Abstract

The three-dimensional Airy beam (AiB) is thoroughly explored from a wave-theory point of view. We utilize the exact spectral integral for the AiB to derive local ray-based solutions that do not suffer from the limitations of the conventional parabolic equation (PE) solution and are valid far beyond the paraxial zone and for longer ranges. The ray topology near the main lobe of the AiB delineates a hyperbolic umbilic catastrophe, consisting of a cusped double-layered caustic. In the far zone this caustic is deformed and the field loses its beam shape. The field in the vicinity of this caustic is described uniformly by a hyperbolic umbilic canonical integral, which is structured explicitly on the local geometry of the caustic. In order to accommodate the finite-energy AiB, we also modify the conventional canonical integral by adding a complex loss parameter. The canonical integral is calculated using a series expansion, and the results are used to identify the validity zone of the conventional PE solution. The analysis is performed within the framework of the nondispersive AiB where the aperture field is scaled with frequency such that the ray skeleton is frequency independent. This scaling enables an extension of the theory to the ultrawideband regime and ensures that the pulsed field propagates along the curved beam trajectory without dispersion, as will be demonstrated in a subsequent publication.

© 2012 Optical Society of America

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References

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  1. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  2. G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  3. M. Bandres and J. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007).
    [CrossRef]
  4. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [CrossRef]
  5. J. Broky, G. Siviloglou, A. Dogariu, and D. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
    [CrossRef]
  6. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
    [CrossRef]
  7. S. Vo, K. Fuerschbach, K. Thompson, M. Alonso, and J. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A 27, 2574–2582 (2010).
    [CrossRef]
  8. Y. Gu and Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35, 3456–3458 (2010).
    [CrossRef]
  9. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [CrossRef]
  10. 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]
  11. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010).
    [CrossRef]
  12. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
    [CrossRef]
  13. D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).
  14. Y. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. 7, 664–673 (1964).
  15. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
    [CrossRef]
  16. Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28, 1243–1255 (2011).
    [CrossRef]
  17. P. Saari, “Laterally accelerating Airy pulses,” Opt. Express 16, 10303–10308 (2008).
    [CrossRef]
  18. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
    [CrossRef]
  19. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
    [CrossRef]
  20. E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
    [CrossRef]
  21. Y. Kaganovsky and E. Heyman are preparing a manuscript to be called “3D Airy pulsed beams.”
  22. H. Trinkaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
    [CrossRef]
  23. M. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, Vol. 18 (Elsevier, 1980), pp. 257–346.
    [CrossRef]
  24. J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, 1999).
  25. J. Nye, “Dislocation lines in the hyperbolic umbilic diffraction catastrophe,” Proc. R. Soc. A 462, 2299–2313 (2006).
    [CrossRef]
  26. M. Berry and C. Howls, “Axial and focal-plane diffraction catastrophe integrals,” J. Phys. A 43, 375206 (2010).
    [CrossRef]
  27. T. Poston and I. Stewart, Catastrophe Theory and Its Applications, Vol. 2 (Dover, 1996).
  28. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, 1993).
  29. Digital Library of Mathematical Functions, National Institute of Standards and Technology (2011), http://dlmf.nist.gov .
  30. M. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  31. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech. 17, 533–559 (1967).
  32. J. Duistermaat, “Oscillatory integrals, Lagrange immersions and unfolding of singularities,” Commun. Pure Appl. Math. 27, 207–281 (1974).
    [CrossRef]
  33. Actually, the peak trajectory is described by replacing the 0 on the right-hand side of Eqs. (4)–(5) by the asymptotically small parameter (kβ)−2/3a1′ where a1′≃−1.0188 is the first zero of Ai′(x) [29, Table 9.9.1].
  34. The PE model is a special case in which the effect of the propagation is only a translation [see Eqs. (A23)–(A24)], and therefore the caustic retains its unstable shape.
  35. M. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
    [CrossRef]
  36. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).
  37. Specifically, in the present case, one applies a change of variables such that the stationary points of the phase function τ in the exact spectral integral in Eq. (24) are mapped to those of the phase Φ in Eq. (32) while satisfying the constraint that the phases at the respective points are equal; i.e., ωτ=Φ.
  38. R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1993).
  39. Y. A. Kravtzov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media, Vol. 6 of the Springer Series on Wave Phenomena (Springer-Verlag, 1990).

