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Seize the Moments for Subdiffraction Incoherent Imaging

Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang

Electrical and Computer Engineering/PhysicsNational University of Singapore

Shanghai, Sep 2018

Lord Rayleigh (1879)

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� Resolved:

� Not resolved:

Telescopes, Fluorescence Microscopy

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(images from the internet)

Photon Shot Noise

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� Random arrival of photons

Parameter Estimation

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� Each photon arrives at camera with position x and probability density f(x|θ)� Noisy data Y : e.g., positions (x1, x2, . . . ) or histogram of photon count

(n1, n2, . . . )� f depends on some unknown parameters θ� Estimator θ(Y ): guess θ from noisy data Y� Mean-square error:

MSE = E[θ(Y )− θ

]2. (1)

Limit to Parameter Estimation

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� Cramer-Rao bound:

MSE(θ) ≥ J(θ)−1, J(θ) = N

∫ ∞

−∞

dx1

f(x|θ)

[∂f(x|θ)

∂θ

]2

. (2)

J : Fisher information� Conventional direct imaging (photon counting on image plane):

J(0) = 0, J(∞) = N4σ2 , σ = λ

NA

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 )

1J(0) = ∞, 1

J(∞) =4σ2

N

� “Rayleigh’s curse”� See, e.g., Ram, Ward, Ober, PNAS 103, 4457 (2006).

Superresolution Microscopy

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� PALM, STORM, etc.: make sparse subsets offluorophores emit

https://cam.facilities.northwestern.edu/588-2/single-molecule-localization-microscopy/

� avoid violating Rayleigh� Need controllable fluorophores� slow, cumbersome� doesn’t work for stars, passive imaging

Fundamental Quantum Limit

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Quantum Measurement

on image plane

� Quantum: Direct imaging/photon counting is justone of the infinitely many possible measure-ments.

� Helstrom (1967), etc.: For any measurement,

MSE ≥ J−1 ≥ K−1, (3)

K(ρ⊗M ) =M trL2ρ, (4)

∂ρ

∂θ=

1

2(Lρ+ ρL) . (5)

� K(ρ) is the quantum Fisher information, the ulti-mate amount of information in the photons.

Quantum Optics for Incoherent Imaging

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� Thermal optical source: average photon number per mode ǫ≪ 1

image plane

� Quantum state in M spectral modes = ρ⊗M ,

ρ = (1− ǫ) |vac〉 〈vac|+ǫ

2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2),

|ψs〉 ≡

∫ ∞

−∞

dxψ(x−Xs) |x〉 .

� derived from zero-mean Gaussian Glauber-Sudarshan function� see, e.g., Tsang, PRL 107, 270402 (2011); Tsang, Nair, Lu, PRX (2016).

Plenty of Room at the Bottom

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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 )

� Tsang, Nair, Lu, PRX 6, 031033 (2016).

Optimal Measurement

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� Sort the photons in Hermite-Gaussian(TEM) modes first, then do photon count-ing

– Tsang, Nair, Lu, PRX (2016); Rehaceket al., OL 42, 231 (2017).

Spatial-Mode Demultiplexing (SPADE)

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image plane

...

...

Estimator

Image

Inversion

� Tsang, Nair, Lu, PRX (2016); Nair and Tsang, OE 24, 3684 (2016)� Many other ways (in optical comm., photonic circuits, etc.)� Classical sources� Far-field linear optics/photon counting� Important applications (astronomy, fluorescence microscopy, etc.)

Elementary Explanation

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� Incoherent sources: energy in 1st-order mode is

∝

(d

2

)2

+

(

−d

2

)2

=d2

2. (6)

� 0th-order mode is just background noise; filtering it improves SNR.

Experimental Demonstrations

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1. Tham, Ferretti, Steinberg (Toronto), PRL118, 070801 (2017).

