Abstract

To speed up the numerical computation of beam shape coefficients in the generalized Lorenz–Mie theory (GLMT) for cylinders, a cylindrical localized approximation has recently been introduced [J. Opt. Soc. Am. A 14, 3014 (1997)], in analogy with the localized approximation used for the GLMT for spheres. A rigorous justification of this cylindrical localized approximation is presented.

© 1998 Optical Society of America

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References

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  1. G. Gouesbet, B. Maheu, 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. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  3. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.
  4. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
    [CrossRef]
  5. F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  6. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  7. E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
    [CrossRef]
  8. K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in GLMT-framework: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  10. G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  11. J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. 1. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  12. G. Gouesbet, J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  13. L. Schwartz, Théorie des distributions (Hermann, Paris, 1951).
  14. F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, New York, 1982).
  15. E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).
  16. J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).
  17. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
    [CrossRef]
  18. K. F. Ren, G. Gréhan, 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]
  19. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  20. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  21. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  22. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  23. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
    [CrossRef]
  24. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
    [CrossRef]
  25. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  26. G. Arlken, Mathematical Methods for Physicists (Academic, New York, 1976).
  27. G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
    [CrossRef] [PubMed]

1997 (3)

1996 (3)

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

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

1995 (3)

1994 (4)

1990 (1)

1989 (1)

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

1988 (2)

G. Gouesbet, B. Maheu, 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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1979 (1)

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

Alexander, D. R.

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

Arlken, G.

G. Arlken, Mathematical Methods for Physicists (Academic, New York, 1976).

Barton, J. P.

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

Butkov, E.

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

Davis, L. W.

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

Gouesbet, G.

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

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in GLMT-framework: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

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

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

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

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, 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, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. 1. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation 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, 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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Gréhan, G.

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in GLMT-framework: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

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

K. F. Ren, G. Gréhan, 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, G. Gréhan, B. Maheu, “A localized interpretation 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, 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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

Lenglart, E.

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

Letellier, C.

Lock, J. A.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation 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, 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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

Onofri, F.

Ren, K. F.

Roddier, F.

F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, New York, 1982).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Schwartz, L.

L. Schwartz, Théorie des distributions (Hermann, Paris, 1951).

van Bladel, J.

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Appl. Opt. (4)

J. Appl. Phys. (1)

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

J. Math. Phys. (1)

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

J. Opt. (Paris) (5)

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

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

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

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

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
[CrossRef]

Phys. Rev. A (1)

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

Other (8)

L. Schwartz, Théorie des distributions (Hermann, Paris, 1951).

F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, New York, 1982).

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

G. Arlken, Mathematical Methods for Physicists (Academic, New York, 1976).

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Figures (7)

Fig. 1
Fig. 1

Geometry under study.

Fig. 2
Fig. 2

Comparisons between BSP’s UTEi versus Z for s=0.01. Imaginary parts are zero.

Fig. 3
Fig. 3

Comparisons between BSP’s UTEi versus Y for s=0.01. Imaginary parts are zero.

Fig. 4
Fig. 4

Comparisons between BSP’s Re(UTEi) versus X for s=0.01. Imaginary parts are not zero.

Fig. 5
Fig. 5

Comparisons between UTEi(3), UTEi(1, s2), and UTEi(1, s10) versus Z for s=0.1. Imaginary parts are zero.

Fig. 6
Fig. 6

Comparisons between UTEi(1) and UTEi(3) for s=0.1. Imaginary parts are zero.

Fig. 7
Fig. 7

Comparisons between UTEi’s for s=0.1. Imaginary parts are zero. A small difference is exhibited between UTEi(1, s10) and UTEi(2, s10) for y>3w0/2.

Equations (158)

