Quantum-optimal direct imaging coronagraph implemented with spatial mode sorters

Abstract

Imaging systems and associated imaging methods are described. One example imaging system includes a lens positioned to receive a light associated with a scene, and a first spatial mode sorter positioned along an optical path to receive the light from the lens and to demultiplex the incident light incident. The imaging system also includes an occulting mask, and a second spatial mode sorter. The occulting mask is positioned between the first spatial mode sorter and the second spatial sorter to inhibit one or more predetermined modes of the first multiplexed light from propagating towards the second spatial sorter, the second spatial mode sorter is configured to recombine light received thereon to produce an output light for detection by an imaging detector, the output light allows for detection by the imaging detector of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit.

Description

QUANTUM-OPTIMAL DIRECT IMAGING CORONAGRAPH IMPLEMENTED WITH SPATIAL MODE SORTERS CROSS-REFERENCE TO RELATED APPLICATION

[0001] This patent document claims priority to and benefits of U.S. Provisional Appl. No. 63 / 673,062, entitled “QUANTUM-OPTIMAL DIRECT IMAGING CORONAGRAPH IMPLEMENTED WITH SPATIAL MODE SORTERS” and filed on July 18, 2024. The entire contents of the before-mentioned patent application are incorporated by reference as part of the disclosure of this document. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under Grant No.: DGE- 2137419 awarded by the National Science Foundation. The government has certain rights in the invention. TECHNICAL FIELD

[0003] This invention relates to a direct imaging coronagraph based on spatial mode sorting. BACKGROUND

[0004] A coronagraph is an optical instrument designed to block or reduce the light from a bright star, enabling direct observation of nearby faint objects such as exoplanets or circumstellar disks. Traditional coronagraphs face challenges in achieving high contrast and precise wavefront control due to diffraction and optical aberrations. Advances in coronagraphic technology aim to enhance imaging performance by incorporating novel optical designs, adaptive elements, and improved starlight suppression techniques. SUMMARY

[0005] Disclosed are systems, methods, and devices related to a direct imaging coronagraph for imaging astronomical objects based on spatial mode sorting.

[0006] The disclosed systems, methods, and devices use spatial mode sorters to reject optical modes of light corresponding to a fundamental mode of an optical imaging -1- 044974.8137.WO00\182811491.1instrument from being received by an imaging detector. By rejecting the fundamental mode, the disclosed embodiments enable imaging of a dim astronomical object in orbit around a brighter astronomical object, including in scenarios in which a substantial brightness contrast exists between the two objects.

[0007] In one example implementation, the direct imaging coronagraph receives light from an illumination source and uses a first moder sorter to demultiplex the light and reject a specific mode of the light. Light in the residual modes then propagates towards a second mode sorter which coherently recombines the demultiplexed light to form an image.

[0008] One advantage of the disclosed embodiments is that the direct imaging coronagraph may be utilized to image astronomical objects below the Rayleigh limit which may aid in the detection and localization of exoplanets. Among other features and benefits, the disclosed technology may be deployed at the focal plane of next- generation space-based telescopes in order to experimentally reach the fundamental limits of exoplanet detection and localization imposed by quantum mechanics.

[0009] In one aspect, an imaging system is disclosed. The imaging system comprises: a lens positioned to receive a light associated with a scene; a first spatial mode sorter positioned, along an optical path, to receive the light from the lens and to demultiplex the light incident thereupon; an occulting mask; and a second spatial mode sorter, wherein: the occulting mask is positioned between the first spatial mode sorter and the second spatial sorter to receive a first multiplexed light and inhibit one or more predetermined modes of the first multiplexed light from propagating towards the second spatial sorter, the second spatial mode sorter is configured to recombine light received thereon to produce an output light for detection by an imaging detector, the output light allows for detection by the imaging detector of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit.

[0010] In another aspect, an imaging system is disclosed. The imaging system comprises: a polarization element; a polarizing beam splitter (PBS); one or more additional polarization elements; a spatial mode sorter; a first mirror; a reflective element; and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and -2- 044974.8137.WO00\182811491.1the second optical path are associated with a scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements, and the PBS to reach the reflective element.

[0011] In yet another aspect, a method for imaging based on spatial mode sorting is disclosed. The method comprises: operating an imaging device to exclude one or more optical modes of an input light from a scene, wherein the imaging device comprises: an imaging detector, a polarization element, a polarizing beam splitter (PBS), one or more additional polarization elements, a spatial mode sorter, a first mirror, a reflective element, and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and the second optical path are associated with the scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements and the PBS to reach the imaging detector; receiving data associated with a measurement acquired by the imaging detector; and obtaining an image of the scene based on the data, wherein the data represents the input light with the one or more modes excluded.

[0012] The above and other aspects and implementations of the disclosed technology are described in more detail in the drawings, the description and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] FIG. 1A shows a schematic of an optical system according to an embodiment of the disclosed technology.

[0014] FIG. 1B shows another schematic of an optical system according to an embodiment of the disclosed technology. -3- 044974.8137.WO00\182811491.1

[0015] FIG. 1C shows example intensity profiles obtained in a demonstration performed in accordance with disclosed techniques.

[0016] FIG. 2A shows a graph of example theoretical photon detection probabilities obtained in a demonstration performed in accordance with disclosed techniques.

[0017] FIG. 2B shows an example distribution of classical Fisher information obtained in a demonstration performed in accordance with disclosed techniques.

[0018] FIGS.3A-3C show examples of calibration data obtained in a demonstration performed in accordance with disclosed techniques.

[0019] FIGS. 4A-4B show additional examples of calibration data obtained in a demonstration performed in accordance with disclosed techniques.

[0020] FIG.5 shows plots to provide a comparison between a simulated and an experimental coronagraph based on the disclosed technology.

[0021] FIGS. 6A-6B show plots of example data obtained in a demonstration performed in accordance with disclosed techniques.

[0022] FIGS.7A-7B show plots of example data obtained in an analysis performed in accordance with disclosed techniques.

[0023] FIG.8 shows plots of example data obtained in an analysis performed in accordance with disclosed techniques.

[0024] FIG.9 shows plots of example data obtained in an analysis performed in accordance with disclosed techniques.

[0025] FIG.10 show plots of example data obtained in an analysis performed in accordance with disclosed techniques.

[0026] FIG. 11 shows a schematic of an optical system according to an embodiment of the disclosed technology.

[0027] FIG.12 shows an example histogram obtained in an analysis performed in accordance with disclosed techniques. -4- 044974.8137.WO00\182811491.1

[0028] FIGS. 13A-13B show plots of example data obtained in an analysis performed in accordance with disclosed techniques.

[0029] FIG. 14 shows an example method according to an embodiment of the disclosed technology. DETAILED DESCRIPTION

[0030] Coronagraphs are astronomical imaging systems designed for imaging dim objects (e.g. exoplanets, zodiacal dust, unbalanced binary star systems) in orbit around a brighter object (e.g., a star). Recently, it was proved that a coronagraph which manages to exclusively reject the fundamental mode of a telescope achieves the quantum limits of exoplanet detection and localization.

[0031] Discovering exoplanets via direct-imaging is fundamentally limited by the extreme brightness contrast between an exoplanet and its host star. Recently, it was theoretically demonstrated that exclusively rejecting the fundamental mode of a telescope prior to imaging yields a quantum-optimal measurement for detecting and localizing an exoplanet. Conceptually, this approach enhances shot-noise limited sensitivity to exoplanet signatures by discarding all light emitted by an on-axis star whilst maximizing throughput for an off-axis exoplanet.