2011 (2)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28, 1243–1255 (2011).
[CrossRef]

2010 (4)

2009 (2)

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

2008 (4)

2007 (3)

2006 (1)

J. Nye, “Dislocation lines in the hyperbolic umbilic diffraction catastrophe,” Proc. R. Soc. A 462, 2299–2313 (2006).
[CrossRef]

1987 (3)

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

1977 (1)

H. Trinkaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[CrossRef]

1976 (1)

M. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

1975 (1)

M. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

1974 (1)

J. Duistermaat, “Oscillatory integrals, Lagrange immersions and unfolding of singularities,” Commun. Pure Appl. Math. 27, 207–281 (1974).
[CrossRef]

1967 (1)

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech. 17, 533–559 (1967).

1965 (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).

1964 (1)

Y. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. 7, 664–673 (1964).

1957 (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
[CrossRef]

Alonso, M.

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]

Bandres, M.

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.

M. Berry and C. Howls, “Axial and focal-plane diffraction catastrophe integrals,” J. Phys. A 43, 375206 (2010).
[CrossRef]

M. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

M. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

M. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, Vol. 18 (Elsevier, 1980), pp. 257–346.
[CrossRef]

Bleistein, N.

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech. 17, 533–559 (1967).

Broky, J.

Chester, C.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
[CrossRef]

Christodoulides, D.

Christodoulides, D. N.

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

Drepper, F.

H. Trinkaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[CrossRef]

Duistermaat, J.

J. Duistermaat, “Oscillatory integrals, Lagrange immersions and unfolding of singularities,” Commun. Pure Appl. Math. 27, 207–281 (1974).
[CrossRef]

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]

Felsen, L. B.

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
[CrossRef]

Fuerschbach, K.

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]

Gbur,

Gilmore, R.

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1993).

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Gu, Y.

Gutiérrez-Vega, J.

Heyman, E.

Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A 28, 1243–1255 (2011).
[CrossRef]

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

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

Y. Kaganovsky and E. Heyman are preparing a manuscript to be called “3D Airy pulsed beams.”

Howls, C.

M. Berry and C. Howls, “Axial and focal-plane diffraction catastrophe integrals,” J. Phys. A 43, 375206 (2010).
[CrossRef]

Kaganovsky, Y.

Kolesik, M.

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

Kravtsov, Y.

Y. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. 7, 664–673 (1964).

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, 1993).

Kravtzov, Y. A.

Y. A. Kravtzov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media, Vol. 6 of the Springer Series on Wave Phenomena (Springer-Verlag, 1990).

Ludwig, D.

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).

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. V.

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

Nye, J.

J. Nye, “Dislocation lines in the hyperbolic umbilic diffraction catastrophe,” Proc. R. Soc. A 462, 2299–2313 (2006).
[CrossRef]

J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, 1999).

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, 1993).

Y. A. Kravtzov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media, Vol. 6 of the Springer Series on Wave Phenomena (Springer-Verlag, 1990).

Polynkin, P.

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

Poston, T.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications, Vol. 2 (Dover, 1996).

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Rolland, J.

Saari, P.

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Siviloglou, G.

Siviloglou, G. A.

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

Stewart, I.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications, Vol. 2 (Dover, 1996).

Thompson, K.

Trinkaus, H.

H. Trinkaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[CrossRef]

Upstill, C.

M. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, Vol. 18 (Elsevier, 1980), pp. 257–346.
[CrossRef]

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
[CrossRef]

Vo, S.

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]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Adv. Phys. (1)

M. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Bull. Amer. Math. Soc. (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).

Commun. Pure Appl. Math. (1)

J. Duistermaat, “Oscillatory integrals, Lagrange immersions and unfolding of singularities,” Commun. Pure Appl. Math. 27, 207–281 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

Izv. VUZ Radiofiz. (1)

Y. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. 7, 664–673 (1964).

J. Math. Mech. (1)

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech. 17, 533–559 (1967).

