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arXiv:1703.08833 · Physical Review A

Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: Semiclassical treatment

Mankei Tsang

Published Feb 21, 2018• 52 citations
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Abstract

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.

DOI
10.1103/physreva.97.023830

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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.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
I derive classical and quantum lower error bounds for moment estimation in subdiffraction incoherent optical imaging. While the end results are similar to earlier ones [Tsang, Phys. Rev. Res. 1, 033006 (2019); Phys. Rev. A 99, 012305 (2019)], the present work avoids the flawed mathematical assumptions in the earlier works by adopting a parametric-submodel approach from semiparametric statistics. With the rigorous bounds at hand and redefining the parameter of interest, I show that an alternative measurement called spatial-mode demultiplexing (SPADE) is superior to ideal direct imaging for a subdiffraction object, even if SPADE can measure only a finite number of modes.
To investigate the fundamental limit to far-field incoherent imaging, the prequels to this work [M. Tsang, Phys. Rev. A 99, 012305 (2019); Phys. Rev. A 104, 052411 (2021)]. have studied a quantum lower bound on the error of estimating an object moment and proved a scaling law for the bound with respect to the object size. As the scaling law was proved only in the asymptotic limit of vanishing object size, this work performs a numerical analysis of the quantum bound to verify that the law works well for nonzero object sizes in reality. We also use the numerical bounds to study the optimality of a measurement called spatial-mode demultiplexing (SPADE), showing that SPADE not only follows the scaling but is also numerically close to being optimal, at least for low-order moments.
Optical measurements that can achieve the fundamental quantum limits have the potential to improve the imaging of subdiffraction objects in important applications, including optical astronomy and fluorescence microscopy. Working towards the goal of implementing such quantum-inspired measurements for real applications, we experimentally demonstrate the localization of two incoherent optical point sources and the semiparametric estimation of object moments in the subdiffraction regime via spatial-mode demultiplexing (SPADE). In the case of two sources, we are able to estimate both of their locations accurately, not just their separation, by exploiting the asymmetric response of our SPADE device. In the case of semiparametric estimation, we demonstrate that, even if the source number is unknown, the moments of the source distribution can still be estimated accurately. Our demonstration paves the way towards the use of SPADE for optical superresolution in practical scenarios, where adaptive measurements are difficult and many parameters are unknown.
In a previous paper [M. Tsang, Phys. Rev. A 99, 012305 (2019)], I proposed a quantum limit to the estimation of object moments in subdiffraction incoherent optical imaging. In this sequel, I prove the quantum limit rigorously by infinite-dimensional analysis. A key to the proof is the choice of an unfavorable parametric submodel to give a bound for the semiparametric problem. By generalizing the quantum limit for a larger class of moments, I also prove that the measurement method of spatial-mode demultiplexing with just one or two modes is able to achieve the quantum limit. For comparison, I derive a classical bound for direct imaging using the parametric-submodel approach, which suggests that direct imaging is substantially inferior.
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.