[0032] The challenge of discovering habitable planets beyond our solar system has motivated astronomers to develop a diverse repertoire of exoplanet detection techniques. Broadly speaking, transit photometry, radial velocity, gravitational microlensing, and astrometry methods all monitor perturbations to the brightness, position, and spectrum of a prospective host star over time to infer the presence and dynamics of a dim orbiting companion. While these methods have found tremendous success in detecting exoplanets, contributing over 5,500 confirmed exoplanet discoveries as of 2024, they fundamentally rely on indirect observations which provide limited insights toward finer characteristics of the planets themselves. Remote sensing of detailed planetary features such as atmospheric composition, weather patterns, surface temperature, surface gravity, and the distribution of liquid and land bodies over the planet's surface are crucial for understanding extrasolar chemical environments and identifying potential biosignatures. -5- 044974.8137.WO00\182811491.1

[0033] By comparison, direct imaging techniques aspire to observe / resolve orbiting exoplanets themselves, thereby providing more comprehensive planetary data. Exo-Earths occupying the habitable zone are often extremely dim and close to their host stars, posing a significant challenge for direct imaging. Under these circumstances, a conventional telescope focuses light from the exoplanet onto the diffraction features of the host star, effectively burying the exoplanet signal within the photon shot noise of the detected starlight. Developments in coronagraph technology have enabled complete nulling of an on-axis point-like star so that residual light arriving on the focal plane originates entirely from the exoplanet.

[0034] Key to reaching these performance levels is optical rejecting of the fundamental mode (i.e., the complex point-spread function (PSF)), of the telescope so as to remove all photons emitted by the star. In doing so, state-of-the-art coronagraphs suppress photon shot noise intrinsic to measurements of the optical field, thereby enhancing the signal-to-noise ratio of exoplanet signatures. Recent prior work proved that a direct-imaging coronagraph which manages to exclusively reject the fundamental mode of the telescope achieves the quantum information limits of exoplanet detection and localization under extreme star-planet contrasts. High-performance coronagraphs such as the phase-induced amplitude apodization complex mask coronagraph (PIAACMC) and the vortex coronagraph excessively attenuate light in other modes beyond the fundamental mode, leading to sub-optimal performance particularly in the sub-Rayleigh regime.

[0035] The present patent document discloses, among other things, a demonstration of a quantum-optimal direct imaging coronagraph (e.g., see FIG. 1A) which employs two cascaded spatial mode sorters to: (1) optically null the fundamental mode, and (2) reconstitute the field for a direct imaging measurement. Using this example system, we are able to localize synthetic exoplanets below the diffraction limit at 103: 1 star-planet brightness contrast.

[0036] In an example embodiment, a direct imaging coronagraph capable of completely attenuating all on-axis starlight while transmitting light from off-axis secondary sources is disclosed. The design makes use of forward and inverse spatial mode sorters implemented with multi-plane light converters. -6- 044974.8137.WO00\182811491.1

[0037] In another example embodiment, a benchtop system that involves a forward and inverse pass through a free-space programmable spatial mode sorter configured to isolate photons in a PSF-adapted basis is disclosed. On the forward pass, the fundamental mode is rejected, eliminating all light from an on-axis point-like star. On the inverse pass, the residual modes are coherently recombined for direct imaging of a dim companion. In one example demonstration performed in accordance with disclosed techniques, we localize an artificial exoplanet separated by distances below the Rayleigh diffraction limit σ from its host star. Under a star-planet contrast ratio of 103: 1, the coronagraph achieves localization accuracy between σ / 50 and σ / 100 within distances ≤ .6σ from the star.

[0038] Various demonstrations of the working principles of an example quantum- optimal direct-imaging coronagraph based on spatial mode (de)multiplexing are disclosed.

[0039] In one example demonstration, a disclosed embodiment of a quantum- optimal direct imaging coronagraph using a spatial mode sorter implemented with a multi-plane light converter (MPLC) was used to demonstrate localization of an artificial exoplanet at sub-diffraction angular separations from an artificial host star under a 103: 1 star-planet brightness contrast. Denoting the Rayleigh diffraction limit of our imaging system as σ, the absolute error of the empirical mean maximum likelihood estimator (MLE) remains below 0.03σ over the separation range [0, 0.6]σ. The empirical precision (standard deviation) of the MLE varies over the sub-diffraction regime between ∼ 0.1σ and ∼ 0.01σ for an exoplanet in the range [0,0.1]σ and [0.1,0.6]σ, respectively. We invoke a probabilistic measurement model to characterize the impact of real-world noise sources and experimental constraints expected in an operating environment which hinder the attainment of quantum-limited performance set by shot noise.

[0040] FIG. 1A shows a schematic of an optical system 100 comprising an example embodiment of a quantum-optimal direct-imaging coronagraph based on spatial mode sorting. The forward mode sorter 110 demultiplexes the field incident on the image plane from the imaging optics 120 into a PSF-matched basis. The field is associated with light from the scene 130. Light in the fundamental mode is rejected by a mask 140 at the sorting plane while light in the remaining modes propagates freely to -7- 044974.8137.WO00\182811491.1an inverse mode sorter 150. The inverse mode sorter 150 coherently recombines the light to form an image at the detector 160. In some implementations of the coronagraph, the light in the fundamental mode is predominantly starlight.

[0041] FIG.1B shows another schematic of an optical system 170 comprising an example embodiment of a quantum-optimal coronagraph. On the forward pass, the MPLC sorts the field into a PSF-adapted basis. The fundamental mode is sent through a pinhole mirror (M4) to a beam dump (BD), while all other modes are reflected back into the system. Upon propagating backward through the MPLC, the mode sorting process is inverted, providing an image of the scene without the on-axis star. A Faraday rotator non-reciprocally alters the polarization to split the optical path depending on the beam propagation direction. In some implementations, the quantum-optimal coronagraph may be implemented by double-passing the field through a Zernike mode sorter.

[0042] The system 170 performs a forward and backward pass through a free- space programmable mode sorter implemented on a 3-plane MPLC. On the forward pass, the MPLC spatially demultiplexes the incident optical field in the Fourier-Zernike basis and focuses light in each mode to a distinct spot on the sorting plane. The spot corresponding to the fundamental mode is directed to the opening of a pinhole mirror and absorbed at a beam dump. The remaining modes reflect off the pinhole mirror and are sent backwards through the mode sorter. The unitary nature of spatial mode sorters ensures that this backward pass inverts the mode transformation. Non-reciprocal polarization elements split the optical path for the forward (pre-nulling) and backward (post-nulling) pass, sending the filtered field to a detector. Through this process, the field at the detector plane is identical to the field at the focal plane minus all contribution from the fundamental mode. Thus, we functionally emulate the cascaded design of FIG. 1A with a single mode sorter. The 4f imaging system used in our setup ischaracterized by a circular entrance pupil of diameter D ^ 400 µm and focal lengthf ^ 200 mm operating at wavelength ^ ^ 532nm , giving a Rayleigh resolution of^ ^ 1.22 ^ fD ^ 324 µm on the object plane.the system 170 comprises a multi-plane light conversion (MPLC) spatial mode sorter retrofitted with polarization elements that enable path splitting between forward-propagating and backward-propagating beams. On the -8- 044974.8137.WO00\182811491.1forward pass, the mode sorter spatially demultiplexes the incident optical field into asubset of Zernike modes ^^0, ^ 1 , ^ 2 , ^ 3 ^ which collectively couple most of the lightemitted by a vertically- (i.e., the planet) in the sub-Rayleigh rangey ^ ^ ^ ^1,1 ^ , where ^ islimit. Light in each mode focuses toa well-separated spot on an intermediate sorting plane, which coincides with the surfaceof a pinhole mirror. The fundamental mode^ 0 , containing all light from on-axis star, isdirected to the opening of the pinhole mirror and absorbed at a beam dump. The remaining modes reflect off the pinhole mirror and propagate backward through the mode sorter. The unitary nature of spatial mode sorting ensures that the backward pass inverts the Zernike basis transformation. Through this process, the field at the detector matches the field at the focal plane, but with the fundamental mode removed.

