Abstract

We investigate both theoretically and experimentally the self-healing properties of accelerating Airy beams. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments is also studied experimentally. Our observations are in excellent agreement with numerical simulations.

© 2008 Optical Society of America

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References

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  1. G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
    [CrossRef] [PubMed]
  4. I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2249 (2007).
    [CrossRef] [PubMed]
  5. M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  6. K. Dholakia, "Optics: Against the spread of the light," Nature 451, 413 (2008).
    [CrossRef] [PubMed]
  7. M. A. Bandres and J. C. Gutiérrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007).
    [CrossRef] [PubMed]
  8. H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008).
    [CrossRef] [PubMed]
  9. M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
    [CrossRef]
  10. A. V. Gorbach and D. V. Skryabin, "Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers," Opt. Express 16, 4858-4865 (2008).
    [CrossRef] [PubMed]
  11. P. Saari, "Laterally accelerating airy pulses," Opt. Express 16, 10303-10308 (2008).
    [CrossRef] [PubMed]
  12. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  13. D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  14. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
    [CrossRef]
  15. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
    [CrossRef]
  16. V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
    [CrossRef] [PubMed]
  17. O. Vallée and M. Soares, Airy Functions and Applications to Physics, (Imperial College Press, London, 2004).
  18. 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]
  19. H. C.  van de Hulst, Light Scattering by Small Particles, (Dover Publication Inc., New York, 1981).
  20. S. Prahl, "Mie Scattering Calculator," (2008). http://omlc.ogi.edu/calc/mie_calc.html
  21. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
    [CrossRef]
  22. A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]

2008 (6)

2007 (4)

2005 (1)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

2003 (1)

2002 (1)

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

1996 (1)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (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 (1)

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

1979 (1)

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

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

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]

Amarande, S.

Asorey, M.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[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.

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]

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Besieris, I. M.

Boothroyd, S. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
[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]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Christodoulides, D. N.

Chrostowski, J.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Dholakia, K.

K. Dholakia, "Optics: Against the spread of the light," Nature 451, 413 (2008).
[CrossRef] [PubMed]

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Dogariu, A.

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]

Facchi, P.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Gorbach, A. V.

Gutiérrez-Vega, J. C.

MacDonald, R. P.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Man'ko, V. I.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

Marmo, G.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

McGloin, D.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Miceli, J. J.

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

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Okamoto, T.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Pascazio, S.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

Saari, P.

Shaarawi, A. M.

Sibbett, W.

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Siviloglou, G. A.

Skryabin, D. V.

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]

Sudarshan, E. C. G.

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

Syrett, B. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Sztul, H. I.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[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]

Am. J. Phys. (1)

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

Contemp. Phys. (1)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Nature (2)

K. Dholakia, "Optics: Against the spread of the light," Nature 451, 413 (2008).
[CrossRef] [PubMed]

V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Opt. Commun. (2)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A. (2)

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]

M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008).
[CrossRef]

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]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Other (3)

H. C.  van de Hulst, Light Scattering by Small Particles, (Dover Publication Inc., New York, 1981).

S. Prahl, "Mie Scattering Calculator," (2008). http://omlc.ogi.edu/calc/mie_calc.html

O. Vallée and M. Soares, Airy Functions and Applications to Physics, (Imperial College Press, London, 2004).

Supplementary Material (2)

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» Media 2: AVI (1329 KB)     

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Figures (15)

Fig. 1.
Fig. 1.

Intensity cross-sections of an experimentally observed two-dimensional Airy beam with x 0=150µm and y 0=77µm and a=0.08 at (a) z=0 and (b) z=20cm.

Fig. 2.
Fig. 2.

Motion of the main lobe of a symmetric (x 0=y 0=77µm) Airy beam when launched at θx =-2mrad and θy =2mrad.

Fig. 3.
Fig. 3.

Schematic representation of the Poynting vector S⃗ in a Cartesian system of coordinates.

Fig. 4.
Fig. 4.

Direction of the Poynting vector S⃗ (red line) associated with an one-dimensional finite energy Airy when a=0.05 and x 0=77µm as a function of distance ξ. The blue line depicts the direction of the tangent unit vector of an ideal Airy beam (a=0 and x 0=77µm).

Fig. 5.
Fig. 5.

(a). Experimental set-up, (b). Two-dimensional cubic phase mask, and (c). experimentally observed Airy beam.

Fig. 6.
Fig. 6.

Self-healing of an Airy beam when its main lobe is blocked. Observed intensity profile at (a) the input z=0, (b) z=11cm, and (c) z=30cm. The corresponding numerical simulations are shown in (d–f).

Fig. 7.
Fig. 7.

(a). Main lobe in isolation is observed at z=0, (b) and after diffraction at z=30cm. (c), (d) Corresponding intensity profiles as obtained from theory.

Fig. 8.
Fig. 8.

Calculated transverse power flow S⃗ at (a) z=1cm, and (b) z=11cm.

Fig. 9.
Fig. 9.

Self-healing of an Airy beam when all the internal lobes are blocked. Observed intensity profiles at (a) the input z=0 and (b) z=16cm. The corresponding numerical simulations are shown in (c) and (d).

Fig. 10.
Fig. 10.

Transverse power flow S⃗ revealing the self-healing mechanism at z=1cm.

Fig. 11.
Fig. 11.

Self-healing of an Airy beam when a region of 9 lobes is blocked. Observed intensity profiles at (a) the input z=0 and (b) z=24cm. The corresponding numerical simulations are shown in (c) and (d). Calculated transverse power flow S⃗ at (e) z=1cm.

Fig. 12.
Fig. 12.

Self-healing of an Airy beam when 3 lobes along y axis are blocked. Observed intensity cross-sections at (a) the input z=0 and (b) z=18cm. The corresponding numerical simulations are shown in (c) and (d).

Fig. 13.
Fig. 13.

Self-healing of an Airy beam when propagating in a suspension of 0.5µm silica micro-spheres in pure water. Observed intensity profiles at (a) the input z=0, (b) z=5cm, and (c) z=10cm.

Fig. 14.
Fig. 14.

Self-healing of an Airy beam in a suspension of 1.5µm silica micro-spheres in pure water after a propagation distance of z=10cm.

Fig. 15.
Fig. 15.

Propagation in a turbulent medium of (a) (Media 1) an optical Airy beam and (b) (Media 2) a comparable Gaussian beam.

Equations (9)

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i ϕ z + 1 2 k 2 ϕ x 2 + 1 2 k 2 ϕ y 2 = 0
ϕ ( x , y , z ) = Π m = x , y u m ( s m , ξ m )
u m ( s m , ξ m ) = Ai [ s m ( ξ m 2 ) 2 v m ξ m + ia m ξ m ] × exp [ a m s m ( a m ξ m 2 2 ) a m v m ξ m ]
× exp [ i ( ( ξ m 3 12 ) + ( ( a m 2 v m 2 + s m ) ξ m 2 ) + v m s m ( v m ξ m 2 2 ) ) ] .
x d = 1 4 k 2 x 0 3 z 2 + θ x z
y d = 1 4 k 2 y 0 3 z 2 + θ y z .
S = S z + S = 1 2 η 0 ϕ 2 z ̂ + i 4 η 0 k [ ϕ ϕ * ϕ * ϕ ]
tan ψ = S y S x = θ y + z 2 k 2 y 0 3 θ x + z 2 k 2 x 0 3 .
tan δ = S x 2 + S y 2 S z = ( θ x + z 2 k 2 x 0 3 ) 2 + ( θ y + z 2 k 2 y 0 3 ) 2 .

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