Quantum Limits of Passive Optical Surface Metrology and Defect Detection

Abstract

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.

Figures (4)

• Figure 1: Representation of a surface crack with width parameter $\delta x$ and depth parameter $\delta z$, modelled by three incoherent point sources.
• Figure 2: Cramér-Rao bounds for estimating surface crack parameters (a) $\delta x$ and (b) $\delta z$ as modelled in Fig. \ref{['fig:3source_layout']}. Red curves correspond to the ultimate quantum limit (QFI), blue curves to Hermite-Gaussian mode sorting (MS), and green curves to classical direct imaging (DI).
• Figure 3: Chernoff information for detecting surface cracks, plotted as a function of: (a) the width $\delta x$ for $\delta z/z_R=1$ (solid) and $0.1$ (dashed); (b) the depth $\delta z$ for $\sqrt{k/z_R}\delta_x=2$ (solid) and $0.5$ (dashed). Red curves correspond to the quantum Chernoff bound, blue curves to Hermite-Gaussian mode sorting, and green ones to direct imaging.
• Figure S1: Estimation of the parameters $\delta x$ [(a),(b)] and $\delta z$ [(c),(d)], for the incoherent three-source crack model depicted in Fig. 1 of the main text. Contributions of Hermite-Gaussian modes to the Fisher Information (Blue - MS), compared to the Quantum Fisher Information (Red - QFI) and the classical intensity measurement (Green - DI) [(a),(c)]. Individual Hermite-Gaussian modes with $(i,j)$ the 2D Hermite polynomial order, where for brevity we combine all the modes where the FIM matrix element is 0 (Brown - ij) and the last column is the sum of all the modes (Blue - $\sum$) [(b),(d)].