Perfectly matched layer method for optical modes in dielectric cavities

The optical resonance problem is similar to but different from time-steady Schr{o}dinger equation. One big challenge is that the eigenfunctions in resonance problem is exponentially growing. We give physical explanation to this boundary condition and introduce perfectly matched layer (PML) method to transform eigenfunctions from exponential-growth to exponential-decay. Based on the complex stretching technique, we construct a non-Hermitian Hamiltonian for the optical resonance problem. We successfully validate the effectiveness of the Hamiltonian by calculate its eigenvalues in the circular cavity and compare with the analytical results. We also use the proposed Hamiltonian to investigate the mode evolution around exceptional points in the quad-cosine cavity.