� ∼ 2× QCRB

2. Tang, Durak, Ling (CQT), OE 24, 22004(2016).

3. Yang, Taschilina, Moiseev, Simon,Lvovsky (Calgary), Optica 3, 1148 (2016).

4. Paur, Stoklasa, Hradil, Sanchez-Soto, Re-hacek (Palacky/Madrid/Max Planck), Op-tica 3, 1144 (2016).

5. Parniak et al. (Warsaw, Poland),arXiv:1803.07096 (2018).

6. Donohue et al. (Paderborn, Germany),PRL 121, 090501 (2018).

7. Paur et al. (Palacky/Madrid/MaxPlanck/ESA), arXiv:1809.00633 (2018).

8. J. Hassett et al. (Rochester), Frontiers inOptics/Laser Science, OSA Technical Di-gest, paper JW4A.124 (2018).

European Space Agency

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https://www.esa.int/gsp/ACT/projects/super_resolution.html

Arbitrary Source Distribution

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� Direct imaging of arbitrary source distribution = F (X|θ):

f(x|θ) =

∫

dX|ψ(x−X)|2F (X|θ), Jµν =

∫ ∞

−∞

dx1

f

∂f

∂θµ

∂f

∂θν. (7)

� Infinite number of sources: Infinite number of parameters

Defining Subdiffraction Regime

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Sparse (Good Regime,PALM, STED, compressedsensing, etc.)

Subdiffraction (Bad Regime)

object width ≡ ∆ ≪ 1. (8)

Cramer-Rao Bound for Direct Imaging

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� Define parameters as object mo-ments:

θµ =

∫

dXXµF (X|θ) (9)

� CRB for ∆ ≪ 1:

(J−1)µν =O(1)

N. (10)

� Tsang, NJP 19, 023054 (2017); PRA97, 023830 (2018).

� Special case: for 2 point sources withseparation d, θ2 = d2/4,

J (d) =

(∂θ2∂d

)2

J22 = NO(d2).

(11)

d0 2 4 6 8

Fisher

inform

ation/(N

/4)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5Classical Fisher information

J(direct)11

J(direct)22

Quantum Limit

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� One-photon density operator:

ρ1(θ) =

∫

dXF (X|θ) |ψX〉 〈ψX | , |ψX〉 =

∫ ∞

−∞

dxψ(x−X) |x〉 . (12)

– mixed state– infinite number of spatial modes– infinite number of parameters

� Quantum Cramer-Rao bound [Tsang, arXiv:1806.02781 (2018)]:

(J−1)µµ ≥ (K−1)µµ ≥O(∆2⌊µ/2⌋)

N. (13)

See also Zhou and Jiang (Yale), arXiv:1801.02917 (2018).� Big enhancements possible when

– Subdiffraction: ∆ ≪ 1– Second or higher moments: µ ≥ 2

SPADE for Moment Estimation in 2D Imaging

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� Gaussian PSF [Tsang, NJP (2017)]:

– For moments with even µ1 & even µ2: TEM basis

⊲ See also Yang et al., Optica (2016)

– For other moments: interference of pairs of TEM modes

More General PSFs

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� Any centrosymmetric and separable PSF [Tsang, PRA (2018)]:

– For moments with even µ1, even µ2: “PSF-adapted” (PAD) basis(Rehacek et al. OL (2017), generalizes TEM)

– For other moments: interference of pairs of PAD modes

PAD

iPAD1 iPAD2 iPAD3

iPAD4 iPAD5 iPAD6

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(10) (50)

(12) (52)

(54)

(56)

(14)

(16)

(01) (21) (41) (61)

(05) (25) (45) (65)

(11) (31) (51)

(15) (35) (55)

(36)

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(32)

(30)

(03) (23) (43) (63) (13) (53)(33)

(00) (20) (40) (60)

(02) (22) (42) (62)

(04) (24) (44) (64)

(06) (26) (46) (66)

Performance of SPADE

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� MSE of SPADE:

MSEµµ =O(∆2⌊µ/2⌋

)

N︸ ︷︷ ︸

variance

+O(∆2µ+4

)

︸ ︷︷ ︸

bias2

.

∼ quantum limit, big enhancement over di-rect imaging when

– ∆ ≪ 1 (subdiffraction)– µ = µX + µY ≥ 2– bias is negligible.

