The transformation of electromagnetic waves into Langmuir oscillations (and vice versa) is explicitly examined in the vicinity where the wave frequency matches the electron plasma frequency in an inhomogeneous plasma. For an unmagnetized plasma with a linear density profile of scale length L, closed-form, analytic expressions are derived, in terms of Airy functions, for the reflection and mode conversion coefficients of Langmuir and electromagnetic waves utilizing a source approximation that is valid when the electromagnetic field scale length is large compared to that of the electrostatic field. The technique developed to determine the energy flux coefficients and the fields is general enough to apply to a plasma with a profile other than a linear one, and should prove useful in other problems where a scale length separation is valid. The reflection coefficient for the ‘‘direct’’ problem (incident electromagnetic wave) is equal in magnitude to that of the ‘‘inverse’’ problem (incident electrostatic wave) and the corresponding mode conversion coefficients satisfy energy conservation. The mode conversion coefficient of the warm, collisionless problem is independent of electron temperature, T, in the limit of small T, and is equal in magnitude to the mode conversion coefficient of the cold, collisional problem, which is likewise independent of the collision frequency. It also depends on the angle of incidence θ and the vacuum scale length k0L, and for T/mc2≪1, agrees with earlier, numerical calculations.

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 w matches the electron plasma frequency wP 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.

This study provides an analytic solution to the general problem of mode conversion in an unmagnetized plasma. Specifically, an electromagnetic wave of frequency ω propagating through a plasma with a parabolic density profile of scale length Lp is examined. The mode conversion points are located a distance Δ0 from the peak of the profile, where the electron plasma frequency ωp(z) matches the wave frequency ω. The corresponding reflection, transmission, and mode conversion coefficients are expressed analytically in terms of parabolic cylinder functions for all values of Δ0. The method of solution is based on a source approximation technique that is valid when the electromagnetic and electrostatic scale lengths are well separated. For large Δ0, i.e., (cLp/ω)1/2≪Δ0<Lp, the appropriately scaled result [D. E. Hinkel-Lipsker et al., Phys. Fluids B 4, 559 (1992)] for a linear density profile is recovered as the parabolic cylinder functions asymptotically become Airy functions. When Δ0→0, the special case of conversion at the peak of the profile [D. E. Hinkel-Lipsker et al., Phys. Fluids B 4, 1772 (1992)] is obtained.

We have derived analytic expressions, in terms of Airy functions, for the reflection and modeconversion coefficients of Langmuir and electromagnetic waves in an inhomogeneous, unmagnetized plasma. The reflection coetflcient for the "direct" problem (incident electromagnetic wave) is equal in magnitude to that for the "inverse" problem (incident electrostatic wave) and the corresponding modeconversion coefficients satisfy energy conservation. Our results, which are valid in the limit of nonrelativistic electron temperature, T/mc « 1, agree with earlier numerical calculations.

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An advanced ray‐tracing numerical method is coupled to an ionospheric plasma transport code to model the self‐consistent propagation of an obliquely incident high‐frequency (HF) beam in the ionosphere. An HF radar propagation scenario is presented as an application. During the daytime, a density increase in the lower ionosphere is predicted causing the defocusing of the incident HF beam.

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