Abstract

Wave dislocation reactions, for example, one dislocation splitting into three, are organized by dislocations of higher order. A reaction can be regarded as an unfolding of a higher-order dislocation, rather as singularities in catastrophe theory unfold into clusters of lower-order ones. The paper lists the forms of the first few higher-order wave dislocations. It shows how the reaction reported by Karman et al. [Opt. Lett. 22, 1503 (1997)], in which an Airy dark ring surrounding a focus splits into three, followed by the annihilation of an inner fourth ring by the central ring, is part of the unfolding of a fourth-order dislocation.

© 1998 Optical Society of America

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References

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  1. 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]
  2. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  3. J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. London, Ser. A 378, 219–239 (1981).
    [CrossRef]
  4. J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983), Sec. 3.
    [CrossRef]
  5. R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models (Benjamin, Reading, Mass., 1975).
  6. V. I. Arnold, “Critical points of smooth functions and their normal forms,” Russian Math. Surveys 30, 1–75 (translated from Uspekhi Math. Nauk 30, 3–65 (1975).
  7. J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
    [CrossRef]
  8. M. V. Berry, “Wave dislocation reactions in nonparaxial Gaussian beams,” J. Mod. Opt. (to be published).
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Fig. 8.46.
  10. J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991), Sec. 5.
    [CrossRef]

1997

1991

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991), Sec. 5.
[CrossRef]

1988

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

1983

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983), Sec. 3.
[CrossRef]

1981

J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. London, Ser. A 378, 219–239 (1981).
[CrossRef]

1974

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

Arnold, V. I.

V. I. Arnold, “Critical points of smooth functions and their normal forms,” Russian Math. Surveys 30, 1–75 (translated from Uspekhi Math. Nauk 30, 3–65 (1975).

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. (to be published).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Fig. 8.46.

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

Hannay, J. H.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

Karman, G. P.

Nye, J. F.

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991), Sec. 5.
[CrossRef]

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983), Sec. 3.
[CrossRef]

J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. London, Ser. A 378, 219–239 (1981).
[CrossRef]

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

Thom, R.

R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models (Benjamin, Reading, Mass., 1975).

van Duijl, A.

Woerdman, J. P.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Fig. 8.46.

J. Mod. Opt.

J. F. Nye, “Diffraction by a small unstopped lens,” J. Mod. Opt. 38, 743–754 (1991), Sec. 5.
[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]

J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. London, Ser. A 378, 219–239 (1981).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983), Sec. 3.
[CrossRef]

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

Russian Math. Surveys

V. I. Arnold, “Critical points of smooth functions and their normal forms,” Russian Math. Surveys 30, 1–75 (translated from Uspekhi Math. Nauk 30, 3–65 (1975).

Other

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Fig. 8.46.

R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models (Benjamin, Reading, Mass., 1975).

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Figures (6)

Fig. 1
Fig. 1

Equiphase lines for the singularity of Eq. (4) at intervals of π/4. The arrows show the direction of increasing phase; full lines denote phase 0. The width of the diagram is 16k-1= 2.6λ. (a) a=0, (b) a positive.

Fig. 2
Fig. 2

Each dot (p, q) represents a term xpyq. An arrow from a term means that this term implies the presence of the term it points to. (a) q even, (b) q odd. (a) With q even also applies to terms zprq.

Fig. 3
Fig. 3

(a) Equiphase lines for the singularity with m=4 in Table 2, at intervals of π/4. Full lines denote phase 0. The width of the diagram is 16k-1=2.6λ.(b) The same partially unfolded to two dislocations of double strength.

Fig. 4
Fig. 4

Unfoldings in the space of c, a of f(x, y) in Eq. (9). Each box is a plot in x, y of the loci of Re f=0 (full curves) and Im f=0 (dashed lines) for the c, a pair indicated. The width of each box is 4k-1=0.64λ.

Fig. 5
Fig. 5

Loci in x, y space of Re f=0 (full curves) and Im f =0 (dashed lines) for f(x, y) in Eqs. (9). c is held constant at -6; a has values 1.5, 2.2, 2.4, respectively in (a)–(c). Filled dots are positive dislocations; open dots are negative dislocations. The width of each box is 4k-1=0.64λ.

Fig. 6
Fig. 6

Corresponds to the lower part of Fig. 5 (the origin is at the top). Equiphase lines from Eqs. (5) and (9), at intervals of π/8. The arrows show the direction of increasing phase. Full lines denote phase 0. The values of the unfolding parameters are the same as in Fig. 5. The width of each diagram is 2k-1= 0.32λ.

Tables (3)

Tables Icon

Table 1 Exact Polynomial Solutions (k=1)

Tables Icon

Table 2 Degenerate dislocations at O

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Table 3 Exact Polynomial Solutions in Cylindrical Polar Coordinates (k=1)

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

(2+k2)ψ=0,
ψ(x, y)=f(x, y)exp(ikx).
fxx+fyy+2ikfx=0,
f(x, y)=y2+ix-a(a=real constant)
ϕ=kx+arg f(x, y)
f(x, y)=y4+6ixy2-3x2-3ix+d(y3+3ixy)+c(y2+ix)+by+a,
f(x, y)=f(-x, y),f(x, y)=f*(-x, y).
f(x, y)=y4+6ixy2-3x2+3y2+d(y3+3ixy)+c(y2+ix)+by+a.
Re f(x, y)=y4-3x2+3y2+dy3+cy2+by+a=0,
Im f(x, y)=6xy2+3dxy+cx=0.
Re f(x, y)=y4-3x2+3y2+cy2+a=0,
Im f(x, y)=x(6y2+c)=0.
ψ(z, r)=f(z, r)exp(ikz).
frr+1rfr+fzz+2ikfz=0.
f(z, r)=r2+2iz-a(a=real constant),
ϕ=kz+arg f(z, r),
f(z, r)=r4+8izr2-8z2+4r2+b(r2+2iz)+c,
Re f(z, r)=r4-8z2+4r2+br2+c=0,
Im f(z, r)=z(8r2+2b)=0.

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