Remarkable Predictive Power of the Modified Long Wavelength Approximation

TL;DR

The paper demonstrates that the modified long-wavelength approximation (MLWA) can accurately predict electromagnetic properties of plasmonic nanoparticles beyond its traditional limits by exploiting a design freedom in the dynamic depolarization term, applicable to various metals and higher multipoles.

Contribution

It reveals a new flexibility in the MLWA functional form that extends its validity range, enabling accurate modeling of plasmonic nanoparticles for larger size parameters and higher multipole orders.

Findings

MLWA yields accurate results for size parameters up to and beyond 1.

MLWA effectively models higher order multipoles ($>1$).

The approach simplifies nanoparticle analysis by avoiding complex functions.

Abstract

The modified long-wavelength approximation (MLWA), a next order approximation beyond the Rayleigh limit, has been applied usually only to the dipole $\ell=1$ contribution and for the range of size parameters $x$ not exceeding $x\lesssim 1$ to estimate far- and near-field electromagnetic properties of plasmonic nanoparticles. Provided that the MLWA functional form for the $T$-matrix in a given channel $\ell$ is limited to the ratio $T\sim iR/(F+D-iR)$, where $F$ is the familiar size-independent Fr\"ohlich term and $R\sim {\cal O}(x^{2\ell+1})$ is a radiative reaction term, there is a one-parameter freedom in selecting the dynamic depolarization term $D\sim {\cal O}(x^2)$ which preserves the fundamental feature of the MLWA that its predictions coincide with those of the Mie theory up to the order ${\cal O}(x^2)$. By exploiting this untapped design freedom, we demonstrate on a number of different metals (Ag, Al, Au, Mg), and using real material data, that the MLWA may surprisingly yield very accurate results for plasmonic spheres both for (i) $x$ up to $\gtrsim 1$ and beyond, and (ii) higher order multipoles ($\ell>1$), essentially doubling its expected range of validity. Because the MLWA obviates the need of using spherical Bessel and Hankel functions and allows for an intuitive description of (nano)particle properties in terms of a driven damped harmonic oscillator parameters, a significantly improved analysis and understanding of nanoparticle scattering and near-field properties can be achieved.