Superoscillations and Physical Applications

TL;DR

Superoscillations describe local oscillations that exceed the Fourier bandwidth of a signal. The chapter surveys physical implementations and applications, emphasizing electromagnetic waves, and connects the concepts to the real part and the imaginary part of the quantum weak value to define local wavenumber and growth rate. It reviews influential experiments—random waves, hot spots, super-resolution spectroscopy, PSF engineering, and noise-robust recovery—and surveys systematic construction methods for super-PSFs, providing practical guidance on phase and amplitude control via spatial light modulators and related hardware. The discussion highlights how superoscillatory and supergrowing fields enable resolution enhancements beyond conventional diffraction limits, while acknowledging sidelobe energy and noise challenges, and points to promising future directions such as superradar and generalized super-phenomena in quantum and other wave systems.

Abstract

This book chapter gives a selective review of physical implementations and applications of superoscillations and associated phenomena. We introduce the field by reviewing simple examples of superoscillations and showing how their existence naturally follows from the real part of the quantum mechanical weak value, which the parallel phenomena of supergrowth naturally follows from the imaginary part. Focusing on electromagnetic applications, we review the topics of superoscillation and supergrowth in speckle, creating superoscillating hot spots with patterned filters, superspectroscopic discrimination of two molecules, noise mitigation and the engineering of super behavior in point spread functions for the purpose of optical superresolution. We also cover a variety of different methods for creating superoscillatory and supergrowing functions, reviewing both mathematical and physical ways to create this class of functions, and beyond. Promising directions for future research, including superoscillations in other wave phenomena, super radar, and generalized super-phenomena in quantum physics, are outlined.

Figures (14)

• Figure 1: The local wavenumber $k$, Eq. (\ref{['localk']}) and the local growth rate $\kappa$, Eq. (\ref{['localkappa']}) are plotted versus $x$. The band limits $k = \pm 1$ are shown as dashed lines. Here we choose $N=20, a=6$. Regions of superoscillation and supergrowth are when the functions stray outside the dashed boundaries, and are generally different. In this example the point of maximum superoscillation at $x=0$ has zero supergrowth.
• Figure 2: Visualization of the joint probability distribution presented in Eq. (\ref{['Dennis1']}). Fast phase variation occurs in regions of small irradiance. Taken from dennis2008superoscillation.
• Figure 3: (a), (b) Two different speckles realizations. The speckle is created by passing a focused laser through ground glass. The laser beam size on the ground glass controls the observed speckle structure. (c),(d) The observed supergowth in the speckle irradiance of panel (a), (b) where the growth parameter $\Gamma>1$ indicates local amplitude variation exceeding that predicted by the bandlimit. Taken from Ref. viteri2024supergrowth.
• Figure 4: Observed photon carpet diffraction patterns at different distances from the hole array. The middle two panel insets show local hotspots. The field-of-view is $\sim$20-$\mu$m by 20-$\mu$m. Taken from Ref. huang2007optical.
• Figure 5: Three examples of superoscillation combinations. [(a)-(c)] show the predicted and measured result field over several oscillations, [(d)-(f)] show the superoscillation region comparing the highest frequency in the sum against the superoscillation frequency and [(g)-(h)] show the local frequencies of the largest wave and superoscillation. Reproduced from Ref. mccaul2023superoscillations. Published by permission from APS.