## Generation of pseudonondiffracting optical beams with superlattice structures |

Optics Express, Vol. 21, Issue 20, pp. 23441-23449 (2013)

http://dx.doi.org/10.1364/OE.21.023441

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### Abstract

We demonstrate an approach to generate a class of pseudonondiffracting optical beams with the transverse shapes related to the superlattice structures. For constructing the superlattice waves, we consider a coherent superposition of two identical lattice waves with a specific relative angle in the azimuthal direction. We theoretically derive the general conditions of the relative angles for superlattice waves. In the experiment, a mask with multiple apertures which fulfill the conditions for superlattice structures is utilized to generate the pseudonondiffracting superlattice beams. With the analytical wave functions and experimental patterns, the pseudonondiffracting optical beams with a variety of structures can be generated systematically.

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## 1. Introduction

14. M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett. **81**(12), 2450–2453 (1998). [CrossRef]

16. H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett. **84**(23), 5316–5319 (2000). [CrossRef] [PubMed]

17. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. **336**(1605), 165–190 (1974). [CrossRef]

18. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

## 2. Theoretical analysis for forming superlattice waves

12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A **83**(5), 053813 (2011). [CrossRef]

*q*is equal to 3, 4, or 6. Considering the coherent superposition of two identical lattice waves with a relative angle in the azimuthal direction, we can obtain the superposed waves aswhere

*q*= 3, 4, and 6, respectively. The solid and dashed vectors represent

*q*= 4, the wave vectors shown in Fig. 1(b) must be satisfied the following conditionswhere

*q*= 4 with spatial periodicity. With the wave vectors in terms of the reciprocal primitive translation vectors, the reciprocal lattice constant can be given byEquation (7) indicates that the spatial period becomes longer when the value of

*q*= 3 and 6 can be obtained. Since the scalar product of the reciprocal primitive translation vectors is −1/2 in the cases of

*q*= 3 and 6, the condition of the relative angles leads to the equationand the reciprocal lattice constant is given byConsequently, the superlattice waves can be constructed by the superposed waves

*q*= 4. It can be seen that a rich variety of superlattice wave patterns can be constructed by controlling the specific relative angle. The numerical patterns for the intensity of superlattice waves with

*q*= 3 and 6 are shown in Figs. 3(a)-3(c) and Figs. 3(d)-3(f), respectively. These numerical patterns show that the specific relative angle can turn into the main parameter for generating nondiffracting optical superlattice beams. In the following section we present an approach to realize the pseudonondiffracting optical superlattice beams.

## 3. Generation of the pseudonondiffracting optical superlattice beams

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

*z*can be expressed aswhere

*f*is the focal length of the lens. For a pseudonondiffracting Bessel beam, the input field is determined by an infinitesimally thin annulus at

12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A **83**(5), 053813 (2011). [CrossRef]

*R*is the radius of the ring where the apertures are located on. Equation (12) implies that the aperture size must be infinitesimal for generating ideal nondiffracting beams with crystalline structures. For generating nondiffracting superlattice beams, the input field is given bywhere Δ

*satisfies the criteria of superlattice patterns in Eq. (6) or (8). However, an infinitesimal aperture is not realistic. Since the aperture sizes cannot be infinitesimal, the generated beams are called the pseudo-nondiffracting beams. Furthermore, the selection of the aperture size determines how many spatial periods can be included in the pseudonondiffracting superlattice beam. Therefore, the analysis for determining the aperture size is of crucial importance. For considering the effect of the aperture size with finite energy, we exploit the multiple Gaussian beams to model the input field just after the apertures. Based on the locations of the pinholes in Eq. (13), the multiple Gaussian waves for describing the input field is given by*

_{q}*a*. Since the output field in the focal plane behind the lens can be found by the Fourier transform of the input field, the substitution of Eq. (14) into Eq. (10) and considering

*z = f*lead to an equation for the output fieldWith transformation of coordinates from polar coordinates to Cartesian coordinates, Eq. (15) can be an analytical integration by utilizing Gaussian integral:By some algebraic operation, the output field in polar coordinates can be derived asIt can be seen that the terms in the summation represent the superlattice waves with

*q*= 4 observed in the experiment under the condition of the optimal alignment. It can be seen that the experimental observations agree very well with the numerical patterns shown in Fig. 2(a)-2(c). The experimental patterns reveal that the relation between the reciprocal lattice constant and the spatial period is satisfied in our theory. Figures 7(a)-7(c) and Figs. 7(d)-7(f) illustrate the experimental results for pseudonondiffracting optical superlattice patterns with

*q*= 3 and 6, respectively. Once again, the experimental results match very well the numerical calculations depicted in Figs. 3(a)-3(c) and Figs. 3(d)-2(f), respectively. It can be experimentally observed that there are honeycomb strucutres in some optical superlattice pattern with

*q*= 3, as shown in Figs. 7(b) and 7(c). Moreover, in Fig. 7(f), the optical superlattice pattern with

*q*= 6 displays the exotic kaleidoscopic structure. The excellent agreement validates the theoretical analysis of superlattice waves and confirms the experimental approach. The experimental patterns also confirm our analysis of the influence of the aperture size on the transverse unit cell.

*q*= 3.

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

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

4. | D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. |

5. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

6. | Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. |

7. | C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun. |

8. | Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys. |

9. | M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A |

10. | M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. |

11. | P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. |

12. | Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A |

13. | A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D |

14. | M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett. |

15. | H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett. |

16. | H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett. |

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

18. | M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. |

**OCIS Codes**

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

(050.4865) Diffraction and gratings : Optical vortices

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 8, 2013

Revised Manuscript: September 16, 2013

Manuscript Accepted: September 17, 2013

Published: September 25, 2013

**Citation**

C. H. Tsou, T. W. Wu, J. C. Tung, H. C. Liang, P. H. Tuan, and Y. F. Chen, "Generation of pseudonondiffracting optical beams with superlattice structures," Opt. Express **21**, 23441-23449 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23441

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### References

- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002). [CrossRef] [PubMed]
- D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003). [CrossRef] [PubMed]
- J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001). [CrossRef]
- Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett.27(4), 243–245 (2002). [CrossRef] [PubMed]
- C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000). [CrossRef]
- Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003). [CrossRef]
- M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011). [CrossRef]
- M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011). [CrossRef]
- P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012). [CrossRef]
- Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011). [CrossRef]
- A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).
- M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998). [CrossRef]
- H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998). [CrossRef]
- H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000). [CrossRef] [PubMed]
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974). [CrossRef]
- M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001). [CrossRef]

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