Abstract

We study the propagation of off-axis vortices in a paraxial beam formed by two collinear Laguerre-Gauss beams. We show that the vortices move about the beam axis as the light propagates resulting in a rotation of the beam’s transverse profile. This rotation is explained by the Gouy phase acquired by the component beams. Experimental measurements of the angular position of the vortices are in good agreement with a two-mode theory.

© 2009 Optical Society of America

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  1. D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press-Elsevier, Burlington, 2008).
    [PubMed]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
    [CrossRef]
  3. M. S. Soskin and M. V. Vasnetsov, "Singular Optics," in Progress in Optics 42, E. Wolf, ed. (Elsevier 2001), pp. 219-276.
  4. G. Indebetow, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
    [CrossRef]
  5. 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]
  6. M. V. Berry and M. R. Dennis "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. Lond. A 4572251-2263 (2001).
    [CrossRef]
  7. M. V. Berry and M. R. Dennis "Knotting and unknotting of phase singularitites: Helmholtz waves, paraxialwaves and waves in 2+1 spacetime," J. Phys. A 348877-8888 (2001).
    [CrossRef]
  8. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 1-11 (2005).
    [CrossRef]
  9. A. I. Bishop, T. A. Niemen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical microrheology using rotating laser-trapped particles," Phys. Rev. Lett. 42, 198104-1-4 (2004).
  10. G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the angular momentum of the light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601-1-4 (2002).
  11. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
    [CrossRef] [PubMed]
  12. E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).
  13. E. J. Galvez and S. M. Baumann, "Composite vortex patterns formed by component light beams with non-integral topological charge, Proc. SPIE 6905, 6905D-1-7 (2008).
  14. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A.J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, "Optical ferris wheel for ultracold atoms," Opt. Express 15, 8619-8625 (2007).
    [CrossRef] [PubMed]
  15. C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).
  16. J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, "Measurement of Gouy phase evolution by use of spatial mode interference," Opt. Lett. 29, 2339-2341 (2004).
    [CrossRef] [PubMed]
  17. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
    [CrossRef]
  18. F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001-1-4 (2004).
    [CrossRef]
  19. J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).
  20. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, "Direct observation of Gouy phase shift in a propagating optical vortex," Opt. Express 14, 8382-8392 (2006).
    [CrossRef] [PubMed]
  21. I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
    [CrossRef]
  22. S. M. Baumann and E. J. Galvez, "Non-integral vortex structures in diffracted light beams," Proc. of SPIE 6483, 64830T-1-8 (2007).
  23. M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
    [CrossRef]
  24. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
    [CrossRef]
  25. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
    [CrossRef]
  26. I. D. Maleev and G. A. Swartzlander, Jr., "Composite optical vortices," J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  27. D. M. Kalb and E. J. Galvez "Composite vortices of displaced Laguerre-Gauss beams," Proc. SPIE 7227, 72270B-1-8 (2009).
  28. J. Courtial, "Self-imaging beams and the Gouy effect," Opt. Commun. 151, 1-4 (1998).
    [CrossRef]
  29. K. Patorski, "The self-imaging phenomenon and its applications" in Progress in Optics 27, E. Wolf, ed. (1989) pp. 1-108.

2007 (1)

2006 (2)

J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, "Direct observation of Gouy phase shift in a propagating optical vortex," Opt. Express 14, 8382-8392 (2006).
[CrossRef] [PubMed]

E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).

2005 (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 1-11 (2005).
[CrossRef]

2004 (2)

2003 (2)

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

I. D. Maleev and G. A. Swartzlander, Jr., "Composite optical vortices," J. Opt. Soc. Am. B 20, 1169-1176 (2003).
[CrossRef]

2001 (2)

M. V. Berry and M. R. Dennis "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. Lond. A 4572251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis "Knotting and unknotting of phase singularitites: Helmholtz waves, paraxialwaves and waves in 2+1 spacetime," J. Phys. A 348877-8888 (2001).
[CrossRef]

1999 (1)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

1998 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

J. Courtial, "Self-imaging beams and the Gouy effect," Opt. Commun. 151, 1-4 (1998).
[CrossRef]

1997 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

1993 (3)

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

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

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

1959 (1)

C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).

Allen, L.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

Arlt, J.

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

Arnold, A. S.

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. Lond. A 4572251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis "Knotting and unknotting of phase singularitites: Helmholtz waves, paraxialwaves and waves in 2+1 spacetime," J. Phys. A 348877-8888 (2001).
[CrossRef]

Carpenter, C. R.

C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).

Chow, J. H.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 1-11 (2005).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

J. Courtial, "Self-imaging beams and the Gouy effect," Opt. Commun. 151, 1-4 (1998).
[CrossRef]

de Vine, G.

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 1-11 (2005).
[CrossRef]

M. V. Berry and M. R. Dennis "Knotting and unknotting of phase singularitites: Helmholtz waves, paraxialwaves and waves in 2+1 spacetime," J. Phys. A 348877-8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. Lond. A 4572251-2263 (2001).
[CrossRef]

Ellinas, D.

Feng, S.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Fernandes, N.

E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).

Franke-Arnold, S.

Galvez, E. J.

E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).

Gibson, G.

Girkin, J. M.

Gray, M. B.

Hamazaki, J.

Heckenberg, N. R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Indebetow, G.

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

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

Leach, J.

Lembessis, V. E.

Maleev, I. D.

McClelland, D. E.

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Mineta, Y.

Morita, R.

Ohberg, P.

Oka, K.

Padgett, M. J.

Pas’ko, V.

Rozas, D.

Rubinztein-Dunlop, H.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Smiley, N.

E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Soskin, N. S.