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UTMi=E0k2m=-+(-i)m exp(imφ) × Im,TM(γ), Jm(R1-γ2)exp(iγZ),
UTEi=H0k2m=-+(-i)m exp(imφ) × Im,TE(γ), Jm(R1-γ2)exp(iγZ),
Z=kz,R=kρ,
Im,TM(γ), (1-γ2)Jm(R1-γ2)exp(iγZ)
=12π(-i)m02π EziE0exp(-imφ)dφ,
Im,TE(γ), (1-γ2)Jm(R1-γ2)exp(iγZ)
=12π(-i)m02π HziH0exp(-imφ)dφ.
(Im,TM(γ), Im,TE(γ))=(Im,TM, Im,TE)δ(γ),
δ(γ), F(γ)=F(0).
UTMi=E0k2m=-+(-i)m exp(imφ)Im,TMJm(R),
UTEi=H0k2m=-+(-i)m exp(imφ)Im,TEJm(R),
Im,TM=0,
Im,TE=(-1)m+1.
Im,X(γ)=-1+1Im,X(γ)δ(γ-γ)dγ,
UTMi=E0k2m=-+(-i)m exp(imφ) × -1+1Im,TM(γ)δ(γ-γ)dγ,Jm(R1-γ2)exp(iγZ).
UTMi=E0k2m=-+(-i)m exp(imφ) × -1+1dγIm,TM(γ)δ(γ-γ),Jm(R1-γ2)exp(iγZ).
UTMi=E0k2m=-+(-i)m exp(imφ) × -1+1dγdγ Im,TM(γ)δ(γ-γ),Jm(R1-γ2)exp(iγZ)=E0k2m=-+(-i)m exp(imφ) × -1+1dγ Im,TM(γ)δ(γ-γ),Jm(R1-γ2)exp(iγZ),
UTMi=E0k2m=-+(-i)m exp(imφ)-1+1Im,TM(γ) × Jm(R1-γ2)exp(iγZ)dγ.
UTEi=H0k2m=-+(-i)m exp(imφ)-1+1Im,TE(γ) × Jm(R1-γ2)exp(iγZ)dγ.
δ(γ-γ)=12π-+ exp[i(γ-γ)Z]dZ,
Im,TM(γ)=im4π2(1-γ2)Jm(R1-γ2) × 02πexp(-imφ)dφ-+ EziE0exp(-iγZ)dZ,
Im,TE(γ)=im4π2(1-γ2)Jm(R1-γ2) × 02π exp(-imφ)dφ-+ HziH0exp(-iγZ)dZ.
krn+1/2,θ=π/2,
kr[(n-1)(n+2)]1/2,θ=π/2.
Eu=E0Ψ0 exp(-ikw),
Ev=0,
Ew=-2QulEu,
Hu=0,
Hv=H0Ψ0 exp(-ikw),
Hw=-2QvlHv,
Ψ0=iQ exp-iQ u2+v2w02,
Q=1i+2(w/l),
l=kw02,
R=kρm,φ=π/2.
Ezi=0,
Hzi=-H0Ψ0 exp(iR cos φ),
Ψ0=iQ exp[-iQs2(R2sin2 φ+Z2)],
Q=1i-2s2R cos φ,
s=w0/l=1/(kw0).
Hzi¯=-H0Gˆ(Ψ0)exp(iR cos φ),
Hzi¯=Hz0i¯ exp(iR cos φ),
Hz0i¯=-H0Gˆ(Ψ0)=-H0Ψ0¯=-H0 exp[-s2(m2+Z2)],
Im,TE(γ)¯=A-+ Hz0i¯H0exp(-iγZ)dZ.
Im,TE(γ)¯=(-1)m2π(1-γ2)-+ Hz0i¯H0exp(-iγZ)dZ.
Im,TM(γ)¯=(-1)m2π(1-γ2)-+ Ez0i¯E0exp(-iγZ)dZ=0.
Im,TE(γ)¯=-(-1)m2π(1-γ2)exp(-m2s2) × -+ exp(-s2Z2)exp(-iγZ)dZ.
I=-+ exp(-s2Z2)exp(-iγZ)dZ=20 exp(-s2Z2)cos(γZ)dZ.
0 exp(-ax2)cos(bx)dx=π2exp[-b2/(4a)]a,
a>0.
I=πsexp[-γ2/(4s2)],
Im,TE(γ)¯=-(-1)m2πs(1-γ2)exp-m2s2-γ24s2.
Hzi=H0[-1+s2(R2sin2 φ+2iR cos φ+Z2)] × exp(iR cos φ).
Im,TE(γ)=(-1)m[-δ(γ)+s2(m2-2)δ(γ)-s2δ(γ)].
I=-+ exp(-iγZ)dZ-s2-+Z2 exp(-iγZ)dZ.
-+ exp(-iγZ)dZ=2πδ(γ);
-+Z2 exp(-iγZ)dZ=-2πδ(γ).
Im,TE(γ)¯=-(-1)m(1-m2s2)δ(γ)1-γ2+s2 δ(γ)1-γ2.
