## Topology of optical vortex lines formed by the interference of three, four, and five plane waves

Optics Express, Vol. 14, Issue 7, pp. 3039-3044 (2006)

http://dx.doi.org/10.1364/OE.14.003039

Acrobat PDF (290 KB)

### 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

## 1. Introduction

1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A **336**, 165–90 (1974). [CrossRef]

*π*occur, increasing in either a clockwise or counterclockwise sense (Fig. 1).

3. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A **456**, 2059–79 (2000). [CrossRef]

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]

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]

*k*-space distribution of paraxial plane waves [13

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]

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

## 2. Numerical calculations of vortex topology

*k*-space, with spacing Δ

*k*. This results in an interference pattern that is periodic both transversally and axially (the Talbot effect [14, 15]), with repeat periods of 2

*π*/Δ

*k*and 4

*πk*

_{0}/Δ

*k*

^{2}respectively. This ‘Talbot cell’ can be tiled to give the interference pattern, and therefore the vortex structure, over all space (Figs. 1, 2, and 3) without ambiguity.

*π*indicating a vortex. Knowledge of whether two vortex points in neighboring planes form part of the same line is limited by the spatial resolution. However, since vortex lines are continuous, ambiguity only arises when two different lines approach (characteristically in an ‘avoided crossing’ or ‘reconnection’ [9

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**, 8877–88 (2001). [CrossRef]

16. J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A: Pure Appl. Opt. **6**, S251–S254 (2004). [CrossRef]

## 3. Phasor argument for vortex topology

*N*waves with wavevectors

*(for*

**k**_{n}*n*= 1,…,

*N*) can be understood using phasors. At a particular position, the interference is calculated in the Argand plane by the vector addition of the individual phasors,

*ψ*

_{n}, of magnitude

*a*

_{n}= |

*ψ*

_{n}| and argument arg

*ψ*

_{n}i.e. the total field at the point is

*ψ*

_{n}. The waves are labeled in order of decreasing magnitude

*a*

_{1}≥

*a*

_{2}≥…≥

*a*

_{N}.

*ψ*

_{1}=-

*ψ*

_{2}. This is a single real condition (i.e. has codimension one), and in three dimensions, occurs on surfaces.

*a*

_{1}≤

*a*

_{2}+

*a*

_{3}, allowing a triangular configuration of the phasors. Under a different ordering of the phasors, the triangle may be reflected or rotated, but its shape cannot change. Along a vortex line, the phasor triangle rotates, so each component plane wave changes by the same phase, which can only be satisfied if the vortex direction is that in which the components of

*are the same. Therefore, the vortex structures arising from three plane waves are straight lines in the same direction,*

**k**_{n}

**k**_{1}×

**k**_{2}+

**k**_{2}×

**k**_{3}+

**k**_{3}×

**k**_{1}, as in Fig. 2(a).

*a*

_{1}≤

*a*

_{2}+

*a*

_{3}+

*a*

_{4}, giving a quadrilateral configuration of the phasors. With the magnitudes and wavevectors of the four waves fixed, the relative phases constitute three additional degrees of freedom. All possible phase relationships are explored throughout the three-dimensional Talbot cell. Consequently, a change to the initial phase of any of the superposed waves results only in a spatial translation, and does not affect any aspect of the vortex geometry.

*ψ*

_{1}=-

*ψ*

_{3}and

*ψ*

_{2}=-

*ψ*

_{4}. The vortex follows the intersection of two planes, determined by the two-wave interference described above (for waves 1 and 3, and 2 and 4), and is therefore again a straight line. However, under different orderings, a rhombus is also possible if

*ψ*

_{1}=-

*ψ*

_{2}and

*ψ*

_{3}=-

*ψ*

_{4}or

*ψ*

_{1}=-

*ψ*

_{4}and

*ψ*

_{2}=-

*ψ*

_{3}. Hence, there are three characteristic straight line vortex directions, along which the rhombus rotates and deforms. As the rhombus deforms, it may pass through a ‘flat’ configuration. At this point, the phasor pairs are identical, and the vortex lines cross [9

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**, 8877–88 (2001). [CrossRef]

