Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for...
Transcript of Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for...
![Page 1: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/1.jpg)
1 / 31
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
![Page 2: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/2.jpg)
Lord Rayleigh (1879)
2 / 31
� Resolved:
� Not resolved:
![Page 3: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/3.jpg)
Telescopes, Fluorescence Microscopy
3 / 31
(images from the internet)
![Page 4: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/4.jpg)
Photon Shot Noise
4 / 31
� Random arrival of photons
![Page 5: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/5.jpg)
Parameter Estimation
5 / 31
� 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)
![Page 6: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/6.jpg)
Limit to Parameter Estimation
6 / 31
� 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).
![Page 7: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/7.jpg)
Superresolution Microscopy
7 / 31
� 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
![Page 8: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/8.jpg)
Fundamental Quantum Limit
8 / 31
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.
![Page 9: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/9.jpg)
Quantum Optics for Incoherent Imaging
9 / 31
� 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).
![Page 10: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/10.jpg)
Plenty of Room at the Bottom
10 / 31
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).
![Page 11: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/11.jpg)
Optimal Measurement
11 / 31
� 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).
![Page 12: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/12.jpg)
Spatial-Mode Demultiplexing (SPADE)
12 / 31
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.)
![Page 13: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/13.jpg)
Elementary Explanation
13 / 31
� 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.
![Page 14: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/14.jpg)
Experimental Demonstrations
14 / 31
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).
![Page 15: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/15.jpg)
European Space Agency
15 / 31
https://www.esa.int/gsp/ACT/projects/super_resolution.html
![Page 16: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/16.jpg)
Arbitrary Source Distribution
16 / 31
� 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
![Page 17: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/17.jpg)
Defining Subdiffraction Regime
17 / 31
Sparse (Good Regime,PALM, STED, compressedsensing, etc.)
Subdiffraction (Bad Regime)
object width ≡ ∆ ≪ 1. (8)
![Page 18: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/18.jpg)
Cramer-Rao Bound for Direct Imaging
18 / 31
� 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
![Page 19: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/19.jpg)
Quantum Limit
19 / 31
� 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
![Page 20: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/20.jpg)
SPADE for Moment Estimation in 2D Imaging
20 / 31
� 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
![Page 21: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/21.jpg)
More General PSFs
21 / 31
� 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
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33 00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
00 10 20 30
01
02
03
11 21 31
12 22 32
13 23 33
(10) (50)
(12) (52)
(54)
(56)
(14)
(16)
(01) (21) (41) (61)
(05) (25) (45) (65)
(11) (31) (51)
(15) (35) (55)
(36)
(34)
(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)
![Page 22: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/22.jpg)
Performance of SPADE
22 / 31
� 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).
![Page 23: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/23.jpg)
Elementary Explanation
23 / 31
� 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)
![Page 24: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/24.jpg)
Publications
24 / 31
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).
![Page 25: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/25.jpg)
Future Directions
25 / 31
� 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.
![Page 26: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/26.jpg)
Quantum Technology 1.5
26 / 31
![Page 27: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/27.jpg)
Wave Nature: Diffraction Limit
27 / 31
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).
![Page 28: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/28.jpg)
Photon Shot Noise
28 / 31
� 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)
![Page 29: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/29.jpg)
Two Point Sources
29 / 31
(a)
(b)
![Page 30: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/30.jpg)
Stellar Interferometry
30 / 31
� 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)
![Page 31: Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang · 1/31 Seize the Moments for Subdiffraction Incoherent Imaging Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, Mankei Tsang](https://reader033.fdocuments.net/reader033/viewer/2022053018/5f1e14f3c093a75a8304bae9/html5/thumbnails/31.jpg)
Misalignment
31 / 31
∆ = centroid displacement+ object size ≪ 1 (16)