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

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References

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  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]

2009 (1)

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]

2008 (3)

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]

2007 (2)

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

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]

2006 (4)

2004 (1)

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]

2003 (2)

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

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]

2001 (1)

1999 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

1996 (1)

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]

1991 (1)

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]

1989 (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]

1983 (1)

1982 (1)

1976 (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976).
[CrossRef]

1952 (1)

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambridge Philos. Soc. 48, 733-741 (1952).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755-776 (1909).
[CrossRef]

Alonso, M. A.

Borghi, R.

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, “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] [PubMed]

M. A. Alonso and R. Borghi, “Complete far-field asymptotic series for free fields,” Opt. Lett. 31, 3028-3030 (2006).
[CrossRef] [PubMed]

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]

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]

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]

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp. 484-498.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp.921-923.

Caliceti, E.

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]

Chew, W. C.

Cìzek, J.

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]

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]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755-776 (1909).
[CrossRef]

Jentschura, U. D.

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]

Jones, D. S.

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambridge Philos. Soc. 48, 733-741 (1952).
[CrossRef]

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 128-141.

Meyer-Hermann, M.

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]

Ribeca, P.

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]

Saghafi, S.

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]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

Sherman, G. C.

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]

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976).
[CrossRef]

Skála, L.

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]

Stamnes, J. J.

J. J. Stamnes, “Uniform asymptotic theory of diffraction by apertures,” J. Opt. Soc. Am. 73, 96-109 (1983).
[CrossRef]

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976).
[CrossRef]

Surzhykov, A.

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]

Vinette, F.

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]

Weniger, E. J.

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]

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]

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]

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.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 128-141.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp.921-923.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp. 484-498.

Zamastil, J.

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]

Ann. Physik (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755-776 (1909).
[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]

J. Math. Phys. (3)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760 (1976).
[CrossRef]

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]

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]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Lett. (4)

Phys. Lett. A (1)

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]

Phys. Rep. (1)

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]

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[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]

Phys. Rev. Lett. (1)

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]

Proc. Cambridge Philos. Soc. (1)

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambridge Philos. Soc. 48, 733-741 (1952).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp.921-923.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 128-141.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp. 484-498.

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Figures (7)

Fig. 1
Fig. 1

Comparison of | A n ( ϕ ) | as a function of n evaluated via Eq. (13) (dots) and through the asymptotic expression in Eq. (39) (solid curves), for ϕ = 0 (a), ϕ = π 4 (b), ϕ = π 2 (c), and ϕ = π (d).

Fig. 2
Fig. 2

Relative truncation error as a function of truncation order N corresponding to a 2D CFF with q = 1 , for r = 1 (open circles), 2 (filled circles), 4 (open squares) and 8 (filled squares), and for ϕ = 0 (solid curves), π 4 (dashed curves), and π 2 (dotted curves). The results for ϕ = π and 3 π 4 are identical to those for 0 and π 4 , respectively.

Fig. 3
Fig. 3

Behavior of the modulus of the functions A n ( θ ) with q = 1 versus n, for θ = 0 (solid curve), θ = π 4 (open circles), θ = π 2 (filled circles), θ = 3 π 4 (open squares), and θ = π (dotted curve).

Fig. 4
Fig. 4

Relative truncation error as a function of truncation order N corresponding to a 3D CFF with q = 1 , for r = 1 (open circles), 2 (filled circles), 4 (open squares) and 8 (filled squares), and for θ = 0 (solid curves), π 4 (dashed curves), and π 2 (dotted curves). The results for θ = π and 3 π 4 are identical to those for 0 and π 4 , respectively.

Fig. 5
Fig. 5

Same as in Fig. 3 but for the axial component of the electric field in Eq. (45). Note that E z does not depend on the azimuthal variable ϕ.

