# Three-dimensional-subwavelength field localization, time reversal of sources, and infinite-asymptotic degeneracy in spherical structures

## TL;DR

This paper introduces a spherical resonant structure that achieves deep subwavelength 3D field localization and time reversal of sources, overcoming traditional resolution limits and revealing infinite eigenvalue degeneracy with significant optical effects.

## Contribution

The work demonstrates a novel spherical layer with resonant permittivity that enables continuous time reversal and subwavelength localization in three dimensions, and uncovers infinite degeneracy in eigenvalues leading to enhanced optical phenomena.

## Key findings

- Achieved subwavelength focal spot of 0.4λ lateral and 0.58λ axial resolution.
- Generated continuous time-reversed signals for multipole sources using spherical structures.
- Discovered infinite eigenvalue degeneracy causing strong scattering and emission enhancement.

## Abstract

High-resolution field localization in three dimensions is one of the main challenges in optics and has immense importance in fields such as chemistry, biology, and medicine. In order to generate the time reversed signal of a monochromatic source \emph{discrete} sources that are modulated according to the wave amplitude on a spherical envelope are required, rendering it applicable only in acoustics. Here we approach these challenges by introducing a spherical layer with a resonant permittivity, which naturally generates the spatially \emph{continuous} time-reversed signal of an atomic and molecular multipole transition at the origin. We start by utilizing a spherical layer with a resonant TM $l=1$ permittivity situated in a uniform medium to generate a free-space-subwavelength focal spot at the origin. We remove the degeneracy of the eigenfunctions of the composite medium by situating a point current source (or polarization) directed parallel to the spherical layer, which generates a focal spot at the origin \emph{independently} of its location. The free-space focal spot has a full width at half maximum of $0.4\lambda$ in the lateral axes and $0.58\lambda$ in the axial axis, which is tighter by a factor of $\sqrt{2}$ in each dimension in excitation-collection mode, overcoming the $\lambda/2$ far-field resolution limit in three dimensions. We then explore two directions to localize electric field with deep-subwavelength resolution in three dimensions using this setup. In addition, we show that spherical structures exhibit a new type of degeneracy in which an infinite number of eigenvalues asymptotically coalesce. This high degeneracy results in a variety of optical phenomena such as strong scattering and enhancement of absorption and emission from an atom or molecule by orders of magnitude compared with a standard resonance.