OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 4 — Apr. 1, 2014
  • pp: 685–690
« Show journal navigation

Beam wander of an Airy beam with a spiral phase

Wei Wen and Xiuxiang Chu  »View Author Affiliations


JOSA A, Vol. 31, Issue 4, pp. 685-690 (2014)
http://dx.doi.org/10.1364/JOSAA.31.000685


View Full Text Article

Acrobat PDF (563 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Beam wander of an Airy beam with a spiral phase in turbulence is investigated. Using the Wigner distribution function, analytical expressions for the second-order moments and second central moments of an Airy beam with a spiral phase in turbulence are derived. A general expression of the beam wander for an Airy beam with a spiral phase is obtained. Based on the derived formula, various factors that impact on the beam wander are illustrated numerically. The results show that increasing the topological charge and the characteristic scale, or decreasing the exponential truncation factor, can be used to decrease the beam wander.

© 2014 Optical Society of America

1. INTRODUCTION

Apart from the broadening effects of diffraction, light beams tend to propagate along straight lines in vacuum. In 2007, Christodoulides and co-workers came up with a new family of nondiffracting beams [1

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

] that strangely appears to curve [2

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

,3

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

]. The so-called Airy beam is asymmetric, with one bright region at the center and a series of progressively dimmer patches on one side of the central spot. However, rather than propagating in a straight line like other beams, the entire pattern of bright and dark patches of an Airy beam curves toward one side. Another remarkable property of the Airy beam is its very ability to self-reconstruct during propagation [4

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

,5

5. X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85, 013815 (2012). [CrossRef]

]. The study of Airy beam is interesting from the viewpoint of fundamental science. Until now, Airy beams were produced by using a phase plate [6

6. D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34, 2634–2636 (2009). [CrossRef]

8

8. Y. Liang, Z. Ye, D. Song, C. Lou, X. Zhang, J. Xu, and Z. Chen, “Generation of linear and nonlinear propagation of three-Airy beams,” Opt. Express 21, 1615–1622 (2013). [CrossRef]

], liquid crystal cells [9

9. D. Luo, H. T. Dai, X. W. Sun, and H. V. Demir, “Electrically switchable finite energy Airy beams generated by a liquid crystal cell with patterned electrode,” Opt. Commun. 283, 3846–3849 (2010). [CrossRef]

], cylindrical lenses [10

10. B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A 27, 2344–2346 (2010). [CrossRef]

], three-wave mixing processes [11

11. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009). [CrossRef]

], microchip laser [12

12. S. Longhi, “Airy beams from a microchip laser,” Opt. Lett. 36, 716–718 (2011). [CrossRef]

], and waveguide arrays [13

13. H. Deng and L. Yuan, “Generation of Airy-like wave with one-dimensional waveguide array,” Opt. Lett. 38, 1645–1647 (2013). [CrossRef]

]. The generation of Airy plasmons [14

14. 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, 116802 (2011). [CrossRef]

16

16. I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109, 203903 (2012). [CrossRef]

], high-power Airy beams [17

17. X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10, 125102 (2013). [CrossRef]

], electron Airy beams [18

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

], and incomplete Airy beams [19

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

] has also been reported. In addition, useful properties of Airy beams make them attractive in applications for clearing optically mediated particles [20

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

], producing self-bending plasma channels [21

21. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). [CrossRef]

], trapping and guiding microparticles [22

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

,23

23. 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, 2883–2885 (2011). [CrossRef]

], electron capture and acceleration driven [24

24. J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. 35, 3258–3260 (2010). [CrossRef]

,25

25. J. X. Li, X. L. Fan, W. P. Zang, and J. G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36, 648–650 (2011). [CrossRef]

], ultrafast self-accelerating pulses [26

26. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

], light bullets accelerating in both transverse dimensions and time [27

27. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010). [CrossRef]

], and all-optical routing [28

28. P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102, 101101 (2013). [CrossRef]

].

