Abstract

Weniger transformation is a powerful nonlinear sequence transformation that, when applied to the sequence of the partial sums of a divergent or a slowly convergent series, can convert it to a fast-converging sequence. Weniger transformation is not yet well known in optics. Diffraction catastrophes are fundamental tools for evaluating an optical field in proximity to caustics and singularities. The action of the Weniger transformation on the power series representation of diffraction catastrophes is numerically studied for two particular cases, corresponding to the Airy and the Pearcey functions. The obtained results clearly show that Weniger transformation could become a computational tool of great importance for summing several types of series expansions in optics.

© 2007 Optical Society of America

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  2. M. V. Berry and C. Upstill, in Progress in Optics XVIII, E.Wolf, ed. (Elsevier, 1980), pp. 257-346.
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  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
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    [CrossRef]
  7. J. N. L. Connor and P. R. Curtis, J. Phys. A 15, 1179 (1982).
    [CrossRef]
  8. J. J. Stamnes and B. Spjelkavik, J. Mol. Spectrosc. 30, 1331 (1983).
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    [CrossRef]
  10. J. N. L. Connor and P. R. Curtis, J. Math. Phys. (Cambridge, Mass.) 25, 2895 (1984).
  11. M. V. Berry and C. Howls, Proc. R. Soc. London, Ser. A 434, 657 (1991).
    [CrossRef]
  12. N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
    [CrossRef]
  13. M. V. Berry and C. Howls, "Integrals with coalescing saddles," in Digital Library of Mathematical Functions, http://dlmf.nist.gov.
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    [CrossRef]
  15. E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
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  16. E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
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  19. E. J. Weniger, J. Math. Phys. 45, 1209 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]

2006

2004

E. J. Weniger, J. Math. Phys. 45, 1209 (2004).
[CrossRef]

2003

R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
[CrossRef] [PubMed]

J. Cízek, J. Zamastil, and L. Skála, J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

2000

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
[CrossRef]

1996

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

1991

E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
[CrossRef]

M. V. Berry and C. Howls, Proc. R. Soc. London, Ser. A 434, 657 (1991).
[CrossRef]

1989

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

1984

J. N. L. Connor, P. R. Curtis, and D. Farrelly, J. Phys. A 17, 281 (1984).
[CrossRef]

J. N. L. Connor and P. R. Curtis, J. Math. Phys. (Cambridge, Mass.) 25, 2895 (1984).

1983

J. J. Stamnes and B. Spjelkavik, J. Mol. Spectrosc. 30, 1331 (1983).

1982

J. N. L. Connor and P. R. Curtis, J. Phys. A 15, 1179 (1982).
[CrossRef]

1981

J. N. L. Connor and D. Farrelly, J. Chem. Phys. 75, 2831 (1981).
[CrossRef]

1976

M. V. Berry, Adv. Phys. 25, 1 (1976).
[CrossRef]

1946

T. Pearcey, Philos. Mag. 37, 311 (1946).

Alonso, M. A.

Berry, M. V.

M. V. Berry and C. Howls, Proc. R. Soc. London, Ser. A 434, 657 (1991).
[CrossRef]

M. V. Berry, Adv. Phys. 25, 1 (1976).
[CrossRef]

M. V. Berry and C. Howls, "Integrals with coalescing saddles," in Digital Library of Mathematical Functions, http://dlmf.nist.gov.

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E.Wolf, ed. (Elsevier, 1980), pp. 257-346.
[CrossRef]

Borghi, R.

Born, M.

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

Cízek, J.

J. Cízek, J. Zamastil, and L. Skála, J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Cìzek, J.

E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
[CrossRef]

Connor, J. N. L.

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
[CrossRef]

J. N. L. Connor, P. R. Curtis, and D. Farrelly, J. Phys. A 17, 281 (1984).
[CrossRef]

J. N. L. Connor and P. R. Curtis, J. Math. Phys. (Cambridge, Mass.) 25, 2895 (1984).

J. N. L. Connor and P. R. Curtis, J. Phys. A 15, 1179 (1982).
[CrossRef]

J. N. L. Connor and D. Farrelly, J. Chem. Phys. 75, 2831 (1981).
[CrossRef]

Curtis, P. R.

J. N. L. Connor, P. R. Curtis, and D. Farrelly, J. Phys. A 17, 281 (1984).
[CrossRef]

J. N. L. Connor and P. R. Curtis, J. Math. Phys. (Cambridge, Mass.) 25, 2895 (1984).

J. N. L. Connor and P. R. Curtis, J. Phys. A 15, 1179 (1982).
[CrossRef]

Farrelly, D.

J. N. L. Connor, P. R. Curtis, and D. Farrelly, J. Phys. A 17, 281 (1984).
[CrossRef]

J. N. L. Connor and D. Farrelly, J. Chem. Phys. 75, 2831 (1981).
[CrossRef]

Hobbs, C. A.

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
[CrossRef]

Howls, C.

M. V. Berry and C. Howls, Proc. R. Soc. London, Ser. A 434, 657 (1991).
[CrossRef]

M. V. Berry and C. Howls, "Integrals with coalescing saddles," in Digital Library of Mathematical Functions, http://dlmf.nist.gov.

