Abstract

It was recently shown that, when a converging spherical wave is focused in a diffraction-limited system of sufficiently low Fresnel numbers, the point of maximum intensity does not coincide with the geometrical focus but is located closer to the exit pupil. In the present paper both qualitative and quantitative arguments are presented that elucidate the modifications that the whole three-dimensional structure of the diffracted field undergoes as the Fresnel number is gradually decreased. Contours of equal intensity in the focal region are presented for systems of selected Fresnel numbers, which focus uniform waves.

© 1984 Optical Society of America

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References

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  1. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [Crossref]
  2. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [Crossref]
  3. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [Crossref]
  4. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [Crossref]
  5. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [Crossref]
  6. Similar results were also found in connection with focused Gaussian beams. For a discussion of this subject and for pertinent references, see Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [Crossref]
  7. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  8. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [Crossref]
  9. G. C. Sherman, W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (1982).
    [Crossref]
  10. In this connection it seems worthwhile to mention that none of the currently available theories of diffraction at an aperture in an opaque screen appears to be adequate to predict the structure of the diffracted field in the near zone of the aperture; nor are any experimental results available to elucidate this question.
  11. For the same reason as indicated in Ref. 10, this assumption may not be valid when the maximum reaches the immediate vicinity of the aperture.
  12. A somewhat paradoxical aspect of this situation should be noted. As the Fresnel number N becomes smaller, with a and λ being kept fixed, the distance f increases, i.e., the geometrical focus F moves farther away from the aperture; however, the point of maximum intensity moves in the opposite direction, i.e., closer to the aperture.
  13. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [Crossref]
  14. The choice of these variables is suggested by detailed calculations given, for example, in Ref. 15. As will be seen shortly, they reduce to the usual dimensionless variables of the classic theory of Lommel in the limit of large Fresnel numbers.
  15. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).
  16. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmchens,” Abh. Bayer. Akad. Math. Naturwiss. Kl. 15, 233–328, (1885). The main part of Lommel’s analysis is presented (in English) in Sec. 8.8 of Ref. 7.
  17. This limiting procedure is equivalent to keeping the field point P(z, r, ψ) as well as the wavelength λ and the angular semiaperture a/f fixed and letting f→ ∞. That the resulting expression is to be interpreted as an asymptotic rather than an ordinary approximation is strongly suggested by recent discussions of the Debye representation of focused fields,8,9 which is known to be equivalent to Lommel’s representation under the usual circumstances.
  18. When the suffix n is a negative integer, Lommel’s original definition of the function Vn differs from that given by Eq. (3.18) by a factor (−1)n.
  19. E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
    [Crossref]

1983 (2)

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

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

1982 (4)

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

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

Similar results were also found in connection with focused Gaussian beams. For a discussion of this subject and for pertinent references, see Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

G. C. Sherman, W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (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. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (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]

1956 (1)

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[Crossref]

1885 (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmchens,” Abh. Bayer. Akad. Math. Naturwiss. Kl. 15, 233–328, (1885). The main part of Lommel’s analysis is presented (in English) in Sec. 8.8 of Ref. 7.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Chew, W. C.

Erkkila, J. H.

Givens, M. P.

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

Li, Y.

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

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

Similar results were also found in connection with focused Gaussian beams. For a discussion of this subject and for pertinent references, see Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

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

Linfoot, E. H.

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[Crossref]

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmchens,” Abh. Bayer. Akad. Math. Naturwiss. Kl. 15, 233–328, (1885). The main part of Lommel’s analysis is presented (in English) in Sec. 8.8 of Ref. 7.

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]

Rogers, M. E.

Sherman, G. C.

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]

Wolf, E.

Similar results were also found in connection with focused Gaussian beams. For a discussion of this subject and for pertinent references, see Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[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. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[Crossref]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Abh. Bayer. Akad. Math. Naturwiss. Kl. (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmchens,” Abh. Bayer. Akad. Math. Naturwiss. Kl. 15, 233–328, (1885). The main part of Lommel’s analysis is presented (in English) in Sec. 8.8 of Ref. 7.

J. Opt. Soc. Am. (3)

Opt. Acta (1)

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

Opt. Commun. (5)

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

Similar results were also found in connection with focused Gaussian beams. For a discussion of this subject and for pertinent references, see Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[Crossref]

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]

Optik (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

Proc. Phys. Soc. London Sect. B (1)

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London Sect. B 69, 823–832 (1956).
[Crossref]

Other (7)

This limiting procedure is equivalent to keeping the field point P(z, r, ψ) as well as the wavelength λ and the angular semiaperture a/f fixed and letting f→ ∞. That the resulting expression is to be interpreted as an asymptotic rather than an ordinary approximation is strongly suggested by recent discussions of the Debye representation of focused fields,8,9 which is known to be equivalent to Lommel’s representation under the usual circumstances.

When the suffix n is a negative integer, Lommel’s original definition of the function Vn differs from that given by Eq. (3.18) by a factor (−1)n.

The choice of these variables is suggested by detailed calculations given, for example, in Ref. 15. As will be seen shortly, they reduce to the usual dimensionless variables of the classic theory of Lommel in the limit of large Fresnel numbers.

In this connection it seems worthwhile to mention that none of the currently available theories of diffraction at an aperture in an opaque screen appears to be adequate to predict the structure of the diffracted field in the near zone of the aperture; nor are any experimental results available to elucidate this question.

For the same reason as indicated in Ref. 10, this assumption may not be valid when the maximum reaches the immediate vicinity of the aperture.

