Abstract

The phase behavior of Airy beams is studied, and their Gouy phase is defined. Analytic expressions for the idealized, infinite-energy type beam are derived. They are shown to be excellent approximations for finite-energy beams generated under typical experimental conditions.

© 2011 Optical Society of America

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  1. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  2. J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  3. J. Turunen and A. T. Friberg, in Progress in Optics, Vol.  54, E.Wolf, ed. (Elsevier, 2010), pp. 1–88.
    [CrossRef]
  4. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  5. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  7. M. A. Bandres, Opt. Lett. 33, 1678 (2008).
    [CrossRef] [PubMed]
  8. J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, Opt. Express 17, 13236 (2009).
    [CrossRef] [PubMed]
  9. S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, J. Opt. Soc. Am. A 27, 2574 (2010).
    [CrossRef]
  10. Y. Kaganovsky and E. Heyman, Opt. Express 18, 8440(2010).
    [CrossRef] [PubMed]
  11. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
    [CrossRef] [PubMed]
  12. Y. L. Gu and G. Gbur, Opt. Lett. 35, 3456 (2010).
    [CrossRef] [PubMed]
  13. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
    [CrossRef] [PubMed]
  14. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
    [CrossRef] [PubMed]
  15. T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
    [CrossRef]
  16. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, Opt. Express 18, 7108 (2010).
    [CrossRef] [PubMed]

2010 (6)

2009 (3)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, Opt. Express 17, 13236 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

1987 (2)

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Alonso, M. A.

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, Opt. Express 17, 13236 (2009).
[CrossRef] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Christodoulides, D. N.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Cizmar, T.

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, Opt. Express 17, 13236 (2009).
[CrossRef] [PubMed]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Friberg, A. T.

J. Turunen and A. T. Friberg, in Progress in Optics, Vol.  54, E.Wolf, ed. (Elsevier, 2010), pp. 1–88.
[CrossRef]

Fuerschbach, K.

Gatto, A.

Gbur, G.

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Gu, Y. L.

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Heyman, E.

Kaganovsky, Y.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Martelli, P.

Martinelli, M.

Mazilu, M.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Moneta, G.

Morris, J. E.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Rolland, J. P.

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Tacca, M.

Thompson, K. P.

Turunen, J.

J. Turunen and A. T. Friberg, in Progress in Optics, Vol.  54, E.Wolf, ed. (Elsevier, 2010), pp. 1–88.
[CrossRef]

Visser, T. D.

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

Vo, S.

Wolf, E.

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Other (1)

J. Turunen and A. T. Friberg, in Progress in Optics, Vol.  54, E.Wolf, ed. (Elsevier, 2010), pp. 1–88.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Normalized intensity distribution of an Airy beam propagating in the positive ξ-direction.

Fig. 2
Fig. 2

Color-coded plot of the Gouy phase of an Airy beam. Only the s ξ -region in which the Airy function has no zeros is shown. Along the two dashed curves, given by Eq. (9), the Gouy phase equals zero. Along the solid curve, given by Eq. (10), the Gouy phase reaches its maximum. The dotted curve is given by Eq. (11).

Fig. 3
Fig. 3

Color-coded plot of the Gouy phase of an Airy beam. A portion of the region in which the function Ai ( x ) has zeros is shown. The solid black line indicates the fifth zero of the Airy function. The Gouy phase jumps by an amount π across this line.

Fig. 4
Fig. 4

Intensity of a finite-energy Airy beam in different cross sections perpendicular to the ξ-axis: the source plane ξ = 0 (black), ξ = 2 (blue), ξ = 4 (red), and ξ = 6 (green).

Fig. 5
Fig. 5

Gouy phase of a finite-energy Airy beam along the curves s = ( 3 + 3 1 / 2 ) / 6 ] ξ 2 (red), and s = ( 3 3 1 / 2 ) / 6 ] ξ 2 (blue).

Equations (15)

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U ( x , z , ω ) = ϕ ( x , z ) e i ( k z ω t ) ,
2 ϕ x 2 + 2 i k ϕ z = 0.
ϕ ( s , ξ ) = Ai [ s ( ξ 2 4 ) ] exp [ i ( s ξ 2 ξ 3 12 ) ] ,
δ ( x , z , ω ) = ψ [ U ( x , z , ω ) ] ψ [ U cyl ( x , z , ω ) ] ,
U cyl ( x , z , ω ) = i C 4 H 0 ( 1 ) ( k ρ ) .
U cyl ( x , z , ω ) C 2 π k ρ e i ( k ρ - π / 4 ) , ( k ρ 1 / 4 ) .
k ρ k z [ 1 + 1 2 ( x z ) 2 ] = k z + 1 2 s 2 ξ .
δ ( s , ξ , ω ) = s ξ 2 ξ 3 12 s 2 2 ξ + ψ Ai ,
δ ( s , ξ , ω ) = 0 , if     s = ( 3 ± 3 1 / 2 ) ξ 2 / 6.
δ ( s , ξ , ω ) = ξ 3 24 , if     s = ξ 2 / 2.
δ ( s , ξ , ω ) = ξ 3 96 , if     s = ξ 2 / 4.
δ ( 0 , ξ , ω ) = ξ 3 12 .
ϕ ( fe ) ( s , 0 ) = Ai ( s ) e a s ,
ϕ ( fe ) ( s , ξ ) = Ai ( s ξ 2 / 4 + i a ξ ) e a s a ξ 2 / 2 × e [ i ( ξ 3 / 12 + a 2 ξ / 2 + s ξ / 2 ) ] .
δ ( fe ) ( s , ξ , ω ) = s ξ 2 ξ 3 12 s 2 2 ξ + a 2 ξ 2 + ψ Ai .

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