## Creation of vortex lattices by a wavefront division

Optics Express, Vol. 15, Issue 8, pp. 5196-5207 (2007)

http://dx.doi.org/10.1364/OE.15.005196

Acrobat PDF (1926 KB)

### Abstract

The Young’s double-slit experiment is one of the most popular stories in the history of physics. This paper, like many others, has emerged from the Young’s idea. It investigates the diffraction of the plane or spherical wave produced by three or four small holes in an opaque screen. It was noticed that the interference field contained a lattice of optical vortices which were equivalent to those produced in optical vortex interferometer. The vortex lattice generated by the three holes possessed some unique properties from which the analytical formulae for vortex points position were derived. We also pointed out the differences between our case and the double-slit experiment. Finally, some remarks on possible applications of our arrangement are discussed briefly. These theoretical considerations are illustrated with the use of experimental results.

© 2007 Optical Society of America

## 1. Introduction

8. J. F. Nye and M. V. Berry, “Dislocations in wave
trains,” Proc. Roy. Soc. Lond. A **336**, 165–189
(1974). [CrossRef]

11. L. Allen, S. M. Barnett, and M. J. Padgett, *Optical angular momentum* (IoP,
Bristol, Philadelphia, 2003). [CrossRef]

12. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt.39, chapter IV (1999). [CrossRef]

## 2. The plane wave illumination

*z*-axis is perpendicular to the plate. In such a coordinate system the wavevector of the incident wave is

**k**(0,0,k). We have assumed that the center of the hole A overlaps with the center of coordinate system. The waves emerging from the holes A, B, C, will be marked by the same letters A, B, C.

*f*

^{2}

_{p}=

*f*

^{2}

_{x}+

*f*

^{2}

_{y},

*f*,

_{x}*f*are spatial frequencies,

_{y}**T**

_{B}(

*T*

_{Bx},

*T*

_{By}),

**T**

_{C}(

*T*

_{Cx},

*T*

_{Cy}) are translation vectors indicating the position of B and C holes, respectively:

_{1}is a Bessel function of the first kind and first order and

*R*is the radius of the hole.

**T**

_{B}we get a sum of the same somb functions. Thus one of them must be multiplied by the phase term exp{2

*πi*(

*T*

_{Bx}*f*+

_{x}*T*

_{By}*f*)} (shift theorem for Fourier transforms). The phase differences between three contributing waves are described by the sum in square parenthesis in Eq. (1). Since we are interested in phase differences between interfering waves we can assume that wave A is our reference wave. This interference field can be represented with the use of phasors. The length of the phasor represents wave amplitude and its angle represents wave phase. Towards zero amplitude the three phasors must form a triangle (must sum up to zero). Since the amplitudes of the three interfering waves are the same at each propagation direction, the resulting triangles must be equilateral. The phases of waves A B, C in vortex points are:

_{y}*m*and

*n*are integers. The relations (3) lead to the following conditions:

*z*is the distance between the plate and the observation plane. The two sets of solutions correspond to two topological charges of the optical vortices. Figure 3 shows two experimental examples as well as comparison to our theory. Figure 4 shows the interference pattern obtained as a result of adding the light emerging from the plate together with an additional plane wave. The fork like fringe pattern in Fig. 4 is an experimental evidence that our plate generates the vortex lattice.

16. J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem
in optical vortices interferometr,” Opt.,
Commun. **234**, 23–28
(2004). [CrossRef]

*x,y*) the point on the screen can be approximated as (Fig. 6)

*q*∊(A, B, C),

*x*,

_{q}*y*are coordinates of the center of the q-hole. We can still assume that the center of the hole A lies at the origin of a coordinate system. We can also assume that the phase of the incident wave equals zero in the center of this hole.

