## Generation of finite power Airy beams via initial field modulation |

Optics Express, Vol. 21, Issue 16, pp. 18797-18804 (2013)

http://dx.doi.org/10.1364/OE.21.018797

Acrobat PDF (2827 KB)

### Abstract

We investigate the finite power Airy beams generated by finite extent input beams such as a Gaussian beam, a uniform beam of finite extent, and an inverse Gaussian beam. Each has different propagation behavior: A finite Airy beam generated by a uniform input beam keeps its Airy profile much longer than the conventional finite Airy beam. Also, an inverse Gaussian beam generates a finite Airy beam with a good bent focusing in free space. In this paper, the analysis and experimental results of finite Airy beams are presented.

© 2013 OSA

## 1. Introduction

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

3. 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]

18. C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. **4**(1), 174–180 (2012). [CrossRef]

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

20. 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]

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

3. 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]

*apodized*finite power Airy beams cannot maintain their shapes for a long time and are gradually spreading out during propagation [3

3. 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]

21. Y. Jiang, K. Huang, and X. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A **29**(7), 1412–1416 (2012). [CrossRef] [PubMed]

22. S. Barwick, “Reduced side-lobe Airy beams,” Opt. Lett. **36**(15), 2827–2829 (2011). [CrossRef] [PubMed]

23. J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. **38**(10), 1639–1641 (2013). [CrossRef]

## 2. Theoretical analysis of Airy beams

*s*and

*ξ*denote the transverse coordinate

*x*scaled by an arbitrary scaling factor

*x*

_{0}and the longitudinal coordinate

*z*scaled by

*k*

_{n}x_{0}

^{2}, respectively, where

*k*( = 2

_{n}*πn*/

*λ*) is a wavenumber in a medium with a refractive index

*n*, and

*λ*is the wavelength of light in free space. By solving Eq. (1), the Airy beam solution

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

_{0}of Eq. (2) at

*ξ*= 0 is given byAs was mentioned in the introduction, Eq. (2) is not square integrable and thus

*as*) is multiplied to

_{1}becomesBased on Eq. (5) and its optical Fourier transform, the first observation of finite Airy beams was conducted: the cubic phase [

**99**(21), 213901 (2007). [CrossRef] [PubMed]

*a*,

*ak*

^{2})] which is originated from the apodization, i.e., exponentially decaying term exp(

*as*). Therefore, Eq. (3) can be understood as a plane wave (of infinite extent) with a cubic phase modulation, suggesting that the finite-extent feature of the input beam results in the finite power Airy beam.

*CASE*I), a uniform distribution of finite extent (

*CASE*II), and an inverse Gaussian distribution (

*CASE*III). Throughout this paper, we assume that the wavelength of an incident beam

*λ*is 633 nm,

*x*

_{0}is 50 μm, the focal length of the lens

*f*is 50 cm, and the SLM has 1080 × 1080 pixels with an 8 μm pixel pitch.

*CASE*I, which adopts a Gaussian beam as an input beam. Due to the finite SLM size, the incident Gaussian beam is truncated. Therefore, Eq. (5) becomeswhere

*l*is the length of the SLM along the one dimension given by the product of the pixel pitch and the number of pixels. The propagation dynamics of the

*CASE*I Airy beam can now be written as follows using the Fresnel diffraction form of Eq. (4):where the value of

*a*is chosen to be 0.1 throughout this paper. Calculation results of Eq. (7) are plotted in Fig. 1(a). In this case, the number of side lobes in the initial plane (

*z*= 0 cm) is decreased because the Gaussian distribution of the input beam cannot fully retain the high spatial frequency components. As a result, unlike the ideal case shown in Fig. 1(d), the Airy beam is diffracted or spreads out due to insufficient power flows from the side lobes.

*CASE*II), In this case, we have

*CASE*II Airy beam can be expressed as:and its calculation results are presented in Fig. 1(b). In Fig. 1(b), we can observe more side lobes at the initial plane compared with the

*CASE*I. Figure 1(b) also shows that the incidence of a truncated plane wave and its Fourier transform after the cubic phase modulation can generate a finite power Airy beam which takes up a bending trajectory with the acceleration toward +

*x*direction. What is interesting is that this finite Airy beam can preserve its Airy profile much farther than that generated by the Gaussian input beam. This is because more high-frequency components are retained in Eq. (8) than in Eq. (6).

