  This work opens a series of papers where we develop a general quasioptical
theory for mode-converting electromagnetic beams in plasma and implement it in
a numerical algorithm. Here, the basic theory is introduced. We consider a
general quasimonochromatic multi-component wave in a weakly inhomogeneous
linear medium with no sources. For any given dispersion operator that governs
the wave field, we explicitly calculate the approximate operator that governs
the wave envelope $\psi$ to the second order in the geometrical-optics
parameter. Then, we further simplify this envelope operator by assuming that
the gradient of $\psi$ transverse to the local group velocity is much larger
than the corresponding parallel gradient. This leads to a parabolic
differential equation for $\psi$ ("quasioptical equation") in the basis of the
geometrical-optics polarization vectors. Scalar and mode-converting vector
beams are described on the same footing. We also explain how to apply this
model to electromagnetic waves in general. In the next papers of this series,
we report successful quasioptical modeling of radiofrequency wave beams in
magnetized plasma based on this theory.
