Abstract

A novel procedure for shaping the axial component of the point spread function of nonparaxial focusing systems by use of phase-only pupil filters is presented. The procedure is based on the Toraldo technique for tailoring focused fields. The resulting pupil filters consist of a number of concentric annular zones with constant real transmittance. The number of zones and their widths can be adapted according to the shape requirements. Our method is applied to design filters that produce axial superresolution in confocal scanning systems.

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References

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  1. P. Jacquinot and B. Rozien-Dossier, "Apodisation" in Progress in Optics, E. Wolf, ed., Vol III (North-Holland, Amsterdam, 1964).
  2. A. Boivin, Th?orie el Calcul des Figures de Diffraction de R?volution (Universit? de Laval, Quebec, 1964).
  3. C. J. R. Sheppard and Z. S. Hegedus, "Axial behavior of pupil-plane filters," J. Opt. Soc. Am. A 5, 643-647 (1988).
    [CrossRef]
  4. M. Mart?nez-Corral, P. Andr?s, J. Ojeda-Casta?eda and G. Saavedra, "Tunable axial superresolution by annular binary filters. Application to confocal microscopy," Opt. Commun. 119, 491-498 (1995).
    [CrossRef]
  5. J. Campos, J. C. Escalera, C. J. R. Sheppard and M. J. Yzuel, "Axially invariant pupil filters," J. Mod. Opt. 47, 57-68 (2000).
  6. C. J. R. Sheppard, M. D. Sharma and A. Arbouet, "Axial apodizing filters for confocal imaging," Optik 111, 347-354 (2000).
  7. C. J. R. Sheppard, "Leaky annular pupils for improved axial imaging," Optik 99, 32-34 (1995).
  8. M. Mart?nez-Corral, P. Andr?s, C. J. Zapata-Rodr?guez and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
    [CrossRef]
  9. S. Grill and E. H. K. Stelzer, "Method to calculate lateral and axial gain factors of optical setups with a large solid angle," J. Opt. Soc. Am. A 16, 2658-2665 (1999).
    [CrossRef]
  10. M. A. A. Neil, R. Juskaitis, T. Wilson, Z. J. Laczik and V. Sarafis, "Optimized pupil-plane filters for confocal microscope point-spread function engineering," Opt. Lett. 25, 245-247 (2000).
    [CrossRef]
  11. M. Mart?nez-Corral, L. Mu?oz-Escriv?, M. Kowalczyk and T. Cichocki, "One-dimensional iterative algorithm for three-dimensional point-spread function engineering," Opt. Lett. 26, 1861-1863 (2001).
    [CrossRef]
  12. G. Toraldo di Francia, "Nuovo pupille superresolventi," Atti Fond. Giorgio Ronchi 7, 366-372 (1952).
  13. Z. S. Hegedus and V. Sarafis, "Superresolving filters in confocally scanned imaging systems," J. Opt. Soc. Am. A 3, 1892-1896 (1986).
    [CrossRef]
  14. T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997).
    [CrossRef]
  15. T. R. M. Sales and G. M. Morris, "Axial superresolution with phase-only pupil filters," Opt. Commun. 156, 227-230 (1998).
    [CrossRef]
  16. C. J. R. Sheppard, G. Calvert and M. Wheatland, "Focal distribution for superresolving toraldo filters," J. Opt. Soc. Am. A 15, 849-856 (1998).
    [CrossRef]
  17. H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-562 (2001).
    [CrossRef]
  18. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, Berlin, 2000).

Other (18)

P. Jacquinot and B. Rozien-Dossier, "Apodisation" in Progress in Optics, E. Wolf, ed., Vol III (North-Holland, Amsterdam, 1964).

A. Boivin, Th?orie el Calcul des Figures de Diffraction de R?volution (Universit? de Laval, Quebec, 1964).

C. J. R. Sheppard and Z. S. Hegedus, "Axial behavior of pupil-plane filters," J. Opt. Soc. Am. A 5, 643-647 (1988).
[CrossRef]

M. Mart?nez-Corral, P. Andr?s, J. Ojeda-Casta?eda and G. Saavedra, "Tunable axial superresolution by annular binary filters. Application to confocal microscopy," Opt. Commun. 119, 491-498 (1995).
[CrossRef]

J. Campos, J. C. Escalera, C. J. R. Sheppard and M. J. Yzuel, "Axially invariant pupil filters," J. Mod. Opt. 47, 57-68 (2000).

C. J. R. Sheppard, M. D. Sharma and A. Arbouet, "Axial apodizing filters for confocal imaging," Optik 111, 347-354 (2000).