J. Opt. Soc. Am. A (2)

J. Phys. A (3)

H. Trinkaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[CrossRef]

M. Berry and C. Howls, “Axial and focal-plane diffraction catastrophe integrals,” J. Phys. A 43, 375206 (2010).
[CrossRef]

M. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

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. Lett. (2)

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

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Proc. Camb. Phil. Soc. Math. Phys. Sci. (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. Math. Phys. Sci. 53, 599–611 (1957).
[CrossRef]

Proc. R. Soc. A (1)

J. Nye, “Dislocation lines in the hyperbolic umbilic diffraction catastrophe,” Proc. R. Soc. A 462, 2299–2313 (2006).
[CrossRef]

Science (1)

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

Other (12)

Y. Kaganovsky and E. Heyman are preparing a manuscript to be called “3D Airy pulsed beams.”

M. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, Vol. 18 (Elsevier, 1980), pp. 257–346.
[CrossRef]

J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, 1999).

T. Poston and I. Stewart, Catastrophe Theory and Its Applications, Vol. 2 (Dover, 1996).

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, 1993).

Digital Library of Mathematical Functions, National Institute of Standards and Technology (2011), http://dlmf.nist.gov .

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Waves (IEEE, 1994).

Specifically, in the present case, one applies a change of variables such that the stationary points of the phase function τ in the exact spectral integral in Eq. (24) are mapped to those of the phase Φ in Eq. (32) while satisfying the constraint that the phases at the respective points are equal; i.e., ωτ=Φ.

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1993).

Y. A. Kravtzov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media, Vol. 6 of the Springer Series on Wave Phenomena (Springer-Verlag, 1990).

Actually, the peak trajectory is described by replacing the 0 on the right-hand side of Eqs. (4)–(5) by the asymptotically small parameter (kβ)−2/3a1′ where a1′≃−1.0188 is the first zero of Ai′(x) [29, Table 9.9.1].

The PE model is a special case in which the effect of the propagation is only a translation [see Eqs. (A23)–(A24)], and therefore the caustic retains its unstable shape.

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Figures (16)

Fig. 1.
Fig. 1.

Field intensity |U|2 at (a) the aperture plane z=0 and (b) the z=0.8β˜ plane. The field is given by the PE solution in Eqs. (2)–(3). Beam parameters are as in Eqs. (8)–(9) with kβ˜=100, α˜=105. The beam propagates in the plane of symmetry y˜=0 in the direction φ=45°. The two coordinate systems (x,z) and (x˜,y˜) are related via Eq. (6). The zeros |U|=0 in (a) along the x and y axes are given, respectively, by x=anβ(βk)2/3, y=anβ(βk)2/3, where an with n=1,2,, are the zeros of Ai given in ([29], Table 9.9.1).

Fig. 2.
Fig. 2.

Projections of the ray directions s^ defined in Eqs. (12) and (18) onto the z=0 plane (white arrows). Also shown for reference the intensity |U|2 from Fig. 1(a). (a)–(d) Four ray species defined in Eq. (18) such that (a) θx,θy[0,π/2], (b) θx[π/2,π] and θy[0,π/2], (c) θx[0,π/2] and θy[π/2,π], (d) θx,θy[π/2,π].

Fig. 3.
Fig. 3.

Caustic of the AiB within the PE approximation (gray surface). The caustic is analyzed in Appendix A.3, defined in Eqs. (A23)–(A24) [dark and lit sheets, respectively]. The sheets are joined at the edge (dashed red line), which delineates the beam propagation trajectory. Cross-sectional cuts are shown in Fig. 4.

Fig. 4.
Fig. 4.

Cross sections of the caustics of the AiB at two different ranges: (a) z/β˜=0.2 and (b) z/β˜=0.25. Black line, the caustic of the PE approximation in Fig. 3. Gray and green lines, the exact caustic, which consists of two surfaces denoted as “caustic 1” and “caustic 2” respectively (see Subsection 3.B.2 and Fig. 5). The x˜ axis in this figure is centered about x1˜(z) the edge of caustic 1 in Eq. (A14). The insets zoom on the area near the origin where caustic 1 has a “cusped edge,” while caustic 2 has a smooth corner. One observes that the distance between caustics 1 and 2 and the PE approximated caustic increases with z (see also Fig. 6).

Fig. 5.
Fig. 5.

