Fundamentals of Plasma Physics

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

3. Integration of Eq.(7.38) across the interface shows that the quantity

P

(

P−k^2 c^2 /ω^2

)− 1

∂Ez/∂r

must be continuous across the interface.

In order to satisfy constraint #1 the parallel electric field in the plasma must be

Ez(r)=Ez(a)

I 0 (κpr)

I 0 (κpa)

(7.41)

and the parallel electric field in the vacuum must be

Ez(r)=Ez(a)

K 0 (κvr)

K 0 (κva)

. (7.42)

The normalization has been set so thatEzis continuous across the interface as required by
constraint #2.
Constraint #3 gives
[(
ω^2
c^2
P−k^2

)− 1

P

∂Ez

∂r

]

r=a−

=

[(

ω^2

c^2

−k^2

)− 1

∂Ez

∂r

]

r=a+

. (7.43)

Inserting Eqs. (7.41) and (7.42) into the respective left and right hand sides of the above
expression gives

(

ω^2

c^2

P−k^2

)− 1

P

κpI 0 ′(κpa)

I 0 (κpa)

=

(

ω^2

c^2

−k^2

)− 1

κvK 0 ′(κva)

K 0 (κva)

(7.44)

where a prime means a derivative with respect to the argument of the function. This expres-
sion is effectively a dispersion relation since it prescribes a functional relationship between
ωandk.It is qualitatively different from the previously discussed uniform plasma disper-
sion relations, because of the dependence on the plasma radiusa,a physical dimension.
This dependence indicates that this mode requires the existence of the plasma-vacuum in-
terface. The mode amplitude is strongest in the vicinity of the interfacebecause both the
plasma and vacuum fields decay exponentially on moving away from the interface.
The surface wave dispersion depends on a combination of Bessel functions and the par-
allel dielectricP.However, a limit exists for which the dispersion relation reduces to a
simpler form, and this limit illustrates important features of these surface waves. Specifi-
cally, if the axial wavelength is sufficiently short fork^2 to be much larger than bothω^2 P/c^2
andω^2 /c^2 then it is possible to approximatek^2 ≃κ^2 v≃κ^2 pso that the dispersion simplifies
to

P

I′ 0 (ka)

I 0 (ka)

≃

K 0 ′(ka)

K 0 (ka)

. (7.45)

If, in addition, the axial wavelength is sufficiently long to satisfyka<< 1 , then the small-
argument limits of the modified Bessel functions can be used, namely,

lim

ξ→ 0

I 0 (ξ) = 1+

ξ^2

4

, (7.46)

lim

ξ→ 0

K 0 (ξ) = −lnξ. (7.47)