Abstract

The Rayleigh criterion specifies the minimum separation between two incoherent point sources that may be resolved into distinct objects. We revisit this problem by examining the Fisher information required for resolving the two sources. The resulting Cramér–Rao bound gives the minimum error achievable for any unbiased estimator. When only the intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect that has been dubbed the Rayleigh curse. Nonetheless, this curse can be lifted with suitable measurements. Here, we work out optimal strategies and present a realization for Gaussian and slit apertures, which is accomplished with digital holographic techniques. Our results confirm immunity to the Rayleigh curse and an unprecedented experimental precision.

© 2016 Optical Society of America

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References

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  1. E. Abbe, Arch. Mikrosk. Anat. 9, 469 (1873).
    [Crossref]
  2. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).
  3. L. Rayleigh, Philos. Mag. 8, 261 (1879).
    [Crossref]
  4. Nat. Photonics3, 361 (2009).
    [Crossref]
  5. S. W. Hell, Science 316, 1153 (2007).
    [Crossref]
  6. M. I. Kolobov, “Quantum limits of optical super-resolution,” in Quantum Imaging (Springer, 2007), pp. 113–138.
  7. S. W. Hell, Nat. Methods 6, 24 (2009).
    [Crossref]
  8. G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
    [Crossref]
  9. A. J. den Dekker and A. van den Bos, J. Opt. Soc. Am. A 14, 547 (1997).
    [Crossref]
  10. C. Cremer and B. R. Masters, Eur. Phys. J. H 38, 281 (2013).
    [Crossref]
  11. M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).
  12. R. Nair and M. Tsang, “Ultimate quantum limit on resolution of two thermal point sources,” arXiv:16004.00937.
  13. S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” arXiv:1606.00603.
  14. L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
    [Crossref]
  15. D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), Vol. 27, pp. 261–281.
  16. C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367.
  17. T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297.
  18. F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.
  19. W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh's curse by imaging using phase information,” arXiv:1606.02666.

2016 (2)

M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

2013 (1)

C. Cremer and B. R. Masters, Eur. Phys. J. H 38, 281 (2013).
[Crossref]

2010 (1)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

2009 (1)

S. W. Hell, Nat. Methods 6, 24 (2009).
[Crossref]

2007 (1)

S. W. Hell, Science 316, 1153 (2007).
[Crossref]

1997 (1)

1879 (1)

L. Rayleigh, Philos. Mag. 8, 261 (1879).
[Crossref]

1873 (1)

E. Abbe, Arch. Mikrosk. Anat. 9, 469 (1873).
[Crossref]

Abbe, E.

E. Abbe, Arch. Mikrosk. Anat. 9, 469 (1873).
[Crossref]

Ang, S. Z.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” arXiv:1606.00603.

Cremer, C.

C. Cremer and B. R. Masters, Eur. Phys. J. H 38, 281 (2013).
[Crossref]

D’Angelo, M.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Davidson, M.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

den Dekker, A. J.

Durak, K.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297.

Facchi, P.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Ferretti, H.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh's curse by imaging using phase information,” arXiv:1606.02666.

Garuccio, A.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Ghinea, C.

D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), Vol. 27, pp. 261–281.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).

Hell, S. W.

S. W. Hell, Nat. Methods 6, 24 (2009).
[Crossref]

S. W. Hell, Science 316, 1153 (2007).
[Crossref]

Hradil, Z.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Kolobov, M. I.

M. I. Kolobov, “Quantum limits of optical super-resolution,” in Quantum Imaging (Springer, 2007), pp. 113–138.

Ling, A.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297.

Lippincott-Schwartz, J.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

Lu, X.-M.

M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).

Lupo, C.

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367.

Lvovsky, A. I.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

Manley, S.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

Masters, B. R.

C. Cremer and B. R. Masters, Eur. Phys. J. H 38, 281 (2013).
[Crossref]

Moiseev, E. S.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

Motka, L.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Nair, R.

M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” arXiv:1606.00603.

R. Nair and M. Tsang, “Ultimate quantum limit on resolution of two thermal point sources,” arXiv:16004.00937.

Pascazio, S.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Patterson, G.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

Pepe, F. V.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Petz, D.

D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), Vol. 27, pp. 261–281.

Pirandola, S.

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367.

Rayleigh, L.

L. Rayleigh, Philos. Mag. 8, 261 (1879).
[Crossref]

Rehacek, J.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Sanchez-Soto, L. L.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Sheng, T. Z.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297.

