Abstract

It is well known that phase singularities, in general lines in space, are topologically stable features of a wave field. An exception is a pointlike singularity, which is unstable and deforms into a ring or disappears when a small perturbation is applied. Recently, Nye showed how such an event can be understood as an unfolding of a higher-order dislocation [J. Opt. Soc. Am. A (to be published)]. We present an optical implementation of this model and show experimentally that the focal region of a lens contains points of zero intensity on the optical axis that deform into rings when a small amount of spherical aberration is applied to the system.

© 1998 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
    [CrossRef]
  3. G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).
  4. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A (to be published).
  5. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).
  6. J. J. Stamnes, Waves in Focal Regions (IOP, Bristol, UK, 1986).
  7. J. F. Nye, J. Mod. Opt. 38, 743 (1991).
    [CrossRef]
  8. M. E. R. Walford and J. F. Nye, J. Glaciol. 37, 107 (1991).
  9. J. J. Stamnes and B. Spjelkavik, Opt. Commun. 40, 81 (1981).
    [CrossRef]
  10. G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
    [CrossRef]

1997 (2)

1991 (2)

J. F. Nye, J. Mod. Opt. 38, 743 (1991).
[CrossRef]

M. E. R. Walford and J. F. Nye, J. Glaciol. 37, 107 (1991).

1981 (1)

J. J. Stamnes and B. Spjelkavik, 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]

Beijersbergen, M. W.

G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
[CrossRef]

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
[CrossRef]

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

Berry, M. V.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

Bouwmeester, D.

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

Karman, G. P.

G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
[CrossRef]

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
[CrossRef]

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

Nye, J. F.

J. F. Nye, J. Mod. Opt. 38, 743 (1991).
[CrossRef]

M. E. R. Walford and J. F. Nye, J. Glaciol. 37, 107 (1991).

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

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A (to be published).

Spjelkavik, B.

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

Stamnes, J. J.

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

J. J. Stamnes, Waves in Focal Regions (IOP, Bristol, UK, 1986).

van Duijl, A.

G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
[CrossRef]

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
[CrossRef]

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

Walford, M. E. R.

M. E. R. Walford and J. F. Nye, J. Glaciol. 37, 107 (1991).

Woerdman, J. P.

G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
[CrossRef]

G. P. Karman, A. van Duijl, M. W. Beijersbergen, and J. P. Woerdman, Appl. Opt. 36, 8091 (1997).
[CrossRef]

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

Appl. Opt. (1)

J. Glaciol. (1)

M. E. R. Walford and J. F. Nye, J. Glaciol. 37, 107 (1991).

J. Mod. Opt. (1)

J. F. Nye, J. Mod. Opt. 38, 743 (1991).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (1)

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 (4)

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A (to be published).

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A (to be published).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986).

J. J. Stamnes, Waves in Focal Regions (IOP, Bristol, UK, 1986).

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Figures (3)

Fig. 1
Fig. 1

Calculated focal field. NA=0.1, N=10, f=1000λ, δ=0. (a) Contours of constant intensity, normalized to 1 in the geometric focal point; adjacent curves differ by a factor of 10. (b) Enlargement of the boxed region of (a), showing in detail an axial zero at z=-166.65λ. The thick curves are intensity contours, and the thin curves are curves of constant phase, spaced by π/4. The arrow indicates the direction of increasing phase.

Fig. 2
Fig. 2

Calculated field in the boxed region of Fig.  1(a) when a small amount of spherical aberration is present. NA=0.1, N=10, f=1000λ, δ=+0.05. The thick curves are intensity contours, normalized to 1 in the geometric focal point for the case δ=0; adjacent curves differ by a factor of 10. The thin curves are phase contours, space by π/4. The arrow indicates the direction of increasing phase. The axial zero has deformed into a ring of radius 0.8λ around the optical axis at z=-173.5λ.

Fig. 3
Fig. 3

Experimentally obtained intensity distribution, with f=9.6 m, a=0.96 mm, δ=+0.3, NA=10-4, N=0.15. Shown are contours of constant intensity in the order 20, 10, 5, 2, . . .  (I1 in the geometric focus for δ=0). (a) ρz plane near the first axial zero point (cf.  Fig.  1). A and B refer to off-axis singularities; C denotes a local minimum (not a zero). (b) CCD image taken at z=-9.07 m. The central bright spot is surrounded by a dark ring that corresponds to singularity A in (a). The intensity of the central bright spot is much lower than that of the bright ring surrounding the dark ring. The picture measures 80×80 pixels, which corresponds to 0.7 mm×0.7 mm.

Equations (2)

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uA=expikρ22f+δλρa4.
uρ, z=ρ2+2izk-αexpikz,

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