Abstract

A so-called elliptical cylinder localized approximation theory allowing one to speed up the evaluation of beam shape distributions in generalized Lorenz–Mie theory for infinitely long cylinders with elliptical cross sections has been previously introduced and, in the case of Gaussian beams, rigorously justified. The validity of this approximation for arbitrary shaped beams is examined.

© 1999 Optical Society of America

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References

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  1. G. Gouesbet, L. Mees, G. Gréhan, “Partial wave description of shaped beams in elliptical cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
    [CrossRef]
  2. G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
    [CrossRef]
  3. G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
    [CrossRef]
  4. G. Gouesbet, L. Mees, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
    [CrossRef]
  5. G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
    [CrossRef]
  6. G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).
  7. G. Gouesbet, “Exact description of arbitrary shaped beams for use in light scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 (1996).
    [CrossRef]
  8. G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
    [CrossRef]
  9. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
    [CrossRef]
  10. R. Campbell, Théorie générale de l’équation de Mathieu (Masson, Paris, 1955).
  11. N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).
  12. G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).
  13. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
    [CrossRef]
  14. C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. 55, 309–314 (1965).
    [CrossRef]
  15. G. Gouesbet, “Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for circular cylinders,” J. Mod. Opt. 46, 1185–1200 (1999).
    [CrossRef]
  16. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  17. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave rep-resentations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  18. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
    [CrossRef]
  19. 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]
  20. J. A. Lock, 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]
  21. G. Gouesbet, 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, 2516–2525 (1994).
    [CrossRef]

1999 (6)

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[CrossRef]

G. Gouesbet, “Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for circular cylinders,” J. Mod. Opt. 46, 1185–1200 (1999).
[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]

G. Gouesbet, L. Mees, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
[CrossRef]

1998 (1)

1997 (1)

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
[CrossRef]

1996 (2)

1995 (2)

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

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

1994 (2)

1979 (1)

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

1965 (1)

1963 (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

Campbell, R.

R. Campbell, Théorie générale de l’équation de Mathieu (Masson, Paris, 1955).

Davis, L. W.

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

Gouesbet, G.

G. Gouesbet, “Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for circular cylinders,” J. Mod. Opt. 46, 1185–1200 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[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]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave description of shaped beams in elliptical cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
[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, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

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

J. A. Lock, 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, 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, 2516–2525 (1994).
[CrossRef]

G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).

Gréhan, G.

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave description of shaped beams in elliptical cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

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

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).

Lock, J. A.

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).

Mees, L.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, L. Mees, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave description of shaped beams in elliptical cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).

Ren, K. F.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).

Yeh, C.

C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. 55, 309–314 (1965).
[CrossRef]

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

Appl. Opt. (1)

J. Math. Phys. (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

J. Mod. Opt. (1)

G. Gouesbet, “Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz–Mie theory for circular cylinders,” J. Mod. Opt. 46, 1185–1200 (1999).
[CrossRef]

J. Opt. (Paris) (2)

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
[CrossRef]

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A Pure Appl. Opt. 1, 121–132 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

Part. Part. Syst. Charact. (2)

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “The structure of generalized Lorenz–Mie theory for elliptical infinite cylinders,” Part. Part. Syst. Charact. 16, 3–10 (1999).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

Phys. Rev. A (1)

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

Other (4)

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Localized approximation for Gaussian beams in elliptical cylinder coordinates,” submitted to Appl. Opt. (preprint available from the author at the address on the title page).

R. Campbell, Théorie générale de l’équation de Mathieu (Masson, Paris, 1955).

N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).

G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).

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Figures (2)

Fig. 1
Fig. 1

Auxiliary Cartesian coordinate system (x, y, z), where v is for the perpendicular case and u is for the parallel case.

Fig. 2
Fig. 2

Definition of the rotation angle β.

