Abstract

Larson and Saleh [Optica 5, 1382 (2018) [CrossRef]  ] suggest that Rayleigh’s curse can recur and become unavoidable if the two sources are partially coherent. Here we show that their calculations and assertions have fundamental problems, and spatial-mode demultiplexing can overcome Rayleigh’s curse even for partially coherent sources.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. W. Larson and B. E. A. Saleh, “Resurgence of Rayleigh’s curse in the presence of partial coherence,” Optica 5, 1382–1389 (2018).
    [Crossref]
  2. M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
    [Crossref]
  3. M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
    [Crossref]
  4. M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
    [Crossref]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  6. J. W. Goodman, Statistical Optics (Wiley, 1985).
  7. J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 2006).
  8. M. Tsang, “Quantum limits to optical point-source localization,” Optica 2, 646–653 (2015).
    [Crossref]
  9. S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
    [Crossref]
  10. M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017).
    [Crossref]
  11. M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99, 012305 (2019).
    [Crossref]

2019 (1)

M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99, 012305 (2019).
[Crossref]

2018 (2)

W. Larson and B. E. A. Saleh, “Resurgence of Rayleigh’s curse in the presence of partial coherence,” Optica 5, 1382–1389 (2018).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
[Crossref]

2017 (2)

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017).
[Crossref]

2016 (2)

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
[Crossref]

2015 (1)

Ang, S. Z.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Larson, W.

Lu, X.-M.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Nair, R.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
[Crossref]

Saleh, B. E. A.

Tsang, M.

M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99, 012305 (2019).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
[Crossref]

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
[Crossref]

M. Tsang, “Quantum limits to optical point-source localization,” Optica 2, 646–653 (2015).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

New J. Phys. (1)

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017).
[Crossref]

Optica (2)

Phys. Rev. A (3)

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95, 063847 (2017).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
[Crossref]

M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99, 012305 (2019).
[Crossref]

Phys. Rev. X (1)

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).
[Crossref]

Proc. SPIE (1)

M. Tsang, R. Nair, and X.-M. Lu, “Quantum information for semiclassical optics,” Proc. SPIE 10029, 1002903 (2016).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 2006).

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Figures (2)

Fig. 1.
Fig. 1. Total Fisher information in Hermite–Gaussian modes up to q = 20 versus the separation for various values of γ (denoted by the legend).
Fig. 2.
Fig. 2. Plots of the degree of coherence γ versus separation s for various values of p (denoted by the legends), according to Eq. (13) in Ref. [1]. The Gaussian point-spread function is assumed.

Equations (8)

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Γ ( x , x ) = N 0 [ h + ( x ) h + * ( x ) + h ( x ) h * ( x ) + γ h + ( x ) h * ( x ) + γ * h ( x ) h + * ( x ) ] ,
n q = d x d x ϕ q * ( x ) Γ ( x , x ) ϕ q ( x ) .
n 1 = 2 N 0 ( 1 Re γ ) s 2 16 σ 2 exp ( s 2 16 σ 2 ) .
F q = 1 n q ( n q s ) 2 ,
F 1 = N 0 2 σ 2 ( 1 Re γ ) ( 1 s 2 16 σ 2 ) 2 exp ( s 2 16 σ 2 ) .
F 1 ( s = 0 ) = N 0 2 σ 2 ( 1 Re γ ) ,
Q ( ρ ) = ϵ Q ( ρ 1 ) + J ( ϵ ) ,
J μ ν ( ϵ ) = 1 ϵ ( 1 ϵ ) ϵ θ μ ϵ θ ν 1 ϵ ϵ θ μ ϵ θ ν .