Kirk, N. P.

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
[CrossRef]

Pearcey, T.

T. Pearcey, Philos. Mag. 37, 311 (1946).

Santarsiero, M.

Skála, L.

J. Cízek, J. Zamastil, and L. Skála, J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Spjelkavik, B.

J. J. Stamnes and B. Spjelkavik, J. Mol. Spectrosc. 30, 1331 (1983).

Stamnes, J. J.

J. J. Stamnes and B. Spjelkavik, J. Mol. Spectrosc. 30, 1331 (1983).

Upstill, C.

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E.Wolf, ed. (Elsevier, 1980), pp. 257-346.
[CrossRef]

Vinette, F.

E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
[CrossRef]

Weniger, E. J.

E. J. Weniger, J. Math. Phys. 45, 1209 (2004).
[CrossRef]

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
[CrossRef]

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

Wolf, E.

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

Zamastil, J.

J. Cízek, J. Zamastil, and L. Skála, J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Adv. Phys.

M. V. Berry, Adv. Phys. 25, 1 (1976).
[CrossRef]

Comput. Phys. Commun.

N. P. Kirk, J. N. L. Connor, and C. A. Hobbs, Comput. Phys. Commun. 132, 142 (2000).
[CrossRef]

Comput. Phys. Rep.

E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989).
[CrossRef]

J. Chem. Phys.

J. N. L. Connor and D. Farrelly, J. Chem. Phys. 75, 2831 (1981).
[CrossRef]

J. Math. Phys.

J. Cízek, J. Zamastil, and L. Skála, J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

E. J. Weniger, J. Math. Phys. 45, 1209 (2004).
[CrossRef]

J. Math. Phys. (Cambridge, Mass.)

J. N. L. Connor and P. R. Curtis, J. Math. Phys. (Cambridge, Mass.) 25, 2895 (1984).

J. Mol. Spectrosc.

J. J. Stamnes and B. Spjelkavik, J. Mol. Spectrosc. 30, 1331 (1983).

J. Phys. A

J. N. L. Connor, P. R. Curtis, and D. Farrelly, J. Phys. A 17, 281 (1984).
[CrossRef]

J. N. L. Connor and P. R. Curtis, J. Phys. A 15, 1179 (1982).
[CrossRef]

Opt. Lett.

Philos. Mag.

T. Pearcey, Philos. Mag. 37, 311 (1946).

Phys. Lett. A

E. J. Weniger, J. Cìzek, and F. Vinette, Phys. Lett. A 156, 169 (1991).
[CrossRef]

Phys. Rev. Lett.

E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

M. V. Berry and C. Howls, Proc. R. Soc. London, Ser. A 434, 657 (1991).
[CrossRef]

Other

M. V. Berry and C. Howls, "Integrals with coalescing saddles," in Digital Library of Mathematical Functions, http://dlmf.nist.gov.

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E.Wolf, ed. (Elsevier, 1980), pp. 257-346.
[CrossRef]

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

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

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Figures (2)

Fig. 1
Fig. 1

Behavior of the relative error as a function x for the values of the Airy function Ai ( x ) , evaluated for different values of the maximum value N max of the index n in Eq. (5) (solid curves). The relative error pertinent to the first term of the asymptotic expansion of the Airy function (dotted curve) is also shown.

Fig. 2
Fig. 2

Modulus of the Pearcey function for x [ 20 , 1 ] and y [ 30 , 30 ] , evaluated on using Weniger transformation. The values of the Pearcey function are evaluated on a grid of evenly distributed 251 × 501 points with N max 140 .

Tables (1)

Tables Icon

Table 1 Action of the Weniger Transformation on the Highly Divergent Sequence of the Partial Sum when Evaluating the Value of the Pearcey Function P ( x , y ) for x = 7 , y = 2 ( 1 + i )

Equations (7)

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

Ψ ( C ) = 1 ( 2 π ) N 2 d N s exp [ i Φ ( s ; C ) ] ,
Φ ( s ; C ) = s n + k = 1 n 2 C k s k ,
Ai ( x ) = 1 2 π + exp [ i ( s 3 3 + x s ) ] d s .
P ( x , y ) = + exp [ i ( s 4 + x s 2 + y s ) ] d s .
δ n = j = 0 n ( 1 ) j ( n j ) ( 1 + j ) n 1 s j b j + 1 j = 0 n ( 1 ) j ( n j ) ( 1 + j ) n 1 1 b j + 1 ,
Ai ( x ) = 1 3 2 3 Γ ( 2 3 ) k = 0 1 ( 2 3 ) k k ! ( x 3 9 ) k x 3 1 3 Γ ( 1 3 ) k = 0 1 ( 4 3 ) k k ! ( x 3 9 ) k ,
P ( x , y ) = exp ( i π 8 ) 2 n = 0 [ x exp ( i π 4 ) ] n ( 2 n ) ! Γ ( 2 n + 1 4 ) H 2 n ( y 2 4 i x ) ,

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