Abstract

The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in three-dimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.

© 2011 Optical Society of America

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References

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    [Crossref]
  2. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
    [Crossref]
  3. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).
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  7. R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1981).
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    [Crossref]
  9. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).
  10. P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
    [Crossref]
  11. J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
    [Crossref]
  12. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [Crossref]
  13. J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
    [Crossref]
  14. J. F. Nye and J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).
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  16. J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846 (1981).
    [Crossref]
  17. J. N. L. Connor and P. R. Curtis, “A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey’s integral and its derivatives,” J. Phys. A 15, 1179–1190 (1982).
    [Crossref]
  18. J. J. Stamnes and B. Spjelkavik, “Evaluation of the field near a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358 (1983).
    [Crossref]
  19. J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
    [Crossref]
  20. J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
    [Crossref]
  21. J. N. L. Connor and P. R. Curtis, “Differential equations for the cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902(1984).
    [Crossref]
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    [Crossref]
  23. M. V. Berry and C. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1991).
    [Crossref]
  24. D. Kaminski, “Asymptotics of the swallowtail integral near the cusp of the caustic,” SIAM J. Math. Anal. 23, 262–285 (1992).
    [Crossref]
  25. N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
    [Crossref]
  26. R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions,” J. Comput. Appl. Math. 190, 437–452 (2006).
    [Crossref]
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    [Crossref]
  28. C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
    [Crossref]
  29. A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).
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    [Crossref]
  31. R. Borghi, “On the numerical evaluation of cuspoid diffraction catastrophes,” J. Opt. Soc. Am. A 25, 1682–1690 (2008).
    [Crossref]
  32. R. Borghi, “On the numerical evaluation of umbilic diffraction catastrophes,” J. Opt. Soc. Am. A 27, 1661–1670 (2010).
    [Crossref]
  33. E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189–371 (1989).
    [Crossref]
  34. Digital Library of Mathematical Functions, National Institute of Standards and Technology (release date 7 May 2010) http://dlmf.nist.gov/.
  35. J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Philos. Trans. R. Soc. A 292, 25–44 (1979).
    [Crossref]
  36. J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
    [Crossref]
  37. J. F. Nye, “Caustics in seismology,” Geophys. J. R. Astron. Soc. 83, 477–485 (1985).
  38. F. Wright, “Earthquake modeling: caustics in seismology,” Nature 319, 720–721 (1986).
    [Crossref]
  39. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial propagation,” Opt. Lett. 28, 774–776 (2003).
    [Crossref]
  40. R. Borghi, “Summing Pauli asymptotic series to solve the wedge problem,” J. Opt. Soc. Am. A 25, 211–218 (2008).
    [Crossref]
  41. R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals,” Phys. Rev. E 78, 026703 (2008).
    [Crossref]
  42. R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations,” Phys. Rev. E 80, 016704 (2009).
    [Crossref]
  43. R. Borghi and M. A. Alonso, “Free-space asymptotic far-field series,” J. Opt. Soc. Am. A 26, 2410–2417 (2009).
    [Crossref]
  44. D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049(2009).
    [Crossref]
  45. J. Li, W. Zang, and J. Tian, “Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method,” Opt. Express 17, 4959–4969 (2009).
    [Crossref]
  46. J.-X. Li, W. Zang, Y.-D. Li, and J. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express 17, 11850–11859 (2009).
    [Crossref]
  47. J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96, 031103–031105 (2010).
    [Crossref]
  48. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Decoding divergent series in nonparaxial optics,” Opt. Lett. 36, 963–965 (2011).
    [Crossref]
  49. E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory,” J. Math. Phys. 45, 1209–1246(2004).
    [Crossref]
  50. E. J. Weniger, “Asymptotic approximations to truncation errors of series representations for special functions,” in Algorithms for Approximation, A.Iske and J.Levesley, eds. (Springer-Verlag, 2007), pp. 331–348.
  51. For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
    [Crossref]
  52. R. Borghi, “Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials,” Appl. Numer. Math. 60, 1242–1250 (2010).
    [Crossref]
  53. A. S. Kryukovskii and D. S. Lukin, “Theoretical calculation of reference focal and diffractional electromagnetic fields based on wave catastrophe special functions,” J. Commun. Technol. Electron. 48, 831–840 (2003).
  54. J. F. Nye, “Diffraction in lips and beak-to-beak caustics,” J. Opt. A Pure Appl. Opt. 11, 065708 (2009).
    [Crossref]
  55. J. F. Nye, “Wave dislocations in the diffraction pattern of a higher-order optical catastrophe,” J. Opt. A Pure Appl. Opt. 12, 015702 (2010).
  56. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

