Abstract

We demonstrate the feasibility of achieving superresolved images by using Fourier-plane phase masks and image multiplication, which together create effective point-spread functions that are not positive definite and therefore cannot be created by any single Fourier plane mask in a linear system. Three different configurations were investigated, all of which gave a spatial resolution exceeding that corresponding to the full open aperture of the optical system. One price that must be paid for the superresolution is inefficient use of the light source.

© 2002 Optical Society of America

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References

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  1. I. Leiserson, S. G. Lipson, V. Sarafis, “Superresolution in far-field imaging,” Opt. Lett. 25, 209–211 (2000).
    [CrossRef]
  2. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  3. P. Jacquinot, “Apodization,” Prog. Opt. 3, 29–186 (1964).
    [CrossRef]
  4. D. Yu. Gal’pern, “Apodization,” Opt. Spectrosc. 9, 291 (1960).
  5. T. Wilson, C. J. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  6. S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics3rd ed. (Cambridge U. Press, Cambridge, UK, 1995).
  7. D. Gorlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
    [CrossRef]
  8. E. Marom, N. Konforti, “Low frequency de-emphasis of the modulation transfer function. One dimensional case,” Opt. Commun. 41, 388–392 (1982).
    [CrossRef]
  9. H. Bartelt, A. W. Lohmann, “Optical processing of one-dimensional signals,” Opt. Commun. 42, 87–91 (1982).
    [CrossRef]
  10. N. Konforti, E. Marom, “Two-dimensional optical low frequency deemphasis of the modulation transfer function,” Opt. Commun. 54, 212–216 (1985).
    [CrossRef]
  11. W. T. Rhodes, “Bipolar point spread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
    [CrossRef] [PubMed]
  13. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
    [CrossRef]
  14. Z. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  15. R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.
  16. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980) p. 440.
  17. M. Paesler, P. Moyer, Near-Field Optics (Wiley Interscience, New York, 1996).

2000 (1)

1986 (1)

1985 (1)

N. Konforti, E. Marom, “Two-dimensional optical low frequency deemphasis of the modulation transfer function,” Opt. Commun. 54, 212–216 (1985).
[CrossRef]

1982 (2)

E. Marom, N. Konforti, “Low frequency de-emphasis of the modulation transfer function. One dimensional case,” Opt. Commun. 41, 388–392 (1982).
[CrossRef]

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

1978 (1)

1977 (2)

D. Gorlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

W. T. Rhodes, “Bipolar point spread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
[CrossRef] [PubMed]

1966 (1)

1964 (1)

P. Jacquinot, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

1960 (1)

D. Yu. Gal’pern, “Apodization,” Opt. Spectrosc. 9, 291 (1960).

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[CrossRef]

Bartelt, H.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980) p. 440.

Gal’pern, D. Yu.

D. Yu. Gal’pern, “Apodization,” Opt. Spectrosc. 9, 291 (1960).

Gorlitz, D.

D. Gorlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

Hegedus, Z.

Jacquinot, P.

P. Jacquinot, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

Konforti, N.

N. Konforti, E. Marom, “Two-dimensional optical low frequency deemphasis of the modulation transfer function,” Opt. Commun. 54, 212–216 (1985).
[CrossRef]

E. Marom, N. Konforti, “Low frequency de-emphasis of the modulation transfer function. One dimensional case,” Opt. Commun. 41, 388–392 (1982).
[CrossRef]

Lanzl, F.

D. Gorlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

Leiserson, I.

Lipson, H.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics3rd ed. (Cambridge U. Press, Cambridge, UK, 1995).

Lipson, S. G.

I. Leiserson, S. G. Lipson, V. Sarafis, “Superresolution in far-field imaging,” Opt. Lett. 25, 209–211 (2000).
[CrossRef]

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics3rd ed. (Cambridge U. Press, Cambridge, UK, 1995).

Lohmann, A. W.

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

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
[CrossRef] [PubMed]

Lukosz, W.

Luneberg, R. K.

R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.

Marom, E.

