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arXiv:2610.39221

Spatial-Mode Selection for Local Mineral Absorption Spectroscopy

Qipeng Qian, Jialiang Jin, Shihao Wang, Yuntao Qian

physics.opticsquant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Absorption-band depth measures the fractional decrease in mineral reflectance below its local continuum. Recovering this feature strength for a small mineral region supports quantitative comparisons between regions, but diffraction mixes its light with the surroundings. Conventional spectral fitting can separate mean contributions; it cannot remove background photon noise already recorded. We develop a quantum-inspired measurement framework that uses spatial-mode demultiplexing (SPADE) to retain the target absorption response while reducing background acceptance before detection. The framework connects finite mineral-region geometry and reference spectra to estimation precision. A loss-dependent design condition identifies when adding SPADE exposure improves on direct imaging (DI) under fixed-response Poisson measurements with unknown scene parameters. Joint inference from initial DI observations and both mode ports estimates target depth together with an unknown, spatially shared background spectrum. In equal-time simulations that include localization uncertainty, alignment time, calibration, and detector costs, this acquisition reduces depth root-mean-square error (RMSE) by 14\%--26\% across five absorption strengths relative to all-pixel DI with the same background freedom. These results use a supplied target feature shape and geometry, with target continuum slope fitted and curvature fixed. The analysis separates optical selectivity from parameter ambiguity and acquisition cost, providing a quantitative basis for spatial-mode receiver design in local mineral spectroscopy.

AI Summary

The paper asks whether optical spatial-mode selection can improve estimation of a small mineral patch’s absorption-band depth when diffraction mixes its light with an extended background. It models finite, spatially incoherent reflecting regions using reference spectra and wavelength-dependent diffraction, comparing all-pixel direct imaging (DI) with initial DI localization followed by binary spatial-mode demultiplexing (SPADE). Both mode ports and the initial DI data are fitted jointly, including an unknown background spectrum shared across space. Fisher-information analysis separates background shot noise from ambiguity involving brightness, position, and spectral parameters. For fixed-response Poisson measurements, it specializes local optimal-design theory to give a necessary and sufficient condition for beneficial SPADE exposure: loss-adjusted SPADE information along DI’s least-distinguishable depth–nuisance direction must exceed the displaced DI information. In the calcite/dolomite example, the transmission threshold is 0.4559; transmission 0.70 permits a maximum 35.8% information increase. Complete simulated 10-second acquisitions include localization uncertainty, alignment time, calibration, and detector noise. Across five depths, 400 trials per receiver give 14–26% lower depth RMSE than DI with identical background freedom. A separate supplied-spectrum benchmark covering 24 material/geometry conditions gives a geometric-mean RMSE ratio of 0.884. The results assume supplied target geometry and feature shape, fitted continuum slope but fixed curvature, accurately executed alignment, and spatially shared background spectra. Unknown curvature substantially reduces absolute information. Spatial composition mismatch can reverse the advantage: one gradient test gives an adaptive/DI RMSE ratio of 1.88. Broadband sorter performance remains experimentally untested; quantum optimality is not established.

