  The properties of the modified Maxwell electrodynamics (ModMax) are
investigated in presence of external and uniform electric and magnetic fields.
We expand the non-linear theory around an electromagnetic background up to
second order in the propagating fields to obtain the permittivity and
permeability tensors, dispersion relations, group velocity and refractive
indices as functions of external fields. The case with perpendicular background
fields is contemplated. The phenomenon of birefringence is discussed and the
fundamental role of the ModMax parameter becomes clear. We calculate the
difference of the refractive indices in terms of this parameter and the
external fields. Finally, we set up a scenario where the axion is present and
compute the interaction energy for the coupled axion-ModMax electrodynamics if
a magnetic background field is considered. This calculation is carried out in
the framework of the gauge-invariant, but path-dependent variables formalism.
Our results show that the interaction energy contains a linear component,
leading to the confinement of static probe charges where the interference
between the ModMax parameter, the axion mass and the axion-photon coupling
constant is pointed out.
