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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 25, Iss. 1 — Jan. 1, 2008
  • pp: 211–218

Summing Pauli asymptotic series to solve the wedge problem

Riccardo Borghi  »View Author Affiliations


JOSA A, Vol. 25, Issue 1, pp. 211-218 (2008)
http://dx.doi.org/10.1364/JOSAA.25.000211


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Abstract

The use of the asymptotic treatment for the wedge diffraction problem established long ago by Pauli [Phys. Rev. 54, 924 (1938)] is here revisited and proposed in the character of a powerful computational tool for accurately retrieving the total electromagnetic field even in the near zone. After proving its factorial divergent character, the Pauli series is summed through the Weniger transformation, a nonlinear resummation scheme particularly efficient in the case of factorial divergence. Numerical results are carried out to show the accuracy and effectiveness of the proposed approach.

© 2008 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(260.0260) Physical optics : Physical optics
(260.1960) Physical optics : Diffraction theory

ToC Category:
Physical Optics

History
Original Manuscript: October 4, 2007
Manuscript Accepted: October 29, 2007
Published: December 20, 2007

Citation
Riccardo Borghi, "Summing Pauli asymptotic series to solve the wedge problem," J. Opt. Soc. Am. A 25, 211-218 (2008)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-1-211


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References

  1. A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).
  2. Y. Umul, "Modified theory of physical optics approach to wedge diffraction problems," Opt. Express 13, 216-224 (2005). [CrossRef] [PubMed]
  3. Y. Umul, "Modified theory of the physical-optics approach to the impedance wedge problem," Opt. Lett. 31, 401-403 (2006). [CrossRef] [PubMed]
  4. M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007). [CrossRef]
  5. V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002). [CrossRef]
  6. N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995). [CrossRef] [PubMed]
  7. H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996). [CrossRef]
  8. J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003). [CrossRef]
  9. W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938). [CrossRef]
  10. P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950). [CrossRef]
  11. M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996). [CrossRef] [PubMed]
  12. M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001). [CrossRef]
  13. Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).
  14. R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).
  15. G. H. Hardy, Divergent Series (AMS, 1991).
  16. 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]
  17. J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003). [CrossRef]
  18. 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]
  19. E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.
  20. For an updated review about methods for summing 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.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1. [CrossRef]
  21. It should be noted that in the original paper, Pauli used the opposite convention; i. e., the temporal factor had the form exp(+iωt). For this reason, the subsequent formulas are the complex conjugate of those written in .
  22. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  23. F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983). [CrossRef]
  24. We use, as far as possible, a notation similar to the original one adopted in .
  25. If, on the contrary, the illumination produces no geometrical shadow, to obtain a regular representation also at the second reflection, it is sufficient to change φ into φ−2πn in all the following equations.
  26. M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).
  27. M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005). [CrossRef]
  28. When z is a real negative, the integral in Eq. must be intended in the Cauchy principal value sense. This is related to the fact that the line arg z=π represents a branch cut for the exponential integral function .
  29. R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003). [CrossRef] [PubMed]
  30. R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007). [CrossRef] [PubMed]
  31. R. Borghi, "Optical asymptotics via Weniger transformation," arXiv.org e-Print archieve, physics/0706.3573, 25 June 2007, http://arxiv.org/abs/0706.3573.
  32. M. A. Alonso and R. Borghi, "Complete far-field asymptotic series for free fields," Opt. Lett. 31, 3028-3030 (2006). [CrossRef] [PubMed]

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