Killing Rayleigh s Criterion by Farfield Linear Photonics

Transcription

1 Killing Rayleigh s Criterion by Farfield Linear Photonics Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, and Mankei Tsang mankei@nus.edu.sg June / 26

2 Summary (a) (b) 2 / 26

3 Imaging of One Point Source 3 / 26

4 Point-Spread Function of Hubble Space Telescope 4 / 26

5 Inferring Position of One Point Source Classical source Given N detected photons, mean-square error: X 2 1 = σ2 N, (1) σ λ sinφ. (2) Helstrom, Lindegren (astrometry), Bobroff,... Full EM, full quantum: Tsang, Optica 2, 646 (215). 5 / 26

6 Superresolution Microscopy PALM, STED, STORM, etc.: isolate emitters. Locate centroids. (25:45) Special fluorophores slow doesn t work for stars e.g., Betzig, Optics Letters 2, 237 (1995). For a review, see Moerner, PNAS 14, (27). 6 / 26

7 Two Point Sources (a) (b) Rayleigh s criterion (1879): requires θ 2 σ (heuristic) 7 / 26

8 Centroid and Separation Estimation Bettens et al., Ultramicroscopy 77, 37 (1999); Van Aert et al., J. Struct. Biol. 138, 21 (22); Ram, Ward, Ober, PNAS 13, 4457 (26). Incoherent sources, Poisson statistics X 1 = θ 1 θ 2 /2, X 2 = θ 1 +θ 2 /2. Cramér-Rao bound for centroid: θ 2 1 σ2 N. (3) CRB for separation estimation: two regimes θ 2 σ: (a) θ 2 σ: θ 2 2 4σ2 N, (4) (b) θ 2 2 4σ2 N (5) Rayleigh s curse PALM/STED/STORM: avoid Rayleigh 8 / 26

9 Rayleigh s Curse Cramér-Rao bounds: θ J (direct) 11 J (direct) is Fisher information for CCD Gaussian PSF, similar behavior for other PSF θ J (direct) 22 (6) Fisher information / (N/4σ 2 ) Classical Fisher information J (direct) 11 J (direct) 22 Mean-square error / (4σ 2 /N) Cramér-Rao bound on separation error Direct imaging (1/J (direct) 22 ) θ 2 /σ θ 2 /σ 9 / 26

10 Quantum Information CCD is just one measurement method. Quantum mechanics allows infinite possibilities. Helstrom: For any measurement (POVM) of an optical state ρ M (M here is number of copies) Ultimate amount of information in the photons Coherent sources: Tsang, Optica 2, 646 (215). Mixed states: Σ J 1 K 1, (7) K µν = M Re(trL µ L ν ρ), (8) ρ θ µ = 1 2 (L µρ+ρl µ ). (9) ρ = n D n e n e n, (1) L µ = 2 n,m;d n +D m ρ e n e θ m µ e n e m. (11) D n +D m 1 / 26

11 Quantum Optics Mandel and Wolf, Optical Coherence and Quantum Optics; Goodman, Statistical Optics Thermal sources, e.g., stars, fluorescent particles. Coherence time 1 fs. Within each coherence time interval, average photon number ǫ 1 at optical frequencies (visible, UV, X-ray, etc.). Quantum state at image plane: ρ = (1 ǫ) vac vac + ǫ 2 ( ψ 1 ψ 1 + ψ 2 ψ 2 )+O(ǫ 2 ) ψ 1 ψ 2, (12) ψ 1 = dxψ(x X 1 ) x, ψ 2 = dxψ(x X 2 ) x. (13) derive from zero-mean Gaussian P function Multiphoton coincidence: rare, little information as ǫ 1 (homeopathy) Similar model for stellar interferometry in Gottesman, Jennewein, Croke, PRL 19, 753 (212); Tsang, PRL 17, 2742 (211). 11 / 26

12 Plenty of Room at the Bottom Fisher information / (N/4σ 2 ) Quantum and classical Fisher information θ 2 /σ K 11 J (direct) 11 K 22 J (direct) 22 Mean-square error / (4σ 2 /N) Tsang, Nair, and Lu, e-print arxiv: Cramér-Rao bounds on separation error Quantum (1/K 22 ) Direct imaging (1/J (direct) 22 ) θ 2 /σ θ2 2 1 = 1 K 22 N k2. (14) Nair and Tsang, e-print arxiv: : thermal sources with arbitrary ǫ Hayashi ed., Asymptotic Theory of Quantum Statistical Inference; Fujiwara JPA 39, (26): there exists a POVM such that θµ 2 1/K µµ, M. 12 / 26

