  Vibrational polaritons form in a planar Fabry-Perot microcavity when a
vibrational mode of a layer of molecules is near resonant with an infrared
cavity mode. Herein, dispersion relations of vibrational polaritons are studied
when the molecular density distribution breaks the macroscopic translational
symmetry along the cavity mirror plane. Both perturbative theory and numerical
calculations show that, if a homogeneous in-plane molecular distribution is
modulated by sinusoidal fluctuations, in addition to a pair of upper and lower
polariton branches, a discrete number of side polariton branches may emerge in
the polariton dispersion relation. Moreover, for a periodic Gaussian molecular
in-plane density distribution, only two, yet significantly broadened polariton
branches exist in the spectra. This polariton linewidth broadening is caused by
the scattering between cavity modes at neighboring in-plane frequencies due to
the symmetry breaking, which is distinguished from known origins of polariton
broadening such as the homogeneous broadening of molecules, the cavity loss, or
the large energetic disorder of molecules. Associated with the broadened
polariton branches, under the periodic Gaussian in-plane inhomogeneity, a
significant number of the VSC eigenstates contain a non-zero contribution from
the cavity photon mode at zero in-plane frequency, blurring the distinction
between the bright and the dark modes. Looking forward, our theoretical
investigation should facilitate the experimental exploration of vibrational
polaritons with patterned in-plane molecular density distributions.