Exact caustic of the AiB, discussed in Subsection 3.B.2 and in Appendix A.2. It consists of two surfaces denoted as “caustic 1” and “caustic 2” (gray and green surfaces, respectively). The distance between these surfaces is hardly noticeable on the scale of this figure, but it is shown in the cross-sectional cut of the caustics in Fig. 4. Each caustic consists of two surfaces joined at an edge (dashed red lines). The ray formations of caustics 1 and 2 are described in Figs. 7 and 8, respectively.

Fig. 6.
Fig. 6.

Difference between the edges of the PE approximated and of the exact caustics. The figure plots zp-zi as a function of x˜, where zp is the coordinate of the paraxially approximated beam trajectory in Eq. (10) and zi, i=1,2 are the coordinates of the exact edges of caustic i, given in Eqs. (A16)–(A17). From Eq. (10), the horizontal range x˜/β˜=0.1 corresponds to the observation range zp/β˜=20.1 and there (zpzi)/zp5%6%.

Fig. 7.
Fig. 7.

Ray formation of caustic 1 (gray surface) of Fig. 5. Each exit point (x˜,y˜) in the aperture (red point) emits four rays (black lines) belonging to species s=1,4 of Fig. 2. Ray 4 does not touch the caustic. Rays 2, 3 touch only caustic 1 (red points). Ray 1 penetrates caustic 1 (through the white circle), touches caustic 2 (green point), penetrates caustic 1 again (through the second white circle), and finally touches caustic 1 (red point), never touching the caustic again. The figure shows traces of the exit points in the aperture (thick lines) and the corresponding traces of the points of tangency on the caustic (thin lines with the arrows indicating the direction of the traces). To emphasize the symmetry, traces with y˜0 appear as full or dashed lines. Lines of constant y˜ and of constant x˜ are plotted in magenta and blue, respectively. The traces corresponding to the edge of caustic 1 are shown as red dashed lines. The parameters used here are x˜1=0.004β˜, x˜2=0.009β˜, y˜1=0.0015β˜, y˜2=0.0065β˜, y˜0=0.

Fig. 8.
Fig. 8.

Ray formation of caustic 2 (green surface) of Fig. 5. The notations are the same as in Fig. 7. Note that caustic 2 is formed only by ray species 1.

Fig. 9.
Fig. 9.

Rays reaching an observation point at r=(x˜,y˜,z)=(0.005,0,0.17)β˜ (blue point) on the lit side of the caustic, tagged by the ray index r=1,2,2,3. The parts of the rays before and after r are depicted as solid and dashed lines, respectively, and the parts between caustic 1 and 2 are depicted as dotted lines. The ray initiation points (x˜r,y˜r)/β˜ are as follows: ray 1, (0.14,0)102; rays 2,2, (0.95,0.81)102; ray 3, (1.8,0)102. The ray direction parameters (ξr,ηr) are as follows: ray 1, (2.7,2.7)102; ray 2, (2.7,9.4)102; ray 2, (9.4,2.7)102; ray 3, (9.5,9.5)102. The paths of rays 2,2 are similar to that of ray s=1 in Fig. 7: they penetrate caustic 1 (through the white circles), touch caustic 2 (green points), penetrate caustic 1 again (upper white circles), and then touch caustic 1 (red points) after passing through r (blue point). Rays 1 and 3 propagate in the y˜=0 plane. Ray 3 touches the caustic after passing through r: it penetrates caustic 1 (through the white circle), touches the edge of caustic 2 (green point), and then penetrates back through the cusped edge of caustic 1, and at the same point it is also tangent to the cusp (red point; see discussion in the last paragraph of Subsection 3.B.2 and also Fig. 10). The path of ray 1 is similar to that of ray 3, except it already touched the caustics before passing through r. When r moves toward the cusped-edge of caustic 1 from its lit side, rays 1,2,2 coalesce. When r is on the shadow side of edges 1 such that it is between the edges of caustics 1 and 2, it is reached by two rays, which penetrate caustic 1, one that results from the coalesced rays 1,2,2 and has already touched caustic 2, and ray 3, which has not touched caustic 2 yet. These two rays coalesce as r moves toward the edge of caustic 2.

Fig. 10.
Fig. 10.

