Abstract

We present the quantum theory of superresolution for discrete subwavelength structures. It allows to formulate, in particular, the standard quantum limit of superresolution achieved for illumination of the structure by light in coherent state. Our theory is based on discrete prolate spheroidal sequences and functions which are the proper basis set of the problem. We demonstrate that the superresolution factor is much higher for discrete structures than for continuous objects for the same signal-to-noise ratio. This result is a clear illustration of the crucial role of a priori information in superresolution problems.

© 2008 Optical Society of America

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References

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  1. F. Thomas, “Multilevel subwavelength optical data storage using reflective nanostructures,” presented at the 2005 INSIC Symposium “Alternative Data Storage,” Monterey, USA, 2005.
  2. H. Kostal, J. J. Wang, and F. Thomas, “Manufacture of multi-level encoded subwavelength optical data storage media,” invited paper at the Topical Meeting “Advanced Imaging Techniques” of the European Optical Society, London, UK, 2005.
  3. A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
    [Crossref]
  4. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
    [Crossref] [PubMed]
  5. V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802(1–10) (2005).
    [Crossref]
  6. M. I. Kolobov and V. N. Beskrovnyy, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun. 264, 9–12 (2006).
    [Crossref]
  7. I. V. Sokolov and M. I. Kolobov, “Squeezed-light source for superresolving microscopy,” Opt. Lett. 29, 703–705 (2004).
    [Crossref] [PubMed]
  8. M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1996), Vol. XXXVI, pp. 129–178.
  9. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [Crossref] [PubMed]
  10. M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
    [Crossref]
  11. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case”, Bell System Tech. J. 57, 1371–1430 (1978).
  12. A. Papoulis and M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory,  CT-19, 674–681 (1972).
    [Crossref]

2006 (2)

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
[Crossref]

M. I. Kolobov and V. N. Beskrovnyy, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun. 264, 9–12 (2006).
[Crossref]

2005 (1)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802(1–10) (2005).
[Crossref]

2004 (1)

2000 (1)

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

1985 (1)

M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
[Crossref]

1978 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case”, Bell System Tech. J. 57, 1371–1430 (1978).

1972 (2)

A. Papoulis and M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory,  CT-19, 674–681 (1972).
[Crossref]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
[Crossref] [PubMed]

Bertero, M.

M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
[Crossref]

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1996), Vol. XXXVI, pp. 129–178.

Bertran, M. S.

A. Papoulis and M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory,  CT-19, 674–681 (1972).
[Crossref]

Beskrovnyy, V. N.

M. I. Kolobov and V. N. Beskrovnyy, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun. 264, 9–12 (2006).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802(1–10) (2005).
[Crossref]

Braat, J. J. M.

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
[Crossref]

De Mol, C.

M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
[Crossref]

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1996), Vol. XXXVI, pp. 129–178.

Fabre, C.

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

Frieden, B. R.

Kolobov, M. I.

M. I. Kolobov and V. N. Beskrovnyy, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun. 264, 9–12 (2006).
[Crossref]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802(1–10) (2005).
[Crossref]

I. V. Sokolov and M. I. Kolobov, “Squeezed-light source for superresolving microscopy,” Opt. Lett. 29, 703–705 (2004).
[Crossref] [PubMed]

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

Kostal, H.

H. Kostal, J. J. Wang, and F. Thomas, “Manufacture of multi-level encoded subwavelength optical data storage media,” invited paper at the Topical Meeting “Advanced Imaging Techniques” of the European Optical Society, London, UK, 2005.

Papoulis, A.

A. Papoulis and M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory,  CT-19, 674–681 (1972).
[Crossref]

Pereira, S. F.

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
[Crossref]

Pike, E. R.

M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
[Crossref]

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case”, Bell System Tech. J. 57, 1371–1430 (1978).

Sokolov, I. V.

Thomas, F.

F. Thomas, “Multilevel subwavelength optical data storage using reflective nanostructures,” presented at the 2005 INSIC Symposium “Alternative Data Storage,” Monterey, USA, 2005.

H. Kostal, J. J. Wang, and F. Thomas, “Manufacture of multi-level encoded subwavelength optical data storage media,” invited paper at the Topical Meeting “Advanced Imaging Techniques” of the European Optical Society, London, UK, 2005.

van de Nes, A. S.

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
[Crossref]

Wang, J. J.

H. Kostal, J. J. Wang, and F. Thomas, “Manufacture of multi-level encoded subwavelength optical data storage media,” invited paper at the Topical Meeting “Advanced Imaging Techniques” of the European Optical Society, London, UK, 2005.

Bell System Tech. J. (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case”, Bell System Tech. J. 57, 1371–1430 (1978).

