Abstract

When three or more plane waves overlap in space, complete destructive interference occurs on nodal lines, also called phase singularities or optical vortices. For super positions of three plane waves, the vortices are straight, parallel lines. For four plane waves the vortices form an array of closed or open loops. For five or more plane waves the loops are irregular. We illustrate these patterns numerically and experimentally and explain the three-, four- and five-wave topologies with a phasor argument.

© 2006 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 336,165-90 (1974).
    [CrossRef]
  2. J. F. Nye, Natural focusing and fine structure of light (Institute of Physics Publishing, 1999).
  3. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London Ser. A 456, 2059-2079 (2000).
    [CrossRef]
  4. J. W. Goodman, Statistical Optics (Wiley, 1985).
  5. D. Rozas, C. T. Law and G. A. Swartzlander, "Propagation dynamics of optical vortices," J. Opt. Soc. Am. B 14, 3054-65 (1997).
    [CrossRef]
  6. M. V. Berry, "Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices," in. Intl. Conf. on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 1-15 (1998).
    [CrossRef]
  7. J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
    [CrossRef]
  8. M. V. Berry and M. R. Dennis, "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. London Ser. A 457, 2251-2263 (2001).
    [CrossRef]
  9. M. V. Berry and M. R. Dennis, "Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime," J. Phys. A: Math. Gen. 34, 8877-8888 (2001).
    [CrossRef]
  10. J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
    [CrossRef] [PubMed]
  11. J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
    [CrossRef]
  12. M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
    [CrossRef]
  13. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005).
    [CrossRef] [PubMed]
  14. W. H. F. Talbot, "Facts relating to optical science, No. IV," London Edinburgh Dublin Philos. Mag. J. Sci. 9, 401-407 (1836).
  15. K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. XXVII, 103-108 (1989).
  16. J. F. Nye, "Local solutions for the interaction of wave dislocations," J. Opt. A: Pure Appl. Opt. 6, S251-S254 (2004).
    [CrossRef]
  17. J. F. Nye, "Dislocation lines in the hyperbolic umbilic diffraction catastrophe," Proc. R. Soc. London Ser. A, in press (2006).
    [CrossRef]
  18. M. R. Dennis, "Local phase structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6, S202-S208 (2004).
    [CrossRef]

2006 (1)

J. F. Nye, "Dislocation lines in the hyperbolic umbilic diffraction catastrophe," Proc. R. Soc. London Ser. A, in press (2006).
[CrossRef]

2005 (2)

2004 (3)

J. F. Nye, "Local solutions for the interaction of wave dislocations," J. Opt. A: Pure Appl. Opt. 6, S251-S254 (2004).
[CrossRef]

M. R. Dennis, "Local phase structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6, S202-S208 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

2001 (3)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. London Ser. A 457, 2251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime," J. Phys. A: Math. Gen. 34, 8877-8888 (2001).
[CrossRef]

2000 (1)

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London Ser. A 456, 2059-2079 (2000).
[CrossRef]

1997 (1)

1989 (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. XXVII, 103-108 (1989).

1979 (1)

M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974).
[CrossRef]

1836 (1)

W. H. F. Talbot, "Facts relating to optical science, No. IV," London Edinburgh Dublin Philos. Mag. J. Sci. 9, 401-407 (1836).

Berry, M. V.

M. V. Berry and M. R. Dennis, "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. London Ser. A 457, 2251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime," J. Phys. A: Math. Gen. 34, 8877-8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London Ser. A 456, 2059-2079 (2000).
[CrossRef]

M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974).
[CrossRef]

Cooper, J.

Courtial, J.

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
[CrossRef]

M. R. Dennis, "Local phase structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6, S202-S208 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. London Ser. A 457, 2251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime," J. Phys. A: Math. Gen. 34, 8877-8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London Ser. A 456, 2059-2079 (2000).
[CrossRef]

Dubik, B.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Jordan, P.

Law, C. T.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

Masajada, J.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Nye, J. F.

J. F. Nye, "Dislocation lines in the hyperbolic umbilic diffraction catastrophe," Proc. R. Soc. London Ser. A, in press (2006).
[CrossRef]

J. F. Nye, "Local solutions for the interaction of wave dislocations," J. Opt. A: Pure Appl. Opt. 6, S251-S254 (2004).
[CrossRef]

M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974).
[CrossRef]

Padgett, M.

Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. XXVII, 103-108 (1989).

Piestun, R.

Rozas, D.

Schonbrun, E.

Swartzlander, G. A.

Talbot, W. H. F.

