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arXiv:2103.12096 · Quantum

Back to sources – the role of losses and coherence in super-resolution imaging revisited

Stanisław Kurdziałek

Published Apr 27, 2022• 20 citations
arXivHTMLPDF

Abstract

Photon losses are intrinsic for any translationally invariant optical imaging system with a non-trivial Point Spread Function, and the relation between the transmission factor and the coherence properties of an imaged object is universal – we demonstrate the rigorous proof of this statement, based on the principles of quantum mechanics. The fundamental limit on the precision of estimating separation between two partially coherent sources is then derived. The careful study of the role of photon losses allows to resolve conflicting claims present in previous works. We compute the Quantum Fisher Information for the generic model of optical 4f imaging system, and use prior considerations to validate the result for a general, translationally invariant imaging apparatus. We prove that the spatial-mode demultiplexing (SPADE) measurement, optimal for non-coherent sources, remains optimal for an arbitrary degree of coherence. Moreover, we show that some approximations, omnipresent in theoretical works about optical imaging, inevitably lead to unphysical, zero-transmission models, resulting in misleading claims regarding fundamental resolution limits.

DOI
10.22331/q-2022-04-27-697

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Photon losses are intrinsic for any translationally invariant optical imaging system with a non-trivial Point Spread Function, and the relation between the transmission factor and the coherence properties of an imaged object is universal – we demonstrate the rigorous proof of this statement, based on the principles of quantum mechanics. The fundamental limit on the precision of estimating separation between two partially coherent sources is then derived. The careful study of the role of photon losses allows to resolve conflicting claims present in previous works. We compute the Quantum Fisher Information for the generic model of optical 4f imaging system, and use prior considerations to validate the result for a general, translationally invariant imaging apparatus. We prove that the spatial-mode demultiplexing (SPADE) measurement, optimal for non-coherent sources, remains optimal for an arbitrary degree of coherence. Moreover, we show that some approximations, omnipresent in theoretical works about optical imaging, inevitably lead to unphysical, zero-transmission models, resulting in misleading claims regarding fundamental resolution limits.
The fundamental limit on the precision of the separation estimation between two partially coherent sources is derived. Photon losses, inevitably associated with imperfect imaging, are properly taken into account. We begin our analysis with a simple and common imaging model based on the 4f system. We then show that our results are valid for any passive, linear, translationally invariant optical imaging system. We obtain the expression for the Quantum Fisher Information, which is consistent with the already computed classical Fisher Information and latest experimental work. We also identify the source of discrepancies in the results obtained earlier and show how to correctly interpret them.
Discriminating two spatially separated sources is one of the most fundamental problems in imaging.Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken.However, in realistic optical systems, there often exists coherence in the imaging light field, and there have been efforts to analyze the optical resolution in the presence of partial coherence.Nevertheless, how the degree of coherence between two point sources affects the resolution has not been fully understood.Here, we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources.In particular, we consider the situation in which coherence varies with the separation.We propose a feasible experiment scheme to realize the nearly optimal measurement, which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging.Our results will have wide applications in imaging involving coherence of light.
Quantum superresolution reveals that the vanishing of separation sensitivity in conventional imaging below the Rayleigh limit does not necessarily indicate a fundamental loss of information in the optical field. However, the quantum limit for estimating the physical distance between two incoherent point sources in three-dimensional imaging systems and its dependence on the spatial structure of the point-spread function remains largely unknown. In this work, we derive the quantum-limited precision for estimating the full distance between two incoherent point sources with arbitrary intensity imbalance in a three-dimensional spatially invariant imaging system. We show that the distance information remains finite in the sub-Rayleigh regime and is governed by the second-order displacement-response tensor of the point-spread function. The eigensystem of this tensor determines the optimal relative orientation between the two sources, and reflection symmetries of the point-spread function can further provide a simplified means of identifying the optimal orientation. This geometric structure is coordinate invariant and provides a direct strategy for improving resolution by physically rotating an anisotropic imaging system to align its optimal principal response direction with the source displacement. For a general three-dimensional Gaussian point-spread function, the response tensor is proportional to the inverse spatial covariance, establishing a direct connection between quantum-limited distance precision and the geometry of Gaussian distribution.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works, analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. However, most previous works have either consider low-intensity, or thermal sources. To broaden the applicability of this approach, it is important to extend these results to more general light sources. To this goal, we will present an analytical expression for the Quantum Fisher Information, determining the ultimate resolution limit, for the separation between two sources in an arbitrary Gaussian state. Applying this result to different quantum states, we can shine some light on some relevant questions. We can for example explore the role of partial coherence considering displaced and correlated thermal states, or investigate the importance of quantum correlations by considering squeezed light. In addition to the ultimate quantum limit, we will discuss a simple estimation technique, requiring access only to the mean value of a linear combination of demultiplexed intensity measurements, which is often sufficient to saturate these limits, and can easily be adapted to incorporate the most common noise sources. Finally, we will present our experimental setup that allows for the generation of the images of two sources with different photon statistics, as well as for spatial mode demultiplexing and we will discuss the first practical implementations if the above mentioned estimation techniques.
We consider imaging of two partially coherent sources and derive the ultimate quantum limits for estimating the separation, location, relative intensity, and coherence factor. We show that super-resolution in the separation is achievable both in the presence of nuisance parameters as well as when some of the parameters are assumed known. Nuisance parameters restrict super-resolution to balanced sources, whereas for known parameters super-resolution persists over a broader range of relative intensities and is lost only for perfectly correlated sources, i.e., γ=1γ=1γ=1. The achievable precision is governed primarily by interference-induced photon statistics and depends strongly on the degree of coherence. In the sub-Rayleigh regime, the imaging problem reduces to an effective two-dimensional Hilbert space description, provided a consistent reference position is used, with all parameters encoded in a Bloch vector representation. Finally, indirect estimation schemes based on the purity of the image-plane state are generally suboptimal for all non-zero coherence and become valid only in the incoherent limit.