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Quantum Theory of Superresolution for Incoherent Optical Imaging ∗
Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, and Mankei Tsang
http://mankei.tsang.googlepages.com/
Jan 2017
∗This work is supported by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2011-07 and an MOETier 1 grant.
Summary
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(a)
(b)
Quantum Measurement
on image plane
1. Tsang, Nair, and Lu,Physical Review X 6,031033 (2016).
2. Nair and Tsang, Op-tics Express 24, 3684(2016).
3. Tsang, Nair, and Lu,Proc. SPIE 10029,1002903 (2016).
4. Nair and Tsang, PRL(Editors’ Suggestion)117, 190801 (2016).
5. Tsang,arXiv:1605.03799(2016).
6. Ang, Nair, and Tsang,arXiv:1606.00603(2016).
7. Tsang,arXiv:1608.03211(2016).
8. Lu, Nair, Tsang,arXiv:1609.03025(2016).
Experiments
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■ Tang, Durak, and Ling, “Fault-tolerant andfinite-error localization for point emitterswithin the diffraction limit,” Optics Express24, 22004 (2016).
■ Yang, Taschilina, Moiseev, Simon, Lvovsky,“Far-field linear optical superresolution viaheterodyne detection in a higher-order localoscillator mode,” Optica 3, 1148 (2016).
■ Tham, Ferretti, Steinberg, “BeatingRayleigh’s Curse by Imaging Using PhaseInformation,” Phys. Rev. Lett., e-printarXiv:1606.02666 (2016).
■ Paur, Stoklasa, Hradil, Sanchez-Soto, Re-hacek, “Achieving the ultimate optical reso-lution,” Optica 3, 1144 (2016).
Imaging of One Point Source
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Diffraction limited
1. Fluorescence microscopy2. Space telescopes (Webb, $8.8 billion)3. Ground-based telescopes:
(a) Large Binocular Telescope (LBT) ($120million)
(b) Giant Magellan Telescope (GMT) (∼$1billion)
(c) Thirty Meter Telescope (TMT) (∼$1.2billion)
(d) European Extremely Large Telescope(E-ELT) (∼$1.2 billion)
LBT Point-Spread Function
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■ Esposito et al., “Large Binocular Telescope Adaptive Optics System: New achievements andperspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011)
■ Strehl ratio > 80% at infrared
Inferring Position of One Point Source
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■ Classical source■ Given N detected photons, mean-square error:
∆X21 =
σ2
N, (1)
σ ∼ λ
sinφ. (2)
■ Farrell (1966), Helstrom (1970), Lindegren(1978), Bobroff (1986), ...
■ Full EM, full quantum: Tsang, Optica 2, 646(2015).
Superresolution Microscopy
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■ PALM, STORM, etc.: isolate emitters. Locate cen-troids.
https://cam.facilities.northwestern.edu/588-2/single-molecule-localization-microscopy/
■ Special fluorophores■ slow■ doesn’t work for stars
Two Point Sources
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(a)
(b)
■ Λ(x) = 12
[
|ψ1(x)|2 + |ψ2(x)|2]
■ Rayleigh’s criterion (1879): requires θ2 &σ (heuristic)
Centroid and Separation Estimation
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■ Bettens et al., Ultramicroscopy 77, 37 (1999);Van Aert et al., J. Struct. Biol. 138, 21 (2002);Ram, Ward, Ober, PNAS 103, 4457 (2006).
■ Incoherent sources, Poisson statistics■ X1 = θ1 − θ2/2, X2 = θ1 + θ2/2.■ Cramer-Rao bound for centroid:
∆θ21 ≥ σ2
N. (3)
■ CRB for separation estimation: two regimes
◆ θ2 ≫ σ:
∆θ22 ≥ 4σ2
N, (4)
◆ θ2 ≪ σ:
∆θ22 → 4σ2
N×∞ (5)
(a)
(b)
Rayleigh’s Curse
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■ Cramer-Rao bound (standard in single-molecule imaging):
∆θ21 ≥ 1
J (direct)11
∆θ22 ≥ 1
J (direct)22
(6)
J (direct) is Fisher information for direct imaging■ Gaussian PSF, similar behavior for other PSF■ Rayleigh’s curse■ PALM/STED/STORM: avoid violating Rayleigh
θ2/σ0 2 4 6 8 10
Fisher
inform
ation/(N
/4σ
2)
0
0.5
1
1.5
2
2.5
3
3.5
4Classical Fisher information
J(direct)11
J(direct)22
θ2/σ0 0.2 0.4 0.6 0.8 1
Mean-squareerror/(4σ2/N
)
0
20
40
60
80
100Cramer-Rao bound on separation error
Direct imaging (1/J(direct)22 )
Quantum Information
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■ Measuring the spatial intensity (direct imaging, CCD) isjust one measurement method. Quantum mechanics al-lows infinite possibilities.
