  The current article shall contribute to understanding the classical analogue
of the minimal photon sector in the Lorentz-violating Standard-Model Extension
(SME). It is supposed to complement all studies performed on classical
point-particle equivalents of SME fermions. The classical analogue of a photon
is not a massive particle being described by a usual equation of motion, but a
geometric ray underlying the eikonal equation. The first part of the paper will
set up the necessary tools to understand this correspondence for interesting
cases of the minimal SME photon sector. In conventional optics the eikonal
equation follows from an action principle, which is demonstrated to work in
most (but not all) Lorentz-violating cases as well. The integrands of the
action functional correspond to Finsler structures, which establishes the
connection to Finsler geometry. The second part of the article treats
Lorentz-violating light rays in a weak gravitational background by implementing
the principle of minimal coupling. Thereby it is shown how Lorentz violation in
the photon sector can be constrained by measurements of light bending at
massive bodies such as the Sun. The phenomenological studies are based on the
currently running ESA mission GAIA and the planned NASA/ESA mission LATOR. The
final part of the paper discusses certain aspects of explicit Lorentz violation
in gravity based on the setting of Finsler geometry.
