Spatial-mode Demultiplexing Research Articles

Multi-aperture telescopes at the quantum limit of superresolution imaging : Detecting subRayleigh object near a star

Angular resolution is a critical aspect of astronomical observation, as it determines the minimum resolvable angle between two objects. This is governed by the Rayleigh criterion, which states that the minimum resolvable angle is proportional to the wavelength (λ) over the diameter (D) of the aperture of a monolithic telescope. While larger aperture telescopes have advantages such as smaller angular resolution and higher sensitivity to dim and small exoplanets, they also have disadvantages such as higher launch cost, design intricacies, and high mission cost. Using multi-aperture telescopes can be a cost-effective alternative as they work on the principles of baseline interferometry, making the minimum resolvable angle proportional to λ/B, where B is the baseline. In this work, we compare the performance of a single monolithic telescope to a multi-aperture alternative with the same effective glass area in the context of hypothesis testing between two scenarios - H1 (a star) and H2 (a star with an exoplanet). We formulate the theory based on likelihood ratio tests and find that the multi-aperture telescope performs better than the monolithic telescope when direct detection is used on the focal plane. While this is expected, the performance can be further improved by using quantum-inspired detecting strategies. We utilize Quantum Binary Spatial Mode Demultiplexing (BSPADE) to process the point spread function (PSF) of the telescopes and find better performance compared to the respective direct detection measurement. Therefore, our efforts can be viewed as one of the initial steps toward employing quantum-inspired detection techniques in sparse aperture configurations for high-contrast imaging applications. In conclusion, our work shows that multi-aperture telescopes are an effective alternative to monolithic telescopes for object discrimination and potentially for super-resolution imaging and their performance can be further improved by using quantum-inspired detection strategies. With their cost-effectiveness (see Appendix C) and potential for high performance, multi-aperture telescopes can significantly advance our ability to observe and study exoplanets and other celestial objects. Future research can explore ways to optimize the performance of multi-aperture telescopes and further improve their capabilities for astronomical observation, potentially facilitating the detection and imaging of Earth-like exoplanets.

Application range of crosstalk-affected spatial demultiplexing for resolving separations between unbalanced sources

Super resolution is one of the key issues at the crossroads of contemporary quantum optics and metrology. Recently, it was shown that for an idealized case of two balanced sources, spatial mode demultiplexing (SPADE) achieves resolution better than direct imaging even in the presence of measurement crosstalk (Gessner et al 2020 Phys. Rev. Lett. 125 100501). In this work, we consider arbitrarily unbalanced sources and provide a systematic analysis of the impact of crosstalk on the resolution obtained from SPADE. As we dissect, in this generalized scenario, SPADE’s effectiveness depends non-trivially on the strength of crosstalk, relative brightness and the separation between the sources. In particular, for any source imbalance, SPADE performs worse than ideal direct imaging in the asymptotic limit of vanishing source separations. Nonetheless, for realistic values of crosstalk strength, SPADE is still the superior method for several orders of magnitude of source separations.

Spatial-mode demultiplexing for enhanced intensity and distance measurement.

Spatial-mode demultiplexing (SPADE) has recently been adopted to measure the separation in the transverse plane between two incoherent point-like sources with sub-wavelength separation. It has been argued that this approach may yield extraordinary performances in the photon-counting regime. Here, we explore SPADE as a tool for precision measurements in the regime of bright, incoherent sources. First we analyze the general problem of estimating the second moments of the source's intensity distribution, for an extended incoherent source of any shape. Our theory predicts a substantial improvement in signal-to-noise ratio (SNR) of SPADE over direct imaging in the sub-wavelength regime. Second, we present an experimental application of SPADE to the case of two point-like, bright sources. We demonstrate the use of this setup for the estimation of the transverse separation and for the estimation of the relative intensity, confirming the expected improvement in SNR.

