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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. 26, Iss. 11 — Nov. 1, 2009
  • pp: 2410–2417

Free-space asymptotic far-field series

Riccardo Borghi and Miguel A. Alonso  »View Author Affiliations


JOSA A, Vol. 26, Issue 11, pp. 2410-2417 (2009)
http://dx.doi.org/10.1364/JOSAA.26.002410


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Abstract

A systematic procedure for deriving the complete asymptotic series in inverse powers of the distance from the origin of free-space 2D and 3D, scalar and vectorial, monochromatic electromagnetic fields is derived here. Each term of the series is expressed in closed form through the use of a differential operator acting on the angular spectrum. A simple recursive routine for computing the derivatives is provided. Examples of application are also given.

© 2009 Optical Society of America

OCIS Codes
(260.0260) Physical optics : Physical optics
(260.1960) Physical optics : Diffraction theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: June 24, 2009
Manuscript Accepted: September 9, 2009
Published: October 19, 2009

Citation
Riccardo Borghi and Miguel A. Alonso, "Free-space asymptotic far-field series," J. Opt. Soc. Am. A 26, 2410-2417 (2009)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-11-2410


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References

  1. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755-776 (1909). [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp. 484-498.
  3. G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976). [CrossRef]
  4. G. C. Sherman and W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076 (1982). [CrossRef]
  5. J. J. Stamnes, “Uniform asymptotic theory of diffraction by apertures,” J. Opt. Soc. Am. 73, 96-109 (1983). [CrossRef]
  6. C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579-1587 (2001). [CrossRef]
  7. M. A. Alonso, R. Borghi, and M. Santarsiero, “Nonparaxial fields with maximum joint spatial-angular localization. I. Scalar case,” J. Opt. Soc. Am. A 23, 691-700 (2006). [CrossRef]
  8. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998). [CrossRef]
  9. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543-1545 (1999). [CrossRef]
  10. M. A. Alonso, R. Borghi, and M. Santarsiero, “Nonparaxial fields with maximum joint spatial-angular localization. II. Vectorial case,” J. Opt. Soc. Am. A 23, 701-712 (2006). [CrossRef]
  11. M. A. Alonso, R. Borghi, and M. Santarsiero, “Joint spatial-directional localization features of wave fields focused at a complex point,” J. Opt. Soc. Am. A 23, 933-939 (2006). [CrossRef]
  12. D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambridge Philos. Soc. 48, 733-741 (1952). [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp.921-923.
  14. M. A. Alonso and R. Borghi, “Complete far-field asymptotic series for free fields,” Opt. Lett. 31, 3028-3030 (2006). [CrossRef] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 128-141.
  16. For an updated review of 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:0707.1596v1. [CrossRef]
  17. 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]
  18. E. J. Weniger, J. Čìzek, and F. Vinette, “Very accurate summation for the infinite coupling limit of the perturbation-series expansions of anharmonic-oscillators,” Phys. Lett. A 156, 169-174 (1991). [CrossRef]
  19. E. J. Weniger, “Construction of the strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator via a renormalized strong coupling expansion,” Phys. Rev. Lett. 77, 2859-2862 (1996). [CrossRef] [PubMed]
  20. J. Cìzek, 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]
  21. E. J. Weniger, “Mathematical properties of a new Levin-type sequence transformation introduced by Cìzek, Zamastil, and Skàla. I. Algebraic theory,” J. Math. Phys. 45, 1209-1246 (2004). [CrossRef]
  22. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial propagation,” Opt. Lett. 28, 774-776 (2003). [CrossRef] [PubMed]
  23. R. Borghi, “Evaluation of diffraction catastrophes by using Weniger transformation,” Opt. Lett. 32, 226-228 (2007). [CrossRef] [PubMed]
  24. R. Borghi, “Summing Pauli asymptotic series to solve the wedge problem,” J. Opt. Soc. Am. A 25, 211-218 (2008). [CrossRef]
  25. R. Borghi, “On the numerical evaluation of cuspoid diffraction catastrophes,” J. Opt. Soc. Am. A 25, 1682-1690 (2008). [CrossRef]
  26. 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]
  27. 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]

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