Supergrowth and sub-wavelength object imaging

TL;DR

This paper explores supergrowth, a phenomenon where local amplitude growth exceeds the maximum wavenumber, and demonstrates its potential for sub-wavelength imaging as an alternative to superoscillation-based methods.

Contribution

It introduces supergrowth as a new concept, compares it with superoscillation, and proposes methods for sub-wavelength imaging using supergrowing regions.

Findings

Supergrowing regions have exponentially larger intensities than superoscillating regions.

Supergrowth can be used for sub-wavelength imaging with potentially better experimental feasibility.

Quantitative analysis of superoscillating and supergrowing regions enhances understanding of their properties.

Abstract

We further develop the concept of supergrowth [Jordan, Quantum Stud.: Math. Found. $\textbf{7}$, 285-292 (2020)], a phenomenon complementary to superoscillation, defined as the local amplitude growth rate of a function being higher than its largest wavenumber. We identify the superoscillating and supergrowing regions of a canonical oscillatory function and find the maximum values of local growth rate and wavenumber. Next, we provide a quantitative comparison of lengths and relevant intensities between the superoscillating and the supergrowing regions of a canonical oscillatory function. Our analysis shows that the supergrowing regions contain intensities that are exponentially larger in terms of the highest local wavenumber compared to the superoscillating regions. Finally, we prescribe methods to reconstruct a sub-wavelength object from the imaging data using both superoscillatory and supergrowing point spread functions. Our investigation provides an experimentally preferable alternative to the superoscillation based superresolution schemes and is relevant to cutting-edge research in far-field sub-wavelength imaging.