Abstract

A novel interferometric method – SLIVER (Super Localization by Image inVERsion interferometry) – is proposed for estimating the separation of two incoherent point sources with a mean squared error that does not deteriorate as the sources are brought closer. The essential component of the interferometer is an image inversion device that inverts the field in the transverse plane about the optical axis, assumed to pass through the centroid of the sources. The performance of the device is analyzed using the Cramér-Rao bound applied to the statistics of spatially-unresolved photon counting using photon number-resolving and on-off detectors. The analysis is supported by Monte-Carlo simulations of the maximum likelihood estimator for the source separation, demonstrating the superlocalization effect for separations well below that set by the Rayleigh criterion. Simulations indicating the robustness of SLIVER to mismatch between the optical axis and the centroid are also presented. The results are valid for any imaging system with a circularly symmetric point-spread function.

© 2016 Optical Society of America

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  1. Lord Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879).
    [Crossref]
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    [Crossref]
  3. E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
    [Crossref]
  4. S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
    [Crossref]
  5. S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
    [Crossref] [PubMed]
  6. H. L. Van Trees, Detection, Estimation, and Modulation Theory : Part I, 1st ed. (Wiley-Interscience1st Ed, 2001).
  7. M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” (2015). http://arxiv.org/abs/1511.00552 .
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    [Crossref]
  9. J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
    [Crossref]
  10. M. Tsang, “Quantum limits to optical point-source localization,” Optica 2, 646–653 (2015).
    [Crossref]
  11. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).
  12. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni Della Normale, 2011).
    [Crossref]
  13. M. G. Paris, “Quantum estimation for quantum technology,” International Journal of Quantum Information 7, 125–137 (2009).
    [Crossref]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  15. J. H. Shapiro, “The quantum theory of optical communications,” IEEE Journal of Selected Topics in Quantum Electronics 15, 1547–1569 (2009).
    [Crossref]
  16. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
    [Crossref]
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    [Crossref]
  27. R. S. Kennedy, “A near-optimum receiver for the binary coherent state channel,” Tech. Rep. 108, MIT RLE Quarterly Progress Report (1973).
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    [Crossref]
  30. U. Kubitscheck, (Ed.), Fluorescence Microscopy: from Principles to Biological Applications (John Wiley & Sons, 2013).
    [Crossref]

2015 (3)

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

S. Weisenburger and V. Sandoghdar, “Light microscopy: an ongoing contemporary revolution,” Contemporary Physics 56, 123–143 (2015).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

2014 (2)

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Physical Review A 89, 032318 (2014).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

2013 (2)

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

2011 (2)

D. Weigel, R. Foerster, H. Babovsky, A. Kiessling, and R. Kowarschik, “Enhanced resolution of microscopic objects by image inversion interferometry,” Opt. Express 19, 26451–26462 (2011).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

2009 (3)

K. Wicker, S. Sindbert, and R. Heintzmann, “Characterisation of a resolution enhancing image inversion interferometer,” Opt. Express 17, 15491–15501 (2009).
[Crossref] [PubMed]

M. G. Paris, “Quantum estimation for quantum technology,” International Journal of Quantum Information 7, 125–137 (2009).
[Crossref]

J. H. Shapiro, “The quantum theory of optical communications,” IEEE Journal of Selected Topics in Quantum Electronics 15, 1547–1569 (2009).
[Crossref]

2008 (1)

S. Kay and Y. C. Eldar, “Rethinking biased estimation,” IEEE Signal Processing Magazine 25, 133–136 (2008).
[Crossref]

2007 (1)

2006 (2)

N. Sandeau and H. Giovannini, “Increasing the lateral resolution of 4pi fluorescence microscopes,” J. Opt. Soc. Am. A 23, 1089–1095 (2006).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
[Crossref]

2002 (1)

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

1999 (1)

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

1879 (1)

Lord Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879).
[Crossref]

Aert, S. V.

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

Babovsky, H.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

D. Weigel, R. Foerster, H. Babovsky, A. Kiessling, and R. Kowarschik, “Enhanced resolution of microscopic objects by image inversion interferometry,” Opt. Express 19, 26451–26462 (2011).
[Crossref]

Bettens, E.

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Chao, J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

den Dekker, A.

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

DenDekker, A.

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Dolinar, S. J.

S. J. Dolinar, “An optimum receiver for the binary coherent state channel,” Tech. Rep. 111, MIT RLE Quarterly Progress Report (1973).

Dyck, D. V.

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

Eldar, Y. C.