[0044] In an example demonstration, disclosed techniques were used to demonstrate exoplanet localization at sub-Rayleigh star-planet separations; the regime where quantum-optimal coronagraphs offer the greatest theoretical advantage over existing high-performance coronagraph designs. To sample this separation regime, we align the coronagraph to a bright on-axis point-source (artificial star) and vertically stepthe position of a second point-source (artificial exoplanet) r^ e^ ^0, y e ^over the discretedomain ye^^ ^ ^ ^.85 : .0215:.85 ^ ^ . The artificial star-was set to103: 1.

[0045] For a circular aperture, light from a sub-Rayleigh exoplanet couples predominantly to lower-order Fourier-Zernike modes. Therefore, only a few modes are relevant at small angular separations. We configured our programmable MPLC to sorta truncated basis consisting of modes ^ ^ ^ ^^^^0 ^r ^, ^ ^ 1 ^ r ^ , ^ ^ 2 ^ r ^ , ^ ^ 3 ^ r ^ ^ shown in FIG. 1Cwhere ^ r^ is the fund^^0 ^ ^ amental r ^ ^ x, y ^ are thecoordinates of the image plane. FIG. 1C shows exampleof the truncated Fourier-Zernike mode subset sorted by the MPLC. Collectively, these modes contain a majority of the energy and the Fisher information in the field generated by a sub-Rayleigh companion as shown in FIG.2, which will be described in further detail.We define the nominal region of support for this truncated basis to be ye ^ .6 ^ .Expanding the basis to more than four modes was found tothe -9- 044974.8137.WO00\182811491.1cross-talk of the mode sorter due to the limited number of phase masks available on our programmable MPLC. In principle, introducing more masks would allow one to sort more modes while maintaining low cross-talk.

[0046] FIG. 2A shows example theoretical photon detection probabilities (i.e., coupling efficiencies) of the Fourier-Zernike modes in the truncated subset as a function of off-axis source location over the sub-Rayleigh regime. The nominal 'region of support' is given by off-axis positions in the domain [0,0.6]σ within which light dominantly couples to the truncated mode basis.

[0047] FIG.2B shows an example distribution of the classical Fisher information in each mode as a function of off-axis source location. The classical Fisher information of the truncated mode basis remains within 10% of the quantum Fisher information limit over the domain ~ [0,0.4]σ. Outside this domain, the information concentrates in higher- order modes that are ignored by our mode-sorter.

[0048] In the description that follows, a measurement model based on disclosed techniques is provided and applied to an example embodiment of a coronagraph.

[0049] We invoke a theoretically and empirically-driven probability distribution forthe direct imaging measurement. Let X ^r ^0 ^ ^ ^ M be a random vector containing thenumber of photons measured atdetector over an integration period T ^ when imaging a single point-source located at position r0. This vector is characterized by a Poisson distribution, X^r^0 ^^ Poiss^^0 q ^ r ^0 ^^ s ^ ^ D 1 ^ (1)where ^ is the ph q r ^^ ^ M0 oton flux entering the pupil from the point source, ^ 0 ^ is(post- photon arrival probabilities at each detector pixelthepupil field through the coronagraph, s^ ^ M is a structured background rate induced bystray light reflections, and ^D1 ^ ^ M is the spatially-uniform dark click rate of ourdetector operating a roomIn our notation, the Poiss function applies element-wise since the photons arrivals at each pixel are independent. For simplicity, all rates are given in photons per integration period T. The post-nulling probability vector can be further decomposed as, -10- 044974.8137.WO00\182811491.1q ^r ^0 ^^ ^^C ^† ^ † z ^ r ^ 20 ^(2) where^ 2 is appliedvectorized optical field at the focal plane of the telescope induced by a single point ^ source at location r0. We have also introduced several system-dependent matrices:^^^ M^ K is a change of basis matrix whose columns are vectorized PSF-^^ve^ kc ^^ ^^ k ^ r ^ ^ ^, ^| | ^^ ^ ^^ ^ ^ ^ (3)

[0050] C^^ K ^ K is the^ 00T ^C^ ^ ^(4)which represents rejection of thetalk matrix of the mode sorter whose entries are determined from calibration measurements.

[0051] A synthetic measurement of a star-planet system Y ^ X s ^ X e isconstructed by adding multiple measurement realizations of the artificial star and planetilluminated independently such thatXNs ^ k ^^ s^^k^1X^ 0 ^ and Xe ^ X ^r^ ^ e^where reis the position of the exoplanet. Thisconstruction is motivated by a technical constraint in our experimental setup; our light source is a coherent laser, requiring us to illuminate and image each point-source independently to avoid interference effects. Adding the measurements in this fashion constitutes an approximation of measuring two incoherently radiating sources. The complete measurement model is then given by, Y^r^e ^^ Poiss^^ ^0 p ^ r e ^ ^ ^ B p B ^ (5)where we have made the -11- 044974.8137.WO00\182811491.1p ^r^ ^e ^^ ^1 ^ b ^ q ^ 0 ^ ^ b q ^ r ^e ^(6a)with relative brightness ofofmeasurement realizations being N ^ Ns ^ 1. The two terms in Equation 5 are the spatialflux rates induced by radiation from the scene ^0 p ^r^ e^ and the background ^ B p B . Forour particular experimental setup, we have K ^ 4 , N 3 2s ^10 , and M ^77 .

[0052] FIGS. 3A-3C show example calibration measurements made for characterizing modal cross-talk through the MPLC mode sorter. We estimate the cross- talk matrix from these calibration measurements by optimizing a least-squares objective function between the measurement model and the data. •Z ^^ K ^ P : Zernike expansion coefficients for each off-axis source(real / imaginary parity removed). •M ^^ K ^ P : Measurements of each mode intensity.• ^^^ K^ K : Cross-talk matrix•M ^ ^ ^ Z ^2 2F : Frobenius norm cost function.• ^ ^ ^ ^ ^ ^ : Gradient of the cost function with respect to the cross-talkmatrix. •G ^^ ^^^ ^ ^ † ^ ^ ^ †^ : Riemannian gradient of the cost function(Hermitian).

[0053] We optimize the cost function with respect to the cross-talk matrix underthe constraint that ^ is unitary. Gradient descent algorithms can be used for suchconstrained optimization problems and are described in greater detail elsewhere. We -12- 044974.8137.WO00\182811491.1deploy the following simple algorithm. Instantiate ^ 0 ^ I and learning rate µ ^ 0. Atiteration t, ^t^1 ^expm ^ ^µG t ^ ^ t (7)where Gt ^ G ^ ^ t ^ . The iterations are performed until convergence. In FIG. 3A, weshow the mode intensity data alongside the theoretical mode intensities afterapplying cross-talk matrix shown in FIG.3B.

