Abstract

We study the Wigner distribution function (WDF) of an Airy beam. The analytical expression of the WDF of an Airy beam is obtained. Numerical and graphical results of the WDF of an Airy beam provide an intuitive picture to explain the intriguing features of an Airy beam, such as weak diffraction, curved propagation, and self-healing. Our results confirm that these novel properties of an Airy beam are attributed to the continuum of sideways contributions to the field.

© 2011 Optical Society of America

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  1. G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. 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]
  6. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
    [CrossRef]
  7. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16, 9411–9416(2008).
    [CrossRef] [PubMed]
  8. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
    [CrossRef]
  9. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009).
    [CrossRef] [PubMed]
  10. S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
    [CrossRef] [PubMed]
  11. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
    [CrossRef] [PubMed]
  12. R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
    [CrossRef]
  13. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
    [CrossRef]
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2010 (3)

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[CrossRef] [PubMed]

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010).
[CrossRef] [PubMed]

2009 (3)

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[CrossRef]

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (3)

2005 (1)

2002 (1)

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

2000 (1)

1997 (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

1996 (1)

1995 (1)

R. Gase, “Representation of Laguerre-Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

1994 (1)

1986 (1)

1979 (2)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–754 (1932).
[CrossRef]

Agarwal, G. S.

Alfano, R. R.

Anderson, D.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[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.

Bastiaans, M. J.

Baumgartl, J.

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

Berry, M. V.

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

Brokly, J.

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

Broky, J.

Chen, R. P.

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Christodoulides, D. N.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[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. Brokly, 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]

Chu, X. X.

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Dholakia, K.

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

Dogariu, A.

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. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Dragoman, D.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[CrossRef]

Fedele, R.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Fleischer, J. W.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[CrossRef] [PubMed]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[CrossRef]

Gase, R.

R. Gase, “Representation of Laguerre-Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

Gutierrez-Vega, J. C.

Hall, B.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Hansson, T.

Hasegawa, A.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Heyman, E.

Jia, S.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[CrossRef] [PubMed]

Kaganovsky, Y.

Kolesik, M.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
[CrossRef] [PubMed]

Lee, J.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[CrossRef] [PubMed]

Lisak, M.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Mazilu, M.

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

Moloney, J.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
[CrossRef] [PubMed]

Mortel, J.

Novitsky, A. V.

Novitsky, D. V.

Österberg, U.

Polynkin, P.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
[CrossRef] [PubMed]

Semenov, V. E.

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Österberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Shukla, P. K.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Simon, R.

Siviloglou, G. A.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[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. Brokly, 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]

Soares, M.

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

Sun, D.

Sztul, H. I.

Vallée, O.

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

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[CrossRef]

Wang, H.

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–754 (1932).
[CrossRef]

Yin, C. F.

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Zhao, D.

Am. J. Phys. (1)

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

IEEE J. Quantum Electron. (1)

R. Gase, “Representation of Laguerre-Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Nat. Photon. (2)

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

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–754 (1932).
[CrossRef]

Phys. Rev. A (1)

R. P. Chen, C. F. Yin, X. X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82, 043832(2010).
[CrossRef]

Phys. Rev. Lett. (3)

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104, 253904 (2010).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103, 123902 (2009).
[CrossRef] [PubMed]

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

Phys. Scr. (1)

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. E. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr. T98, 12–17 (2002).
[CrossRef]

Prog. Opt. (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Wigner distribution function of an Airy beam in the ( x , p x ) plane of the phase space, W ( x , p x ; z = 0 ) , in the plane z = 0 for several values of the modulation parameter: (a)  a = 0.02 , (b)  a = 0.1 , (c)  a = 0.3 , (d)  a = 0.5 .

Fig. 2
Fig. 2

Wigner distribution function of an Airy beam, W ( x , p x ; z = 5 z 0 ) , in the propagation distance z = 5 z 0 , the parameters are same as in Fig. 1.

Fig. 3
Fig. 3

Wigner distribution function of an Airy beam, W ( x , y , p x , p y ; z = 0 ) , with a = 0.02 in the plane z = 0 for several different positions: (a)  x = 0 , y = 0 ; (b)  x = 0 , y = x 0 ; (c)  x = x 0 , y = 2 x 0 ; (d)  x = 2 x 0 , y = 5 x 0 .

Fig. 4
Fig. 4

Wigner distribution function of an Airy beam, W ( x , y , p x , p y ; z = 5 z 0 ) , in the propagation distance z = 5 z 0 , the parameters are same as in Fig. 3.

Equations (8)

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φ ( x ; z ) = A i [ x x 0 ( z 2 z 0 ) 2 + i a z z 0 ] exp [ a x x 0 a 2 ( z z 0 ) 2 i 1 12 ( z z 0 ) 3 + i a 2 2 z z 0 + i x 2 x 0 z z 0 ] ,
W ( x , p ; z ) = φ ( x + x / 2 , z ) φ * ( x - x / 2 , z ) exp ( i p x ) d x ,
W ( x , p ; z ) = exp [ 2 a x x 0 a ( z z 0 ) 2 ] A i [ x + x / 2 x 0 ( z 2 z 0 ) 2 + i a z z 0 ] A i * [ x x / 2 x 0 ( z 2 z 0 ) 2 + i a z z 0 ] exp ( i p x ) d x .
A i ( x ) = 1 2 π exp ( i u 3 / 3 + i x u ) d u .
W ( x , p ; z ) = 1 4 π 2 exp [ 2 a x x 0 a ( z z 0 ) 2 ] × exp [ i ( u 3 + v 3 3 + 2 u x + u x + 2 v x v x 2 x 0 u z 2 + v z 2 4 z 0 2 p x ) + a v z a u z z 0 ] d x d u d v .
W ( x , p ; z ) = 1 2 π exp ( 2 a x x 0 a z 2 z 0 2 2 p a x 0 z z 0 ) × δ ( u v 2 x 0 p ) exp [ i ( u 3 3 + v 3 3 + u x x 0 u z 2 4 z 0 2 + v x x 0 v z 2 4 z 0 2 ) ] d u d v = 1 2 π exp ( 2 a x x 0 a z 2 z 0 2 2 p a x 0 z z 0 + 4 p 2 x 0 2 + i 2 p x + i 8 p 3 x 0 3 3 i p x 0 z 2 2 z 0 2 ) × exp [ i ( 2 v 3 3 + 2 p v 2 x 0 + 2 v x x 0 v z 2 2 z 0 2 ) ] d v .
W ( x , p ; z ) = 2 2 / 3 exp ( 2 a x x 0 a z 2 z 0 2 2 p a x 0 z z 0 ) A i [ 2 2 / 3 ( x x 0 z 2 4 z 0 2 + p 2 x 0 2 ) ] .
W ( x , y , p x , p y ; z ) = 2 4 / 3 exp ( 2 a x x 0 + 2 a y y 0 2 a z 2 z 0 2 2 p x a x 0 z z 0 2 p y a y 0 z z 0 ) × A i [ 2 2 / 3 ( x x 0 z 2 4 z 0 2 + p x 2 x 0 2 ) ] A i [ 2 2 / 3 ( y y 0 z 2 4 z 0 2 + p y 2 y 0 2 ) ] ,

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