# Space-time superoscillations

## TL;DR

This paper introduces space-time superoscillations, a new phenomenon where light pulses can focus beyond classical limits in both space and time.

## Contribution

The novel concept of simultaneous spatial and temporal superoscillations is demonstrated using structured light pulses.

## Key findings

- Space-time superoscillations occur in band-limited supertoroidal light pulses.
- The phenomenon enables extreme subwavelength and ultrafast focusing of electromagnetic waves.
- This opens new possibilities for ultrafast metrology and light-matter interactions.

## Abstract

Superoscillation (SO) refers to the phenomenon in which a wavefield locally oscillates at a rate exceeding its highest spatial or temporal Fourier component. SO has enabled light to be focused into arbitrarily small hotspots, forming the basis of superresolution imaging and metrology far beyond the Abbe-Rayleigh diffraction limit. Here we show that spatial and temporal superoscillations can occur simultaneously at the same point in space-time, a phenomenon we term space-time superoscillation (STSO). We demonstrate STSOs in a band-limited version of supertoroidal light pulses, a recently introduced family of space–time nonseparable finite-energy solutions of Maxwell’s equations. Our results reveal a new regime of extreme spatiotemporal field structuring, with implications for ultrafast metrology, light–matter interactions, and deep-subwavelength control of electromagnetic waves.

Superoscillations enable waves to oscillate faster beyond classical limits. Here, the authors demonstrate simultaneous spatial and temporal superoscillations in structured light pulses, achieving extreme both subwavelength and ultrafast focusing in space-time.