Equations (176)

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

(Eu, Ev, Ew)=E0 exp(-iW)p=0q=0l=0×(Epqlu, Epqlv, Epqlw)UpVqWl,
(U, V, W)=(ku, kv, kw),
(q+1)Epq+1lw-(l+1)Epql+1v+i(Epqlv+Hpqlu)=0,
(l+1)Epql+1u-(p+1)Ep+1qlw+i(Hpqlv-Epqlu)=0,
(p+1)Ep+1qlv-(q+1)Epq+1lu+iHpqlw=0,
(p+1)Hp+1qlu+(q+1)Hpq+1lu
+(l+1)Hpql+1w-iHpqlw=0,
(q+1)Hpq+1lw-(l+1)Hpql+1v+i(Hpqlv-Epqlu)=0,
(l+1)Hpql+1u-(p+1)Hp+1qlw-i(Hpqlu+Epqlv)=0,
(p+1)Hp+1qlv-(q+1)Hpq+1lu-iEpqlw=0,
(p+1)Ep+1qlu+(q+1)Epq+1lv+(l+1)Epql+1w
-iEpqlw=0.
Cpqli=0,p+q+l>N,
C=cos Γ,S=sin Γ
(X, Y, Z)=k(x, y, z)
U=Y,
V=CX-SZ,
W=-SX-CZ,
Ez=-SEv-CEw,
Ez=-E0 exp(iSX)exp(iCZ)p=0q=0l=0(-1)l(SEpqlv+CEpqlw)Yp(CX-SZ)q(SX+CZ)l,
U=CX-SZ,
V=-Y,
W=-SX-CZ,
Ez=-SEu-CEw,
Ez=-E0 exp(iSX)exp(iCZ)×p=0q=0l=0(-1)q+l(SEpqlu+CEpqlw)×(CX-SZ)pYq(SX+CZ)l.
x2a2+y2b2=1,a>b,
L=ka1-ba21/2.
X=-X cos β-Y sin β,
Y=X sin β-Y cos β,
X=L cosh μ cos θ,
Y=L sinh μ sin θ,
Z=Z,
cosh(iμ)=cos μ,
sinh(iμ)=i sin μ,
Ez(z, iμ, θ)=-E0E1 exp(iCZ)p=0q=0l=0Lp(-1)q×(SEpqlv+CEpqlw)τp(LCη+SZ)q×(LSη-CZ)l
Ez(z, iμ, θ)=-E0E1 exp(iCZ)p=0q=0l=0(-1)p+qLq×(SEpqlu+CEpqlw)×(LCη+SZ)pτq(LSη-CZ)l
E1=exp(-iLSη),
η=cos μ cos θ cos β+i sin μ sin θ sin β,
τ=-ηβ=cos μ cos θ sin β-i sin μ sin θ cos β.
q2=L24 (1-γ2),
q02=q2(γ=C)=L2S24.
E1=exp(-2iq0η),
E1β=iLSτE1,
2E1β2=(iLSη-L2S2τ2)E1.
UTM=E0k2 n=0[An,TM(γ), cehn(μ, q2)cen(θ, q2)×exp(ikγ z)+Bn,TM(γ), sehn(μ, q2)×sen(θ, q2)exp(ikγ z)],
UTE=H0k2 n=0[An,TE(γ), cehn(μ, q2)cen(θ, q2)×exp(ikγ z)+Bn,TE(γ), sehn(μ, q2)×sen(θ, q2)exp(ikγ z)],
UTM=E0k2 n=0An,TM(γ)cehn(μ, q2)cen(θ, q2)×exp(ikγ z)dγ+Bn,TM(γ)sehn(μ, q2)×sen(θ, q2)exp(ikγ z)dγ,
UTE=H0k2 n=0An,TE(γ)cehn(μ, q2)cen(θ, q2)×exp(ikγ z)dγ+Bn,TE(γ)sehn(μ, q2)×sen(θ, q2)exp(ikγ z)dγ .
An,TM(γ)=12π3(1-γ2) 02π02π-+Ez+cen(μ, q2)×cen(θ, q2)exp(-iγZ)dμ dθ dZ,
Bn,TM(γ)=12π3(1-γ2) 02π02π-+Ez-sen(μ, q2)×sen(θ, q2)exp(-iγZ)dμ dθ dZ,
Ez+=12E0 [Ez(z, iμ, θ)+Ez(z, iμ, -θ)],