2011 (1)

2010 (4)

R. Borghi, “On the numerical evaluation of umbilic diffraction catastrophes,” J. Opt. Soc. Am. A 27, 1661–1670 (2010).
[Crossref]

R. Borghi, “Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials,” Appl. Numer. Math. 60, 1242–1250 (2010).
[Crossref]

J. F. Nye, “Wave dislocations in the diffraction pattern of a higher-order optical catastrophe,” J. Opt. A Pure Appl. Opt. 12, 015702 (2010).

J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96, 031103–031105 (2010).
[Crossref]

2009 (7)

J. F. Nye, “Diffraction in lips and beak-to-beak caustics,” J. Opt. A Pure Appl. Opt. 11, 065708 (2009).
[Crossref]

J. Li, W. Zang, and J. Tian, “Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method,” Opt. Express 17, 4959–4969 (2009).
[Crossref]

J.-X. Li, W. Zang, Y.-D. Li, and J. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express 17, 11850–11859 (2009).
[Crossref]

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049(2009).
[Crossref]

R. Borghi and M. A. Alonso, “Free-space asymptotic far-field series,” J. Opt. Soc. Am. A 26, 2410–2417 (2009).
[Crossref]

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations,” Phys. Rev. E 80, 016704 (2009).
[Crossref]

A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).

2008 (3)

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals,” Phys. Rev. E 78, 026703 (2008).
[Crossref]

R. Borghi, “Summing Pauli asymptotic series to solve the wedge problem,” J. Opt. Soc. Am. A 25, 211–218 (2008).
[Crossref]

R. Borghi, “On the numerical evaluation of cuspoid diffraction catastrophes,” J. Opt. Soc. Am. A 25, 1682–1690 (2008).
[Crossref]

2007 (4)

R. Borghi, “Evaluation of diffraction catastrophes by using Weniger transformation,” Opt. Lett. 32, 226–228 (2007).
[Crossref]

E. J. Weniger, “Asymptotic approximations to truncation errors of series representations for special functions,” in Algorithms for Approximation, A.Iske and J.Levesley, eds. (Springer-Verlag, 2007), pp. 331–348.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
[Crossref]

2006 (3)

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

R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions,” J. Comput. Appl. Math. 190, 437–452 (2006).
[Crossref]

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

2004 (1)

E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory,” J. Math. Phys. 45, 1209–1246(2004).
[Crossref]

2003 (2)

A. S. Kryukovskii and D. S. Lukin, “Theoretical calculation of reference focal and diffractional electromagnetic fields based on wave catastrophe special functions,” J. Commun. Technol. Electron. 48, 831–840 (2003).

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial propagation,” Opt. Lett. 28, 774–776 (2003).
[Crossref]

2000 (1)

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
[Crossref]

1999 (3)

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

1992 (2)

D. Kaminski, “Asymptotics of the swallowtail integral near the cusp of the caustic,” SIAM J. Math. Anal. 23, 262–285 (1992).
[Crossref]

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
[Crossref]

1991 (1)

M. V. Berry and C. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1991).
[Crossref]

1989 (2)

R. Thom, Structural Stability and Morphogenesis (Westview, 1989).

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189–371 (1989).
[Crossref]

1986 (3)

