Electromagnetic wavefunctions for parabolic plasma density profiles

The one‐dimensional Helmholtz equation ${\mathrm{w″\hspace{0.17em}+\hspace{0.17em}k}}_0^2\mathrm{g(z)w(z)\hspace{0.17em}=\hspace{0.17em}}$ , $k_0{\text{\hspace{0.17em}=\hspace{0.17em}2π/λ}}_0$ , is solved making use of an extension of Langer's method. The function $\mathrm{g(z)}$ is restricted to the class of Epstein profiles, ${\text{g(z)\hspace{0.17em}=\hspace{0.17em}g(0)\hspace{0.17em}+\hspace{0.17em}[1\hspace{0.17em}−\hspace{0.17em}g(0)]\hspace{0.17em}tanh}}^2\text{(z/2λ)}$ , where $\lambda$ is the scale length, corresponding to no turning points, a quadratic turning point at $\text{z\hspace{0.17em}=\hspace{0.17em}0}$ , or a pair of turning points, depending on whether $\text{g(0)}$ is positive, zero, or negative. This method yields uniformly valid solutions in terms of confluent hypergeometric functions. First, a solution $w_{+}\mathrm{(z)}$ is obtained for $\text{0\hspace{0.17em}≤\hspace{0.17em}z}$ which, as $\mathrm{z\hspace{0.17em}\to \hspace{0.17em}\infty }$ , becomes the transmitted plane wave. Then, applying the method of analytic continuation, a linear combination is obtained $w_{-}{\mathrm{(z)\hspace{0.17em}=\hspace{0.17em}w}}_i{\mathrm{(z)\hspace{0.17em}+\hspace{0.17em}w}}_r\mathrm{(z)}$ valid for $\text{z\hspace{0.17em}≤\hspace{0.17em}0}$ which, as $\mathrm{z\hspace{0.17em}\to \hspace{0.17em}-\hspace{0.17em}\infty }$ , yields the superposition of the incident and reflected plane waves. As an illustration of this method and as an application to parabolic density profiles, the mean energy density of the electric field is obtained numerically, in the vicinity of $\text{z\hspace{0.17em}=\hspace{0.17em}0}$ , when the parameter ratio ${\mathrm{\lambda /\lambda }}_0\text{\hspace{0.17em}≥\hspace{0.17em}100}$ , and for the case of a single quadratic turning point and also for the case in which there are two neighboring turning points. With this choice of parameters, the results yield useful information on the tuning criterion for parametric excitation of the ionosphere.