16. J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A: Pure Appl. Opt. **6**, S251–S254 (2004). [CrossRef]

*a*

_{1}+

*a*

_{4}=

*a*

_{2}+

*a*

_{3}. The vortex lines are curved, but at particular points within the Talbot cell the four phasors are co-linear. As with the equal magnitude case, this corresponds to a vortices crossing, as in Fig. 3(a). More generally, either

*a*

_{1}+

*a*

_{4}<

*a*

_{2}+

*a*

_{3}or

*a*

_{1}+

*a*

_{4}>

*a*

_{2}+

*a*

_{3}. The former gives rise to an array of helical vortex lines (Fig. 3 (b), 4 (a)) whereas the latter gives an array of identical vortex loops [Figs. 3(c) and 4(b)], some of which may fall across the Talbot cell boundary. This difference is illustrated in Fig. 5, where phasor configurations along vortex lines in the two cases are shown. When

*a*

_{1}+

*a*

_{4}>

*a*

_{2}+

*a*

_{3}(loops) the smallest magnitude cannot execute a full 2

*π*rotation about the largest; this geometric constraint limits the maximum phase difference between any two phasors and hence the spatial extent of the vortex line, which must therefore be a loop. By contrast, when

*a*

_{1}+

*a*

_{4}<

*a*

_{2}+

*a*

_{3}(lines) the relative phase difference between the smallest amplitude phasor and the remaining three can cycle unidirectionally through the full 2

*π*range. In this case the vortex line can extend indefinitely (essentially a perturbed three wave superposition) [2].

## 4. Discussion

12. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A **291**, 453–84 (1979). [CrossRef]

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

3. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A **456**, 2059–79 (2000). [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]

## Acknowledgments

## References and links

1. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A |

2. | J. F. Nye, |

3. | M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. A |

4. | J. W. Goodman, |

5. | D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B |

6. | M. V. Berry, “Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices,” in . |

7. | J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. |

8. | M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. A |

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. |

10. | J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Knotted threads of darkness,” Nature |

11. | J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. |

12. | M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A |

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 |

14. | W. H. F. Talbot, “Facts relating to optical science, No. IV,” London Edinburgh Dublin Philos. Mag. J. Sci. |

15. | K. Patorski, “The self-imaging phenomenon and its applications,” Progress in Optics |

16. | J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A: Pure Appl. Opt. |

17. | J. F. Nye, “Dislocation lines in the hyperbolic umbilic diffraction catastrophe,” Proc. R. Soc. 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. |

**OCIS Codes**

(230.6120) Optical devices : Spatial light modulators

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 27, 2006

Revised Manuscript: March 27, 2006

Manuscript Accepted: March 28, 2006

Published: April 3, 2006

**Citation**

Kevin O'Holleran, Miles J. Padgett, and Mark R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express **14**, 3039-3044 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-3039

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### References

- J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974). [CrossRef]
- J. F. Nye, Natural focusing and fine structure of light (Institute of Physics Publishing, 1999).
- 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. W. Goodman, Statistical Optics (Wiley, 1985).
- D. Rozas, C. T. Law and G. A. Swartzlander, "Propagation dynamics of optical vortices," J. Opt. Soc. Am. B 14, 3054-65 (1997). [CrossRef]
- 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. 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]
- J. Leach, M. R. Dennis, J. Courtial and M. J. Padgett, "Knotted threads of darkness," Nature 432,165 (2004). [CrossRef] [PubMed]
- J. Leach, M. R. Dennis, J. Courtial and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005). [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]
- 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]
- W. H. F. Talbot, "Facts relating to optical science, No. IV," London Edinburgh Dublin Philos. Mag. J. Sci. 9, 401-407 (1836).
- K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. XXVII, 103-108 (1989).
- J. F. Nye, "Local solutions for the interaction of wave dislocations," J. Opt. A: Pure Appl. Opt. 6, S251-S254 (2004). [CrossRef]
- J. F. Nye, "Dislocation lines in the hyperbolic umbilic diffraction catastrophe," Proc. R. Soc. London Ser. A, in press (2006). [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]

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