■ Helstrom (1967): For any measurement
Σ ≥ J−1 ≥ K−1, (7)
Kµν =M Re (trLµLνρ) , (8)
∂ρ
∂θµ=
1
2(Lµρ+ ρLµ) . (9)
■ K(ρ) is the quantum Fisher information, the ultimateamount of information in the photons.
■ Coherent sources: Tsang, Optica 2, 646 (2015).■ Mixed states:
ρ =∑
n
Dn |en〉 〈en| , (10)
Lµ = 2∑
n,m;Dn+Dm 6=0
〈en| ∂ρ∂θµ
|em〉Dn +Dm
|en〉 〈em| . (11)
Quantum Measurement
on image plane
Quantum Optics
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■ Mandel and Wolf, Optical Coherence and Quantum Optics; Goodman, Statistical Optics■ Thermal sources, e.g., stars, fluorescent particles.■ Average photon number per mode ǫ≪ 1 at optical frequencies (visible, UV, X-ray, etc.).■ ǫ ∼ 0.01 for the sun at visible, ǫ ∼ 10−6 for fluorophores.
image plane
■ Quantum state in M temporal modes on image plane is ρ⊗M , where
ρ = (1− ǫ) |vac〉 〈vac|+ ǫ
2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2) 〈ψ1|ψ2〉 6= 0, (12)
|ψ1〉 =∫ ∞
−∞
dxψ(x−X1) |x〉 , |ψ2〉 =∫ ∞
−∞
dxψ(x−X2) |x〉 . (13)
■ derive from zero-mean Gaussian P function, mutual coherence■ Multiphoton coincidence: rare, little info as ǫ≪ 1 (homeopathy)■ Similar model for stellar interferometry in Gottesman, Jennewein, Croke, PRL 109, 070503
(2012); Tsang, PRL 107, 270402 (2011).
Plenty of Room at the Bottom
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θ2/σ0 2 4 6 8 10
Fisher
inform
ation
/(N
/4σ2)
0
1
2
3
4Quantum and classical Fisher information
K11
J(direct)11
K22
J(direct)22
θ2/σ0 0.2 0.4 0.6 0.8 1
Mean-squareerror/(4σ2/N
)
0
20
40
60
80
100Cramer-Rao bounds on separation error
Quantum (1/K22)
Direct imaging (1/J(direct)22 )
■ Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016)
∆θ22 ≥ 1
K22=
1
N∆k2. (14)
■ Nair and Tsang, PRL (Editors’ Suggestion) 117, 190801 (2016): thermal sources with arbitrary ǫ(see also Lupo and Pirandola, PRL (Editors’ Suggestion) 117, 190802 (2016))
■ Hayashi ed., Asymptotic Theory of Quantum Statistical Inference; Fujiwara JPA 39, 12489(2006): there exists a POVM such that ∆θ2µ → 1/Kµµ, N → ∞.
Hermite-Gaussian Basis
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■ project in Hermite-Gaussian basis:
E1(q) = |φq〉 〈φq | , (15)
|φq〉 =∫ ∞
−∞
dxφq(x) |x〉 , (16)
φq(x) =
(
1
2πσ2
)1/4
Hq
(
x√2σ
)
exp
(
− x2
4σ2
)
. (17)
■ Assume PSF ψ(x) is Gaussian (common).
1
J (HG)22
=1
K22=
4σ2
N. (18)
■ Maximum-likelihood estimator can saturate the classical bound asymptotically for large N .
Spatial-Mode Demultiplexing (SPADE)
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image plane
...
...
image plane
...
...
■ Many other ways (optical comm.), e.g., DAB Miller, “Self-configuring universal linear opticalcomponents,” Photonics Research 1, 1 (2013).
Elementary Explanation
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■ Incoherent sources: energy in first-order mode is ∝ (d/2)2 + (−d/2)2 = d2/2■ Zeroth-order mode is just background noise, removing it improves SNR.■ Why quantum formalism?