_{q}*z*from the plate is:

_{b}δ

_{c}are the phase differences between the center of the holes B, C and the hole A for the given incident wave. The value of δ

_{b}δ

_{c}is important when the incident plane wave is inclined in respect to the plate; in the case of normal illumination δ

_{b}= δ

_{c}= 0. Using the parabolic approximation we get:

24. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines
formed by the interference of three, four and five plane
waves,” Opt. Express **14**, 3039 – 3044
(2006). [CrossRef] [PubMed]

**k**

_{2}(

*k*,

_{2x}*k*,

_{2y}*k*). The far field solution is

_{2z}*f*′

^{2}

_{p}= (

*f*-

_{x}*f*

_{0x})

^{2}+ (

*f*-

_{y}*f*

_{0y})

^{2},

*f*

_{0x}=

*k*

_{2x}/2π,

*f*

_{0y}=

*k*

_{2y}/2π . We can read from this solution that although nothing changes the whole pattern is shifted in the frequency domain, which means that the vortex lattice moves like a rigid body. The same conclusion can be drawn from formula (8). There we have a very simple method for measuring the small tilt angles of the incident wave. Instead of tracing the shift of the whole spot the vortex lattice shift can be traced. The small shift of vortex points can be determined with higher accuracy than the whole light spot. Moreover, the determination of the tilt axis orientation in a single measurement is possible. Figure 9 shows an example of vortex lattice reaction to the wave tilt (experiment). The wave was tilted by introducing 90arcsec wedge in front of the plate. The wave tilt is half of the wedge angle – 45arcsec in our case. Knowing the distance between the plate and the screen the tilt angle can be determined. In the experiment the vortex lattice shift equal to 193 μm (30 pixels) was measured; the

*z*distance was 890mm. If we recalculate these data to the tilt angle we get the value of 44.8 ±1 arcsec. All this procedures are correct if the incidence angle is small enough (we still work in far or near field approximations).

## 3. The case of illumination with the spherical wave

*z*-axis we can scan larger area of the wavefront. With the use of this method the changes in the wavevector direction the can be measured. This is an important advantage of our method. In the standard interferometry we measure the phase differences which are recomputed to wavefront slope on the basis of some additional assumptions. Figure 12(a) shows the comparison between two vortex lattices. Both were obtained using spherical illumination. After registering the first interferogram the plate was slightly shifted (in the direction perpendicular to the z-axis), so the wavevectors in the center of the holes changed their orientation. As a result the geometry of the positive and negative sublattices was the same but the distance between these two sublattices changed. The results of our calculations are in qualitative agreement with the estimation of the wavefront geometry. However, more precise experiments are necessary to judge the value of this method.

## 4. Conclusions and remarks

## References and links

1. | T. Young, “Experimental demonstration of the General Law of the Interference of Light,” Philos. Trans. R. Soc. London94, (1804). |

2. | R. Welti, “Light transmission through two
slits: the Young experiment revisited,”
J. Opt. A: Pure Appl. Opt. |

3. | C. Jönsson, “Electron diffraction at multiple
slits,” Am. J. Phys. |

4. | A. Zeilinger, R. Gähler, C.G. Shull, W. Treimer, and W. Mampe, “Single and double-slit diffraction
of neutrons,” Rev. Mod. Phys. |

5. | O. Carnal and J. Mlynek, “Young’s double-slit
experiment with atoms: A simple atom
interferometer,” Phys. Rev. Lett. |

6. | W. Schöllkopf and J. Toennies, “The nondestructive mass selection of
small van der Waals clusters,” Science |

7. | C. K. Hong and T. G. Noh, “Two-photon double slit interference
experiment,” J. Opt. Soc. Am. B |

8. | J. F. Nye and M. V. Berry, “Dislocations in wave
trains,” Proc. Roy. Soc. Lond. A |

9. | J. F. Nye, |

10. | M. Vasnetsov and K. Staliunas, eds., |

11. | L. Allen, S. M. Barnett, and M. J. Padgett, |

12. | L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt.39, chapter IV (1999). [CrossRef] |

