Conversion of electrostatic and electromagnetic waves in a plasma at the peak of a parabolic density profile

Analytic expressions for the reflection, transmission, and mode-conversion coefficients for electromagnetic and Langmuir waves in an unmagnetized plasma with a parabolic density profile are found for both the ''direct'' problem (incident electromagnetic wave) and the ''inverse'' problem (incident Langmuir wave). In contrast to the linear-profile problem, the absorption depends explicitly on the value of the collision frequency (cold model) or temperature (warm plasma), but a transformation of parameters relates the results for these two limits