## Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge |

JOSA A, Vol. 31, Issue 6, pp. 1295-1302 (2014)

http://dx.doi.org/10.1364/JOSAA.31.001295

Acrobat PDF (951 KB)

### Abstract

We embed a pair of vortices with different topological charges in a Gaussian beam and study its evolution through an astigmatic optical system, a tilted lens. The propagation dynamics are explained by a closed-form analytical expression. Furthermore, we show that a careful examination of the intensity distribution at a predicted position past the lens can determine the charge present in the beam. To the best of our knowledge, our method is the first noninterferometric technique to measure the charge of an arbitrary vortex pair. Our theoretical results are well supported by experimental observations.

© 2014 Optical Society of America

## 1. INTRODUCTION

1. 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]

2. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef]

4. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef]

5. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science **340**, 1545–1548 (2013). [CrossRef]

7. G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. **3**, 305–310 (2007). [CrossRef]

8. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

10. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. **76**, 916–921 (2008). [CrossRef]

11. J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science **285**, 230–233 (1999). [CrossRef]

12. R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A **3**, 527–532 (2001). [CrossRef]

13. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. **39**, 985–990 (1992). [CrossRef]

19. H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik **124**, 4201–4205 (2013). [CrossRef]

*et al.*have experimentally generated the pair of vortices in a single beam by using diffraction gratings for the first time [13

13. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. **39**, 985–990 (1992). [CrossRef]

14. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. **40**, 73–87 (1993). [CrossRef]

17. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A **25**, 1279–1286 (2008). [CrossRef]

18. Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A **375**, 2958–2963 (2011). [CrossRef]

19. H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik **124**, 4201–4205 (2013). [CrossRef]

19. H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik **124**, 4201–4205 (2013). [CrossRef]

20. D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. **46**, 419–423 (2008). [CrossRef]

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A **377**, 1154–1156 (2013). [CrossRef]

28. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express **16**, 4991–4999 (2008). [CrossRef]

29. A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A **86**, 013825 (2012). [CrossRef]

30. A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. **35**, 3841–3843 (2010). [CrossRef]

31. A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express **19**, 6182–6190 (2011). [CrossRef]

34. 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]

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

36. J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. **91**, 244302 (2003). [CrossRef]

## 2. THEORY

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A **377**, 1154–1156 (2013). [CrossRef]

**124**, 4201–4205 (2013). [CrossRef]

27. P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A **377**, 1154–1156 (2013). [CrossRef]

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. **344**, 408–416 (2008). [CrossRef]

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. **344**, 408–416 (2008). [CrossRef]

### A. Determination of Net Topological Charge

**377**, 1154–1156 (2013). [CrossRef]

#### 1. Vortices with Topological Charges of the Same Sign

#### 2. Vortices with Topological Charges of Opposite Signs

### B. Propagation Dynamics Away from z = z c

39. S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. **344**, 408–416 (2008). [CrossRef]

42. G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. **284**, 447–454 (2003). [CrossRef]

### C. Propagation Dynamics When the Lens Is Not Tilted (θ = 0 )

## 3. EXPERIMENT

10. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. **76**, 916–921 (2008). [CrossRef]

## 4. INTENSITY PATTERN AT z = z c AND DETERMINATION OF NET TOPOLOGICAL CHARGE

### A. Vortices with Topological Charges of the Same Sign

### B. Vortices with Topological Charges of Opposite Signs

*and*

## 5. PROPAGATION DYNAMICS AWAY FROM z = z c

**377**, 1154–1156 (2013). [CrossRef]

## 6. CONCLUSIONS

## REFERENCES

1. | 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 |

2. | K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. |

3. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

4. | D. G. Grier, “A revolution in optical manipulation,” Nature |

5. | N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science |

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

7. | G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

8. | L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. |

9. | S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. |

10. | A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. |

11. | J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science |

12. | R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A |

13. | V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. |

14. | G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. |

15. | F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. |

16. | F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B |

17. | M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A |

18. | Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A |

19. | H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik |

20. | D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. |

21. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. |

22. | A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, “Information content of optical vortex fields,” Opt. Lett. |