Fig. 6
Fig. 6

2D map of the logarithm of the relative error obtained for a scalar 2D CFF with q = 1 via a 10th-order WT. The abscissas correspond to the values of x, while the ordinates to the values of y. The fillled disk of radius π 2 (corresponding to a physical size of λ 4 ) represents the zone where the relative error turns out to be greater than 10 2 .

Fig. 7
Fig. 7

2D map of the logarithm of the relative error obtained for a scalar 3D CFF with q = 1 via a 10th-order WT. The abscissas correspond to the values of z the ordinates to the values of x 2 + y 2 . The filled disk of radius π 2 (corresponding to a physical size of λ 4 ) represents the zone where the relative error turns out to be greater than 10 2 .

Equations (54)

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

U ( r ) = 1 ( 4 π ) ( D 1 ) 2 S A ( u ̂ ) exp ( i k u ̂ r ) d Ω ,
U ( r ) A ( r ̂ ) exp ( i r ) ( 2 i r ) ( D 1 ) 2 + A ( r ̂ ) exp ( i r ) ( 2 i r ) ( D 1 ) 2 .
U ( r ) A ( r ̂ ) exp ( i r ) ( 2 i r ) ( D 1 ) 2 , z > 0 .
2 U ( r ) + U ( r ) = 0 ,
2 = r 2 + 1 r 2 Ω 2 .
r 2 = 1 r D 1 r ( r D 1 r ) ,
Ω 2 = { 2 ϕ 2 , D = 2 1 sin θ θ ( sin θ θ ) + 1 sin 2 θ 2 ϕ 2 , D = 3 } , ,
U ( r ) = n = 0 exp ( i r ) ( 2 i r ) n + D 1 2 A n ( r ̂ ) + n = 0 exp ( i r ) ( 2 i r ) n + D 1 2 A n ( r ̂ ) ,
r 2 [ exp ( i r ) r n + D 1 2 ] = exp ( i r ) r n + D 1 2 [ ( n D 3 2 ) ( n + D 1 2 ) r 2 2 i n r 1 ] .
A n ( r ̂ ) = [ ( n + D 3 2 ) ( n D 1 2 ) + Ω 2 ] n A n 1 ( r ̂ ) ,
A n ( r ̂ ) = Q n A ( r ̂ ) ,
Q n = 1 n ! j = 0 n 1 [ ( j D 3 2 ) ( j + D 1 2 ) L 2 ] ,
A n ( ϕ ) = 1 n ! j = 0 n 1 [ ( j + 1 2 ) 2 + d 2 d ϕ 2 ] A ( ϕ ) .
A n ( θ , ϕ ) = 1 n ! j = 0 n 1 [ j ( j + 1 ) + Ω 2 ] A ( θ , ϕ ) .
Q n = ( 1 ( D 2 ) + 4 L 2 2 ) n ( 1 + ( D 2 ) 2 + 4 L 2 2 ) n n ! ,
Q n exp ( n ) ( 1 n ) ( 1 2 n ) 2 π cos [ π 2 ( D 2 ) 2 + 4 L 2 ] ( n 1 ) ! π cos [ π 2 ( D 2 ) 2 + 4 L 2 ] , n 1 ,
Q n ( n 1 ) ! π cos ( π L 2 ) , n 1 .
A ( ϕ ) = m a m exp ( i m ϕ ) .
A n ( ϕ ) ( n 1 ) ! π m a m cos ( π | m | ) exp ( i m ϕ ) = ( n 1 ) ! π m a m ( 1 ) m exp ( i m ϕ ) = ( n 1 ) ! π m a m exp [ i m ( ϕ ± π ) ] = ( n 1 ) ! π A ( ϕ ± π ) ,
Q n ( n 1 ) ! π cos ( π 2 1 + 4 L 2 ) , n 1 .
