Abstract

It has been argued that, for a spatially invariant imaging system, the information one can gain about the separation of two incoherent point sources decays quadratically to zero with decreasing separation. The effect is termed Rayleigh’s curse. Contrary to this belief, we identify a class of point-spread functions (PSFs) with a linear information decrease. Moreover, we show that any well-behaved symmetric PSF can be converted into such a form with a simple nonabsorbing signum filter. We experimentally demonstrate significant superresolution capabilities based on this idea.

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

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References

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  4. M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).
    [Crossref]
  5. R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).
    [Crossref]
  6. R. Nair and M. Tsang, Opt. Express 24, 3684 (2016).
    [Crossref]
  7. M. Tsang, New J. Phys. 19, 023054 (2017).
    [Crossref]
  8. D. Petz and C. Ghinea, Introduction to Quantum Fisher Information (World Scientific, 2011), pp. 261–281.
  9. C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).
    [Crossref]
  10. J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, Opt. Lett. 42, 231 (2017).
    [Crossref]
  11. M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, Optica 3, 1144 (2016).
    [Crossref]
  12. F. Yang, A. Taschilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Optica 3, 1148 (2016).
    [Crossref]
  13. F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, Phys. Rev. A 96, 063829 (2017).
    [Crossref]
  14. W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
    [Crossref]
  15. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2004).
  16. S. Ram, E. Sally Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
    [Crossref]
  17. H. L. Royden, Real Analysis (Prentice Hall, 1988).
  18. A. Kastler, Rev. Opt. 29, 308 (1950).
  19. H. Wolter, Ann. Phys. 442, 341 (1950).
    [Crossref]
  20. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Opt. Lett. 21, 281 (1996).
    [Crossref]
  21. F. W. King, Hilbert Transforms (Cambridge University, 2009).
  22. N. M. Temme, “Error functions, Dawson’s and Fresnel integrals,” in NIST Handbook of Mathematical Functions (Cambridge University, 2010).
  23. R. J. Wells, J. Quant. Spectrosc. Radiat. Transfer 62, 29 (1999).
    [Crossref]

2017 (4)

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]

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

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

2016 (6)

2012 (1)

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

2006 (1)

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

1999 (1)

R. J. Wells, J. Quant. Spectrosc. Radiat. Transfer 62, 29 (1999).
[Crossref]

1997 (1)

1996 (1)

1950 (2)

A. Kastler, Rev. Opt. 29, 308 (1950).

H. Wolter, Ann. Phys. 442, 341 (1950).
[Crossref]

den Dekker, A. J.

Ferretti, H.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Ghinea, C.

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

Goodman, J. W.

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

Hemmer, P. R.

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

Hradil, Z.

Kastler, A.

A. Kastler, Rev. Opt. 29, 308 (1950).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice Hall, 1993), Vol. 1.

King, F. W.

F. W. King, Hilbert Transforms (Cambridge University, 2009).

Lohmann, A. W.

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]

Mendlovic, D.

Moiseev, E. S.

Nair, R.

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

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

Ober, R. J.

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

Paur, M.

Paúr, M.

Petz, D.

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

Pirandola, S.

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

Ram, S.

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

Rehacek, J.

Royden, H. L.

H. L. Royden, Real Analysis (Prentice Hall, 1988).

Sally Ward, E.

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

Sanchez-Soto, L. L.

Sánchez-Soto, L. L.

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]

Steinberg, A. M.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Stoklasa, B.

Taschilina, A.

Temme, N. M.

N. M. Temme, “Error functions, Dawson’s and Fresnel integrals,” in NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Tham, W. K.

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Tsang, M.

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]

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

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

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

van den Bos, A.

Wells, R. J.

R. J. Wells, J. Quant. Spectrosc. Radiat. Transfer 62, 29 (1999).
[Crossref]

Wolter, H.

H. Wolter, Ann. Phys. 442, 341 (1950).
[Crossref]

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]

Zalevsky, Z.

Zapata, T.

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

Ann. Phys. (1)

H. Wolter, Ann. Phys. 442, 341 (1950).
[Crossref]

J. Opt. (1)

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

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

J. Quant. Spectrosc. Radiat. Transfer (1)

R. J. Wells, J. Quant. Spectrosc. Radiat. Transfer 62, 29 (1999).
[Crossref]

New J. Phys. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Optica (2)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (3)

W. K. Tham, H. Ferretti, and A. M. Steinberg, Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

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

R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (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. Sally Ward, and R. J. Ober, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
[Crossref]

Rev. Opt. (1)

A. Kastler, Rev. Opt. 29, 308 (1950).

Other (6)

H. L. Royden, Real Analysis (Prentice Hall, 1988).

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

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice Hall, 1993), Vol. 1.

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

F. W. King, Hilbert Transforms (Cambridge University, 2009).

N. M. Temme, “Error functions, Dawson’s and Fresnel integrals,” in NIST Handbook of Mathematical Functions (Cambridge University, 2010).

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Figures (6)

Fig. 1.
Fig. 1. Scheme of an optical coherent 4f processor, with a signum mask in the Fourier plane.
Fig. 2.
Fig. 2. Detection probabilities (blue) and Fisher information density [i.e., the integrand in the definition (2)] (red) corresponding to a Gaussian PSF modified by the signum filter for separations 0.2σ (solid lines) and 0.4σ (dashed lines). Here, and in all the figures, length is in units of σ.
Fig. 3.
Fig. 3. Fisher information about separation for imaging with a Gaussian PSF with (red concave line) and without (blue convex line) the signum filter. The asymptotic behavior of the superresolution given by the right-hand side of Eq. (12) is also shown (dashed red line).
Fig. 4.
Fig. 4. Experimental setup. The notations used are as follows: DMD, digital micromirror chip; BS, beam splitter; AS, aperture stop at the Fourier plane of the lens; and SLM, spatial light modulator. In the inset, we show a typical intensity scan recorded with zero separation setting and a total detection count of 434,000. The separation is estimated from the total number of detections registered in the central pixel column.
Fig. 5.
Fig. 5. Experimental variances of the separation estimator (blue dots) compared with the direct detection (blue dashed line) and the signum-enhanced limit (solid blue line). The latter is corrected for the finite pixel size of 7.6 μm. For completeness, the reciprocal of the variances, called the precisions, are shown in red.
Fig. 6.
Fig. 6. Estimation of the separation from signum-enhanced imaging. Estimator means (dots) and standard deviations (error bars) are shown. The same statistics are provided for the best unbiased estimators from direct (blue lines) and signum-enhanced (red lines) imaging as given by the CRLB. The latter takes into account the finite pixel size used in the experiment.

Equations (12)

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p(x|s)=12[I(xs/2)+I(x+s/2)],
F(s)=N[sp(x|s)]2p(x|s)dx,
F(s)=s216{[I(x)]2I(x)+O(s2)}dx.
p(x|s)α(x2+s2/4),x,s1.
F(s)α2s24x2+s2dx.
F(s)λs,
Ψ±sgn(x,s)=iπΨ(x±s/2)xxdx,
Ψ±sgn(x,s)iπ(x±s/2)Ψ(ξ)ξdξ,x,s1.
psgn(x|s)=12[|Ψsgn(x,s)|2+|Ψ+sgn(x,s)|2]α(x2+s2/4),
Fdirect(s)(s/σ)28σ2,
|Ψ±sgn(x,s)|2=22  D(x±s/22σ)2π3/2σ,
Fsgn(s)(s/σ)22πσ2.

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