F. Wright, “Earthquake modeling: caustics in seismology,” Nature 319, 720–721 (1986).
[Crossref]

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[Crossref]

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

1985 (3)

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[Crossref]

J. F. Nye, “Caustics in seismology,” Geophys. J. R. Astron. Soc. 83, 477–485 (1985).

E. B. Ipatov, D. S. Lukin, and E. A. Palkin, “Numerical methods of computing special functions of wave catastrophes,” USSR Comput. Math. Math. Phys. 25, 144–153 (1985).
[Crossref]

1984 (5)

J. F. Nye and J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
[Crossref]

J. N. L. Connor and P. R. Curtis, “Differential equations for the cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902(1984).
[Crossref]

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
[Crossref]

1983 (2)

J. J. Stamnes and B. Spjelkavik, “Evaluation of the field near a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358 (1983).
[Crossref]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
[Crossref]

1982 (1)

J. N. L. Connor and P. R. Curtis, “A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey’s integral and its derivatives,” J. Phys. A 15, 1179–1190 (1982).
[Crossref]

1981 (2)

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846 (1981).
[Crossref]

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1981).

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

1979 (2)

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Philos. Trans. R. Soc. A 292, 25–44 (1979).
[Crossref]

1976 (1)

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

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Alonso, M. A.

Berry, M. V.

M. V. Berry and C. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1991).
[Crossref]

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).

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

Borghi, R.

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Decoding divergent series in nonparaxial optics,” Opt. Lett. 36, 963–965 (2011).
[Crossref]

R. Borghi, “On the numerical evaluation of umbilic diffraction catastrophes,” J. Opt. Soc. Am. A 27, 1661–1670 (2010).
[Crossref]

R. Borghi, “Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials,” Appl. Numer. Math. 60, 1242–1250 (2010).
[Crossref]

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations,” Phys. Rev. E 80, 016704 (2009).
[Crossref]

R. Borghi and M. A. Alonso, “Free-space asymptotic far-field series,” J. Opt. Soc. Am. A 26, 2410–2417 (2009).
[Crossref]

R. Borghi, “Summing Pauli asymptotic series to solve the wedge problem,” J. Opt. Soc. Am. A 25, 211–218 (2008).
[Crossref]

R. Borghi, “On the numerical evaluation of cuspoid diffraction catastrophes,” J. Opt. Soc. Am. A 25, 1682–1690 (2008).
[Crossref]

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals,” Phys. Rev. E 78, 026703 (2008).
[Crossref]

R. Borghi, “Evaluation of diffraction catastrophes by using Weniger transformation,” Opt. Lett. 32, 226–228 (2007).
[Crossref]

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial propagation,” Opt. Lett. 28, 774–776 (2003).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

Caliceti, E.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Connor, J. N. L.

C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
[Crossref]

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
[Crossref]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
[Crossref]

J. N. L. Connor and P. R. Curtis, “Differential equations for the cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902(1984).
[Crossref]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
[Crossref]

J. N. L. Connor and P. R. Curtis, “A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey’s integral and its derivatives,” J. Phys. A 15, 1179–1190 (1982).
[Crossref]

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846 (1981).
[Crossref]

Curtis, P. R.

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
[Crossref]

J. N. L. Connor and P. R. Curtis, “Differential equations for the cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902(1984).
[Crossref]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
[Crossref]

J. N. L. Connor and P. R. Curtis, “A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey’s integral and its derivatives,” J. Phys. A 15, 1179–1190 (1982).
[Crossref]

Deng, D.

Farrelly, D.

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
[Crossref]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
[Crossref]

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846 (1981).
[Crossref]

Gilmore, R.

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1981).

Gori, F.

Guattari, G.

Guo, Q.

Hannay, J. H.

J. F. Nye and J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).

Hobbs, C. A.

C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
[Crossref]

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
[Crossref]

Howls, C.

M. V. Berry and C. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1991).
[Crossref]

Ipatov, E. B.

