Breaking the Diffraction Barrier for Passive Sources: Parameter-Decoupled Superresolution Assisted by Physics-Informed Machine Learning

TL;DR

This work tackles sub-diffraction estimation of separations between passive sources by introducing a parameter-decoupled framework that uses the ratio of Hermite-Gaussian mode probabilities $r_{mn}$, which depends only on the separation $s$ and is independent of nuisance parameters like coherence $|\gamma|$, balanceness $b$, relative phase $\varphi$, and photon statistics. A physics-informed CNN is trained on synthetic data to map mode-ratio information to an estimate $s_E$ while explicitly accounting for background noise, photon loss, and misalignment. On computer-generated and lab-generated data, the method achieves fidelity $F > 0.82$ for separations down to about $0.06\sigma$ (roughly $10$–$14$ nm in optical contexts), rivaling state-of-the-art active-source SRM techniques and showing strong robustness to parameter variability. By bridging theoretical superresolution limits with practical imperfections, this approach enables high-resolution passive-imaging applications in astrophysics, live-cell microscopy, and quantum metrology, using common hardware and efficient training.

Abstract

We present a parameter-decoupled superresolution framework for estimating sub-wavelength separations of passive two-point sources without requiring prior knowledge or control of the source. Our theoretical foundation circumvents the need to estimate multiple challenging parameters such as partial coherence, brightness imbalance, random relative phase, and photon statistics. A physics-informed machine learning (ML) model (trained with a standard desktop workstation), synergistically integrating this theory, further addresses practical imperfections including background noise, photon loss, and centroid/orientation misalignment. The integrated parameter-decoupling superresolution method achieves resolution 14 and more times below the diffraction limit (corresponding to ~ 13.5 nm in optical microscopy) on experimentally generated realistic images with >82% fidelity, performance rivaling state-of-the-art techniques for actively controllable sources. Critically, our method's robustness against source parameter variability and source-independent noises enables potential applications in realistic scenarios where source control is infeasible, such as astrophysical imaging, live-cell microscopy, and quantum metrology. This work bridges a critical gap between theoretical superresolution limits and practical implementations for passive systems.

Figures (3)

• Figure 1: Parameter-decoupled superresolution framework for passive sources. (a) Conceptual workflow of the parameter-decoupled protocol for estimating and evaluating two passive point-source separation. (b) Architecture of the physics-informed CNN, trained to infer the separation $s$ from input images. (c) Example training images with varying sets of parameters ($s$, $|\gamma|$, $b$) with the separation $s$ in the unit of $\sigma$. (d) Fisher information $I^{HG}$ for estimating $s$. Unlike conventional methods, $I^{HG}$ remains finite at $s=0$ (except for the idealized case of perfect destructive interference, red curve), ensuring robust superresolution across realistic experimental conditions.
• Figure 2: Separation estimation fidelity of the physics-informed CNN model for computer-generated data. Effects of source-dependent parameters: (a) illustrates the case when only $s$ is unknown while other parameters are fixed $\gamma=0$, $b=1$, $\varphi=0$, (b) represents when $s$ and $b$ are unknown, while $\gamma=0$, $\varphi=0$ are fixed, (c) shows the case when both $s$ and $\gamma$ are unknown, while $b=1$ and $\varphi=0$ are fixed, and (d) depicts when all four parameters are unknown and randomized. Effects of source-independent factors: (e) and (f) illustrate the fidelity with respect to the number of polluted pixels with luminosity noise for separations $s=0.1\sigma$ and $0.5 \sigma$ respectively. A sample image with 800 noisy pixels is shown for separation $s=0.1\sigma$. Panels (g) and (h) illustrate the fidelity with respect to the number of polluted pixels with photon loss for separations $s=0.1\sigma$ and $0.5 \sigma$ respectively. A sample image with 1000 affected pixels is shown for separation $s=0.1\sigma$. Panel (j) illustrates the fidelity with respect to $s$ for data images with 5% centroid misalignment and a sample data image, and panel (k) illustrates fidelity with respect to $s$ for data images with arbitrary orientation misalignment and a sample data image. Effects of training properties: Panel (i) illustrates two data images for separation $s=0.1\sigma$ with low resolutions of ($10\times10$) and ($20\times20$) pixels, respectively. Panels (l-n) illustrate fidelity behaviors for different number of training images and different number of pixels per image.
• Figure 3: Separation estimation fidelity of the physics-informed CNN model for experimental data. MZ Interferometer Experiments: (a) schematic illustration of the MZ interferometer setup to generate two-point source data images; (b) experimental raw data images for varying sets of parameters ($s$, $|\gamma|=0$, $b$), with fixed phase $\varphi=0$; (c) and (d) illustrate fidelity behaviors for incoherent ($|\gamma|=0$) data under balanced ($b=1$) and partially balanced ($b=0.56$) conditions, respectively. SLM Setup Experiments: (e) schematic illustration of the SLM setup to generate two-point source data images; (f) experimental raw data images for varying sets of parameters ($s$, $|\gamma|=0$, $b$) with fixed phase $\varphi=0$; (g) shows the fidelity behaviors for balanced ($b=1$) and partially coherent ($|\gamma|=0.7$) SLM testing data; (h) illustrates the fidelity behaviors for partially balanced ($b=0.56$) and partially coherent ($|\gamma|=0.8$) SLM testing data.