N. Konforti, E. Marom, “Two-dimensional optical low frequency deemphasis of the modulation transfer function,” Opt. Commun. 54, 212–216 (1985).
[CrossRef]

E. Marom, N. Konforti, “Low frequency de-emphasis of the modulation transfer function. One dimensional case,” Opt. Commun. 41, 388–392 (1982).
[CrossRef]

Moyer, P.

M. Paesler, P. Moyer, Near-Field Optics (Wiley Interscience, New York, 1996).

Paesler, M.

M. Paesler, P. Moyer, Near-Field Optics (Wiley Interscience, New York, 1996).

Rhodes, W. T.

Sarafis, V.

Sheppard, C. J.

T. Wilson, C. J. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Tannhauser, D. S.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics3rd ed. (Cambridge U. Press, Cambridge, UK, 1995).

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[CrossRef]

Wilson, T.

T. Wilson, C. J. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980) p. 440.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[CrossRef]

Opt. Commun. (4)

D. Gorlitz, F. Lanzl, “Methods of zero-order non-coherent filtering,” Opt. Commun. 20, 68–72 (1977).
[CrossRef]

E. Marom, N. Konforti, “Low frequency de-emphasis of the modulation transfer function. One dimensional case,” Opt. Commun. 41, 388–392 (1982).
[CrossRef]

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

N. Konforti, E. Marom, “Two-dimensional optical low frequency deemphasis of the modulation transfer function,” Opt. Commun. 54, 212–216 (1985).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

D. Yu. Gal’pern, “Apodization,” Opt. Spectrosc. 9, 291 (1960).

Prog. Opt. (1)

P. Jacquinot, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

Other (5)

T. Wilson, C. J. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics3rd ed. (Cambridge U. Press, Cambridge, UK, 1995).

R. K. Luneberg, The Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1966), pp. 344–359.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980) p. 440.

M. Paesler, P. Moyer, Near-Field Optics (Wiley Interscience, New York, 1996).

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Figures (9)

Fig. 1
Fig. 1

Experimental system used to implement the phase mask synthesis and interferometric multiplication. The explanation of the labels is given in the text.

Fig. 2
Fig. 2

a, Optimized mask with circular symmetry: White indicates transmission amplitude 1, gray is 0, and black is -1. b, Diametric profile of the mask’s diffraction pattern. c, The product of the diffraction pattern and the Airy-disk function. The Fourier-plane units are arbitrary.

Fig. 3
Fig. 3

Contour maps showing sections in the xz plane of PSFs: a, for the superresolving mask after multiplication; b, for the full circular aperture. The latter result reproduces the classic result (Ref. 10 p. 440); the contour spacing in the central region is 10 times greater than in the wings.

Fig. 4
Fig. 4

Simulated images of double point sources in the focal plane and at a defocus of 0.3λ comparing (a–d) the superresolving mask and multiplication and (e–h) the full circular aperture. Both gray-scale images and profiles through the centers of the two images are shown.

Fig. 5
Fig. 5

Simulated image of 20 point sources with variable distances between them, in the region of the Sparrow resolution distance. This image shows that there is no significant degradation of the superresolution when the technique is applied to a wide incoherently illuminated field: a, with the superresolution mask and image multiplication; b, with a full circular aperture of the same dimensions.

Fig. 6
Fig. 6

Synthetic phase mask with square symmetry used in the experiments: a, positive part M2a; b, negative part M2b; c, d, observed diffraction pattern of the antiphase superposition of a and b; e, f, the interferometric product of c and the Airy-disk function.

Fig. 7
Fig. 7

Experimental profiles of images of two incoherent point sources imaged through the full aperture and through the phase mask: a, b, the separation is 1.13°, c, d, it is 0.75 in the units of 0.95λf#, which has unit value when the separation has the Sparrow resolution limit of a circular aperture.

Fig. 8
Fig. 8

PSF for the annular aperture (thin curve) and its product with that of the full aperture (thick curve): a, experiment; b, theory.

Fig. 9
Fig. 9

Measured profiles of image of two point sources: a, unresolved, with the full aperture, b, superresolved in one dimension, with the overlapping-aperture configuration. The abscissa units are as in Fig. 6.

Equations (1)

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|p1+p2|2-|p1-p2|2=4p1p2.

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