Abstract

The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. Lord Rayleigh, “LVI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(51), 477–486 (1879).
    [Crossref]
  2. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. für Mikrosk. Anat. 9, 413–418 (1873).
  3. C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
    [Crossref]
  4. A. J. D. Dekker and A. V. D. Bos, “Resolution: a survey,” J. Opt. Soc. Am. A 14, 547–557 (1997).
    [Crossref]
  5. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1471 (1966).
    [Crossref]
  6. C. Rushforth and R. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [Crossref]
  7. J. G. Walker, “Optical imaging with resolution exceeding the Rayleigh criterion,” Opt. Acta 30, 1197–1202 (1983).
    [Crossref]
  8. E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
    [Crossref]
  9. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [Crossref]
  10. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
    [Crossref]
  11. S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
    [Crossref]
  12. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, 1989), Vol. 14.
  13. C. W. Helstrom, “Resolvability of objects from the standpoint of statistical parameter estimation,” J. Opt. Soc. Am. 60, 659–666 (1970).
    [Crossref]
  14. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  15. J. Zmuidzinas, “Cramér-Rao sensitivity limits for astronomical instruments: implications for interferometer design,” J. Opt. Soc. Am. A 20, 218–233 (2003).
    [Crossref]
  16. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
    [Crossref]
  17. C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurement of very small displacements in optical images,” Opt. Lett. 25, 76–78 (2000).
    [Crossref]
  18. M. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
    [Crossref]
  19. M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
    [Crossref]
  20. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  21. A. Holevo, Probabilistic and Statistical Aspects of Quantum Mechanics (North-Holland, 1982).
  22. M. Tsang, “Quantum limits to optical point-source localization,” Optica 2, 646–653 (2015).
    [Crossref]
  23. Z. S. Tang, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” Opt. Express 24, 22004–22012 (2016).
    [Crossref]
  24. M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
    [Crossref]
  25. R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (2016).
    [Crossref]
  26. 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]
  27. 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]
  28. W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
    [Crossref]
  29. R. Kerviche, S. Guha, and A. Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in IEEE International Symposium on Information Theory (2017), pp. 441–445.
  30. M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
    [Crossref]
  31. J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
    [Crossref]
  32. M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
    [Crossref]
  33. V. P. Nayyar and N. K. Verma, “Two-point resolution of Gaussian aperture operating in partially coherent light using various resolution criteria,” Appl. Opt. 17, 2176–2180 (1978).
    [Crossref]
  34. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  35. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Vol. 64.
  36. R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
    [Crossref]
  37. R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117, 190801 (2016).
    [Crossref]
  38. S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
    [Crossref]
  39. K. Matsumoto, “A new approach to the Cramer-Rao-type bound of the pure-state model,” J. Phys. A 35, 3111–3123 (2002).
    [Crossref]
  40. P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
    [Crossref]
  41. L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
    [Crossref]

2018 (1)

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

2017 (4)

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]

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

2016 (7)

R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
[Crossref]

Z. S. Tang, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” Opt. Express 24, 22004–22012 (2016).
[Crossref]

M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
[Crossref]

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (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]

2015 (1)

2014 (2)

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

2012 (1)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

2009 (1)

M. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

2006 (2)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

2003 (1)

2002 (2)

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

K. Matsumoto, “A new approach to the Cramer-Rao-type bound of the pure-state model,” J. Phys. A 35, 3111–3123 (2002).
[Crossref]

2000 (2)

1997 (1)

1994 (1)

1983 (1)

J. G. Walker, “Optical imaging with resolution exceeding the Rayleigh criterion,” Opt. Acta 30, 1197–1202 (1983).
[Crossref]

1978 (1)

1970 (1)

1968 (1)

1966 (1)

1916 (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

1879 (1)

Lord Rayleigh, “LVI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(51), 477–486 (1879).
[Crossref]

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. für Mikrosk. Anat. 9, 413–418 (1873).

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. für Mikrosk. Anat. 9, 413–418 (1873).

Abdullah, M. S.

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

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]

Ashok, A.

R. Kerviche, S. Guha, and A. Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in IEEE International Symposium on Information Theory (2017), pp. 441–445.

Barbieri, M.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Baumgratz, T.

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
[Crossref]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Bos, A. V. D.

Candès, E. J.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

Ciampini, M. A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Crowley, P. J.

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Datta, A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Dekker, A. J. D.

Demkowicz-Dobrza, R.

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Durak, K.

Fabre, C.

Fernandez-Granda, C.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

Ferretti, H.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Fouet, J. B.

Girirajan, T. P.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Grover, J.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

Guha, S.

R. Kerviche, S. Guha, and A. Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in IEEE International Symposium on Information Theory (2017), pp. 441–445.

