# Point dipole and quadrupole scattering approximation to collectively responding resonator systems

## TL;DR

This paper develops a comprehensive theoretical framework for modeling collective electromagnetic responses of resonator systems, incorporating electric dipole, magnetic dipole, and electric quadrupole interactions, applicable across various frequency regimes.

## Contribution

It introduces a detailed formalism including electric quadrupole effects for systems of coupled resonators, extending previous dipole-only models to more accurately describe complex interactions.

## Key findings

- Derived equations for quadrupole contributions in resonator interactions
- Validated the model with examples of electric dipole pairs
- Showed the importance of quadrupole effects in collective responses

## Abstract

We develop a theoretical formalism for collectively responding point scatterers where the radiating electromagnetic fields from each emitter are considered in the electric dipole, magnetic dipole, and electric quadrupole approximation. The contributions of the electric quadrupole moment to electromagnetically-mediated interactions between the scatterers are derived in detail for a system where each scatterer represents a linear $RLC$ circuit resonator, representing common metamaterial resonators in radiofrequency, microwave, and optical regimes. The resulting theory includes a closed set of equations for an ensemble of discrete resonators that are radiatively coupled to each other by propagating electromagnetic fields, incorporating potentially strong interactions and recurrent scattering processes. The effective model is illustrated and tested for examples of pairs of interacting point electric dipoles, where each pair can be qualitatively replaced by a model point emitter with different multipole radiation moments.