Abstract

New types of finite energy Airy beams are proposed. We consider two different types of beams, namely, beams that are obtained as blocked and exponentially attenuated versions of Airy functions Ai and Bi, and beams of finite width but having the Airy functions typical phase. All of them show very interesting properties, such as parabolic trajectories for longer propagation distances, profile evolution exhibiting less diffraction, or better definiteness of the main peak, when compared with other finite energy Airy beams studied before.

© 2010 Optical Society of America

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  1. M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  2. J. Durin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  4. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  5. M. A. Bandres, "Accelerating parabolic beams," Opt. Lett. 33, 1678-1680 (2008).
    [CrossRef] [PubMed]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, "Observation of accelerating parabolic beams," Opt. Express 16, 12866-12871 (2008).
    [CrossRef] [PubMed]
  7. G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  9. I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
    [CrossRef] [PubMed]
  10. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
    [CrossRef] [PubMed]
  11. I. Dolev, T. Ellenbogen, and A. Arie, "Switching the acceleration direction of Airy beams by a nonlinear optical process," Opt. Lett. 35, 1581-1583 (2010).
    [CrossRef] [PubMed]
  12. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, "Optimal control of the ballistic motion of Airy beams," Opt. Lett. 35, 2260-2262 (2010).
    [CrossRef] [PubMed]
  13. 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] [PubMed]
  14. M. A. Bandres and J. C. Gutierrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007).
    [CrossRef] [PubMed]
  15. H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008).
    [CrossRef] [PubMed]
  16. L. Carretero, P. Acebal, S. Blaya, C. Garcia, A. Fimia, R. Madrigal, and A. Murciano, "Nonparaxial diffraction analysis of Airy beams and SAiry beams," Opt. Express 17, 22432-22441 (2009).
    [CrossRef]
  17. 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]
  18. J. Baumgartl,M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008).
    [CrossRef]
  19. A. Salandrino and D. N. Christodoulides, "Airy plasmon: a nondiffracting surface wave," Opt. Lett. 35, 2082-2084 (2010).
    [CrossRef] [PubMed]
  20. 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]
  21. 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, 8185-8189 (1992).
    [CrossRef] [PubMed]
  22. K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy packet in quantum mechamics," Am. J. Phys. 64, 1034-1035 (1996).
    [CrossRef]
  23. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, 1965).
  24. D. M. Cottrell and A. R. P. Rau, "Direct generation of accelerating Airy beams using a 3/2 phase-only pattern," Opt. Lett. 34, 2634-2636 (2009).
    [CrossRef] [PubMed]

2010 (4)

2009 (3)

2008 (6)

2007 (4)

1996 (1)

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy packet in quantum mechamics," Am. J. Phys. 64, 1034-1035 (1996).
[CrossRef]

1992 (1)

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, 8185-8189 (1992).
[CrossRef] [PubMed]

1987 (3)

J. Durin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

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

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Acebal, P.

Alfano, R. R.

Allen, L.

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Arie, A.

I. Dolev, T. Ellenbogen, and A. Arie, "Switching the acceleration direction of Airy beams by a nonlinear optical process," Opt. Lett. 35, 1581-1583 (2010).
[CrossRef] [PubMed]

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]

Balazs, N. L.

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl,M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008).
[CrossRef]

Beijersbergen, M. W.

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Besieris, I. M.

Blaya, S.

Broky, J.

Carretero, L.

Chen, Z.

Chong, A.

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]

Christodoulides, D. N.

Cottrell, D. M.

Davis, J. A.

Dholakia, K.

J. Baumgartl,M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008).
[CrossRef]

Dogariu, A.

Dolev, I.

Durin, J.

Durnin, J.

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

Eberly, J. H.

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

Ellenbogen, T.

I. Dolev, T. Ellenbogen, and A. Arie, "Switching the acceleration direction of Airy beams by a nonlinear optical process," Opt. Lett. 35, 1581-1583 (2010).
[CrossRef] [PubMed]

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]

Fimia, A.

Ganany-Padowicz, A.

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]

Garcia, C.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gutierrez-Vega, J. C.

Hu, Y.

Huang, S.

Lou, C.

Madrigal, R.