As we know, the instantaneous centers of a laser beam will randomly displace in the receiver plane when it propagates through the turbulent atmosphere, producing what is called beam wander [47

47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

]. Beam wander is an important characteristic of laser beams, which determines their utility for practical applications, such as global quantum communication [48

48. D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012). [CrossRef]

]. In the past years, beam wander of many structured light fields have been studied [49

49. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007). [CrossRef]

54

54. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010). [CrossRef]

]. Berman et al. have discussed the influence of the initial spatially coherent length on the beam wander [49

49. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007). [CrossRef]

]. Hereafter, Chen et al. and Yu et al. have investigated the temporal spectrum of beam wander for Gaussian Schell-model beams and the beam wander of electromagnetic Gaussian–Schell beams [50

50. C. Chen and H. Yang, “Temporal spectrum of beam wander for Gaussian Shell-model beams propagating in atmospheric turbulence with finite outer scale,” Opt. Lett. 38, 1887–1889 (2013). [CrossRef]

,51

51. S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian–Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51, 7581–7585 (2012). [CrossRef]

]. Eyyuboğlu and Cil have studied the beam wander of dark hollow beam, flat-topped beam and annular beam [52

52. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008). [CrossRef]

]. Cil and co-workers have studied the beam wander of cosh-Gaussian beam and Bessel beam [53

53. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009). [CrossRef]

,54

54. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010). [CrossRef]

]. Recently, beam wander of twin thin beam and quantization Gaussian beam have been reported also [55

55. D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20, 27766–27780 (2012). [CrossRef]

,56

56. C. Si and Y. Zhang, “Beam wander of quantization beam in a non-Kolmogorov turbulent atmosphere,” Optik 124, 1175–1178 (2013). [CrossRef]

]. However, to the best of our knowledge, the beam wander of an Airy beam with a spiral phase in turbulence has not been reported.

In this paper, we derive the general expression of the beam wander for an Airy beam with a spiral phase based on the central second-order moment. Using numerical analysis, some factors which impact on the beam wander of an Airy beam with a spiral phase are discussed.

2. FORMULATION

The optical field distribution of Airy beams with a spiral phase at the plane of z=0 in the Cartesian coordinate system is given by [46

46. X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37, 5202–5204 (2012). [CrossRef]

]
u0(x0,y0)=[(x0xd)+i(y0yd)]m[(x0xd)2+(y0yd)2]m/2×Ai(x0/w0)Ai(y0/w0)exp[a(x0+y0)/w0],
(1)
where Ai(·) is the Airy function and m is the topological charge, respectively. The parameter w0 is transverse scale, and a is the exponential truncation factor.

If we assume that a laser beam propagates along the z-axis, then the average intensity at the receiver plane can be expressed as [57

57. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

]
I(x,y)=(k2πz)2H(p2,q2)×exp[12D(p2,q2)]exp[ikx(xp2+yq2)]dp2dq2,
(2)
where
H(p2,q2)=(k2πz)2u0(p1+p22,q1+q22)×uo*(p1p22,q1q22)exp[ikx(p1p2+q1q2)]dp1dq1,
(3)
and D(p2,q2) is wave structure function, which can be denoted as [47

47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

]
D(p2,q2)=8π2k2z001[1J0(κξp22+q22)]Φ(κ)κdξdκ.
(4)
The function Φ(κ) is the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere and the parameter κ is spatial frequency. The parameter ξ is the normalized distance variable and ξ=1z/L(parameter z is the propagation distance and L is the turbulence distance). The Tatarskii spectrum is adopted in this paper, i.e. [47

47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

,58

58. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

],
Φn(k)=0.033Cn2κ11/3exp(κ2κm2),
(5)
where Cn2 is the structure constant of the refractive index fluctuations of the turbulence and κm=5.92/l0 (l0 is the inner scale of the turbulence). In Eqs. (2)–(4), p1=(x01+x02)/2, p2=x01x02, q1=(y01+y02)/2, q2=y01y02.

The second moment of Airy beam with a spiral phase at the receiver plane can be expressed as [46

46. X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37, 5202–5204 (2012). [CrossRef]

]
(x2y2)=1P(k2πz)2(x2y2)exp[12D(p2,q2)]×exp[ikz(xp2+yq2)]H(p2,q2)dp2dq2dxdy,
(6)
where P is the total power of the input beam and can be read as [58

58. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

]
P=w028πaexp(4a3/3).
(7)
Applying the properties of the Dirac delta function,
δ(n)(t)=12π(ix)nexp(itx)dx;n=0,1,2,
(8)
δ(n)(ax)=an1δ(n)(x);n=0,1,2,
(9)
F(x)δ(n)(x)dx=(1)nF(n)(0);n=0,1,2,
(10)
and performing the integration, we get
x2=y2=A+8πaw02exp(4a3/3)(Bm2z2k2w042Cmzkw02)+D,
(11)
where
A=z24ak2w02+3w0216a2+aw022+a4w02,
(12)
B=(q1p12+q12)2exp[2a(p1+q1)]Ai2(p1)Ai2(q1)dp1dq1,
(13)
C=(p1q1p12+q12)exp[2a(p1+q1)]Ai2(p1)Ai2(q1)dp1dq1,
(14)
and
D=23π2z30κ3Φn(κ)dκ=0.066π2z3Cn2κm1/3Γ(7/6).
(15)
It can be seen from Eq. (11) that the second moment of the Airy beam with a spiral phase is dependent on the topological charge. In general, the central moments are more interesting than the moments about the origin. The second central moment of an Airy beam with a spiral phase in atmospheric turbulence can be expressed as
μ2=x2xc2=A+8πaw02exp(4a3/3)(Bm2z2k2w042Cmzkw02)+D[(4a31)w04a8πamzEkw0exp(4a3/3)]2,
(16)
where
E=(q1p12+q12)exp[2a(p1+q1)]Ai2(p1)Ai2(q1)dp1dq1.
(17)
Here, xc=(a21/4a)/w0 is the first central moment of an Airy beam [58

58. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

]. Based on the second-order central moment, one can get that the long-term width of an Airy beam with a spiral phase in turbulent atmosphere can be expressed as
WLT(z)=2r2I(r,z)d2r/I(r,z)d2r=2[A+8πaw02exp(4a3/3)(Bm2z2k2w042Cmzkw02)+D[(4a31)w04a8πamzEkw0exp(4a3/3)]2]1/2.
(18)
From Eq. (18) one can readily obtain the beam width WFS by setting Cn2=0 and z=L. The model of beam wander that is valid under all turbulence conditions is given by Andrews and Phillips as [47

47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

]
rc2=4πk2WFS20L0κΦn(κ)exp(κ2WLT2)×{1exp[2L2κ2(1z/L)2k2WFS2]}dκdz.
(19)
Submitting Eq. (5) into Eq. (19), one can get that the beam wander of an Airy beam with a spiral phase as
rc2=0.066πΓ(5/6)k2Cn2WFS2×0L{(1κm2+WLT2)5/6[1κm2+WLT2+2L2(1z/L)2k2WFS2]5/6}dz.
(20)
Equation (20) is the main analytical result of this study. It shows that the beam wander of an Airy beam with a spiral phase varies with changes in the propagation distance, topological charge, wavenumber, refractive index structure constant, inner scale of turbulence, long-term beam width, and the beam width.

3. NUMERICAL RESULTS AND ANALYSIS

In this section, we investigate the beam wander of an Airy beam with a spiral phase numerically. For the dimensionless quantity, BW=rc2/WLT2 is more informative than merely rc2 about the practical significance of the beam wander; BW is used to investigate the beam wander of an Airy beam with a spiral phase by number calculation.

The beam wander of an Airy beam with a spiral phase as a function of the propagation distance for different exponential truncation factors and topological charges is plotted in Fig. 1. The calculation parameters are w0=5mm, Cn2=1014m2/3 and l0=1mm. The exponential truncation factors in Fig. 1 are 0.05, 0.11, 0.63, and 1.00, respectively [1

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

,23

23. 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, 2883–2885 (2011). [CrossRef]

,46

46. X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37, 5202–5204 (2012). [CrossRef]

,58

58. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

]. From Fig. 1, one finds that smaller exponential truncation factors have weaker beam wander. One can also find that the beam wander of an Airy beam with a spiral phase is less significant than that of an Airy beam without spiral phase for all exponential truncation factors. Moreover, the difference of beam wander between an Airy beam with and without spiral phase is more significant as the exponential truncation factor increases [Figs. 1(c) and 1(d)]. It should be pointed out that the beam wanders with positive topological charge and negative topological charge do not have the same value in the strict sense from the analytical results. From a practical point of view, however, the difference of beam wander between an Airy beam with and without spiral phase is not obvious from the results of Fig. 1.

Fig. 1. Dimensionless quantity BW of an Airy beam with a spiral phase versus propagation distance for different exponential truncation factors and topological charges. The calculation parameters are w0=5mm, Cn2=1014m2/3, and l0=1mm.

