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

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

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.

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

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.
Paper Structure (17 sections, 167 equations, 3 figures)

This paper contains 17 sections, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Quantum Metrology Theory
  4. Quantum parameter estimation and the Fisher information
  5. Classical and quantum semi-parametric estimation
  6. Imaging theory
  7. Ultimate limits: Computing the Quantum Fisher Information
  8. Achieving the limits: Measurement Techniques
  9. Failure of Direct Imaging
  10. Triumph of SPADE
  11. Discussion
  12. Derivation of the C Matrix for Direct Imaging
  13. Direct Imaging Influence Function Efficiency
  14. Derivation of the Direct imaging CRB
  15. Derivation of the W Matrix for SPADE
  16. ...and 2 more sections

Figures (3)

  • Figure 1: A schematic displaying the focusing of a small profile of a rough surface through a lens. The image of the rough surface can then be attained via different measurement schemes.
  • Figure 2: Panel (a): A schematic of the full imaging system consisting of a rough surface, modelled as $N$ discrete light sources displaced axially and radially from the focal point of the lens characterized by a point spread function $\psi$. The incoming field is then measured using an appropriate measurement scheme (represented as the field of cameras in the image) yielding data in the representation $\zeta$ with intensity $f(\zeta)$. Panel (b): A special case of the model in (a), consisting of sources only axially displaced from focus.
  • Figure 3: Intensity plots of the first four radially symmetric Laguerre-Gauss modes for $\ell=0,p=0,1,2,3$ in the $(x,y)$ plane centered at $(0,0)$. The $\ell=0,p=0$ mode is simply the PSF of an emitter in focus.