## Spiral autofocusing Airy beams carrying power-exponent-phase vortices |

Optics Express, Vol. 22, Issue 7, pp. 7598-7606 (2014)

http://dx.doi.org/10.1364/OE.22.007598

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### Abstract

We propose a new type of noncanonical optical vortex, named “power-exponent-phase vortex (PEPV)”. The spiral focusing of the autofocusing Airy beams carrying PEPVs are experimentally demonstrated, and the physical mechanism is theoretically analyzed by using the energy flow and far field mapping. In addition, the influences of the parameters of PEPVs on the focal fields and orbital angular momenta are also discussed. It is expected that the proposed PEPVs and the corresponding conclusions can be useful for the extension applications of optical vortices, especially for particle trapping and rotating.

© 2014 Optical Society of America

## 1. Introduction

1. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. **35**(23), 4045–4047 (2010). [CrossRef] [PubMed]

9. J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express **20**(12), 13302–13310 (2012). [CrossRef] [PubMed]

1. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. **35**(23), 4045–4047 (2010). [CrossRef] [PubMed]

3. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. **36**(10), 1842–1844 (2011). [CrossRef] [PubMed]

8. S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. **38**(14), 2416–2418 (2013). [CrossRef] [PubMed]

10. P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. **4**, 2622 (2013). [CrossRef] [PubMed]

5. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. **36**(15), 2883–2885 (2011). [CrossRef] [PubMed]

11. Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle,” Opt. Express **21**(20), 24413–24421 (2013). [CrossRef] [PubMed]

12. 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**(11), 8185–8189 (1992). [CrossRef] [PubMed]

15. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

16. M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**(6), 343–348 (2011). [CrossRef]

18. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. **21**(11), 827–829 (1996). [CrossRef] [PubMed]

19. X. Gan, J. Zhao, S. Liu, and L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett. **7**(12), 1142–1145 (2009). [CrossRef]

21. isX. Gan, P. Zhang, S. Liu, F. Xiao, and J. Zhao, “Beam steering and topological transformations driven by interactions between a discrete vortex soliton and a discrete fundamental soliton,” Phys. Rev. A **89**(1), 013844 (2014). [CrossRef]

22. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. **26**(3), 163–165 (2001). [CrossRef] [PubMed]

23. G.-H. Kim, H. J. Lee, J.-U. Kim, and H. Suk, “Propagation dynamics of optical vortices with anisotropic phase profiles,” J. Opt. Soc. Am. B **20**(2), 351–360 (2003). [CrossRef]

24. N. Hermosa, C. Rosales-Guzmán, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. **38**(3), 383–385 (2013). [CrossRef] [PubMed]

26. N. P. Hermosa II and C. O. Manaois, “Phase structure of helicon-conical optical beams,” Opt. Commun. **271**(1), 178–183 (2007). [CrossRef]

27. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express **16**(2), 993–1006 (2008). [CrossRef] [PubMed]

28. S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express **13**(20), 7726–7731 (2005). [CrossRef] [PubMed]

29. H. Li and J. Yin, “Generation of a vectorial Mathieu-like hollow beam with a periodically rotated polarization property,” Opt. Lett. **36**(10), 1755–1757 (2011). [CrossRef] [PubMed]

30. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. **28**(11), 872–874 (2003). [CrossRef] [PubMed]

6. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express **20**(17), 18579–18584 (2012). [CrossRef] [PubMed]

9. J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express **20**(12), 13302–13310 (2012). [CrossRef] [PubMed]

6. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express **20**(17), 18579–18584 (2012). [CrossRef] [PubMed]

8. S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. **38**(14), 2416–2418 (2013). [CrossRef] [PubMed]

## 2. Power-exponent-phase vortex (PEPV)

*ψ*), where the phase function is expressed bywhere,

*θ*is the azimuthal angle, ranging from 0 to 2π;

*l*is the topological charge, which determines the number of 2π-phase shifts that occurs across one revolution of

*θ*, and the sign of

*l*determines the handedness of helix.

