  In this work we study how CPT-odd Maxwell-Carroll-Field-Jackiw (MCFJ)
electrodynamics as well as a dimension-5 extension of it affect the optical
activity of continuous media. The starting point is dimension-3 MCFJ
electrodynamics in matter whose modified Maxwell equations, permittivity
tensor, and dispersion relations are recapitulated. Corresponding refractive
indices are achieved in terms of the frequency and the vector-valued background
field. For a purely timelike background, the refractive indices are real. Their
associated propagation modes are circularly polarized and exhibit
birefringence. For a purely spacelike background, one refractive index is
always real and the other can be complex. The circularly polarized propagating
modes may exhibit birefringence and dichroism (associated with absorption).
Subsequently, we examine a dimension-five MCFJ-type electrodynamics, previously
scrutinized in the literature, in a continuous medium. Following the same
procedure, we find the refractive indices from a sixth-order dispersion
equation. For a purely timelike background, three distinct refractive indices
are obtained, one of them being real and two being complex. They are associated
with two circularly polarized propagating modes that exhibit birefringence or
dichroism, depending on the frequency range. Scenarios of propagation and
absorption analogous to those found in dispersive dielectrics are also observed
for spacelike background configurations. We conclude by comparing the
dimension-three and five results and by emphasizing the richer phenomenology of
the propagating modes in the higher-derivative model. Our results are
applicable in the realm of Weyl semimetals.
