Abstract

We derive a general form of Airy wave function which satisfies paraxial equation of diffraction. Based on this, we propose a new form of Airy beam, which is composed of two symmetrical Airy beams which accelerate mutually in the opposite directions. This ‘dual’ Airy beam shows several distinguishing features: it has a symmetric transverse intensity pattern and improved self-regeneration property. In addition, we can easily control the propagation direction. We also propose ‘quad’ Airy beam, which forms a rectangular shaped optical array of narrow beams that travel along a straight line. We can control its propagation direction without changing transverse intensity patterns. These kinds of superposed optical beams are expected to be useful for various applications with their unique properties.

©2010 Optical Society of America

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References

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  • Publication
  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  2. 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]
  3. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010).
    [Crossref] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
  5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref] [PubMed]
  7. 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(13), 2260–2262 (2010).
    [Crossref] [PubMed]
  8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  9. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
    [Crossref] [PubMed]
  10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  11. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
    [Crossref]
  13. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [Crossref]
  14. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010).
    [Crossref] [PubMed]
  15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
    [Crossref] [PubMed]
  16. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref] [PubMed]

2010 (5)

2008 (3)

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

2000 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Broky, J.

Chávez-Cerda, S.

Chen, Z.

Cheng, H.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

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(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Dogariu, A.

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Gutiérrez-Vega, J. C.

Heyman, E.

Hu, Y.

Huang, S.

Iturbe-Castillo, M. D.

Kaganovsky, Y.

Lou, C.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

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(2), 103–106 (2010).
[Crossref]

Salandrino, A.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Siviloglou, G. A.

Tian, J.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[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(2), 103–106 (2010).
[Crossref]

Xu, J.

Zang, W.

Zhang, P.

Zhou, W.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Nat. Photonics (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 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(2), 103–106 (2010).
[Crossref]

Nature (2)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Supplementary Material (3)

Media 1: AVI (1435 KB)     
Media 2: AVI (795 KB)     
Media 3: AVI (873 KB)     

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Figures (8)
Fig. 1
Fig. 1 (a) Definition of (launch) angle and the coordinate system. Dual Airy beam profiles when (b) α = − 1.45 , γ = − 8 and (c) α = − 1.65 , γ = − 10 .

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Fig. 2
Fig. 2 Numerical results for the self-regeneration property: (a) single Airy beam and (b) dual Airy beam.

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Fig. 3
Fig. 3 Numerical results for the directional property of dual Airy beams: (a) β = − 2 (Media 1, 1.44 MB) and (b) β = 2 .

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Fig. 4
Fig. 4 Intensity patterns are not affected by the beam propagation direction: (a) β = 0 and (b) β > 0 .

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Fig. 5
Fig. 5 Numerical results of the intensity patterns at z = 80 μ m : (a) β = − 2 and (b) β = 2 .

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Fig. 6
Fig. 6 Transverse intensity patterns of quad Airy beams, changing the propagation distance from z = 0 μ m to plane (a) z = 40 μ m (Media 2, 794 KB), (b) z = 80 μ m , (c) z = 120 μ m , and (d) z = 160 μ m .

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Fig. 7
Fig. 7 Directional property of quad Airy beams at z = 80 μ m under the changes in two directional factors (a) β x = 1 , β y = 0 , (b) β x = 0 , β y = 1 , (c) β x = 1 , β y = 1 , and (d) β x = − 2 , β y = − 2 (Media 3, 873 KB). These results prove the invariant property of transverse intensity patterns of quad Airy beams.

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Fig. 8
Fig. 8 Magnified image of the transverse intensity pattern of quad Airy beam shown in Fig. 7. In our numerical results, radius of the individual optical beam spot is about 0.5 μ m . This uniform optical array is almost maintained in some range of propagation distance. In this range, we may use it as a multiple optical tweezer.