13 Hermite-Gaussian Basis project the photon in Hermite-Gaussian basis: E 1 (q) = φ q φ q, (15) φ q = φ q (x) = ( 1 2πσ 2 Assume PSF ψ(x) is Gaussian (common). dxφ q (x) x, (16) ) 1/4 ( ) x H q )exp ( x2 2σ 4σ 2. (17) 1 J (HG) 22 = 1 K 22 = 4σ2 N. (18) Maximum-likelihood estimator can saturate the classical bound asymptotically for large N. 13 / 26

14 Spatial-Mode Demultiplexing (SPADE) image plane image plane 14 / 26

15 Binary SPADE leaky modes image plane leaky modes Fisher information / (N/4σ 2 ) image plane Classical Fisher information θ 2 /σ J (HG) 22 = K 22 J (direct) 22 J (b) 22 Fisher information / (π 2 N/3W 2 ) Fisher information for sinc PSF θ 2 /W K 22 J (direct) 22 J (b) / 26

16 Numerical Performance of Maximum-Likelihood Estimators 2 Simulated errors for SPADE 2 Simulated errors for binary SPADE Mean-square error / (4σ 2 /L) θ 2 /σ L = number of detected photons biased, < 2 CRB. 1/J (HG) 22 = 1/K 22 L = 1 L = 2 L = 1 Mean-square error / (4σ 2 /L) θ 2 /σ 1/J (b) 22 L = 1 L = 2 L = 1 16 / 26

17 Minimax/Bayesian Van Trees inequality for any biased/unbiased estimator (e-print arxiv: ) Quantum/SPADE: sup θ Σ 22 (θ) 4σ2 N, Direct imaging: supσ (direct) 22 (θ) σ2. (19) θ N 2 (a) SPADE with ML 2 (b) SPADE with modified ML MSE / (4σ 2 /L) MSE / (4σ 2 /L) CRB L = 1.5 L = 2 L = 5 L = θ/σ (c) Direct imaging with ML CRB L = 1 L = 2 L = 5 L = θ/σ MSE / (4σ 2 /L) supθ MSE / σ CRB L = 1.5 L = 2 L = 5 L = θ/σ (d) Worst-case error Quantum/SPADE limit SPADE (ML) SPADE (modified ML) Direct-imaging limit Direct imaging (ML) Photon Number L 17 / 26

18 E SLIVER o m i e Inversion m I SuperLocalization via Image-inVERsion interferometry Classical theory/experiment of image-inversion interferometer: Wicker, Heintzmann, Opt. Express 15, 1226 (27); Wicker, Sindbert, Heintzmann, Opt. Express 17, (29) Nair and Tsang, Opt. Express 24, 3684 (216). Similar to binary SPADE Laser Focus World, Feb 216 issue. 18 / 26

19 Misalignment 1 Fisher information for misaligned SPADE object plane image plane Fisher information / (N/4σ 2 ) mean-square error/(4σ 2 /L).8.6 J (HG) 22 (ξ = ) J (HG) 22 (ξ =.1).4 J (HG) 22 (ξ =.2) J (HG) 22 (ξ =.3).2 J (HG) 22 (ξ =.4) J (HG) 22 (ξ =.5) J (direct) θ 2 /σ Simulated errors for misaligned binary SPADE 12 1/J (b) 22 (ξ =.1) 1 1/J (direct) 22 L = 1 8 L = 1 L = image plane ξ ˇθ 1 θ /σ 1 Overhead photons N 1 1/ξ 2 ξ =.1, N 1 1. CRB for X s = θ 1 ±θ 2 /2 photon-counting array Mean-square error / (2σ 2 /N) θ 2 /σ 1 2 Cramér-Rao bounds on localization error 1 1 Hybrid (ξ = ) Hybrid (ξ =.1) Hybrid (ξ =.2) Hybrid (ξ =.3) Hybrid (ξ =.4) Hybrid (ξ =.5) Direct imaging / 26