E. B. Ipatov, D. S. Lukin, and E. A. Palkin, “Numerical methods of computing special functions of wave catastrophes,” USSR Comput. Math. Math. Phys. 25, 144–153 (1985).
[Crossref]

Jentschura, U. D.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Kaminski, D.

R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions,” J. Comput. Appl. Math. 190, 437–452 (2006).
[Crossref]

D. Kaminski, “Asymptotics of the swallowtail integral near the cusp of the caustic,” SIAM J. Math. Anal. 23, 262–285 (1992).
[Crossref]

Kirk, N. P.

C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
[Crossref]

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
[Crossref]

Kravtsov, Yu. A.

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Kryukovskii, A. S.

A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

A. S. Kryukovskii and D. S. Lukin, “Theoretical calculation of reference focal and diffractional electromagnetic fields based on wave catastrophe special functions,” J. Commun. Technol. Electron. 48, 831–840 (2003).

Li, J.

Li, J.-X.

J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96, 031103–031105 (2010).
[Crossref]

J.-X. Li, W. Zang, Y.-D. Li, and J. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express 17, 11850–11859 (2009).
[Crossref]

Li, Y.-D.

Lukin, D. S.

A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

A. S. Kryukovskii and D. S. Lukin, “Theoretical calculation of reference focal and diffractional electromagnetic fields based on wave catastrophe special functions,” J. Commun. Technol. Electron. 48, 831–840 (2003).

E. B. Ipatov, D. S. Lukin, and E. A. Palkin, “Numerical methods of computing special functions of wave catastrophes,” USSR Comput. Math. Math. Phys. 25, 144–153 (1985).
[Crossref]

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

Marston, P. L.

P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[Crossref]

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

Meyer-Hermann, M.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Nye, J. F.

J. F. Nye, “Wave dislocations in the diffraction pattern of a higher-order optical catastrophe,” J. Opt. A Pure Appl. Opt. 12, 015702 (2010).

J. F. Nye, “Diffraction in lips and beak-to-beak caustics,” J. Opt. A Pure Appl. Opt. 11, 065708 (2009).
[Crossref]

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

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
[Crossref]

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[Crossref]

J. F. Nye, “Caustics in seismology,” Geophys. J. R. Astron. Soc. 83, 477–485 (1985).

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
[Crossref]

J. F. Nye and J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Philos. Trans. R. Soc. A 292, 25–44 (1979).
[Crossref]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Palkin, E. A.

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

E. B. Ipatov, D. S. Lukin, and E. A. Palkin, “Numerical methods of computing special functions of wave catastrophes,” USSR Comput. Math. Math. Phys. 25, 144–153 (1985).
[Crossref]

Paris, R. B.

R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions,” J. Comput. Appl. Math. 190, 437–452 (2006).
[Crossref]

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

Rastyagaev, D. S.

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

Rastyagaev, D. V.

A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).

Ribeca, P.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Santarsiero, M.

Spjelkavik, B.

J. J. Stamnes and B. Spjelkavik, “Evaluation of the field near a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358 (1983).
[Crossref]

Stamnes, J. J.

J. J. Stamnes and B. Spjelkavik, “Evaluation of the field near a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358 (1983).
[Crossref]

Surzhykov, A.

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Thom, R.

R. Thom, Structural Stability and Morphogenesis (Westview, 1989).

Tian, J.

Trinh, E. H.

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

Weniger, E. J.

E. J. Weniger, “Asymptotic approximations to truncation errors of series representations for special functions,” in Algorithms for Approximation, A.Iske and J.Levesley, eds. (Springer-Verlag, 2007), pp. 331–348.

E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory,” J. Math. Phys. 45, 1209–1246(2004).
[Crossref]

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189–371 (1989).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Wright, F.

F. Wright, “Earthquake modeling: caustics in seismology,” Nature 319, 720–721 (1986).
[Crossref]

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).

Zang, W.

Zang, W.-P.