[0054] More specifically, FIG.3A shows example cross-talk characterization data (dotted) of relative mode intensity in each channel as a function of off-axis source position. The relative intensities were measured at the sorting plane of the MPLC. The solid curve corresponds to the theoretical mode intensities under a least-squares fit of the unitary cross-talk matrix to the data. FIG. 3B shows example magnitude of the entries in the cross-talk matrix. The relative power leakage from the fundamental mode into all other modes sums to ~ 10−3, setting the resolvable star-planet contrast limit of our system. FIG. 3C shows example measurements at the sorting plane with region segmentation for measuring energy in each mode.

[0055] Construction of the noise model for the experimental system required several calibration measurements. FIGS. 4A-4B show example calibration data forestimating the dark noise rate ^ D and structured background rate s of equation 1. FIG.4A shows an example Poisson fit to detector dark-noise histogram. The dark rate ofthe detector was found to be ^D ^ 246 photons per image integration period T. Therate dark noise rate ^D ^ 246 [photons / T] was found by fitting a Poisson distribution toa histogram of photon counts recorded by our camera operating in a dark environment at room temperature. FIG.4B shows an example of a structured background profile s induced by stray light and undesired reflections in the experimental setup. The structured background rate was found by averaging multiple images of integration time T while the system was illuminated by an on-axis source. We then subtracted off the dark noise rate from the average image. ^ s^1 n ^X^i^^^ 0^(8)-13- 044974.8137.WO00\182811491.1

[0056] The background contribution term employed in our single-shot measurement model arises from a combination of undesired stray light reflections and the dark noise rate of the pixels in the camera. We thus decompose the background into two terms as, ^Bp B ^ s ^ ^ D 1 (C3)

[0057] wheres^ ^ Mrate observed across ourexperimental measurements, and ^ D ^ ^ is the spatially-uniform dark click rate of eachpixel operating a roomFurthermore,1^ ^ Mis a vector of ones, and wehave impose the constraint 1TpB ^ 1 such that p B is a discrete probability distribution.FIGS. 4A-4B show calibration data for estimating the dark noise rate ^ D and structured^background rate s of Equation 1. An estimate of the dark noise rate ^D ^ 246[photons / T] was found by fitting a Poisson distribution to a histogram ofcounts recorded by our camera operating in a dark environment at room temperature. The structured background rate was estimated by averaging n = 103images of the an on- axis source as viewed through the coronagraph, ^ s^1 n ^[X^i^^ 0^^^^ D1]where we havedetector due to imperfect nulling of the fundamental mode (i.e. modal cross-talk) are negligible compared to the ^ dark noise rate^0q ^ 0 ^^^ ^ D.

[0058] of exoplanet localization using disclosed techniques are described herein.

[0059] An example comparison between theoretically predicted and measured intensity distributions for different sub-diffraction off-axis planet locations is shown in FIG.5. In each image, the exoplanet location is at the intersection of the crosshairs. The intensities measurements shown are after subtraction of the structured background and the uniform detector dark noise rate. Asymmetries in the intensity distributions forequidistant off-axis ^y e exoplanet locations stem from modal cross-talk as shown in-14- 044974.8137.WO00\182811491.1FIGS. 3A-C. In FIG. 5, we compare simulated (ideal) and experimental coronagraphimages of the star-planet scene at 103 : 1 contrast. For separations above ye ^ 0.25 ^ ,we observe similar qualitative structures between simulated and image intensity profiles. Below this separation threshold, where much of thelight couples fundamental mode, the noise of our experimental system visually dominates over the signal of the exoplanet. A qualitative asymmetry is also observed between images of the exoplanet positioned at equal distances above and below the optical axis^y e due to modal cross-talk.

[0060] For each exoplanet location y e ^^ , we collected repeated measurementsY ^i ^ ^y e ^ for i ^ 1,..., ^ (with ^^ 100 ) of equal exposure time T = 0.1s. To localize thewe ran a maximum likelihood estimator (MLE),y^ ^ i ^e ^argmax log P^ Y ^ i ^ ^ y e ^(9) y ^^e ^where P^Y ^ ye ^ ^ is given by the measurement model in equation 5. In FIG. 6A, wemapof the estimator, averaged over all experimental trials, as a function of the ground truth exoplanet location. Specifically, FIG. 6A shows the likelihood map over sub-diffraction exoplanet locations averaged over measurements of various ground-truth exoplanet positions. The ridge of peak likelihoods (dark red regions) follow the ground truth exoplanet location. However certain regions near the axis and outside the support of the truncated mode basis exhibit a weakly peaked likelihood map indicating greater uncertainty in these regions. In general, the likelihood map exhibits strong correlation between the maximum likelihood position and the ground-truth position of the exoplanet, though regions near the optical axis appear to have greater estimator uncertainty (weakly peaked likelihood). FIG.6B shows the MLE outputs over all repeated measurements over the exoplanet translation scan in the form of a scatter plot of the maximum likelihood estimates of the exoplanet position at each ground truth position. The colorbar indicates the frequency with which different locations were estimated. Outliers were removed for clarity. The estimator effectivelylocalizes the exoplanet within the nominal region of support ye ^ .6 ^ of the truncated-15- 044974.8137.WO00\182811491.1mode basis. Outside of this domain, the estimator experiences a constant bias due to the finite support of the truncated experimental mode set.

[0061] In the description that follows, a statistical performance analysis of an example embodiment of a coronagraph based on the disclosed technology is disclosed.

[0062] To quantify the efficacy of our coronagraph, we analyze the statistical error and imprecision of the MLE running on repeated experimental trials. For a given, ground truth exoplanet position, we denote the mean and variance of the MLE to be y^e ^ ^P^ Y ^ ye ^ ^ ^ y e ^ and ^2 ye ^ ^P^Y^ y e ^ ^ ^ y ^e ^ respectively. The unbiased empiricalare given by,^y 1 ^ ^^i^e^y(10a) ^ ^e^ 2 ^ (10b) ^

[0063] FIG. 7A shows they e ^ y eover the domain of the exoplanet position scan (i.e., the average exoplanet localization error as a function of off-axis position). The average error was calculated from the MLEs of 100 independent images of the exoplanet collected through the coronagraph at each position. Within the region of support, the error is nearly zero everywhere (i.e.,unbiased) with an imprecision of ^^ye ^ ^ 100 over the domain ye ^ ^0.1,0.6 ^ ^ . In thedeeply sub-Rayleigh regime ye ^ ^.01,.1 ^ ^ the imprecision peaks at approximately^ 50. The coronagraphachieves sub-diffraction localization with low errorwithin the region of support for the truncated mode set.

[0064] FIG.7B shows the estimated imprecision of the MLE as a function of the exoplanet position. Analysis of the statistical variance of the MLEs as a function of exoplanet position is compared to the Cramer-Rao lower bound (CRLB) corresponding to the measurement model. Uncertainty bars on the imprecision estimator were calculated using delete-one jackknife resampling. We find that this imprecision curve roughly aligns with the classical Cramer-Rao Lower Bound (CRLB) computed for our experimental noise model. In particular, we observe a central spike in the imprecision -16- 044974.8137.WO00\182811491.1near the optical axis as most of the exoplanet photons are being discarded in this regime. Additionally, two secondary peaks in imprecision appear further away from the optical axis where the field at the image plane falls outside the support of the truncated mode basis.