S. Kay and Y. C. Eldar, “Rethinking biased estimation,” IEEE Signal Processing Magazine 25, 133–136 (2008).
[Crossref]

Elsmann, T.

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

Foerster, R.

Giovannini, H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

Guha, S.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Physical Review A 89, 032318 (2014).
[Crossref]

Heintzmann, R.

Helstrom, C. W.

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

Holevo, A. S.

A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni Della Normale, 2011).
[Crossref]

Kay, S.

S. Kay and Y. C. Eldar, “Rethinking biased estimation,” IEEE Signal Processing Magazine 25, 133–136 (2008).
[Crossref]

Kennedy, R. S.

R. S. Kennedy, “A near-optimum receiver for the binary coherent state channel,” Tech. Rep. 108, MIT RLE Quarterly Progress Report (1973).

Kiessling, A.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

D. Weigel, R. Foerster, H. Babovsky, A. Kiessling, and R. Kowarschik, “Enhanced resolution of microscopic objects by image inversion interferometry,” Opt. Express 19, 26451–26462 (2011).
[Crossref]

Kowarschik, R.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

D. Weigel, R. Foerster, H. Babovsky, A. Kiessling, and R. Kowarschik, “Enhanced resolution of microscopic objects by image inversion interferometry,” Opt. Express 19, 26451–26462 (2011).
[Crossref]

Kubitscheck, U.

U. Kubitscheck, (Ed.), Fluorescence Microscopy: from Principles to Biological Applications (John Wiley & Sons, 2013).
[Crossref]

Mandel, L.

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

Nair, R.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Physical Review A 89, 032318 (2014).
[Crossref]

Ober, R. J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
[Crossref]

Paris, M. G.

M. G. Paris, “Quantum estimation for quantum technology,” International Journal of Quantum Information 7, 125–137 (2009).
[Crossref]

Ram, S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
[Crossref]

Rayleigh, Lord

Lord Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879).
[Crossref]

Sandeau, N.

Sandoghdar, V.

S. Weisenburger and V. Sandoghdar, “Light microscopy: an ongoing contemporary revolution,” Contemporary Physics 56, 123–143 (2015).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “The quantum theory of optical communications,” IEEE Journal of Selected Topics in Quantum Electronics 15, 1547–1569 (2009).
[Crossref]

Sijbers, J.

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Sindbert, S.

Tan, S.-H.

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Physical Review A 89, 032318 (2014).
[Crossref]

Tsang, M.

van den Bos, A.

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Van Dyck, D.

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory : Part I, 1st ed. (Wiley-Interscience1st Ed, 2001).

Ward, E. S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
[Crossref]

Weigel, D.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

D. Weigel, R. Foerster, H. Babovsky, A. Kiessling, and R. Kowarschik, “Enhanced resolution of microscopic objects by image inversion interferometry,” Opt. Express 19, 26451–26462 (2011).
[Crossref]

Weisenburger, S.

S. Weisenburger and V. Sandoghdar, “Light microscopy: an ongoing contemporary revolution,” Contemporary Physics 56, 123–143 (2015).
[Crossref]

Wicker, K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

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

Contemporary Physics (1)

S. Weisenburger and V. Sandoghdar, “Light microscopy: an ongoing contemporary revolution,” Contemporary Physics 56, 123–143 (2015).
[Crossref]

IEEE Journal of Selected Topics in Quantum Electronics (1)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE Journal of Selected Topics in Quantum Electronics 15, 1547–1569 (2009).
[Crossref]

IEEE Signal Processing Magazine (1)

S. Kay and Y. C. Eldar, “Rethinking biased estimation,” IEEE Signal Processing Magazine 25, 133–136 (2008).
[Crossref]

International Journal of Quantum Information (1)

M. G. Paris, “Quantum estimation for quantum technology,” International Journal of Quantum Information 7, 125–137 (2009).
[Crossref]

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

Journal of Structural Biology (1)

S. V. Aert, A. den Dekker, D. V. Dyck, and A. van den Bos, “High-resolution electron microscopy and electron tomography: resolution versus precision,” Journal of Structural Biology 138, 21–33 (2002).
[Crossref] [PubMed]

Nat. Meth. (1)

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Meth. 10, 335–338 (2013).
[Crossref]

Opt. Commun. (4)

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011).
[Crossref]

D. Weigel, T. Elsmann, H. Babovsky, A. Kiessling, and R. Kowarschik, “Combination of the resolution enhancing image inversion microscopy with digital holography,” Opt. Commun. 291, 110–115 (2013).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Imaging properties of different types of microscopes in combination with an image inversion interferometer,” Opt. Commun. 332, 301–310 (2014).
[Crossref]

