Abstract

Superresolution permits fine details to be observed that cannot be resolved by standard optical instruments. We demonstrate that superresolution is nothing more than an adaptation of degrees of freedom converted from the spatial domain to some other domains and vice versa. The tool used for the required conversion and adaptation is the Wigner chart. In addition, we show that the Wigner chart may be naturally applied for analyzing and understanding complex optical setups used for obtaining superresolution.

© 2000 Optical Society of America

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References

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  1. O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, Germany, 1910).
  2. G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  3. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef]
  4. M. Françon, “Amelioration de resolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).
  5. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  6. 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); D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36, 6687–6691 (1997).
    [CrossRef] [PubMed]
  7. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).
  8. H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
    [CrossRef]
  9. W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–21 (1963).
  10. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  11. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  12. 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]
  13. D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]
  14. J. G. Proakis, Digital Communications, 2nd ed. (McGraw-Hill, New York, 1989).
  15. D. Mendlovic, Z. Zalevsky, “Definition and properties of the generalized temporal–spatial Wigner distribution function,” Optik (Stuttgart) 107, 49–56 (1997).
  16. H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971); H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  17. D. Mendlovic, J. Garcia, Z. Zalevsky, E. Marom, D. Mas, C. Ferreira, A. W. Lohmann, “Wavelength multiplexing system for a single-mode image transmission,” Appl. Opt. 36, 8474–8480 (1997).
    [CrossRef]
  18. Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Super resolution optical systems using fixed gratings,” Opt. Commun. 163, 79–85 (1999).
    [CrossRef]
  19. K. B. Wolf, D. Mendlovic, Z. Zalevsky, “Generalized Wigner function for the analysis of superresolution systems,” Appl. Opt. 37, 4374–4379 (1998).
    [CrossRef]

1999 (1)

Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Super resolution optical systems using fixed gratings,” Opt. Commun. 163, 79–85 (1999).
[CrossRef]

1998 (1)

1997 (5)

1993 (1)

1982 (1)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

1971 (1)

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971); H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1969 (1)

1966 (1)

1963 (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–21 (1963).

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

1955 (1)

1952 (1)

M. Françon, “Amelioration de resolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Bartelt, H.

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Dammann, H.

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971); H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Ferreira, C.

Françon, M.

M. Françon, “Amelioration de resolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Garcia, J.

Gartner, W.

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–21 (1963).

Gortler, K.

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971); H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Kiryuschev, I.

Konforti, N.

Lohmann, A. W.

Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Super resolution optical systems using fixed gratings,” Opt. Commun. 163, 79–85 (1999).
[CrossRef]

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); D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, D. Farkas, “Two-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36, 6687–6691 (1997).
[CrossRef] [PubMed]

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]

D. Mendlovic, J. Garcia, Z. Zalevsky, E. Marom, D. Mas, C. Ferreira, A. W. Lohmann, “Wavelength multiplexing system for a single-mode image transmission,” Appl. Opt. 36, 8474–8480 (1997).
[CrossRef]

D. Mendlovic, A. W. Lohmann, Z. Zalevsky, “Space–bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–21 (1963).

Lukosz, W.

Lummer, O.

O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, Germany, 1910).

Marom, E.

Mas, D.

Mendlovic, D.

Proakis, J. G.

J. G. Proakis, Digital Communications, 2nd ed. (McGraw-Hill, New York, 1989).

Reiche, F.

O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, Germany, 1910).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Toraldo di Francia, G.

Wolf, K. B.

Zalevsky, Z.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Nuovo Cimento Suppl. (1)

M. Françon, “Amelioration de resolution d’optique,” Nuovo Cimento Suppl. 9, 283–290 (1952).

Opt. Commun. (3)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971); H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Super resolution optical systems using fixed gratings,” Opt. Commun. 163, 79–85 (1999).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Optik (Stuttgart) (1)

D. Mendlovic, Z. Zalevsky, “Definition and properties of the generalized temporal–spatial Wigner distribution function,” Optik (Stuttgart) 107, 49–56 (1997).

Z. Phys. (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Phys. 174, 18–21 (1963).

Other (3)

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

J. G. Proakis, Digital Communications, 2nd ed. (McGraw-Hill, New York, 1989).

O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, Germany, 1910).

