RLE Progress Report No - Research Laboratory of Electronics - MIT

In Eq. (2), α = −k x , ( )
z o
β = k
z
d − x ,
o
t = tanh[ k
z
( a − d)]
, and ( I
o(1)
, K
o(1)
) are the modified
Bessel functions of the first kind and second kind. This dispersion relation has an infinite number
of roots for a resonance ω at any x in the density gradient. Figure 2(a) shows the contour plots
of Re { D(
k ) = 0}
and Im { D(
k ) = 0}
, as well as the location of the first roots, for ω /
2
= 0. 7 ,
z
z
n
1
/ n2
= 0.2 and d = 0. 25 . Figure 2(b) shows the dispersion relation ( k z
vs. Ω = ω / ω ) for the
pe2
first root.
Based on Ricati’s transformation, we developed a numeric routine to solve for the electrostatic
potential, assuming a weak collisional frequency ( ν