Quantitative Analysis of Second-Harmonic Generation in Plasmonic Nanowires: Symmetry Breaking and a Topological Defect

This paper develops a quantitative theory of second-harmonic generation (SHG) in hollow plasmonic nanowires, distinguishing geometric symmetry breaking from cavity effects. We consider a centrosymmetric material with a concentric circular cavity and a slightly perturbed outer boundary. Relative to the solid nanowire in~\cite{liu2026quantitative}, the cavity introduces a topological defect by changing the cross-section from a disk to a doubly connected domain, whereas the outer-boundary perturbation breaks geometric symmetry. Local inversion-symmetry breaking at the interfaces induces nonlinear surface polarization. Using layer potentials and boundary-perturbation asymptotics, we derive explicit leading- and first-order formulas for the fundamental and second-harmonic fields. For a circular outer boundary, the leading-order second-harmonic field is purely quadrupolar; its amplitude differs from the solid-wire case by $O(γ^2)$ as $γ=r_i/r_e\to0$ and decays as $O(r_e-r_i)$ in the nonresonant thin-shell limit. For perturbations proportional to $\cos(nθ)$, $n\ge3$, the first-order second-harmonic correction contains only modes $n-2$, $n$, and $n+2$, yielding dipolar, quadrupolar, or higher-order radiation for $n=3$, $n=4$, or $n\ge5$, respectively, provided the lowest-order modal coefficient is nonzero. We derive plasmonic resonance conditions at both frequencies and estimate the singular orders of single and simultaneous resonances, including possible higher-order singularities from coincident resonance factors. Thus geometric symmetry determines the multipolar modes, whereas the cavity radius controls their amplitudes and resonant enhancement.