*θ*, as shown in Fig. 1(a). Now we construct a new type of noncanonical OV by writing the phase function aswhere,

*l*still denotes the topological charge of the vortex, and

*n*determines the power order of the spiral phase, which can be either an integer or a fraction. Such an OV can be named as PEPV. Figure 1(b) shows a PEPV with

*l =*7 and

*n =*4. In contrast to the canonical vortex, the phase variation of PEPV with

*θ*is gradually intensified due to the power-exponent phase term. To introduce PEPV to the incident beam, the most convenient way is to imprint the phase directly by using a phase spatial light modulator (PSLM) to form a helical phase plate, the same way as that has been done for noncanonical OVs [24

24. N. Hermosa, C. Rosales-Guzmán, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. **38**(3), 383–385 (2013). [CrossRef] [PubMed]

25. C.-A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**(5), 1749–1760 (2005). [CrossRef] [PubMed]

## 3. AABs carrying PEPVs

*r,θ*) denote the polar coordinates,

*a*is the decaying parameter,

*r*

_{0}and

*ω*are the radius and width of the main lobe of AABs, respectively;

*ψ*denotes the phase function given by Eq. (2).

8. S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. **38**(14), 2416–2418 (2013). [CrossRef] [PubMed]

19. X. Gan, J. Zhao, S. Liu, and L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett. **7**(12), 1142–1145 (2009). [CrossRef]

*u*(

*r,θ*) + exp(i

*fx*)|

^{2}, where

*f*determines the spatial frequency of the CGH. After passing through the PSLM, the expanded beam is filtered via a 4f system consisting of lenses L1 and L2 and forms the desired AABs. Placing the CCD at different distances, we observe the propagation characteristics of AABs carrying PEPVs.

*l = n =*0), which consists of a bright spot similar to the zero-order Bessel-like pattern [5

5. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. **36**(15), 2883–2885 (2011). [CrossRef] [PubMed]

*l = n =*1, as shown in Fig. 3(c), the AABs are nested with a single charged vortex, and evolved into a hollow spot [6

6. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express **20**(17), 18579–18584 (2012). [CrossRef] [PubMed]

**38**(14), 2416–2418 (2013). [CrossRef] [PubMed]

*l*= 8,

*n*= 2], the autofocusing process will be totally changed, and it forms the spiral spot. This resembles to an Archimedes spiral at the focal region, which is much similar to the focal field of Helico-conical optical beams [24

24. N. Hermosa, C. Rosales-Guzmán, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. **38**(3), 383–385 (2013). [CrossRef] [PubMed]

25. C.-A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**(5), 1749–1760 (2005). [CrossRef] [PubMed]

*y*-

*z*plane. It can be seen that the light energy gradually converged and the intensity presents a bit oscillation when closing to the focal point. This can reveal the spiral focusing to some extent. Actually, the formed spiral focal spot will rotate during propagating at the focal region, resembling the cork-screw path. To demonstrate clearly this phenomenon, we simulate the propagation process of the AAB carrying a PEPV near the focal point (the parameters are selected as

*l*= 12, and

*n*= 2), as shown in Fig. 4, where Figs. 4(a)–4(d) correspond to the intensity profiles at different propagation distances. The rotation direction is indicated by the white arrow head in Fig. 4(a). It can be clearly seen that the intensity profile of AAB follows the cork-screw path, and evolves into a spiral spot.

31. 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]

32. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

*l = n =*0), the light energy flow to the central region whether at the initial position or just before the focal point, behaving as focusing. For the AAB with a single charged vortex (

*l = n =*1), as shown in Fig. 5(b), the light energy circulates along a ring while shrinks gradually during propagation. This illustrates that the light spirals around the propagation axis while focusing. Due to the balanced energy cycle, the light intensity distribution keeps axisymmetric, i.e. a donut-shaped spot. For the AABs with a PEPV, the cycle of the energy flow is broken, and the energy will concentratedly flow to the certain region [see the regions with weak energy flow in the top of Figs. 5(c) and 5(d)]. On the other hand, it can be seen that the energy flow tends to shrink into the center because of the focusing effect. These lead to that the energy behaviors a spiral-like flow. Due to the spiral energy flow of PEPV, the light gradually evolves to a spiral-shape spot.

