Abstract

When a Gaussian beam with two oppositely charged vortices propagates in free space, these two vortices will move around on the transverse beam plane. They may either move toward each other and annihilate each other spontaneously or survive all the way depending on the conditions. Here, we investigate how to force vortex dipoles to annihilate. We find that the background phase function created by two oppositely charged vortices during beam propagation can cause the vortices to move together and annihilate each other. The background phase function on a transverse plane just beyond the point where a dipole annihilated is continuous and retains the potential that forces a dipole to annihilate. We use this background phase function to accelerate the annihilation of vortex dipoles. Numerical results are provided to show the acceleration of dipole annihilation in a Gaussian beam, using such a background phase function.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2007

2004

F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B 21, 655-663 (2004).
[CrossRef]

F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. B 21, 664-670 (2004).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S318-S322 (2004).
[CrossRef]

F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433-440 (2004).
[CrossRef]

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55 (2004).
[CrossRef]

2003

2001

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163-165 (2001).
[CrossRef]

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71-90 (2001).
[CrossRef]

2000

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19-33 (2000).
[CrossRef]

1998

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759-2768 (1998).
[CrossRef]

1997

1996

Y. Y. Schechner and J. Shamir, “Parameterization and orbital angular momentum of anisotropic dislocations,” J. Opt. Soc. Am. A 13, 967-973 (1996).
[CrossRef]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

1995

1993

1992

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123-127 (1992).
[CrossRef]

1989

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1983

1974

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

1973

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765-786 (1973).
[CrossRef]

Assa, M.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

Baranova, N.

Berry, M.

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

Chen, M.

Christou, J.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Devaney, A. J.

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765-786 (1973).
[CrossRef]

Dinev, S.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

Dreischuh, A.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

Freund, I.

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71-90 (2001).
[CrossRef]

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19-33 (2000).
[CrossRef]

Fried, D. L.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Indebetouw, G.

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

Kessler, D. A.

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71-90 (2001).
[CrossRef]

Kim, G.-H.

Kim, J.-U.

Kivshar, Y. S.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

Law, C. T.

Lee, H. J.

Luther-Davies, B.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

Mamaev, A.

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S318-S322 (2004).
[CrossRef]

Molina-Terriza, G.

Neshev, D.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

Nye, J.

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

Olivier, J. C.

Pilipetsky, N.

Pismen, L. M.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Roux, F. S.

Rozas, D.

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054-3065 (1997).
[CrossRef]

D. Rozas, Generation and propagation of optical vortices,” Ph.D. dissertation (Worcester Polytechnic Institute, 1999).

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S318-S322 (2004).
[CrossRef]

Schechner, Y. Y.

Shamir, J.

Sherman, G. C.

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765-786 (1973).
[CrossRef]

Shkunov, V.

Staliunas, K.

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123-127 (1992).
[CrossRef]

M. Vasnetsov and K. Staliunas, Optical Vortices, (Nova Science, 1999), Vol. 228.

Suk, H.

Swartzlander, G. A.

Tikhonenko, V.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

Torner, L.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

Vasnetsov, M.

M. Vasnetsov and K. Staliunas, Optical Vortices, (Nova Science, 1999), Vol. 228.

Velchev, I.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Wright, E. M.

Zel'dovich, B.

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S318-S322 (2004).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

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

J. Opt. B: Quantum Semiclassical Opt.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclassical Opt. 6, S318-S322 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413-421 (1998).
[CrossRef]

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123-127 (1992).
[CrossRef]

F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433-440 (2004).
[CrossRef]

F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45-55 (2004).
[CrossRef]

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19-33 (2000).
[CrossRef]

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71-90 (2001).
[CrossRef]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385-392 (1996).
[CrossRef]

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198-206 (1998).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London, Ser. A

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

SIAM Rev.

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765-786 (1973).
[CrossRef]

Other

M. Vasnetsov and K. Staliunas, Optical Vortices, (Nova Science, 1999), Vol. 228.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

D. Rozas, Generation and propagation of optical vortices,” Ph.D. dissertation (Worcester Polytechnic Institute, 1999).

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Figures (5)

Fig. 1
Fig. 1

Phase functions of a Gaussian beam before and after a vortex dipole annihilation. (a) Shows the phase function with two oppositely charge optical vortices, which are located at the ends of the short black line. (b) Shows the background phase function extracted from the phase function in (a). (c) Shows the continuous phase function after the dipole annihilation. (d) Shows the continuous background phase function extracted from the phase function in (c).