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Absorption-band depth measures the fractional decrease in mineral reflectance below its local continuum. Recovering this feature strength for a small mineral region supports quantitative comparisons between regions, but diffraction mixes its light with the surroundings. Conventional spectral fitting can separate mean contributions; it cannot remove background photon noise already recorded. We develop a quantum-inspired measurement framework that uses spatial-mode demultiplexing (SPADE) to retain the target absorption response while reducing background acceptance before detection. The framework connects finite mineral-region geometry and reference spectra to estimation precision. A loss-dependent design condition identifies when adding SPADE exposure improves on direct imaging (DI) under fixed-response Poisson measurements with unknown scene parameters. Joint inference from initial DI observations and both mode ports estimates target depth together with an unknown, spatially shared background spectrum. In equal-time simulations that include localization uncertainty, alignment time, calibration, and detector costs, this acquisition reduces depth root-mean-square error (RMSE) by 14%--26% across five absorption strengths relative to all-pixel DI with the same background freedom. These results use a supplied target feature shape and geometry, with target continuum slope fitted and curvature fixed. The analysis separates optical selectivity from parameter ambiguity and acquisition cost, providing a quantitative basis for spatial-mode receiver design in local mineral spectroscopy.
The paper asks whether optical spatial-mode selection can improve estimation of a small mineral patch’s absorption-band depth when diffraction mixes its light with an extended background. It models finite, spatially incoherent reflecting regions using reference spectra and wavelength-dependent diffraction, comparing all-pixel direct imaging (DI) with initial DI localization followed by binary spatial-mode demultiplexing (SPADE). Both mode ports and the initial DI data are fitted jointly, including an unknown background spectrum shared across space.
Fisher-information analysis separates background shot noise from ambiguity involving brightness, position, and spectral parameters. For fixed-response Poisson measurements, it specializes local optimal-design theory to give a necessary and sufficient condition for beneficial SPADE exposure: loss-adjusted SPADE information along DI’s least-distinguishable depth–nuisance direction must exceed the displaced DI information. In the calcite/dolomite example, the transmission threshold is 0.4559; transmission 0.70 permits a maximum 35.8% information increase. Complete simulated 10-second acquisitions include localization uncertainty, alignment time, calibration, and detector noise. Across five depths, 400 trials per receiver give 14–26% lower depth RMSE than DI with identical background freedom. A separate supplied-spectrum benchmark covering 24 material/geometry conditions gives a geometric-mean RMSE ratio of 0.884.
The results assume supplied target geometry and feature shape, fitted continuum slope but fixed curvature, accurately executed alignment, and spatially shared background spectra. Unknown curvature substantially reduces absolute information. Spatial composition mismatch can reverse the advantage: one gradient test gives an adaptive/DI RMSE ratio of 1.88. Broadband sorter performance remains experimentally untested; quantum optimality is not established.
Small mineral patches can be spectrally distinct yet difficult to detect when diffraction mixes their light with the surrounding rock. Conventional hyperspectral detectors analyze recorded intensities, leaving an opportunity to recover target evidence through the optical measurement itself. We develop a quantum-assisted framework for exploiting this opportunity in prospective unmanned aerial vehicle (UAV) hyperspectral sensing. A common radiometric and diffraction model describes direct imaging and spatial-mode measurements of the same light. The theory identifies target changes that survive background fitting and explains when mode measurements retain more detection information. It quantifies losses from mode mismatch and unresolved outputs, linking the quantum benchmark to finite receiver design. A common-basis result identifies when material spectra can share a spatial measurement at fixed geometry. An adaptive receiver uses an imaging pilot to select candidate modes and combines both stages in a full-region presence test. Simulations based on measured mineral spectra and published UAV acquisition scales demonstrate gains over complete direct imaging at a shared active-exposure budget, including pilot cost and optical loss. At matched empirical 1\% region-level false alarms, detection increases from 62.95\% to 69.65\% for a representative target. Paired analysis supports gains with registration uncertainty and, separately, under controlled nonuniform backgrounds. Calibrated mode errors reveal where measurement losses weaken or reverse the benefit. These results connect background identifiability to receiver choices and their operating conditions. They provide a framework for designing spatial measurements alongside spectral channels to detect small material features.