Quantum noise spectroscopy as an incoherent imaging problem

I point out the mathematical correspondence between an incoherent imaging model proposed by my group in the study of quantum-inspired superresolution [Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016)] and a noise spectroscopy model also proposed by us [Tsang and Nair, Physical Review A 86, 042115 (2012); Ng et al., Physical Review A 93, 042121 (2016)]. Both can be regarded as random displacement models, where the probability measure for the random displacement depends on unknown parameters. The spatial-mode demultiplexing (SPADE) method proposed for imaging is analogous to the spectral photon counting method proposed in Ng et al. (2016) for optical phase noise spectroscopy -- Both methods are discrete-variable measurements that are superior to direct displacement measurements (direct imaging or homodyne detection) and can achieve the respective quantum limits. Inspired by SPADE, I propose a modification of spectral photon counting when the input field is squeezed -- simply unsqueeze the output field before spectral photon counting. I show that this method is quantum-optimal and far superior to homodyne detection for both parameter estimation and detection, thus solving the open problems in Tsang and Nair (2012) and Ng et al. (2016).

Quantum-limited resolution of partially coherent sources

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.

Resolving mutually-coherent point sources of light with arbitrary statistics

We analyze the problem of resolving two mutually coherent point sources with arbitrary quantum statistics, mutual phase, and relative and absolute intensity. We use a sensitivity measure based on the method of moments and compare direct imaging with spatial-mode demultiplexing (SPADE), analytically proving advantage of the latter. We show that the moment-based sensitivity of SPADE saturates the quantum Fisher information for all known cases, even for non-Gaussian states of the sources.

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

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.

State-vector geometry and guided-wave physics behind optical super-resolution.

We examine the state-vector geometry and guided-wave physics underpinning spatial super-resolution, which can be attained in far-field linear microscopy via a combination of statistical analysis, quantum optics, and spatial mode demultiplexing. A suitably tailored guided-wave signal pickup is shown to provide an information channel that can distill the super-resolving spatial modes, thus enabling an estimation of sub-Rayleigh space intervals ξ. We derive closed-form analytical expressions describing the distribution of the ξ-estimation Fisher information over waveguide modes, showing that this information remains nonvanishing as ξ → 0, thus preventing the variance of ξ estimation from diverging at ξ → 0. We demonstrate that the transverse refractive index profile nQ(r) tailored to support the optimal wave function ψQ(r) for super-resolving ξ estimation encodes the same information about ξ as the entire manifold of waveguide modes needed to represent ψQ(r). Unlike ψQ(r), nQ(r) does not need a representation in a lengthy manifold of eigenmodes and can be found instead via adaptive feedback-controlled learning.

Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach

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 (SPADE) 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.

Moment-based superresolution: Formalism and applications

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 on 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.

Attaining quantum limited precision of localizing an object in passive imaging

We investigate our ability to determine the mean position, or centroid, of a linear array of equally-bright incoherent point sources of light, whose continuum limit is the problem of estimating the center of a uniformly-radiating object. We consider two receivers: an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) Spatial-mode Demultiplexing (SPADE) in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher Information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the Quantum Fisher Information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly sub-optimal, although not by a large margin. We also find that the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object) is not only suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous, uniformly-bright object in terms of its length, and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.

Resolution limits of spatial mode demultiplexing with noisy detection

We consider the problem of estimating the spatial separation between two mutually incoherent point light sources using the super-resolution imaging technique based on spatial mode demultiplexing (SPADE) with noisy detectors. We show that in the presence of noise, the resolution of the measurement is limited by the signal-to-noise ratio (SNR) and the minimum resolvable spatial separation has a characteristic dependence of [Formula: see text]. Several detection techniques, including direct photon counting, as well as homodyne and heterodyne detection are considered.

Quantum limits of localisation microscopy

Localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two—implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits.

Semiparametric estimation for incoherent optical imaging

The theory of semiparametric estimation offers an elegant way of computing the Cram\'er-Rao bound for a parameter of interest in the midst of infinitely many nuisance parameters. Here I apply the theory to the problem of moment estimation for incoherent imaging under the effects of diffraction and photon shot noise. Using a Hilbert-space formalism designed for Poisson processes, I derive exact semiparametric Cram\'er-Rao bounds and efficient estimators for both direct imaging and a quantum-inspired measurement method called spatial-mode demultiplexing (SPADE). The results establish the superiority of SPADE even when little prior information about the object is available.