A ( θ , ϕ ) = l , m a l , m Y l , m ( θ , ϕ ) ,
A n ( r ̂ ) = Q n A ( r ̂ ) ( n 1 ) ! π l , m a l , m Y l , m ( θ , ϕ ) cos ( π 2 1 + 4 l ( l + 1 ) ) = 0 , n 1 .
Q n = 1 n ! j = 0 n 1 [ j ( j + 1 ) L 2 ] .
A n ( r ̂ ) = 1 n ! l , m a l , m Y l , m ( θ , ϕ ) j = 0 n 1 ( j l ) ( j + l + 1 ) .
n ! Q n = j = 0 n 1 ( b j L 2 ) ,
b j = { ( j + 1 2 ) 2 , D = 2 j ( j + 1 ) , D = 3 . .
n ! Q n = j = 0 n 1 ( b j L 2 ) = j = 0 n c j ( n ) ( L 2 ) ( n j ) ,
c 1 ( n ) = j = 0 n 1 b j ,
c 2 ( n ) = j = 0 n 1 k = j + 1 n 1 b j b k ,
c 3 ( n ) = j = 0 n 1 k = j + 1 n 1 l = k + 1 n 1 b j b k b l ,
c n ( n ) = b 0 b 1 b n 1 .
j = 0 n 1 ( b j + a ) = j = 0 n c j ( n ) a n j ,
j = 0 n ( b j + a ) = j = 0 n + 1 c j ( n + 1 ) a n + 1 j .
j = 0 n + 1 c j ( n + 1 ) a n + 1 j = j = 1 n + 1 b n c j 1 ( n ) a n j + 1 + j = 0 n c j ( n ) a n + 1 j ,
c j ( n + 1 ) = { 1 , j = 0 c j ( n ) + b n c j 1 ( n ) , j [ 0 , n ] b n c n ( n ) , j = n + 1 . .
c n ( n ) = { Γ 2 ( n + 1 2 ) π , D = 2 δ n , 0 , D = 3 , ,
U ( r ) = { H 0 ( r ) , D = 2 h 0 ( r ) , D = 3 ,
A ( u ̂ ) = { 2 π , D = 2 2 , D = 3 } ,
Γ 2 ( n + 1 2 ) n ! ( n 1 ) ! ,
A ( u ̂ ) = { 1 π exp ( q cos ϕ ) , D = 2 exp ( q cos θ ) , D = 3 } ,
U ( r ) = { J 0 ( ( x i q ) 2 + y 2 ) , D = 2 sinc ( x 2 + y 2 + ( z i q ) 2 ) , D = 3 . ,
A n ( ϕ ) ( n 1 ) ! π π exp ( q cos ϕ ) .
U ( r ) = sin ( z i q ) z i q ,
U ( r ) = exp ( i r ) 2 i r exp ( q ) 1 i q r exp ( i r ) 2 i r exp ( q ) 1 i q r ,
A n ( 0 ) = exp ( q ) ( 2 q ) n ,
A n ( π ) = exp ( q ) ( 2 q ) n ,
Ω 2 = d d u z [ ( 1 u z 2 ) d d u z ] ,
A ( u ̂ ) = ( z ̂ u z u ̂ ) exp ( q u z ) ,
E ( r ) = ( 2 x z , 2 y z , 1 + 2 z 2 ) sinc ( R ) ,
E z ( 0 , 0 , z ) = ( 1 + 2 z 2 ) sin ( z i q ) z i q = 2 sin ( z i q ) ( z i q ) 3 2 cos ( z i q ) ( z i q ) 2 ,
E z ( 0 , 0 , z ) = exp ( i z ) exp ( q ) [ 1 i ( z i q ) 3 1 ( z i q ) 2 ] exp ( i z ) exp ( q ) [ 1 i ( z i q ) 3 + 1 ( z i q ) 2 ] .
A n ( z ) ( 0 π ) = exp ( ± q ) × { 0 , n = 0 4 , n = 1 4 n ( n ± 2 q 1 ) ( 2 q ) n 2 , n > 1 ,
δ k = j = 0 k ( 1 ) j ( k j ) ( 1 + j ) k 1 s j a j + 1 j = 0 k ( 1 ) j ( k j ) ( 1 + j ) k 1 1 a j + 1 .

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