Ez-=12iE0 [Ez(z, iμ, θ)-Ez(z, iμ, -θ)],
Ez0(z, iμ, θ)=Ez(z, iμ, θ)/E1.
E1¯=1π (-i)ppn cen(β, q2),
An,TM¯(γ)
=E1¯1-γ2 -+ 1E0 Ez0(z, iμ0, θ0)exp(-iγZ)dZ,
θ0=β-π/2
E1¯=1π (-i)psn sen(β, q2),
Bn,TM¯(γ)
=E1¯1-γ2 -+ 1E0 Ez0(z, iμ0, θ0)exp(-iγZ)dZ,
Cpqli=0,p+q+l>0,
E000v+H000u=0,
H000v-E000u=0,
E000w=H000w=0,
Ez(z, iμ, θ)=-E0E1 exp(iCZ)SE000v,
Ez+=-12 exp(iCZ)S(E1+E˜1)E000v,
Ez-=i2 exp(iCZ)S(E1-E˜1)E000v,
A˜(θ)=A(-θ)
-+ exp(iCZ-iγ Z)d Z=2πδ(γ, C).
f(q2)=cen(μ, q2)cen(θ, q2)
δ(γ, C) f(q2)1-γ2,φ(γ)
=δ(γ, C), f(q2)1-γ2 φ(γ)=f(q2)1-γ2 φ(γ)γ=C=1S2 f(q02)φ(C)=1S2 f(q02)δ(γ, C),φ(γ),
δ(γ, C) f(q2)1-γ2=1S2 f(q02)δ(γ, C).
An(γ)=-12π2S E000v In0δ(γ, C),
In0=02π02π(E1+E˜1)cen(μ, q02)cen(θ, q02)dμ dθ.
In0=202π02πE1cen(μ, q02)cen(θ, q02)dμ dθ.
E1=exp(-2iq0η)=2n=0(-i)ppn cen(μ, q02)cen(θ, q02)cen(β, q02)+i (-i)psn sen(μ, q02)sen(θ, q02)sen(β, q02),
02πcen(x, q2)cem(x, q2)d x=πδnm,
02πsen(x, q2)sem(x, q2)d x=πδnm,
02πcen(x, q2)sem(x, q2)d x=0,
In0=4π2 (-i)ppn cen(β, q02).
An(γ)=-2 (-i)ppn 1S cen(β, q02)E000vδ(γ, C).
Tn0=202π02πE1sen(μ, q02)sen(θ, q02)dμ dθ=4iπ2 (-i)psn sen(β, q02);
Bn(γ)=-2 (-i)psn 1S sen(β, q02)E000vδ(γ, C).
δ(γ, C)1-γ2=1S2 δ(γ, C),
An¯(γ)=An(γ),
Bn¯(γ)=Bn(γ).
Ez(z, iμ, θ)=-E0E1 exp(iCZ)SE000u,
Cpqli=0,p+q+l>1.
(Epqlw)N=1=E001u=E001v=0.
-+Z exp(iCZ-iγZ)d Z=2πiδ(γ, C).
δ(j)(γ, C), φ(γ)=(-1)jddγ φ(γ)γ=C
δ(γ, C) f(q2)1-γ2
=L2C2S2 cen(μ, q2)q2γ=Ccen(θ, q02)+cen(μ, q02)cen(θ, q2)q2γ=Cδ(γ, C)-2CS4 cen(μ, q02)cen(θ, q02)δ(γ, C)+1S2 cen(μ, q02)cen(θ, q02)δ(γ, C).
An(γ)=-12π2 1S E000v+CS2 E000wIn0δ(γ, C)+LS E100v In1δ(γ, C)-LCS E010v In2δ(γ, C)-i L2C2 E010v In3δ(γ, C)+i 2CS2 E010v In0δ(γ, C)-iE010v In0δ(γ, C),
In1=202π02πE1τcen(μ, q02)cen(θ, q02)dμ dθ,
In2=202π02πE1ηcen(μ, q02)cen(θ, q02)dμ dθ,
In3=202π02πE1cen(μ, q2)q2γ=Ccen(θ, q02)+cen(μ, q02)cen(θ, q2)q2γ=Cdμ dθ.
In1=1iLS dIn0dβ=-i 4π2LS (-i)ppn ddβ cen(β, q02).
