Abstract

The output field of a uniformly illuminated lens contains points of zero intensity on the optical axis and zero-intensity Airy rings in the focal plane, as formed by diffraction. These intensity zeros have been recognized as phase singularities or wave dislocations. Recently it was shown that, under the influence of a perturbation, the axial singularities may transform into rings or disappear and that the Airy rings may split into triplets. Starting from optical diffraction theory, we identify the physical perturbations that can induce such topological transformations. The basic perturbations are phase and amplitude aberrations of the wave front that is incident on the lens; we show that their different natures have consequences for the possible dislocation reactions.

© 1998 Optical Society of America

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References

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  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. G. P. Karman, M. W. Beijersbergen, A. van Duijl, J. P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22, 1503–1505 (1997).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1986), Subsec. 8.8.
  4. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
    [CrossRef]
  5. G. P. Karman, A. van Duijl, J. P. Woerdman, “Unfolding of an unstable singularity point into a ring,” Opt. Lett. 23, 403–405 (1998).
    [CrossRef]
  6. G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A 15, 884–899 (1998).
    [CrossRef]
  7. M. V. Berry, “Wave dislocation reactions in nonparaxial Gaussian beams,” J. Mod. Opt.45 (to be published).
  8. M. E. R. Walford, J. F. Nye, “Measuring the dihedral angle of water at a grain boundary in ice by an optical diffraction method,” J. Glaciol. 37, 107–112 (1991).
  9. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  10. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  11. Exact expressions for the roots of a polynomial of degree four exist, but they are involved. See, for instance, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1970), p. 17.

1998

1997

1991

M. E. R. Walford, J. F. Nye, “Measuring the dihedral angle of water at a grain boundary in ice by an optical diffraction method,” J. Glaciol. 37, 107–112 (1991).

1981

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

1974

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Abramowitz, M.

Exact expressions for the roots of a polynomial of degree four exist, but they are involved. See, for instance, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1970), p. 17.

Beijersbergen, M. W.

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

M. V. Berry, “Wave dislocation reactions in nonparaxial Gaussian beams,” J. Mod. Opt.45 (to be published).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1986), Subsec. 8.8.

Bouwmeester, D.

Karman, G. P.

Li, Y.

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

Nye, J. F.

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
[CrossRef]

M. E. R. Walford, J. F. Nye, “Measuring the dihedral angle of water at a grain boundary in ice by an optical diffraction method,” J. Glaciol. 37, 107–112 (1991).

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Stamnes, J. J.

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

Stegun, I. A.

Exact expressions for the roots of a polynomial of degree four exist, but they are involved. See, for instance, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1970), p. 17.

van Duijl, A.

Walford, M. E. R.

M. E. R. Walford, J. F. Nye, “Measuring the dihedral angle of water at a grain boundary in ice by an optical diffraction method,” J. Glaciol. 37, 107–112 (1991).

Woerdman, J. P.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1986), Subsec. 8.8.

J. Glaciol.

M. E. R. Walford, J. F. Nye, “Measuring the dihedral angle of water at a grain boundary in ice by an optical diffraction method,” J. Glaciol. 37, 107–112 (1991).

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Lett.

Proc. R. Soc. London Ser. A

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1986), Subsec. 8.8.

M. V. Berry, “Wave dislocation reactions in nonparaxial Gaussian beams,” J. Mod. Opt.45 (to be published).

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

Exact expressions for the roots of a polynomial of degree four exist, but they are involved. See, for instance, M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1970), p. 17.

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Figures (9)

Fig. 1
Fig. 1

Focusing configuration. The origin of the coordinate system is placed in the geometrical focal point of the lens L, which has a focal length f ;  z is the longitudinal coordinate, and ρ is the transverse coordinate. The circular aperture A has a radius a and is located at z=-f ; θ is the half-aperture angle.

Fig. 2
Fig. 2

Intensity profiles in the aperture, for different values of .

Fig. 3
Fig. 3

Field distribution near the first axial zero for the case f=1000λ, a=100λ, θ=0.1, δ=0, as calculated with Eq. (2). Thick curves are contours of constant intensity, normalized to 1 in the geometrical focal point; adjacent curves correspond to intensities that differ by a factor 10; thin curves are phase contours, spaced by π/4; and the horizontal arrow indicates the direction of increasing phase. (a) =0, (b) =+0.01, (c) =-0.01. S is a phase saddle point.

Fig. 4
Fig. 4

Diagram showing conditions under which the axial zeros in front of the focal point (n<0) transform into rings (light region) or disappear (dark region). The δ plane is divided into two regions by the diagonal, on which Re α=0 and the axial singularity remains a point.

Fig. 5
Fig. 5

Wave fronts (solid curves) in front of and beyond the focal point when δ0. For comparison, the dashed curves are spherical surfaces, centered at the focal point z=0.

Fig. 6
Fig. 6

Intensity distribution near the first two Airy rings for the case f=1000λ, a=100λ, θ=0.1, δ=0, as calculated with Eq. (2). Shown are contours of constant intensity, normalized to 1 in the geometrical focal point; adjacent curves correspond to intensities that differ by a factor 10. (a) =0. Two Airy rings (labeled A and B) can be seen in the focal plane, (b) =2.5. The (former) two Airy rings are labeled A and B, whereas C and D denote extra created singularities.

Fig. 7
Fig. 7

As in Fig. 6(b), i.e., =2.5, but for different values of δ. (a) δ=0.02 (B and C approach), (b) δ=0.04 (B and C have been annihilated).

Fig. 8
Fig. 8

Diagram of the δ plane showing conditions under which the Airy rings are involved in a triplet event (f=1000λ,a=100λ, θ=0.1); the labeling is as in Figs. 6 and 7. The triangular-shaped dark region is a region in which four singularities are present (A, B, C, D). Outside the triangle, only two singularities are present. Arrows indicate pair events, and the corner points of the triangle correspond to triplet events. The dashed lines indicate soft boundaries across which the labeling of the singularities changes.

Fig. 9
Fig. 9

Schematic picture of the positions of the singularities when triplet formation occurs, i.e., for points in parameter space just inside the corner points of the triangle in Fig. 8. (a) lower-middle corner point, (b) left-corner point, (c) right-corner point.

Tables (1)

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Table 1 Effect of Different Perturbations on the Airy Rings and on the Axial Singularitiesa

Equations (16)

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uA=exp+ik ρ22 f- ρ2a2+ikδ λ ρ4a4.
u(ρ, z)exp(ikz)01 exp(A1t2+A2t4)J0(A3t)td t,
A1--12ikθ 2z,
A2-2πiδ,
A3kθρ
u(ρ=0, z)exp[ik(1-14θ 2)z]sinc(14kθ 2z),
zn2λθ 2n,withn=±1,±2,,
u(ρ, z=0)J1(kθρ)kθρ,
kθρm=3.832,7.016,10.173,
(m=1, 2, 3,).
||1,| δ |1,
ρ1/kθ,
z=zn+z,with|z|1/kθ2.
u(ρ, z=zn+z)ρ2+2izk-αexp(ikz),
α-4k2θ 2+2δn+i2πδ.
u(ρ, z)ρ4+8izkρ2-8k2z2+4k2ρ2+bρ2+2izk+cexp(ikz),

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