Abstract

We report on the propagation dynamics of Airy light beams under nonparaxial conditions. The partial waves forming the Airy beam can be divided into two parts, the first of which contains only propagating waves, while the second part consists of evanescent waves. In this Letter we propose the concept of the evanescent Airy beam. We analyze the structure of the ideal evanescent Airy beam, the initial profile of which has the Airy form, while its spectral decomposition consists of only evanescent partial waves. Also, we discuss the refraction of the Airy beam through an interface and investigate the field of the transmitted Airy beam.

© 2009 Optical Society of America

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References

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  1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  2. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  3. I. M. Besieris and A. M. Shaarawi, Opt. Lett. 32, 2447 (2007).
    [CrossRef] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
    [CrossRef] [PubMed]
  5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Photonics News 19(12), 21 (2008).
    [CrossRef]
  6. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, Opt. Express 16, 12880 (2008).
    [CrossRef] [PubMed]
  7. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  8. J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
    [CrossRef]

2008 (4)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Photonics News 19(12), 21 (2008).
[CrossRef]

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

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

2007 (3)

1979 (1)

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

Balazs, N. L.

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

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

Berry, M. V.

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

Besieris, I. M.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Photonics News 19(12), 21 (2008).
[CrossRef]

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.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

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, Opt. Lett. 33, 207 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Photonics News 19(12), 21 (2008).
[CrossRef]

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

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

Shaarawi, A. M.

Siviloglou, G. A.

Am. J. Phys. (1)

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

Nat. Photonics (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photonics 2, 675 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Photonics News (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Photonics News 19(12), 21 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

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

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Figures (4)

Fig. 1
Fig. 1

Distribution of normalized intensity of (a) paraxial Airy beam calculated using Eq. (4) ( χ = 100 , a = 0.1 ) and (b) evanescent Airy beam calculated using Eq. (6) ( a = 0.05 ) . Refractive index is n = 1 . Mathematica 5 packet is used to calculate the integral (4) having infinite limits.

Fig. 2
Fig. 2

Distribution of normalized intensity of the nonparaxial Airy beam calculated using Eq. (4) at (a) χ = 5 , (b) χ = 2 , (c) χ = 1 , (d) χ = 0.5 . Parameters: a = 0.1 and n = 1 .

Fig. 3
Fig. 3

Transmission of the Airy beam through the interface between dielectric media with n = 3.5 ( 0 < z ̃ < 2 ) and n = 1 ( z ̃ > 2 ) . (a) Intensity I = n | E y A | 2 of the incident Airy beam. (b) Transmission of the incident Airy beam through the interface. (c) Transmitted propagating waves. (d) Transmitted evanescent waves. For the (b)–(d), the quantity I is plotted. Parameters: a = 0.1 , χ = 1 .

Fig. 4
Fig. 4

(a) Propagation of the Airy beam without refraction. (b) Transmission of the Airy beam through the interface between dielectric media with n = 3.5 ( 0 < z ̃ < 200 ) and n = 1 ( z ̃ > 200 ) . Parameters: a = 0.1 , χ = 100 .

Equations (9)

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u z + 1 2 k 2 u x 2 = 0 ,
E y ( x , z , k x ) = c ( k x ) e i k x x + i k z z ,
c ( q ) = 1 2 π A e a q 2 e i ( q 3 3 a 2 q i a 3 ) 3 ,
E y ( x , z ) = c ( q ) e i q x ̃ + i χ 2 ϵ μ q 2 z ̃ d q .
E y ( x ̃ , z ̃ ) = A e i n χ z ̃ Ai [ x ̃ ( z ̃ 2 ) 2 + i a z ̃ ] × exp ( a x ̃ a z ̃ 2 2 i ( z ̃ 3 12 a 2 z ̃ 2 x ̃ z ̃ 2 ) ) ,
E y ( x , z ) = A 2 π e i q x ̃ | q | z ̃ e a q 2 e i ( q 3 3 a 2 q i a 3 ) 3 d q .
E y ( x , z ) = n χ n χ c ( q ) e i q x ̃ + i χ 2 n 2 q 2 z ̃ d q .
τ ( q ) = 2 ζ 1 ( q ) ζ 1 ( q ) + ζ 2 ( q ) e i ( ζ 1 ( q ) ζ 2 ( q ) ) z ̃ 0 ,
E y ( x , z ) = n χ n χ τ ( q ) c ( q ) e i q x ̃ + i χ 2 q 2 z ̃ d q .

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