20 Follow-up 1. Tsang, Nair, and Lu, e-print arxiv: Nair and Tsang, Optics Express 24, 3684 (216) 3. Tsang, Nair, and Lu, e-print arxiv: Nair and Tsang, e-print arxiv: Ang, Nair, and Tsang, e-print arxiv: Dimensions Sources Theory Experimental proposals 1D Weak thermal Quantum SPADE (optical frequencies and above) 2D Thermal (any frequency) Semiclassical SLIVER 1D Weak thermal, lasers Semiclassical 1D Thermal Quantum SLIVER SPADE, SLIVER 2D Weak thermal Quantum SPADE, SLIVER Lupo and Pirandola, e-print arxiv: Bayesian/minimax: M. Tsang, e-print arxiv: More to come! 2 / 26

21 Experiments Tang, Durak, and Ling (CQT Singapore), Fault-tolerant and finite-error localization for point emitters within the diffraction limit, e-print arxiv: (216). SLIVER Laser, classical noise Stay tuned! 21 / 26

22 Quantum Computational Imaging Design quantum computer to Maximize information extraction Reduce classical computational complexity 22 / 26

23 Quantum Metrology with Classical Sources Thermal/fluorescent/laser sources, linear optics, photon counting Compare with other QIP applications: QKD (Bennett et al.) Nonclassical-state metrology (Yuen, Caves) Shor s algorithm Quantum simulations (Feynman, Lloyd) Boson sampling (Aaronson) Microscopy (physics, chemistry, biology, engineering), telescopy (astronomy), radar/lidar (military), etc. Classical sources are more robust to losses. Current microscopy limited by photon shot noise in EMCCD (see, e.g., Pawley ed., Handbook of Biological Confocal Microscopy). Ground telescopes limited by atmospheric turbulence, space telescopes are diffraction/shot-noise-limited. Linear optics/photon-counting technology is mature Although any given scheme can be explained by a semiclassical model, quantum metrology remains a powerful tool for exploring the ultimate performance. 23 / 26

24 Quantum Metrology Kills Rayleigh s Criterion Mean-square error / (4σ 2 /N) Cramér-Rao bounds on separation error Quantum (1/K 22 ) Direct imaging (1/J (direct) 22 ) θ 2 /σ imator i ne FAQ: mankei@nus.edu.sg Image Inversion 24 / 26

25 ǫ 1 Approximation Chap. 9, Goodman, Statistical Optics: If the count degeneracy parameter is much less than 1, it is highly probable that there will be either zero or one counts in each separate coherence interval of the incident classical wave. In such a case the classical intensity fluctuations have a negligible bunching effect on the photo-events, for (with high probability) the light is simply too weak to generate multiple events in a single coherence cell. Zmuidzinas ( JOSA A 2, 218 (23): It is well established that the photon counts registered by the detectors in an optical instrument follow statistically independent Poisson distributions, so that the fluctuations of the counts in different detectors are uncorrelated. To be more precise, this situation holds for the case of thermal emission (from the source, the atmosphere, the telescope, etc.) in which the mean photon occupation numbers of the modes incident on the detectors are low, n 1. In the high occupancy limit, n 1, photon bunching becomes important in that it changes the counting statistics and can introduce correlations among the detectors. We will discuss only the first case, n 1, which applies to most astronomical observations at optical and infrared wavelengths. Hanbury Brown-Twiss (post-selects on two-photon coincidence): poor SNR, obsolete for decades in astronomy. See also Labeyrie et al., An Introduction to Optical Stellar Interferometry, etc. Fluorescent particles: Pawley ed., Handbook of Biological Confocal Microscopy, Ram, Ober, Ward (26), etc., may have antibunching, but Poisson model is fine because of ǫ / 26

26 Quantum Metrology for Dynamical Systems, e.g., Gravitational-Wave Detectors, Atomic Clocks Tsang, Wiseman, and Caves, Fundamental Quantum Limit to Waveform Estimation, PRL 16, 941 (211). Bayesian QCRB for stochastic process (infinite number of parameters), non-commuting Heisenbergpicture generators, sequential/continuous measurements Tsang and Nair, Fundamental quantum limits to waveform detection, PRA 86, (212). Helstrom bounds on waveform detection/false-alarm errors Tsang, Quantum metrology with open dynamical systems, NJP 15, 735 (213). Iwasawa et al., Quantum-limited mirror-motion estimation, PRL 111, (213). Berry et al., The Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation, PRX 5, 3118 (215). Ng et al., Spectrum analysis with quantum dynamical systems, PRA 93, (216). More on 26 / 26