A somewhat paradoxical aspect of this situation should be noted. As the Fresnel number N becomes smaller, with a and λ being kept fixed, the distance f increases, i.e., the geometrical focus F moves farther away from the aperture; however, the point of maximum intensity moves in the opposite direction, i.e., closer to the aperture.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (4)

Fig. 1
Fig. 1

Illustrating some qualitative differences in the structure of the far field generated by the diffraction of a uniform, converging monochromatic spherical wave at an aperture in an opaque screen in systems of large Fresnel numbers (upper figures) and low Fresnel numbers (lower figures).

Fig. 2
Fig. 2

Illustrating the notation used in determining the structure of the focal region. The point Q is located on a spherical wave front W of radius f, centered on F and passing through the center O of the aperture.

Fig. 3
Fig. 3

Illustrating consequences of the nonlinear transformation [Eqs. (3.3)]. Some of the coordinate lines uN = constant (vertical) and vN = constant (horizontal or inclined to the z axis), plotted in the z, r plane, for systems of different Fresnel numbers. Effects of the transformation on the isophote diagrams that are included in the top left-hand figure (N → ∞) are shown in Fig. 4.

Fig. 4
Fig. 4

Isophotes (contours of the intensity) in a meridional plane in the neighborhood of the geometrical focus of a uniform, converging, monochromatic spherical wave diffracted at a circular aperture in an opaque screen, in systems of different Fresnel number N. The intensity is normalized to unity at the geometrical focus. The dotted lines represent the boundary of the geometrical shadow. The first isophote diagram (for N = 100) is essentially the same as that obtained from Lommel’s classic theory (Fig. 2 of Ref. 19 and Fig. 8.41 of Ref. 7).

Equations (39)

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a λ ,             ( a f ) 2 1 ,
N = a 2 / λ f
Δ max = a 2 / 2 f .
N = Δ max / ( λ / 2 ) ,
ϕ ~ θ = a / f
ϕ λ / a .
a / f λ / a .
N 1.
V ( i ) ( Q , t ) = A e - i k f f e - i ω t
U N ( P ) = - i λ A e - i k f f W e i k s s d S ,
u N = 2 π N z / f 1 + z / f ,
v N = 2 π N r / a 1 + z / f .
U N ( P ) = B N ( u N ) exp [ i Φ N ( u N , v N ) ] × 0 1 J 0 ( v N ρ ) exp ( - i U N ρ 2 / 2 ) ρ d ρ ,
B N ( u N ) = - 2 π i λ ( a f ) 2 ( 1 - u N 2 π N ) A ,
Φ N ( u N , v N ) = 1 1 - u N / 2 π N [ ( f a ) 2 u N + 1 4 π N v N 2 ] .
U ( P ) = B exp [ i Φ ( u ) ] 0 1 J 0 ( v ρ ) exp ( - i u ρ 2 / 2 ) ρ d ρ ,
u = 2 π λ ( a f ) 2 z ,
v = 2 π λ ( a f ) r ,
B = - 2 π i λ ( a f ) 2 A ,
Φ ( u ) = ( f a ) 2 u .
u N = u 1 + u / 2 π N ,
v N = v 1 + u / 2 π N ,
B N ( u N ) = ( 1 - u N 2 π N ) B ,
Φ N ( u N , v N ) = 1 1 - u N / 2 π N [ Φ ( u N ) + 1 4 π N v N 2 ] .
u N ~ u ,             v N ~ v ,
B N ( u N ) ~ B ,             Φ N ( u N , v N ) ~ Φ ( u ) ,
U N ( P ) ~ U ( P ) ,
U N ( P ) = ( ½ ) B N ( u N ) { exp [ i Φ N ( u N , v N ) ] } × [ C ( u N , v N ) - i S ( u N , v N ) ] ,
C N ( u N , v N ) = cos ( u N / 2 ) ( u N / 2 ) U 1 ( u N , v N ) + sin ( u N / 2 ) ( u N / 2 ) U 2 ( u N , v N ) ,
S N ( u N , v N ) = sin ( u N / 2 ) ( u N / 2 ) U 1 ( u N , v N ) - cos ( u N / 2 ) ( u N / 2 ) U 2 ( u N , v N ) ,
U n ( u N , v N ) = s = 0 ( - 1 ) s ( u N v N ) n + 2 s J n + 2 s ( v N ) ,
C ( u N , v N ) = 2 u N sin v N 2 2 u N + sin ( u N / 2 ) ( u N / 2 ) V 0 ( u N , v N ) - cos ( u N / 2 ) ( u N / 2 ) V 1 ( u N , v N ) ,
S ( u N , v N ) = 2 u N cos v N 2 2 u N - cos ( u N / 2 ) ( u N / 2 ) V 0 ( u N , v N ) - sin ( u N / 2 ) ( u N / 2 ) V 1 ( u N , v N ) ,
V n ( u N , v N ) = s = 0 ( - 1 ) s ( v N u N ) n + 2 s J n + 2 s ( v N ) .
u N = u = 0 ,             v N = v ,
B N ( 0 ) = B ,             Φ N ( 0 , v N ) = Φ ( 0 ) + 1 4 π N v 2 .
U N ( P ) z = 0 = U ( P ) z = 0 exp ( i v 2 4 π N ) .
I N ( P ) z = 0 U N ( P ) z = 0 2 = [ 2 J 1 ( v ) v ] 2 I 0 ,
I 0 = ( π a 2 A λ f 2 ) 2

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