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

Opt. Express (3)

Optica (1)

Physical Review A (1)

R. Nair, S. Guha, and S.-H. Tan, “Realizable receivers for discriminating coherent and multicopy quantum states near the quantum limit,” Physical Review A 89, 032318 (2014).
[Crossref]

Proceedings of the National Academy of Sciences of the United States of America (1)

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proceedings of the National Academy of Sciences of the United States of America 103, 4457–4462 (2006).
[Crossref]

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (1)

Lord Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879).
[Crossref]

Ultramicroscopy (1)

E. Bettens, D. Van Dyck, A. DenDekker, J. Sijbers, and A. Van den Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
[Crossref]

Other (11)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

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

A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni Della Normale, 2011).
[Crossref]

H. L. Van Trees, Detection, Estimation, and Modulation Theory : Part I, 1st ed. (Wiley-Interscience1st Ed, 2001).

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” (2015). http://arxiv.org/abs/1511.00552 .

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

U. Kubitscheck, (Ed.), Fluorescence Microscopy: from Principles to Biological Applications (John Wiley & Sons, 2013).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

R. S. Kennedy, “A near-optimum receiver for the binary coherent state channel,” Tech. Rep. 108, MIT RLE Quarterly Progress Report (1973).

S. J. Dolinar, “An optimum receiver for the binary coherent state channel,” Tech. Rep. 111, MIT RLE Quarterly Progress Report (1973).

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (9)

Fig. 1
Fig. 1

A rendering of the focused image of two point sources 1 and 2 located at the tail and tip respectively of the vector d. The blue spots indicate the approximate extent of the point-spread function (PSF) centered at each source. The centroid of the sources is assumed to coincide with the intersection O of the optical axis with the image plane. The focus of this paper is the regime in which d = |d| is smaller than the width of the PSF, in which conventional image-plane photon counting performs poorly in estimating d.

Fig. 2
Fig. 2

The SLIVER method for separation estimation:– The input field E(ρ) is separated into its symmetric (Es(ρ)) and antisymmetric (Ea(ρ)) components with respect to inversion about the centroid of the sources, which is also the optical axis. This separation is accomplished by an image inversion interferometer as shown, where the image inversion box can be implemented by any of a variety of methods (see Sec. 3). The components impinge upon separate bucket detectors that collect spatially-unresolved photocounts Ns and Na. The photocount record over a series of M detection windows of Ns and (or) Na is processed to yield an estimate d ^ of the separation d.

Fig. 3
Fig. 3

Fisher information obtainable using the SLIVER method: - The Fisher information for photon counting J d asym pc at the symmetric output port of the image inversion interferometer (blue curve), photon counting at the antisymmetric output port of the image inversion interferometer ( J d asym pc – dashed green curve) and for on-off detection at the antisymmetric output port ( J d asym oo – dotted red curve) are plotted against the source separation d. The plots are normalized with respect to the maximum value 2εavek)2 of J d asym pc and J d asym oo, attained at d = 0. The superlocalization effect, consistent with the quantum bound of [7], is evident for d ~ 0. The circular Gaussian PSF of Eq. (41) has been assumed and the plots are independent of the value of the half-width σ. The average source strength εave = 0.5 photons.

Fig. 4
Fig. 4

Fisher information obtainable using the SLIVER method: - The Fisher information for photon counting J d sym pc at the symmetric output port of the image inversion interferometer (blue curve), photon counting at the antisymmetric output port of the image inversion interferometer ( J d asym pc – dashed green curve) and for on-off detection at the antisymmetric output port ( J d asym oo – dotted red curve) are plotted against the source separation d. The plots are normalized with respect to the maximum value 2εave (∆k)2 of J d asym pc and J d asym oo, attained at d =0. The superlocalization effect, consistent with the quantum bound of [7], is evident for d ∼ 0. The circular Gaussian PSF of Eq. (41) has been assumed and the plots are independent of the value of the half-width σ. The average source strength εave = 0.5 photons. so that superresolution persists outside the regime εave ≪ 1 and is more marked for number-resolved measurements.

Fig. 5
Fig. 5

Simulated mean squared errors for the maximum likelihood estimator d ^ ML of Eq. (45) as a function of the source separation for M = 100,200,400 measurements. Each measurement consists of a binary value indicating whether or not an on-off detector in the antisymmetric output port of the interferometer registered a click. The MSE curves are each scaled by [ 2 M ε ave ( Δ k ) 2 ] 1, the value of the Cramér-Rao bound for that M at d = 0. Each data point was obtained as an average of 105 Monte-Carlo runs. The Cramér-Rao bound for on-off detection in the antisymmetric output port (solid blue curve) normalized to unity at its minimum value of [ 2 M ε ave ( Δ k ) 2 ] 1 is also shown. A circular Gaussian PSF (Eq. (41)) was assumed and the source strength εave = 0.2 photons. The simulated results are independent of the half-width σ.