[0065] As points of comparison, we numerically evaluate the performance of experimental coronagraph against the PIAACMC, Vortex (charge-2) coronagraph, and the Perfect coronagraph. The Perfect coronagraph is the idealized version of our experimental coronagraph in the absence of mode truncation or cross-talk. To level the playing field between theoretical and experimental performance, we assign the samebackground illumination profile espoused in our experiment ^ B p B to the measurementmodel of each coronagraph. What changes across coronagraphs is the spatial photonflux from the scene ^0 p ^r^ e^. This now depends on singular-value decomposition particular to each which appears in,q(r ^ † ^20) ^ UDV z ^ r 0 ^(11)

[0066] where U , V^^ M ^ Mmatrix of singularto be described inthe Cramer- Rao lower bounds to the exoplanet localization imprecision for each coronagraph. We see that the idealized version of our experimental setup (Perfect coronagraph) outperforms both the PIAACMC and Vortex in the absence of noise. Additionally, we see that the idealized version of our experimental setup (Experiment 4-Mode) outperforms both the PIAACMC and Vortex coronagraph for over sub-diffraction exoplanet locations in the range which corresponds to where the CFI of the truncated mode basis begins to depart from the QFI limit as shown in FIG.2B.

[0067] FIG. 8 shows the comparison of the Cramer-Rao Lower Bounds on imprecision of an unbiased estimator of the exoplanet position for different coronagraphs operating under the same empirical background model found in our experimental system. The quantum limit curve (black dashed) is for an ideal situation without any background contribution. Surprisingly, the imprecision of our experimentalcoronagraph outperforms the other coronagraphs at separations below ye ^ ^3 ^ 10 ^ 3.At moderate contrasts like those considered here (b = 10−3), information still persists in -17- 044974.8137.WO00\182811491.1the fundamental mode. Imperfections in the cross-talk matrix allows some of this information to leak into the other modes.

[0068] FIG. 9 shows the example experimental coronagraph throughput as a function of the scanned source position. As shown in FIG. 9, alignment and magnification discrepancies in a 4f system can be accounted for by fitting the experimental throughput to theoretically-predicted values. Comparing the measured throughput to the theoretical throughput curve of our system revealed a scaling c andbias a between the true source displacements y e relative to the diffraction limit and thedisplacements computed from our system specifications ye^ ^ c ^ y e ^ a ^ . The scalingand bias terms arise from experimental error in magnification and alignment of the optical axis. Accounting for these discrepancies revealed the true rangeye ^ ^ ^ ^.85,.85 ^ over which we scanned our point source.

[0069] The mean photon rate entering the pupil from the star-planet scene is approximated using the Best Linear Unbiased Estimator (BLUE). Let the randomvariable Y ^i ^j ( r^ e)represent the photons collected in the jthpixel of the ithmeasurement ^ for alocation re. The residual between the measurement outcome and its expected value is given by, r^ i ^j ^^0 ^^Y ^ i ^j ^r ^e ^ ^ N( ^ ^0 p j ^ r e ^ ^ ^ Bj )(12)

[0070] The BLUE is^^i 2 ^ r^j^^ 0^ ^^0^^0^where the ^ 2 ^thj ^ ^0 ^ ^ N^ ^ 0 p j ^ r e ^ ^ ^ B j is jpixel.problem over the entire collected data set ^and estimate the mean photon rate to be ^ 0 ^ 3,020,568[photons / T].

[0071] A general discussion ofZernike Modes is now described. Consider an imaging system with a circular pupil of radius R, focal length f, and operatingwavelength ^ . Let ^Xa, Y a ^ and ^Xb, Y b ^ denote the coordinate space of the pupil-18- 044974.8137.WO00\182811491.1plane and focal plane respectively. We define the dimensionless pupil plane and focal plane coordinate vectors, u ^ ^1^ X , Y ^(14a) Ra ar^ R ^^ Xb, Y ^(14b) ^ fb

[0072] The Zernike modes ^^^nm (pupil. In polar coordinates, they are given by, ^^^nm( u , ^ )^ Rnm ^ u ^ ^ m ^ ^ ^ circ ^ u ^ (15a)where theradial index is m^ Sn ^ ^ ^ n, ^ n ^ 2,^ , n ^ 2, n ^ . These modes are defined to satisfy theorthonormality condition, ^^^ ^^nm ^u, ^ ^ ^ ^ ^ n ^ m ^ ^ u , ^ ^ udud ^ ^ ^ nn ^ ^ mm ^

[0073] The Fourierpupil are found to be, ^J^^ in ^2 m n ^1 n ^ 1 ^ 2^r ^^ ^^^which we call theresults, we sort the truncated mode basis ^^^0, ^ ^ 1 , ^ ^ 2 , ^ ^ 3 ^ corresponding to modes-19- 044974.8137.WO00\182811491.1^^^00, ^ ^ 1,^ 1 , ^ ^ 2,0 , ^ ^ 2,2 ^ with order preserved. The squared magnitude of these modes isshown in FIG.1C.

[0074] In the description that follows, an example measurement model of an experimental coronagraph design is provided.

[0075] When imaging a single point-source located at position y on the vertical axis, the number of photons (intensity) measured at each pixel of the detector ismodeled as a random vector X ^y ^ ^ ^ M . We invoke a theoretically and empirically-driven probability described by,X^y ^^ Poiss^^0 p ^ y ^^ s ^ ^ D 1 ^ (17)where ^ is the photon flux at the M0 system pupil generated by the point source, s^ ^is a structured background flux caused by imperfections in oursetup, ^ D is the dark count rate of the detector operating at room temperature, andp ^y ^ ^ ^ M is the non-normalized photon arrival probability distribution over thegiven by: ^ ^ ^† 2p y ^ ^ C ^ z ^ y ^ . Here z ^y ^ ^ ^ 4 is a vector ofexpansion coefficients forin the subset of experimentallysorted Zernike modes. We have also introduced several system matrices: ^^^M^ 4is a truncated change-of-basis matrix whose columns are the vectorized Zernike modes, C^^ 4 ^ 4 is the diagonal coronagraph matrix responsible for nulling the fundamentalmode, and ^^^ 4^ 4 is the Hermitian cross-talk matrix of the mode sorter. The system-specific parameters s, ^ D , and ^ are characterized empirically. A measurement modelof a star-planet system Y ^ X s ^ X e is constructed by adding multiple measurementrealizations of equal integration time T for different source locations such that s^^N3X s^10 ^Xk ^0^and ^ X ^y ^ . Thus the star-planet measurement model isY^ Poiss^N^0^^ ^ 1^ b ^ p ^ 0 ^ ^ b p ^ y e ^ ^ ^ ^ N ^ s ^ ^ D 1 ^ ^ (18)where N ^ Ns ^ 1 is the total number of integrated images and b ^ 1 N is the relativebrightness of the exoplanet to the star. -20- 044974.8137.WO00\182811491.1

[0076] We collect and synthesize 100 experimental measurements of a star- exoplanet system at 103: 1 contrast for 80 equally-spaced exoplanet positions in therange ye ^ ^ ^ ^1,1 ^ . For each measurement, we numerically calculate the maximum-likelihood estimator for the exoplanet position y ˆe . FIG. 10 shows example theoretical(top) and experimental (bottom) direct-imaging measurements for different off-axis exoplanet locations (designated by crosshairs). Structured background and detector dark click rates have been subtracted from the experimental images.

[0077] The experimental results above constitute a proof-of-principle for quantum- optimal direct-imaging coronagraphs implemented with spatial mode sorters. Such systems may provide a new modality for discovering Earth-like exoplanets hidden below the diffraction limit of space-based telescopes.

[0078] As evident from the foregoing, we have developed a quantum-optimal direct imaging coronagraph with a spatial mode sorter. We demonstrate high-precision localization of an artificial exoplanet at sub-Rayleigh distances from its host star under 103: 1 star-planet brightness contrast. The results substantiate the potential value of mode-sorting solutions for astronomical imaging tasks. Among other features and benefits, one pragmatic advantage of this mode sorting coronagraph is that it may be deployed at the focal plane of existing telescopes in a modular fashion - there is no need to integrate new optics.

