Abstract

The Debye formulation of focused fields has been systematically used to evaluate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmetries about the geometrical focus. However, certain discrepancies arise when the Fresnel number, as viewed from focus, is close to unity. In that case, we should use the Kirchhoff formulation to evaluate accurately the three-dimensional amplitude distribution of the field in the focal region. We make some important remarks regarding both diffraction theories. In the end we demonstrate that, in the paraxial regime, given a defocused transverse pattern in the Debye approximation, it is possible to find a similar pattern but magnified and situated at another plane within the Kirchhoff theory. Moreover, we may evaluate this correspondence as the action of a virtual thin lens located at the focal plane and whose focus is situated at the axial point of the aperture plane. As a result, we give a geometrical interpretation of the focal-shift effect and present a brief comment on the problem of the best-focus location.

© 2000 Optical Society of America

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References

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  1. P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.
  2. M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef] [PubMed]
  3. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1979), Sec. 8.8.
  4. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  5. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, Bristol, England, 1986), Part IV.
  6. E. Collet, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
    [CrossRef]
  7. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [CrossRef]
  8. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  9. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  10. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  11. G. P. Karman, A. van Kuijl, M. W. Beijersbergen, J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. 36, 8091–8095 (1997).
    [CrossRef]
  12. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  13. M. Martı́nez-Corral, C. J. Zapata-Rodrı́guez, P. Andrés, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
    [CrossRef]
  14. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 8, 801–808 (1984).
    [CrossRef]
  15. W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
    [CrossRef]
  16. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.
  18. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980).
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    [CrossRef]
  20. T. Wilson, ed., Confocal Microscopy (Academic, London, 1990).
  21. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996), Chap. 2.
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    [CrossRef] [PubMed]
  23. For an extensive discussion on spatial invariance and thecorrect use of the point-spread function in optical systems of finite Fresnel number, see C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  24. Y. Li, “A high-accuracy formula for fast evaluation of the effect of focal shift,” J. Mod. Opt. 38, 1815–1819 (1991).
    [CrossRef]
  25. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [CrossRef]
  26. M. Parker Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [CrossRef]
  27. C. J. R. Sheppard, Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]

1998 (2)

W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
[CrossRef]

M. Martı́nez-Corral, C. J. Zapata-Rodrı́guez, P. Andrés, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
[CrossRef]

1997 (1)

1994 (1)

1991 (1)

Y. Li, “A high-accuracy formula for fast evaluation of the effect of focal shift,” J. Mod. Opt. 38, 1815–1819 (1991).
[CrossRef]

1990 (1)

1988 (1)

1986 (2)

1985 (1)

1984 (1)

1983 (2)

V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
[CrossRef] [PubMed]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1982 (1)

M. Parker Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

1981 (4)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

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

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

1980 (1)

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Andrés, P.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Beijersbergen, M. W.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1979), Sec. 8.8.

Collet, E.

Furlan, W. D.

W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980).

Gu, M.

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

Hegedus, Z. S.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.

Karman, G. P.

Li, Y.

Y. Li, “A high-accuracy formula for fast evaluation of the effect of focal shift,” J. Mod. Opt. 38, 1815–1819 (1991).
[CrossRef]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 8, 801–808 (1984).
[CrossRef]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

Mahajan, V. N.

Marti´nez-Corral, M.

Ojeda-Castañeda, J.

Parker Givens, M.

M. Parker Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Rhodes, W. T.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980).

Saavedra, G.

W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
[CrossRef]

Sheppard, C. J. R.

Silvestre, E.

W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
[CrossRef]

M. Martı́nez-Corral, C. J. Zapata-Rodrı́guez, P. Andrés, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
[CrossRef]

Sitter, D. N.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, Bristol, England, 1986), Part IV.

Streibl, N.

van Kuijl, A.

Woerdman, J. P.

Wolf, E.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 8, 801–808 (1984).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

E. Collet, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1979), Sec. 8.8.

Zapata-Rodri´guez, C. J.

Appl. Opt. (4)

J. Mod. Opt. (2)

Y. Li, “A high-accuracy formula for fast evaluation of the effect of focal shift,” J. Mod. Opt. 38, 1815–1819 (1991).
[CrossRef]

W. D. Furlan, G. Saavedra, E. Silvestre, M. Martı́nez-Corral, “On-axis irradiance for spherical aberrated optical systems with obscured rectangular apertures: a study using the Wigner distribution function,” J. Mod. Opt. 45, 69–77 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (2)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Opt. Commun. (4)

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

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

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

M. Parker Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Opt. Lett. (1)

Other (7)

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 29–186.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, Bristol, England, 1986), Part IV.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1979), Sec. 8.8.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980).

T. Wilson, ed., Confocal Microscopy (Academic, London, 1990).

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

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the focusing setup.

Fig. 2
Fig. 2

Three-dimensional mapping of the focal volume provided by (a) the paraxial Debye formulation, thus giving (b) the Fresnel–Kirchhoff representation of the focal wave field.

Fig. 3
Fig. 3

Diagram of isophotes corresponding to the impulse response of an optical imaging system with a circular clear pupil of radius a=1 mm when the wavelength is given by λ=500 nm and the Fresnel number is (a) high (N=500), (b) moderate (N=10), and (c) low (N=3). The continuous white line passes through the axial point of the pupil plane, which gives a rough idea of the relative distance between the focal plane and the aperture plane.

Equations (17)

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U(P)=exp(-ikf )iλf WA(S) exp(iks)s dS,
aλ,(a/f )21.
sf-ηx0+ξy0f+x02+y022 f.
U0(x0, y0)=1iλf2 expi k2 f (x02+y02)×-A(η, ξ)exp-i kf (ηx0+ξy0)dηdξ.
U(P)=-U0(x0, y0) 1iλz×expi k2z [(x-x0)2+(y-y0)2]dx0dy0,
12π - exp(iax2+iζx)dx=12a expi ζ24a.
U(P)=1iλf(f+z) expi k2(f+z) (x2+y2)×-A(η, ξ)exp-i k2 f zf+z (η2+ξ2)×exp-i kf+z (xη+yζ)dηdξ.
U0D(x0, y0)=1iλf2 -A(η, ξ)×exp-i kf (ηx0+ξy0)dηdξ;
UD(P)=1iλf2 -A(η, ξ)exp-i k2 f zf (η2+ξ2)×exp-i kf (xη+yξ)dηdξ.
N=a2/(λf )
Δz=λ4 sin2(α/2)λfa2,
N=f/(Δz),
zD=ff+z z
M(z)=f+zf.
U(x, y, z)=expi k2(f+z) (x2+y2)×1M UDxM, yM, zM.
Δff=-11+(π2/12)N2.
zD=-12π2 ΔzN,

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