## 4. Theoretical explanation

*x*,

*y*) and (

*x'*,

*y'*) are the Cartesian coordinates,

*k*is the wave number,

*z*is the coordinate along the propagation direction, and

*u*(

*x'*,

*y'*) corresponds to the light field given by Eq. (3), of which the phase term is

*ψ*(

*r*,

*θ*) described by Eq. (2). Equation (4) can be considered as the Fourier transform of

*u*(

*x'*,

*y'*) if the propagation distance is far enough [25

25. C.-A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**(5), 1749–1760 (2005). [CrossRef] [PubMed]

**13**(5), 1749–1760 (2005). [CrossRef] [PubMed]

30. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. **28**(11), 872–874 (2003). [CrossRef] [PubMed]

*f*,

_{x}*f*) is the orthogonal coordinates in frequency domain. Applying this theory to Eq. (2), we obtain

_{y}*f*,

_{x}*f*) to far field can be plotted, as shown in Fig. 7(a), where the parameters are selected as

_{y}*n =*3,

*l =*7, and

*r =*1. The curve represents an Archimede spiral, which meets the shape of the corresponding focal field of AABs with PEPV [see Fig. 7(b)].

## 5. Influences of the parameters on the focal fields

*l*and

*n*on the focal fields of the AABs carrying PEPVs. Figure 8 demonstrates the experimental results of the focal fields (the intensity distributions at the plane 30 cm away from the lens L2) of AABs carrying PEPVs (

*l =*8), where Figs. 8(a)–8(h) correspond to

*n =*2, 3, 4, …, 9, respectively, with the corresponding spatial frequency mappings inserted as white lines. It reveals that as the power order

*n*increases, the focal spot shrinks with the tail shortened and the curved line stretches itself gradually. This phenomenon can be also seen from Eq. (2): with the increase of

*n*, the phase gradients concentrate to the interval of

*θ*near 2π, leading to the locally high gradient of phase. As a result, the light energy flow more quickly to the certain region. With increasing the parameters

*n*, the light concentrates faster, and the focal spot became more shrinked. The experimental results are coincident with the corresponding frequency maps calculated from Eq. (6).

*n*= 2), where Figs. 9(a)–9(h) correspond to

*l =*3, 4, 5, …, 10, respectively, with the corresponding spatial frequency mappings inserted as white lines. It can be seen that with the increase of the topological charge

*l*, the spiral focal spot is wholly enlarged, but the shape remains the same. From Eq. (2), it can be also seen that the topological charge

*l*never influences the distribution of the phase function, but changes the whole phase gradient. Thus, the energy flow distribution keeps wholly unchanged. While the rotating flow is speeded up with increasing

*l*, and the focusing flow is accordingly reduced. These result in the wholly enlarged spiral focal spot. The conclusions of PEPVs are same as that of the canonical vortices [33

33. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**(13), 133901 (2003). [CrossRef] [PubMed]

*l*. It means that

*l*merely changes the size of the frequency mapping, as also shown in Fig. 9.

## 6. OAM of AABs carrying PEPVs

34. S. A. C. Baluyot and N. P. Hermosa 2nd, “Intensity profiles and propagation of optical beams with bored helical phase,” Opt. Express **17**(18), 16244–16254 (2009). [CrossRef] [PubMed]

*z*axis in spatial space can be expressed aswhere,

**E**and

**H**are the electric and magnetic fields, respectively;

*r =*(

*x*

^{2}+

*y*

^{2})

^{1/2};

*S*and

_{x}*S*are the components along the

_{y}*x*and

*y*axes of the Poynting vector (

**S**

*=*

**E**×

**H)**, respectively.

## 7. Conclusions

## Acknowledgments

## References and links

1. | N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. |

2. | I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. |

3. | D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. |

4. | I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. |

5. | P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. |

6. | Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express |

7. | S. Liu, P. Li, M. Wang, P. Zhang, and J. Zhao, “Observation of abrupt polarization transitions associated with spin-orbit interaction of vector autofocusing Airy beams,” in |