13. | M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, chapter IV (2001). |

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

15. | J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using
optical vortices as a phase markers,”
Opt. Commun. |

16. | J. Masajada, A. Popiołek Masajada, E. Frączek, and W. Frączek, “Vortex points localization problem
in optical vortices interferometr,” Opt.,
Commun. |

17. | J. Masajada, “Small rotation-angle measurement
with optical vortex interferometer,” Opt.
Commun. |

18. | A. PopioВek-Masajada, M Borwińska, and W Frączek, “Testing a new method for small-angle
rotation measurements with Optical Vortices
Interferometer,” Meas. Sci. Technol. |

19. | M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using Optical Vortex Interferometer,” Opt. Eng. (to be published). |

20. | J. Masajada, “The interferometry based on regular net of optical vortices,” Opt. Appl. to be published in vol.37, (2007). |

21. | J. Masajada, “The optical vortex interferometer,
theory, technology and applications,”
Proc. SPIE |

22. | P. Kurzynowski, A. WoŴniak, and E. Frączek, “Optical vortices generation using
the Wollaston prism,” Appl. Opt. |

23. | P. Kurzynowski and M. Borwińska, “Generation of the vortex type
markers in a one wave setup,” Appl. Opt. |

24. | K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines
formed by the interference of three, four and five plane
waves,” Opt. Express |

25. | J. Courtial, R. Zambrini, M. Dennis, and M. Vasnetsov, “Angular momentum of optical vortex
arrays”, Opt. Express |

26. | W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic
signal of white-light speckle pattern with application to micro-displacement
measurement,” Opt. Commun. |

27. | J. Primot and L. Sogno, “Achromatic three-wave (or more)
lateral shearing interferometer,” J. Opt.
Soc. Am. A |

28. | J. S. Darlin, P. Senthilkumaran, S. Bhattacharaya, M. P. Kothiyal, and R. S. Sirohi, “Fabrication of an array illuminator
using tandem Michelson interferometers,”
Opt. Commun. |

29. | N. Guerineau and J. Primot, “Nondiffracting array generation
using an N-wave interferometer,” J. Opt.
Soc. Am. A |

30. | S. Velghe, J. Primot, N. Guerineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from
multidirectional phase derivatives generated by multilateral shearing
interferometers,” Opt. Lett. |

31. | A. S. Patra and A. Khare, “Interferometric array
generation,” Opt. Laser Techn. |

32. | W. Singer, M. Totzeck, and H. Gross, |

33. | W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. Hanson, “Optical vortex metrology for
nanometric speckle displacement measurement,”
Opt. Express |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(260.3160) Physical optics : Interference

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 21, 2007

Revised Manuscript: March 19, 2007

Manuscript Accepted: March 28, 2007

Published: April 13, 2007

**Citation**

J. Masajada, A. Popiolek-Masajada, and M. Leniec, "Creation of vortex lattices by a wavefront division," Opt. Express **15**, 5196-5207 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5196