Quantum-limited estimation of the axial separation of two incoherent point sources

Improving axial resolution is crucial for three-dimensional optical imaging systems. Here we present a scheme of axial superresolution for two incoherent point sources based on spatial mode demultiplexing. A radial mode sorter is used to losslessly decompose the optical fields into a radial mode basis set to extract the phase information associated with the axial positions of the point sources. We show theoretically and experimentally that, in the limit of a zero axial separation, our scheme allows for reaching the quantum Cram\'er-Rao lower bound and thus can be considered as one of the optimal measurement methods. Unlike other superresolution schemes, this scheme does not require neither activation of fluorophores nor sophisticated stabilization control. Moreover, it is applicable to the localization of a single point source in the axial direction. Our demonstration can be useful to a variety of applications such as far-field fluorescence microscopy.

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

Larson and Saleh [Optica 5, 1382 (2018)] suggest that Rayeleigh's curse can recur and become unavoidable if the two sources are partially coherent. Here we show that their calculations and assertions have fundamental problems, and spatial-mode demultiplexing (SPADE) can overcome Rayleigh's curse even for partially coherent sources.

Quantum limit to subdiffraction incoherent optical imaging

The application of quantum estimation theory to the problem of imaging two incoherent point sources has recently led to new insights and better measurements for incoherent imaging and spectroscopy. To establish a more general limit beyond the case of two sources, here I evaluate a quantum bound on the Fisher information that can be extracted by any far-field optical measurement about the moments of a subdiffraction object. The bound matches the performance of a spatial-mode-demultiplexing (SPADE) measurement scheme in terms of its scaling with the object size, indicating that SPADE is close to quantum-optimal. Coincidentally, the result is also applicable to the estimation of diffusion parameters with a quantum probe subject to random displacements.

Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation

We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent on-axis source with brightness epsilon or by two epsilon {mathrm{/}}2-brightness incoherent sources that are symmetrically disposed about the optical axis. Using the exact thermal-state model of the field, we derive the quantum Chernoff bound for the detection problem, which specifies the optimum rate of decay of the error probability with increasing number of collected photons that is allowed by quantum mechanics. We then show that recently proposed linear-optic schemes approach the quantum Chernoff bound—the method of binary spatial-mode demultiplexing (B-SPADE) is quantum-optimal for all values of separation, while a method using image inversion interferometry (SLIVER) is near-optimal for sub-Rayleigh separations. We then simplify our model using a low-brightness approximation that is very accurate for optical microscopy and astronomy, derive quantum Chernoff bounds conditional on the number of photons detected, and show the optimality of our schemes in this conditional detection paradigm. For comparison, we analytically demonstrate the superior scaling of the Chernoff bound for our schemes with source separation relative to that of spatially resolved direct imaging. Our schemes have the advantages over the quantum-optimal (Helstrom) measurement in that they do not involve joint measurements over multiple modes, and that they do not require the angular separation for the two-source hypothesis to be given a priori and can offer that information as a bonus in the event of a successful detection.

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

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 the second and higher-order moments.

Volume holographic spatial mode demultiplexer with a dual-wavelength method.

Volume holographic demultiplexers (VHDMs) provide spatial mode demultiplexing using simple optical systems. However, applying VHDM to practical optical communication systems is difficult, as typical holographic media have no sensitivity in the infrared region, which includes optical transmission bands. In this paper, we propose a VHDM scheme combined with a dual-wavelength method (DWM). Using the DWM, VHDMs are able to perform mode demultiplexing in the optical transmission bands. We experimentally demonstrated the basic operation of our proposal using experiments performed at an 850-nm wavelength. In addition, we performed numerical simulations to investigate the application of VHDM to the C-band.