Abstract

We implement a self-interference technique for determining the separation between two incoherent point sources. This method relies on image inversion interferometry and when used with the appropriate data analytics, it yields an estimate of the separation with finite-error, including the case when the sources overlap completely. The experimental results show that the technique has a good tolerance to noise and misalignment, making it an interesting consideration for high resolution instruments in microscopy or astronomy.

© 2016 Optical Society of America

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References

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  1. L. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
    [Crossref]
  2. C. M. Sparrow, “On spectroscopic resolving power,” Astro Phys. 44, 76 (1916).
    [Crossref]
  3. A. Pearlman, A. Ling, E. A. Goldschmidt, C. F. Wildfeuer, J. Fan, and A. Migdall, “Enhancing image contrast using coherent states and photon number resolving detectors,” Opt. Express 18, 6033 (2010).
    [Crossref] [PubMed]
  4. E. Bettens, D. V. Dyck, A. J. D. Dekker, J. Sijbers, and A. V. d. Bos, “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
    [Crossref]
  5. S. W. Hell and J. Wichman, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [Crossref] [PubMed]
  6. T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
    [Crossref] [PubMed]
  7. F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
    [Crossref]
  8. M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
    [Crossref]
  9. J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
    [Crossref]
  10. G. F. Schröder, M. Levitt, and A. T. Brunger, “Super-resolution biomolecular crystallography with low-resolution data,” Nature 464, 1218–1222 (2010).
    [Crossref] [PubMed]
  11. E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
    [Crossref]
  12. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
    [Crossref] [PubMed]
  13. M. Tsang, “Quantum limits to optical point-source localization,” Optica 2, 646–653 (2015).
    [Crossref]
  14. M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” arXiv:1511.00552 (2015).
  15. R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3584 (2016).
    [Crossref]
  16. K. Wicker and R. Heintzmann, “Interferometric resolution improvement for confocal microscopes,” Opt. Express 15, 12206 (2007).
    [Crossref] [PubMed]
  17. K. Wicker, S. Sindbert, and R. Heintzmann, “Characterization of a resolution enhancing image inversion interferometer,” Opt. Express 17, 15491 (2009).
    [Crossref] [PubMed]
  18. H. H. Ku, “Notes on the use of propagation of error formulas,” J. Res. National Bureau of Standards 70, 4 (1966).
  19. P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (John Wiley & Sons, 2011).
  20. D. S. Moore, The Basic Practice of Statistics (W.H. Freeman, 2007, Vol. II).
  21. M. Tsang, “Conservative error measures for classical and quantum metrology,” arXiv:1605.03799 [quant-ph] (2016).
  22. C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367v3 (2016).
  23. 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:1606.02662v2 (2016).
  24. W. Kian Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” arXiv:1606.02666 (2016).
  25. M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

2016 (1)

2015 (1)

2010 (2)

2009 (2)

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

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

2008 (1)

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

2007 (1)

2006 (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

2000 (1)

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

1999 (2)

M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
[Crossref]

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

1994 (1)

1993 (1)

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

1966 (1)

H. H. Ku, “Notes on the use of propagation of error formulas,” J. Res. National Bureau of Standards 70, 4 (1966).

1916 (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astro Phys. 44, 76 (1916).
[Crossref]

1879 (1)

L. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
[Crossref]

Agard, D. A.

M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
[Crossref]

Bailey, B.

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Bettens, E.

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

Biteen, J. S.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Bos, A. V. d.

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

Bowman, G. R.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Brunger, A. T.

G. F. Schröder, M. Levitt, and A. T. Brunger, “Super-resolution biomolecular crystallography with low-resolution data,” Nature 464, 1218–1222 (2010).
[Crossref] [PubMed]

Dekker, A. J. D.

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

Dyba, M.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

Dyck, D. V.

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

Eggeling, C.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

Egner, A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

Fan, J.

Farkas, D. L.

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

Ferretti, H.

W. Kian Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” arXiv:1606.02666 (2016).

Goldschmidt, E. A.

Gustafsson, M .G. L.

M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
[Crossref]

Han, K. Y.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

Heintzmann, R.

Hell, S. W.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

S. W. Hell and J. Wichman, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
[Crossref] [PubMed]

Hradil, Z.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

Huber, P. J.