J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96, 031103–031105 (2010).
[Crossref]

Adv. Phys. (1)

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

Appl. Numer. Math. (1)

R. Borghi, “Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials,” Appl. Numer. Math. 60, 1242–1250 (2010).
[Crossref]

Appl. Phys. Lett. (1)

J.-X. Li, W.-P. Zang, and J. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96, 031103–031105 (2010).
[Crossref]

Comput. Phys. Commun. (1)

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, “An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives,” Comput. Phys. Commun. 132, 142–165 (2000).
[Crossref]

Comput. Phys. Rep. (1)

E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10, 189–371 (1989).
[Crossref]

Geophys. J. R. Astron. Soc. (1)

J. F. Nye, “Caustics in seismology,” Geophys. J. R. Astron. Soc. 83, 477–485 (1985).

J. Chem. Phys. (1)

J. N. L. Connor and D. Farrelly, “Theory of cusped rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s integral,” J. Chem. Phys. 75, 2831–2846 (1981).
[Crossref]

J. Commun. Technol. Electron. (2)

A. S. Kryukovskii, D. S. Lukin, E. A. Palkin, and D. S. Rastyagaev, “Wave catastrophes: types of focusing in diffraction and propagation of electromagnetic waves,” J. Commun. Technol. Electron. 51, 1087–1125 (2006).
[Crossref]

A. S. Kryukovskii and D. S. Lukin, “Theoretical calculation of reference focal and diffractional electromagnetic fields based on wave catastrophe special functions,” J. Commun. Technol. Electron. 48, 831–840 (2003).

J. Comput. Appl. Math. (2)

C. A. Hobbs, J. N. L. Connor, and N. P. Kirk, “Theory and numerical evaluation of oddoids and evenoids: oscillatory cuspoid integrals with odd and even polynomial phase functions,” J. Comput. Appl. Math. 207, 192–213 (2007).
[Crossref]

R. B. Paris and D. Kaminski, “Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions,” J. Comput. Appl. Math. 190, 437–452 (2006).
[Crossref]

J. Math. Phys. (2)

J. N. L. Connor and P. R. Curtis, “Differential equations for the cuspoid canonical integrals,” J. Math. Phys. 25, 2895–2902(1984).
[Crossref]

E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory,” J. Math. Phys. 45, 1209–1246(2004).
[Crossref]

J. Mod. Opt. (1)

J. J. Stamnes and B. Spjelkavik, “Evaluation of the field near a cusp of a caustic,” J. Mod. Opt. 30, 1331–1358 (1983).
[Crossref]

J. Opt. A Pure Appl. Opt. (2)

J. F. Nye, “Diffraction in lips and beak-to-beak caustics,” J. Opt. A Pure Appl. Opt. 11, 065708 (2009).
[Crossref]

J. F. Nye, “Wave dislocations in the diffraction pattern of a higher-order optical catastrophe,” J. Opt. A Pure Appl. Opt. 12, 015702 (2010).

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

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

J. Phys. A (2)

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points,” J. Phys. A 17, 283–310 (1984).
[Crossref]

J. N. L. Connor and P. R. Curtis, “A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey’s integral and its derivatives,” J. Phys. A 15, 1179–1190 (1982).
[Crossref]

Mol. Phys. (1)

J. N. L. Connor, P. R. Curtis, and D. Farrelly, “A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives,” Mol. Phys. 48, 1305–1330 (1983).
[Crossref]

Nature (3)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

J. F. Nye, “Rainbow scattering from spheroidal drops: an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
[Crossref]

F. Wright, “Earthquake modeling: caustics in seismology,” Nature 319, 720–721 (1986).
[Crossref]

Opt. Acta (1)

J. F. Nye and J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 116–130 (1984).

Opt. Express (2)

Opt. Lett. (4)

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Philos. Trans. R. Soc. A (1)

J. F. Nye, “Optical caustics from liquid drops under gravity: observations of the parabolic and symbolic umbilics,” Philos. Trans. R. Soc. A 292, 25–44 (1979).
[Crossref]

