Abstract

We theoretically analyze Airy beams by solving the exact vectorial Helmholtz equation using boundary conditions at a diffraction aperture. As result, the diffracted beams are obtained in the whole space; thus, we demonstrate that the parabolic trajectories are larger than those previously reported, showing that the Airy beams start to form before the Fourier plane. We also demonstrate the possibility of using a new type of Airy beams (SAiry beams) with finite energy that can be generated at the focal plane of the lens due to diffraction by a circular aperture of a spherical wave modified by a cubic phase. The finite energy ensured by the principle of conservation of energy of a diffracted 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, “Observation of accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 ( 2007).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 ( 2007).
    [Crossref] [PubMed]
  3. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 ( 2008).
    [Crossref]
  4. J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
    [Crossref] [PubMed]
  5. P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
    [Crossref] [PubMed]
  6. 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]
  7. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 ( 2008).
    [Crossref] [PubMed]
  8. 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]
  9. M. A. Bandres and J. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 ( 2007).
    [Crossref] [PubMed]
  10. H. Sztul and R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16, 9411–9416 ( 2008).
    [Crossref] [PubMed]
  11. R. K. Luneburg, Mathematical theory of optics (University California Press, Berkeley, California, 1964).
  12. J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
    [Crossref]

2009 (4)

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[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]

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
[Crossref]

2008 (4)

2007 (3)

Alfano, R.

Arie, 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]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

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

Broky, J.

Christodoulides, D. N.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 ( 2008).
[Crossref] [PubMed]

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]

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

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

Day, D.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

Dholakia, K.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

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

Dogariu, A.

Ellenbogen, T.

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]

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]

Gu, M.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

Guo, J.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
[Crossref]

Gutierrez-Vega, J.

Hannappel, G.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

Koleskik, M.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

Luneburg, R. K.

R. K. Luneburg, Mathematical theory of optics (University California Press, Berkeley, California, 1964).

Mazilu, M.

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

Min, Y.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
[Crossref]

Moloney, J. V.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

Polynkin, P.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

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]

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]

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

Stevenson, D. J.

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

Sztul, H.

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]

Zhao, X.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
[Crossref]

Lab on a Chip (1)

J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip 9, 1334–1336 ( 2009).
[Crossref] [PubMed]

Nat. Photonics (2)

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]

Opt. Comm. (1)

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. 282, 1511–1515 ( 2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Science (1)

P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science 324, 229–232 ( 2009).
[Crossref] [PubMed]

Other (1)

R. K. Luneburg, Mathematical theory of optics (University California Press, Berkeley, California, 1964).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the generation of Airy beams.

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Fig. 2.
Fig. 2.

Normalized intensity distribution for Airy beams at different Z=(z-f) planes of PRF region.

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Fig. 3.
Fig. 3.

Normalized intensity distribution for Airy beams at different Z=(z-f) planes of PF region.

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Fig. 4.
Fig. 4.

Normalized intensity distribution for Airy beams at different Z=(z-f) planes of the PF region numerically calculated (a),(c),(e),(g) and analytically obtained by means of the paraxial Helmholtz solutions [1],(b),(d),(f), (h).

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Fig. 5.
Fig. 5.

Normalized maximum intensities of Airy beams (blue) and SAiry Beams (red) at PRF and PF regions

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Fig. 6.
Fig. 6.

Transverse acceleration of Airy beams (blue) and SAiry Beams (red) at PRF and PF regions.

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Fig. 7.
Fig. 7.

Normalized intensity distribution for SAiry beams at different planes Z=(z-f) of PRF region.

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Fig. 8.
Fig. 8.

Normalized intensity distribution for SAiry beams at different planes Z=(z-f) of PF region.

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Fig. 9.
Fig. 9.

Real (a)–(b) and Imaginary (c)–(d) part of t(xo ,yo ) for SAiry and Airy beams respectively

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Tables (1)

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Table 1. Values of parameters used in the numerical simulations

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Equations (11)

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∂ 2 E → ∂ x 2 + ∂ 2 E → ∂ y 2 + ∂ 2 E → ∂ z 2 + k 2 E → = 0
E → o x o y o O = E x x o y o 0 x ̂ + E y x o y o 0 y ̂
E x x, y z = z 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ E x x o y o 0 ikR − 1 R 3 exp ( ik R ) dx o dy o
E y x y z = − z 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ E y x o y o 0 ikR − 1 R 3 exp ( ik R ) dx o dy o
E z ( x , y , z ) = − x z E x x y z − y z E y x y z −
1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ ( x o E x x o y o 0 + y o E y x o y o 0 ) = i k R − 1 R 3 e x p ( i k R ) dx o dy o
E → o x o y o 0 = t ( x o , y o ) exp ( − ik ( xo 2 + yo 2 + zo 2 ) 1 2 ) ( xo 2 + yo 2 + zo 2 ) 1 2 [ cos ( α ) x ̂ + sin ( α ) y ̂ ]
I x y z = ∣ E x y z ∣ 2 = ∣ E x x y z ∣ 2 + ∣ E y x y z ∣ 2 + ∣ E z x y z ∣ 2
E x x y z = − z 2 π ∬ D E o x o y o 0 ikR − 1 R 3 exp ( ikR ) dx o dy o
E z ( x , y , z ) = − x z E x x y z − 1 2 π
∬ D x o E o x o y o 0 ikR − 1 R 3 exp ( ikR ) dx o dy o

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