## Generation of arbitrary complex quasi-non-diffracting optical patterns |

Optics Express, Vol. 21, Issue 19, pp. 22221-22231 (2013)

http://dx.doi.org/10.1364/OE.21.022221

Acrobat PDF (1333 KB)

### Abstract

Due to their unique ability to maintain an intensity distribution upon propagation, non-diffracting light fields are used extensively in various areas of science, including optical tweezers, nonlinear optics and quantum optics, in applications where complex transverse field distributions are required. However, the number and type of rigorously non-diffracting beams is severely limited because their symmetry is dictated by one of the coordinate system where the Helmholtz equation governing beam propagation is separable. Here, we demonstrate a powerful technique that allows the generation of a rich variety of quasi-non-diffracting optical beams featuring nearly arbitrary intensity distributions in the transverse plane. These can be readily engineered via modifications of the angular spectrum of the beam in order to meet the requirements of particular applications. Such beams are not rigorously non-diffracting but they maintain their shape over large distances, which may be tuned by varying the width of the angular spectrum. We report the generation of unique spiral patterns and patterns involving arbitrary combinations of truncated harmonic, Bessel, Mathieu, or parabolic beams occupying different spatial domains. Optical trapping experiments illustrate the opto-mechanical properties of such beams.

© 2013 OSA

## 1. Introduction

1. M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev. **4**(4), 529–547 (2010). [CrossRef]

2. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A **60**(3), 2438–2441 (1999). [CrossRef]

3. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**(2), 023902 (2003). [CrossRef] [PubMed]

8. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. **52**, 63–148 (2009). [CrossRef]

9. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. **78**(1), 179–215 (2006). [CrossRef]

11. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. **86**(22), 5188–5191 (2001). [CrossRef] [PubMed]

12. D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett. **93**(15), 150501 (2004). [CrossRef] [PubMed]

13. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. **1**(1), 23–30 (2005). [CrossRef]

14. S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys. **80**(4), 1215–1274 (2008). [CrossRef]

15. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

16. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

17. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

19. J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express **18**(16), 17130–17140 (2010). [CrossRef] [PubMed]

20. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature **394**(6691), 348–350 (1998). [CrossRef]

21. J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express **12**(1), 220–226 (2004). [CrossRef] [PubMed]

22. P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys. **114**(17), 7570–7573 (2001). [CrossRef]

23. A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A. **99**(25), 16024–16028 (2002). [CrossRef] [PubMed]

25. D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics **2**(11), 682–692 (2009). [CrossRef] [PubMed]

26. C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol. **1**(2), 130–136 (2000). [CrossRef] [PubMed]

27. T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita Jr., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature **377**(6546), 251–254 (1995). [CrossRef] [PubMed]

15. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

7. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. **463**(1-3), 1–126 (2008). [CrossRef]

8. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. **52**, 63–148 (2009). [CrossRef]

28. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

29. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651 (1987). [CrossRef]

1. M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev. **4**(4), 529–547 (2010). [CrossRef]

28. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

29. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651 (1987). [CrossRef]

31. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**(20), 1493–1495 (2000). [CrossRef] [PubMed]

32. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. **195**(1-4), 35–40 (2001). [CrossRef]

33. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**(1), 44–46 (2004). [CrossRef] [PubMed]

34. C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express **13**(7), 2364–2369 (2005). [CrossRef] [PubMed]

35. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264 (1979). [CrossRef]

36. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

37. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett. **105**(1), 013902 (2010). [CrossRef] [PubMed]

38. Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express **19**(10), 9505–9511 (2011). [CrossRef] [PubMed]

39. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. **35**(8), 1215–1217 (2010). [CrossRef] [PubMed]

37. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett. **105**(1), 013902 (2010). [CrossRef] [PubMed]

## 2. Construction of quasi-non-diffracting beams

*truly*non-diffracting beam propagating parallel to the

34. C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express **13**(7), 2364–2369 (2005). [CrossRef] [PubMed]

*any*functional shape of infinitely narrow angular spectrum

40. J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. **10**(1), 015009 (2008). [CrossRef]

41. D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett. **32**(3), 298–300 (2007). [CrossRef] [PubMed]

37. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett. **105**(1), 013902 (2010). [CrossRef] [PubMed]

38. Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express **19**(10), 9505–9511 (2011). [CrossRef] [PubMed]

42. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**(15), 2758–2769 (1982). [CrossRef] [PubMed]

44. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**(10), 7264–7278 (2008). [CrossRef] [PubMed]

45. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**(2), 289–298 (2005). [CrossRef] [PubMed]

46. C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. **45**(6), 068001 (2006). [CrossRef]

## 3. Experimental generation of quasi-non-diffracting patterns

47. R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A **15**(8), 2146–2155 (1998). [CrossRef]

48. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A **17**(2), 294–303 (2000). [CrossRef] [PubMed]

49. Z. K. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. **27**(16), 1376–1378 (2002). [CrossRef] [PubMed]

39. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. **35**(8), 1215–1217 (2010). [CrossRef] [PubMed]

## 4. Angular spectrum of quasi-non-diffracting patterns and their diffraction rate

51. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. **13**(2), 79–80 (1988). [CrossRef] [PubMed]

## 5. Self-healing properties and complex quasi-non-diffracting optical patterns

52. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. **122**(4-6), 169–177 (1996). [CrossRef]

54. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A **28**(5), 837–843 (2011). [CrossRef] [PubMed]

## 6. Trapping of micro-particles

15. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

16. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

## 7. Conclusions

55. X. D. He, P. Xu, J. Wang, and M. S. Zhan, “Rotating single atoms in a ring lattice generated by a spatial light modulator,” Opt. Express **17**(23), 21007–21014 (2009). [CrossRef] [PubMed]

56. S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. **2**, 692 (2012). [CrossRef] [PubMed]

57. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature **467**(7313), 301–304 (2010). [CrossRef] [PubMed]

59. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature **494**(7437), 331–335 (2013). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev. |

2. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A |

3. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

4. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

5. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

6. | H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. |

7. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

8. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. |

9. | O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. |

10. | L. Pitaevskii and S. Stringari, |

11. | R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. |

12. | D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett. |

13. | I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. |

14. | S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys. |

15. | K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics |

16. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

17. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

18. | J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip |

19. | J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express |

20. | M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature |

21. | J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express |

22. | P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys. |

23. | A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A. |

24. | D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express |

25. | D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics |

26. | C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol. |

27. | T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita Jr., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature |

28. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. |

29. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

30. | E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys. |

31. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

32. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. |

33. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

34. | C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express |

35. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

36. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. |

37. | S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett. |

38. | Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express |

39. | C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. |

40. | J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt. |

41. | D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett. |

42. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

43. | Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. |

44. | M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express |

45. | J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A |

46. | C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. |

47. | R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A |

48. | R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A |

49. | Z. K. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. |

50. | J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. |

51. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. |

52. | R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. |

53. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

54. | S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A |

55. | X. D. He, P. Xu, J. Wang, and M. S. Zhan, “Rotating single atoms in a ring lattice generated by a spatial light modulator,” Opt. Express |

56. | S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. |

57. | J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature |

58. | M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature |

59. | N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(070.3185) Fourier optics and signal processing : Invariant optical fields

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 5, 2013

Revised Manuscript: August 30, 2013

Manuscript Accepted: September 2, 2013

Published: September 12, 2013

**Citation**

Antonio Ortiz-Ambriz, Servando Lopez-Aguayo, Yaroslav V. Kartashov, Victor A. Vysloukh, Dmitri Petrov, Hipolito Garcia-Gracia, Julio C. Gutiérrez-Vega, and Lluis Torner, "Generation of arbitrary complex quasi-non-diffracting optical patterns," Opt. Express **21**, 22221-22231 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22221

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### References

- M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010). [CrossRef]
- J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999). [CrossRef]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003). [CrossRef] [PubMed]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature422(6928), 147–150 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett.28(9), 710–712 (2003). [CrossRef] [PubMed]
- H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004). [CrossRef] [PubMed]
- F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009). [CrossRef]
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