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (John Wiley & Sons, 2011).

Irvine, S. E.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

Jakobs, S.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

Klar, T. A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

Ku, H. H.

H. H. Ku, “Notes on the use of propagation of error formulas,” J. Res. National Bureau of Standards 70, 4 (1966).

Lanni, F.

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

Levitt, M.

G. F. Schröder, M. Levitt, and A. T. Brunger, “Super-resolution biomolecular crystallography with low-resolution data,” Nature 464, 1218–1222 (2010).
[Crossref] [PubMed]

Ling, A.

Lu, X. M.

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” arXiv:1511.00552 (2015).

Lupo, C.

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367v3 (2016).

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:1606.02662v2 (2016).

Migdall, A.

Moerner, W. E.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[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:1606.02662v2 (2016).

Moore, D. S.

D. S. Moore, The Basic Practice of Statistics (W.H. Freeman, 2007, Vol. II).

Nair, R.

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3584 (2016).
[Crossref]

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” arXiv:1511.00552 (2015).

Paur, M.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

Pearlman, A.

Pirandola, S.

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367v3 (2016).

Rayleigh, L.

L. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
[Crossref]

Rehacek, J.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

Rittweger, E.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

Ronchetti, E. M.

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (John Wiley & Sons, 2011).

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Sanchez-Soto, L. L.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

Schröder, G. F.

G. F. Schröder, M. Levitt, and A. T. Brunger, “Super-resolution biomolecular crystallography with low-resolution data,” Nature 464, 1218–1222 (2010).
[Crossref] [PubMed]

Sedat, J. W.

M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
[Crossref]

Shapiro, L.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Sijbers, J.

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

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:1606.02662v2 (2016).

Sindbert, S.

Sparrow, C. M.

C. M. Sparrow, “On spectroscopic resolving power,” Astro Phys. 44, 76 (1916).
[Crossref]

Steinberg, A. M.

W. Kian Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” arXiv:1606.02666 (2016).

Stoklasa, B.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

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:1606.02662v2 (2016).

Taylor, D. L.

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

Tham, W. Kian

W. Kian Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” arXiv:1606.02666 (2016).

Thompson, M. A.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Tsang, M.

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3584 (2016).
[Crossref]

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

M. Tsang, “Conservative error measures for classical and quantum metrology,” arXiv:1605.03799 [quant-ph] (2016).

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” arXiv:1511.00552 (2015).

Tselentis, N. K.

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Wichman, J.

Wicker, K.

Wildfeuer, C. F.

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:1606.02662v2 (2016).

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Astro Phys. (1)

C. M. Sparrow, “On spectroscopic resolving power,” Astro Phys. 44, 76 (1916).
[Crossref]

Bioimaging (1)

F. Lanni, B. Bailey, D. L. Farkas, and D. L. Taylor, “Excitation field synthesis as a means for obtaining enhanced axial resolution in fluorescence microscope,” Bioimaging 1, 187–196 (1993).
[Crossref]

J. Microscopy (1)

M .G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5m: 3D widefield light microscopy with better than 100nm axial resolution,” J. Microscopy 195, 10–16 (1999).
[Crossref]

J. Res. National Bureau of Standards (1)

H. H. Ku, “Notes on the use of propagation of error formulas,” J. Res. National Bureau of Standards 70, 4 (1966).

Nat Meth (1)

J. S. Biteen, M. A. Thompson, N. K. Tselentis, G. R. Bowman, L. Shapiro, and W. E. Moerner, “Super-resolution imaging in live Caulobacter crescentus cells using photoswitchable EYFP,” Nat Meth 5, 947–949 (2008).
[Crossref]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Nat. Photon. (1)

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photon. 3, 144–147 (2009).
[Crossref]

Nature (1)

G. F. Schröder, M. Levitt, and A. T. Brunger, “Super-resolution biomolecular crystallography with low-resolution data,” Nature 464, 1218–1222 (2010).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Philos. Mag. (1)

L. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
[Crossref]

Proc. Nat. Acad. Sci. (1)

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Nat. Acad. Sci. 97, 8206–8210 (2000).
[Crossref] [PubMed]

Ultramicroscopy (1)

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

Other (8)

P. J. Huber and E. M. Ronchetti, Robust Statistics, 2nd ed. (John Wiley & Sons, 2011).

D. S. Moore, The Basic Practice of Statistics (W.H. Freeman, 2007, Vol. II).

M. Tsang, “Conservative error measures for classical and quantum metrology,” arXiv:1605.03799 [quant-ph] (2016).

C. Lupo and S. Pirandola, “Ultimate precision limits for quantum sub-Rayleigh imaging,” arXiv:1604.07367v3 (2016).

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:1606.02662v2 (2016).

