Abstract

Arbitrary electromagnetic shaped beams may be described by using expansions over a set of basis functions, with expansion coefficients containing subcoefficients named “beam shape coefficients” (BSCs). When BSCs cannot be obtained in closed form, and/or when the beam description does not exactly satisfy Maxwell’s equations, the most efficient method to evaluate the BSCs is to rely on localized approximations. One of them, named the second modified localized approximation, has been presented in a way that may be found ambiguous in some cases. The aim of the present paper is to remove any ambiguity on the use of the second modified localized approximation.

© 2013 Optical Society of America

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  1. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443(1988).
    [CrossRef]
  2. G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer-Verlag, 2011).
  3. G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” [Invited Review Paper] J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
    [CrossRef]
  4. G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
    [CrossRef]
  5. G. Gouesbet and J. A. Lock, “A list of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus,” [Commemorative Invited Review for the 50th Anniversary of Applied Optics] Appl. Opt. 52, 897–916 (2013).
    [CrossRef]
  6. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
    [CrossRef]
  7. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beams,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  8. G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef]
  9. G. Gouesbet, G. Gréhan, and B. Maheu, “A localized approximation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  10. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  11. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  12. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
    [CrossRef]
  13. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
    [CrossRef]
  14. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
    [CrossRef]
  15. G. Gouesbet, “Exact description of arbitrary shaped beams for use in light scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 (1996).
    [CrossRef]
  16. J. J. Wang and G. Gouesbet, “Note on the use of localized beam models for light scattering theories in spherical coordinates,” Appl. Opt. 51, 3832–3836 (2012).
    [CrossRef]
  17. G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
    [CrossRef]
  18. G. Gouesbet, “Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series,” Phys. Rev. A 43, 5321–5331 (1991).
    [CrossRef]
  19. G. Gouesbet, S. Meunier-Guttin-Cluzel, and O. Ménard, eds., Chaos and Its Reconstruction (Novascience, 2003).
  20. K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
    [CrossRef]
  21. F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
    [CrossRef]
  22. P. L. Marston, “Quasi-Gaussian Bessel-beam superposition: application to the scattering of focused waves by spheres,” J. Acoust. Soc. Am. 129, 1773–1782 (2011).
    [CrossRef]
  23. L. A. Ambrosio and H. E. Hernandez-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. 50, 4489–4498 (2011).
    [CrossRef]
  24. L. A. Ambrosio and H. E. Hernandez-Figueroa, “Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces,” Biomed. Opt. Express 2, 1893–1906 (2011).
    [CrossRef]
  25. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef]
  26. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]

2013

2012

2011

L. A. Ambrosio and H. E. Hernandez-Figueroa, “Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces,” Biomed. Opt. Express 2, 1893–1906 (2011).
[CrossRef]

L. A. Ambrosio and H. E. Hernandez-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. 50, 4489–4498 (2011).
[CrossRef]

G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

P. L. Marston, “Quasi-Gaussian Bessel-beam superposition: application to the scattering of focused waves by spheres,” J. Acoust. Soc. Am. 129, 1773–1782 (2011).
[CrossRef]

2009

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” [Invited Review Paper] J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

1999

1998

1996

1995

1994

1993

1991

G. Gouesbet, “Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series,” Phys. Rev. A 43, 5321–5331 (1991).
[CrossRef]

F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

1990

1989

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beams,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beams,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Ambrosio, L. A.

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beams,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Corbin, F.

F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

Gouesbet, G.

G. Gouesbet and J. A. Lock, “A list of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus,” [Commemorative Invited Review for the 50th Anniversary of Applied Optics] Appl. Opt. 52, 897–916 (2013).
[CrossRef]

J. J. Wang and G. Gouesbet, “Note on the use of localized beam models for light scattering theories in spherical coordinates,” Appl. Opt. 51, 3832–3836 (2012).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” [Invited Review Paper] J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

G. Gouesbet, “Exact description of arbitrary shaped beams for use in light scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 (1996).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

G. Gouesbet, “Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series,” Phys. Rev. A 43, 5321–5331 (1991).
[CrossRef]

F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “A localized approximation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443(1988).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer-Verlag, 2011).

Gréhan, G.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “A localized approximation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443(1988).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer-Verlag, 2011).

Han, Y. P.

G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
[CrossRef]

Hernandez-Figueroa, H. E.

Lock, J. A.

G. Gouesbet and J. A. Lock, “A list of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus,” [Commemorative Invited Review for the 50th Anniversary of Applied Optics] Appl. Opt. 52, 897–916 (2013).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef]

J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

Maheu, B.

Marston, P. L.

