Super-resolution with Fourier measurements

TL;DR

Measuring intensity in the Fourier plane allows super-resolution imaging beyond the diffraction limit without requiring phase information or perfect alignment, making it more practical and robust.

Contribution

The paper demonstrates that Fourier measurements can achieve super-resolution at the quantum limit, overcoming limitations of traditional image-based methods.

Findings

Fourier-based measurements saturate the quantum limit for fully coherent sources.

Super-resolution can be achieved for sub-Rayleigh source configurations.

The approach simplifies and enhances robustness of super-resolution imaging techniques.

Abstract

Resolving sources beyond the diffraction limit is important in imaging, communications, and metrology. Current image-based methods of super-resolution require phase information (either of the source points or an added filter) and perfect alignment with the centroid of the object. Both inhibit the practical application of these methods, as uniform motion and/or relative jitter destroy their assumptions. Here, we show that measuring intensity in the Fourier plane enables super-resolution without any of the issues of image-based methods. We start with the shift-invariance of the Fourier transform and the observation that the two-point position problem ${x_1,x_2}$ in the near field corresponds to the single-point wavenumber problem $k\ =2\pi/(x_2-x_1)$ in the far field. We consider the full range of mutual coherence and show that for fully coherent sources, the Fourier method saturates the quantum limit, i.e. it gives the best possible measurement. Similar results hold for sub-Rayleigh constellations of $N$ sources, which can act collectively as a spatially averaged metasurface and/or individually as elements of a phased-array antenna. The theory paves the way to merge Fourier optics with super-resolution techniques, enabling experimental devices that are both simpler and more robust than previous designs.