Optimal plasmonic multipole resonances of a sphere in lossy media

TL;DR

This paper establishes fundamental upper bounds for plasmonic multipole resonances of a sphere in lossy media, providing explicit solutions and optimal material conditions, with applications to metal nanospheres.

Contribution

It develops a specialized Mie theory and optimization framework to determine the maximum absorption and scattering of dielectric spheres in lossy media, including explicit asymptotic expressions.

Findings

Optimal passive material approximates the complex conjugate of pole positions.

Explicit asymptotic formulas for dipole and quadrupole resonances.

Numerical validation with gold, silver, and aluminum nanospheres.

Abstract

Fundamental upper bounds are given for the plasmonic multipole absorption and scattering of a rotationally invariant dielectric sphere embedded in a lossy surrounding medium. A specialized Mie theory is developed for this purpose and when combined with the corresponding generalized optical theorem, an optimization problem is obtained which is explicitly solved by straightforward analysis. In particular, the absorption cross section is a concave quadratic form in the related Mie (scattering) parameters and the convex scattering cross section can be maximized by using a Lagrange multiplier constraining the absorption to be non-negative. For the homogeneous sphere, the Weierstrass preparation theorem is used to establish the existence and the uniqueness of the plasmonic singularities and explicit asymptotic expressions are given for the dipole and the quadrupole. It is shown that the optimal passive material for multipole absorption and scattering of a small homogeneous dielectric sphere embedded in a dispersive medium is given approximately as the complex conjugate and the real part of the corresponding pole positions, respectively. Numerical examples are given to illustrate the theory, including a comparison with the plasmonic dipole and quadrupole resonances obtained in gold, silver and aluminum nanospheres based on some specific Brendel-Bormann (BB) dielectric models for these metals. % of gold, silver and aluminum. Based on these BB-models, it is interesting to note that the metal spheres can be tuned to optimal absorption at a particular size at a particular frequency.