Abstract

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described h e by the UGO, fully agrees with the Airy beam solution. These observations explain that the “weak-diffraction” and the “self healing” properties are generated, in fact, by a continuum of sideways contributions to the field, and not by local self-curving dynamics. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

M. A. Bandres, "Accelerating beams," Opt. Lett. 34, 3791-3793 (2009).
[CrossRef]

2008 (4)

2007 (4)

2001 (1)

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

1979 (1)

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

1976 (1)

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

1969 (1)

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

1965 (1)

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

1964 (1)

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

1957 (1)

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 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.

Berry, M. V.

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

Besieris, I. M.

Boersma, J.

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Brokly, J.

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

Broky, J.

Chester, C.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Christodoulides, D. N.

Deschamps, G. A.

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

Dogariu, A.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Felsen, L. B.

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Gutierrez-Vega, J. C.

Heyman, E.

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

Lee, S. W.

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

Lewis, R. M.

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Ludwig, D.

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

Shaarawi, A. M.

Siviloglou, G. A.

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Am. J. Phys. (1)

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

Bull. Amer. Math. Soc. (1)

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

J. Opt. Soc. Am. (1)

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Math. Phys. (1)

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Nature Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

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

Proc. of Cambridge Phi.Soc. (1)

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

Other (1)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

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