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arXiv:1908.10949 · Physica Scripta

Quantum limited super-resolution of an unequal-brightness source pair in three dimensions

Sudhakar Prasad

Published Nov 13, 2019• 25 citations
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Abstract

Abstract This paper extends our previous quantum Fisher information (QFI) based analysis of the problem of separating a pair of equal-brightness incoherent point sources in three dimensions to the case of a pair of sources that are unequally bright. When the pair's geometric center is perfectly known in advance, QFI with respect to the estimation of the three separation coordinates remains independent of the degree of brightness asymmetry. For the experimentally more relevant case of perfect prior knowledge of the pair's brightness centroid, however, such QFI becomes dependent on the pair separation vector in a way that is controlled by the degree of its brightness asymmetry. This study yields potentially useful insights into the analysis of a more general superresolution imaging problem involving extended incoherent sources with nontrivial brightness distributions.

DOI
10.1088/1402-4896/ab573d

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Abstract This paper extends our previous quantum Fisher information (QFI) based analysis of the problem of separating a pair of equal-brightness incoherent point sources in three dimensions to the case of a pair of sources that are unequally bright. When the pair's geometric center is perfectly known in advance, QFI with respect to the estimation of the three separation coordinates remains independent of the degree of brightness asymmetry. For the experimentally more relevant case of perfect prior knowledge of the pair's brightness centroid, however, such QFI becomes dependent on the pair separation vector in a way that is controlled by the degree of its brightness asymmetry. This study yields potentially useful insights into the analysis of a more general superresolution imaging problem involving extended incoherent sources with nontrivial brightness distributions.
The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.
A recent paper [Yu and Prasad Phys. Rev. Lett. 121, 180504 (2018)] considered the problem of quantum-limited estimation of the separation vector for a pair of incoherent point sources in all three dimensions. Here we extend our analysis to treat the problem of simultaneously estimating the location of the centroid and the separation of the source pair, which is equivalent to localizing both sources simultaneously. We first calculate the quantum Fisher information for simultaneous pair centroid-separation estimation and then discuss the fundamental, estimation-theoretic tradeoffs between the two tasks, which we confirm using simulations.
Resolving the separation between two incoherent optical sources with high precision is of great significance for fluorescence imaging and astronomical observations. In this paper, we focus on a more general scenario where two sources have unequal brightnesses. We give the ultimate precision limit with respect to separation by using the quantum Fisher information. Through the calculation of the classical Fisher information, we analyze and compare several specific measurement schemes including direct measurement, Gaussian mode measurement and zero-photon measurement. The results indicate that Gaussian mode measurement is the nearly optimal for a small separation. On this basis, the effects of relative brightness fluctuation are addressed. Our work provides a positive complement to the aspect of superresolution imaging of incoherent sources.
We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed midpoint and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wave functions (PSWFs) on a centered line segment and their generalized versions on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.
We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of their separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost importance in realistic scenarios, such as observational astronomy.
Quantum superresolution promises to resolve closely spaced sources beyond the diffraction limit, but existing approaches generally rely on idealized source properties or require simultaneous estimation of multiple coupled parameters of realistic sources. Here we introduce a parameter-decoupled measurement that determines the separation of realistic passive sources without estimating their unknown brightness imbalance, mutual coherence, or relative phase. A complete four-parameter quantum Fisher information analysis identifies the separation information that remains accessible in the presence of these nuisance parameters. We then construct a directly measurable log-probability invariant from projection channels whose conditional statistics depend only on the separation. Broad classes of channel pairs satisfy the resulting decoupling condition. For a Gaussian point-spread function, the lowest-order implementation asymptotically attains the four-parameter quantum limit per incident signal in the sub-Rayleigh regime. This approach converts a realistic multiparameter imaging problem into an operationally single-parameter measurement, providing a practical route toward quantum-enhanced resolution of passive spatial, temporal, and spectral signals.