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Optics Express

Single-shot spatial coherence metrology using grating diffraction broadening with Fisher-limited performance

Juan Sumaya-Martinez

Published Mar 5, 2026• 0 citations
Publisher

Abstract

We present a grating-based, single-shot method to estimate the transverse spatial coherence width $\delta$ of a partially coherent beam from a single far-field diffraction image. Building on closed-form far-zone grating theory \cite{Li2025FarZoneGrating}, we cast the measurement as a statistical inverse problem by combining a Gaussian Schell-model forward model with detector noise (Poisson shot noise and Gaussian read noise). A global likelihood-based fit of the full diffraction pattern yields $\hat{\delta}$ in one exposure, including regimes with significant diffraction-order overlap. We further derive Fisher information and the Cram\'er--Rao lower bound (CRLB) \cite{Kay1993} to predict achievable precision and to guide experimental design (choice of grating duty cycle and the number of diffraction orders used). At a peak level of $\sim5\times10^3$ counts, Monte Carlo errors closely follow CRLB trends across a broad range of coherence widths.

DOI
10.1364/oe.589841

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We present a grating-based, single-shot method to estimate the transverse spatial coherence width δ\deltaδ of a partially coherent beam from a single far-field diffraction image. Building on closed-form far-zone grating theory \cite{Li2025FarZoneGrating}, we cast the measurement as a statistical inverse problem by combining a Gaussian Schell-model forward model with detector noise (Poisson shot noise and Gaussian read noise). A global likelihood-based fit of the full diffraction pattern yields δ^\hat{\delta}δ^ in one exposure, including regimes with significant diffraction-order overlap. We further derive Fisher information and the Cram'er--Rao lower bound (CRLB) \cite{Kay1993} to predict achievable precision and to guide experimental design (choice of grating duty cycle and the number of diffraction orders used). At a peak level of ∼5×103\sim5\times10^3∼5×103 counts, Monte Carlo errors closely follow CRLB trends across a broad range of coherence widths.
The role of spatial coherence on diffraction intensity is investigated for a partially coherent incident Gaussian Schell-model beam which is diffracted from a two-level atomic grating. It is shown that the performance of the atomic grating is greatly influenced by the spectral coherence width of the partially coherent fields. It is observed that relatively large intensity of the diffracted light can be obtained via spatial coherence, beam width, interaction length, and mode index of partially coherent incident light. The scheme provides possibilities for the potential applications of atomic grating in lensless imaging using the partially coherent light field.
Precision measurement has been an important research area in sensing and metrology. In classical physics, the Fisher information determines the maximum extractable information from statistically unknown signals, based on a joint probability density function of independently and identically distributed random variables. The Cramer-Rao lower bound (CRLB) indicates the minimum error of the Fisher information, generally known as the shot-noise limit. On the other hand, coherence has pushed the resolution limit further overcoming the diffraction limit using many-wave interference strictly confined to the first-order intensity correlation. However, practical implementation is limited by the lithographic constraints in, e.g., optical gratings. Recently, a coherence technique of superresolution has been introduced to overcome the diffraction limit in phase sensitivity using higher-order intensity correlations of a phase-controlled output field from an interferometer. Here, the superresolution is adopted for precision metrology in an optical spectrometer, whose enhanced frequency resolution is linearly proportional to the intensity-product order, overcoming CRLB. Unlike quantum sensing using entangled photons, this technique is purely classical and offers robust performance against environmental noises, benefiting from the interferometer's scanning mode for fringe counting.
I present a semiclassical analysis of a spatial-mode demultiplexing (SPADE) measurement scheme for far-field incoherent optical imaging under the effects of diffraction and photon shot noise. Building on previous results that assume two point sources or the Gaussian point-spread function, I generalize SPADE for a larger class of point-spread functions and evaluate its errors in estimating the moments of an arbitrary subdiffraction object. Compared with the limits to direct imaging set by the Cram\'er-Rao bounds, the results show that SPADE can offer far superior accuracy in estimating second- and higher-order moments.
We examine how partial coherence and wavefront curvature of beams affect interference fringes behind diffraction gratings. We simulate (1) the Talbot effect, (2) far-field diffraction, (3) Mach-Zehnader interferometers, (4) Talbot-Lau interferometers, and (5) Lau interferometers using a numerically efficient expression. We show how interference fringes in each case depend on the beam's initial width, its coherence width (transverse coherence length), and its wavefront curvature in directions both parallel and perpendicular to the grating bars. The separation between gratings and the rotational alignment of the gratings about the optical axis are also considered. The formula used for our simulations is derived using the mutual intensity function of a Gaussian Schell-model beam.
Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.
Interference frequency estimation is essential in spectral-domain interferometric sensing and imaging, and its performance determines system sensitivity. To date, an objective and practical criterion is still absent for the proper evaluation of fundamental sensitivity limit in a given system. Here, we report the derivation of Cramer-Rao bound (CRB) for unbiased estimators of spectral interference frequency, which imposes a theoretical limit on measurement sensitivity. Results based on a shot noise-limited model are presented. A more complete model, including dark and read noises, is also discussed and its approximate CRB is obtained. Poisson statistics were used in both cases. Asymptotic behaviors and simplified forms of CRB are studied for fringe visibility approaching 0 or 1. Further, we show that the current Fourier transform-based estimation algorithm achieves CRB only for low visibility, but is inferior by as much as √2 times for high visibility. This performance gap may potentially permit a sensitivity gain if better algorithms can be devised. Finally, we introduce a practical means to accurately estimate CRB from experimentally acquired spectral data. The results are verified in both simulation and experiments. Although spectral interference is discussed here, the same derivation can be applied to spatial or temporal interference as well, provided they can be similarly modelled.