Abstract

Investigations of the three-dimensional intensity distribution near the focus are extended here to systems with Fresnel numbers N < 0.5. Isophotes (contours of equal intensity) are presented that demonstrate the structure of the field deep within the region of Fresnel diffraction.

© 1987 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, 1980), Sec. 8.8, pp. 435–449.
  2. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  3. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,”J. Opt. Soc. Am. 71, 904–905.
  4. J. J. Stamnes, H. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  5. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  6. Y. Li, “Dependence of the focal shift on Fresnel number and f number,”J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  7. 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]
  8. B. N. Gorbachev, “Three-dimensional energy distribution of a converging spherical wave for arbitrary Fresnel numbers,” Opt. Spectrosc. (USSR) 44, 204–206 (1978).
  9. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  10. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,”J. Opt. Am. A 1, 801–808 (1984).
    [CrossRef]
  11. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1962), Sec. 16.5, pp. 537–550. In this book the definition of Lommel’s V functions differs by a factor (−1) from that given in Refs. 1 and 10.
  12. Y. Li, “Focal shift formulae,” Optik 69, 41–42 (1984).
  13. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 66, 207–218 (1983).
  14. A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1954), p. 217. Under the condition of an incident plane wave, Sommerfeld’s Eq. (19) on p. 217 should read as J=J0 sin2(ka2/4ρ),which is in agreement with Eq. (3.6) of this paper.

1986 (1)

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]

1984 (2)

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

Y. Li, “Focal shift formulae,” Optik 69, 41–42 (1984).

1983 (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 66, 207–218 (1983).

1982 (1)

Y. Li, “Dependence of the focal shift on Fresnel number and f number,”J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

1981 (3)

J. J. Stamnes, H. 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]

1978 (1)

B. N. Gorbachev, “Three-dimensional energy distribution of a converging spherical wave for arbitrary Fresnel numbers,” Opt. Spectrosc. (USSR) 44, 204–206 (1978).

1972 (1)

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

Avizonis, P. V.

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8, pp. 435–449.

Erkkila, J. H.

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,”J. Opt. Soc. Am. 71, 904–905.

Gorbachev, B. N.

B. N. Gorbachev, “Three-dimensional energy distribution of a converging spherical wave for arbitrary Fresnel numbers,” Opt. Spectrosc. (USSR) 44, 204–206 (1978).

Holmes, D. A.

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

Korka, J. E.

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

Li, Y.

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

Y. Li, “Focal shift formulae,” Optik 69, 41–42 (1984).

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 66, 207–218 (1983).

Y. Li, “Dependence of the focal shift on Fresnel number and f number,”J. Opt. Soc. Am. 72, 770–774 (1982).
[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]

Mahajan, V. N.

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]

Rogers, M. E.

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,”J. Opt. Soc. Am. 71, 904–905.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1954), p. 217. Under the condition of an incident plane wave, Sommerfeld’s Eq. (19) on p. 217 should read as J=J0 sin2(ka2/4ρ),which is in agreement with Eq. (3.6) of this paper.

Spjelkavik, H.

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

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1962), Sec. 16.5, pp. 537–550. In this book the definition of Lommel’s V functions differs by a factor (−1) from that given in Refs. 1 and 10.

Wolf, E.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,”J. Opt. Am. A 1, 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]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8, pp. 435–449.

Appl. Opt. (1)

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

J. Opt. Am. A (1)

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

J. Opt. Soc. Am. (2)

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,”J. Opt. Soc. Am. 71, 904–905.

Y. Li, “Dependence of the focal shift on Fresnel number and f number,”J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

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

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]

Opt. Commun. (3)

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, H. 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]

Opt. Spectrosc. (USSR) (1)

B. N. Gorbachev, “Three-dimensional energy distribution of a converging spherical wave for arbitrary Fresnel numbers,” Opt. Spectrosc. (USSR) 44, 204–206 (1978).

Optik (2)

Y. Li, “Focal shift formulae,” Optik 69, 41–42 (1984).

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 66, 207–218 (1983).

Other (3)

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1954), p. 217. Under the condition of an incident plane wave, Sommerfeld’s Eq. (19) on p. 217 should read as J=J0 sin2(ka2/4ρ),which is in agreement with Eq. (3.6) of this paper.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1962), Sec. 16.5, pp. 537–550. In this book the definition of Lommel’s V functions differs by a factor (−1) from that given in Refs. 1 and 10.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8, pp. 435–449.

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Figures (3)

Fig. 1
Fig. 1

Isophotes (contours of equal intensity) in a section through the three-dimensional intensity distribution in low-Fresnel-number focusing systems. The intensity is normalized to unity at (a), (b) the geometrical focus or (c) the point of the axial maximum. The dotted lines represent the boundary of the geometrical shadow. The last isophote diagram shows the intensity distribution in system of N = 0, a case that corresponds to the diffraction of a uniform, monochromatic plane wave at a circular aperture in an opaque screen.

Fig. 2
Fig. 2

An enlarged view of Fig. 1(c). Details of the isophote diagram in the region deep within the field of Fresnel diffraction are shown.

Fig. 3
Fig. 3

Intensity distributions in various planes perpendicular to the axis in the system of N = 0.

Equations (16)

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N = a 2 / λ f ,
u N = 2 π N z / f 1 + z / f
v N = 2 π N r / a 1 + z / f .
I N ( P ) = const .
I N ( P ) = I F ( 1 - u N 2 π N ) 2 [ U 1 2 ( u N , v N ) + U 2 2 ( u N , v N ) ] / ( u N / 2 ) 2
I N ( P ) = I F ( 1 - u N 2 π N ) 2 × { [ V 0 ( u N , v N ) - cos 1 2 ( u N + v N 2 u N ) ] 2 + [ V 1 ( u N , v N ) + sin 1 2 ( u N + v N 2 u N ) ] 2 } / ( u N / 2 ) 2 ,
I N ( P ) = I F ( 1 - u N 2 π N ) 2 1 - 2 J 0 ( u N ) cos u N + J 0 2 ( u N ) u N 2 .
ζ = f + z
u N ~ u 0 = - 2 π ( g / ζ )
v N ~ v 0 = 2 π ( g / ζ ) ( r / a ) ,
u 0 = - 2 π N ( ζ ) ,             v 0 = 2 π N ( ζ ) ( r / a ) .
A / f A 0 .
I 0 ( P ) = A 0 2 [ U 1 2 ( u 0 , v 0 ) + U 2 2 ( u 0 , v 0 ) ] ,
I 0 ( P ) = A 0 2 { [ V 0 ( u 0 , v 0 ) - cos 1 2 ( u 0 + v 0 2 u 0 ) ] 2 + [ V 1 ( u 0 , v 0 ) + sin 1 2 ( u 0 + v 0 2 u 0 ) ] 2 } ,
I 0 ( P ) = A 0 2 1 - 2 J 0 ( u 0 ) cos u 0 + J 0 2 ( u 0 ) 4 .
I 0 ( u 0 , 0 ) = I G sin 2 ( u 0 4 ) = I G sin 2 ( π 2 / ζ g ) ,

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