  We discuss the theoretical foundations for testing non-linear vacuum
electrodynamics with Michelson interferometry. Apart from some non-degeneracy
conditions to be imposed, our discussion applies to all non-linear
electrodynamical theories of the Pleba\'nski class, i.e., to all Lagrangians
that depend only on the two Lorentz-invariant scalars quadratic in the field
strength. The main idea of the experiment proposed here is to use the fact
that, according to non-linear electrodynamics, the phase velocity of light
should depend on the strength and on the direction of an electromagnetic
background field. There are two possible experimental set-ups for testing this
prediction with Michelson interferometry. The first possibility is to apply a
strong electromagnetic field to the beam in one arm of the interferometer and
to compare the situation where the field is switched on with the situation
where it is switched off. The second possibility is to place the whole
interferometer in a strong electromagnetic field and to rotate it. If an
electromagnetic field is placed in one arm, the interferometer could have the
size of a gravitational wave detector, i.e., an arm-length of several hundred
meters. If the whole interferometer is placed in an electromagnetic field, one
would have to do the experiment with a table-top interferometer. [...] We
specify the general results to some particular theories of the Pleba\'nski
class; in particular, we give numerical estimates for Born, Born-Infeld and
Heisenberg-Euler theories.
