Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties

TL;DR

This work addresses how symmetry governs the eigenmodes and their multipole content in non-spherical dielectric resonators. It develops a group-theory–driven framework with two complementary methods, identifying which vector spherical harmonics participate in each mode and mapping mode types to specific multipole series across several symmetry groups. Key contributions include ready-made tables linking irreducible representations to multipole content for $D_{\infty h}$, $D_{\infty v}$, $D_{3h}$, $D_{4h}$, $O_h$, and $C_{4h}$, along with a demonstration of a quasi-BIC in a triangular prism arising from symmetry-enabled suppression of a main radiative channel. The approach enables symmetry-based design of photonic and microwave devices without full numerical simulations and broadens the applicability of multipole analysis beyond spherical resonators.

Abstract

The resonator is one of the main building blocks of a plethora of photonic and microwave devices from nanolasers to compact biosensors and magnetic resonance scanners. The symmetry of the resonators is tightly related to their mode structure and multipole content which determines the linear and non-linear response of the resonator. Here, we develop the algorithm for the classification of eigenmodes in resonators of the simplest shapes depending on their symmetry group. For each type of mode, we find its multipole content. As an illustrative example, we apply the developed formalism to the analysis of dielectric triangular prism and demonstrate the formation of high-Q resonances originated due to suppression of the scattering through the main multipole channel. The developed approach one to engineer, predict, and explain scattering phenomena and optical properties of resonators and meta-atoms basing only on their symmetry without the need for numerical simulations and it can be used for the design of new photonic and microwave devices.

Figures (8)

• Figure 1: Sketch showing the main idea of the paper. The resonators of different shapes can be classified according to their group symmetry and their eigenmodes are classified according to the irreducible representations of the resonator's symmetry group. Each irreducible representation can be characterized by a set of vector spherical harmonics defining the multipole content of each mode.
• Figure 2: (a) Examples of different resonators of the same symmetry group $D_{\infty h}$. (b) Transformation the sphere [$O(3)$ group] with dielectric susceptibility $\varepsilon$ to the resonator ($D_{\infty h}$ group) with dielectric susceptibility $\varepsilon + \Delta \varepsilon$.
• Figure 3: The classification of modes by representations and the multipole composition of eigenmode for cone ($D_{\infty v}$ group), cylinder ($D_{\infty h}$ group). For each mode type showed the dimension of irreducible representations and found specific series of VSHs $\mathbf{N}_{pml}$ and $\mathbf{M}_{pml}$. Index $l$ expressed in $k$, which is a positive integer.
• Figure 4: The classification of modes by representations and the multipole composition of each mode for cone ($D_{\infty v}$ group), cylinder ($D_{\infty h}$ group) and triangular prism ($D_{3h}$ group).
• Figure 5: The classification of modes by representations and the multipole composition of eigenmode for chiral resonator ($C_{4 h}$ group) and chiral resonator for slab ($C_{4}$ group). For each mode type showed the dimension of irreducible representations and found specific series of VSHs $\mathbf{N}_{pml}$ and $\mathbf{M}_{pml}$. Indices $m$ and $l$ expressed in $s$ and $k$, which are positive integer, respectively.