Ultrafast Dipolar Electrostatic Modeling of Plasmonic Nanoparticles with Arbitrary Geometry

TL;DR

The paper tackles fast and accurate modeling of localized surface plasmon resonances (LSPR) for metallic nanoparticles with arbitrary geometry by projecting the electrostatic boundary problem onto a Cartesian dipole subspace, yielding a compact 3×3 geometry matrix M. It separates geometry-dependent quantities from material dispersion by deriving geometry-only spectral descriptors ε_n from projected Neumann–Poincaré eigenvalues κ_n and reconstructing the dipolar polarizability via V_n^eff, with retardation included through MLWA to extend accuracy into the weakly retarded regime. The framework enables ultrafast, parametric spectral evaluations once geometry is preprocessed, and it is validated against full BEM solutions for near-field and extinction spectra across representative shapes, coatings, and environmental changes. This approach provides a physically transparent and computationally efficient tool for rapid design and optimization of plasmonic nanoparticles in sensing and nano-optics, while clearly delineating its regime of validity and potential extensions to layered environments.

Abstract

Accurate and fast calculations of localized surface plasmon resonances (LSPR) in metallic nanoparticles is essential for applications in sensing, nano-optics, and energy harvesting. Although full-wave numerical techniques such as the boundary element method (BEM) or the discrete dipole approximation (DDA) provide high accuracy, their computational cost often hinders rapid parametric studies. Here it is presented an ultrafast method that avoids solving large eigenproblems. Instead, only the dipolar component of the induced surface charge density \((σ_{dipolar})\) is retained through a expansion into Cartesion dipole basis, yielding a compact $3\times3$ geometric formulation that avoids full boundary-integral solves. The spectral response is obtained in a similar way, by projecting the Neumann--Poincaré surface operator onto the dipole subspace and evaluating a Rayleigh quotient, giving geometry-only eigenvalues again without an $N\times N$ eigenproblem. A major advantage of this method is that all geometry-dependent quantities are computed once per nanoparticle, while material dispersion and environmental changes enter only through simple algebraic expressions for the polarizability, enabling rapid evaluation across wavelengths. Retardation effects are incorporated through the modified long-wavelength approximation (MLWA), extending accuracy into the weakly retarded regime. The resulting framework provides a valuable tool for fast modelling and optimization of plasmonic nanoparticles at a significant lesser computational cost than BEM, DDA, and other standard tools.

Figures (9)

• Figure 1: Normalized surface charge density comparison between the geometric-moment approach and the BEM for three nanoparticle geometries: $\mathbf{a}$ Nanosphere (r = 10 nm) with the external field along the z-axis; $\mathbf{b}$ Nanorod (w = 10 nm, l = 60 nm) with the external field applied along the x-axis; $\mathbf{c}$ nanodisk (r = 10 nm, h = 2 nm) with the field along the x-axis.
• Figure 2: Near-field magnitude (color map) and vector field (arrows) computed in the xy-plane for a gold nanobipyramid ($\mathbf{a}$) and a gold nanoring immersed in water ($\mathbf{b}$), under linearly polarized excitation with the incident electric field oriented along the x-axis. The fields are reconstructed from $\sigma(s)$ and evaluated within the quasistatic approximation. The arrows indicate the local field direction, while the color scale represents the field magnitude $|\mathbf{E}|$ in arbitrary units.
• Figure 3: Geometry-dependent dipolar K eigenvalues $\kappa_n$ as a function of aspect ratio for representative nanoparticle shapes. $\mathbf{a}$ Prolate ellipsoids: comparison between analytical eigenvalues obtained from the classical depolarization factors (solid lines) and eigenvalues calculated from the dipole-subspace K projection (symbols). $\mathbf{b}$ Nanorods with spherical caps: calculated dipolar eigenvalues obtained from the K projection, $\mathbf{c}$ Geometric amplification factor $g_n$ as a function of the dipolar eigenvalue $\kappa_n$. The nonlinear increase of $g_n$ as $\kappa_n \to 1/2$ highlights the strong enhancement of the optical response for geometries supporting large dipolar eigenvalues. Representative values for a sphere ($\kappa=1/6$), nanocube ($\kappa\approx0.21$), nanobipyramids, and nanorods of aspect ratios of 2 and 10 are indicated. $\mathbf{d}$ Comparison between the real permittivity components (top panel), and their derivatives (bottom panel) for Au and Ag. The permittivity data is obtained from the Drude model fitted to experimental data from Johnson and Christy.
• Figure 4: Extinction spectra of gold nanorods in water ($\varepsilon_d = 1.77$) for excitation polarized along the long axis for a $10\times10\times60$ nm nanorod with this method ($\mathbf{a}$) and with BEM ($\mathbf{b}$). Similarly for $20\times20\times160$ nm nanorod with this method ($\mathbf{c}$) and with BEM ($\mathbf{d}$).
• Figure 5: $\mathbf{a}$ Integrated spatial decay profile of the electric field magnitude $|\mathbf{E}|$ with respect to distance to the surface for a gold nanorod ($20\times20\times160$ nm), extracted from numerical simulations and fitted with the power-law decay [Eq. (\ref{['eq. fit']})]. $\mathbf{b}$ Top: effective refractive index $n_{\mathrm{eff}}$ sensed by the gold nanorods with aspect ratios between 2 and 8, at constant width of 20 nm, calculated using Eq. (\ref{['eq:neff_powerlaw']}). Bottom: resulting LSPR wavelength shift $\Delta\lambda$ as a function of aspect ratio, for several PVP coating thicknesses.