23. | P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. |

24. | S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. |

25. | J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. |

26. | A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express |

27. | P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A |

28. | M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express |

29. | A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A |

30. | A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. |

31. | A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express |

32. | A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. |

33. | A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. |

34. | 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. |

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

36. | J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. |

37. | S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B |

38. | A. E. Siegman, |

39. | S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. |

40. | G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in |

41. | W. Magnus, F. Oberhettinger, and R. P. Soni, |

42. | G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. |

43. | R. S. Sirohi, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 6, 2014

Revised Manuscript: April 14, 2014

Manuscript Accepted: April 22, 2014

Published: May 22, 2014

**Citation**

Salla Gangi Reddy, Shashi Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, "Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge," J. Opt. Soc. Am. A **31**, 1295-1302 (2014)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-6-1295

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### References

- 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]
- K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]
- N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]
- L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
- G. M. Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]
- L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]
- S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000). [CrossRef]
- A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]
- J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef]
- R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A 3, 527–532 (2001). [CrossRef]
- V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]
- G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]
- F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236, 433–440 (2004). [CrossRef]
- F. S. Roux, “Canonical vortex dipole dynamics,” J. Opt. Soc. Am. B 21, 655–663 (2004). [CrossRef]
- M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25, 1279–1286 (2008). [CrossRef]
- Z. Chen, J. Pu, and D. Zhao, “Tight focusing properties of linearly polarized Gaussian beam with a pair of vortices,” Phys. Lett. A 375, 2958–2963 (2011). [CrossRef]
- H. Chen, Z. Gao, H. Yang, F. Wang, and X. Huang, “Propagation of a pair of vortices through a tilted lens,” Optik 124, 4201–4205 (2013). [CrossRef]
- D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Opt. Lasers Eng. 46, 419–423 (2008). [CrossRef]
- J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). [CrossRef]
- A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, “Information content of optical vortex fields,” Opt. Lett. 36, 1161–1163 (2011). [CrossRef]
- P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37, 1301–1303 (2012). [CrossRef]
- S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36, 4398–4400 (2011). [CrossRef]
- J. Hickmann, E. Fonseca, W. Soares, and S. Chvez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010). [CrossRef]
- A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011). [CrossRef]
- P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013). [CrossRef]
- M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16, 4991–4999 (2008). [CrossRef]
- A. Kumar, J. Banerji, and R. P. Singh, “Hanbury BrownTwiss-type experiments with optical vortices and observation of modulated intensity correlation on scattering from rotating ground glass,” Phys. Rev. A 86, 013825 (2012). [CrossRef]
- A. Kumar, J. Banerji, and R. P. Singh, “Intensity correlation properties of high-order optical vortices passing through a rotating ground-glass plate,” Opt. Lett. 35, 3841–3843 (2010). [CrossRef]
- A. Kumar, P. Vaity, and R. P. Singh, “Crafting the core asymmetry to lift the degeneracy of optical vortices,” Opt. Express 19, 6182–6190 (2011). [CrossRef]
- A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focussing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004). [CrossRef]
- A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008). [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]
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]
- J.-L. Thomas and R. Marchiano, “Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices,” Phys. Rev. Lett. 91, 244302 (2003). [CrossRef]
- S. Prabhakar, R. P. Singh, S. Gautam, and D. Angom, “Annihilation of vortex dipoles in an oblate Bose-Einstein condensate,” J. Phys. B 46, 125302 (2013).
- A. E. Siegman, Lasers (University Science Books, 1986).
- S. Khan, M. A. Pathan, N. A. M. Hassan, and G. Yasmin, “Implicit summation formulae for Hermite and related polynomials,” J. Math. Anal. Appl. 344, 408–416 (2008). [CrossRef]
- G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a byproduct of the monomiality principle,” in Advanced Special Functions and Applications (Melfi, 1999), pp. 147–164.
- W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer-Verlag, 1966).
- G. Dattoli, “Incomplete 2D Hermite polynomials: properties and applications,” J. Math. Anal. Appl. 284, 447–454 (2003). [CrossRef]
- R. S. Sirohi, A Course of Experiments with He-Ne Lasers (New Age International, 1991).

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