A microwave beam in the ordinary (O) mode at frequency enters a plasma slab in which…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

A microwave beam in the ordinary (O) mode at frequency

f = 35\,\mathrm{GHz}

enters a plasma slab in which the electron density rises smoothly, over many wavelengths, from

n_1 = 1.0\times10^{18}\,\mathrm{m^{-3}}

n_2 = 6.0\times10^{18}\,\mathrm{m^{-3}}

. The field obeys

E'' + k^2(x)E = 0

with

k^2(x) = \left(\omega^2 - \omega_{pe}^2(x)\right)/c^2

, and the WKB condition

|k'| \ll k^2

holds throughout the region. ((a)) Starting from the constancy of the wave's energy flux, derive the general WKB relation between the field amplitude

|E(x)|

and the local wavenumber

k(x)

. ((b)) Use it to find the ratio

|E_2|/|E_1|

of the field amplitude at the point where

n_e = n_2

to that where

n_e = n_1