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Optica

Resurgence of Rayleigh’s curse in the presence of partial coherence

Walker D. Larson, Bahaa E. A. Saleh

Published Oct 24, 2018• 73 citations
Publisher

Abstract

The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.

DOI
10.1364/optica.5.001382

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The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.
Tsang et al. have shown that the Fisher information of the two incoherent point source separation, below the Rayleigh limit, is finite and achievable using optical modes measurements.1 However, recent claims regarding partial coherence of sources, no matter how small, leads to necessarily zero Fisher information as the source separation decreases below the Rayleigh limit approaching zero have proved to be controversial.2, 3 Thus, the impact of partial coherence on the photon counting optical modal measurements merits further exploration. In this work, we derive the mutual coherence function (image plane) of two partially coherent point sources and find the classical Fisher information of the source separation using both direct image plane and photon counting modal measurements. A classical Fisher information analysis of partially coherent source(s) leads to some rather surprising results for two-point source resolution as the source separation approaches zero. We find that the magnitude of the Fisher information strongly depends on the degree of (positive/negative) partial coherence, which can be understood using an intuitive semi-classical analysis of direct image plane and photon counting modal measurements. We also provide an error analysis of the maximum likelihood estimators for both measurements.
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.
The quantum Fisher information (FI), when applied to the estimation of the separation of two point sources, has been shown to be non-zero in cases where the coherence between the sources is known. Although it has been claimed that ignorance of the coherence causes the quantum FI to vanish (a resurgence of Rayleigh’s curse), a more complete analysis including both the magnitude and phase of the coherence parameter is given here. Partial ignorance of the coherence is shown to potentially break Rayleigh’s curse, whereas complete ignorance guarantees its resurgence.
The quantum Fisher information (FI), when applied to the estimation of the separation of two point sources, has been shown to be non-zero in cases where the coherence between the sources are known. Although it has been claimed that ignorance of the coherence causes the quantum FI to vanish (a resurgence of Rayleigh's curse), a more complete analysis including both the magnitude and phase of the coherence parameter is given here. Partial ignorance of the coherence is shown to potentially break Rayleigh's curse, whereas complete ignorance guarantees its resurgence.
We devise a systematic method to determine the Fisher information required for resolving two incoherent point sources with a diffraction-limited linear imaging device. The resulting Cramér-Rao bound gives the lowest variance achievable for an unbiased estimator. When only intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect which has been dubbed as Rayleigh's curse. However, this curse can be lifted using suitable coherent measurements. In particular, we determine a class of measurements that can be easily implemented as projections on the orthogonalised derivatives of the point spread function of the system.
In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.