Abstract

The scalar field in the focal volume of nontelecentric apodized focusing systems cannot be accurately described by the Debye integral representation. By use of the Fresnel–Kirchhoff diffraction formula it is found that, if the aperture stop is axially displaced, the focal-volume structure is tuned. We analyze the influence of the apodizing function and find that, whereas axially superresolving pupil filters are highly sensitive to the focal-volume reshaping effect, axially apodizing filters are more inclined to the focal-shift effect.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 8.
  2. J. J. Stammes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  3. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, London, 1996), Chap. 2.
  4. P. Jacquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3.
    [CrossRef]
  5. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  6. S. De Nicola, “On-axis focal shift effects in focused truncated Jo Bessel beams,” Pure Appl. Opt. 5, 827–831 (1996).
    [CrossRef]
  7. R. Borghi, M. Santarsiero, F. Gori, “Axial intensity of apertured Bessel beams,” J. Opt. Soc. Am. A 14, 23–26 (1997).
    [CrossRef]
  8. M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
    [CrossRef]
  9. M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
    [CrossRef]
  10. E. H. K. Stelzer, S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Opt. Commun. 173, 51–56 (2000).
    [CrossRef]
  11. U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
    [CrossRef]
  12. I. G. Palchikova, S. G. Rautian, “The diffractive optical power of a diaphragm,” Opt. Commun. 174, 1–5 (2000).
    [CrossRef]
  13. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  14. R. G. Wenzel, “Effect of the aperture-lens separation on the focal shift in large-F-number systems,” J. Opt. Soc. Am. A 4, 340–345 (1987).
    [CrossRef]
  15. C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  17. C. J. Zapata-Rodríguez, P. Andrés, M. Martínez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
    [CrossRef]
  18. M. Martínez-Corral, V. Climent, “Focal switch: a new effect in low Fresnel-number systems,” Appl. Opt. 35, 24–26 (1996).
    [CrossRef] [PubMed]
  19. Y. Li, “Focal shift and focal switch in dual-focus systems,” J. Opt. Soc. Am. A 14, 1297–1304 (1997).
    [CrossRef]
  20. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749–753 (1960).
    [CrossRef]

2000 (3)

E. H. K. Stelzer, S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Opt. Commun. 173, 51–56 (2000).
[CrossRef]

I. G. Palchikova, S. G. Rautian, “The diffractive optical power of a diaphragm,” Opt. Commun. 174, 1–5 (2000).
[CrossRef]

C. J. Zapata-Rodríguez, P. Andrés, M. Martínez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
[CrossRef]

1999 (1)

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

1998 (3)

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

1997 (2)

1996 (2)

M. Martínez-Corral, V. Climent, “Focal switch: a new effect in low Fresnel-number systems,” Appl. Opt. 35, 24–26 (1996).
[CrossRef] [PubMed]

S. De Nicola, “On-axis focal shift effects in focused truncated Jo Bessel beams,” Pure Appl. Opt. 5, 827–831 (1996).
[CrossRef]

1987 (1)

1984 (1)

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1960 (1)

Andrés, P.

C. J. Zapata-Rodríguez, P. Andrés, M. Martínez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
[CrossRef]

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Blattner, P.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 8.

Climent, V.

Dändliker, R.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

De Nicola, S.

S. De Nicola, “On-axis focal shift effects in focused truncated Jo Bessel beams,” Pure Appl. Opt. 5, 827–831 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gori, F.

Grill, S.

E. H. K. Stelzer, S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Opt. Commun. 173, 51–56 (2000).
[CrossRef]

Gu, M.

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, London, 1996), Chap. 2.

Herzig, H. P.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

Kowalczyk, M.

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Li, Y.

Martínez-Corral, M.

C. J. Zapata-Rodríguez, P. Andrés, M. Martínez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
[CrossRef]

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

M. Martínez-Corral, V. Climent, “Focal switch: a new effect in low Fresnel-number systems,” Appl. Opt. 35, 24–26 (1996).
[CrossRef] [PubMed]

Muñoz-Escrivá, L.

Palchikova, I. G.

I. G. Palchikova, S. G. Rautian, “The diffractive optical power of a diaphragm,” Opt. Commun. 174, 1–5 (2000).
[CrossRef]

Rautian, S. G.

I. G. Palchikova, S. G. Rautian, “The diffractive optical power of a diaphragm,” Opt. Commun. 174, 1–5 (2000).
[CrossRef]

Rozien-Dossier, B.

P. Jacquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

Stammes, J. J.

J. J. Stammes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Stelzer, E. H. K.

E. H. K. Stelzer, S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Opt. Commun. 173, 51–56 (2000).
[CrossRef]

Török, P.

Vokinger, U.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Welford, W. T.

Wenzel, R. G.

Wolf, E.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 8.

Zapata-Rodríguez, C. J.

C. J. Zapata-Rodríguez, P. Andrés, M. Martínez-Corral, L. Muñoz-Escrivá, “Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system,” J. Opt. Soc. Am. A 17, 1185–1191 (2000).
[CrossRef]

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (5)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

E. H. K. Stelzer, S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Opt. Commun. 173, 51–56 (2000).
[CrossRef]

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

I. G. Palchikova, S. G. Rautian, “The diffractive optical power of a diaphragm,” Opt. Commun. 174, 1–5 (2000).
[CrossRef]

Pure Appl. Opt. (1)

S. De Nicola, “On-axis focal shift effects in focused truncated Jo Bessel beams,” Pure Appl. Opt. 5, 827–831 (1996).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 8.

J. J. Stammes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, London, 1996), Chap. 2.

P. Jacquinot, B. Rozien-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the focusing setup. A monochromatic converging spherical wave, with focus at F, illuminates a circular aperture of radius r max.

Fig. 2
Fig. 2

Schematic layout of a nontelecentric focusing system. At the image plane, O′, the amplitude distribution is given by the product between the geometrical-optics image of the screen and a quadratic phase factor with focus at F′.

Fig. 3
Fig. 3

Halftoning representation of the numerically evaluated irradiance distribution in the meridian plane corresponding to the axially apodizing filter of Eq. (12): (a) telecentric system and (b) nontelecentric system. The parameters for the calculation were N L = 12.25 and ζ N = 0 (telecentric) or ζ N = -0.853 (nontelecentric).

Fig. 4
Fig. 4

Halftoning representation of the numerically evaluated irradiance distribution in the meridian plane corresponding to the axially superresolving filter of Eq. (13): (a) telecentric system and (b) nontelecentric system. The parameters for the calculation were the same than those of Fig. 3

Fig. 5
Fig. 5

Actual two-dimensional representation of the binary filters: (a) axially apodizing and (b) axially superresolving.

Equations (13)

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IP=I01-u2πNL2×01exp-i 12 uρ2J0(vρ)ρdρ2,
u=2πNLz/f1+z/f,
v=2πNLr/rmax1+z/f.
U0r0=exp-i k2ζ r021|m0| tr0|m0|,
Uu, v=2πNL1+ζN2πNL u01 tρ×exp-i 12 uρ2J0vρρdρ,
u=2πNLzN1-zNζN.
v=2πNLrN1-zNζN,
Iu, v=I01+ζN2πNL u2×01tρexp-i 12 uρ2Jovρρdρ2,
zN=u2πNL+uζN,
rN=v2πNL+uζN.
M=1+ζN2πNL u.
taρ=0if0  ρ  2/21if2/2 < ρ  1.
tsρ=1if0  ρ  1/20if1/2 < ρ  3/21if3/2 < ρ  1.

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