# Paradox of multiple plasmonic resonances at light scattering by a cylinder of infinitesimal radius

## TL;DR

This paper investigates the paradoxical divergence in light scattering by an infinitesimally small cylinder with specific permittivity, clarifying the conditions under which the scattering cross section vanishes and revealing new effects at small sizes.

## Contribution

It resolves the paradox of divergent scattering cross section by refining the analysis to include linewidth and limit processes, showing the cross section vanishes as radius approaches zero.

## Key findings

- Divergence caused by resonance overlap at specific permittivity.
- Proper limit process removes the divergence, leading to zero cross section.
- Unusual size-dependent scattering effects at small but finite radii.

## Abstract

The paradox of the divergence of the resonant scattering cross section of a cylinder with the permittivity equals minus unity and vanishing radius (R) irradiated by a monochromatic electromagnetic wave is discussed. Within the framework of the exact solution of the Maxwell equations, the divergence at the specified conditions is caused by the overlap of all but one multipolar resonances. It is shown that the paradox is caused by the too straightforward analysis of the expression for the cross section, which has a singularity at this point. To resolve the singularity, one must, first, generalize the problem formulation, taking into account the final linewidth of the incident wave, and then perform the correct sequence of limit transitions. The application of this approach gives rise to the vanishing cross section at the vanishing R. It ruins the expectations to employ such a cylinder as a superscatterer but simultaneously open a door to counterintuitive effects both in far and near field zones related to unusual size dependences of the scattered fields at small but finite R.