ANALYTIC EXPRESSIONS FOR MODE CONVERSION IN A PLASMA AT THE PEAK OF A PARABOLIC DENSITY PROFILE

For mode conversion in an unmagnetized plasma with a parabolic density profile of scale length L, analytic expressions, in terms of parabolic cylinder functions, for the energy flux coefficients (reflection, transmission, and mode conversion) and the fields for both the "direct" problem (incident electromagnetic wave converting to a Langmuir wave) and the "inverse" problem (incident Langmuir wave converting to an electromagnetic wave) are derived for the case where the incident wave frequency-omega matches the electron plasma frequency-omega(p) at the peak of the density profile. The mode conversion coefficient for the direct problem is equal in magnitude to that of the inverse problem, and the corresponding reflection and transmission coefficients satisfy energy conservation. In contrast to the linear profile problem, the conversion efficiency depends explicitly on the value of the collision frequency (in the cold, collisional limit) or electron temperature (in the warm, collisionless limit), but a transformation of parameters relates the results for these two limits.