The beam wander of an Airy beam with a spiral phase as a function of the exponential truncation factor for different topological charge is shown in Fig. 2. The other parameters are w0=5mm, L=1000m, Cn2=1014m2/3, and l0=1mm. One can observe that the beam wander of a vortex Airy beam increases with the increase of the exponential truncation factor. As with the result in Fig. 1, one can find from Fig. 2 that the beam wander of an Airy beam with a spiral phase is less significant than that of an Airy beam without spiral phase. This situation will appear later in this paper.

Fig. 2. Dimensionless quantity BW of an Airy beam with a spiral phase versus exponential truncation factors for different topological charges. The calculation parameters are w0=5mm, Cn2=1014m2/3, L=1000m, and l0=1mm.

The effect of structure constant of turbulence on the beam wander is shown in Fig. 3. The other parameters are a=0.11, w0=5mm, and l0=1mm. Physically, the refractive index structure constant is a measure of the strength of the fluctuations in the refractive index. Therefore, Fig. 3 reveals that raising the turbulence level will increase the beam wander of an Airy beam with a spiral phase. According to Eq. (20), the beam wander of an Airy beam with a spiral phase is not directly proportional to the structure constant. However, the results of Fig. 3 display that the relationship between the beam wander and structure constant is linearly correlated. Therefore, the weight of structure constant in the integral kernel of the Eq. (20) is low.

Fig. 3. Dimensionless quantity BW of an Airy beam with a spiral phase versus propagation distance for different structure constants. The calculation parameters are a=0.11, w0=5mm, and l0=1mm.

The effect of characteristic scales on the beam wander of turbulence is shown in Fig. 4, where a=0.11, Cn2=1014m2/3, and l0=1mm. The results show that increasing the characteristic scales of vortex Airy beam can be used to decrease the wander of an Airy beam.

Fig. 4. Dimensionless quantity BW of an Airy beam with a spiral phase versus propagation distance for different transverse scales. The calculation parameters are a=0.11, Cn2=1014m2/3, and l0=1mm.

The beam wander of an Airy beam with a spiral phase as a function of the propagation distance for four different inner scales is shown in Fig. 5. The other parameters are a=0.11, w0=5mm, and Cn2=1014m2/3. It is generally admitted that the inner scales of atmosphere turbulence are several millimeters [47

47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

]; therefore, in this paper, we select inner scales that are 1, 3, 5, and 7 mm. It can be seen that the beam wander of an Airy beam with a spiral phase remains nearly constant for different inner scales. This phenomenon agrees with the results of Ref. [52

52. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008). [CrossRef]

]. Consequently, the effect of inner scales seems to be negligible.

Fig. 5. Dimensionless quantity BW of an Airy beam with a spiral phase versus propagation distance for different inner scales. The calculation parameters are a=0.11, w0=5mm, and Cn2=1014m2/3.

4. CONCLUSION

We investigated in detail the beam wander of an Airy beam with a spiral phase in a turbulent atmosphere. The expressions for the second-order moment, the second central moments, and the beam wander of an Airy beam with a spiral phase in turbulence are derived. The analytical results indicated that the factors influencing the beam wander of an Airy beam with a spiral phase are the propagation distance, wavenumber, refractive index structure constant, inner scale of turbulence, long-term beam width, topological charge, exponent truncation factor, characteristic scales, and the beam width without turbulence at the receiver plane of an Airy beam with a spiral phase. In addition, these factors are illustrated numerically. The results demonstrated that the exponent truncation factor of a vortex Airy beam and structure constant of turbulence are positively correlated to the beam wander. However, the topological charge and characteristic scales are negatively correlated to the beam wander. As with other beams, the inner scale of turbulence does not affect the beam wander of an Airy beam with a spiral phase. Our analytical formulas provide an effective and convenient way to analyze the propagation of an Airy beam with a spiral phase in a turbulent atmosphere, and can be used to effectively control the beam wander of a vortex Airy beam in practice.

APPENDIX A

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11374264, 11274005, 11174100 and 11374222, and the Innovation Plan for Graduate Students in the University of Jiangsu Province under Grant No. CXLX13_80(133).

REFERENCES

1.

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

2.

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

3.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

4.

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

5.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85, 013815 (2012). [CrossRef]

6.

D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34, 2634–2636 (2009). [CrossRef]

7.

E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef]

8.

