Guiding polaritonic energy and momentum through two-dimensional Bravais lattices

The strong exciton absorption in monolayer transition metal dichalcogenides provides a promising platform for studying polaritons with tunable dispersions, which are crucial for controlling polaritonic energy and momentum, but remain underexplored. In this work, monolayer MoS$_2$ is coupled with a Fabry-P\'erot microcavity to form polaritons. Five types of Bravais lattices with sub-wavelength periods, based on polymethyl methacrylate (PMMA) nanopillars, are intentionally designed. The energy overlap between the periodic PMMA scattering wave and the polariton establishes a coupling channel that controls the directional flow of polaritonic energy, as demonstrated through angle-resolved reflectance measurements. Back-space image measurements further demonstrate that the dispersion in reciprocal space can be directly and manually tuned, allowing for control over their number and their positions. The coupling between the polariton and PMMA scattering wave is further demonstrated by analyzing the reflectance using the two-port two-mode model. The symmetries of 2D Bravais lattices allow the angle between energy and momentum flow to vary widely, from 90{\deg}, 60{\deg}, 45{\deg}, and 30{\deg} to arbitrary values. By adjusting the lattice vector lengths, the position of the dispersion branch in a specific direction can be fine-tuned, enabling full-range control over polariton dispersion. This work presents the first theoretical and experimental demonstrations of guiding the direction of polaritonic energy and momentum through Bravais lattice design.