Schematic view of the edges of caustics 1 and 2 (dashed and solid red lines, respectively) in the symmetry plane y˜=0. The edges are given in Eqs. (A16)–(A17). As shown in Fig. 4, caustic 1 is actually a cuspoid but only its edge is shown here. A ray (dashed black line) penetrates through caustic 1 before the point of tangency with caustic 2. After that point it penetrates again through caustic 1 (point z^=δ^ in the figure). At that point it is also tangent to the cusp of caustic 1 (see discussion in the last paragraph of Subsection 3.B.2). Also shown, the local Cartesian coordinate system (x^,y^,z^) centered around a point on caustic 2, such that z^ is the tangent to the edge at that point, while x^ and y^=y˜ are the normal and the binormal there, respectively. This system is parameterized by the angle θz of the z^ axis with respect to the z axis, and it is used to construct the canonical integral of the field in Section 5.

Fig. 11.
Fig. 11.

Intensity |U|2 of the field in the planes (a) y¯=0 and (b) y¯=1, plotted along x¯, the normalized x^ axis of Fig. 10, which is normal to the beam axis, and centered at a point on this axis; here this point is (x˜,z)=β˜(0.006,0.16). The fields were calculated via the catastrophe theory (green line), the PE approximation (blue dots) and GO (red line). The intensity is normalized with respect to max|U0|2=Ai2(0) of Eq. (1). Parameters: kβ˜=104, α˜=104.

Fig. 12.
Fig. 12.

Same as Fig. 11 but for kβ=106.

Fig. 13.
Fig. 13.

Global structure of caustic 2 (green surface). “Caustic 2A” is the part that was shown in Figs. 5 and 8, whose edge follows the beam axis. This part of the caustic is formed by rays leaving the aperture at points close to the caustic. “Caustic 2B” is an additional part of caustic 2 not shown in Fig. 5, and it is formed by rays leaving the aperture sideways from faraway points (see Fig. 16). At some range, caustic 2 terminates at a pyramidlike edge. Red dashed line, cross section of the caustic at y˜=0, which is also shown as red dashed line in Fig. 16.

Fig. 14.
Fig. 14.

Global structure of caustic 1. Because the structure is complex, we show separately the parts formed by ray species s=1 and by species s=2,3,4 [Figs. (a) and (b), respectively]. In (a), caustic 1A is the caspoid shown in Figs. 5 and 7, which is formed by rays leaving the aperture at points near the beam axis, while caustic 1B is formed by rays leaving the aperture sideways at faraway points. In (b), caustic 1C (blue) and caustic 1D (magenta) are two overlapping sheets connect to caustic 1B at margins 3 and 4 and to caustic 1A at margins 1 and 2 (see also cross-sectional cuts of the caustic in Fig. 15).

Fig. 15.
Fig. 15.

Cross sections of caustic 1 in Fig. 14 at z constant planes.

Fig. 16.
Fig. 16.

Complete cross section of caustics 1 and 2 (gray and green lines, respectively) at the plane of symmetry y˜=0, also shown as red dashed lines in Figs. 13 and 14. These lines are defined in Eqs. (A14)–(A15) [or alternatively by Eqs. (A16)–(A17)]. Black solid lines, rays emanating sideways from faraway points, forming caustics 1B and 2B. Black dashed line, a typical ray near the axis forming caustic 1A and 2A (indistinguishable within the scale of the figure). Caustics 1C and 1D of Fig. 14(b) are not formed by rays contained in the y˜=0 plane and are therefore omitted here.

Equations (98)