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### References

- T. Young, "Experimental demonstration of the General Law of the Interference of Light," Philos. Trans. R. Soc. London 94, (1804).
- R. Welti, "Light transmission through two slits: the Young experiment revisited," J. Opt. A: Pure Appl. Opt. 8, 606-609 (2006). [CrossRef]
- C. Jönsson, "Electron diffraction at multiple slits," Am. J. Phys. 42, 4-11 (1974). [CrossRef]
- A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe, "Single and double-slit diffraction of neutrons," Rev. Mod. Phys. 60, 1067-1073 (1988). [CrossRef]
- O. Carnal and J. Mlynek, "Young’s double-slit experiment with atoms: A simple atom interferometer," Phys. Rev. Lett. 66, 2689-2692 (1991). [CrossRef] [PubMed]
- W. Schöllkopf and J. Toennies, "The nondestructive mass selection of small van der Waals clusters," Science 226, 1345-1348 (1994). [CrossRef]
- C. K. Hong and T. G. Noh, "Two-photon double slit interference experiment," J. Opt. Soc. Am. B 15, 1192-1197 (1997). [CrossRef]
- J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. Roy. Soc. Lond. A 336, 165-189 (1974). [CrossRef]
- J. F. Nye, Natural focusing and fine structure of light (IoP, Bristol and Philadelphia, 1999).
- M. Vasnetsov, K. Staliunas, eds., Optical vortices (Nova Science Publishers, 1999).
- L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (IoP, Bristol, Philadelphia, 2003). [CrossRef]
- L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, chapter IV (1999). [CrossRef]
- M. S. Soskin and M. V. Vasnetsov, "Singular Optics," Prog. Opt. 42, chapter IV (2001).
- J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001). [CrossRef]
- J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, "The interferometric system using optical vortices as a phase markers," Opt. Commun. 207, 85-93 (2002). [CrossRef]
- J. Masajada, A. Popiołek Masajada, E. Frączek and W. Frączek, "Vortex points localization problem in optical vortices interferometr," Opt. Commun. 234, 23-28 (2004). [CrossRef]
- J. Masajada, "Small rotation-angle measurement with optical vortex interferometer," Opt. Commun. 234, 373-381 (2004). [CrossRef]
- A. Popiołek-Masajada, M Borwińska and W Frączek, "Testing a new method for small-angle rotation measurements with Optical Vortices Interferometer," Meas. Sci. Technol. 17, 653-658 (2006). [CrossRef]
- M. Borwińska, A. Popiołek-Masajada and B. Dubik, "Reconstruction of the plane wave tilt and its orientation using Optical Vortex Interferometer," Opt. Eng. (to be published).
- J. Masajada, "The interferometry based on regular net of optical vortices," Opt. Appl. to be published in vol. 37, (2007).
- J. Masajada, "The optical vortex interferometer, theory, technology, and applications," Proc. SPIE 6254, 62540C1-10 (2006).
- P. Kurzynowski, A. Woźniak and E. Frączek, "Optical vortices generation using the Wollaston prism," Appl. Opt. 45, 7898-7903 (2006). [CrossRef] [PubMed]
- P. Kurzynowski and M. Borwińska, "Generation of the vortex type markers in a one wave setup," Appl. Opt. 46, 676-679 (2007). [CrossRef] [PubMed]
- K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 14, 3039 - 3044 (2006). [CrossRef] [PubMed]
- J. Courtial, R. Zambrini, M. Dennis, and M. Vasnetsov, "Angular momentum of optical vortex arrays", Opt. Express 14, 938-949 (2006) [CrossRef] [PubMed]
- W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005). [CrossRef]
- J. Primot and L. Sogno, "Achromatic three-wave (or more) lateral shearing interferometer," J. Opt. Soc. Am. A 12, 2679-2685 (1995). [CrossRef]
- J. S. Darlin, P. Senthilkumaran, S. Bhattacharaya, M. P. Kothiyal and R. S. Sirohi, "Fabrication of an array illuminator using tandem Michelson interferometers," Opt. Commun. 123, 1-4 (1996). [CrossRef]
- N. Guerineau and J. Primot, "Nondiffracting array generation using an N-wave interferometer," J. Opt. Soc. Am. A 16, 293-298 (1999). [CrossRef]
- S. Velghe, J. Primot, N. Guerineau, M. Cohen and B. Wattellier, "Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers," Opt. Lett. 30, 245-247 (2005). [CrossRef] [PubMed]
- A. S. Patra and A. Khare, "Interferometric array generation," Opt. Laser Techn. 38, 37-45 (2006). [CrossRef]
- W. Singer, M. Totzeck and H. Gross, Handbook of Optical System, (Wiley-VCH, Berlin, 2005) Vol. 2
- W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda and S. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006). [CrossRef] [PubMed]

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