  We theoretically investigate coupling interaction between molecule excitons
and whispering gallery modes (WGMs) that approaches the limit of single exciton
strong coupling in hyperbolic metamaterial (HMM) cavity, composed of a
dielectric core wrapped by several stacks of alternative layers of metal and
dielectric. We demonstrate that associated with the excitation of the
whispering gallery mode, the electric fields with resonance wavelengths that
are much larger than the cavity size can be highly localized within a
dielectric layer, leading to extremely small mode volumes. By using temporal
coupled mode theory to model the interaction in the proposed WGMs-exciton
system, we are able to demonstrate that the coupling between molecular excitons
and hyperbolic cavity WGMs can reach the strong coupling regime. Furthermore,
we also prove that changing both the thickness of inner core and outermost
layer can lead to invariant resonance frequency while the threshold number of
coupled excitons to fulfill the criteria for strong coupling remains ~ 4 in
various sets of tunable HMM cavities.