Y. Liang, Z. Ye, D. Song, C. Lou, X. Zhang, J. Xu, and Z. Chen, “Generation of linear and nonlinear propagation of three-Airy beams,” Opt. Express 21, 1615–1622 (2013). [CrossRef]

9.

D. Luo, H. T. Dai, X. W. Sun, and H. V. Demir, “Electrically switchable finite energy Airy beams generated by a liquid crystal cell with patterned electrode,” Opt. Commun. 283, 3846–3849 (2010). [CrossRef]

10.

B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A 27, 2344–2346 (2010). [CrossRef]

11.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009). [CrossRef]

12.

S. Longhi, “Airy beams from a microchip laser,” Opt. Lett. 36, 716–718 (2011). [CrossRef]

13.

H. Deng and L. Yuan, “Generation of Airy-like wave with one-dimensional waveguide array,” Opt. Lett. 38, 1645–1647 (2013). [CrossRef]

14.

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, 116802 (2011). [CrossRef]

15.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011). [CrossRef]

16.

I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109, 203903 (2012). [CrossRef]

17.

X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10, 125102 (2013). [CrossRef]

18.

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

19.

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

20.

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

21.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). [CrossRef]

22.

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

23.

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, 2883–2885 (2011). [CrossRef]

24.

J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. 35, 3258–3260 (2010). [CrossRef]

25.

J. X. Li, X. L. Fan, W. P. Zang, and J. G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36, 648–650 (2011). [CrossRef]

26.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

27.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010). [CrossRef]

28.

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102, 101101 (2013). [CrossRef]

29.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]

30.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). [CrossRef]

31.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]

32.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

33.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]

34.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013). [CrossRef]

35.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]

36.

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35, 4075–4077 (2010). [CrossRef]

37.

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” Opt. Lett. 36, 1617–1619 (2011). [CrossRef]

38.

M. Mazilu, J. Baumgartl, T. i már, and K. Dholakia, “Accelerating vortices in Airy beams,” Proc. SPIE 7430, 74300C (2009). [CrossRef]

39.

K. Cheng, X. Zhong, and A. Xiang, “Propagation dynamics, Poynting vector and accelerating vortices of afocused Airy vortex beam,” Opt. Laser Technol. 57, 77–83 (2014). [CrossRef]

40.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013). [CrossRef]

41.

R. Chen and C. H. Raymond Ooi, “Nonclassicality of vortex Airy beams in the Wigner representation,” Phys. Rev. A 84, 043846 (2011). [CrossRef]

42.

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44, 2015–2019 (2012). [CrossRef]

43.

R. Chen and K. Chew, “Far-field properties of a vortex Airy beam,” Laser Part. Beams 31, 9–15 (2013). [CrossRef]

44.

R. Chen, K. Chew, and S. He, “Dynamic control of collapse in a vortex Airy beam,” Sci. Rep. 3, 1–9 (2013).

45.

C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15, 044001 (2013). [CrossRef]

46.

X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37, 5202–5204 (2012). [CrossRef]

47.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

48.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012). [CrossRef]

49.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007). [CrossRef]

50.

C. Chen and H. Yang, “Temporal spectrum of beam wander for Gaussian Shell-model beams propagating in atmospheric turbulence with finite outer scale,” Opt. Lett. 38, 1887–1889 (2013). [CrossRef]

51.

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian–Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51, 7581–7585 (2012). [CrossRef]

52.

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008). [CrossRef]

53.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009). [CrossRef]

54.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010). [CrossRef]

55.

D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20, 27766–27780 (2012). [CrossRef]

56.

C. Si and Y. Zhang, “Beam wander of quantization beam in a non-Kolmogorov turbulent atmosphere,” Optik 124, 1175–1178 (2013). [CrossRef]

57.

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

58.

X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

OCIS Codes
(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(080.4865) Geometric optics : Optical vortices

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: December 17, 2013
Revised Manuscript: January 23, 2014
Manuscript Accepted: January 24, 2014
Published: March 6, 2014

Citation
Wei Wen and Xiuxiang Chu, "Beam wander of an Airy beam with a spiral phase," J. Opt. Soc. Am. A 31, 685-690 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-4-685


