  Exciton-polariton modes arising from interaction between bound excitons in
monolayer thin semiconductor sheets and photons in a Fabry-Perot microcavity
are considered theoretically. We calculate the dispersion curves, mode
lifetimes, Rabi splitting, and Hopfield coefficients of these structures for
two nearly 2D semiconductor materials, MoS2 and WS2, and suggest that they are
interesting for studying the rich physics associated with the Bose-Einstein
condensation of exciton-polaritons. The large exciton binding energy and dipole
allowed exciton transitions, in addition to the relatively easily controllable
distance between the semiconductor sheets are the advantages of this system in
comparison with traditional GaAs or CdTe based semiconductor microcavities. In
particular, in order to mimic the rich physical properties of the quantum
degenerate mixture of two bosonic species of dilute atomic gases with tunable
inter-species interaction , we put forward a structure containing two
semiconductor sheets separated by some atomic-scale distance (l) using a nearly
2D dielectric (e.g. h-BN), which offers the possibility of tuning the
interaction between two exciton-polariton Bose-Enstein condensates. We show
that the dynamics of this novel structure are ruled by two coupled
Gross-Pitaevskii equations with the coupling parameter proportional to 1/l.