Phys. Rep. (1)

For an updated review about methods for decoding diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, “From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions,” Phys. Rep. 446, 1–96 (2007), arXiv:0707.1596v1.
[Crossref]

Phys. Rev. E (2)

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals,” Phys. Rev. E 78, 026703 (2008).
[Crossref]

R. Borghi, “Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations,” Phys. Rev. E 80, 016704 (2009).
[Crossref]

Proc. R. Soc. London Ser. A (5)

J. F. Nye, “The catastrophe optics of liquid drop lenses,” Proc. R. Soc. London Ser. A 403, 1–26 (1986).
[Crossref]

M. V. Berry and C. Howls, “Hyperasymptotics for integrals with saddles,” Proc. R. Soc. London Ser. A 434, 657–675 (1991).
[Crossref]

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

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Proc. R. Soc. London Ser. A 291, 453–484 (1979).

J. F. Nye, “Rainbows from ellipsoidal water droplets,” Proc. R. Soc. London Ser. A 438, 397–417 (1992).
[Crossref]

Prog. Opt. (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

Russ. J. Math. Phys. (1)

A. S. Kryukovskii, D. S. Lukin, and D. V. Rastyagaev, “Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory,” Russ. J. Math. Phys. 16, 251–264 (2009).

SIAM J. Math. Anal. (1)

D. Kaminski, “Asymptotics of the swallowtail integral near the cusp of the caustic,” SIAM J. Math. Anal. 23, 262–285 (1992).
[Crossref]

USSR Comput. Math. Math. Phys. (1)

E. B. Ipatov, D. S. Lukin, and E. A. Palkin, “Numerical methods of computing special functions of wave catastrophes,” USSR Comput. Math. Math. Phys. 25, 144–153 (1985).
[Crossref]

Other (8)

Digital Library of Mathematical Functions, National Institute of Standards and Technology (release date 7 May 2010) http://dlmf.nist.gov/.

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

R. Thom, Structural Stability and Morphogenesis (Westview, 1989).

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Dover, 1981).

E. J. Weniger, “Asymptotic approximations to truncation errors of series representations for special functions,” in Algorithms for Approximation, A.Iske and J.Levesley, eds. (Springer-Verlag, 2007), pp. 331–348.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Vol.  I of Integrals and Series (Gordon and Breach, 1986).

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Figures (6)

Fig. 1
Fig. 1 Representation, in the plane ( C 1 , C 2 ) , of the caustic distributions associated to A 5 , for (a)  ( C 3 , C 4 ) = ( 0 , 3 ) , (b)  ( 3 , 0 ) , (c)  ( 0 , 3 ) , and (d)  ( 1 , 3 ) . Note that the scales are different.
Fig. 2
Fig. 2 Representation, in the plane ( C 2 , C 1 ) , of the caustic distributions associated to D 5 , for (a)  ( C 3 , C 4 ) = ( 4 / 7 , 1 ) , (b)  ( 0 , 5 / 4 ) , (c)  ( 169 / 100 , 106 / 100 ) , (d)  ( 4 , 64 / 100 ) , (e)  ( 3 , 1 ) , and (f)  ( 5 , 1 / 10 ) . Note that the scales are different.
Fig. 3
Fig. 3 2D maps of the modulus, in the plane ( C 1 , C 2 ) , of the diffraction catastrophes associated to the caustic distributions depicted in Fig. 1. (a)  ( C 3 , C 4 ) = ( 0 , 3 ) , (b)  ( 3 , 0 ) , (c)  ( 0 , 3 ) , and (d)  ( 1 , 3 ) . The WT order has been kept fixed for each plot, and, precisely, it was set to 25 for (a), 50 for (b), and 40 for (c) and (d).
Fig. 4
Fig. 4 2D maps of the modulus, in the plane ( C 2 , C 1 ) , of the diffraction patterns associated to the caustic distributions depicted in Fig. 2. (a)  ( C 3 , C 4 ) = ( 4 / 7 , 1 ) , (b)  ( 0 , 5 / 4 ) , (c)  ( 169 / 100 , 106 / 100 ) , (d)  ( 4 , 64 / 100 ) , (e)  ( 3 , 1 ) , and (f)  ( 5 , 1 / 10 ) .
Fig. 5
Fig. 5 2D contour map of the modulus distribution of a parabolic umbilic diffraction catastrophe with ( C 3 , C 4 ) = ( 7 , 1 / 2 ) . The choice of the WT order has been chosen in order to reduce the computational time. The accuracy has been chosen up to the second digit.
Fig. 6
Fig. 6 Contour path for the integral in Eq. (D5).

