Quantitative symmetry-breaking and nonlinear harmonic generation in plasmonics

TL;DR

This paper presents a rigorous mathematical framework for understanding how symmetry breaking influences nonlinear harmonic generation in plasmonic nanostructures, especially focusing on second harmonic generation.

Contribution

It introduces a quantitative theory linking symmetry properties to nonlinear harmonic responses, extending to higher-order harmonics and considering shape, size, and defects.

Findings

Quantifies second harmonic generation using group theory and multipolar analysis.

Shows nonlinear optical efficiency depends on symmetry, shape, size, and defects.

Provides a framework applicable to various plasmonic structures and higher harmonics.

Abstract

We develop a quantitative mathematical theory that offers new perspectives on nonlinear harmonic generation in plasmonic structures arising from symmetry breaking. Focusing on second harmonic generation--the most fundamental process and the most extensively studied owing to its practical significance--we establish a theoretical framework that can be readily extended to higher-order harmonics. We investigate the plasmonic system in the static regime using a columnar nanowire with \(n\)-fold rotational symmetry (\(n \in \mathbb{N}\)) and construct a phenomenological model in which the second harmonic response originates from nonlinear sources confined to a selvedge region near the surface. By introducing a notion of symmetry degree grounded in group theory, we precisely quantify the second harmonic generation in terms of multipolar contributions. Our theory complements existing physical descriptions of this practically important phenomenon and provides a rigorous account of how nonlinear optical efficiency depends on shape, size, symmetry, and defects in plasmonic structures.