Quantum mechanical-like approach with non-Hermitian effective Hamiltonians in spin-orbit coupled optical cavities

We present a comprehensive analytical model of resonant states in birefringent microcavities with permeable mirrors. We derive an effective, non-Hermitian photonic Hamiltonian describing cavity mode dispersion and modal lifetimes by applying the Green's function technique based on the Mittag-Leffler expansion with respect to the resonant states and the $k·p$ perturbation theory known from semiconductor physics. Using this formalism we obtained results which can be interpreted as effective cavity mode spin-orbit coupling. We use this method to derive the two-mode Hamiltonian to describe the properties of light in the cavity, which significantly reduces the computational effort and properly captures the polarization of the eigenmodes. This is done by introducing the necessary corrections to the Hamiltonian matrix and modifying the basis modes resulting from the coupling with other optical modes present in the system. This simplified Hamiltonian allows us to determine the positions of exceptional points in momentum space. These points are connected by Fermi arcs and appear due to non-Hermiticity and $\mathrm{PT}$ symmetry.