*CASE*III, we use a beam with an inverse Gaussian intensity distribution as an input beam. We haveIn this case, whose results are shown in Fig. 1(c), the main lobe is suppressed at the initial plane because the inverse Gaussian distribution can retain only high spatial frequency components. However, as can be found in Fig. 1(c), the resultant finite power beam generates the main lobe after some propagation distance and is accelerated along the +

*x*direction. That is, Eq. (11) also describes a finite power Airy beam. Actually, this case can be taken as an extreme example of the self-healing property [7

7. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**(17), 12880–12891 (2008). [CrossRef] [PubMed]

*z*. In Fig. 2, we plotted the variation of the maximum value of this cross-correlation, i.e.,

*C*

_{1}(

*z*) defined byGreen dashed, blue solid, and red dash-dotted lines correspond to the

*CASE*I, II, and III, respectively. The cross-correlation values in Fig. 2 are normalized by those at the initial plane (

*z*= 0 cm). For the

*CASE*I and II, the cross-correlation becomes maximum at the initial plane and gradually decreases. This means that the finite power Airy beams spread out or are diffracted so that they lose their initial Airy shape during propagation. However, we would like to point out that the cross-correlation value of the

*CASE*II Airy beam is always higher than that of the

*CASE*I at every

*z*. This means that the

*CASE*II Airy beam can preserve its original shape much farther than the

*CASE*I Airy beam. In the

*CASE*III, however, the cross-correlation increases after some propagation distance. This indicates the recovery process of the Airy profile: the regeneration of the main lobe and its acceleration along the transverse coordinate.

*CASE*I, II, and III Airy beams, respectively, where

*l*and

_{x}*l*are the horizontal and vertical lengths of the SLM and

_{y}*y*

_{0}is an arbitrary scaling factor along the

*y*coordinate.

*CASE*I), Figs. 3(c) and 3(d) (

*CASE*II), and Figs. 3(e) and 3(f) (

*CASE*III). Figures 3(a), 3(c), 3(e) and 3(b), 3(d), 3(f) are the results at

*z*= 0 cm and

*z*= 15 cm, respectively. Comparing Figs. 3(a), 3(c) and 3(e), we can find more side lobes in Fig. 3(c) than in Fig. 3(a) while the main lobe disappears in Fig. 3(e). After some propagation (

*z*= 15 cm), the finite power Airy beam generated by a Gaussian beam (

*CASE*I) does not preserve its initial Airy profile anymore. On the other hand, the finite power Airy beam generated by a truncated plane wave (or a uniform beam of finite extent;

*CASE*II) maintains its initial profile although the beam is broadened due to the diffraction. Therefore, the uniform beam of the

*CASE*II is more advantageous than the Gaussian beam of the

*CASE*I. In the case of the inverse Gaussian beam (

*CASE*III), although the main lobe is missing at the initial plane (Fig. 3(e)), side lobes recover the main lobe after some propagations as can be found in Fig. 3(f).

## 3. Experiments of the finite power Airy beams

24. J. Hahn, H. Kim, and B. Lee, “Optimization of the spatial light modulation with twisted nematic liquid crystals by a genetic algorithm,” Appl. Opt. **47**(19), D87–D95 (2008). [CrossRef] [PubMed]

*CASE*I), a uniform distribution of finite extent (

*CASE*II), or an inverse Gaussian distribution (

*CASE*III). Passing these beams through the optical Fourier transform (2-

*f*) system, we can obtain finite power Airy beams at the initial plane (

*z*= 0 cm). The optical Fourier transform system consists of a lens (L1;

*f =*50 cm) and a phase-only SLM 2 (Holoeye Pluto with 1920 × 1080 pixels of 8 μm pixel pitch) which is placed in front of the lens and imposes cubic (

*k*

^{3}) phase. The same cubic phase mask is used to all

*CASES*because the higher-order phase terms can be ignored due to the relatively small constant

*a*. Another polarizer P3 is used to attenuate the output beam so that a twin image can be eliminated.

*z*= 0 cm and

*z*= 15 cm and shown in Figs. 5(a) - 5(f). All images are obtained under the same conditions that no adjustments in intensities were made. From these results, we can conclude that the

*CASE*II Airy beam [Figs. 5(c) and 5(d)] retains the Airy profile much longer than the

*CASE*I Airy beam [Figs. 5(a) and 5(b)]. Meanwhile, the

*CASE*III Airy beam recovers the main lobe at

*z*= 15 cm as shown in Figs. 5(e) and 5(f). These experimental results coincide well with the calculation results shown in Figs. 3(a) - 3(f).

