  We provide a systematic theoretical, experimental, and historical critique of
the standard derivation of Fresnel's equations, which shows in particular that
these well-established equations actually contradict the traditional,
macroscopic approach to electrodynamics in media. Subsequently, we give a
rederivation of Fresnel's equations which is exclusively based on the
microscopic Maxwell equations and hence in accordance with modern
first-principles materials physics. In particular, as a main outcome of this
analysis being of a more general interest, we propose the most general boundary
conditions on electric and magnetic fields which are valid on the microscopic
level.