E1*=exp(-2iqη),
E1=E1*(γ=C),
ηE1=iLS2 E1*q2γ=C.
In2=2iLSπ2(-i)pq2 cen(β, q2)pnγ=C+4iLSπ (-i)ppn cen(β, q02)02πcen(x, q2)q2γ=C×cen(x, q02)dx.
02πcen(x, q2)q2γ=Ccen(x, q02)dx=0.
In2=2iLSπ2(-i)pq2 cen(β, q2)pnγ=C.
In3=0.
An(γ)=(-i)ppn0 -2S E000v-2 CS2 E000w-4i CS2 E010v×cen(β, q02)δ(γ, C)+2iS2 E100v ddβ×cen(β, q02)δ(γ, C)+iL2CE010v×pn0q2 cen(β, q2)pnγ=Cδ(γ, C)+2iE010vcen(β, q02)δ(γ, C).
Bn(γ)=(-i)psn0 -2S E000v-2 CS2 E000w-4i CS2 E010v×sen(β, q02)δ(γ, C)+2iS2 E100v ddβ×sen(β, q02)δ(γ, C)+iL2CE010vsn0q2 sen(β, q2)snγ=Cδ(γ, C)+2iE010vsen(β, q02)δ(γ, C),
δ(γ, C)1-γ2 cen(β, q2)pn
=L2C2S2 q2 cen(β, q2)pnγ=Cδ(γ, C)-2 CS4 cen(β, q02)pnδ(γ, C)+1S2 cen(β, q02)pnδ(γ, C),
An¯(γ)=(-i)ppn0 -2S E000v-2 CS2 E000w-4 iCS2 E010v×cen(β, q02)δ(γ, C)+iL2CE010v pn0q2 cen(β, q2)pnγ=Cδ(γ, C)+2iE010v cen(β, q02)δ(γ, C)+2LS (CηE¯010v-τE¯100v)cen(β, q02)δ(γ, C),
Bn¯(γ)=(-i)psn0 -2S E000v-2 CS2 E000w-4 iCS2 E010v×sen(β, q02)δ(γ, C)+iL2CE010vsn0q2 sen(β, q2)snγ=Cδ(γ, C)+2iE010vsen(β, q02)δ(γ, C)+2LS (CηE¯010v-τE¯100v)sen(β, q02)δ(γ, C).
(A)ddβ cen(β, q02)=iLSτc¯en(β, q02),
(B)ddβ sen(β, q02)=iLSτs¯en(β, q02),
(C)LCηE¯010v=0.
τ¯=cos μ0 sin2 β+i sin μ0 cos2 β, 
η¯=(sin β cos β)(cos μ0-i sin μ0).
μ0=A+iB,A, BR,
cosh B sin2 β-sinh B cos2 β=0,
(sin A)(cosh B cos2 β-sinh B sin2 β)
=-1LScen(β, q02) ddβ cen(β, q02),
dτ¯dβ=2η¯,
2iLSηc¯en(β, q02)-L2S2τ¯2cen(β, q02)
=d2dβ2 cen(β, q02).
Cpqli=0,p+q+l>2.
(Epqlw)N=2=E110u=E101u=E011u=E002u=E110v=E101v=E011v=E002v=E001w=0.
Ez(z, iμ, θ)=-E0E1 exp(iCZ)(A+BZ+DZ2),
A=A1+A2τ+A3η+A4τ2+A5η2,
A1=SE000v+CE000w,
A2=L(SE100v+CE100w),
A3=-LC(SE010v+CE010w)+LS2E001v,
A4=L2SE200v,
A5=L2C2SE020v,
B=B1+B2η,
B1=-S(SE010v+CE010w)-CSE001v,
B2=2LCS2E020v,
D=S3E020v.
-+Z2 exp(iCZ-iγ Z)d Z=-2πδ(γ, C).