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

Ψ ( C ) = 1 ( 2 π ) m / 2 d m s exp [ i Φ ( s ; C ) ] .
Φ ( s ; C ) = s 6 + j = 1 4 C j s j ,
Ψ A 5 ( C ) = 1 2 π B ( C 4 , C 3 , C 2 , C 1 ) ,
B ( x , y , z , w ) = + d s exp [ i ( s 6 + x s 4 + y s 3 + z s 2 + w s ) ] .
B ( x , y , z , w ) = B 1 / 2 ( x , y , z , w ) + B 1 / 2 ( x , y , z , w ) ,
B 1 / 2 ( x , y , z , w ) = 0 d s exp [ i ( s 6 + x s 4 + y s 3 + z s 2 + w s ) ] .
( i ) 1 / 6 B 1 / 2 ( x , y , z , w ) = k = 0 i 7 k / 6 k ! j = 0 k = 0 j ( k j ) ( j ) y z j w k j J k + j + ( x i 5 / 3 ) ,
J n ( μ ) = 1 6 k = 0 2 μ k k ! Γ ( n + 4 k + 1 6 ) × 3 F 3 [ 1 , Δ ( 2 , n + 4 k + 1 6 ) ; Δ ( 3 , 1 + k ) ; 4 μ 3 27 ] .
Δ ( q , a ) = { a q , a + 1 q , , a + q 1 q } .
Φ ( s ; C ) = s 1 4 + s 1 s 2 2 + C 4 s 2 2 + C 3 s 1 2 + C 2 s 2 + C 1 s 1 ,
Ψ D 5 ( C ) = 1 2 π P ( C 4 , C 3 , C 2 , C 1 ) ,
P ( x , y , z , w ) = IR 2 d ξ d η exp [ i ( ξ 2 η + η 4 + x ξ 2 + y η 2 + z ξ + w η ) ] .
P ( x , y , z , w ) = + d η exp [ i ( η 4 + y η 2 + w η ) ] × + d ξ exp { i [ ( x + η ) ξ 2 + z ξ ] } ,
P ( x , y , z , w ) = { x d η exp [ i ( η 4 + y η 2 + w η ) ] × + d ξ exp { i [ ( x + η ) ξ 2 + z ξ ] } + x + d η exp [ i ( η 4 + y η 2 + w η ) ] + d ξ exp { i [ ( x + η ) ξ 2 + z ξ ] } } .
+ d ξ exp [ ± i s ξ 2 + i z ξ ] = π s exp ( ± i π 4 i z 2 4 s ) ,
P ( x , y , z , w ) = i π exp [ i ( x 4 + x 2 y x w ) ] × [ P 1 / 2 ( X , Y , Z , W ) i P 1 / 2 ( X , Y , Z , W ) ] ,
P 1 / 2 ( X , Y , Z , W ) = 0 d s s exp [ i ( s 4 + X s 3 + Y s 2 + Z s + W s ) ] ,
P 1 / 2 ( X , Y , Z , W ) = k = 0 i k k ! j = 0 k = 0 j ( k j ) ( j ) X Y j Z k j I k + j + ( W ) ,
I n ( μ ) = i n + 1 / 2 4 { ( i 3 / 4 μ ) n + 1 2 Γ ( n 1 2 ) × F 0 4 ( ; 3 + 2 n 8 , 5 + 2 n 8 , 7 + 2 n 8 , 9 + 2 n 8 ; i μ 4 256 ) + 1 4 k = 0 3 ( i 3 / 4 μ ) k k ! Γ ( n k 4 + 1 8 ) × F 1 5 [ 1 ; Δ ( 1 , 1 + k n 1 / 2 4 ) , Δ ( 4 , 1 + k ) ; i μ 4 256 ] } ,