[0079] In the description that follows, additional details related to the example analysis described above are provided.

[0080] ’Fourier-Zernike’ modes throughout the main text can be obtained as follows. For an imaging system with circular pupil of radius R, focal length f, andoperating wavelength ^ , let ^Xa, Y a ^ and ^Xb, Y b ^ denote the coordinate space of thepupil plane and focal planethe dimensionless pupil plane and focal plane coordinate vectors, u ^ ^1^ X , Y ^

[0083] The Zernike modes ^^^nm ( u ^) constitute a PSF-matched basis over a circularpupil. In polar coordinates, are by,

[0084] ^^^nm( u , ^ )^ Rnm ^ u ^ ^ m ^ ^ ^ circ ^ u ^ (A2a)rangefor a given radial index is m^ Sn ^ ^ ^ n, ^ n ^ 2,^ , n ^ 2, n ^ . These modes are defined tosatisfy the^^^ u, ^ ^

[0089] ^^ nm ^ ^ ^ ^ n ^ m ^ ^ u , ^ ^ udud ^ ^ ^ nn ^ ^ mm ^over the pupil are found to be, ^^ n J ^ 2^r ^nm ^r,^^^ i ^2 m n ^1 n ^ 1 ^ m ^^^the main text. In this work, we sort the truncated mode basis ^^^0, ^ ^ 1 , ^ ^ 2 , ^ ^ 3 ^ corresponding to modes^^^00, ^ ^ 1,^ 1 , ^ ^ 2,0 , ^ ^ 2,2 ^ . The squared magnitudeis shown in FIG. 1C.shot measurement model of Equation 5 can be expanded as follows. The post-nulling optical intensity distribution on the detector is modeled as, ^† † ^ 2q ^r 0 ^^ ^^C ^ ^ z ^ r 0 ^which provides a matrixthe MPLC, nulling the fundamental mode, back-propagating the field to an image plane, -22- 044974.8137.WO00\182811491.1and measuring the intensity on a detector array. Here, ψ0(⃗r0) = vec[ψ(⃗r−⃗r0)] is the vectorized form of the shifted PSF satisfying ψ0(⃗r0)†ψ0(⃗r0) = 1. Ψ is the truncated change-of-basis matrix for which the columns are the vectorized Fourier-Zernike modes [ψ = vek c[ψk(⃗r)] ^| | ^^ ^ ^^^^^ ^^ ^

[0094]

[0095] ^ † ^ ^ I K where I K is the identityon ^ K . The coronagraph matrix is given by,^ 00T ^C^ ^ ^

[0096]

[0097] the fundamental mode (in the absence of cross-talk). While we treat the optical field as a scalar quantity (as the field is always is in a definite linear polarization state throughout our experiment), details on the experimental polarization manipulation for path splitting are illustrated in FIG.11.

[0098] Specifically, FIG. 11 shows another example embodiment of an optical system based on the disclosed technology. The optical system can include, among other components, a detector, beam splitter, optical polarization elements, and an occulting mask (e.g, a mirror comprising a pinhole). FIG.11 shows an unfolded beam path illustrating polarization control for path splitting of forward (black arrows) and backward (red arrows) propagating beams. Variations of FIG. 11 can be made to achieve an optical system, in accordance with the disclosed technology, which utilizes a single mode sorter (e.g., FIG.2B).

[0099] The mean photon rate entering the pupil from the star-planet scene is approximated using the Best Linear Unbiased Estimator (BLUE). Let the randomvariable Y ^i ^^ j^ r e ^ rep-resent the photons collected in the j th pixel of the i th^ measurement for a given exoplanet location re. In accordance with equation 3, each pixel follows distribution -23- 044974.8137.WO00\182811491.1Y ^i ^ r^^ Poiss ^ p r ^^ ^

[0100] j ^ e ^ ^ 0 j ^ e ^ B p B j ^term linear in ^ 0 with zero-meanadditive Gaussian noise,

[0102] Y ^i^j ^ r ^ ^e ^^ ^0 p j ^ r e ^ ^ ^ B p B ^ ^ j(C5) residuals weighted bythe variance of the photon arrivals in each pixel ^ i 2 ^ ^^^M^^ r ^j ^^ 0 ,y e ^ ^

[0107]

[0108] over the entire collected data set and estimate the mean photon rate entering the pupil from a single point source ^^to be ^0 ^ ^ 60 N ^ 3.17 ^ 10[photons / T].

[0109] FIG. 12 shows a histogram over estimates of the scene photon emission rate λ0from which we compute the best linear unbiased estimator (dashed) ˇλ0= 3.17 × 106[Photons ^T−1]. The edges of the FWHM are used in FIGS.7A-7B to determine the range of Cramer-Rao bounds associated with our experiment.

[0110] FIG. 13A shows an example comparison between the Chernoff exponent for various coronagraph designs relative to the quantum limits (black) of exoplanet detection and localization at high contrasts. FIG. 13B shows an example comparison between the Fisher information per photon for the various coronagraph designs relative to the quantum limits (black) of exoplanet detection and localization at high contrasts. A disclosed coronagraph embodiment, represented in FIGS. 13A-13B as “Perfect Coronagraph” (dashed yellow), is shown to theoretically saturate quantum limits. State- of-the-art systems such as the phase-induced amplitude apodization complex mask -24- 044974.8137.WO00\182811491.1coronagraph (PIAACMC) and the vortex coronagraph are notably sub-optimal in the sub-diffraction regime.

[0111] FIG.14 shows a flow diagram of an example method 1400. At step 1410, the method 1400 includes operating an imaging device to exclude one or more optical modes of an input light from a scene. At step 1420, the method 1400 includes receiving data associated with a measurement acquired by the imaging detector. At step 1430, the method 1400 includes obtaining an image of the scene based on the data.

[0112] Various implementations of features of the disclosed technology can be made based on the above disclosure, including the examples listed below. Example 1. An imaging system, comprising: a lens positioned to receive a light associated with a scene; a first spatial mode sorter positioned, along an optical path, to receive the light from the lens and to demultiplex the light incident thereupon; an occulting mask; and a second spatial mode sorter, wherein: the occulting mask is positioned between the first spatial mode sorter and the second spatial sorter to receive a first demultiplexed light and inhibit one or more predetermined modes of the first demultiplexed light from propagating towards the second spatial sorter, the second spatial mode sorter is configured to recombine light received thereon to produce an output light for detection by an imaging detector, the output light allows for detection by the imaging detector of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit.

[0113] Example 2. The imaging system of example 1, wherein the occulting mask comprises a mirror with a pinhole, wherein the mirror is positioned such that the one or more predetermined modes propagate through the pinhole.

[0114] Example 3. The imaging system of example 2, comprising a beam dump configured to absorb the one or more predetermined modes after propagation of the one or more predetermined modes through the pinhole.

[0115] Example 4. The system of example 1, comprising: the imaging detector; a polarizing beam splitter (PBS); a polarization element; and a plurality of mirrors including a first mirror positioned to direct light incident thereupon from the PBS towards a position of the imaging detector, wherein the polarization element and the PBS are -25- 044974.8137.WO00\182811491.1positioned along the optical path, wherein the PBS is positioned between the first mirror and the polarization element.

[0116] Example 5. An imaging system, comprising: a polarization element; a polarizing beam splitter (PBS); one or more additional polarization elements; a spatial mode sorter; a first mirror; a reflective element; and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and the second optical path are associated with a scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements, and the PBS to reach the reflective element.

