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Electromagnetic Scattering by a Cluster of Hybrid Dielectric-Plasmonic Dimers

Xinlin Cao, Ahcene Ghandriche, Mourad Sini

TL;DR

The paper derives a rigorous Foldy–Lax reduction for electromagnetic waves scattered by a large cluster of hybrid dielectric-plasmonic dimers, each formed by a high-contrast dielectric particle and a plasmonic particle at a subwavelength gap. By scaling intra- and inter-dimer spacing and tuning frequency near resonances, the scattered field and far-field admit expansions in four moments per dimer, governed by a finite, explicit 12N×12N linear system that is invertible under quantified smallness conditions; the leading-order dynamics reduce to co-located electric and magnetic dipoles with a 6×6 polarizability per dimer. This discrete model enables fast forward simulations, supports inverse design, and provides a foundation for effective-medium descriptions that can yield double negativity, bi-anisotropy, and hyperbolic regimes. The results illuminate parameter windows where clusters of hybrid dimers act as metamaterials with engineered electric/magnetic responses and magneto-electric coupling, informing both theory and applications in metamaterial design.

Abstract

We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in $\mathbb{R}^3$. Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose size $a$ is small compared to the wavelength, with intra-dimer and inter-dimer distances scaling like $a^{t_1}$ and $a^{t_2}$, and the frequency is tuned near suitable electric and magnetic resonances of the associated Newtonian and magnetization operators on the reference shapes. Under these geometric, contrast and spectral assumptions, we derive a Foldy--Lax type approximation for the Maxwell system. We show that the scattered field and its far field admit asymptotic expansions in terms of four moments attached to each dimer, which solve an explicit finite-dimensional linear system. We prove invertibility of this system under quantitative smallness conditions on the contrast and the dimer density, and we obtain error estimates uniform in the number of dimers. By extracting the dominant components, we further show that each hybrid dimer behaves, at leading order, as a co-located electric and magnetic dipole driven by the local fields, and we identify the corresponding $6\times 6$ polarizability matrix. This provides a discrete model for clusters of hybrid dimers that is suitable for fast forward simulations, inverse schemes, and as input for effective-medium descriptions. In particular, it suggests parameter regimes where clusters of hybrid dimers can generate (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.

Electromagnetic Scattering by a Cluster of Hybrid Dielectric-Plasmonic Dimers

TL;DR

The paper derives a rigorous Foldy–Lax reduction for electromagnetic waves scattered by a large cluster of hybrid dielectric-plasmonic dimers, each formed by a high-contrast dielectric particle and a plasmonic particle at a subwavelength gap. By scaling intra- and inter-dimer spacing and tuning frequency near resonances, the scattered field and far-field admit expansions in four moments per dimer, governed by a finite, explicit 12N×12N linear system that is invertible under quantified smallness conditions; the leading-order dynamics reduce to co-located electric and magnetic dipoles with a 6×6 polarizability per dimer. This discrete model enables fast forward simulations, supports inverse design, and provides a foundation for effective-medium descriptions that can yield double negativity, bi-anisotropy, and hyperbolic regimes. The results illuminate parameter windows where clusters of hybrid dimers act as metamaterials with engineered electric/magnetic responses and magneto-electric coupling, informing both theory and applications in metamaterial design.

Abstract

We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in . Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose size is small compared to the wavelength, with intra-dimer and inter-dimer distances scaling like and , and the frequency is tuned near suitable electric and magnetic resonances of the associated Newtonian and magnetization operators on the reference shapes. Under these geometric, contrast and spectral assumptions, we derive a Foldy--Lax type approximation for the Maxwell system. We show that the scattered field and its far field admit asymptotic expansions in terms of four moments attached to each dimer, which solve an explicit finite-dimensional linear system. We prove invertibility of this system under quantitative smallness conditions on the contrast and the dimer density, and we obtain error estimates uniform in the number of dimers. By extracting the dominant components, we further show that each hybrid dimer behaves, at leading order, as a co-located electric and magnetic dipole driven by the local fields, and we identify the corresponding polarizability matrix. This provides a discrete model for clusters of hybrid dimers that is suitable for fast forward simulations, inverse schemes, and as input for effective-medium descriptions. In particular, it suggests parameter regimes where clusters of hybrid dimers can generate (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.
Paper Structure (13 sections, 6 theorems, 92 equations)

This paper contains 13 sections, 6 theorems, 92 equations.

Table of Contents

  1. Introduction and the main results
  2. Background and motivation
  3. Main results
  4. Discussion of the results and possible applications
  5. Some preliminaries and a-priori estimates
  6. Decomposition of $\mathbb{L}^2$.
  7. Lippmann-Schwinger integral formulation of the solutions.
  8. A-priori estimates.
  9. Linear algebraic system and its invertibility
  10. Proof of the main results
  11. Proof of Theorem \ref{['main-1']}
  12. Proof of Corollary \ref{['corollary-main']}
  13. Appendix: Structure of the effective susceptibilities

Key Result

Theorem 1.4

Let the Assumptions ASass on the electromagnetic scattering problem model-m, which is generated by a cluster of dimers $D \, := \, \overset{\aleph}{\underset{m=1}{\cup}} D_{m} \, = \, \overset{\aleph}{\underset{m=1}{\cup}} \left( D_{m_{1}} \cup D_{m_{2}} \right)$, be satisfied. For $0<t_2\leq t_1< then the scattered wave admits the following expansion and the corresponding far field admits the

Theorems & Definitions (16)

  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 2.1
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 6 more