  Before an exoplanet transit, atmospheric refraction bends light into the line
of sight of an observer. The refracted light forms a stellar mirage, a
distorted secondary image of the host star. I model this phenomenon and the
resultant out-of-transit flux increase across a comprehensive exoplanetary
parameter space. At visible wavelengths, Rayleigh scattering limits the
detectability of stellar mirages in most exoplanetary systems with semi-major
axes $\lesssim$6 AU. A notable exception is almost any planet orbiting a late M
or ultra-cool dwarf star at $\gtrsim$0.5 AU, where the maximum relative flux
increase is greater than 50 parts-per-million. Based partly on previous work, I
propose that the importance of refraction in an exoplanet system is governed by
two angles: the orbital distance divided by the stellar radius and the total
deflection achieved by a ray in the optically thin portion of the atmosphere.
Atmospheric lensing events caused by non-transiting exoplanets, which allow for
exoplanet detection and atmospheric characterization, are also investigated. I
derive the basic formalism to determine the total signal-to-noise ratio of an
atmospheric lensing event, with application to Kepler data. It is unlikely that
out-of-transit refracted light signals are clearly present in Kepler data due
to Rayleigh scattering and the bias toward short-period exoplanets. However,
observations at long wavelengths (e.g., the near-infrared) are significantly
more likely to detect stellar mirages. Lastly, I discuss the potential for the
Transiting Exoplanet Survey Satellite to detect refracted light and consider
novel science cases enabled by refracted light spectra from the James Webb
Space Telescope.