[0117] Example 6. The imaging system of example 5, wherein the reflective element comprises at least two reflective surfaces, each of the at least two reflective surfaces positioned to allow propagation of the first light along the forward optical path and propagation of the second light along the backward optical path.

[0118] Example 7. The imaging system of example 5, comprising: a lens system comprising at least two lenses, wherein the polarization element is positioned produce a polarized light associated with the scene, wherein a first of the at least two lenses is positioned to direct the polarized light therethrough towards a second of the at least two lenses.

[0119] Example 8. The imaging system of example 7, wherein the lens system is configured as a 4f imaging system.

[0120] Example 9. The imaging system of example 5, comprising a beam dump positioned to receive a mode of the first light that propagates through the opening of the second mirror.

[0121] Example 10. The imaging system of example 5, wherein the polarization element is configured to produce linearly polarized light. -26- 044974.8137.WO00\182811491.1

[0122] Example 11. The imaging system of example 5, comprising an additional mirror positioned to allow a folded optical path.

[0123] Example 12. The imaging system of example 5, wherein the one or more additional polarization elements include a half-wave plate and a Faraday rotator.

[0124] Example 13. A method for imaging based on spatial mode sorting, comprising: operating an imaging device to exclude one or more optical modes of an input light from a scene, wherein the imaging device comprises: an imaging detector, a polarization element, a polarizing beam splitter (PBS), one or more additional polarization elements, a spatial mode sorter, a first mirror, a reflective element, and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and the second optical path are associated with the scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements and the PBS to reach the imaging detector; receiving data associated with a measurement acquired by the imaging detector; and obtaining an image of the scene based on the data, wherein the data represents the input light with the one or more modes excluded.

[0125] Example 14. The method of example 13, wherein the image enables detection of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit

[0126] Example 15. The method of example 13, wherein the one or more optical modes corresponds to a fundamental optical mode of an optical imaging instrument.

[0127] Example 16. The method of example 13, wherein the image enables detection of a first astronomical object separated by a distance from a second astronomical object, wherein the second astronomical object is brighter than the first astronomical object. -27- 044974.8137.WO00\182811491.1

[0128] Example 17. The method of example 16, wherein the distance is below a Rayleigh diffraction limit of an optical imaging instrument implementing the method.

[0129] Example 18. The method of example 13, wherein the imaging device comprises a beam dump positioned to receive a mode of the first light that propagates through the opening of the second mirror.

[0130] Example 19. An imaging system, comprising: an imaging detector; a lens configured to receive light from a real scene; a first spatial mode sorter configured to demultiplex light from the real scene that is incident thereupon from the lens according to modes of the incident light; a second spatial mode sorter configured to recombine the demultiplexed light; and an occulting mask configured to inhibit one or more predetermined modes of the demultiplexed light from propagating towards the imaging detector, wherein the imaging detector is configured to produce an image of the real scene based on the recombined light.

[0131] Example 20. The system of example 19, wherein the first spatial mode sorter comprises a multi-plane light converter configured to demultiplex the light from the real scene into a subset of Zernike modes.

[0132] Example 21. The system of example 19, further comprising: a beam dump configured to absorb light incident thereupon; and a mirror, comprising a pinhole, configured to direct light in the one or more predetermined modes that passes through the pinhole towards the beam dump.

[0133] Example 22. The system of example 19, further comprising: a polarizing beam splitter (PBS); and a plurality of mirrors including at least one mirror configured to direct light incident thereupon from the PBS towards the imaging detector.

[0134] Example 23. The system of example 19, wherein the recombined light comprises light in modes which are different from the one or more predetermined modes.

[0135] Example 24. The system of example 19, wherein the first spatial mode sorter is further configured to focus the demultiplexed light onto regions of a sorting plane, each region corresponding to the modes of the incident light. -28- 044974.8137.WO00\182811491.1

[0136] Example 25. The system of example 19, wherein the occulting mask is positioned at the sorting plane at a region associated with at least one of the one or more predetermined modes.

[0137] Example 26. A quantum optimal imaging system, comprising: a polarization element configured to produce horizontal linearly polarized light at an output thereof based on light received thereupon from an illumination source; an optical subsystem comprising at least two lenses configured to receive light from the output of the polarization element; a multi-plane light converter (MPLC); a half-wave plate (HWP); a Faraday rotator, positioned between the HWP and the MPLC, configured to alter a polarization of light incident thereupon based on a propagation direction of the incident light; a polarizing beam splitter (PBS); an imaging detector configured to produce images based on light received thereon; and a beam dump configured to absorb light incident thereupon, wherein at least a portion of the light absorbed by the beam dump corresponds to light from the illumination source that propagates in one or more predetermined modes.

[0138] Example 27. The system of example 26, further comprising: a first mirror configured to reflect light received thereupon from the optical subsystem in a direction of the MPLC; a second mirror coupled to the MPLC; a third mirror comprising a pinhole; and a fourth mirror configured to: direct light received from the PBS that is incident thereupon from a first direction towards the imaging detector; direct light received from the MPLC that is incident thereupon from a second direction towards the third mirror; and reflect at least a portion of light received from the third mirror that is incident thereupon from a third direction towards the MPLC, wherein at least a portion of the light directed by the fourth mirror towards the third mirror passes through the pinhole and corresponds to light from the illumination source that propagates in the one or more predetermined modes, wherein the third mirror is configured to reflect light from the illumination source that propagates in one or more optical modes that are different from the one or more predetermined modes towards the imaging detector.

[0139] Example 28. The system of example 26, wherein the PBS is positioned to receive light from the HWP and to direct light received thereon in a direction of the MPLC. -29- 044974.8137.WO00\182811491.1

[0140] Example 29. The system of example 26, wherein the imaging detector is further configured to capture an image corresponding to a real scene based on light from the illumination source that propagates in one or more optical modes that are different from the one or more predetermined modes.

[0141] Example 30. The system of example 26, wherein light from the illumination source propagating in one or more optical modes that are different from the one or more predetermined modes interacts with the MPLC before reaching the imaging detector.

[0142] Example 31. The system of example 26, wherein the Faraday rotator is further configured to cause light incident thereupon from along a first optical path to propagate along a second optical path that is different from the first optical path based on the propagation direction of the incident light.

[0143] Example 32. The system of example 26, wherein the optical subsystem is a 4f imaging system.

[0144] Example 33. A method for quantum-optimal direct imaging, comprising: demultiplexing, by a first spatial mode sorter, light from a real scene that is incident upon the first spatial mode sorter according to modes of the incident light; focusing the demultiplexed light at regions on a sorting plane, each of the regions corresponding, respectively, to the modes; inhibiting the demultiplexed light corresponding to at least one of the modes from propagating towards a second spatial mode sorter; recombining, at the second spatial mode sorter, the demultiplexed light that is received by the second spatial mode sorter; and obtaining an image of the real scene from an imaging detector configured to receive the recombined light.

[0145] Example 34. The method of example 33, wherein the light from the real scene that is incident upon the first spatial mode sorter is demultiplexed by the first spatial mode sorter into a subset of Zernike modes.

[0146] Example 35. The method of example 33, wherein the at least one of the modes corresponds to a fundamental optical mode of an optical imaging instrument.

[0147] Example 36. The method of example 35, wherein the first mode sorter demultiplexes the light from the real scene based on a point-spread function of the optical imaging instrument. -30- 044974.8137.WO00\182811491.1

[0148] Example 37. The method of example 36, wherein the recombining, at the second spatial mode sorter, the demultiplexed light comprises recombining light in modes orthogonal to the point-spread function of the optical imaging instrument.

