Mixed Fields Formulation for Electromagnetic Waves Confined in Dielectric Rings

TL;DR

This work presents a robust, easy-to-implement 2D cylindrical finite-difference solver for electromagnetic waves confined in dielectric rings, formulating three Maxwell-based variants (E-field, H-field, and a mixed E-H approach) and reducing Maxwell's equations to a generalized eigenproblem for the propagation constant $\beta$ under azimuthal dependence $e^{-jm\phi}$. A two-step azimuthal-mode determination algorithm combines an approximate $m_{\rm approx}$ with a refinement using the computed $\beta$ to accurately recover resonant modes, while a direct approach provides an alternative augmented-system route. Validation against COMSOL Multiphysics and other commercial tools across a buried ring, a ring on a substrate, and a torus shows typical effective-index differences below $0.3\%$, confirming accuracy in field confinement, curvature effects, and dispersion. The method is demonstrated on an optical-frequency problem with a high-quality microresonator, predicting a free spectral range of $99.6$ GHz and a loaded quality factor of $1.6\times10^6$, in agreement with experimental measurements, highlighting its practical utility for designing integrated photonic devices and optical frequency comb systems.

Abstract

We present an easy-to-implement numerical method for analyzing electromagnetic wave propagation in dielectric rings. Our approach employs a finite-difference-based solver in cylindrical coordinates, solving a mixed electric-magnetic field formulation to accurately enforce boundary conditions and compute resonant modes. The method avoids geometric transformations; instead, it directly discretizes the Helmholtz wave equation in cylindrical coordinates and solves the resulting generalized eigenvalue problem. We validate our model against commercial solvers for various structures, including a Si3N4 ring embedded in SiO2, a ring on a thin-film-coated substrate, and a torus, achieving agreement in effective refractive indices within 0.3%. The formulation accurately captures field confinement, curvature effects, and dispersion, enabling precise determination of propagation constants and mode profiles. As an application, we model optical frequency comb generation in a high-Q microresonator, predicting a free spectral range of 99.6 GHz and a loaded quality factor of 1.6 million, corroborated by experimental measurements.

Figures (8)

• Figure 1: A dielectric ring with a width of $w_r$, a height of $h_r$, and a central radius of $R_c$ on a substrate: (a) three-dimensional and (b) two-dimensional views. $R_i$ and $R_o$ are the inner and outer radii of the ring, i.e., $R_i = R_c -w_r/2 = R_o - w_r$. By assuming the material properties are uniform axially around the $z$-axis, the problem can be solved in two dimensions using differential vector operators in the cylindrical coordinate system.
• Figure 2: Magnitude of the electric field's $\rho$ (top), $\phi$ (middle), and $z$ (bottom) components for the first resonant mode of the electromagnetic waves computed with our solver (left) and COMSOL (right) in a Si$_3$N$_4$ ring with a central radius, width, and height of 23 $\mu$m, 1.5 $\mu$m, and 0.7 $\mu$m.The white dashed lines outline the boundaries of the ring, giving insight into how the electromagnetic fields are confined but not completely symmetric.
• Figure 3: (Left) Real and (right) imaginary parts of the electric field's $\rho$ (top row), $\phi$ (middle row), and $z$ (bottom row) components on the $xy$-plane at $z=h_r/2$. The white dashed lines outline the inner and outer boundaries of the ring.
• Figure 4: Effective index vs. central radius for the first resonant mode.
• Figure 5: (a) 3D and (b) 2D views of the geometry studied in the second example: a Si$_3$N$_4$ ring grown on a Si$_3$N$_4$ thin film-coated glass substrate.