Guta, M.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Harris, R.

Hayat, M. M.

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

Hell, S. W.

Helstrom, C. W.

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Hess, S. T.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

Holevo, A.

A. Holevo, Probabilistic and Statistical Aspects of Quantum Mechanics (North-Holland, 1982).

Hradil, Z.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
[Crossref]

Humphreys, P. C.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, 1989), Vol. 14.

Jarzyna, M.

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

Joobeur, A.

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Kerviche, R.

R. Kerviche, S. Guha, and A. Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in IEEE International Symposium on Information Theory (2017), pp. 441–445.

Kolobov, M. I.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref]

Kolodynski, J.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Krzic, A.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Ling, A.

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Lord Rayleigh,

Lord Rayleigh, “LVI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(51), 477–486 (1879).
[Crossref]

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]

Lukosz, W.

Maître, A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Vol. 64.

Mason, M. D.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

Matsumoto, K.

K. Matsumoto, “A new approach to the Cramer-Rao-type bound of the pure-state model,” J. Phys. A 35, 3111–3123 (2002).
[Crossref]

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]

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (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]

R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

Nayyar, V. P.

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Paris, M.

M. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Paur, M.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

Paúr, M.

Pezzè, L.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Ragy, S.

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

Rehacek, J.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
[Crossref]

Rushforth, C.

Saleh, B. E.

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

Sanchez-Soto, L. L.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

Sánchez-Soto, L. L.

Sciarrino, F.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Smerzi, A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Spagnolo, N.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

Sparrow, C. M.

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

Steinberg, A. M.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Stoklasa, B.

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
[Crossref]

Szczykulska, M.

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
[Crossref]

Tang, Z. S.

Tham, W. K.

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Tsang, M.

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]

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (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]

R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

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

Verma, N. K.

Walker, J. G.

J. G. Walker, “Optical imaging with resolution exceeding the Rayleigh criterion,” Opt. Acta 30, 1197–1202 (1983).
[Crossref]

Walmsley, I. A.

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Wichmann, J.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Vol. 64.

Zmuidzinas, J.

Adv. Phys. (1)

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Phys. 1, 621–639 (2016).
[Crossref]

Appl. Opt. (1)

Arch. für Mikrosk. Anat. (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. für Mikrosk. Anat. 9, 413–418 (1873).

Astrophys. J. (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76–86 (1916).
[Crossref]

Biophys. J. (1)

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

Commun. Pure Appl. Math. (1)

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure Appl. Math. 67, 906–956 (2014).
[Crossref]

IEEE Trans. Image Process. (1)

M. M. Hayat, M. S. Abdullah, A. Joobeur, and B. E. Saleh, “Maximum-likelihood image estimation using photon-correlated beams,” IEEE Trans. Image Process. 11, 838–846 (2002).
[Crossref]

Int. J. Quantum Inf. (1)

M. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 7, 125–137 (2009).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

K. Matsumoto, “A new approach to the Cramer-Rao-type bound of the pure-state model,” J. Phys. A 35, 3111–3123 (2002).
[Crossref]

Nat. Commun. (1)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

Opt. Acta (1)

J. G. Walker, “Optical imaging with resolution exceeding the Rayleigh criterion,” Opt. Acta 30, 1197–1202 (1983).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Optica (2)

Philos. Mag. (1)

Lord Rayleigh, “LVI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(51), 477–486 (1879).
[Crossref]

Phys. Rev. A (5)

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza, “Compatibility in multiparameter quantum metrology,” Phys. Rev. A 94, 052108 (2016).
[Crossref]

P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[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]

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, and L. L. Sanchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

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

Phys. Rev. Lett. (4)

W. K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref]

L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, “Optimal measurements for simultaneous quantum estimation of multiple phases,” Phys. Rev. Lett. 119, 130504 (2017).
[Crossref]

R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117, 190801 (2016).
[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]

Science (1)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Other (7)

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, 1989), Vol. 14.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

A. Holevo, Probabilistic and Statistical Aspects of Quantum Mechanics (North-Holland, 1982).

R. Kerviche, S. Guha, and A. Ashok, “Fundamental limit of resolving two point sources limited by an arbitrary point spread function,” in IEEE International Symposium on Information Theory (2017), pp. 441–445.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Vol. 64.

Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. CFI precision Hs1/Var(s) for estimates of the two-point separation s (normalized to the width σ of the PSF), based on direct intensity measurement in the image plane, assuming that the coherence parameter p is known, in the correlated case (solid) and the anti-correlated case (dashed). Here, p takes values between 0 and 1, with intermediate values 1/3 and 2/3. For p=0 the correlated and anti-correlated cases are identical.
Fig. 2.
Fig. 2. Same as in Fig. 1, except that the coherence parameter p is estimated concurrently.
Fig. 3.
Fig. 3. Quantum Fisher information precision bound Hs=1/Var(s) on estimates of the separation s between two point sources by use of optimal measurement in the image plane. The coherence parameter p is assumed to be perfectly known. Correlated (solid) and anti-correlated (dashed) coherence are assumed. Here, p takes values between 0 and 1, with intermediate values 1/3 and 2/3. For p=0 the correlated and anti-correlated cases are identical.
Fig. 4.
Fig. 4. Same as in Fig. 3, but assuming that the coherence parameter p is unknown and is concurrently estimated with the separation s using the multi-parameter QFIM. When p is not precisely known, Rayleigh’s curse persists.
Fig. 5.
Fig. 5. Ratio of the precision bounds for the optimal strategy against the precision bounds afforded to intensity imaging for the correlated (solid) and anti-correlated (dashed) cases, and p=0, 1/4, 2/4, 3/4, 1. For p=0, the correlated and anti-correlated cases are identical.
Fig. 6.
Fig. 6. Precision bound Hp=1/Var(p) on estimates of the coherence parameter p based on direct intensity measurement (top) and quantum optimal measurement (bottom) for concurrent estimates of p and the two-point separation s in the correlated case (solid) and the anti-correlated case (dashed), for p=0, 1/3, 2/3, 1.

Equations (43)

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E(x)=E0[h+(x)±h(x)],
I(x)=|E(x)|2=I0|h+(x)±h(x)|2,
|ψc=12(1±d)[|ψ+±|ψ],
ψc(x)=12(1±d)[h+(x)±h(x)].
d=Re(ψ|ψ+)=Re(dxh*(x+s2)h(xs2))
f(x)=|ψc(x)|2=12(1±d)|h+(x)±h(x)|2.
E(x)=E+h+(x)+Eh(x)
I(x)=|E(x)|2=I0[|h+(x)|2+|h(x)|2]+2I0Re[γh+*(x)h(x)],
ρ^i=12(|ψ+ψ+|+|ψψ|).
ρ^=pρ^c+(1p)ρ^i,
f(x)=I0(|h+(x)|2+|h(x)|2)+2I0Re(γh+*(x)h(x)),
I0=12[p1±d+(1p)],
γ=±p1±(1p)d.
F=dx(sf(x))21f(x).
h2(x)=12πσex22σ2,
Fjk=dx(jf(x))(kf(x))1f(x),
Qi,j=12Tr[(L^iL^j+L^jL^i)ρ^],
12(L^iρ^+ρ^L^i)=iρ^,
Var(s)Qss1=QppQppQssQsp2,
Var(p)Qpp1=QssQppQssQsp2.
Hs=QssQsp2Qpp,
Hp=QppQsp2Qss.
Tr[ρ^[Ls,Lp]]=0,
|e1=12(1d)(|ψ+|ψ),
|e2=12(1+d)(|ψ++|ψ),
λ1±=12[(1p)(1d)+pp],
λ2±=12[(1p)(1+d)+p±p],
ρ^±=λ1|e1e1|+λ2|e2e2|.
pρ^±=λ1p|e1e1|+λ2p|e2e2|,
sρ^±=λ1s|e1e1|+λ2s|e2e2|+2[λ1α3|e3e1|+λ2α4|e4e2|+H.C.],
α3|e3=s|e1,
α4|e4=s|e2,
L^i±=l,k=142λk+λl(ek|iρ^(±)|el)|ekel|.
[Ls±]11=1λ1±sλ1±=(1p)λ1±Γ,[Ls±]22=1λ2±sλ2±=(1p)2λ2±Γ,[Ls±]31=[Ls±]13=2α3,[Ls±]42=[Ls±]24=2α4,[Lp±]11=1λ1±pλ1±=(d1)2λ1±,[Lp±]22=1λ2±pλ2±=(d1)2λ2±.
α3=12a2+b21dΓ2(1d)2,
α4=12a2b21+dΓ2(1+d)2,
a2=dx[h(x)]2,
b2=dxh(xs2)h(x+s2),
Γ=ds=dxh(x)h(xs).
Qss±=λ1±([Ls±]112+[Ls±]132)+λ2±([Ls±]222+[Ls±]132),
Qpp±=λ1±[Ls±]112+λ2±[Ls±]222,
Qsp±=λ1±[Ls±]11[Ls±]11+λ2±[Ls±]22[Ls±]22.
d=es2/8σ2,a2=14σ2,Γ=ds4σ2,b2=d4σ2(s24σ21).

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