Fig. 2
Fig. 2

Three-dimensional view of the background phase function after vortex dipole annihilation.

Fig. 3
Fig. 3

(a) Shows the amplitude of a Gaussian beam with two oppositely charged vortices, shown as two dark cores, separated by a distance of d = 0.91 . The Gaussian beam waist size is ω 0 = 640 μ m . (b) Shows a background phase function created according to the locations of the vortices in (a).

Fig. 4
Fig. 4

(a) Amplitude and (b) phase of a Gaussian beam with two positive canonical vortices ( V 1 and V 3 ) and two negative canonical vortices ( V 2 and V 4 ). The Gaussian beam waist size is ω 0 = 640 μ m . (c) Amplitude and (d) phase after the Gaussian beam, as shown in (a) and (b), propagated in free space over a distance of 3.1 z R . Those two dipoles still exist in the beam with their separation distance being enlarged.

Fig. 5
Fig. 5

(a) Background phase function created to force the vortex dipoles in Fig. 4a to annihilate. (c) Amplitude and (d) phase of the Gaussian beam after the beam propagated with the background phase as shown in (a) in free space over a distance of 0.15 z R . All four vortices have disappeared through dipole annihilations.

Equations (29)

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

E ( u , v , t ) = P ( u , v , t ) E g ( u , v , t ) ,
E g ( u , v , t ) = exp ( u 2 + v 2 1 i t 2 )
P ( u , v ) = 1 2 [ ξ ( u + i v ) + ζ ( u i v ) ] ,
ξ = cos ( ψ 2 ) exp ( i ϕ 2 ) ,
ζ = sin ( ψ 2 ) exp ( i ϕ 2 ) .
P ( u , v , t ) = P e j θ ,
θ = n Φ ( u u n , v v n ; ψ n , ϕ n ) + Φ C ,
P ( u , v , t 0 ) = [ ( u Δ u u 0 ) + i ( v v 0 ) ] [ ( u Δ u + u 0 ) i ( v v 0 ) ] .
P ( u , v , t ) = [ ( u a α Δ u ) + i ( v b + β Δ u ) ] [ ( u + a α Δ u ) i ( v b β Δ u ) ] σ β ( β + i α ) ,
a = u 0 α + v 0 β , b = v 0 α u 0 β ,
α = ( 1 + t t 0 ) σ , β = ( t t 0 ) σ , σ = 1 + t 0 2 .
u 1 , 2 ( t ) = β 2 Δ u σ α ± a D 1 2 2 ( β 2 Δ u 2 + a 2 ) + α Δ u ,
v 1 , 2 ( t ) = β ( a σ α Δ u D 1 2 ) 2 ( β 2 Δ u 2 + a 2 ) + b ,
D = [ β 2 ( 2 Δ u 2 + σ ) + 2 a 2 ] 2 σ 2 β 2 ( α 2 + β 2 ) .
ψ = arccos ( f 2 f + 2 f 2 + f + 2 ) ,
ϕ = i 2 ln ( f + f * f + * f ) ,
f ± = 1 2 ( U u ± i U v ) .
cos ( ψ ) = ± D 1 2 β 2 ( σ + 2 Δ u 2 ) + 2 a 2 ,
tan ( ϕ ) = ± α D 1 2 β ( 2 a 2 + 2 β 2 Δ u 2 α 2 σ ) ,
c Φ ( u , v ) d s = ν 2 π ,
V d = 1 2 [ ( ξ 1 + ζ 1 ) ( u u 1 ) + i ( ξ 1 ζ 1 ) ( v v 1 ) ] [ ( ξ 2 + ζ 2 ) ( u u 2 ) + i ( ξ 2 ζ 2 ) ( v v 2 ) ] [ ( ξ j + ζ j ) ( u u j ) + i ( ξ j ζ j ) ( v v j ) ] ,
Φ b g = arg { E t 1 E g V d } ,
G 0 = [ u + i v + d 2 exp ( i α ) ] [ u i v d 2 exp ( i α ) ] exp ( u 2 v 2 ) .
t a = d 2 2 1 d 2 ,
u a = d ( d 2 2 ) sin ( α ) 4 1 d 2 ,
v a = d ( d 2 2 ) cos ( α ) 4 1 d 2 .
Φ a = arg { G a exp [ u 2 + v 2 1 i ( t a + Δ ) 2 ] } .
Φ a = Φ a ( u Δ u , v Δ v ) ,
G = G exp ( i Φ a ) .

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