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 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 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 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 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 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 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 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]

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 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 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 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.

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

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

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).

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

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

Fig. 8.
Fig. 8.

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

Fig. 9.
Fig. 9.

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

Tables (1)

Tables Icon

Table 1. Values of parameters used in the numerical simulations

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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