Expert: Philip A. Stahl - 2/11/2008

Question
For parallel propagation in an EM field, how can one *quantitatively* distinguish the cut-off plasma frequency from the resonant one?

Answer
Hello,

In your parallel propagation (3D) case, we assume: w >> W _i

(e.g. plasma electron frequency much higher than ion cyclotron frequency) and for our fields:

B1  =  k X  E1  = (B_x, B_y, 0)


E1 =  (E_x, E_y, 0)

Curl B = 4 pi J1/c  +  E1'/c


where E1' denotes a time derivative and in the high frequency regime one has: J1 = - en_o v1

In terms of the E-field components (E_x, E_y) one can set out a matrix:

( [1 + k^2c^2/w^2 - w^2/w_e^2) (i W_e/w_e^2(w - k^2c^2/w))

(-i W_e/w_e^2(w - k^2c^2/w)  (1 + k^2c^2/w_e^2 - w^2/w_e^2))

=  0


from which we can simplify key quanitites in each of the brackets, viz.

w* = w/W_e,   b^2 = (k^2c^2/w_e^2), a^2 = W_e^2/w_e^2

From this we can write the matrix eqn. out:


a^2 w*^2 (w* +/- 1) - w* - b^2(w* +/- 1)  = 0


It can easily be seen from the above that one gets an equation in w* with order 3 (w*^3) so there must be three roots for the plasma frequency w* and hence for w.

The one called "cut-off frequency" is that for which k = 0, this implies b = 0 (since as we saw: b^2 = (k^2c^2/w_e^2)) so: the root for plasma frequency obtained (call it w_k) is the cut-off frequency. (For the other roots, one finds w* = 0, and w* = w_L denoting left hand circular polarized wave)

The "resonant frequency" for the same conditions (parallel propagation in E, B field as defined) can simply be found by noting that the same disperson relation (eg. in w, k) applies but in the limit k -> oo. For which one will find:

w =  W_e

Hope this helps, and isn't too complicated!