� Caveat: θ2µ = O(∆2µ), fractional error:

MSE

θ2µ=

O(∆−2⌈µ/2⌉)

N+O(∆4).

Need many photons, especially for large µ.

|θ1| / θ0

10−2 10−1

MSE

/[θ

2 0(∆

/2)2

µ]

×10−3

1

2

3

4

567

µ = 1

θ2 / θ0 ×10−32 4 6

MSE

/[θ

2 0(∆

/2)2

µ]

10−3

10−2

10−1

100µ = 2

θ2 / θ0 ×10−32 4 6

MSE

/[θ

2 0(∆

/2)2

µ]

100

102

µ = 3

θ4 / θ0

10−6 10−5

MSE

/[θ

2 0(∆

/2)2

µ]

100

105µ = 4

Direct imagingDirect imaging (CRB)SPADESPADE (theory)

Tsang, NJP (2017); PRA(2018).

Elementary Explanation

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� Wavefunction from each point source:

� For one point source,

Energy in first-order mode ∝ X2. (14)

� A distribution of incoherent sources:

Total energy in first-order mode ∝

∫

dXF (X|θ)X2. (15)

Publications

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1. Tsang, Nair, Lu, PRX 6, 031033 (2016).2. Nair, Tsang, OE 24, 3684 (2016).3. Tsang, Nair, Lu, SPIE 10029, 1002903 (2016).4. Nair, Tsang, PRL 117, 190801 (2016).5. Ang, Nair, Tsang, PRA 95, 063847 (2017).6. Tsang, NJP 19, 023054 (2017).7. Yang, Nair, Tsang, Simon, Lvovsky, PRA 96, 063829 (2017).8. Tsang, JMO 65, 104 (2018).9. Tsang, PRA 97, 023830 (2018).

10. Lu, Krovi, Nair, Guha, Shapiro, arXiv:1802.02300 (2018).11. Tsang, arXiv:1806.02781 (2018).

Future Directions

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� 3D?

– Backlund, Shechtman, Walsworth (Harvard/Technion), PRL 121, 023904(2018);

– Yu and Prasad (New Mexico, USA), arXiv:1805.09227 (2018);– Napoli, Tufarelli, Piano, Leach, Adesso (Nottingham, UK), arXiv:1805.04116

(2018).

� Experiments: the rest is engineering.� FAQ:

– https://sites.google.com/site/mankeitsang/news/rayleigh/faq

– Mirror: https://www.ece.nus.edu.sg/stfpage/tmk/faq.html

� Thank you.

Quantum Technology 1.5

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Wave Nature: Diffraction Limit

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Diffraction-limited (λ/NA)

1. Fluorescence microscopy2. Space telescopes (Webb, $10 billion)3. Ground-based telescopes (corrected by adap-

tive optics):

(a) Large Binocular Telescope (LBT) (Strehl ra-tio > 80%, $120 million)

(b) Giant Magellan Telescope (GMT)(c) Thirty Meter Telescope (TMT)(d) European Extremely Large Telescope (E-

ELT) (>$1 billion each)

Esposito et al., SPIE 8149, 814902 (2011).

Photon Shot Noise

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� Thermal sources (stars, etc.)

– Poisson, bunching negligible at optical– Goodman, Statistical Optics; Zmuidzinas, JOSA A

20, 218 (2003)

� Fluorophores (GFP, dye molecules, quantum dots, etc.)

– Poisson, negligible anti-bunching– Pawley ed., Handbook of Biological Confocal Mi-

croscopy ; Ram, Ober, Ward, PNAS (2006)

Two Point Sources

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(a)

(b)

Stellar Interferometry

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� Astrophotonics: photonic cir-cuits for stellar interferometry

� Conventional wisdom: lesssensitive to atmospheric turbu-lence

� Our work: Fundamental ad-vantage with diffraction +photon shot noise

� Singapore: fluorescence mi-croscopy “Dragonfly,” Jovanovic et al.,

Mon. Not. R. Astron. Soc. 427, 806(2012)

Misalignment

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∆ = centroid displacement+ object size ≪ 1 (16)