W. Kian Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” arXiv:1606.02666 (2016).

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332v1 (2016).

M. Tsang, R. Nair, and X. M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” arXiv:1511.00552 (2015).

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Figures (6)

Fig. 1:
Fig. 1:

The image inversion interferometer is essentially a balanced Mach-Zehnder interferometer with an appropriate delay, ϕ, in one arm, and an image inverter in the other arm. The output power in the arm associated with destructive interference for on-axis light is monitored. This output arm is sensitive to the separation of the point sources. When the sources completely overlap on the optical axis, the power at the image plane is at its weakest value. This interferometer can be constructed into a microscope or telescope (shown in dashed boxes) such that the light from the two sources enter at the first beamsplitter (BS).

Fig. 2:
Fig. 2:

Trends in error of estimated distance using conventional error propagation and the root mean square error (RMSE) technique. The vertical line represents the Sparrow limit.

Fig. 3:
Fig. 3:

Schematic diagram of the experimental setup demonstrating the localization of the light sources for a range of separation values. The two test sources, with orthogonal polarizations, are obtained from a laser-illuminated single-mode optical fibre whose output is separated by a polarizing beam splitter (PBS). The separation of the sources is controlled by translating mirrors mounted on linear-motion stages, and can achieve complete overlap of the two light “sources”. The image inversion interferometer then detects the interference value at detector D2. In an actual instrument, only the image inverter would be present, with the input beams coming from the microscopic sample, or astronomical objects.

Fig. 4:
Fig. 4:

(a) Experimental observation of β values when d=0 μm. Clearly, there is a floor to the value of β with interferometer instability causing outliers in the observational data. To filter out the outliers, a quartile sorting technique was adopted with accepted data points presented in blue. (b) The sorted data.

Fig. 5:
Fig. 5:

(a) Measured values of residual power β against set separation for two different degrees of interference. The blue (red) data points have a minimal value of 0.44 (0.22) when the two point sources overlap. For comparison, the theoretical expectation of β for a perfect instrument is provided (solid line). The black vertical line indicates Sparrow’s limit in our experiment. (b) The same data points after subtracting for background. These plots serve as a calibration curve for the instrument. (c) The error in estimated separation, when using conventional error propagation, diverges for small separation. (d) The error in estimated separation, when using the RMSE technique, remains finite even for very small separation.

Fig. 6:
Fig. 6:

Estimated separation between two sources is measured as a function of actual separation adjusted in experimental setup with two different degrees of alignment accuracy. Dashed line indicates the Sparrow’s limit and the solid line is the ideal case where the estimated separation is equal to the actual separation.

Equations (12)

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E i ( r , d ) = E i , A ( r + d ) E i , B ( r d ) , i { red , blue } ,
P = d A | E blue ( r ) | 2 + d A | E red ( r ) | 2
E ( r , r 0 , w ) = E 0 exp ( | r r 0 | 2 w ) 2 ,
P total = P A + P B 2 P A P B exp ( d 2 2 w 2 ) .
β = P total P A + P B = 1 2 P A P B P A + P B exp ( d 2 2 w 2 ) 1 exp ( d 2 2 w 2 ) | P A ~ P B .
d est = w 2 log ( 1 1 β ) if 0 β 1 .
( σ f ( x i x n ) ) 2 = i n ( f x i ) 2 ( σ x i ) 2 .
σ d est 2 = | σ β 2 w 2 2 ( 1 β 2 ) log ( 1 1 β ) | + | 2 σ w 2 log ( 1 1 β ) | .
σ d est = d β f ( β ) ( d est ( β ) d ) 2 .
P i = < P D 2 , i > < P D 1 > P D 1 , i { A , B } .
| β i Q 2 | < 1 2 ( Q 3 Q 1 ) .
β = P D 2 F ( P A + P B ) F ,

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