Mazilu, M.

J. Baumgartl,M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008).
[CrossRef]

Miceli, J. J.

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

Mitry, M. J.

Murciano, A.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Rau, A. R. P.

D. M. Cottrell and A. R. P. Rau, "Direct generation of accelerating Airy beams using a 3/2 phase-only pattern," Opt. Lett. 34, 2634-2636 (2009).
[CrossRef] [PubMed]

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy packet in quantum mechamics," Am. J. Phys. 64, 1034-1035 (1996).
[CrossRef]

Renninger, W. H.

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]

Salandrino, A.

Shaarawi, A. M.

Siviloglou, G. A.

Spreeuw, R. J. C.

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Sztul, H. I.

Unnikrishnan, K.

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy packet in quantum mechamics," Am. J. Phys. 64, 1034-1035 (1996).
[CrossRef]

Voloch-Bloch, N.

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]

Wise, F.W.

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]

Woerdman, J. P.

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Xu, J.

Zhang, P.

Am. J. Phys. (2)

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy packet in quantum mechamics," Am. J. Phys. 64, 1034-1035 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Photonics (3)

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]

J. Baumgartl,M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008).
[CrossRef]

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]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Express (5)

Opt. Lett. (8)

Phys. Rev. A (1)

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

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

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

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, 1965).

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Figures (10)

Fig. 1
Fig. 1

Propagation dynamics of of (a) Ai0z and (b) Ai3z. Amplitude profiles of both beams at (c) ξ = 2.5, (d) ξ = 5.0 and (e) ξ = 7.5. Normalized transverse component of the average Poynting vector at (f) ξ = 2.5, (g) ξ = 5.0 and (h) ξ = 7.5. Both input profiles have a = 0.1.

Fig. 2
Fig. 2

Trajectories of different inputs, all having a = 0.1.

Fig. 3
Fig. 3

Maximum ξ reached by the profiles while maintaining their peak at the parabolic trajectory, for different initial inputs.

Fig. 4
Fig. 4

Intensity profiles at (a) input and at (b) ξfinal. The Ai3z versions have the same input profile but are blocked to the right of vertical line.

Fig. 5
Fig. 5

Intensity profiles at origin and at ξfinal for Ai0z and for the non-oscillatory attenuated envelopes with a = 0.1. Since Ai0z-iBi1z does not exhibit a parabolic trajectory, its final profile is shown at ξfinal,Ai0z +iBi1z

Fig. 6
Fig. 6

(a) Trajectories and (b) intensity profiles at ξfinal for several input profiles and respective input profiles in inset.

Fig. 7
Fig. 7

Trajectory of the amplitude maxima for right blocked exponential amplitude beams with a = 0.04 and a phase approximately equal to the phase of Ai0z +iBi1z. The degree of blocking is indicated by s0.

Fig. 8
Fig. 8

Trajectories of finite width constant amplitude beams, extending from se to 0, with phase similar to Ai0z +iBi1z.

Fig. 9
Fig. 9

Evolution of a unitary amplitude beam of width 60 and with phase similar to Ai0z +iBi1z.

Fig. 10
Fig. 10

Evolution of the finite width Ai beam with se = −58.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

i u ξ + 1 2 2 u s 2 = 0 ,
W g η W A 2 W 3 = 0 ,
θ ( η , ξ ) = A 0 η d η W 2 + ( g 2 ξ b ) η + g 2 24 ξ 3 g b 4 ξ 2 + b 2 2 ξ ,
F ζ F = 0
u ( s , ξ = 0 ) = Ai ( s ) exp ( a s )
u ( s , ξ ) = Ai ( η + i a ξ ) e a η a 4 ξ 2 e i ( η 2 + a 2 2 + ξ 2 24 ) ξ
Ai n z ( s ) = { Ai ( s ) exp ( a s ) , s < a n 0 , otherwise
Bi n z ( s ) = { Bi ( s ) exp ( a s ) , s < b n 0 , otherwise
S a v = 1 2 μ / ɛ | u | 2 [ z ^ + 1 k α x x ^ ]
u ( s , ξ = 0 ) = Ai 0 z ( s ) ± i Bi 1 z ( s )

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