Measuring the Oscillation Frequency Beyond the Diffraction Limit

TL;DR

The paper addresses the challenge of measuring motion characteristics of a diffraction-limited optical point source amid excess noise. It develops a Fisher-information framework, deriving the Quantum Fisher Information under ideal conditions and the Classical/Fisher Information under excess noise, showing how robustness to background noise emerges via mode design. By comparing direct imaging with Hermite-Gaussian SPADE and the two-mode PM-SPADE, the work demonstrates that PM-SPADE concentrates information into two spatial modes and outperforms direct imaging in sub-Rayleigh regimes with noise, approaching the quantum limits for small displacements. Experimentally, PM-SPADE is implemented with a phase-only SLM and a two-pixel CMOS readout to estimate micro-oscillation frequency, confirming enhanced noise robustness and high estimation precision with a simple, two-detector architecture.

Abstract

High-resolution array detectors are widely used in single-particle tracking, but their performance is limited by excess noise from background light and dark current. As pixel resolution increases, the diminished signal per pixel exacerbates susceptibility to noise, degrading tracking accuracy. To overcome this limitation, we use spatial-mode demultiplexing (SPADE) as a noise-robust approach for estimating the motion characteristics of an optical point-like source. We show that SPADE efficiently concentrate the information into a few key spatial modes, drastically reducing the number of detectors while maintaining high estimation precision. Furthermore, we enhance the robustness of the estimation against excess noise by elaborately designing the modes to be decomposed. We demonstrate, both theoretically and experimentally, that a SPADE with two specific modes outperforms direct imaging in estimating the micro-oscillation frequency of an optical point source in the presence of excess noise.

Figures (9)

• Figure 1: Illustration of the dynamic single point source.
• Figure 2: Experimental setup for PM-SPADE. The notations used are as follows: SLM, spatial light modulator; DMD, digital micromirror device; CMOS, complementary metal oxide semiconductor; CW, continuous wave. In this configuration, only the first-order diffracted light from the SLM is directed towards the CMOS camera. The estimation process relies on intensity measurements from two specific pixels, marked with red crosses in the captured image.
• Figure 3: Means and rescaled variances of the PM-SPADE and direct imaging for frequency estimation in the absence of background noise. Here, the variance is multiplied by the average photon number $\nu$ per frame. The left two panels and the right two panels correspond to the oscillation amplitude of $0.28\sigma$ and $0.47\sigma$, respectively. The QCRB is plotted according to Eq. \ref{['eq:QCRB']} with $N=50$.
• Figure 4: Mean and rescaled variance of direct imaging, PM-SPADE, and HG-SPADE versus the excess noise induced by background light. The results for HG-SPADE are obtained from simulated data for measuring the first 21 order modes SupplementalMaterial. Here, the true value of the dimensionless frequency is $f=0.2$ and the oscillation amplitude is $A=0.47\sigma$. The Cramér-Rao bounds (CRB) are numerically evaluated according to Eq. \ref{['eq:CFI']} and Eq. \ref{['eq:gamma']}, while the QCRB is evaluated according to Eq. \ref{['eq:noisy_qfi']}.
• Figure S1: Values of $\gamma$ for direct imaging, HG-SPADE, and PM-SPADE. Here, the pixel size for direct imaging is set to be $a=4.6$ µm.