Fig. 6
Fig. 6

Simulated mean squared errors for the maximum likelihood estimator d ^ ML of Eq. (46) as a function of the source separation for M = 50,100,150 measurements. Each measurement counts the number of photons in the antisymmetric port, i.e., is a measurement of Na of Sec. 2. The MSE curves are each scaled by [ 2 M ε ave ( Δ k ) 2 ] 1, the value of the Cramér-Rao bound for that M at d = 0. Each data point was obtained as an average of 105 Monte-Carlo runs. The Cramér-Rao bound for photon counting in the antisymmetric output port (solid blue curve) normalized to unity at its minimum value of [ 2 M ε ave ( Δ k ) 2 ] 1 is also shown. A circular Gaussian PSF (Eq. (41)) was assumed, the source strength εave = 5 photons, and the simulated results are independent of the half-width σ.

Fig. 7
Fig. 7

The case of centroid mismatch considered in Section 5.3: The vector OM = c from the optical axis to the centroid is nonzero but parallel to the separation vector 12 = d. The ordering of the points (1−O−M−2) shown is illustrative – the simulation results are valid even if O does not lie between 1 and 2.

Fig. 8
Fig. 8

Simulated mean squared errors for the estimator Eq. (45) for a misaligned SLIVER system using on-off detection in the antisymmetric port shown as a function of the source separation for M = 100,200,400 measurements. We have assumed that the centroid is a distance |c| ≡ξσ away from the inversion axis, that c and d are parallel, and that ξ = 0.1. The MSE curves are each scaled by [ 2 M ε ave ( Δ k ) 2 ] 1, the value of the Cramér-Rao bound for the ideal aligned measurement (ξ = 0) and that M at d = 0. Each data point was obtained as an average of 105 Monte-Carlo runs. The Cramér-Rao bound for on-off detection in the antisymmetric output port of an aligned SLIVER system (solid blue curve) normalized to unity at its minimum value of [ 2 M ε ave ( Δ k ) 2 ] 1 is also shown. A circular Gaussian PSF (Eq. (41)) was assumed and the source strengths ε1 = ε2 = εave = 0.2 photons. The simulated results are independent of the half-width σ.

Fig. 9
Fig. 9

Simulated mean squared errors for the estimator Eq. (45) for misaligned SLIVER as a function of the source separation for M = 100 measurements and for misalignment factors ξ = 0.1,0.2,0.3 and 0.4. The MSE curves are each scaled by [ 2 M ε ave ( Δ k ) 2 ] 1, the value of the Cramér-Rao bound for the ideal aligned measurement (ξ = 0) at d = 0. Each data point was obtained as an average of 105 Monte-Carlo runs. The Cramér-Rao bound for on-off detection in the antisymmetric output port of an aligned SLIVER system normalized to unity at its minimum value is also shown. A circular Gaussian PSF (Eq. (41)) was assumed and ε1 = ε2 = εave = 0.2 photons. The simulated results are independent of the half-width σ.

Equations (48)

