  Monolayers of transition metal dichalcogenides (TMDC) are direct-gap
semiconductors with strong light-matter interactions featuring tightly bound
excitons, while plasmonic crystals (PCs), consisting of metal nanoparticles
that act as meta-atoms, exhibit collective plasmon modes and allow one to
tailor electric fields on the nanoscale. Recent experiments show that TMDC-PC
hybrids can reach the strong-coupling limit between excitons and plasmons
forming new quasiparticles, so-called plexcitons. To describe this coupling
theoretically, we develop a self-consistent Maxwell-Bloch theory for TMDC-PC
hybrid structures, which allows us to compute the scattered light in the near-
and far-field explicitly and provide guidance for experimental studies. Our
calculations reveal a spectral splitting signature of strong coupling of more
than $100\,$meV in gold-MoSe$_2$ structures with $30\,$nm nanoparticles,
manifesting in a hybridization of exciton and plasmon into two effective
plexcitonic bands. In addition to the hybridized states, we find a remaining
excitonic mode with significantly smaller coupling to the plasmonic near-field,
emitting directly into the far-field. Thus, hybrid spectra in the strong
coupling regime can contain three emission peaks.
