Abstract

The limitations of superresolving filters in imaging systems are investigated. The constraints on such filters in the nonscanning imaging mode are discussed. The possible advantages of such filters in confocal scanning imaging are highlighted. It is shown theoretically and verified experimentally that simply designed complex-amplitude filters can be used effectively to double the exit pupil of a confocal imaging system and thus improve resolution. Superresolution can be achieved with acceptable energy losses and manufacturing tolerances.

© 1986 Optical Society of America

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References

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  1. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo cimento Suppl. 9, 426–435 (1952).
    [CrossRef]
  2. A. Lohmann, “Optische Einseitenbandubertragung angewandt auf das Gabor-Mikroskop,” Opt. Acta 3, 97–99 (1956).
    [CrossRef]
  3. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. (USSR) 9, 204–206 (1960).
  4. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  5. I. J. Cox, C. J. R. Sheppard, T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
    [CrossRef]
  6. D. Yu. Gal’pern, “Apodization,” Opt. Spectrosc. (USSR) 9, 291 (1960).
  7. J. E. Wilkins, “Comment on a paper on apodization by D. Yu. Gal’pern,” J. Opt. Soc. Am. 51, 911–912 (1961).
    [CrossRef]
  8. G. R. Boyer, “Pupil filters for moderate superresolution,” Appl. Opt. 15, 3089–3093 (1976).
    [CrossRef] [PubMed]
  9. G. Toraldo di Francia, “Nuovo pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).
  10. B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
    [CrossRef]
  11. R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
    [CrossRef]
  12. D. E. Yansen, A. E. Smith, “Experimental study of superresolving pupil functions,” J. Opt. Soc. Am. 61, 688 (A) (1971).
  13. G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
    [CrossRef]
  14. J. E. Wilkins, “The resolving power of a coated objective II.,” J. Opt. Soc. Am. 40, 222–224 (1950).
    [CrossRef]
  15. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  16. C. J. R. Sheppard, T. Wilson, “Imaging properties of annular lenses,” Appl. Opt. 18, 3764–3769 (1979).
    [PubMed]
  17. G. J. Brakenhoff, “Imaging modes in confocal scanning light microscopy (CLM),” in Proceedings of the Eleventh Congress of the International Commission for Optics (Instituto de Optica “Daza de Valdes,” C.S.I.C. Sociedad Espanolade Optica, 1978), pp. 219–222.
  18. Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
    [CrossRef]

1985 (1)

Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
[CrossRef]

1983 (1)

G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
[CrossRef]

1982 (1)

1980 (1)

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

1979 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1976 (1)

1971 (1)

D. E. Yansen, A. E. Smith, “Experimental study of superresolving pupil functions,” J. Opt. Soc. Am. 61, 688 (A) (1971).

1969 (1)

B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

1966 (1)

1961 (1)

1960 (2)

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

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

1956 (1)

A. Lohmann, “Optische Einseitenbandubertragung angewandt auf das Gabor-Mikroskop,” Opt. Acta 3, 97–99 (1956).
[CrossRef]

1952 (2)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo cimento Suppl. 9, 426–435 (1952).
[CrossRef]

G. Toraldo di Francia, “Nuovo pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

1950 (1)

Boivin, A.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boivin, R.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boyer, G. R.

G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
[CrossRef]

G. R. Boyer, “Pupil filters for moderate superresolution,” Appl. Opt. 15, 3089–3093 (1976).
[CrossRef] [PubMed]

Brakenhoff, G. J.

G. J. Brakenhoff, “Imaging modes in confocal scanning light microscopy (CLM),” in Proceedings of the Eleventh Congress of the International Commission for Optics (Instituto de Optica “Daza de Valdes,” C.S.I.C. Sociedad Espanolade Optica, 1978), pp. 219–222.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Cox, I. J.

Frieden, B. R.

B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

Gal’pern, D. Yu.

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

Hegedus, Z. S.

Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
[CrossRef]

Kartashev, A. I.

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

Lohmann, A.

A. Lohmann, “Optische Einseitenbandubertragung angewandt auf das Gabor-Mikroskop,” Opt. Acta 3, 97–99 (1956).
[CrossRef]

Lukosz, W.

Sheppard, C. J. R.

Smith, A. E.

D. E. Yansen, A. E. Smith, “Experimental study of superresolving pupil functions,” J. Opt. Soc. Am. 61, 688 (A) (1971).

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo cimento Suppl. 9, 426–435 (1952).
[CrossRef]

G. Toraldo di Francia, “Nuovo pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

Wilkins, J. E.

Wilson, T.

Yansen, D. E.

D. E. Yansen, A. E. Smith, “Experimental study of superresolving pupil functions,” J. Opt. Soc. Am. 61, 688 (A) (1971).

Appl. Opt. (2)

Atti Fond. Giorgio Ronchi (1)

G. Toraldo di Francia, “Nuovo pupille superrisolventi,” Atti Fond. Giorgio Ronchi 7, 366–372 (1952).

J. Opt. Soc. Am. (5)

Nuovo cimento Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo cimento Suppl. 9, 426–435 (1952).
[CrossRef]

Opt. Acta (6)

A. Lohmann, “Optische Einseitenbandubertragung angewandt auf das Gabor-Mikroskop,” Opt. Acta 3, 97–99 (1956).
[CrossRef]

B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
[CrossRef]

Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
[CrossRef]

Opt. Spectrosc. (USSR) (2)

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

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

Other (1)

G. J. Brakenhoff, “Imaging modes in confocal scanning light microscopy (CLM),” in Proceedings of the Eleventh Congress of the International Commission for Optics (Instituto de Optica “Daza de Valdes,” C.S.I.C. Sociedad Espanolade Optica, 1978), pp. 219–222.

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Figures (6)

Fig. 1
Fig. 1

Determination of the pupil’s dividing radius R that maximizes the central intensity of the diffraction pattern. The radius of the central disk is varied between u = 0 and u = 2.4. The dotted curve corresponds to the case when the size of the central spot is half that of the Airy disk.

Fig. 2
Fig. 2

Principle of confocally scanned imaging: S, point source; PA, PB, pupils of systems A and B; M, pinhole mask; D, detector.

Fig. 3
Fig. 3

Calculated point-spread functions associated with a maximum-efficiency two-zone superresolving pupil: (a) conventional imaging and (b) confocally scanned system with a clear second pupil. The dotted curves represent the Airy pattern.

Fig. 4
Fig. 4

Calculated confocal point-spread functions for a superresolving pupil with absorption: (a) first zero at u = 1.9 and (b) first zero at u = 2.0. Dotted curves are the Airy patterns corresponding to conventional imaging with pupils having twice the radius of the pupils used in the confocal system.

Fig. 5
Fig. 5

Comparison between the power spectra of the superresolving pupil (top) and a clear pupil of the same size (bottom).

Fig. 6
Fig. 6

Measured point-spread function of (a) full, unobstructed pupil in conventional imaging (dots) and confocal imaging with identical clear pupils in both systems (crosses) and (b) two-zone superresolving pupil in conventional imaging (dots) and imaging when this pupil is used together with a clear pupil (crosses).

Equations (7)

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a i ( u ) = R i 2 α ( R i u ) R i 1 2 α ( R i 1 u ) ,
A j = i = 1 n a i j k i ,
k 1 = R 2 α ( R u ) α ( u ) R 2 [ α ( R u ) α ( u ) ] ,
k 2 = α ( R u ) α ( R u ) α ( u ) ,
R = J 1 ( u ) 2 J 1 ( R u ) ,
P = P A * P B .
i = | t * h A h B | 2 ,

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