{ s Φ = 0 , s 2 Φ = 0 ,
{ 6 s 5 + 4 C 4 s 3 + 3 C 3 s 2 + 2 C 2 s + C 1 = 0 , 15 s 4 + 6 C 4 s 2 + 3 C 3 s + C 2 = 0 ,
{ C 1 = s 2 ( 8 C 4 s + 3 C 3 + 24 s 3 ) , C 2 = 3 s ( 2 C 4 s + C 3 + 5 s 3 ) ,
( i ) 1 / 6 B 1 / 2 ( x , y , z , w ) = 0 d t exp ( t 6 + ξ t 4 + η t 3 + ζ t 2 + ω t ) ,
( i ) 1 / 6 B 1 / 2 ( x , y , z , w ) = k = 0 1 k ! 0 d t exp [ ( t 6 + ξ t 4 ) ( η t 3 + ζ t 2 + ω t ) k ,
( i ) 1 / 6 B 1 / 2 ( x , y , z , w ) = k = 0 i 7 k / 6 k ! × j = 0 k = 0 j i + j 6 ( k j ) ( j ) y z j w k j J k + j + ( x i 5 / 3 ) ,
J n ( μ ) = 0 t n exp ( t 6 + μ t 4 ) .
J n ( μ ) = 1 4 0 d τ τ n 3 4 exp ( τ 3 2 + μ τ ) ,
{ 1 Φ = 0 , 2 Φ = 0 , 11 2 Φ 22 2 Φ = ( 12 2 Φ ) 2 ,
{ C 1 = 2 C 3 s 1 4 s 1 3 s 2 2 , C 2 = 2 s 2 ( C 4 + s 1 ) , s 2 2 = ( C 4 + s 1 ) ( C 3 + 6 s 1 2 ) .
{ C 1 = [ 4 s 3 + 2 s C 3 + ( 6 s 2 + C 3 ) ( s + C 4 ) ] , C 2 = ± 2 ( 6 s 2 + C 3 ) 1 / 2 ( s + C 4 ) 3 / 2 ,
P 1 / 2 ( X , Y , Z , W ) = 0 d s s exp [ i ( s 4 + W s ) ] exp [ i ( X s 3 + Y s 2 + Z s ) ] ,
P 1 / 2 ( X , Y , Z , W ) = k = 0 i k k ! 0 d s s exp [ i ( s 4 + W s ) ] ( X s 3 + Y s 2 + Z s ) k .
( X s 3 + Y s 2 + Z s ) k = j = 0 k = 0 j ( k j ) ( j ) X Y j Z k j s k + j + ,
I n ( μ ) 0 d s s s n exp [ i ( s 4 + μ s ) ] ,
C d s s s n exp [ i ( s 4 + μ s ) ] ,
C ϵ d s s s n exp [ i ( s 4 + μ s ) ] = i ϵ n + 1 / 2 0 π / 8 d φ exp [ i ( n + 1 2 ) ] × exp [ i ϵ 3 exp ( i 4 φ ) ] exp [ i μ ϵ exp ( i φ ) ] .
Re { i μ exp ( i φ ) } 0
π 8 < arg { μ } < π ,
I n ( μ ) = i n + 1 / 2 4 0 d t t n 1 2 exp ( t 4 + i 3 / 4 μ t ) .
I n ( μ ) = i n + 1 / 2 4 0 d ξ ξ n 3 2 exp ( ξ 4 + i 3 / 4 μ ξ ) ,

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