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d ρ | ψ ( ρ ) | 2 = 1
E ( ρ ) = A 1 ψ ( ρ + d 2 ) + A 2 ψ ( ρ d 2 ) .
E [ A μ ] = 0
E [ A μ A ν ] = 0
E [ A μ * A μ ] = ε μ
E [ A 1 * A 2 ] = 0
E s ( ρ ) : = E ( ρ ) + E ( ρ ) 2 = S 2 [ ψ ( ρ + d 2 ) + ψ ( ρ d 2 ) ] E a ( ρ ) : = E ( ρ ) E ( ρ ) 2 = D 2 [ ψ ( ρ + d 2 ) ψ ( ρ d 2 ) ] .
S = A 1 + A 2
D = A 1 A 2
E [ S ] = E [ D ] = 0 E [ S 2 ] = E [ D 2 ] = E [ S D ] = 0 E [ S * S ] = E [ D * D ] = ε 1 + ε 2 E [ S * D ] = ε 1 ε 2
I s ( ρ s ) = | S | 2 4 [ | ψ ( ρ s + d 2 ) | 2 + | ψ ( ρ s d 2 ) | 2 + 2 I int ( ρ s , d ) ] ,
I a ( ρ a ) = | D | 2 4 [ | ψ ( ρ a + d 2 ) | 2 + | ψ ( ρ a d 2 ) | 2 2 I int ( ρ a , d ) ] ,
I int ( ρ , d ) = Re ψ * ( ρ + d 2 ) ψ ( ρ d 2 )
E [ N s | S ] = | S | 2 2 d ρ s I s ( ρ s ) = | S | 2 2 [ 1 + δ ( d ) ] ,
E [ N a | D ] = | D | 2 2 d ρ a I a ( ρ a ) = | D | 2 2 [ 1 δ ( d ) ] ,
d ρ I int ( ρ , d ) = Re d ρ ψ * ( ρ ) ψ ( ρ d )
= Re d x d y ψ * ( x , y ) ψ ( x d , y )
δ ( d ) .
N ¯ s = ε 1 + ε 2 2 [ 1 + δ ( d ) ] = ε ave [ 1 + δ ( d ) ] ,
N ¯ a = ε 1 + ε 2 2 [ 1 δ ( d ) ] = ε ave [ 1 δ ( d ) ] ,
P s ( n ) : = Pr [ N s = n ] = 1 N ¯ s + 1 ( N ¯ s N ¯ s + 1 ) n ,
J d ( sym pc ) = E N s [ log P s ( n ) d ] 2 .
J d ( sym pc ) = n = 0 P s ( n ) [ log P s ( n ) d ] 2
= [ N ¯ s d ] 2 n = 0 P s ( n ) [ log P s ( n ) N ¯ s ] 2
= [ N ¯ s d ] 2 1 N ¯ s ( N ¯ s + 1 )
= ε ave γ 2 ( d ) 1 + δ ( d ) { 1 + ε ave [ 1 + δ ( d ) ] } 1 .
γ ( d ) = δ ( d ) = Re d x d y ψ * ( x , y ) ψ ( x d , y ) x .
J d ( asym pc ) = ε ave γ 2 ( d ) 1 δ ( d ) { 1 + ε ave [ 1 δ ( d ) ] } 1 .
E [ d ^ ( N s ) d ] 2 1 J d ( sym pc ) = 1 + δ ( d ) ε ave γ 2 ( d ) + [ 1 + δ ( d ) γ ( d ) ] 2 ,
E [ d ^ ( N a ) d ] 2 1 J d ( asym pc ) = 1 δ ( d ) ε ave γ 2 ( d ) + [ 1 δ ( d ) γ ( d ) ] 2 .
lim d 0 1 + δ ( d ) γ 2 ( d ) = ,
lim d 0 1 δ ( d ) γ 2 ( d ) = δ ( 0 ) 2 γ ( 0 ) γ ( 0 ) = 1 2 γ ( 0 ) 1 2 ( Δ k ) 2 .
γ ( d ) = Re d x d y ψ * ( x + d , y ) x ψ ( x , y ) x ,
γ ( 0 ) = d x d y | ψ ( x , y ) x | 2
( Δ k ) 2 ,
1 M J 0 ( asym pc ) = 1 2 M ε ave ( Δ k ) 2 1 N ( Δ k ) 2
N = 2 M ε ave
P a ( 0 ) = 1 1 + ε ave [ 1 δ ( d ) ]
P a ( > 0 ) = ε ave [ 1 δ ( d ) ] 1 + ε ave [ 1 δ ( d ) ] .
J d ( asym oo ) = ε ave γ 2 ( d ) 1 δ ( d ) { 1 + ε ave [ 1 δ ( d ) ] } 2 < J d ( asym pc ) ,
ψ G ( ρ ) = 1 ( 2 π σ 2 ) 1 / 2 exp ( | ρ | 2 4 σ 2 ) .
δ G ( d ) = exp ( d 2 8 σ 2 ) ,
γ G ( d ) = d 4 σ 2 exp ( d 2 8 σ 2 ) ,
( Δ k ) G 2 = 1 4 σ 2 .
d ^ ML = { 2 σ 2 ln ( 1 K ( M K ) ε ave ) if K M K < ε ave . 2 σ otherwise .
d ^ ML = { 2 σ 2 ln ( 1 P M ε ave ) if P M < ε ave . 2 σ otherwise ,
P a ( 0 ) = [ 1 + α ε 1 + β ε 2 + ( α β γ 2 ) ε 1 ε 2 ] 1 , P a ( > 0 ) = 1 P a ( 0 ) ,
α = 1 δ ( | 2 c d | ) 2 , β = 1 δ ( | 2 c + d | ) 2 , γ = δ ( | d | ) δ ( | 2 c | ) 2 .

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