Abstract

The generalized Wigner function is able to represent light distributions that contain spatial and temporal information. The use of such a generalized Wigner distribution function for analysis and understanding of temporally restricted superresolving systems is demonstrated. These systems gain spatial resolution by conversion of the temporal degrees of freedom to spatial degrees of freedom.

© 1998 Optical Society of America

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References

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  1. D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36, 2353–2359 (1997).
    [CrossRef] [PubMed]
  2. K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
    [CrossRef]
  3. D. Mendlovic, Z. Zalevsky, “Definition and properties of the generalized temporal-spatial Wigner distribution function,” Optik (Stuttgart) 107, 49–61 (1997).
  4. E. P. Wigner, “On quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  5. H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  6. M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
    [CrossRef]
  7. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  8. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186, (1993).
    [CrossRef]
  9. O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” Vol. 250 of Springer-Verlag Lecture notes in Physics, (Springer Veilag, Heidellerg, 1986), pp. 159–182.
  10. D. Mendlovic, A. W. Lohmann, “Space–bandwidth product adaptation and its application to superresolution: fundamentals” J. Opt. Soc. Am. A 14, 558–562 (1997).
    [CrossRef]
  11. D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]

1997

1996

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

1995

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

1993

1979

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

1966

1932

E. P. Wigner, “On quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

Castanos, O.

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” Vol. 250 of Springer-Verlag Lecture notes in Physics, (Springer Veilag, Heidellerg, 1986), pp. 159–182.

Kiryuschev, I.

Konforti, N.

Lee, H. W.

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Lohmann, A. W.

Lopez-Moreno, E.

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” Vol. 250 of Springer-Verlag Lecture notes in Physics, (Springer Veilag, Heidellerg, 1986), pp. 159–182.

Lukosz, W.

Mendlovic, D.

Wigner, E. P.

E. P. Wigner, “On quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, K. B.

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” Vol. 250 of Springer-Verlag Lecture notes in Physics, (Springer Veilag, Heidellerg, 1986), pp. 159–182.

Zalevsky, Z.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

Opt. Commun.

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

Optik (Stuttgart)

D. Mendlovic, Z. Zalevsky, “Definition and properties of the generalized temporal-spatial Wigner distribution function,” Optik (Stuttgart) 107, 49–61 (1997).

Phys. Rep.

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Phys. Rev.

E. P. Wigner, “On quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” Vol. 250 of Springer-Verlag Lecture notes in Physics, (Springer Veilag, Heidellerg, 1986), pp. 159–182.

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Figures (6)

Fig. 1
Fig. 1

Lukosz multiplexer.

Fig. 2
Fig. 2

Transformation of the Wigner function under relative motion between the signal and the grating by γ = v/ c (the value of γ ≪ 1 is grossly exaggerated). It constitutes a shift in p by γ and a projection in λ by Δλ = λ. A single-level curve is plotted.

Fig. 3
Fig. 3

Five replicas of a Wigner function produced by a cosinusoidal grating Γ c (q) = 1/2[1 -cos(2πq/ L)] of period L in q. The replicas stand apart in the angle p ≈ θ. Above are the maxima of the coefficients W n Γ(q) of Eq. (4) (they sum to unity). Below are their rms values 〈W n Γ(q, t)〉 of Eq. (9).

Fig. 4
Fig. 4

Multiple copies of an originally monochromatic Wigner function separated by wavelength.

Fig. 5
Fig. 5

Slitting of a Gaussian: (a) the Gaussian, (b) the slit, (c) the slitted Gaussian, (d)–(f) their corresponding Wigner functions.

Fig. 6
Fig. 6

Multiple copies of the original Wigner function after the plane of the slit. In this way each copy bears a different portion of the signal and is capable of passing the information through the spatial neck of the finite-width aperture.

Equations (12)

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W f | q ,   p ,   λ = 1 λ - d xf q - x 2 ,   λ * × exp - 2 π ixp / λ f q + x 2 ,   λ .
p p + γ ,     λ λ 1 + γ p , W f | q ,   p ,   λ W f | q ,   p + γ ,   λ 1 + p γ ,
Γ q = 1 L m   Γ m exp 2 π imq / L , Γ m = 1 L - L / 2 L / 2 d q Γ q exp - 2 π imq / L .
W Γ f | q ,   p ,   λ = 1 λ R d x f q - x 2 ,   λ 1 L m   Γ m × exp 2 π im q - x / 2 / L * × exp - 2 π ixp / λ f q + x 2 ,   λ × 1 L m   Γ m exp 2 π im q + x / 2 / L = 1 λ L m , m   Γ m * Γ m R d xf q - x 2 ,   λ * × exp 2 π i m - m L   q - p λ - m + m 2 L x f q + x 2 ,   λ = n 1 L m   Γ m * Γ n - m × exp 2 π i n - 2 m q / L × W f | q ,   p - n λ 2 L ,   λ , = n   W n Γ q W f | q ,   p - n λ 2 L ,   λ ,
Γ c q = 1 2 1 + cos 2 π q L
Γ - 1 c = 1 4 L ,     Γ 0 c = 1 2 L ,     Γ 1 c = 1 4 L ,
W Γ c f | q ,   p ,   λ = 1 16   W f | q ,   p - λ L ,   λ + 1 4 cos 2 π q L   W f | q ,   p - λ 2 L ,   λ + 1 4 + 1 8 cos 4 π q L W f | q ,   p ,   λ + 1 4 cos 2 π q L   W f | q ,   p + λ 2 L ,   λ + 1 16   W f | q ,   p + λ L ,   λ .
W n Γ q ,   t = 1 L m   Γ m * Γ n - m × exp 2 π i n - 2 m q - vt / L .
W n Γ q ,   t = 1 L m   Γ m * Γ n - m δ D n - 2 m = 1 L   | Γ n / 2 | 2
W Γ t f | q ,   p ,   λ = 1 L k   | Γ k | 2 W f | q ,   p + γ - k λ L , λ 1 + p - k λ L γ ,
Δ λ / λ = - k   λ L v c ,     k = 0 ,   ± 1 ,   ± 2 , ,
R w q = 1 - w / 2 < q < w / 2 0 otherwise ,

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