  The polarization dependence of magnon-photon scattering in an optical
microcavity is reported. Due to the short cavity length, the mode-matching
conditions found in previously explored, large path-length whispering gallery
resonators are absent. Nonetheless, for cross-polarized scattering a strong and
broadband suppression of one side-band is observed. This arises due to an
interference between the Faraday and second-order Cotton-Mouton effects. To
fully account for the suppression of the cross-polarized scattering, it is
necessary to consider the squeezing of magnon modes intrinsic to thin-film
geometry. A co-polarized scattering due to Cotton-Mouton effect is also
observed. In addition, the magnon modes involved are identified as
Damon-Eshbach surface modes, whose non-reciprocal propagation could be
exploited in devices applications. This work experimentally demonstrates the
important role of second order Cotton-Mouton effect for optomagnonic devices.
