Super-resolution with Fourier measurements

TL;DR

The paper shows that measuring intensity in the Fourier plane enables sub-Rayleigh super-resolution of coherent sources by converting a two-point problem in image space into a single-wavenumber localization in $k$-space. It derives the Fourier-plane intensity $I(k,\delta)=M\left(\frac{2\sigma^2}{\pi}\right)^{1/2}e^{-2k^2\sigma^2}\left(1+\gamma\cos{(k\delta+\phi)}\right)$ and demonstrates that, for fully coherent sources ($\gamma=1$) and $\phi\neq 0,\pi$, the spectral centroid shifts with the separation $\delta$, enabling sub-Rayleigh estimation. The Fisher information analysis shows a nonzero FI in the sub-Rayleigh limit, saturating the quantum limit for coherent sources and generalizing to $N$ sources where FI scales linearly with $N$ under certain phase structures. The work argues for robust, phase-sensitive Fourier measurements as practical, scalable super-resolution tools with wide applicability in imaging and metrology, including potential temporal-domain extensions.

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 $k\ =2π/(x_2-x_1)$ 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.

Figures (4)

• Figure 1: Basic idea of Fourier super-resolution. (a) Two point sources with separation $\delta$ imaged by a $4-f$ imaging system with a Gaussian aperture in the Fourier plane, and a spatially resolving CCD in the image plane. (b) Imaging two point sources with $\delta=5\sigma$ showing well-resolved direct imaging in the image plane (d). (e) For sub-Rayleigh $\delta=0.5\sigma$, the two-point intensity in the image plane (g) is sub-resolving, and estimating $\delta$ with DI is sub-optimal. The two point intensity in (d, g) is denoted by the solid red line, while the individual wavepacket intensities are denoted by dashed black lines. The vertical dashed lines in the image plane denote the peak location $\pm\delta/2$ for the two Gaussians. In this paper, we propose measuring intensity in Fourier space (c, f). The Fourier intensity in (c) shows the $k-$space oscillations (dashed-dotted red line) due to two-point interference. (f) shows the Fourier intensity for the the sub-Rayleigh regime of (g). While the $k-$space oscillations are much broader than the Gaussian pupil function, they still cause a finite shift of the spectrum peak from the origin in $k-$space. Localizing this peak can provide sub-Rayleigh resolution. For this figure, $\gamma=1$ and the relative phase $\phi=\pi/2$.
• Figure 2: (a) Modal decomposition of shifted Gaussians. For $\delta/\sigma\ll1$, a shifted Gaussian can be represented as sum of an on-axis Gaussian and a first order Hermite--Gauss mode whose amplitude is proportional to the shift. Because the two modes are in quadrature, the perturbative first order mode does not contribute to the intensity, at least to first order in $\delta$. In the Fourier domain, due to the Gouy phase shift, the modes are in phase and the spectrum is shifted. The $k-$space shift varies with $\phi$, and (b) shows the spectral centroid versus $\phi/\pi$ for $\delta=0.5\sigma$. Note that for all phases except $\phi=0,\pi$, the centroid is shifted from $k=0$. For this figure, $\gamma=1$.
• Figure 3: FI (Eq. (\ref{['eq:FIversusdelta']})) for the Fourier plane measurement in units of $M/4\sigma^2$ and for different $\phi$ values. Solid lines indicate the FI for Fourier intensity measurement, and dashed lines indicate the FI for DI. Non-zero FI values for $\phi\neq0$ in the limit $\delta\to0$ show evasion of Rayleigh's curse. The Fourier curves retain the same structure for any misalignment of the centroid from the optical axis. For $\phi=\{0,\pi\}$, the DI curves coincide with the solid Fourier curves, and are slightly blurred for distinguishability.
• Figure 4: Model constellation of $N=8$ point sources. The distance between each source on one side is $\delta$. (a) For a knife-edge style phase structure shown in dashed-dotted blue line, the sources on the right each have a phase of $\phi$ relative to the sources on the left. (b) For a ramp phase, the phase accrues linearly across the constellation given by $\exp{[i\phi x]}$. Note that there is no source at the origin.