Perfectly-matched-layer method for optical modes in dielectric cavities

The optical resonance problem is similar to but different from the time-steady Schrödinger equation to the point that eigenfunctions in resonance problems are exponentially growing. We introduce the perfectly-matched-layer method and the complex stretching technique to transform eigenfunctions from exponential growth to exponential decay. Accordingly, we construct a Hamiltonian operator to calculate eigenstates of optical resonance systems. We successfully apply our method to calculate the eigenvalues for whispering-gallery modes and the results perfectly agree with existing theory that is developed only for regularly shaped cavities. We also apply the method to investigate the mode evolution near exceptional points - a special phenomenon that only happens in non-Hermitian systems. The presenting method is applicable to optical resonance systems with arbitrary dielectric distributions.