Computation of whispering gallery modes for spherical symmetric, heterogeneous Helmholtz problems with piecewise smooth refractive index

In this paper, we develop a numerical method for the computation of (quasi-)resonances in spherical symmetric, heterogeneous Helmholtz problems with piecewise smooth refractive index. Our focus lies in resonances very close to the real axis, which characterize the so-called whispering gallery modes. Our method involves a modal equation incorporating fundamental solutions to decoupled problems, extending the known modal equation to the case of piecewise smooth coefficients. We first establish the well-posedness of the fundamental system, then we formulate the problem of resonances as a nonlinear eigenvalue problem, whose determinant will be the modal equation in the piecewise smooth case. In combination with the numerical approximation of the fundamental solutions using a spectral method, we propose a Newton method to solve the nonlinear modal equation with a proper scaling. We prove the local convergence of the algorithm in the piecewise constant case by showing the simplicity of the roots. We confirm our approach through a series of numerical experiments in the piecewise constant as well as in the variable case.