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Optics Express

Far-field superresolution of thermal sources by double homodyne or double array homodyne detection

Yanxiang Xie, Hongbo Liu, Hengxin Sun, Kui Liu, Jiangrui Gao

Published May 6, 2024• 9 citations
Publisher

Abstract

We propose two schemes for estimating the separation of two thermal sources via double homodyne and double array homodyne detection considering the joint measurement of conjugate quadratures of the image plane field.By using the Cramér–Rao bound, we demonstrate that the two schemes can estimate the separation well below the Rayleigh limit and have an advantage over direct imaging when the average photon number per source exceeds five.For arbitrary source strengths, double homodyne detection is superior to homodyne detection when the separation is above 2 5/4 σ / N s , σ is the beam width, N s is the average photon number per source.A larger separation can be estimated better via double array homodyne detection with the superiority of flexible operation compared with other schemes. High-speed and high-efficiency detection enables the measurement schemes with potential practical applications in fluorescence microscopy, astronomy and quantum imaging.

DOI
10.1364/oe.523046

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We propose two schemes for estimating the separation of two thermal sources via double homodyne and double array homodyne detection considering the joint measurement of conjugate quadratures of the image plane field.By using the Cramér–Rao bound, we demonstrate that the two schemes can estimate the separation well below the Rayleigh limit and have an advantage over direct imaging when the average photon number per source exceeds five.For arbitrary source strengths, double homodyne detection is superior to homodyne detection when the separation is above 2 5/4 σ / N s , σ is the beam width, N s is the average photon number per source.A larger separation can be estimated better via double array homodyne detection with the superiority of flexible operation compared with other schemes. High-speed and high-efficiency detection enables the measurement schemes with potential practical applications in fluorescence microscopy, astronomy and quantum imaging.
We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.
The distance between two point light sources is difficult to estimate if that distance is below the diffraction (Rayleigh's) resolution limit of the imaging device. A recently proposed technique enhances the precision of this estimation by exploiting the source-separation-dependent coupling of light into higher-order transverse-electromagnetic (TEM) modes, particularly the TEM01{\mathrm{TEM}}_{01}TEM01​ mode of the image. We theoretically analyze the estimation of the source separation by means of homodyne or heterodyne detection with a local oscillator in the TEM01{\mathrm{TEM}}_{01}TEM01​ mode, which is maximally sensitive to the separation in the sub-Rayleigh regime. We calculate the Fisher information associated with this estimation and compare it with direct imaging. For thermal sources, the Fisher information in any mode of the image plane depends nonlinearly on the average received photon number. We show that the per-photon Fisher information surpasses that of direct imaging (in the interesting sub-Rayleigh regime) when the average received photon number per source exceeds two for homodyne detection and four for heterodyne detection.
It has been shown through quantum estimation theory that Rayleigh’s limit can be avoided for single-lens imaging, referred to as superresolution. The quantum estimation approach has recently been used to show superresolution is also possible for imaging based on interferometer arrays in the weak source limit. Following this line of discussion, we consider the resolution limit of estimating the separation between two point sources of arbitrary strength using interferometer arrays. By carefully designing the measurement, we find it is possible to overcome the well-known resolution limit of interferometer arrays as determined by the longest baseline. We construct an optimal measurement to achieve superresolution using linear beam-splitters and photon-number resolved detection.
Imaging using interferometer arrays based on the Van Cittert--Zernike theorem has been widely used in astronomical observation. Recently it was shown that superresolution can be achieved in this system for imaging two weak thermal point sources. Using quantum estimation theory, we consider the fundamental quantum limit of resolving the transverse separation of two strong thermal point sources using interferometer arrays, and show that the resolution is not limited by the longest baseline. We propose measurement techniques using linear beam splitters and photon-number-resolving detection to achieve our bound. Our results demonstrate that superresolution for resolving two thermal point sources of any strength can be achieved in interferometer arrays.
Sensitivity limits are usually determined using the Cram\'er-Rao bound. Recently, this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits usually require the full measurement statistics, which can be challenging to access. In this work we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this technique may be constructed for arbitrary thermal sources and we study their sensitivities when one has access to spatially resolved intensity measurements (direct imaging) and photon counting after spatial-mode demultiplexing. For demultiplexing, our method is optimal, i.e., it saturates the quantum Cram\'er-Rao bound. We also investigate the impact of noise on the optimal observables, their measurement sensitivity, and the scaling with the number of detected photons of the smallest resolvable separation. For low signals in the image plane, we demonstrate that our method saturates the Cram\'er-Rao bound even in the presence of noise.