Roadmap on superoscillations

TLDR

Superoscillations are band‑limited functions that can oscillate faster than their highest Fourier component over long intervals, a phenomenon first noted in quantum theory and later in radar and optics, with mathematical analysis showing they become extremely small where they superoscillate, impacting information theory and enabling practical applications in optical vortices, sub‑wavelength microscopy, and nanoscience. This Roadmap surveys these areas, providing background, current research, and anticipating future developments.

Abstract

Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding—still being explored—recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments.