂D
∂x=0. (7.13)
The general condition for satisfying Eq.(7.13) can be established by assumingthat bothk
andx depend on some parameter which increases monotonically along the trajectory of
the wave, for example the distancesalong the wave trajectory.The wave trajectory itself
can also be parametrized as a function ofs.Then using bothk=k(s)andx=x(s), it is
seen that moving a distanceδscorresponds to respective incrementsδk=δsdk/dsand
δx=δsdx/ds.This means that Eq. (7.13) can be expressed as
[
dk
ds
·
∂D
∂k+
dx
ds·
∂D
∂x]
δs=0. (7.14)The general solution to this equation are the two coupled equations
dk
ds= −
∂D
∂x, (7.15)
dx
ds=
∂D
∂k. (7.16)
These are just Hamilton’s equations with the dispersion relationDacting as the Hamil-
tonian, the path lengthsacting like the time,xacting as the position, andkacting as the
momentum. Thus, given the initial momentum at an initial position, the wavenumber evo-
lution and wave trajectory can be calculated using Eqs.(7.15) and (7.16) respectively. The
close relationship between wavenumber and momentum fundamental to quantummechan-
ics is plainly evident here. Snell’s law states that the wavenumber in a particular direction
remains invariant if the medium is uniform in that direction;this is clearly equivalent (cf.
Eq.(7.15)) to the Hamilton-Lagrange result that the canonical momentum in a particular
direction is invariant if the system is uniform in that direction.
This Hamiltonian point of view provides a useful way for interpreting cutoffs and reso-
nances. Suppose thatDis the dispersion relation for a particular mode and suppose thatD
can be written in the form
D(k,x)=∑
ijαijkikj+g(ω,n(x),B(x))=0. (7.17)