◆ Fundamental quantum limit◆ Ensures measurement is physical◆ Discover without prejudice
SLIVER
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Estimator
Image
Inversion
■ SuperLocalization via Image-inVERsion interferometry■ Nair and Tsang, Opt. Express 24, 3684 (2016).■ Laser Focus World, Feb 2016 issue.■ Classical theory/experiment of image-inversion interferometer: Wicker, Heintzmann,
Opt. Express 15, 12206 (2007); Wicker, Sindbert, Heintzmann, Opt. Express 17, 15491 (2009)
Theoretical Follow-up
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Dimensions Sources Theory Experimental proposals1. Tsang, Nair, and Lu,
Phys. Rev. X 6, 031033(2016)
1D Weak thermal (opticalfrequencies and above)
Quantum SPADE
2. Nair and Tsang, Op-tics Express 24, 3684(2016)
2D Thermal (any fre-quency)
Semiclassical SLIVER
3. Tsang, Nair, and Lu,Proc. SPIE 10029,1002903 (2016)
N/A Weak thermal, lasers Semiclassical N/A
4. Nair and Tsang, PRL(Editors’ Suggestion)117, 190801 (2016)
1D Thermal Quantum SLIVER
5. Tsang,arXiv:1605.03799
1D Weak thermal Quantum, Bayesian,Minimax
SPADE
6. Ang, Nair, and Tsang,arXiv:1606.00603
2D Weak thermal Quantum SPADE, SLIVER
7. Tsang,arXiv:1608.03211
2D Weak thermal, multiplesources
Quantum SPADE
8. Lu, Nair, Tsang,arXiv:1609.03025
2D Weak thermal, one-versus-two
Quantum, binary hy-pothesis testing
SPADE, SLIVER
Other groups:
■ Lupo and Pirandola, PRL (Editors’ Suggestion) 117, 190801 (2016).■ Rehacek et al., Optics Letters 42, 231 (2017).■ Krovi, Guha, Shapiro, arXiv:1609.00684.
Experiments
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■ Tang, Durak, and Ling, “Fault-tolerant and finite-error localization for point emitters within thediffraction limit,” Optics Express 24, 22004 (2016).
◆ SLIVER◆ Laser, classical noise
■ Yang, Taschilina, Moiseev, Simon, Lvovsky, “Far-field linear optical superresolution viaheterodyne detection in a higher-order local oscillator mode,” Optica 3, 1148 (2016).
◆ Mode heterodyne◆ Laser
■ Tham, Ferretti, Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,”Phys. Rev. Lett., e-print arXiv:1606.02666 (2016).
◆ variation of SPADE◆ single-photon sources, close to quantum limit
■ Paur, Stoklasa, Hradil, Sanchez-Soto, Rehacek, “Achieving the ultimate optical resolution,”Optica 3, 1144 (2016).
◆ variation of SPADE◆ laser, close to quantum limit
Popular Coverage
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■ Viewpoint in APS Physics■ IoP Physics World and nanotechweb.org:
Seth Lloyd of the Massachusetts Instituteof Technology in the US is impressed. ‘This isawesome work and I am amazed that it hasn’tbeen done before,’ he says. ‘Perhaps everyonethought it was too good to be true.’
■ APS Physics Central■ Phys.org
Arbitrary Source Distributions
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3
iTEM
21
3
2
1
×104
0
2
1.5
1
0.5
|θ′10|
10−3 10−2 10−1
MSE
forθ′ 10
×10−5
2
4
6
8X moment
|θ′01|
10−3 10−2 10−1
MSE
forθ′ 01
×10−5
2
4
6
8Y moment
θ′20
10−2 10−1
MSE
forθ′ 20
10−6
10−5
10−4X2 moment
θ′02
10−2 10−1
MSE
forθ′ 02
10−6
10−5
10−4Y 2 moment
β(10, 01) ×10−3
5 10 15
MSE
forθ′ 11
10−6
10−5
XY moment
SPADE (simulated)
SPADE (theory)
direct imaging (simulated)
direct imaging (theory)
■ Yang et al., Optica 3, 1148 (2016): SPADE (Hermite-Gaussian Microscopy): even moments■ M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,”
arXiv:1608.03211: Generalized SPADE: enhanced estimation of second or higher moments
Breakout Session: “Quantum Metrology for SuperresolutionImaging”
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1. Lars Skovgaard Madsen, The University of Queensland,
■ “Quantum-limited single molecule biosensing: probing nanoscale biological machinery in itsnative state”
2. Alexander Lvovsky, University of Calgary, Russian Quantum Center,
■ “Super-Resolution Microscopy with Heterodyne Detection”
3. Hugo Ferretti, University of Toronto,
■ “Beating Rayleighs Curse Using SPLICE”
4. Jaroslav Rehacek, Palacky University Olomouc,
■ “Achieving quantum-limited optical resolution”
5. Zdenek Hradil, Palacky University Olomouc,
■ “Fischer information and resolution beyond the Rayleigh limit”
Quantum Metrology Kills Rayleigh’s Criterion
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θ2/σ0 0.2 0.4 0.6 0.8 1
Mean-squareerror/(4σ2/N
)
0
20
40
60
80
100Cramer-Rao bounds on separation error
Quantum (1/K22)
Direct imaging (1/J(direct)22 )
image plane
...