δ(γ)1-γ2, F(γ)=δ(γ), F(γ)1-γ2=F(γ)1-γ2γ=0=F(0),
δ(γ)1-γ2=δ(γ).
δ(γ)1-γ2, F(γ)=δ(γ), F(γ)1-γ2=2γ2F(γ)1-γ2γ=0=F(0)+2F(0),
δ(γ)1-γ2=δ(γ)+2δ(γ).
Im,TE(γ)¯=-(-1)m(1-m2s2)[δ(γ)+s2δ(γ)+2s2δ(γ)],
Im,TE(γ)¯=(-1)m[-δ(γ)+s2(m2-2)δ(γ-s2δ(γ)],
UTEi=H0k2m=-+im exp(imφ){Jm(R)[-1+s2(m2-2)]+s2[RJm(R)+Z2Jm(R)]}.
UTEi¯=-H02k2πsm=-+im exp(imφ)exp(-m2s2) × -1+1 exp[-γ2/(4s2)]  × Jm(R1-γ2)1-γ2exp(iγZ)dγ.
UTEi¯=-H02k2πsm=-+im exp(imφ) × (1-m2s2)I1-12Z2I2,
I1=-1+1 exp[-γ2/(4s2)] Jm(R1-γ2)1-γ2dγ,
I2=-1+1 exp[-γ2/(4s2)] γ2Jm(R1-γ2)1-γ2dγ.
I=πsexp[-γ2/(4s2)]=-+ exp(-s2Z2)exp(-iγZ)dZ=2π[δ(γ)+s2δ(γ)+].
πsexp[-γ2/(4s2)], F(γ)
=2πδ(γ), F(γ)+2πs2δ(γ), F(γ)
=2πF(0)+2πs2 2F(γ)γ2γ=0.
πsexp[-γ2/(4s2)], F(γ)
=-1+1 πsexp[-γ2/(4s2)]F(γ)dγ,
-1+1 exp[-γ2/(4s2)]F(γ)dγ
=2sπF(0)+s2 2F(γ)γ2γ=0.
I1=2sπ[Jm(R)-s2RJm(R)+2s2Jm(R)],
I2=4πs3Jm(R).
UTEi¯=H0k2m=-+im exp(imφ) × [-Jm(R)+s2(m2-2)Jm(R)+s2RJm(R)+s2Z2Jm(R)],
I=n=0 1n!(-s2)n-+Z2n exp(-iγZ)dZ.
-+Z2n exp(-iγZ)dZ=(-1)n2πδ(2n)(γ),
Im,TE(γ)¯=-(-1)m exp(-m2s2)n=0 1n!s2n δ(2n)(γ)1-γ2.
δ(2n)(γ)1-γ2, F(γ)=δ(2n)(γ), F(γ)1-γ2=2nγ2nF(γ)1-γ2γ=0
Im,TE(γ)¯=-(-1)mk=05a2k¯s2k,
a0¯=δ(γ),
a2¯=(2-m2)δ(γ)+δ(γ),
a4¯=12-2m2+12m4δ(γ)+(6-m2)δ(γ)+12δ4(γ),
a6¯=120-12m2+m4-16m6δ(γ)+60-6m2+12m4δ(γ)+5-12m2δ(4)(γ)+16δ(6)(γ),
a8¯=1680-120m2+6m4-13m6+124m8δ(γ)+840-60m2+3m4-16m6δ(γ)+70-5m2+14m4δ(4)(γ)+73-16m2 × δ(6)(γ)+124δ(8)(γ),
a10¯=30,240-1680m2+60m4-2m6+112m8-1120m10δ(γ)+15, 120-840m2+30m4-m6+124m8δ(γ)+1260-70m2+52m4-112m6δ(4)(γ)+42-73m2+112m4δ(6)(γ)+34-124m2δ(8)(γ)+1120δ(10)(γ).
Im,TE(γ)=-(-1)mk=05a2ks2k,
a0=δ(γ),
a2=(2-m2)δ(γ)+δ(γ),
a4=6-5m2+12m4δ(γ)+(3-m2)δ(γ)+12δ4(γ),
a6=24-803m2+133m4-16m6δ(γ)+12-8m2+12m4δ(γ)+1-12m2δ(4)(γ)+16δ(6)(γ),
a8=120-4963m2+1033m4-136m6+124m8δ(γ)+60-1823m2+416m4-16m6δ(γ)+5-92m2+14m4δ(4)(γ)+-16-16m2δ(6)(γ)+124δ(8)(γ),
a10=720-59845m2+8743m4-1175m6+34m8-1120m10δ(γ)+360-500m2+2333m4-103m6+124m8δ(γ)+30-1213m2+256m4-112m6δ(4)(γ)+1-43m2+112m4δ(6)(γ)+-14-124m2δ(8)(γ)+1120δ(10)(γ).
a0=a0¯,a2=a2¯,
L(a4)=L(a4¯)=12m4δ(γ)-m2δ(γ)+12δ(4)(γ).