P. L. Marston, “Quasi-Gaussian Bessel-beam superposition: application to the scattering of focused waves by spheres,” J. Acoust. Soc. Am. 129, 1773–1782 (2011).
[CrossRef]

Ren, K. F.

Wang, J. J.

J. J. Wang and G. Gouesbet, “Note on the use of localized beam models for light scattering theories in spherical coordinates,” Appl. Opt. 51, 3832–3836 (2012).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

Xu, F.

G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
[CrossRef]

Appl. Opt.

Biomed. Opt. Express

J. Acoust. Soc. Am.

P. L. Marston, “Quasi-Gaussian Bessel-beam superposition: application to the scattering of focused waves by spheres,” J. Acoust. Soc. Am. 129, 1773–1782 (2011).
[CrossRef]

J. Appl. Phys.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beams,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Opt. Soc. Am. A

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[CrossRef]

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

G. Gouesbet, “Exact description of arbitrary shaped beams for use in light scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 (1996).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443(1988).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “A localized approximation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” [Invited Review Paper] J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

G. Gouesbet, F. Xu, and Y. P. Han, “Expanded description of electromagnetic arbitrary shaped beam in spheroidal coordinates for use in light scattering theories: a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 2249–2267 (2011).
[CrossRef]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

Opt. Commun.

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate system. V. Localized beam models,” Opt. Commun. 284, 411–417 (2011).
[CrossRef]

Part. Part. Syst. Charact.

F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Phys. Rev.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. A

G. Gouesbet, “Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series,” Phys. Rev. A 43, 5321–5331 (1991).
[CrossRef]

Other

G. Gouesbet, S. Meunier-Guttin-Cluzel, and O. Ménard, eds., Chaos and Its Reconstruction (Novascience, 2003).

G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer-Verlag, 2011).

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Equations (29)

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Er=mErm,
Erm={E0exp(iRcosθ)sinθexp(imϕ)}Erm(R,θ).
g¯nm=(iL1/2)|m|1Erm(L1/2,π/2),
L=(n|m|)(n+|m|+1)=(n+12)2(|m|+12)2.
g¯nm={(iL1/2)|m|1Erm(L1/2,π/2)}ε=0,
L=(n|m|)(n+|m|+1)+ε=(n+12)2(|m|+12)2+ε,ε0.
(Ex,Ey,Ez)=E0exp(iZ)p=0q=0l=0(Epqlx,Epqly,Epqlz)XpYqZl,
Cpqli=0,M>N,
(Er±1)0(θ=π/2)=12[E000xiE000y].
(Er0)1(θ=π/2)=R2[E100x+E010y],
(Er±1)1(θ=π/2)=(Er±1)0(θ=π/2),
(Er±2)1(θ=π/2)=R4[E100xiE010xiE100yE010y].
(Er0)2(θ=π/2)=(Er0)1(θ=π/2),
(Er±1)2(θ=π/2)=(Er±1)0(θ=π/2)+R28[iE110x+3E200x+E020x+E110yiE200y3iE020y],
(Er±2)2(θ=π/2)=(Er±2)1(θ=π/2),
(Er±3)2(θ=π/2)=R28[iE110x+E200xE020xE110yiE200y±iE020y].
(Er0)3(θ=π/2)=(Er0)1(θ=π/2)+R38[E120x+3E300x+E210y+3E030y],
(Er±1)3(θ=π/2)=(Er±1)2(θ=π/2),
(Er±2)3(θ=π/2)=(Er±2)1(θ=π/2)+R38[iE210x+2E300xiE030xiE120yiE300y2E030y],
(Er±3)3(θ=π/2)=(Er±3)2(θ=π/2),
(Er±4)3(θ=π/2)=R316[E120xiE210x+E300x±iE030x±iE120yE210yiE300y+E030y].
(Er0)4(θ=π/2)=(Er0)3(θ=π/2),
(Er±1)4(θ=π/2)=(Er±1)2(θ=π/2)+R416[E040xiE130x+E220xiE310x+5E400xiE220yiE400y5iE040y],
(Er±2)4(θ=π/2)=(Er±2)3(θ=π/2),
(Er±3)4(θ=π/2)=(Er±3)2(θ=π/2)+R432[3E040xiE130xE220x3iE310x+5E400xiE220y3iE400y±5iE040y],
(Er±4)4(θ=π/2)=(Er±4)3(θ=π/2),
(Er±5)4(θ=π/2)=R432[E040x±iE130xE220xiE310x+E400x±iE220yiE400yiE040y].
(g¯n0)3=i8n2(n+1)2[E120x+3E300x+E210y+3E030y]+i2n(n+1)[E100x+E010y],
(gn0)3=i8(n2)n(n+1)(n+3)[E120x+3E300x+E210y+3E030y]+i2n(n+1)[E100x+E010y].

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