8. | S. Liu, M. Wang, P. Li, P. Zhang, and J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. |

9. | J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express |

10. | P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. |

11. | Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle,” Opt. Express |

12. | 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 |

13. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. |

14. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

15. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

16. | M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics |

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

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

19. | X. Gan, J. Zhao, S. Liu, and L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett. |

20. | W. Zhang, S. Liu, P. Li, X. Jiao, and J. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express |

21. | isX. Gan, P. Zhang, S. Liu, F. Xiao, and J. Zhao, “Beam steering and topological transformations driven by interactions between a discrete vortex soliton and a discrete fundamental soliton,” Phys. Rev. A |

22. | G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. |

23. | G.-H. Kim, H. J. Lee, J.-U. Kim, and H. Suk, “Propagation dynamics of optical vortices with anisotropic phase profiles,” J. Opt. Soc. Am. B |

24. | N. Hermosa, C. Rosales-Guzmán, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. |

25. | C.-A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express |

26. | N. P. Hermosa II and C. O. Manaois, “Phase structure of helicon-conical optical beams,” Opt. Commun. |

27. | J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express |

28. | S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express |

29. | H. Li and J. Yin, “Generation of a vectorial Mathieu-like hollow beam with a periodically rotated polarization property,” Opt. Lett. |

30. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. |

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

32. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

33. | J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. |

34. | S. A. C. Baluyot and N. P. Hermosa 2nd, “Intensity profiles and propagation of optical beams with bored helical phase,” Opt. Express |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(350.5500) Other areas of optics : Propagation

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 20, 2014

Revised Manuscript: March 2, 2014

Manuscript Accepted: March 6, 2014

Published: March 25, 2014

**Citation**

Peng Li, Sheng Liu, Tao Peng, Gaofeng Xie, Xuetao Gan, and Jianlin Zhao, "Spiral autofocusing Airy beams carrying power-exponent-phase vortices," Opt. Express **22**, 7598-7606 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7598

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### References

- N. K. Efremidis, D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]
- I. Chremmos, N. K. Efremidis, D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011). [CrossRef] [PubMed]
- D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]
- I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]
- P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]
- Y. Jiang, K. Huang, X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef] [PubMed]
- S. Liu, P. Li, M. Wang, P. Zhang, and J. Zhao, “Observation of abrupt polarization transitions associated with spin-orbit interaction of vector autofocusing Airy beams,” in Frontiers in Optics (2013).
- S. Liu, M. Wang, P. Li, P. Zhang, J. Zhao, “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. 38(14), 2416–2418 (2013). [CrossRef] [PubMed]
- J. A. Davis, D. M. Cottrell, D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express 20(12), 13302–13310 (2012). [CrossRef] [PubMed]
- P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013). [CrossRef] [PubMed]
- Y. Jiang, K. Huang, X. Lu, “Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle,” Opt. Express 21(20), 24413–24421 (2013). [CrossRef] [PubMed]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
- M. J. Padgett, R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]
- K. T. Gahagan, G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
- X. Gan, J. Zhao, S. Liu, L. Fang, “Generation and motion control of optical multi-vortex,” Chin. Opt. Lett. 7(12), 1142–1145 (2009). [CrossRef]
- W. Zhang, S. Liu, P. Li, X. Jiao, J. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express 21(1), 974–983 (2013). [CrossRef] [PubMed]
- isX. Gan, P. Zhang, S. Liu, F. Xiao, J. Zhao, “Beam steering and topological transformations driven by interactions between a discrete vortex soliton and a discrete fundamental soliton,” Phys. Rev. A 89(1), 013844 (2014). [CrossRef]
- G. Molina-Terriza, E. M. Wright, L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26(3), 163–165 (2001). [CrossRef] [PubMed]
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