W. H. F. Talbot, "Facts relating to optical science, No. IV," London Edinburgh Dublin Philos. Mag. J. Sci. 9, 401-407 (1836).

Wright, F. J.

M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
[CrossRef]

Wulff, K. D.

J. Opt. A: Pure Appl. Opt. (2)

J. F. Nye, "Local solutions for the interaction of wave dislocations," J. Opt. A: Pure Appl. Opt. 6, S251-S254 (2004).
[CrossRef]

M. R. Dennis, "Local phase structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6, S202-S208 (2004).
[CrossRef]

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

J. Phys. A: Math. Gen. (1)

M. V. Berry and M. R. Dennis, "Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime," J. Phys. A: Math. Gen. 34, 8877-8888 (2001).
[CrossRef]

London Edinburgh Dublin Philos. Mag. J. Sci. (1)

W. H. F. Talbot, "Facts relating to optical science, No. IV," London Edinburgh Dublin Philos. Mag. J. Sci. 9, 401-407 (1836).

Nature (1)

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Knotted threads of darkness," Nature 432,165 (2004).
[CrossRef] [PubMed]

New J. Phys. (1)

J. Leach, M. R. Dennis, J. Courtial and M. J.  Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005).
[CrossRef]

Opt. Commun. (1)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Opt. Express (1)

Phil. Trans. R. Soc. London Ser. A (1)

M. V. Berry, J. F. Nye and F. J. Wright, "The elliptic umbilic diffraction catastrophe," Phil. Trans. R. Soc. London Ser. A 291, 453-84 (1979).
[CrossRef]

Proc. R. Soc. London Ser. A (4)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974).
[CrossRef]

M. V. Berry and M. R. Dennis, "Knotted and linked phase singularities in monochromatic waves," Proc. R. Soc. London Ser. A 457, 2251-2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London Ser. A 456, 2059-2079 (2000).
[CrossRef]

J. F. Nye, "Dislocation lines in the hyperbolic umbilic diffraction catastrophe," Proc. R. Soc. London Ser. A, in press (2006).
[CrossRef]

Prog. Opt. (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. XXVII, 103-108 (1989).

Other (3)

M. V. Berry, "Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices," in. Intl. Conf. on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 1-15 (1998).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. F. Nye, Natural focusing and fine structure of light (Institute of Physics Publishing, 1999).

Supplementary Material (7)

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Figures (6)

Fig. 1.
Fig. 1.

Optical vortex line in an interference field. Left: periodic cell containing a vortex loop (dotted lines are parts of the loop outside the primitive cell). The intensity on the front and back planes is also represented. Right: phase cross section of the front/back face of the cell. The phase circulates in opposite directions at different points on the vortex line.

Fig. 2.
Fig. 2.

Calculated vortex structure for three, four and five equal-amplitude plane waves (a, b, c) and their respective k-space configuration (d, e, f). Each is calculated on a 256×256×256 Talbot cell.

Fig. 3.
Fig. 3.

The calculated vortex structures for four plane wave superpositions with amplitudes: a), a 1 + a 4 = a 2 + a 3 (b), a 1+a 4<a 2+a 3 (c), a 1+a 4 >a 2+a 3. The choice of k-vectors is the same as Fig. 2 (b), given in Fig. 2 (e). Attached multimedia shows rotation of cells around the optical axes (a, b, c) (1.4Mb, 1.3Mb, 1.1Mb).

Fig. 4.
Fig. 4.

Experimentally measured vortex structures from the superposition of 4 plane waves with amplitudes giving rise to a) twisted vortex lines and b) vortex loops. The choice of k-vectors differs slightly from that in Fig. 3(b) and Fig. 3(c). Attached multimedia shows rotation of each cell around the optical axis (1.9Mb, 1.8Mb).

Fig. 5.
Fig. 5.

Left: Vortex line in (x, y, z). Right: phasor diagrams at the points highlighted on a vortex a) loop and b) a line. For clarity, the orientation of the largest magnitude phasor has been fixed. The red arrows show the angular range of the smallest phasor, with respect to the largest. Note that in (a) this is restricted to less than 2π. Attached multimedia shows phasor configuration changing as the vortex paths are followed (2.3Mb, 2.1Mb).

Fig. 6.
Fig. 6.

Examples of different vortex topologies that can be obtained from five waves of the same magnitude but different relative phase. Fig. 6 (a) is the same Talbot cell as Fig. 2(c). Figs. 6(b) and 6(c) are Talbot cells from the same wave magnitudes as (a) but with one wave advanced in phase by π/2 and π/3. Figs. 6(d),6(e), and 6(f) are their unwrapped counterparts.

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