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Figures (15)

Fig. 1
Fig. 1

Effects of elementary optical modules on the Wigner chart of a signal.

Fig. 2
Fig. 2

Block diagram of the SW adaptation process with an example for demonstration.

Fig. 3
Fig. 3

The bulky SW adaptation process, illustrating the dynamic range trade-off operation: (a) SWI, (b) SWY, (c) SWI after grating, (d) SWI after time multiplexing.

Fig. 4
Fig. 4

The bulky SW adaptation process, illustrating the subpixeling operation: (a) SWI, (b) SWY, (c) SWY after each pixel has been divided into three regions.

Fig. 5
Fig. 5

Suggested setup for obtaining time-multiplexing superresolution.

Fig. 6
Fig. 6

1-D experimental results for time-multiplexing superresolution. Output captured by the CCD camera (a) with a clear aperture, (b) with a closed aperture without the rotation rosette, (c) after addition of the rosette.

Fig. 7
Fig. 7

Adequate gratings for 2-D superresolution.

Fig. 8
Fig. 8

2-D experimental results for time-multiplixing superresolution. Output plane with clear aperture: (a) 20° tilted Ronchi grating. (b) 2-D board of squares. Output plane with aperture and without time-multiplexing superresolution. (c) 20° tilted Ronchi grating. (d) 2-D board of squares. Output results obtained with an aperture and with time-multiplexing superresolution. (e) 20° tilted Ronchi grating. (f) 2-D board of squares.

Fig. 9
Fig. 9

Suggested setup for image transmission by use of wavelength multiplexing: (a) encoding, (b) decoding.

Fig. 10
Fig. 10

Wavelength-multiplexing experimental results for 1-D objects: (a) Reconstruction obtained in the output plane, (b) information carried in the green spectral range, (c) information carried in the yellow spectral range, (d) information carried in the orange spectral range, (e) information carried in the red spectral range.

Fig. 11
Fig. 11

Multiplexing by diffraction on static gratings: Obj, object; Ap., aperture; G’s, gratings; Img., image.

Fig. 12
Fig. 12

Principle of operation for the setup of Fig. 11.

Fig. 13
Fig. 13

Setup used to obtain polarization-multiplexed superresolution.

Fig. 14
Fig. 14

Essence of the polarization-multiplexing superresolution.

Fig. 15
Fig. 15

Overall polarization-multiplexing superresolution analysis in Wigner space.

Equations (27)

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δxDIF1.22λF#,
SW=ΔxΔν.
W(x, ν)=-ux+x2u*x-x2exp(-2πiνx)dx,
W(x, ν)dν=|u(x)|2,
W(x, ν)dx=|u˜(ν)|2,
u˜(ν)=u(x)exp(-2πiνx)dx.
SWB(x, ν)=1W(x, ν)>Wthresh0otherwise.
SW(x, ν)=STSWB(x, ν),
Totalenergy=SWB(x, ν)SW(x, ν)dxdν=SW(x, ν)dxdν.
ST=SWB(x, ν)W(x, ν)dxdνSWB(x, ν)dxdν.
N=Δxδx=Δνδν=ΔxΔν.
SWV(x, ν)=W(x, ν)W(x, ν)>Wthresh0otherwise,
C=N log(M+1).
NOUTNIN.
NOUT=NIN.
SWIV(x, ν)SWYV(x, ν),
(Volume{SWIV}=)NSignalNSystem(=Volume{SWYV}).
SWIV(x, ν)SWYV(x, ν).
NSignalNSystem,SWIV(x, ν)SWYV(x, ν).
ν0=D/λF,
τ=1/ν0V,
λm=λ0+mδ λ,m=1, 2,M,Mδ λ=Δλ.
WG(x, ν; λ)=nWnWx, ν-nν02; λ,
W(x, ν; λ)=ux+x2, λu*x-x2, λ×exp(-2πixν)dx.
u¯(x0)=xˆu(x0)+yˆu(x0).
u+(x)+u-(x)=u0(x).
W(x, ν)W(x, ν+Δνw) & W(x, ν-Δνw).

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