In the previous sections, we focused on the QED Hamiltonians under the long wavelength approximation and the single photonic mode approximation. However, these approximations are not adequate to accurately describe experiments conducted with Fabry-Pérot cavities. In this manner, we must start with the most general Hamiltonian in Eq. 35 and derive the convenient expressions for model Hamiltonians that can accurately describe many molecules interacting with many cavity modes. Specifically, many modes are considered with many molecules, and we partially relax the long wavelength approximation such that Â is no longer spatially invariant while the matter interactions are still approximated as dipoles. Such a Hamiltonian is necessary to describe many molecules coupled to a Fabry-Pérot cavity, depicted in Fig. .