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
  4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008). [CrossRef]
  5. X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85, 013815 (2012). [CrossRef]
  6. D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34, 2634–2636 (2009). [CrossRef]
  7. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef]
  8. Y. Liang, Z. Ye, D. Song, C. Lou, X. Zhang, J. Xu, and Z. Chen, “Generation of linear and nonlinear propagation of three-Airy beams,” Opt. Express 21, 1615–1622 (2013). [CrossRef]
  9. D. Luo, H. T. Dai, X. W. Sun, and H. V. Demir, “Electrically switchable finite energy Airy beams generated by a liquid crystal cell with patterned electrode,” Opt. Commun. 283, 3846–3849 (2010). [CrossRef]
  10. B. Yalizay, B. Soylu, and S. Akturk, “Optical element for generation of accelerating Airy beams,” J. Opt. Soc. Am. A 27, 2344–2346 (2010). [CrossRef]
  11. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009). [CrossRef]
  12. S. Longhi, “Airy beams from a microchip laser,” Opt. Lett. 36, 716–718 (2011). [CrossRef]
  13. H. Deng and L. Yuan, “Generation of Airy-like wave with one-dimensional waveguide array,” Opt. Lett. 38, 1645–1647 (2013). [CrossRef]
  14. 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, 116802 (2011). [CrossRef]
  15. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011). [CrossRef]
  16. I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109, 203903 (2012). [CrossRef]
  17. X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. 10, 125102 (2013). [CrossRef]
  18. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013). [CrossRef]
  19. J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. 38, 1639–1641 (2013). [CrossRef]
  20. J. Baumgart, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]
  21. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). [CrossRef]
  22. Z. Zheng, B. F. Zhang, H. Chen, J. Ding, and H. T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). [CrossRef]
  23. 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, 2883–2885 (2011). [CrossRef]
  24. J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. 35, 3258–3260 (2010). [CrossRef]
  25. J. X. Li, X. L. Fan, W. P. Zang, and J. G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36, 648–650 (2011). [CrossRef]
  26. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]
  27. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010). [CrossRef]
  28. P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102, 101101 (2013). [CrossRef]
  29. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]
  30. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). [CrossRef]
  31. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]
  32. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]
  33. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]
  34. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013). [CrossRef]
  35. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]
  36. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35, 4075–4077 (2010). [CrossRef]
  37. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” Opt. Lett. 36, 1617–1619 (2011). [CrossRef]
  38. M. Mazilu, J. Baumgartl, T. i már, and K. Dholakia, “Accelerating vortices in Airy beams,” Proc. SPIE 7430, 74300C (2009). [CrossRef]
  39. K. Cheng, X. Zhong, and A. Xiang, “Propagation dynamics, Poynting vector and accelerating vortices of afocused Airy vortex beam,” Opt. Laser Technol. 57, 77–83 (2014). [CrossRef]
  40. D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013). [CrossRef]
  41. R. Chen and C. H. Raymond Ooi, “Nonclassicality of vortex Airy beams in the Wigner representation,” Phys. Rev. A 84, 043846 (2011). [CrossRef]
  42. R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44, 2015–2019 (2012). [CrossRef]
  43. R. Chen and K. Chew, “Far-field properties of a vortex Airy beam,” Laser Part. Beams 31, 9–15 (2013). [CrossRef]
  44. R. Chen, K. Chew, and S. He, “Dynamic control of collapse in a vortex Airy beam,” Sci. Rep. 3, 1–9 (2013).
  45. C. Rosales-Guzmán, M. Mazilu, J. Baumgartl, V. Rodríguez-Fajardo, R. Ramos-García, and K. Dholakia, “Collision of propagating vortices embedded within Airy beams,” J. Opt. 15, 044001 (2013). [CrossRef]
  46. X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37, 5202–5204 (2012). [CrossRef]
  47. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  48. D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012). [CrossRef]
  49. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007). [CrossRef]
  50. C. Chen and H. Yang, “Temporal spectrum of beam wander for Gaussian Shell-model beams propagating in atmospheric turbulence with finite outer scale,” Opt. Lett. 38, 1887–1889 (2013). [CrossRef]
  51. S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian–Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51, 7581–7585 (2012). [CrossRef]
  52. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008). [CrossRef]
  53. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009). [CrossRef]
  54. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010). [CrossRef]
  55. D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20, 27766–27780 (2012). [CrossRef]
  56. C. Si and Y. Zhang, “Beam wander of quantization beam in a non-Kolmogorov turbulent atmosphere,” Optik 124, 1175–1178 (2013). [CrossRef]
  57. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]
  58. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

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.

CrossCheck Deposited