Spectral theory of the Neumann-Poincaré operator on multi-layer structures and analysis of plasmon mode splitting

TL;DR

The paper develops a comprehensive boundary-integral and spectral-theoretic framework for the Neumann-Poincaré operator on general multi-layered structures, including metamaterial layers, to analyze plasmon resonances and mode splitting. By constructing a matrix-type NP operator and exploiting Calderón-type identities, it proves the spectrum lies in [-1/2,1/2] and shows plasmon modes correspond to real eigenvalues that split into even and odd families due to symmetry. For multi-layered confocal ellipses, explicit Fourier-based representations yield exact GPM matrices and determinant formulas, enabling closed-form or numeric computation of plasmon modes and their asymptotic profiles in concentric-disk and thin-layer limits. The work provides an algebraic design principle linking material truncation and symmetry breaking to controlled plasmon resonances, with numerical evidence illustrating mode splitting and surface localization. This framework lays groundwork for designing layered metamaterials with customized resonant behavior across multiple operating frequencies.

Abstract

In this paper, we develop a general mathematical framework for analyzing electostatics within multi-layered metamaterial structures. The multi-layered structure can be designed by nesting complementary negative and regular materials together, and it can be easily achieved by truncating bulk metallic material in a specific configuration. Using layer potentials and symmetrization techniques, we establish the perturbation formula in terms of Neumann-Poincaré (NP) operator for general multi-layered medium, and obtain the spectral properties of the NP operator, which demonstrates that the number of plasmon modes increases with the number of layers. Based on Fourier series, we present an exact matrix representation of the NP operator in an apparently unsymmetrical structure, exemplified by multi-layered confocal ellipses. By highly intricate and delicate analysis, we establish a handy algebraic framework for studying the splitting of the plasmon modes within multi-layered structures. Moreover, the asymptotic profiles of the plasmon modes are also obtained. This framework helps reveal the effects of material truncation and rotational symmetry breaking on the splitting of the plasmon modes, thereby inducing desired resonances and enabling the realization of customized applications.

Figures (7)

• Figure 4.1: Schematic illustration of an $N$-layer confocal ellipse.
• Figure 7.2: Graph on the left shows the values of $f^+_{15}(\lambda)$ and plot on the right shows the values of $f^-_{15}(\lambda)$, respectively, in the setup \ref{['eq:str01']} with $n=1$.
• Figure 7.3: Graph on the left shows the values of $f^+_{16}(\lambda)$ and plot on the right shows the values of $f^-_{16}(\lambda)$, respectively, in the setup \ref{['eq:str02']} with $n=2$.
• Figure 7.4: The value of $|\lambda_{+}-\lambda_{-}|$ in the setup \ref{['eq:str02']} with $\xi_1 = L*N$, $s = 0.8$, $N=17$ and $n=1$.
• Figure 7.5: The real part of the perturbed electric fields $u^{\pm}_1,u^{\pm}_2,u^{\pm}_3,u^{\pm}_4$ with $n=6$ for the four-layer confocal ellipses designed by \ref{['SMED']} represented as solid black lines. Each plot corresponds to one of the eight plasmon resonance frequencies. The upper plot displays the even, the lower plot the odd plasmon modes.

Theorems & Definitions (34)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Remark 3.1
• Proposition 3.1
• Remark 3.2
• proof : Proof of Proposition \ref{['NPspectral']}
• Theorem 4.1
• proof : Proof
• Theorem 4.2