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arXiv:2603.16418 · Quantum Science and Technology

Looking down the rabbit hole: Towards quantum optimal estimation of surface roughness

Quentin Muller, Tommaso Tufarelli, Madalin Guta, Katherine Inzani, Samanta Piano, Gerardo Adesso

quant-phSubmitted Mar 17, 2026• Announced Mar 18, 2026• 0 citations
v1 · Mar 17, 2026v2 · Aug 11, 2026
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Abstract

Surface roughness is an important quantity to many engineering and precision manufacturing disciplines. In this paper we investigate the problem of estimating the root-mean-square roughness of a sample by passive linear optical methods. By adopting quantum parameter estimation techniques, we determine the ultimate precision limits on roughness estimation. In particular, we show that the information on the first moment (mean height) and standard deviation (roughness) of the axial profile distribution of multiple incoherent point sources is bounded by a constant. While classical imaging techniques fail to achieve this bound, a quantum inspired imaging technique based on spatial mode demultiplexing is proven to be optimal for estimating the axial standard deviation. Combined with analogous recently investigated methods for estimating radial profiles, this can provide a powerful technique for measuring roughness of nearly smooth surface patches beyond the diffraction limit.

DOI
10.1088/2058-9565/ae97ff

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Surface roughness is an important quantity to many engineering and precision manufacturing disciplines. In this paper we investigate the problem of estimating the root-mean-square roughness of a sample by passive linear optical methods. By adopting quantum parameter estimation techniques, we determine the ultimate precision limits on roughness estimation. In particular, we show that the information on the first moment (mean height) and standard deviation (roughness) of the axial profile distribution of multiple incoherent point sources is bounded by a constant. While classical imaging techniques fail to achieve this bound, a quantum inspired imaging technique based on spatial mode demultiplexing is proven to be optimal for estimating the axial standard deviation. Combined with analogous recently investigated methods for estimating radial profiles, this can provide a powerful technique for measuring roughness of nearly smooth surface patches beyond the diffraction limit.
We develop a quantum statistical framework for passive optical surface metrology. Modelling a surface as an incoherent ensemble of point emitters imaged through a diffraction-limited system, we employ techniques from quantum parameter estimation and hypothesis testing to derive ultimate bounds for jointly estimating geometrical features and for deciding the presence or absence of surface defects, and we identify optimal measurements from the geometry of the point-spread-function manifold. As a representative application, we analyse a minimal surface crack model based on three point sources and show that spatial mode sorting can simultaneously enable near-quantum-limited estimation of crack width and depth and markedly enhanced detectability of the crack, compared with direct imaging. Our results pave the way towards enhanced optical inspection and characterisation of sub-diffraction surface features by probing a limited number of spatial modes without any illumination control.
Optical precision measurement is evolving from conventional intensity and phase readout toward a paradigm centered on mode selection, statistical estimation, and information extraction. Conventional intensity- and phase-based schemes can achieve high sensitivity, but their performance is often constrained by finite-aperture diffraction, reference stability, optical path control, and environmental perturbations in systems. Quantum estimation theory provides a physical framework for optical resolution and measurement precision by optimizing the measurement basis. Inspired by spatial mode demultiplexing, this paper extends quantum parameter estimation from incoherent point-source resolution to small-displacement measurement. Metasurface enabled mode control converts displacement induced optical-field variations into measurable changes in orthogonal modal coefficients, providing a route for extracting sub-diffraction-scale displacement information. This paper introduces Rayleigh’s curse, Fisher information, and spatial mode projection, discusses their applications in one-dimensional displacement measurement and two-dimensional super-resolution imaging, and outlines future directions in digital optics, Engineering Optics 2.0, mode-basis optimization, two-dimensional information reconstruction, and intelligent optical measurement systems for engineering applications.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1–3]. Our experimental setup allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1]–[3]. Our experimental setup (see Fig. 1 (left)) allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].