δ(γ, C) f(q2)1-γ2=2 3C2+1S6 cen(μ, q02)cen(θ, q02)-L22 3C2+1S4cen(μ, q2)q2 cen(θ, q02)+cen(μ, q02) cen(θ, q2)q2γ=C+L44 C2S2 2cen(μ, q2)2q2 cen(θ, q02)+2 cen(μ, q2)q2 cen(θ, q2)q2+cen(μ, q02) 2cen(θ, q2)2q2γ=Cδ(γ, C)+L2 CS2 cen(μ, q2)q2 cen(θ, q02)+cen(μ, q02) cen(θ, q2)q2γ=C-4 CS4 cen(μ, q02)cen(θ, q02)δ(γ, C)+1S2 cen(μ, q02)cen(θ, q02)δ(γ, C).
An(γ)=1π2 -12S2 A1+i CS4 B1+3C2+1S6 DIn0-12S2 A2 In1+-12S2 A3+i CS4 B2×In2-12S2 A4 In4-12S2 A5 In5-iL2C4S2 B2 In6+L4C28S2 DIn7δ(γ, C)+-i2S2 B1-2 CS4 DIn0-i2S2 B2 In2×δ(γ, C)+12S2 DIn0δ(γ, C),
In4=202π02πE1τ2cen(μ, q02)cen(θ, q02)dμ dθ,
In5=202π02πE1η2cen(μ, q02)cen(θ, q02)dμ dθ,
In6=202π02πE1ηcen(μ, q2)q2 cen(θ, q02)+cen(μ, q02) cen(θ, q2)q2γ=C dμ dθ,
In7=202π02πE12cen(μ, q2)2q2 cen(θ, q02)+2 cen(μ, q2)q2 cen(θ, q2)q2+cen(μ, q02) 2cen(θ, q2)2q2γ=C dμ dθ.
In4=-2π2 (-i)p q2 cen(β, q2)pnγ=C+2L2S2pn0 d2dβ2 cen(β, q02).
η2E1=-12 E1*q2γ=C-L2S24 2E1*2q2γ=C.
In5=iLS In2-L2S22 In5,
In5=02π02π2E1*2q2γ=Ccen(μ, q02)cen(θ, q02)dμ dθ.
Qnmij=02πicen(x, q2)iq2 jcem(x, q2)jq2γ=C dx.
In5=-π2 (-i)p 2 q2 cen(β, q2)pn+L2S2 22q2 cen(β, q2)pnγ=C-2πL2S2 (-i)ppn Qnn20cen(β, q02)-2L2S2×m=0 (-i)ppm cem(β, q02)(Qmn10)2.
In6=4iLSπ (-i)ppn cen(β, q02)Qnn11+m=0 (-i)ppm cem(β, q02)Qmn10Qnm10.
In7=8π (-i)ppn cen(β, q02)Qnn20+m=0 (-i)ppm cem(β, q02)(Qnm10)2.
An(γ)=(-i)ppn0 cen(β, q02)δ(γ, C)-2S2 A1+4i CS4 B1+4 3C2+1S6 D+L2π A5Qnn20+L3CπS B2Qnn11+L4C2πS2 DQnn20+(-i)ppn0 δ(γ, C)2iLS3 A2 ddβ cen(β, q02)+2L2S4 A4 d2dβ2 cen(β, q02)+-i LS A3-2L CS3 B2+A4+A5S2 pn0 q2 cen(β, q2)pn+12 L2A5pn0 22q2 cen(β, q2)pnγ=C+(-i)ppn0 δ(γ, C)-2iS2 B1+8CS4 Dcen(β, q02)+LS B2 pn0q2 cen(β, q2)pnγ=C+2S2 (-i)ppn Dcen(β, q02)δ(γ, C)+δ(γ, C)π2 m=0 (-i)ppm cem(β, q02)×L2A5(Qmn10)2+L3CS B2Qmn10Qnm10+L4C2S2 D(Qnm10)2.
I=L4C2SE020vm=0 (-i)ppm cem(β, q02)[(Qmn10)2+2Qmn10Qnm10+(Qnm10)2].
An(γ)=(-i)ppn0 cen(β, q02)δ(γ, C)-2S2 A1+4i CS4 B1+4 3C2+1S6 D+(-i)ppn0 δ(γ, C)2iLS3 A2 ddβ cen(β, q02)+2L2S4 A4 d2dβ2 cen(β, q02)+-i LS A3-2L CS3 B2+A4+A5S2pn0 q2 cen(β, q2)pn+12 L2A5 pn0 22q2 cen(β, q2)pnγ=C+(-i)ppn0 δ(γ, C)-2iS2 B1+8CS4 Dcen(β, q02)+LS B2 pn0 q2 cen(β, q2)pnγ=C+2S2 (-i)ppn0 Dcen(β, q02)δ(γ, C).