[0149] Example 38. The method of example 33, wherein the image enables detection of a first astronomical object separated by a distance from a second astronomical object, wherein the second astronomical object is brighter than the first astronomical object.

[0150] Example 39. The method of example 38, wherein the distance is below a Rayleigh diffraction limit of an optical imaging instrument implementing the method.

[0151] Example 40. The method of example 38, wherein the second astronomical object is a star and the first astronomical object is an exoplanet in orbit around the star.

[0152] Example 41. The method of example 33, wherein the image is based on light from a first astronomical object received by an optical imaging instrument along an optical axis of the optical imaging instrument and light from a second astronomical object received by the optical imaging instrument off the optical axis of the optical imaging instrument.

[0153] Example 42. The method of example 33, wherein the imaging detector is a camera comprising a charge-coupled device.

[0154] Example 43. The method of example 33, further comprising: absorbing the demultiplexed light corresponding to the at least one of the modes at a beam dump, wherein an occulting mask positioned at the sorting plane is used to inhibit the demultiplexed light corresponding to the at least one of the modes from propagating towards the second spatial mode sorter.

[0155] Various operations disclosed herein can be implemented using a processor / controller configured to include, or be coupled to, a memory that stores processor executable code that causes the processor / controller carry out various computations and processing of information. The processor / controller can further generate and transmit / receive suitable information to / from the various system components, as well as suitable input / output (IO) capabilities (e.g., wired or wireless) to transmit and receive commands and / or data. The processor / controller may, for -31- 044974.8137.WO00\182811491.1example, provide signals to control the operation of various components such as excitation sources and detectors that are disclosed herein. The processor / controller may be further configured to perform various method steps and computations that are disclosed in this patent document.

[0156] Various information and data processing operations described herein may be implemented in one embodiment by a computer program product, embodied in a computer-readable medium, including computer-executable instructions, such as program code, executed by computers in networked environments. A computer- readable medium may include removable and non-removable storage devices including, but not limited to, Read Only Memory (ROM), Random Access Memory (RAM), compact discs (CDs), digital versatile discs (DVD), etc. Therefore, the computer-readable media that is described in the present application comprises non- transitory storage media. Generally, program modules may include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Computer-executable instructions, associated data structures, and program modules represent examples of program code for executing steps of the methods disclosed herein. The particular sequence of such executable instructions or associated data structures represents examples of corresponding acts for implementing the functions described in such steps or processes.

[0157] A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. -32- 044974.8137.WO00\182811491.1

[0158] The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

[0159] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random-access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of nonvolatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

[0160] While this patent document contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this patent document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub- combination. -33- 044974.8137.WO00\182811491.1

[0161] From the foregoing, it will be appreciated that specific embodiments of the invention have been described herein for purposes of illustration, but that various modifications may be made without deviating from the scope of the invention. Accordingly, the invention is not limited except as by the appended claims. -34- 044974.8137.WO00\182811491.1

Claims

CLAIMS 1. An imaging system, comprising: a lens positioned to receive a light associated with a scene; a first spatial mode sorter positioned, along an optical path, to receive the light from the lens and to demultiplex the light incident thereupon; an occulting mask; and a second spatial mode sorter, wherein: the occulting mask is positioned between the first spatial mode sorter and the second spatial sorter to receive a first demultiplexed light and inhibit one or more predetermined modes of the first demultiplexed light from propagating towards the second spatial sorter, the second spatial mode sorter is configured to recombine light received thereon to produce an output light for detection by an imaging detector, the output light allows for detection by the imaging detector of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit.

2. The imaging system of claim 1, wherein the occulting mask comprises a mirror with a pinhole, wherein the mirror is positioned such that the one or more predetermined modes propagate through the pinhole.

3. The imaging system of claim 2, comprising a beam dump configured to absorb the one or more predetermined modes after propagation of the one or more predetermined modes through the pinhole.

4. The system of claim 1, comprising: the imaging detector; a polarizing beam splitter (PBS); a polarization element; and a plurality of mirrors including a first mirror positioned to direct light incident thereupon from the PBS towards a position of the imaging detector, -35- 044974.8137.WO00\182811491.1wherein the polarization element and the PBS are positioned along the optical path, wherein the PBS is positioned between the first mirror and the polarization element.

5. An imaging system, comprising: a polarization element; a polarizing beam splitter (PBS); one or more additional polarization elements; a spatial mode sorter; a first mirror; a reflective element; and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and the second optical path are associated with a scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements, and the PBS to reach the reflective element.

6. The imaging system of claim 5, wherein the reflective element comprises at least two reflective surfaces, each of the at least two reflective surfaces positioned to allow propagation of the first light along the forward optical path and propagation of the second light along the backward optical path.

7. The imaging system of claim 5, comprising: a lens system comprising at least two lenses, -36- 044974.8137.WO00\182811491.1wherein the polarization element is positioned produce a polarized light associated with the scene, wherein a first of the at least two lenses is positioned to direct the polarized light therethrough towards a second of the at least two lenses.

8. The imaging system of claim 7, wherein the lens system is configured as a 4f imaging system.

9. The imaging system of claim 5, comprising a beam dump positioned to receive a mode of the first light that propagates through the opening of the second mirror.

10. The imaging system of claim 5, wherein the polarization element is configured to produce linearly polarized light.

11. The imaging system of claim 5, comprising an additional mirror positioned to allow a folded optical path.

12. The imaging system of claim 5, wherein the one or more additional polarization elements include a half-wave plate and a Faraday rotator.

13. A method for imaging based on spatial mode sorting, comprising: operating an imaging device to exclude one or more optical modes of an input light from a scene, wherein the imaging device comprises: an imaging detector, a polarization element, a polarizing beam splitter (PBS), one or more additional polarization elements, a spatial mode sorter, a first mirror, a reflective element, and a second mirror having an opening therein, wherein: the PBS is positioned to receive light from the polarization element, -37- 044974.8137.WO00\182811491.1the PBS enables propagation of the light along a forward optical path and a backward optical path, the forward optical path and the second optical path are associated with the scene, the forward optical path is associated with a first light that is incident upon the spatial mode sorter after propagation through the one or more additional polarization elements and is reflected by the first mirror and the reflective element to reach the second mirror, the backward optical path is associated with a second light that is reflected by the second mirror, the reflective optical element, and the first mirror, and propagates through the one or more additional polarization elements and the PBS to reach the imaging detector; receiving data associated with a measurement acquired by the imaging detector; and obtaining an image of the scene based on the data, wherein the data represents the input light with the one or more modes excluded.

14. The method of claim 13, wherein the image enables detection of features in the scene that are separated by a distance that is below a Rayleigh diffraction limit.

15. The method of claim 13, wherein the one or more optical modes corresponds to a fundamental optical mode of an optical imaging instrument.

16. The method of claim 13, wherein the image enables detection of a first astronomical object separated by a distance from a second astronomical object, wherein the second astronomical object is brighter than the first astronomical object.

17. The method of claim 16, wherein the distance is below a Rayleigh diffraction limit of an optical imaging instrument implementing the method.

18. The method of claim 13, wherein the imaging device comprises a beam dump positioned to receive a mode of the first light that propagates through the opening of the second mirror. -38- 044974.8137.WO00\182811491.

119. The method of claim 13, wherein the input light is linearly polarized.

20. The method of claim 13, wherein the one or more additional polarization elements include a half-wave plate and a Faraday rotator. -39- 044974.8137.WO00\182811491.1

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