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

Swartzlander, G. A.

van der Veen, H.E.L.O.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Winful, H. G.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

Wright, A.J.

Am. J. Phys. (1)

C. R. Carpenter, "Gouy phase advance with microwaves," Am. J. Phys. 27, 98-100 (1959).

J. Mod. Opt. (2)

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

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

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

J. Phys. A (1)

M. V. Berry and M. R. Dennis "Knotting and unknotting of phase singularitites: Helmholtz waves, paraxialwaves and waves in 2+1 spacetime," J. Phys. A 348877-8888 (2001).
[CrossRef]

New J. Phys. (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 1-11 (2005).
[CrossRef]

Opt. Commun. (4)

I. V. Basistiy, V. Yu. Bazhenov, N. S. Soskin, and M. V. Vasnetsov, "Optics of light beams with screw dislocations," Opt. Commun. 103, 422-428 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Opt. Commun. 112, 321-324 (1994).
[CrossRef]

J. Courtial, "Self-imaging beams and the Gouy effect," Opt. Commun. 151, 1-4 (1998).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, 951-962 (1998).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gauss modes," Phys. Rev. A 92, 8185-8189 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, "Direct observation of the Gouy phase shift with single-cycle terahertz pulses," Phys. Rev. Lett. 83, 3410-3413 (1999).
[CrossRef]

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. Lond. A 4572251-2263 (2001).
[CrossRef]

Proc. SPIE (1)

E. J. Galvez, N. Smiley, and N. Fernandes, "Composite optical vortices formed by collinear Laguerre-Gauss beams," Proc. SPIE 6131, 19-26 (2006).

Other (9)

E. J. Galvez and S. M. Baumann, "Composite vortex patterns formed by component light beams with non-integral topological charge, Proc. SPIE 6905, 6905D-1-7 (2008).

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Gouy phase shift for few-cycle laser pulses," Phys. Rev. Lett. 92, 113001-1-4 (2004).
[CrossRef]

S. M. Baumann and E. J. Galvez, "Non-integral vortex structures in diffracted light beams," Proc. of SPIE 6483, 64830T-1-8 (2007).

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," in Progress in Optics 42, E. Wolf, ed. (Elsevier 2001), pp. 219-276.

A. I. Bishop, T. A. Niemen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical microrheology using rotating laser-trapped particles," Phys. Rev. Lett. 42, 198104-1-4 (2004).

G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the angular momentum of the light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601-1-4 (2002).

K. Patorski, "The self-imaging phenomenon and its applications" in Progress in Optics 27, E. Wolf, ed. (1989) pp. 1-108.

D. M. Kalb and E. J. Galvez "Composite vortices of displaced Laguerre-Gauss beams," Proc. SPIE 7227, 72270B-1-8 (2009).

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press-Elsevier, Burlington, 2008).
[PubMed]

Supplementary Material (4)

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Figures (6)

Fig. 1.
Fig. 1.

Numerical calculation of the composite-vortex pattern generated by combining two Laguerre-Gauss beams with 1=−1 and 2=+4. In (a) the computed composite vortex pattern can be viewed by the color-encoded phase map of the light pattern. Circular symbols denote the position of the vortices, where a positive charge denotes a counter-clockwise increase in the phase. The intensity pattern is shown in (b) in grey scale.

Fig. 2.
Fig. 2.

Schematic of the apparatus used to study composite vortices. Polarizers [P(α)] and half-wave plates [W λ/2(α)] oriented an angle α relative to the horizontal are used to control the relative intensity of the interfering beams. Two forked binary gratings [G()] of charge are used to prepare the component beams. A beam expander (BE) and neutral density filter (F) are used to prepare a reference beam. Non-polarizing beams splitters (BS) are used to split and combine the light.

Fig. 3.
Fig. 3.

Image of composite beam patterns as a function of experimental parameters. The first row and (Media 1) show the composite beam as a function of the relative amplitude α of the component beams with 1=−1 and 2=+4. α is specified by the θ via α=tanθ. The second row and (Media 2) show the composite as a function of the relative phase δ of the component beams with 1=−1 and 2=+4, and α=1. The third row and (Media 3) show the composite beam as a function of θ for 1=+3 and 2=−3. The fourth row and (Media 4) show the composite beam as a function of δ for 1=+3 and 2=−3, and α=1.

Fig. 4.
Fig. 4.

Schematic of the section of the apparatus used to measure the rotation of the beam as a function of the propagation distance. The composite beam is focused by a lens (L). A reference beam is expanded by a beam expander (BE) and combined with the composite beam via a beam splitter (BS) so to form a fringe pattern on the CCD camera.

Fig. 5.
Fig. 5.

Graph of the data taken with the setup of Fig. 4. It corresponds to the rotation of the beam with two peripheral vortices created by the superposition of 1=0 and =2 modes as a function of propagation distance when the light goes through a focal point. The vortices were located by the forks in the interferograms of the composite beam. Inserts show samples of the beam at different locations.

Fig. 6.
Fig. 6.

Images testing the predictions for the rotation of the beam profiles as the light went through a focal point. The top row corresponds to a case where | 1|ℓ=| 2|, while the second row corresponds to the case where 1=− 2.

Equations (12)

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μ=(2π!)121w(r2w)er2/w2eiϕei[kzkr2/(2R)]e,
ρ=2w.
μT=sinθμ1+cosθμ2e,
α=I1I2 = tan θ ,
rv=w2(2!1!α2)12(21)
ϕv=δ+21,
φ=(2p++1)ξ,
ϕv=δ++(21)ξ21.
Δϕv=2121Δξ
Δ ϕv=σ2Δξ.
Δ ϕv=212+1σ2Δξ.
Δ ϕv = 0 .

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