Abstract

We show that, for optical systems whose point spread functions exhibit isolated zeros, the information one can gain about the separation between two incoherent point light sources does not scale quadratically with the separation (which is the distinctive dependence causing Rayleigh’s curse) but only linearly. Moreover, the dominant contribution to the separation information comes from regions in the vicinity of these zeros. We experimentally confirm this idea, demonstrating significant superresolution using natural or artificially created spectral doublets.

© 2019 Optical Society of America

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References

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  1. L. Rayleigh, Philos. Mag. 8, 261 (1879).
    [Crossref]
  2. A. J. den Dekker and A. van den Bos, J. Opt. Soc. Am. A 14, 547 (1997).
    [Crossref]
  3. P. R. Hemmer and T. Zapata, J. Opt. 14, 083002 (2012).
    [Crossref]
  4. L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, J. Rehacek, and L. L. Sanchez-Soto, Eur. Phys. J. Plus 131, 130 (2016).
    [Crossref]
  5. M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).
    [Crossref]
  6. S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
    [Crossref]
  7. R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
    [Crossref]
  8. R. Nair and M. Tsang, Opt. Express 24, 3684 (2016).
    [Crossref]
  9. M. Tsang, New J. Phys. 19, 023054 (2017).
    [Crossref]
  10. M. Tsang, J. Mod. Opt. 65, 1385 (2018).
    [Crossref]
  11. D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), Vol. 27, pp. 261–281.
  12. C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
    [Crossref]
  13. R. Kerviche, S. Guha, and A. Ashok, in IEEE International Symposium on Information Theory (ISIT) (2017), pp. 441–445.
  14. J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, Opt. Lett. 42, 231 (2017).
    [Crossref]
  15. Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
    [Crossref]
  16. M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, Optica 3, 1144 (2016).
    [Crossref]
  17. T. Z. Sheng, K. Durak, and A. Ling, Opt. Express 24, 22004 (2016).
    [Crossref]
  18. F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
    [Crossref]
  19. W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
    [Crossref]
  20. F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
    [Crossref]
  21. M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
    [Crossref]
  22. F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).
  23. M. Paúr, B. Stoklasa, J. Grover, A. Krzic, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, Optica 5, 1177 (2018).
    [Crossref]
  24. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  25. D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
    [Crossref]
  26. S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
    [Crossref]
  27. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  28. L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
    [Crossref]
  29. J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000).

2018 (4)

Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
[Crossref]

M. Tsang, J. Mod. Opt. 65, 1385 (2018).
[Crossref]

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

M. Paúr, B. Stoklasa, J. Grover, A. Krzic, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, Optica 5, 1177 (2018).
[Crossref]

2017 (3)

J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, Opt. Lett. 42, 231 (2017).
[Crossref]

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

M. Tsang, New J. Phys. 19, 023054 (2017).
[Crossref]

2016 (10)

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

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

S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
[Crossref]

R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

R. Nair and M. Tsang, Opt. Express 24, 3684 (2016).
[Crossref]

C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
[Crossref]

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, Optica 3, 1144 (2016).
[Crossref]

T. Z. Sheng, K. Durak, and A. Ling, Opt. Express 24, 22004 (2016).
[Crossref]

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
[Crossref]

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
[Crossref]

2012 (1)

P. R. Hemmer and T. Zapata, J. Opt. 14, 083002 (2012).
[Crossref]

2006 (1)

S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

2004 (1)

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

1997 (1)

1996 (1)

L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
[Crossref]

1879 (1)

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

Ang, S. Z.

S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
[Crossref]

Ashok, A.

R. Kerviche, S. Guha, and A. Ashok, in IEEE International Symposium on Information Theory (ISIT) (2017), pp. 441–445.

Banaszek, K.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

Bernardo, J. M.

J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boroszko, K.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

Borówka, S.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

Boyd, S.

L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
[Crossref]

D’Angelo, M.

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

Demkowicz-Dobrzanski, R.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

den Dekker, A. J.

Durak, K.

Facchi, P.

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

Ferretti, H.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
[Crossref]

Fuhrmann, D. R.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

Garuccio, A.

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

Ghinea, C.

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

Grover, J.

Guha, S.

R. Kerviche, S. Guha, and A. Ashok, in IEEE International Symposium on Information Theory (ISIT) (2017), pp. 441–445.

Helstrom, C. W.

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

Hemmer, P. R.

P. R. Hemmer and T. Zapata, J. Opt. 14, 083002 (2012).
[Crossref]

Hradil, Z.

Huang, Y.

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

Kerviche, R.

R. Kerviche, S. Guha, and A. Ashok, in IEEE International Symposium on Information Theory (ISIT) (2017), pp. 441–445.

Krzic, A.

Kuang, C.

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

Ling, A.

Liu, S.

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

Liu, X.

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

Lu, X.-M.

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

Lupo, C.

C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
[Crossref]

Lvovsky, A. I.

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
[Crossref]

Moiseev, E. S.

Motka, L.

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

Nair, R.

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

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

S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
[Crossref]

R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

R. Nair and M. Tsang, Opt. Express 24, 3684 (2016).
[Crossref]

O’Sullivan, J. A.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

Ober, R. J.

S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

Parniak, M.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

Pascazio, S.

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

Paur, M.

Paúr, M.

Pepe, F. V.

