Two-component wave formalism in spherical open systems

We study wave evolution in open dielectric spheres by expanding the wave field and its conjugate momentum-the two components-in terms of relevant quasi-normal modes (QNMs), which are complete under appropriate conditions. We first establish a novel outgoing boundary condition at the surface of a sphere for waves emanating from its interior. A proper definition of inner product for two-Component outgoing wavefunctions, involving only the waves inside the sphere and a surface term, can then be defined in general. The orthogonality relation of QNMs and hence a unique expansion in terms of the QNM basis are found, which can be applied to solve for the evolution of waves inside open dielectric cavities. Furthermore, a time-independent perturbation for QNMs can also be developed.