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U0(x,y;ω)=exp[k(αxx+αyy)]Ai(βx1/3k2/3x)Ai(βy1/3k2/3y),
U(x,y,z)=F(x,z;ω,βx,αx)F(y,z;ω,βy,αy)exp(ikz),
F(ρ,z;ω,β,α)=Ai[(kβ)2/3(ρ/β(z/2β)2+iαz/β)]exp[ik(ρz/2βz3/12β2+α2z/2)]exp[kα(ρz2/2β)].
x/βx(z/2βx)2=0,
y/βy(z/2βy)2=0.
(x˜,y˜)T=Rφ(x,y)T,
Rφ=(cosφsinφsinφcosφ).
βx=βyββ˜2,
αx=αyαα˜/2,
x˜/β˜(z/2β˜)2=0.
r=r+s^σ,
s^(ξ,η,ζ)T(cosθx,cosθy,cosθz)T,
ζ=(1ξ2η2)1/2.
Ai(x)(xπ2)1/4sin[23(x)3/2+π4],x1,
U0(x,y)s=14A0sexp[ikψ0s],
ψ0s(x,y)=2[±(x)3/2±(y)3/2]/3β1/2,
A0s(x,y)=iβ1/64πk1/3ekα(x+y)(x)1/4(y)1/4eiπM0s/2,
[ξs,ηs]=[x,y]ψ0s(x,y)=[±(x/β)1/2,±(y/β)1/2],
J=(x,y,z)(σ,ξ,η)=rσrξrη,
J=ξσ2βξ0η0σ2βηζσξ/ζση/ζ.
U0(x,y;ω)=ω2/3(2π)2dξdηAeiωτ0(ξ,η)eiω(ξx+ηy)/c,
τ0(ξ,η)=β[(ξ+iα)3+(η+iα)3]/3c,
A=(β/c)2/3.
U(r;ω)=ω2/3(2π)2dξdηAeiωτ(ξ,η),
τ(ξ,η;r)=[τ0(ξ,η)+ξx+ηy+ζz]/c,
ξτ=0,ητ=0,
Uω1/3A2πrei(π/4)μr|Hr|1/2eiωτr,
H(ξ2τξητηξτη2τ),
μr=sgnh1r+sgnh2r.
τr=τ0r+σr/c,
ω1/3A2πei(π/4)μr|Hr|1/2=eiMrπ/2A0r|Jr(0)Jr(σr)|,
U¯(x¯,y¯,z¯)=C0(2π)2dξ¯dη¯eiΦ(ξ¯,η¯;x¯,y¯,z¯),
(x^,z^)T=Rθz(x˜x˜2,zz2)T,y^=y˜,
U(r˜;ω)=ω2/3(2π)2dξ˜dη˜A˜eiωτ(ξ˜,η˜),
τ(ξ˜,η˜)=β˜[(ξ˜+iα˜)3/3+(ξ˜+iα˜)η˜2]/c+ξ˜x˜+η˜y˜+ζ˜z,
A˜=(β˜/2c)2/3,
(ξ^,ζ^)T=Rθz(ξ˜,ζ˜)T,η^=η˜,
τ=τa+β˜D1ξ^3/c+β˜D2ξ^η^2/c+ξ^x^/c+η^y^/c+z^/cξ^2z^/2cη^2(z^δ^)/2c,
τa=2β˜sinθz(14sin2θz/3)/c,δ^=2β˜sin3θz,
D1=cosθz(5cos2θz3)/6,D2=cosθz(1+cos2θz)/2.
U=A¯eiωτaU¯(x¯,y¯;δ¯)cosθz,
Φ(ξ¯,η¯;x¯,y¯;δ¯)=ξ¯3+ξ¯η¯2+ξ¯x¯+η¯y¯+η¯2δ¯,
C0=31/624/3/[Ai(0)]2.
x¯=(kβ˜)2/3D11/3x^/β˜,y¯=(kβ˜)2/3D11/6D21/2y^/β˜,
ξ¯=(kβ˜)1/3D11/3ξ^,η¯=(kβ˜)1/3D11/6D21/2η^,
δ¯=(kβ˜)1/3D11/3D21δ^/2β˜=(kβ˜)1/3D11/3D21sin3θz(kβ˜/3)1/3θz3,
A¯=D11/6D21/2/C0,
τ=τaα+β˜D1αξ^3/c+β˜D2αξ^η^2/c+ξ^x^α/c+η^y^/c+z^/cξ^2(z^ϵ)/2cη^2(z^δ^α)/2c,