## 4. Conclusion

*CASE*I), a uniform beam of finite extent (

*CASE*II), and an inverse Gaussian beam (

*CASE*III) were discussed. The propagation dynamics of resultant finite power Airy beams were analyzed and compared. We showed both theoretically and experimentally that the finite Airy beam generated by the use of a uniform input beam (

*CASE*II) retains the Airy profile much longer than the conventional finite Airy beam (

*CASE*I). Also, the finite Airy beam via an inverse Gaussian beam (

*CASE*III) builds up a focused-bending beam. We expect that our works in this paper can be utilized to particle tweezing and optical trapping [4

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

6. Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. **50**(1), 43–49 (2011). [CrossRef] [PubMed]

## Acknowledgment

## References and links

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

2. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

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

5. | D. N. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photonics |

6. | Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. |

7. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

8. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

9. | T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics |

10. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

11. | C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express |

12. | A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. |

13. | A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. |

14. | Y. Lim, S.-Y. Lee, K.-Y. Kim, J. Park, and B. Lee, “Negative refraction of Airy plasmons in a metal–insulator–metal waveguide,” IEEE Photon. Technol. Lett. |

15. | D. Choi, Y. Lim, I.-M. Lee, S. Roh, and B. Lee, “Airy beam excitation using a subwavelength metallic slit array,” IEEE Photon. Technol. Lett. |

16. | K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express |

17. | C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Bessel-like beam generation by superposing multiple Airy beams,” Opt. Express |

18. | C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J. |

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

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

21. | Y. Jiang, K. Huang, and X. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A |

22. | S. Barwick, “Reduced side-lobe Airy beams,” Opt. Lett. |

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

24. | J. Hahn, H. Kim, and B. Lee, “Optimization of the spatial light modulation with twisted nematic liquid crystals by a genetic algorithm,” Appl. Opt. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(140.3300) Lasers and laser optics : Laser beam shaping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 3, 2013

Revised Manuscript: July 26, 2013

Manuscript Accepted: July 27, 2013

Published: July 31, 2013

**Citation**

Dawoon Choi, Kyookeun Lee, Keehoon Hong, Il-Min Lee, Kyoung-Youm Kim, and Byoungho Lee, "Generation of finite power Airy beams via initial field modulation," Opt. Express **21**, 18797-18804 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-18797

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
- 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]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008). [CrossRef]
- D. N. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photonics2(11), 652–653 (2008). [CrossRef]
- Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt.50(1), 43–49 (2011). [CrossRef] [PubMed]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324(5924), 229–232 (2009). [CrossRef] [PubMed]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3(7), 395–398 (2009). [CrossRef]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010). [CrossRef]
- C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express18(22), 23504–23516 (2010). [CrossRef] [PubMed]
- A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35(12), 2082–2084 (2010). [CrossRef] [PubMed]
- A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107(11), 116802 (2011). [CrossRef] [PubMed]
- Y. Lim, S.-Y. Lee, K.-Y. Kim, J. Park, and B. Lee, “Negative refraction of Airy plasmons in a metal–insulator–metal waveguide,” IEEE Photon. Technol. Lett.23(17), 1258–1260 (2011). [CrossRef]
- D. Choi, Y. Lim, I.-M. Lee, S. Roh, and B. Lee, “Airy beam excitation using a subwavelength metallic slit array,” IEEE Photon. Technol. Lett.24(16), 1440–1442 (2012). [CrossRef]
- K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express19(3), 2286–2293 (2011). [CrossRef] [PubMed]
- C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Bessel-like beam generation by superposing multiple Airy beams,” Opt. Express19(8), 7356–7364 (2011). [CrossRef] [PubMed]
- C.-Y. Hwang, K.-Y. Kim, and B. Lee, “Dynamic control of circular Airy beams with linear optical potentials,” IEEE Photon. J.4(1), 174–180 (2012). [CrossRef]
- J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- 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]
- Y. Jiang, K. Huang, and X. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A29(7), 1412–1416 (2012). [CrossRef] [PubMed]
- S. Barwick, “Reduced side-lobe Airy beams,” Opt. Lett.36(15), 2827–2829 (2011). [CrossRef] [PubMed]
- J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett.38(10), 1639–1641 (2013). [CrossRef]
- J. Hahn, H. Kim, and B. Lee, “Optimization of the spatial light modulation with twisted nematic liquid crystals by a genetic algorithm,” Appl. Opt.47(19), D87–D95 (2008). [CrossRef] [PubMed]

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