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Physical Review A

Weak-measurement method for surface-roughness estimation

Hui-Chao Qu, Ya Xiao, Zhao-Qi-Zhi Han, Shanchuan Dong, Yongjian Gu

Published Jun 10, 2022• 3 citations
Publisher

Abstract

High-precision surface-roughness estimation plays an important role in many applications. However, the classical estimating methods are limited by shot noise and can only achieve a precision of 0.1 nm with a white-light interferometer. Here, we propose two weak-measurement schemes to estimate surface roughness through a spectrum analysis and intensity analysis. The estimating precision with the spectrum analysis is about ${10}^{\ensuremath{-}5}$ nm by using a currently available spectrometer with a resolution of $\mathrm{\ensuremath{\Delta}}\ensuremath{\lambda}=0.04$ pm and the corresponding sensitivity is better than 0.1 THz/nm. The precision and sensitivity of the light intensity analysis scheme achieved is as high as 0.07 nm and 1/nm, respectively. By introducing a modulated phase, we show that the sensitivity and precision achieved in our schemes can be effectively retained in a wider dynamic range. We further provide the experimental design of a surface profiler based on our schemes. It simultaneously meets the requirements of high precision, high sensitivity, and a wide measurement range, making it a promising practical tool.

DOI
10.1103/physreva.105.063708

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High-precision surface-roughness estimation plays an important role in many applications. However, the classical estimating methods are limited by shot noise and can only achieve a precision of 0.1 nm with a white-light interferometer. Here, we propose two weak-measurement schemes to estimate surface roughness through a spectrum analysis and intensity analysis. The estimating precision with the spectrum analysis is about 10\ensuremath−5{10}^{\ensuremath{-}5}10\ensuremath−5 nm by using a currently available spectrometer with a resolution of \ensuremathΔ\ensuremathλ=0.04\mathrm{\ensuremath{\Delta}}\ensuremath{\lambda}=0.04\ensuremathΔ\ensuremathλ=0.04 pm and the corresponding sensitivity is better than 0.1 THz/nm. The precision and sensitivity of the light intensity analysis scheme achieved is as high as 0.07 nm and 1/nm, respectively. By introducing a modulated phase, we show that the sensitivity and precision achieved in our schemes can be effectively retained in a wider dynamic range. We further provide the experimental design of a surface profiler based on our schemes. It simultaneously meets the requirements of high precision, high sensitivity, and a wide measurement range, making it a promising practical tool.
High-precision surface roughness estimation plays an important role in many applications. However, the classical estimating methods are limited by shot noise and only can achieve the precision of 0.1 nm with white light interferometer. Here, we propose two weak measurement schemes to estimate surface roughness through spectrum analysis and intensity analysis. The estimating precision with spectrum analysis is about 10−510 ^{-5}10−5 nm by using a currently available spectrometer with the resolution of Δλ=0.04Δλ= 0.04Δλ=0.04 pm and the corresponding sensitivity is better than 0.1 THz/nm. And the precision and sensitivity of the light intensity analysis scheme achieve as high as 0.07 nm and 1/nm, respectively. By introducing a modulated phase, we show that the sensitivity and precision achieved in our schemes can be effectively retained in a wider dynamic range. We further provide the experimental design of the surface profiler based on our schemes. It simultaneously meets the requirements of high precision, high sensitivity, and wide measurement range, making it to be a promising practical tool.
We propose a scheme called difference weak measurement, and we illustrate its advantages for measuring a small longitude phase shift in high precision. Compared to the standard interferometry and standard weak measurement schemes, the proposed scheme has a much higher resolution in the presence of various practical imperfections, such as alignment error and light intensity variation error. Moreover, we highlight the advantage of utilizing a complex weak value, where its imaginary part can reduce the harmful effect induced by channel decoherence. Finally, we propose a closed-loop scenario to solve the narrow dynamic range problem that exists in the current weak measurement schemes. The difference weak measurement scheme simultaneously fulfills the requirements of high precision, wide dynamic range, and strong robustness, which makes it a powerfully practical tool for phase-shift measurement and other metrological tasks.
Weak measurement enables the amplification of weak physical effects via post-selection and has become an important tool in precision optical metrology; however, conventional schemes based on mean-pointer shifts suffer from response saturation, limited linear range, and stringent stability requirements. Here, we propose and experimentally demonstrate a weak-measurement scheme based on spectral-interference trough shifts, where the zero-intensity points of the post-selected spectrum act as the measurement pointer, establishing an analytical mapping between the trough displacement and the target phase or time delay. Theoretical analysis shows that, under detector resolution limits, the measurement resolution depends solely on the frequency of extinction point and is independent of weak-value singular amplification or bias-phase modulation, thereby maintaining high sensitivity while avoiding pointer saturation. Experiments demonstrate that the trough-shift scheme achieves significantly better agreement between measured and theoretical sensitivities than biased weak measurement and provides a stable linear response without additional bias-compensation structures, reaching a minimum resolvable phase variation at the 10−7 level. Moreover, the approach intrinsically supports multi-period traceable measurements and exhibits strong robustness against intensity fluctuations and spectral distortions, offering a promising route toward high-sensitivity, large-dynamic-range, and stable weak measurement-based optical sensing.
Phase estimation has attracted much attention in the field of sensing, where weak measurement has become an effective solution for the study of small physical quantities because of its anomalous amplification property. However, it is difficult for weak measurement to cope with large low-frequency noise signals in the environment due to its extremely narrow linear range, which makes the extension of weak measurement to practical applications a challenging task. In order to solve this issue, we propose and experimentally verify a phase demodulation scheme based on weak measurement, which solves the problem of working point offset caused by the noise through artificially introducing a high-frequency carrier and performing demodulation operations on the output light. The experimental results show that the scheme can improve the signal-to-noise ratio by 17 dB compared to standard weak measurement and ensures low distortion in phase demodulation. Moreover, the scheme is not limited to a specific modulation depth, which greatly improves the robustness of the scheme.
We theoretically propose and experimentally demonstrate a modulated weak measurement scheme that permits a precise estimation of the extended phase shift. By choosing the frequency and the total post-selected light intensity as the input and output meter variables, respectively, a linear weak measurement model is established that is especially applicable to precise phase estimation. By performing a time-delay-dependent modulation strategy based on the feedback modulation of the experimental outcome, both the high phase-sensitivity and measurement precision can be retained. We have achieved a precision at the order of ∼10−5 rad that can be effectively retained with an increase in the measured phase.
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 optics. By adopting quantum parameter estimation methods, we determine the ultimate precision limits for estimating spatial moments of a general three-dimensional distribution of incoherent point sources in the sub-diffraction regime. Specializing this result to the axial profile, we show that the information on the first moment (mean height) and standard deviation (roughness) 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. This provides a powerful and experimentally accessible route to measuring roughness of nearly smooth surface patches beyond the diffraction limit.