x^α=x^+β˜[iα˜sin(2θz)α˜2cosθz],
τaα=τa+β˜[iα˜sin2θzα˜2sinθziα˜3/3]/c,
δ^α=δ^+β˜[iα˜+iα˜cos2θz+α˜2sinθz],
D1,2α=D1,2iα˜sinθzcosθz/c,
ϵ=β˜(iα˜+α˜2sinθz/2iα˜sin2θz/2).
Φ(ξ¯,η¯;x¯,y¯;δ¯;ϵ¯)=ξ¯3+ξ¯η¯2+ξ¯x¯+η¯y¯+ξ¯2ϵ¯+η¯2δ¯,
ϵ¯=(kβ˜)1/3(D1α)2/3ϵ/β˜,
U=A¯eiωτaαU¯(x¯,y¯;δ¯;ϵ¯)cosθz,
U¯(x¯,y¯;δ¯;ϵ¯)=C0(2π)2p,m(i)(p+m)(p!m!)1ϵ¯pδ¯mdξ¯dη¯(iξ¯)2p(iη¯)2meiΦ¯(ξ¯,η¯;x¯,y¯),
Φ¯(ξ¯,η¯;x¯,y¯)=Φ(ξ¯,η¯;x¯,y¯;δ¯;ϵ¯)ξ¯2ϵ¯η¯2δ¯=ξ¯3+ξ¯η¯2+ξ¯x¯+η¯y¯.
U¯(x¯,y¯,z¯)=C0p,m(i)(p+m)(p!m!)1ϵ¯pδ¯mx¯2py¯2mI(x¯,y¯),
I(x¯,y¯)1(2π)2dξ¯dη¯eiΦ¯(ξ¯,η¯;x¯,y¯)=31/624/3Ai(u¯)Ai(v¯),
(u¯,v¯)T=(3/2)1/6R45°T(31/2x¯,y¯)T,
x¯p31/3(kβ˜)2/3θz4.
ξτ=ξτ0+x/c(z/c)(ξ/ζ)=0,
ητ=ητ0+y/c(z/c)(η/ζ)=0,
ξτ0=x/c=βξ2/c,
ητ0=y/c=βη2/c,
x=x+ξ(z/ζ),
y=y+η(z/ζ),
(c/β)ξ2τ=2ξ(z/β)(1η2)/ζ3,
(c/β)η2τ=2η(z/β)(1ξ2)/ζ3,
(c/β)ηξτ=(c/β)ξητ=(z/β)ξη/ζ3,
H=[C1z2+C2βz+C3β2]/c2|,
C1(ξ,η)=1/ζ4,C3(ξ,η)=4ξη,
C2(ξ,η)=2[ξ(1ξ2)+η(1η2)]/ζ3.
zc1,2/β=(C2±C224C1C3)/2C1.
(x˜1,z1)β˜(sin2θz,sin2θz),
(x˜2,z2)β˜(sin2θzcos2θz,sin2θz,cos2θz),
z1/β˜=2(x˜/β˜)1/2(1x˜/β˜)1/2,
z2/β˜=2q1/2(1q)3/2,q=(1(18x˜/β˜)1/2)/4,
z1/β˜2(x˜/β˜)1/2(1x˜/4β˜),
z2/β˜2(x˜/β˜)1/2(13x˜/2β˜),
J(σ)=C1ζ3σ2+βC2ζ2σ+β2C3ζ,
ξ=z/2β±(z/2β)2x/β,
η=z/2β±(z/2β)2y/β.
S1:(z/2β)2x/β=0andy/β(z/2β)2,
S2:(z/2β)2y/β=0andx/β(z/2β)2,
H(σ)=J(σ)/ζc2=(σσc1)(σσc2)/c2,
h1,2=(σσc1,2)/c.
μr=sgn{h1r}+sgn{h2r}=2(1+M0s+Mr),
|H|1/2=c21β1/2(x)1/4(y)1/4|J(0)/J(σ)|1/2.
ζζr+ζrξ(ξξr)+ζrη(ηηr),
ζrξ=ξr/ζr,ζrη=ηr/ζr.
(ξ,η)=(ξr,ηr)i(α,α).
cτr=cτ0r+σriα(xr+yr).
Φ(ξ̌,η̌;,;δ̌)=ξ̌3+η̌3+ξ̌η̌δ̌+ξ̌+η̌.
(,)T=21/6R45°T([x¯+3δ¯2/4],31/2y¯)T,
(ξ̌,η̌)T=21/6R45°T([ξ¯+δ¯/2],31/2η¯)T,
δ̌=3δ¯/21/3,

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