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

Petz, D.

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

Pirandola, S.

C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
[Crossref]

Prasad, S.

Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
[Crossref]

Preza, C.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

Ram, S.

S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

Rayleigh, L.

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

Rehacek, J.

J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, Opt. Lett. 42, 231 (2017).
[Crossref]

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, Optica 3, 1144 (2016).
[Crossref]

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

Rehácek, J.

Sanchez-Soto, L. L.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, Optica 3, 1144 (2016).
[Crossref]

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

Sánchez-Soto, L. L.

Sheng, T. Z.

Simon, C.

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
[Crossref]

Smith, A. F. M.

J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000).

Smith, W. H.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

Snyder, D. L.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

Steinberg, A. M.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
[Crossref]

Stoklasa, B.

Taschilina, A.

Teo, Y. S.

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

Tham, W. K.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
[Crossref]

Tsang, M.

M. Tsang, J. Mod. Opt. 65, 1385 (2018).
[Crossref]

M. Tsang, New J. Phys. 19, 023054 (2017).
[Crossref]

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

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

R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
[Crossref]

R. Nair and M. Tsang, Opt. Express 24, 3684 (2016).
[Crossref]

van den Bos, A.

Vandenberghe, L.

L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
[Crossref]

Ward, E. S.

S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

Wasilewski, W.

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yang, F.

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
[Crossref]

Yu, Z.

Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
[Crossref]

Yue, F.

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

Zapata, T.

P. R. Hemmer and T. Zapata, J. Opt. 14, 083002 (2012).
[Crossref]

Eur. Phys. J. Plus (1)

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

IEEE Trans. Signal Process. (1)

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, D. L. Snyder, and W. H. Smith, IEEE Trans. Signal Process. 52, 950 (2004).
[Crossref]

J. Mod. Opt. (1)

M. Tsang, J. Mod. Opt. 65, 1385 (2018).
[Crossref]

J. Opt. (1)

P. R. Hemmer and T. Zapata, J. Opt. 14, 083002 (2012).
[Crossref]

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

New J. Phys. (1)

M. Tsang, New J. Phys. 19, 023054 (2017).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optica (3)

Philos. Mag. (1)

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

Phys. Rev. A (2)

S. Z. Ang, R. Nair, and M. Tsang, Phys. Rev. A 95, 063847 (2016).
[Crossref]

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
[Crossref]

Phys. Rev. Lett. (5)

M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 121, 250503 (2018).
[Crossref]

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2016).
[Crossref]

R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
[Crossref]

Z. Yu and S. Prasad, Phys. Rev. Lett. 121, 180504 (2018).
[Crossref]

C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
[Crossref]

Phys. Rev. X (1)

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

Proc. Natl. Acad. Sci. USA (1)

S. Ram, E. S. Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

SIAM Rev. (1)

L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
[Crossref]

Other (6)

J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

F. Yue, Y. Huang, S. Liu, C. Kuang, and X. Liu, Microscopy Methods in Nanomaterials Characterization (Elsevier, 2017).

R. Kerviche, S. Guha, and A. Ashok, in IEEE International Symposium on Information Theory (ISIT) (2017), pp. 441–445.

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

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Figures (5)

Fig. 1.
Fig. 1. Experimental setup for measuring the spectral separation of lines generated by a swept-wavelength tunable laser.
Fig. 2.
Fig. 2. Sinc PSF, as given in Eq. (6), depicting the sensor areas that define global (light gray) and local (dark gray) measurements of spectral separation. The broken line represents a Gaussian PSF with the same normalization and the same peak value. Horizontal axis is drawn in units of w.
Fig. 3.
Fig. 3. Experimentally determined precisions for the separation of two incoherent objects with a sinc PSF. The separation estimation was done globally (red solid symbols) from 322 pixels (4.7w) centered on the centroid of the intensity pattern encompassing four PSF zeros (k=±1,±2) or locally (blue open symbols) from 22 pixels (0.3w) cropped around a single k=+1 intensity minimum. The symbols are capped by the theoretical upper limits given by the Fisher information. The red solid (blue broken) line shows the corresponding theoretical limit of such global (local) measurement applied to a normalized Gaussian PSF of the same peak and total intensities. Variances are included in the inset for a clearer picture of what happens at very small separations. Precisions, variances, and separations are drawn in units of w2, w2, and w, respectively, and variances and precisions are normalized per detection in the full uncropped 1D image.
Fig. 4.
Fig. 4. Experimental setup for resolving the longitudinal cavity modes of a HeNe laser.
Fig. 5.
Fig. 5. Histograms of experimentally determined frequency differences for single-line (open bars) and double-line (solid bars) HeNe laser operation. The true separation of 1090 MHz of the spectral doublet is marked by the vertical broken line. The corresponding Cramér-Rao lower bound is also shown.

Equations (7)

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p(x|s)=12[I(xs/2)+I(x+s/2)].
F(s)=F(x|s)dx,F(x|s)=[sp(x|s)]2p(x|s),
F(x|s)=[I(x)]2s216I(x)+2I(x)s2,
F(x|s)18I(xk)s2(xxk+is/2)(xxkis/2),s1.
F(s)=[π4kI(xk)]s.
I(x)=1wsin2(πx/w)(πx/w)2,
F(s)=(π2w3k01k2)s=π36w3s,

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