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

Bouchra Bensiali, Stefan Sauter

TL;DR

This work tackles the accurate computation of whispering gallery mode resonances in spherical, radially heterogeneous Helmholtz problems with piecewise smooth refractive index. It recasts resonances as a nonlinear eigenvalue problem via a modal determinant built from a pair of fundamental solutions on subdomains and a transmission interface, then solves det$(T^{m}_{+}(k))=0$ with a complex Newton method. A spectral method (ApproxFun) is employed to compute the required fundamental solutions and their $k$-derivatives in the variable-index case, enabling robust local convergence in the piecewise-constant setting (where roots are simple) and effective exploration of variable-index configurations. Numerical experiments verify convergence, describe the resonance structure (including quasi-resonances near the real axis) and demonstrate how index contrast and mode number $m$ influence the location and confinement of WGMs. The approach provides a scalable, high-accuracy framework for predicting WGMs in radially varying media and lays groundwork for extension to more interfaces and alternative NEP solvers.

Abstract

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-posedeness 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 derive a Newton method to solve the nonlinear modal equation with a proper scaling. We show the local convergence of the algorithm in the piecewise constant case by proving the simplicity of the roots. We confirm our approach through a series of numerical experiments in the piecewise constant and variable case.

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

TL;DR

This work tackles the accurate computation of whispering gallery mode resonances in spherical, radially heterogeneous Helmholtz problems with piecewise smooth refractive index. It recasts resonances as a nonlinear eigenvalue problem via a modal determinant built from a pair of fundamental solutions on subdomains and a transmission interface, then solves det with a complex Newton method. A spectral method (ApproxFun) is employed to compute the required fundamental solutions and their -derivatives in the variable-index case, enabling robust local convergence in the piecewise-constant setting (where roots are simple) and effective exploration of variable-index configurations. Numerical experiments verify convergence, describe the resonance structure (including quasi-resonances near the real axis) and demonstrate how index contrast and mode number influence the location and confinement of WGMs. The approach provides a scalable, high-accuracy framework for predicting WGMs in radially varying media and lays groundwork for extension to more interfaces and alternative NEP solvers.

Abstract

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-posedeness 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 derive a Newton method to solve the nonlinear modal equation with a proper scaling. We show the local convergence of the algorithm in the piecewise constant case by proving the simplicity of the roots. We confirm our approach through a series of numerical experiments in the piecewise constant and variable case.

Paper Structure

This paper contains 51 sections, 12 theorems, 130 equations, 15 figures, 23 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Polar coordinates and fundamental systems
  4. Well-posedness of ODEs for the fundamental system
  5. Quasi-resonances
  6. A Newton method for determining critical $k$'s
  7. Newton's algorithm
  8. Starting values
  9. Local convergence
  10. Piecewise constant case
  11. Convergence of the Newton method: simplicity of the roots
  12. Experiments
  13. Computation of zeros
  14. Verifying $\Im(k_{m,j})<\Im(k_{m,1})<0$ for $j>1$
  15. Verifying $\Im(k_{m,1})\underset{m\to\infty}{\longrightarrow} 0$
  16. ...and 36 more sections

Key Result

Proposition 1

All solutions (resonances) of problem eq:pdenep satisfy $k\in\mathbb{C}_{<0}$.

Figures (15)

  • Figure 1: Some features of resonances in the piecewise constant case for $n_2=1$ and $\xi=1$.
  • Figure 2: Comparison between $\frac{\partial_k \mathrm{det}_2(k)}{\mathrm{det}_2(k)}$ and $\frac{\partial_k \mathrm{det}_1(k)}{\mathrm{det}_1(k)}$ for $\xi=0.5$, $n_1=1.5$, $n_2=1$ and $m=10$. Here $k$ is varying over the complex rectangle with vertices $-50i$ and $100+400i$.
  • Figure 3: Plot of $\Im\bigl(\frac{\det(k_0)}{\partial_k \mathrm{det}(k_0)}\bigr)$ on the real axis for $\xi=0.5$, $n_1=1.5$, $n_2=1$ and $m=10$.
  • Figure 4: Errors between numerical solutions using ApproxFun and exact solutions for $\xi=0.5$, $n_1(r)=2$, $n_2(r)=1$, $m=40$ and $k=100$.
  • Figure 5: Plot of $n_1(r)$\ref{['eq:constant+bump']}.
  • ...and 10 more figures

Theorems & Definitions (27)

  • Proposition 1
  • Remark 1
  • Example 3
  • Lemma 4
  • proof
  • Example 5
  • Remark 2
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 17 more