Simon, C.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

Steinberg, A. M.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh's curse by imaging using phase information,” arXiv:1606.02666.

Stoklasa, B.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Taschilina, A.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

Teo, Y. S.

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

Tham, W. K.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh's curse by imaging using phase information,” arXiv:1606.02666.

Tsang, M.

M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).

R. Nair and M. Tsang, “Ultimate quantum limit on resolution of two thermal point sources,” arXiv:16004.00937.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” arXiv:1606.00603.

van den Bos, A.

Yang, F.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

Annu. Rev. Phys. Chem. (1)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, Annu. Rev. Phys. Chem. 61, 345 (2010).
[Crossref]

Arch. Mikrosk. Anat. (1)

E. Abbe, Arch. Mikrosk. Anat. 9, 469 (1873).
[Crossref]

Eur. Phys. J. H (1)

C. Cremer and B. R. Masters, Eur. Phys. J. H 38, 281 (2013).
[Crossref]

Eur. Phys. J. Plus (1)

L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Methods (1)

S. W. Hell, Nat. Methods 6, 24 (2009).
[Crossref]

Philos. Mag. (1)

L. Rayleigh, Philos. Mag. 8, 261 (1879).
[Crossref]

Phys. Rev. X (1)

M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).

Science (1)

S. W. Hell, Science 316, 1153 (2007).
[Crossref]

Other (10)

M. I. Kolobov, “Quantum limits of optical super-resolution,” in Quantum Imaging (Springer, 2007), pp. 113–138.

D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), Vol. 27, pp. 261–281.

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297.

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” arXiv:1606.02662.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh's curse by imaging using phase information,” arXiv:1606.02666.

R. Nair and M. Tsang, “Ultimate quantum limit on resolution of two thermal point sources,” arXiv:16004.00937.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” arXiv:1606.00603.

Nat. Photonics3, 361 (2009).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004).

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Figures (2)

Fig. 1.
Fig. 1. Schematic diagram of the experimental setup. Two incoherent point sources are created with a high frequency switched digital micromirror chip (DMD) illuminated with an intensity-stabilized He–Ne laser. The sources are imaged by a low-aperture lens. In the image plane, projection onto different modes is performed with a digital hologram created with an amplitude spatial light modulator (SLM). Information about the desired projection is carried by the first-order diffraction spectrum, which is mapped by a lens onto an EMCCD camera.
Fig. 2.
Fig. 2. Mean-square error of the estimated separation for Gaussian (left panel) and sinc (right panel) PSFs. Separations are expressed in units of PSF widths σ and w and the MSE in units of the qCRLB. The main graph compares the performance of our experimental method (blue symbols) with the theoretical lower bound for the CCD measurement (thin red curve) and the ultimate limit (thick red line). The vertical dotted lines delimit the 10% of the Rayleigh limit for each PSF. The insets show the statistics of the experimental estimates. Mean values are plotted in blue dots with standard deviation bars around. The true values are inside the standard deviation intervals for all separations and the estimator bias is negligible. For the two largest measured separations, the experimental MSE nicely follows the CRLB calculated for the experimentally realized antisymmetric projection (orange dots).

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

ϱs=12(|ψ1ψ1|+|ψ2ψ2|),
|ψ1=N(1+iPs/2)|ψ,|ψ2=N(1iPs/2)|ψ,
|ψ+=C+(|ψ1+|ψ2)|ψ,|ψ=C(|ψ1|ψ2)P|ψψ|P2|ψ,
F=2[1pψ|ϱss|ψ+1p+ψ+|ϱss|ψ+]ψ|P2|ψ,
(Δs^)21F=1ψ|P2|ψ.
Fstd=1ϱs(x)[ϱs(x)s]2dx.
Fstds2[I(x)]2I(x)dx.
ψopt(x)=x|ψ=ψ(x)F,
F=ψ|P2|ψ=[ψ(x)]2dx.
ψG(x)=1(2πσ2)14exp(x24σ2),ψS(x)=1wsin(πx/w)πx/w,
ψoptG(x)=1(2π)14σ32xexp(x24σ2),ψoptS(x)=3[w12πxcos(πxw)w32π2x2sin(πxw)].
|ψopt*(x)ψ(x+s/2)dx|2+|ψopt*(x)ψ(xs/2)dx|2.

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