  The propagation of electromagnetic waves in a linearly-varying index of
refraction is a fundamental problem in wave physics, being relevant in fusion
science for describing certain wave-based heating and diagnostic schemes. Here,
an exact solution is obtained for a given incoming wavefield specified on the
boundary transverse to the direction of inhomogeneity by performing a spectral,
rather than asymptotic, matching. Two case studies are then presented: a
Gaussian beam at oblique incidence and a speckled wavefield at normal
incidence. For the Gaussian beam, it is shown that when the waist $W$ is
sufficiently large, oblique incidence manifests simply as rigid translation and
focal shift of the corresponding diffraction pattern at normal incidence. The
destruction of the hyperbolic umbilic caustic (corresponding to a critically
focused beam) as $W$ is reduced is then demonstrated. The caustic disappears
once $W \lesssim \delta_a \sqrt{L}$ ($L$ being the medium lengthscale
normalized by the Airy skin depth $\delta_a$), at which point the wave behavior
is increasingly described by Airy functions, but experiences less focusing as a
result. To maximize the intensity of a launched Gaussian beam at a turning
point, one should therefore minimize the imaginary part of the launched complex
beam parameter while having the real part satisfy critical focusing. For the
speckled wavefield, it is shown that the transverse speckle pattern only
couples to the Airy longitudinal pattern when the coupling parameter $\eta =
\sqrt{L}/f_\#$ is large, with $f_\#$ being the f-number of the launching
aperture. When $\eta \ll 1$, a reduced description of the total wavefield can
be obtained by simply multiplying the incoming speckle pattern with the Airy
swelling.