IEEE Trans. Circuit Theory (1)

A. Papoulis and M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory,  CT-19, 674–681 (1972).
[Crossref]

Inverse Problems (1)

M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems 1, 301–330 (1985).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

M. I. Kolobov and V. N. Beskrovnyy, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun. 264, 9–12 (2006).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802(1–10) (2005).
[Crossref]

Phys. Rev. Lett. (1)

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

Rep. Prog. Phys. (1)

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006).
[Crossref]

Other (3)

F. Thomas, “Multilevel subwavelength optical data storage using reflective nanostructures,” presented at the 2005 INSIC Symposium “Alternative Data Storage,” Monterey, USA, 2005.

H. Kostal, J. J. Wang, and F. Thomas, “Manufacture of multi-level encoded subwavelength optical data storage media,” invited paper at the Topical Meeting “Advanced Imaging Techniques” of the European Optical Society, London, UK, 2005.

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1996), Vol. XXXVI, pp. 129–178.

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Figures (2)

Fig. 1.
Fig. 1.

Optical scheme for the far-field imaging of a discrete subwavelength optical structure.

Fig. 2.
Fig. 2.

Superresolution factor SF as a function of log〈〉, 〈〉 is the mean photon number, for superresolution of discrete structures (bold line) and continuous signals (dotted line).

Equations (28)

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[ a ̂ ( s ) , a ̂ ( s ) ] = δ ( s s ) , [ f ̂ ( ξ ) , f ̂ ( ξ ) ] = δ ( ξ ξ ) .
f ̂ ( ξ ) = ( T a ̂ ) ( ξ ) = c 2 π a ̂ ( s ) e ics ξ ds ,
a ̂ ( m ) = 1 Δ Δ ( m 1 2 ) Δ ( m + 1 2 ) a ̂ ( s ) ds .
[ a ̂ ( m ) , a ̂ ( m ) ] = δ mm ,
A ̂ ( s ) = m = a ̂ ( m ) p ( s Δ m ) ,
F ̂ ( ξ ) = P ( ξ ) c 2 π m = a ̂ ( m ) e ic Δ ξ m ,
f ̂ ( ξ ) = W m = a ̂ ( m ) e 2 π iW ξ m ,
m = M M sin [ 2 π W ( m m ) ] π ( m m ) ψ μ ( m ) = λ μ ψ μ ( m ) ,
m = M M ψ μ ( m ) ψ ν ( m ) = λ μ m = ψ μ ( m ) ψ ν ( m ) = λ μ δ μν .
1 1 sin [ π KW ( ξ ξ ) ] sin [ π W ( ξ ξ ) ] Ψ μ ( ξ ) d ξ = λ μ Ψ μ ( ξ ) ,
1 1 Ψ μ ( ξ ) Ψ ν ( ξ ) d ξ = λ μ 1 2 W 1 2 W Ψ μ ( ξ ) Ψ ν ( ξ ) d ξ = λ μ δ μν .
m = M M ψ μ ( m ) e 2 π iW ξ m = ( i ) μ λ μ W Ψ μ ( ξ ) ,
φ μ ( m ) = { 1 λ μ ψ μ ( m ) m M , 0 m > M , χ μ ( m ) = { 0 m M , 1 1 λ μ ψ μ ( m ) m > M .
Φ μ ( ξ ) = { 1 λ μ Ψ μ ( ξ ) ξ 1 , 0 ξ > 1 , X μ ( ξ ) = { 0 ξ 1 , 1 1 λ μ Ψ μ ( ξ ) ξ > 1 .
a ̂ ( m ) = μ = 0 2 M a ̂ μ φ μ ( m ) + μ = 0 2 M b ̂ μ χ μ ( m ) + a ̂ ( m ) ,
f ̂ ( ξ ) = μ = 0 2 M f ̂ μ Φ μ ( ξ ) + μ = 0 2 M g ̂ μ X μ ( ξ ) + f ̂ ( ξ ) .
f ̂ μ = ( i ) μ ( λ μ a ̂ μ + 1 λ μ b ̂ μ ) .
a ̂ μ ( r ) = f ̂ μ ( i ) μ λ μ = a ̂ μ + 1 λ μ λ μ b ̂ μ ,
e ̂ ( m ) = m = h ( m , m ) a ̂ ( m ) ,
e ̂ ( m ) = m = sin [ 2 π W ( m m ) ] π ( m m ) a ̂ ( m ) + e ̂ ( m ) ,
h ( m , m ) = sin [ 2 π W ( m m ) ] π ( m m ) .
a ̂ ( r ) ( m ) = μ = 0 2 M a ̂ μ ( r ) φ μ ( m ) ,
a ̂ ( r ) ( m ) = m = M M h ( r ) ( m , m ) a ̂ ( m ) + μ = 0 2 M 1 λ μ λ μ b ̂ μ φ μ ( m ) ,
h ( r ) ( m , m ) = μ = 0 2 M φ μ ( m ) φ μ ( m ) ,
μ = 0 2 M φ μ ( m ) φ μ ( m ) = δ m m ,
h ( r ) ( m , m ) = δ m m ,
a ̂ ( r ) ( m ) = a ̂ ( m ) + μ = 0 2 M 1 λ μ λ μ b ̂ μ φ μ ( m ) .
N ̂ = m = M M a ̂ m a ̂ m .

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