δ(γ, C)1-γ2 cen(β, q2)pn=2 3C2+1S6 cen(β, q02)pn-L22 3C2+1S4 q2 cen(β, q2)pn+L44 C2S2 22q2 cen(β, q2)pnγ=Cδ(γ, C)+-4CS4 cen(β, q02)pn+L2 CS2 q2 cen(β, q2)pnγ=Cδ(γ, C)+1S2 cen(β, q02)pnδ(γ, C),
An¯(γ)=(-i)ppn0 cen(β, q02)δ(γ, C)-2S2 A1+4iCS4 B1+4 3C2+1S6 D+(-i)ppn0 δ(γ, C)-2S2 A2τc¯en(β, q02)-2S2 A3ηc¯en(β, q02)+4 iCS4 B2ηc¯en(β, q02)-2S2 A4τ2¯cen(β, q02)-2S2 A5η2¯cen(β, q02)+-i L2CS2 B1-L2 3C2+1S4 D-i L2CS2 B2η¯pn0q2 cen(β, q2)pn+L42 C2S2 D pn0 22q2 cen(β, q2)pnγ=C+(-i)ppn0 δ(γ, C)-2iB1S2-8CS4 D-2iS2 B2η¯cen(β, q02)+2 L2CS2 Dpn0 q2 cen(β, q2)pnγ=C+(-i)ppn0 2DS2 cen(β, q02)δ(γ, C).
12 L4C2S2 D=12 L2A5,
LS B2=2 L2CS2 D.
iE001v=E200v+E020v,
-i L2CS2 B1-L2 3C2+1S4 D=-i LS A3-2L CS3 B2+A4+A5S2.
An¯(γ)=(-i)ppn0 cen(β, q02)δ(γ, C)×-2S2 A1+4iCS4 B1+4 3C2+1S6 D+(-i)ppn0 δ(γ, C)2iLS3 A2 ddβ cen(β, q02)+2L2S4 A4 d2d β2 cen(β, q02)+-i LS A3-2L CS3 B2+A4+A5S2×pn0 q2 cen(β, q2)pn+12 L2A5 pn0 22q2 cen(β, q2)pnγ=C+(-i)ppn0 δ(γ, C)-2iS2 B1+8CS4 D×cen(β, q02)+LS B2 pn0 q2 cen(β, q2)pnγ=C+2S2 (-i)ppn0 Dcen(β, q02)δ(γ, C)+Δn,
Δn=(-i)ppn0 -2S2 A3η¯-4iLS3 A4η¯+4iCS4 B2η¯-2S2 A5η2¯cen(β, q02)δ(γ, C)-iL2CS2 B2ηp¯n0q2 cen(β, q2)pnγ=Cδ(γ, C)-2iS2 B2ηc¯en(β, q02)δ(γ, C).
Δn=(-i)ppn0 2LS2 [CSE010v+C2E010w+iC2(3E020v-E200v)+i(E020v-E200v)]ηc¯en(β, q02)δ(γ, C)-2 L2C2S E020vη2¯cen(β, q02)δ(γ, C)-2iL3C2E020vηp¯n0q2 cen(β, q2)pnγ=Cδ(γ, C)-4iLCE020vηc¯en(β, q02)δ(γ, C).
E010v=E010w=E020v=E200v=0.
Δn=2iL (-i)ppn (E020v-E200v)ηc¯en(β, q02)δ(γ, C),
E020v=E200v.
E000vE000u, E100v-E010u, E100w-E010w, E010vE100u, E010wE100w,
E001vE001u, E200vE020u, E020vE200u
Eu=E0 exp(-iW)[1+s2(2iW-U2-V2)],
Ev=0,
Ew=2is2E0U exp(-iW),
E000u=1,
E001u=2is2,
E200u=E020u=-s2,
E100w=2is2.

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