Abstract

A dipole of optical vortices propagating in a Gaussian beam can produce a variety of possible trajectories. Equations are provided for the trajectories of vortices launched as a canonical dipole from arbitrary locations in a Gaussian beam. These equations are used to compute the trajectories for two examples, which are compared with numerical simulations. The critical parameter values that would produce annihilations and revivals are computed as a function of the propagation distance. This provides a method to identify different types of vortex dipole trajectory.

© 2004 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. F. S. Roux, “Branch-point diffractive optics,” J. Opt. Soc. Am. A 11, 2236–2243 (1994).
    [CrossRef]
  3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [CrossRef] [PubMed]
  4. C. T. Law, X. Zhang, and G. A. Swartzlander, Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25, 55–57 (2000).
    [CrossRef]
  5. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  6. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  7. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).
  8. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  9. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  10. I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
    [CrossRef]
  11. F. S. Roux, “Paraxial modal analysis technique for optical vortex trajectories,” J. Opt. Soc. Am. B 20, 1575–1580 (2003).
    [CrossRef]
  12. G. Molina-Terriza, L. Torner, E. M. Wright, J. J. García-Ripoll, and V. M. Pérez-García, “Vortex revivals with trapped light,” Opt. Lett. 26, 1601–1603 (2001).
    [CrossRef]
  13. D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  14. L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
    [CrossRef]

2003 (1)

2002 (1)

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

1997 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (2)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992 (1)

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

1974 (1)

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

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Berry, M. V.

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

Crasovan, L.-C.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

García-Ripoll, J. J.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Law, C. T.

McDuff, R.

Mihalache, D.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

Molina-Terriza, G.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

G. Molina-Terriza, L. Torner, E. M. Wright, J. J. García-Ripoll, and V. M. Pérez-García, “Vortex revivals with trapped light,” Opt. Lett. 26, 1601–1603 (2001).
[CrossRef]

Nye, J. F.

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

Pérez-García, V. M.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

G. Molina-Terriza, L. Torner, E. M. Wright, J. J. García-Ripoll, and V. M. Pérez-García, “Vortex revivals with trapped light,” Opt. Lett. 26, 1601–1603 (2001).
[CrossRef]

Roux, F. S.

Rozas, D.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Smith, C. P.

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Swartzlander Jr., G. A.

Torner, L.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

G. Molina-Terriza, L. Torner, E. M. Wright, J. J. García-Ripoll, and V. M. Pérez-García, “Vortex revivals with trapped light,” Opt. Lett. 26, 1601–1603 (2001).
[CrossRef]

Torres, J. P.

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

White, A. G.

Wright, E. M.

Zhang, X.

J. Mod. Opt. (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Opt. Commun. (2)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

L.-C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Pérez-García, and D. Mihalache, “Globally linked vortex clusters in trapped wave fields,” Phys. Rev. E 66, 036612 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

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

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

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Figures (8)

Fig. 1
Fig. 1

Critical parameter value, R0c, for t0=-2 when u0=0. The area above (below) the curve represents regions in which vortices are present (absent). The threshold value, R0T=1.048, is represented by the horizontal dotted line. For R0>R0T (R0<R0T), the vortices form two open loops excluding the β plane (one closed loop around the α plane). The horizontal dashed line represents a specific example with R0=0.5. The vortex dipole exists only in the range of t between A and A, where R0c<R0.

Fig. 2
Fig. 2

Critical parameter value, u0c, for t0=-2 when Δu=v0=0. The area above (below) the curve represents regions in which vortices are present (absent). The critical value vanishes at the β plane, and it has a maximum value of u0M=0.866 at the α plane. The threshold value, u0T=0.741, is represented by the horizontal dotted line. For u0<u0T, the vortices form one closed loop around the β plane; for u0T<u0<u0M, the vortices form two open loops that exclude the α plane; and, for u0>u0M, there are vortices all the way from t=- to t=.

Fig. 3
Fig. 3

Vortex trajectories for Δu=v0=0 and u0>u0M, showing how the vortices flip around in the α plane to exchange their topological charges. The arrows denote the flow of positive topological charge.

Fig. 4
Fig. 4

Critical parameter value, u0c, for t0=-2 when Δu=0 and v0=-u0t0. The area above (below) the curve represents regions in which vortices are present (absent). The function vanishes at both the α and the β planes (i.e., the α and the β planes always have vortices) and diverges at t±. Between the α and the β planes, the critical value reaches a peak value, u0P. For u0<u0P (u0>u0P), there are two closed loops (is one closed loop).

Fig. 5
Fig. 5

Critical analysis for the first example. The critical parameter value, u0c, for t0=-2, η=v0/u0=2/3, and δ=Δu/u0=2 is given by the solid curve. The dashed line, which represents the chosen parameter value of u0=0.3, cuts the critical curve in four places, denoted by the four black dots. This indicates that there are two closed loops, which, respectively, include the α and β planes. The threshold value, u0T=0.49, is represented by the horizontal dotted line.

