On the theory of pairwise coupling embedded in more general local dispersion relations

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Abstract
Earlier work on the mode conversion theory by Fuchs, Ko, and Bers is detailed and expanded upon, and its relation to energy conservation is discussed. Given a local dispersion relation, D(ω; k, z)=0, describing stable waves excited at an externally imposed frequency ω, a pairwise mode‐coupling event embedded therein is extracted by expanding D(k, z) around a contour k=kc(z) given by ∂D/∂k=0. The branch points of D(k, z)=0 are the turning points of a second‐order differential‐equation representation. In obtaining the fraction of mode‐converted energy, the connection formula and conservation of energy must be used together. Also, proper attention must be given to distinguish cases for which the coupling disappears or persists upon confluence of the branches, a property which is shown to depend on the forward (vgvph>0) or backward (vgvph<0) nature of the waves. Examples occurring in ion‐cyclotron and lower‐hybrid heating are presented, illustrating the use of the theory.