Im,TE(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)
=12π(-i)m02π Hzi¯H0exp(-imφ)dφ.
Hzi¯=Hz0i¯ exp(iR cos φ),
Hz0i¯=H0[-1+s2(m2+Z2)].
Im,TE(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)
=12π(-i)m[-1+s2(m2+Z2)]
×02π exp(iR cos φ)exp(-imφ)dφ.
02π exp(iR cos φ)exp(-imφ)dφ=2π(-i)mJm(R),
Im,TE(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)
=(-1)m[-1+s2(m2+Z2)]Jm(R).
Im,TE(γ)¯=Im,TE0(γ)¯+s2Im,TE2(γ)¯,
Im,TE0(γ)¯,(1-γ2)Jm(R1-γ2)exp(iγZ)
=(-1)m+1Jm(R),
Im,TE2(γ)¯,(1-γ2)Jm(R1-γ2)exp(iγZ)
=(-1)m(m2+Z2)Jm(R).
Im,TE0(γ)¯=(-1)m+1δ(γ).
Im,TE2(γ)¯=Am0¯δ(γ)+Am1¯δ(γ)+Am2¯δ(γ),
Im,TE2(γ)¯,(1-γ2)Jm(R1-γ2)exp(iγZ)
=Am0¯Jm(R)-Am1¯iZJm(R)
-Am2¯[(Z2+2)Jm(R)+RJm(R)].
Am1¯=0,
Am2¯=(-1)m+1,
Am0¯=(-1)m(m2-2)-(-1)m RJm(R)Jm(R).
Im,TE(γ)¯=(-1)m-δ(γ)+s2(m2-2)δ(γ)-s2δ(γ)-s2 RJm(R)Jm(R)δ(γ).
Im,TE(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)f(R)dR=12π(-i)mf(R)dR02π Hzi¯H0exp(-imφ)dφ,
Im,TE0(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)f(R)dR
=(-1)m+1f(R)Jm(R)dR.
Im,TE0(γ)¯=Kδ(γ),
Kf(R)Jm(R)dR=(-1)m+1f(R)Jm(R)dR,
Am0¯Jm(R)f(R)dR=(-1)m(m2-2)Jm(R)f(R)dR-(-1)mRJm(R)f(R)dR.
Am0¯=(-1)m(m2-2)-(-1)m RJm(R)f(R)dRJm(R)f(R)dR.
0RμJm(R)dR=2μ Γm+μ+12Γm-μ+12,
-m-1<μ<12,
·f(R)dR=0·Rμ dR.
E=RJm(R)f(R)dRJm(R)f(R)dR=INID,
IN=0Rμ+1Jm(R)dR,
ID=0RμJm(R)dR.
Jm(R)=12[Jm-1(R)-Jm+1(R)],
IN=12(I1-I2),
I1=0Rμ+1Jm-1(R)dR,
I2=0Rμ+1Jm+1(R)dR.
-m-1<μ<-12.
μ=α-m
-1<α<m-12.
E=Γm-α-12Γm-α+12-Γα+32Γα+12=m-α-1.
Jm(R)=(-1)mJ-m(R),
E=Γ|m|-α-12Γ|m|-α+12-Γα+32Γα+12=|m|-α-1
-1<α<|m|-12.
α=|m|-1,|m|={1, 2, },
α-1,m=0,
Im,TE(γ)¯, (1-γ2)Jm(R1-γ2)exp(iγZ)f(R)dR
=(-1)m Hz0i¯H0f(R)Jm(R)dR.
f(R)dR-1+1dγIm,TE(γ)¯δ(γ-γ),(1-γ2) × Jm(R1-γ2)exp(iγZ)
=f(R)dR-1+1dγdγ Im,TE(γ)¯ × δ(γ-γ)(1-γ2)Jm(R1-γ2) × exp(iγZ)=f(R)dR-1+1dγ Im,TE(γ)¯(1-γ2) × Jm(R1-γ2)exp(iγZ),
f(R)dR-1+1Im,TE(γ)¯(1-γ2)×Jm(R1-γ2)exp(iγZ)dγ
=(-1)m Hz0i¯H0f(R)Jm(R)dR.
Im,TE(γ)¯=(-1)m2π(1-γ2)-+ Hz0i¯H0exp(-iγZ)dZ × f(R)Jm(R)dRf(R)Jm(R1-γ2)dR.
0RμJm(aR)dR=2μa-μ-1 Γm+μ+12Γm-μ+12.
Im,TE(γ)¯=(-1)m2π(1-γ2)-+ Hz0i¯H0exp(-iγZ)dZ,

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