Fig. 6
Fig. 6

Stereographic projection of the vortex dipole trajectories for t0=-2, u0=0.3, v0=0.2, and Δu=0.6. The vortices propagate from left to right. The solid curves represent the analytically predicted trajectories. The plus symbols represent numerical results. The axis of the beam is shown as a straight line, and the 1/e2 radii of the beam in the waist and the front and back Rayleigh planes are represented by gray circles. Connecting lines are used to indicate the locations of the vortices in the launch plane (black line) and in the waist and the two Rayleigh planes (gray lines).

Fig. 7
Fig. 7

Critical analysis for the second example. The critical parameter value, u0c, for t0=-2, η=0, and δ=1/12 is given by the solid curve. The dashed line, which represents the chosen parameter value of u0=6/7, runs completely above the critical curve. There are therefore no critical points, and the vortices are present over the entire propagation range. The threshold value, u0T=0.74, is represented by the horizontal dotted line.

Fig. 8
Fig. 8

Stereographic projection of the vortex dipole trajectories for t0=-2, u0=6/7, v0=0, and Δu=1/14. The vortices propagate from left to right. The solid curves represent the analytically predicted trajectories. The plus symbols represent numerical results. The axis of the beam is shown as a straight line, and the 1/e2 radii of the beam in the waist and the front and back Rayleigh planes are represented by gray circles. Connecting lines are used to indicate the locations of the vortices in the launch plane (black line) and in the waist and the two Rayleigh planes (gray lines).

Equations (46)

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uv2U-i4 Ut=0,
uv=uxˆ+vyˆ,
u=xω,v=yω,t=zλπω2,
Um±(u, v, t)=2m+1πm!1/2 (u±iv)m(1-it)m+1 exp-(u2+v2)(1-it),
--|Um±(u, v, t)|2dxdy=1.
Mm±(u, v, t)=2mm!1/2u±iv1-itm.
Mm,n±(u, v, t)=[U0(u, v, t)]-1 ntnUm±(u, v, t).
M0,1(u, v, t)=i11-it-u2+v2(1-it)2.
1-it1-it0=(1+tt0)-i(t-t0)1+t02=α-iβ.
σ=1+t02.
α=1+tt0σ,β=t-t0σ.
p1(u, v, t)=AM0(u, v, t)+BM1+(u, v, t)=A+B 2(u+iv)1-it.
A=-(u0+iv0),B=1-it02.
p1(u, v, t)=(u+iv)-(u0+iv0)(α-iβ).
u=u0α+v0β,v=v0α-u0β.
a=u0α+v0β,b=v0α-u0β
p1(u, v, t)=(u-a)+i(v-b).
p2(u, v, t0)=[(u-Δu-u0)+i(v-v0)]×[(u-Δu+u0)-i(v-v0)].
pc(u, v, t)=-σβ(β+iα).
p2(u, v, t0)=[(u-a-αΔu)+i(v-b+βΔu)]×[(u+a-αΔu)-i(v-b-βΔu)]-σβ(β+iα),
(u-αΔu)2+(v-b)2=a2+β2(σ+Δu2),
βΔu(u-αΔu)+(v-b)a-12σαβ
=0.
R=[a2+β2(σ+Δu2)]1/2.
u1,2=β2Δuσα±a{[β2(2Δu2+σ)+2a2]2-σ2β2(α2+β2)}1/22(β2Δu2+a2)+αΔu,
v1,2=β2(aσα±{[β2(2Δu2+σ)+2a2]2-σ2β2(α2+β2)}1/2)2(β2Δu2+a2)+b,
D=[β2(2Δu2+σ)+2a2]2-σ2β2(α2+β2).
α=t0β+1,
C4=[2Δu2+σ+2(u0t0+v0)2]2-σ3.
D=β4[2(Δu2+v02)+σ]2-σ2β2(α2+β2).
D=β4(2R02+σ)2-σ2β2(α2+β2).
R0c=σ2[1+(α/β)2-1]1/2.
R0T=σ2(σ-1)1/2.
D=(2u02α2+σβ2)2-σ2β2(α2+β2).
u0c=σ2(β/α)2[1+(α/β)2-1]1/2.
u0M=u0c(α=0)=σ2.
u0T=σ2(σ+1)1/2.
C4=σ2-σ3<0,
D=(2u02+σβ2)2-σ2β2(α2+β2).
u0c=σ2β2[1+(α/β)2-1]1/2.
δ=Δuu0,η=v0u0.
D={2[δ2β2+(α+ηβ)2]u02+σβ2}2-σ2β2(α2+β2).
u0c=σ2 1+(α/β)2-1δ2+[η+(α/β)]21/2.
(u0c)2=12σβ+O(β2).
β=α-1t0,
(u0c)2=14 σ2-σδ2+η2α2+O(α3).

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