  We study optical excitations of an ultracold gas in an optical lattice in the
Mott insulator case and with anisotropic properties. The anisotropy is induced
by an oriented transition dipole generated by optical pumping and external
static fields, and is of the type of uniaxial crystals. Within a cavity, the
electronic excitations and the cavity photons of both TE and TM polarizations
are coherently mixed to form two polariton branches. Also a photonic branch
appears which decouples from the electronic transitions. The photon
polarization mixing can be observed by linear optical spectra. For example, for
an incident field of TE polarization we obtain TE and TM transmitted and
reflected fields. Also we calculate the phase shift in the TE and TM
transmitted and reflected fields relative to the incident field phase, which
lead to polarization rotations.
