Abstract

It is well known that uniform illumination of a lens leads to a focal field with a pattern of dark Airy rings in the focal plane, whereas this is not the case for Gaussian illumination. We show theoretically and experimentally that in the transition between the two cases the Airy rings, being phase singularities, reorganize themselves by means of a creation–annihilation process leading to new dark rings outside the focal plane.

© 1997 Optical Society of America

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References

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  1. A. Boivin, J. Dow, and E. Wolf, J. Opt. Soc. Am. 57, 1171 (1967).
    [CrossRef]
  2. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
    [CrossRef]
  3. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  4. J. J. Stamnes and B. Spelkavik, Opt. Commun. 40, 81 (1981).
    [CrossRef]
  5. J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).
  6. Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
    [CrossRef]
  7. M. V. Berry, J. Phys. A 27, L391 (1994).
    [CrossRef]
  8. G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

1994 (1)

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

1984 (1)

1981 (1)

J. J. Stamnes and B. Spelkavik, Opt. Commun. 40, 81 (1981).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

1967 (1)

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Beijersbergen, M. W.

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

Berry, M. V.

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Boivin, A.

Dow, J.

Karman, G. P.

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

Li, Y.

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Soskin, M. S.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Spelkavik, B.

J. J. Stamnes and B. Spelkavik, Opt. Commun. 40, 81 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes and B. Spelkavik, Opt. Commun. 40, 81 (1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).

van Duijl, A.

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

Vasnetsov, M. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Woerdman, J. P.

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

Wolf, E.

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, J. Phys. A 27, L391 (1994).
[CrossRef]

Opt. Commun. (2)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

J. J. Stamnes and B. Spelkavik, Opt. Commun. 40, 81 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Other (2)

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, “Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus,” Appl. Opt. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Focusing configuration. Lens L, assumed to be aberration free, has focal length f; aperture A has radius a; θ is the half-aperture angle. The origin of the coordinate system is placed in the geometric focal point of lens L; z is the longitudinal coordinate, and ρ is the transverse coordinate. The incoming beam propagates in the positive z direction, and its waist is located at z=-f.

Fig. 2
Fig. 2

Field distribution in the focal region near the first two Airy rings, calculated with relation  (1). NA=0.1, N=7. Thick curves, contour lines of constant intensity, normalized to 1 in the geometric focal point; adjacent curves differ by a factor of 10. (a) a/w=1.37. A and B are the first two dark Airy rings; ring A is shown enlarged in (b), where the thin curves are phase contours spaced by π/4. (c) a/w=1.587. (d) a/w=1.695.

Fig. 3
Fig. 3

Experimental result: positions of the singularities (distance ρ in pixels to the optical axis) in the focal plane as a function of the ratio a/w (with w kept constant). λ=632.9 nm, f=1±0.02 m, w=1.74±0.04 mm, a=0.83 mm, NA=0.8×10-33×10-3, N=1–14. The circles represent the experimental result, and the curves the theoretical result. The first two Airy rings are labeled A and B.

Fig. 4
Fig. 4

Experimental result showing the intensity distribution near the first two Airy rings in the form of contour lines of constant intensity, normalized to 1.0 in the geometric focal point. f=1±0.02 m, w=1.46±0.05 mm, NA2.3×10-3, N7. (a) For a/w=1.44 a=2.1±0.05 mm four singularities (A, B, C, and D) can be seen. (b) For a/w=1.64 a=2.39±0.05 mm A and B have annihilated and only singularities C and D can be seen, 40 mm outside the focal plane z=0.

Equations (5)

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uρ, z1-u2πNexpiΦ01J0vtexp-γt2tdt, 
Φ1-u2πN-1f2a2u+v24πN,
γa2w2+12iu,
u2πNz/f1+z/f,
v2πNρ/a1+z/f.

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