Single vs double anti-crossing in the strong coupling between surface plasmons and molecular excitons

Strong coupling between surface plasmons and molecular excitons may lead to the formation of new hybrid states-polaritons-that are part light and part matter in character. A key signature of this strong coupling is an anti-crossing of the exciton and surface plasmon modes on a dispersion diagram. In a recent report on strong coupling between the plasmon modes of a small silver nano-rod and a molecular dye, it was shown that when the oscillator strength of the exciton is large enough, an additional anti-crossing feature may arise in the spectral region where the real part of the permittivity of the excitonic material is zero. However, the physics behind this double anti-crossing feature is still unclear. Here, we make use of extensive transfer matrix simulations to explore this phenomenon. We show that for low oscillator strengths of the excitonic resonance, there is a single anti-crossing arising from strong coupling between the surface plasmon and the excitonic resonance, which is associated with the formation of upper and lower plasmon-exciton polaritons. As the oscillator strength is increased, we find that a new mode emerges between these upper and lower polariton states and show that this new mode is an excitonic surface mode. Our study also features an exploration of the role played by the orientation of the excitonic dipole moment and the relationship between the modes we observe and the transverse and longitudinal resonances associated with the excitonic response. We also investigate why this type of double splitting is rarely observed in experiments.