We theoretically predict and experimentally demonstrate that a reduced spatial-mode demultiplexing (SPADE) measurement using only the two lowest-order Hermite-Gaussian modes exhibits remarkable robustness against background noise in discriminating between a single source and two incoherent point sources. We establish a theoretical framework incorporating uniform background noise and derive an analytical Chernoff exponent expression, showing that SPADE consistently outperforms direct imaging (DI) across all source separations. Experimental results confirm that SPADE-based hypothesis testing approaches the quantum limit even when the background-to-signal photon ratio per pixel is 0.11. This advantage stems from SPADE's ability to concentrate source information into minimal detection modes, reducing the cumulative background noise impact. Our findings provide a practical detection scheme for applications where background noise is inevitable, such as astronomical observations and quantum sensing.
We study far-field discrimination between one and two incoherent point sources in the singular regime of weak and closely spaced emitters. Under ideal alignment, spatial-mode demultiplexing (SPADE) attains the quantum-optimal large-sample Stein exponent, but the finite-photon behavior near the one-source boundary and the effect of realistic imperfections remain less understood. Using singular learning theory, we analyze both the aligned and misaligned problems. In the aligned Gaussian case, for prior densities that are smooth and strictly positive in the physical (ε,s)(ε,s)(ε,s) coordinates near the singularity of the aligned model, direct imaging and SPADE share the same real log canonical threshold λ=1/2λ=1/2λ=1/2 but have different multiplicities, yielding distinct Bayes free-energy asymptotics. A fixed nonzero misalignment removes the exact support mismatch of ideal SPADE: locally, the fixed-offset binary-SPADE Kullback--Leibler function has the normal-crossing form K≍ε2s2K\asympε^2s^2K≍ε2s2, giving (λ,m)=(1/2,2)(λ,m)=(1/2,2)(λ,m)=(1/2,2) under the same prior class, as for direct imaging. The local separation scale nevertheless depends on the source-position convention. Moreover, full Hermite--Gaussian mode counting about an offset sorter axis has a reflection-induced KL-zero branch at s∗=2θs^\ast=2θs∗=2θ, where the two-source alternative becomes indistinguishable from the null. Pointwise finite-nnn binary-SPADE calculations exhibit the corresponding power collapse. These results identify measurement support, nuisance geometry, prior weighting, and identifiability as structural ingredients that must be tracked in finite-photon quantum discrimination.
The resolution of direct imaging, which is the most widespread technique for remote observation in the optical domain, is fundamentally limited by diffraction. It has been recently shown that the diffraction limit can be overcome, even with passive optics, using interferometric far-field techniques like spatial mode demultiplexing (SPADE). We present our results on sub-diffraction imaging and spectroscopy of a pair of incoherent point-like emitters using SPADE in the single-photon regime. A theoretical analysis provides a comparison with direct imaging and explains the improvement in terms of signal-to-noise ratio. The application of single-photon SPADE for detection of more complex objects beyond a pair of point-emitters is underway for deployment in earth observation and object detection. The results are presented in the framework of the Quantum Imaging for Exoplanet detection (QUEXO) project.
I point out the mathematical correspondence between an incoherent imaging model proposed by my group in the study of quantum-inspired superresolution [M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016)] and a noise spectroscopy model also proposed by us [M. Tsang and R. Nair, Phys. Rev. A 86, 042115 (2012); S. Ng et al., Phys. Rev. A 93, 042121 (2016)]. Both can be regarded as random displacement models, where the probability measure for the random displacement depends on unknown parameters. The spatial-mode demultiplexing (SPADE) method proposed for imaging is analogous to the spectral photon counting method proposed by Ng et al. for optical phase noise spectroscopy: Both methods are discrete-variable measurements that are superior to direct displacement measurements (direct imaging or homodyne detection) and can achieve the respective quantum limits. Inspired by SPADE, I propose a modification of spectral photon counting when the input field is squeezed: The output field is simply unsqueezed before spectral photon counting. I show that this method is quantum optimal and far superior to homodyne detection for both parameter estimation and detection, thus solving the open problems in the work of Tsang and Nair and Ng et al.
In optical imaging, diffraction strongly degrades the performance of conventional intensity-based estimation once the separation between two sources is below the Rayleigh-Abbe limit. Recent developments showed that this limitation can be surpassed using spatial-mode demultiplexing (SPADE), and this was demonstrated in several experiments for two equal-intensity spots of unknown separation. When there are multiple unknown parameters, as in the case of several unequal sources with unknown intensities and unknown separation, cross-talk among the parameters makes the multi-parameter estimation problem significantly more challenging. In this paper, we adapt super-resolved position localization by inversion of coherence along an edge (SPLICE) to estimate both the separation and relative intensity of two incoherent sources simultaneously. We demonstrate a clear advantage over direct imaging (DI), achieving a root mean squared error (RMSE) approximately 50%50 \%50% larger than the quantum limit and a sixfold improvement over DI within the range of parameters we tested. This improvement can be even greater for smaller separations and larger intensity imbalances when crosstalk is suppressed.