  We experimentally resolve the dispersion of multiple vibro-polariton modes
issued from the strong coupling of different vibrational bands of the methylene
group (CH2) in a 2.56$\mu$m thick polyethylene film with the confined modes of
a mid-infrared Fabry-Perot micro-cavity. We measure a Rabi frequency of 111
cm$^{-1}$ for the stretching doublet around 2950 cm$^{-1}$ and a Rabi frequency
of 29 cm$^{-1}$ for the scissoring doublet around 1460 cm$^{-1}$. This simple
experimental approach offers the possibility to accurately fit the measured
molecular film dielectric function. We show that the polariton dispersion and
Rabi splitting can be precisely predicted from numerical simulations, offering
a valuable tool for the design of strongly coupled system and the development
of novel molecular films with crystalline organization.
