Abstract

We predict new features for the evolution of noncanonical screw wave-front dislocations nested in diffracting light beams. The evolution equations for the location of single-charge dislocations are derived, and their consequences are elaborated by use of illustrative examples. It is shown that the dynamic trajectories of the vortices can be controlled by adjustment of the noncanonical strength of the input dislocations.

© 2001 Optical Society of America

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
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  2. A. Ashkin, Opt. Photon. News 10(5), 41 (1999).
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  3. M. Padgett and L. Allen, Phys. World 10(9), 35 (1997).
  4. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
    [CrossRef]
  5. I. Freund, Opt. Commun. 159, 99 (1999).
    [CrossRef]
  6. I. Freund, Opt. Commun. 181, 19 (2000).
    [CrossRef]
  7. G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
    [CrossRef]
  8. B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 22, 1816 (1994).
    [CrossRef]
  9. J. Christou, V. Tikhonenko, Y. S. Kivshar, and B. Luther-Davies, Opt. Lett. 21, 1649 (1996).
    [CrossRef] [PubMed]
  10. D. Rozas, C. T. Law, and G. A. Swartzlander, J. Opt. Soc. Am. B 14, 3054 (1997).
    [CrossRef]
  11. Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
    [CrossRef]
  12. D. Rozas and G. A. Swartzlander, Opt. Lett. 25, 126 (2000).
    [CrossRef]
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  14. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  15. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
    [CrossRef]

2000 (2)

1999 (3)

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

A. Ashkin, Opt. Photon. News 10(5), 41 (1999).
[CrossRef]

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

1998 (1)

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

1997 (2)

1996 (1)

1994 (1)

B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 22, 1816 (1994).
[CrossRef]

1993 (1)

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

1974 (1)

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

Allen, L.

M. Padgett and L. Allen, Phys. World 10(9), 35 (1997).

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Anderson, B. P.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin, Opt. Photon. News 10(5), 41 (1999).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Berry, M. V.

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

Christou, J.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

J. Christou, V. Tikhonenko, Y. S. Kivshar, and B. Luther-Davies, Opt. Lett. 21, 1649 (1996).
[CrossRef] [PubMed]

Cornell, E. A.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

Freund, I.

I. Freund, Opt. Commun. 181, 19 (2000).
[CrossRef]

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

Haljan, P. C.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

Hall, D. S.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

Heckenberg, N. R.

Indebetouw, G.

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

J. Christou, V. Tikhonenko, Y. S. Kivshar, and B. Luther-Davies, Opt. Lett. 21, 1649 (1996).
[CrossRef] [PubMed]

Law, C. T.

Luther-Davies, B.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

J. Christou, V. Tikhonenko, Y. S. Kivshar, and B. Luther-Davies, Opt. Lett. 21, 1649 (1996).
[CrossRef] [PubMed]

B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 22, 1816 (1994).
[CrossRef]

Matthews, M. R.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

McDuff, R.

Nye, J. F.

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

Padgett, M.

M. Padgett and L. Allen, Phys. World 10(9), 35 (1997).

Pismen, L. M.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

Powles, R.

B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 22, 1816 (1994).
[CrossRef]

Rozas, D.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Smith, C. P.

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tikhonenko, V.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

J. Christou, V. Tikhonenko, Y. S. Kivshar, and B. Luther-Davies, Opt. Lett. 21, 1649 (1996).
[CrossRef] [PubMed]

B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 22, 1816 (1994).
[CrossRef]

Weiman, C. E.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

White, A. G.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

G. Indebetouw, J. Mod. Opt. 40, 73 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, Opt. Commun. 152, 198 (1998).
[CrossRef]

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

I. Freund, Opt. Commun. 181, 19 (2000).
[CrossRef]

Opt. Lett. (4)

Opt. Photon. News (1)

A. Ashkin, Opt. Photon. News 10(5), 41 (1999).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Weiman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999).
[CrossRef]

Phys. World (1)

M. Padgett and L. Allen, Phys. World 10(9), 35 (1997).

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 (1)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (4)

Fig. 1
Fig. 1

Contour plots of (a) the amplitude and (b) the phase of the core of a single-charge vortex with noncanonical strength A=1+i. Contour lines are spaced 0.2 unit in (a) and π/8 units in (b).

Fig. 2
Fig. 2

Evolution of the location of different noncanonical vortices nested in Gaussian beams diffracting in free space from z=0 to z=3. (a) The noncanonical strength of the vortex is A0=1+i, and the pattern of the input beam is also shown. The white line shows the vortex location predicted by the model; the circles indicate the exact location calculated numerically with the full evolution equations. (b) Trajectories of vortices with different noncanonical strengths. The values of A0 have been chosen so that at z=3 all vortices are located on the circumference of the unit radius. The arrows indicate the orientation of the vortex trajectory. Trajectories pointing upward (downward) correspond to vortices with charge +1-1. In all cases w0=1 and x1,0=0.5.

Fig. 3
Fig. 3

Evolution of the vortex location, similar to that in Fig.  2 but for a host beam propagating in a graded-index medium. The labels in (b) stand for the values of the noncanonical strength A0,r,A0,i. In (a) A0,r=1, A0,i=1, and the pattern of input light is shown. In all cases n2=2, w0=1, and x1,0=0.5.

Fig. 4
Fig. 4

Evolution of the vortex location similar to that in Fig.  2 but for elliptical host beams. The input vortices feature a canonical shape A0,r=1, A0,i=0. The labels in (b) indicate the ellipticities w0x:w0y of the input beams. In (a) w0x=1, w0y=4; the pattern of input light is shown. The solid curve and circles show the actual vortex location; the dashed curve shows the vortex trajectory that is obtained when the noncanonical nature of the vortices is ignored. In all cases x1,0=0.5.

Equations (12)

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

iEz+122E+gE,E*;x,y,z=0,
E=x-x1z+iAzy-y1zFx,y,z,
dx1dz+iAdy1dz=-iL1InFx=x1,y=y1,z,
dAdz+(L2L1InFx,y,z+L2gE,E*;x,y,zFx,y,z)x=x1,y=y1=0,
dx1dz+iAdy1dz=2x1qx+iA2y1qy,
dAdz=2A1qx-1qy.
x1z=x1,01+2Ai,0w02Ar,0z,
y1z=2x1,0w02Ar,0z.
x1z=x1,0cos2z/w02+Ai,0Ar,0sin2z/w02,
y1z=x1,0Ar,0sin2z/w02.
Az=A0w0y2w0x2iw0x2-2ziw0y2-2z,
x1zqx+iA0w0y2w0x2y1zqy=ix1,0w0x2.

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