## Engineering a square truncated lattice with light's orbital angular momentum |

Optics Express, Vol. 19, Issue 21, pp. 20616-20621 (2011)

http://dx.doi.org/10.1364/OE.19.020616

Acrobat PDF (1266 KB)

### Abstract

We engineer an intensity square lattice using the Fraunhofer diffraction of a Laguerre-Gauss beam by a square aperture. We verify numerically and experimentally that a perfect optical intensity lattice takes place only for even values of the topological charge. We explain the origin of this behavior based on the decomposition of the patterns. We also study the evolution of the lattice formation by observing the transition from one order to the next of the orbital angular momentum varying the topological charge in fractional steps.

© 2011 OSA

## 1. Introduction

3. V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B **12**(11), 2046–2052 (1995). [CrossRef]

5. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. **23**(18), 1444–1446 (1998). [CrossRef] [PubMed]

6. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science **292**(5516), 476–479 (2001). [CrossRef] [PubMed]

7. J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express **15**(8), 5196–5207 (2007). [CrossRef] [PubMed]

9. J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. **207**(1-6), 85–93 (2002). [CrossRef]

10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. **105**(5), 053904 (2010). [CrossRef] [PubMed]

11. R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A **79**(4), 043809 (2009). [CrossRef]

13. 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, Pure Appl. Opt. **11**(9), 094021 (2009). [CrossRef]

10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. **105**(5), 053904 (2010). [CrossRef] [PubMed]

## 2. Theoretical results

*N*and the TC, namely

*m*.

*m*= 5 and

*m*= 6, respectively. Note that the phase has a uniform distribution, mainly in the central region, only in Fig. 2(b). Phase jumps, which form a square shape, are clearly observed in Fig. 2(b). In contrast, in some parts in the center of Fig. 2(a) (in which

*m*has an odd value), the phase distribution is sufficiently smooth that it is impossible to clearly identify the phase jumps. Similar behavior is observed for all phase diagrams for the various values of

*m*, in accordance with the amount and the parity of OAM.

## 3. Experimental setup

*m*, we used square masks with sides varying from 1.8 mm to 3.3 mm for different LG beams. These masks can be superimposed over the LG hologram in the SLM. The effect is similar to an LG beam incident in a square aperture as illustrated in Fig. 3(b). Note that we have carefully aligned the beam at the center of the aperture to avoid asymmetries in the patterns. We have used a hologram type 1 for coding phase and amplitude as first proposed by Kirk et al [15] and first used for LG beams by Leach et al [16

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

## 4. Experimental results and discussion

*m*from 0 to 12, −1, −2, −7, and −8. We observe a well-formed square optical lattice only for even values of

*m*, confirming the numerical results. The sign of

*m*does not change globally the shape of the pattern because the phase is invariant by

*m*= 0 we have the usual square aperture diffraction pattern.

17. U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science **329**(5990), 418–421 (2010). [CrossRef] [PubMed]

*m*from 2 to 6. The overall pattern

17. U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science **329**(5990), 418–421 (2010). [CrossRef] [PubMed]

*m*increases the number of fringes increases and all patterns are shifted as expected [18

18. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. **36**(16), 3106–3108 (2011). [CrossRef] [PubMed]

*m*but without any TC parity dependence. However, for the parallel slit configuration, it can be seen that there are two types of patterns, one for even values of

*m*and the other for odd values of

*m*. Such dependence on the TC parity can be associated to the fact that there is an odd or even multiple of

*m*.

*m*= 5 and

*m*= 6. In the first case, we observe that the intensity peaks in

*m*= 6 the peaks in the center of each pattern coincide. For even values of

*m*this effect produces a shaped optical intensity lattice when all contributions of all patterns are taken into account. For odd values of

*m*this effect smears out the optical lattice.

*m*in increments of 0.1. Note that the use of an azimuthal profile instead of a true LG beam is a good approximation because the potential functions and the Laguerre have approximately the same behavior at the edges of the square. In the panels of Fig. 7 , we see the results of such simulation. It is clear that there is no well-formed lattice pattern between two patterns from two consecutive even values of

*m*for any fractional TC.

## 5. Conclusions

*m*, and there is no intermediate lattice between them for fractional TC.

## Acknowledgments

## References and links

1. | L. M. Pismen, |

2. | Y. S. Kivshar, and G. P. Agrawal, Optical solitons: from fibers to photonic crystals (Academic Press, Amsterdam; Boston, 2003). |

3. | V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B |

4. | V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. |

5. | D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. |

6. | J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science |

7. | J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express |

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

9. | J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun. |

10. | J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. |

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

12. | 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. |

13. | 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, Pure Appl. Opt. |

14. | J. W. Goodman, |

15. | J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B |

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

17. | U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science |

18. | Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1940) Diffraction and gratings : Diffraction

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 11, 2011

Revised Manuscript: September 19, 2011

Manuscript Accepted: September 19, 2011

Published: October 3, 2011

**Citation**

Pedro H. F. Mesquita, Alcenísio J. Jesus-Silva, Eduardo J. S. Fonseca, and Jandir M. Hickmann, "Engineering a square truncated lattice with light's orbital angular momentum," Opt. Express **19**, 20616-20621 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20616

Sort: Year | Journal | Reset

### References

- L. M. Pismen, Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, from Non-Equilibrium Patterns to Cosmic Strings (Oxford University Press, Oxford, 1999).
- Y. S. Kivshar, and G. P. Agrawal, Optical solitons: from fibers to photonic crystals (Academic Press, Amsterdam; Boston, 2003).
- V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B12(11), 2046–2052 (1995). [CrossRef]
- V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett.76(15), 2698–2701 (1996). [CrossRef] [PubMed]
- D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett.23(18), 1444–1446 (1998). [CrossRef] [PubMed]
- J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose-Einstein condensates,” Science292(5516), 476–479 (2001). [CrossRef] [PubMed]
- J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express15(8), 5196–5207 (2007). [CrossRef] [PubMed]
- J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun.198(1-3), 21–27 (2001). [CrossRef]
- J. Masajada, A. Popiolek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Opt. Commun.207(1-6), 85–93 (2002). [CrossRef]
- J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett.105(5), 053904 (2010). [CrossRef] [PubMed]
- R. W. Schoonover and T. D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A79(4), 043809 (2009). [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(10), 100801 (2008). [CrossRef] [PubMed]
- 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, Pure Appl. Opt.11(9), 094021 (2009). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
- J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,”, J. Opt. Soc. Am. B61, 1023–1028 (1971).
- J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys.7, 55 (2005). [CrossRef]
- U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, “Ruling out multi-order interference in quantum mechanics,” Science329(5990), 418–421 (2010). [CrossRef] [PubMed]
- Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett.36(16), 3106–3108 (2011). [CrossRef] [PubMed]

## 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.