C. J. R. Sheppard, "Leaky annular pupils for improved axial imaging," Optik 99, 32-34 (1995).

M. Mart?nez-Corral, P. Andr?s, C. J. Zapata-Rodr?guez and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999).
[CrossRef]

S. Grill and E. H. K. Stelzer, "Method to calculate lateral and axial gain factors of optical setups with a large solid angle," J. Opt. Soc. Am. A 16, 2658-2665 (1999).
[CrossRef]

M. A. A. Neil, R. Juskaitis, T. Wilson, Z. J. Laczik and V. Sarafis, "Optimized pupil-plane filters for confocal microscope point-spread function engineering," Opt. Lett. 25, 245-247 (2000).
[CrossRef]

M. Mart?nez-Corral, L. Mu?oz-Escriv?, M. Kowalczyk and T. Cichocki, "One-dimensional iterative algorithm for three-dimensional point-spread function engineering," Opt. Lett. 26, 1861-1863 (2001).
[CrossRef]

G. Toraldo di Francia, "Nuovo pupille superresolventi," Atti Fond. Giorgio Ronchi 7, 366-372 (1952).

Z. S. Hegedus and V. Sarafis, "Superresolving filters in confocally scanned imaging systems," J. Opt. Soc. Am. A 3, 1892-1896 (1986).
[CrossRef]

T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997).
[CrossRef]

T. R. M. Sales and G. M. Morris, "Axial superresolution with phase-only pupil filters," Opt. Commun. 156, 227-230 (1998).
[CrossRef]

C. J. R. Sheppard, G. Calvert and M. Wheatland, "Focal distribution for superresolving toraldo filters," J. Opt. Soc. Am. A 15, 849-856 (1998).
[CrossRef]

H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-562 (2001).
[CrossRef]

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, Berlin, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Normalized axial intensity PSF corresponding to the circular pupil and to the three-zone Toraldo filter; (b) contour plot of the 3D intensity PSF in the meridian plane corresponding to the circular pupil; (c) as b) but for the three-zone Toraldo filter. The normalized radial coordinate is rN =(n/λ)r sin2 α.

Fig. 2.
Fig. 2.

Three-zones Toraldo filter: (a) mapped transmittance; (b) actual 2D representation for the cases of α = 10° (1) and α = 67.5° (2). Note that the form of the pupil filter strongly depends on the value of α.

Fig. 3.
Fig. 3.

(a) Seven-zone Toraldo filter designed for obtaining axial superresolution in confocal fluorescence microscopy. The filter was calculated for the case of α = 67.5° ; (b) axial intensity PSF.

Fig. 4.
Fig. 4.

(a) Seven-zone Toraldo filter designed to obtain axial superresolution in reflection confo-cal microscopy. The filter was calculated for the case of α = 67.5° ; (b) 3D intensity PSF of a reflection confocal microscope with two circular pupils; (c) as b) but with the seven-zone Toraldo filter in the illumination system.

Equations (11)

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h ( z ) = 0 α A ( θ ) exp ( i 2 π n cos θ λ z ) sin θ ,
ζ = cos θ cos α 1 cos α 0.5 ; q ( ζ ) = A ( θ )
h ( z N ) = ( 1 cos α ) exp ( i π 1 + cos α 1 cos α z N ) 0.5 0.5 q ( ζ ) exp ( i 2 πζ z N ) ,
z N = n λ ( 1 cos α ) z .
q ( ζ ) = i = 1 m [ k i a i ( ζ ) k i 1 a i 1 ( ζ ) ] , where a i ( ζ ) = { rect ( ζ / Δ i ) if i = 1 , , m 0 if i = 0 .
h ( z N ) = i = 1 m k i a ˜ i ( z N ) , where a ˜ i ( z N ) = Δ i sin c ( Δ i z N ) Δ i 1 sin c ( Δ i 1 z N ) .
h ( z N ) = ( k 1 k 2 ) Δ sin c ( Δ z N ) + k 2 sin c ( z N ) , with Δ = Δ 1 .
( k 1 k 2 ) Δ + k 2 = 1 .
( k 1 k 2 ) Δ sin c ( Δ z 1 ) + k 2 sin c ( z 1 ) = 0 .
k 1 = sin ( πΔ z 1 ) sin ( π z 1 ) sin ( πΔ z 1 ) Δ sin ( π z 1 ) and k 2 = sin ( πΔ z 1 ) sin ( πΔ z 1 ) Δ sin ( π z 1 ) .
Δ = sin c ( z 1 ) 2 sin c ( Δ z 1 ) .

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