...
Estimator
Image
Inversion
■ FAQ: https://sites.google.com/site/mankeitsang/news/rayleigh/faq■ email: [email protected]
ǫ ≪ 1 Approximation
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■ 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 (https://pma.caltech.edu/content/jonas-zmuidzinas), JOSA A 20, 218 (2003):
“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, homeopathy): 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 (2006), etc., may have antibunching, but Poisson model is fine and standard because ofǫ≪ 1.
Binary SPADE
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image plane
image plane
leaky modes
leaky modes
θ2/σ0 2 4 6 8 10
Fisher
inform
ation/(N
/4σ
2)
0
0.2
0.4
0.6
0.8
1Classical Fisher information
J(HG)22 = K22
J(direct)22
J(b)22
θ2/W0 1 2 3 4 5
Fisher
inform
ation/(π
2N/3W
2)
0
0.2
0.4
0.6
0.8
1Fisher information for sinc PSF
K22
J(direct)22
J(b)22
Numerical Performance of Maximum-Likelihood Estimators
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θ2/σ0 0.5 1 1.5 2
Mean-squareerror/(4σ2/L
)
0
0.5
1
1.5
2Simulated errors for SPADE
1/J′(HG)22 = 1/K′
22
L = 10L = 20L = 100
θ2/σ0 0.5 1 1.5 2
Mean-squareerror/(4σ2/L
)
0
0.5
1
1.5
2Simulated errors for binary SPADE
1/J′(b)22
L = 10L = 20L = 100
■ L = number of detected photons■ biased (violate CRB), < 2×CRB.
Minimax/Bayesian
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Van Trees inequality for any biased/unbiased estimator (e-print arXiv:1605.03799)
Quantum/SPADE: supθ
Σ22(θ) ≥4σ2
N, Direct imaging: sup
θΣ
(direct)22 (θ) ≥ σ2
√N. (19)
θ/σ0 0.2 0.4 0.6 0.8 1
MSE
/(4σ2/L)
0
0.5
1
1.5
2(a) SPADE with ML
CRBL = 100L = 200L = 500L = 1000
θ/σ0 0.2 0.4 0.6 0.8 1
MSE
/(4σ2/L)
0
0.5
1
1.5
2(b) SPADE with modified ML
CRBL = 100L = 200L = 500L = 1000
θ/σ0 0.2 0.4 0.6 0.8 1
MSE
/(4σ2/L)
0
5
10
15
20(c) Direct imaging with ML
CRBL = 100L = 200L = 500L = 1000
Photon Number L102 103
supθMSE
/σ2
10−2
10−1
(d) Worst-case error
Quantum/SPADE limitSPADE (ML)SPADE (modified ML)Direct-imaging limitDirect imaging (ML)
2D SPADE and SLIVER
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Ang, Nair, Tsang, e-print arXiv:1606.00603
Misalignment
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SPADE
photon-counting
array
image
plane
image
plane
object
plane
■ ξ ≡ |θ1 − θ|/σ ≪ 1■ Overhead photons N1 ∼ 1/ξ2
■ ξ = 0.1, N1 ∼ 100.■ CRB for Xs = θ1 ± θ2/2
θ2/σ0 2 4 6 8 10
Fisher
inform
ation/(N
/4σ2)
0
0.2
0.4
0.6
0.8
1Fisher information for misaligned SPADE
J(HG)22 (ξ = 0)
J(HG)22 (ξ = 0.1)
J(HG)22 (ξ = 0.2)
J(HG)22 (ξ = 0.3)
J(HG)22 (ξ = 0.4)
J(HG)22 (ξ = 0.5)
J(direct)22
θ2/σ0 0.5 1 1.5 2
mean-square
error/(4σ2/L)
0
2
4
6
8
10
12Simulated errors for misaligned binary SPADE
1/J′(b)22 (ξ = 0.1)
1/J′(direct)22
L = 10L = 100L = 1000
10−1 100 101
Mean-square
error/(2σ2/N
)
100
101
102Cramer-Rao bounds on localization error
Hybrid (ξ = 0)Hybrid (ξ = 0.1)Hybrid (ξ = 0.2)Hybrid (ξ = 0.3)Hybrid (ξ = 0.4)Hybrid (ξ = 0.5)Direct imaging
Quantum Computational Imaging
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■ Design quantum computer to
◆ Maximize information extraction◆ Reduce classical computational complexity
Quantum Technology 1.5
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