## Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer

Optics Express, Vol. 18, Issue 13, pp. 13836-13841 (2010)

http://dx.doi.org/10.1364/OE.18.013836

Acrobat PDF (1168 KB)

### Abstract

We show that it is possible to find and characterise optical vortices in a speckle pattern using a multi-pinhole interferometer. This measurement does not require an additional flat wave front to interfere with the speckle, providing great experimental ease. In addition, a multi-pinhole interferometer can be made arbitrarily large and can therefore be adjusted to the expected speckle size. We present experimental results confirming our understanding.

© 2010 OSA

## 1. Introduction

1. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. **66**, 1145–1150 (1976). [CrossRef]

*ilϕ*), or, in other words, the phase varies in an azimuthal fashion, forming an optical vortex [3

3. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

6. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. **101**, 247–264 (1993). [CrossRef]

7. 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]

8. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. **26**, 163–165 (2001). [CrossRef]

9. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. **94**, 103902 (2005). [CrossRef] [PubMed]

10. K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. **100**, 053902 (2008). [CrossRef] [PubMed]

6. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. **101**, 247–264 (1993). [CrossRef]

11. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A **11**, 1644–1652 (1994). [CrossRef]

13. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. **31**, 2865–2882 (1992). [CrossRef] [PubMed]

14. D. L Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**, 2759–2768 (1998). [CrossRef]

15. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. **597**, 1266–1270 (2003). [CrossRef]

16. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. **492**, 883–922 (2008). [CrossRef]

15. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. **597**, 1266–1270 (2003). [CrossRef]

17. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

18. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

19. R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A **79**, 043809 (2009). [CrossRef]

## 2. Experiment

*x*and

*y*-direction. In this paper we only present the results for a scan in one direction since we believe that this gives the clearest demonstration of the principle.

*N*pinholes uniformly distributed on a circle with radius

*b*. As shown in [17

17. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

*N*≥ 4 and the number of vortex modes that can be detected depends on

*N*. For an odd number of pinholes,

*N*different vortex modes can be distinguished, while this reduces to

*N*/2 + 1 modes for an even number of pinholes as one cannot distinguish between positive and negative values of

*l*in this case. In a speckle pattern, one only expects vortices with

*l*= ±1. For the above reasons, a multi-pinhole interferometer with

*N*= 5 is used in this experiment, which allows detection of optical vortices with

*l*= 0, ±1, ±2.

*X*~ 1mm. The radius of the interferometer is

_{sp}*a*= 100µm, which is much smaller than the average speckle size to reduce the possibility that two vortices impinge the multi-pinhole interferometer at the same time. The diameter of the pinholes is

*b*= 50µm, which is the largest pinhole diameter available to garuantee as much throughput as possible. In this experiment, the translation stage is moved in the

*x*-direction over 3mm and the interference pattern is recorded after every 50µm. Around the positions where the interference pattern changes because of the presence of an optical vortex, the distance between consecutive two measurements is reduced to 10µm to improve the resolution.

## 3. Results

17. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. **101**, 100801 (2008). [CrossRef] [PubMed]

*l*= 0, while Fig. 2 (b) resembles the interference pattern for an optical vortex with

*l*= −1. The orientation of the patterns is determined by the orientation of the multi-pinhole interferometer with respect to the CCD camera. The Fourier transform analysis as presented by Guo et al. allows a quantitative analysis of these interference patterns [18

18. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

*ψ*at the pinholes can be determined from the phase of the Fourier transform of the interference pattern behind a multi-pinhole interferometer. They showed that the relative phases can be extracted from the vertices of a polygon, that is a scaled and shifted copy of the multi-pinhole interferometer. Since we use a multi-pinhole interferometer with five pinholes, the polygon is in fact a pentagon.

*ϕ*denotes the azimuthal angle, and gives a comparable result to Fig. 3 (b) in [18

18. C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**, 231104 (2009). [CrossRef]

*l*= −1. The vorticity of the field impinging the multi-pinhole interferometer is determined by fitting a line through the averaged phase profile and determining the value of the fit at

*ϕ*= 2

*π*.

*x*= 1mm corresponds to the image shown in Fig. 2 (b). From Fig. 4 (b) it is clear that this position can be associated with an optical vortex of topological charge

*l*= −1. The minimum at

*x*= 2.6mm proves to be associated with an optical vortex of topological charge

*l*= +1 (see Fig. 4 (b)). The width of the plateau of both peaks in Fig. 4 is ~ 100µm, which confirms the fact that an optical vortex can be observed as long as its axis lies well within the multi-pinhole interferometer.

## 4. Discussion

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. |

2. | J. W. Goodman, |

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

4. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

5. | N. B. Baranova, V. I. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkunov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Letters |

6. | I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. |

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

8. | G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. |

9. | W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. |

10. | K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. |

11. | I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A |

12. | M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A |

13. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. |

14. | D. L Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

15. | M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. |

16. | N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. |

17. | G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. |

18. | C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. |

19. | R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A |

20. | G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(050.1220) Diffraction and gratings : Apertures

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: April 22, 2010

Revised Manuscript: June 4, 2010

Manuscript Accepted: June 8, 2010

Published: June 11, 2010

**Citation**

Gregorius C. G. Berkhout and Marco W. Beijersbergen, "Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer," Opt. Express **18**, 13836-13841 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13836

Sort: Year | Journal | Reset

### References

- J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976). [CrossRef]
- J. W. Goodman, Speckle phenomena in optics (Roberts & Company, 2006).
- J. F. Nye, and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336, 165–190 (1974). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]
- N. B. Baranova, V. I. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkunov, “Dislocations of the wavefront of a speckle-inhomogeneous fiel (theory and experiment),” JETP Lett. 33, 195–199 (1981).
- I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993). [CrossRef]
- 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]
- G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001). [CrossRef]
- W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005). [CrossRef] [PubMed]
- K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008). [CrossRef] [PubMed]
- I. Freund, “Optical vortices in Gaussian random wave fields statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994). [CrossRef]
- M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A 6, S202–S208 (2004).
- D. L. Fried, and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998). [CrossRef]
- M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597, 1266–1270 (2003). [CrossRef]
- N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492, 883–922 (2008). [CrossRef]
- G. C. G. Berkhout, and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008). [CrossRef] [PubMed]
- C.-S. Guo, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94, 231104 (2009). [CrossRef]
- R. W. Schoonover, and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009). [CrossRef]
- G. C. G. Berkhout, and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A 11, 094021 (2009).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.