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Tables (4)

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Table 1 Summary of roles played by the constants of integration

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Table 2 Simulation parameters used in Fig. 1(b) and (c)

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Table 3 Simulation parameters used in Fig. 3

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Table 4 Parameters for the ideal dual Airy beam

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Equations (37)

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i ∂ ϕ ∂ ξ + 1 2 ∂ 2 ϕ ∂ s 2 = 0 ,
ϕ ( s , ξ ) = A i [ a s + p ( ξ ) ] exp [ q ( ξ ) s + r ( ξ ) ] ,
p ( ξ ) = − 1 4 a 4 ξ 2 + i c 1 a ξ + c 2 ,
q ( ξ ) = i 1 2 a 3 ξ + c 1 ,
r ( ξ ) = − i 1 12 a 6 ξ 3 − 1 2 c 1 a 3 ξ 2 + i 1 2 ( a 2 c 2 + c 1 2 ) ξ + c 3 ,
ϕ ( s , ξ ) = A i [ k 1 ( s , ξ ) + i k 2 ( ξ ) ] exp [ Re { c 1 } s + i ( 1 2 a 3 ξ + Im { c 1 } ) s + k 3 ( ξ ) + i k 4 ( ξ ) ] ,
k 1 ( s , ξ ) = a s − 1 4 a 4 ξ 2 − Im { c 1 } a ξ + Re { c 2 } ,
k 2 ( ξ ) = Re { c 1 } a ξ + Im { c 2 } ,
k 3 ( ξ ) = − 1 2 Re { c 1 } a 3 ξ 2 − 1 2 ( Im { c 2 } a 2 + 2 Re { c 1 } Im { c 1 } ) ξ ,
k 4 ( ξ ) = − 1 12 a 6 ξ 3 − 1 2 Im { c 1 } a 3 ξ 2 + 1 2 ( Re { c 2 } a 2 + Re 2 { c 1 } − Im 2 { c 1 } ) ξ .
s = 1 4 a 3 ξ 2 + Im { c 1 } ξ − Re { c 2 } a = 1 4 a 3 ( ξ + 2 Im { c 1 } a 3 ) 2 − Im 2 { c 1 } a 3 − Re { c 2 } a .
θ i = tan − 1 ( Im { c 1 } k x 0 ) .
x i = − x 0 Re { c 2 } a .
z p = − ( k x 0 2 a ) ( Im { c 2 } Re { c 1 } ) .
Im { c 1 , 1 } = + α + β ,
Im { c 1 , 2 } = − α + β .
θ D = tan − 1 ( β k x 0 ) ,
ϕ Q ( s x , s y , ξ x , ξ y ) = ϕ x , D ( s x , ξ x ) ϕ y , D ( s y , ξ y ) ,
θ Q = tan − 1 ( 1 k ( β x x 0 ) 2 + ( β y y 0 ) 2 ) .
ϕ D ( s , ξ ) = ϕ 1 ( s , ξ ) + ϕ 2 ( s , ξ ) ,
ϕ 1 ( s , ξ ) = A i [ s + p 1 ( ξ ) ] exp [ q 1 ( ξ ) s + r 1 ( ξ ) ] ,
p 1 ( ξ ) = − 1 4 ξ 2 − ( α + β ) ξ + γ ,
q 1 ( ξ ) = i 1 2 ξ + i ( α + β ) ,
r 1 ( ξ ) = − i 1 12 ξ 3 − i 1 2 ( α + β ) ξ 2 + i 1 2 ( γ − α 2 − β 2 − 2 α β ) ξ .
ϕ 2 ( s , ξ ) = A i [ − s + p 2 ( ξ ) ] exp [ q 2 ( ξ ) s + r 2 ( ξ ) ] ,
p 2 ( ξ ) = − 1 4 ξ 2 + ( − α + β ) ξ + γ ,
q 2 ( ξ ) = − i 1 2 ξ + i ( − α + β ) ,
r 2 ( ξ ) = − i 1 12 ξ 3 + i 1 2 ( − α + β ) ξ 2 + i 1 2 ( γ − α 2 − β 2 + 2 α β ) ξ .
| ϕ 1 + ϕ 2 | 2 = | A i [ h 1 ( s , ξ ) ] | 2 + | A i [ h 2 ( s , ξ ) ] | 2
+ 2 A i [ h 1 ( s , ξ ) ] A i [ h 2 ( s , ξ ) ] cos [ ( ξ + 2 α ) ( s − β ξ ) ] ,
h 1 ( s , ξ ) = ( s − β ξ ) − 1 4 ξ 2 − α ξ + γ ,
h 2 ( s , ξ ) = − ( s − β ξ ) − 1 4 ξ 2 − α ξ + γ .
∂ s x , d ∂ ξ x = β x ,
∂ s y , d ∂ ξ y = β y .
∂ x d ∂ z = β x k x 0 ,
∂ y d ∂ z = β y k y 0 .
∂ | r d → | ∂ z = 1 k ( β x x 0 ) 2 + ( β y y 0 ) 2 ,

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