Abstract

A cylindrical localized approximation to speed up numerical computations in generalized Lorenz–Mie theory for cylinders, in a special case of perpendicular illumination, was recently introduced and rigorously justified. We generalize this approximation to the case when the cylinder is arbitrarily located and arbitrarily oriented in a Gaussian beam.

© 1999 Optical Society of America

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References

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  1. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  2. N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).
  3. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” J. Phys. 33, 189–195 (1955).
  9. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  10. J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–661 (1997).
    [CrossRef]
  11. K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
    [CrossRef]
  12. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  13. 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]
  14. J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  15. 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]
  16. G. Gouesbet, K. F. Ren, G. Gréhan, “Rigorous justification of the cylindrical approximation to speed up computations in generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
    [CrossRef]
  17. F. Roddier, Distributions et Transformation de Fourier (McGraw-Hill, New York, 1982).
  18. L. Schwartz, Théorie des Distributions (Hermann, Paris, 1951).
  19. E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).
  20. 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]
  21. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  22. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
    [CrossRef]
  23. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J of Opt. (Paris) 28, 45–65 (1997).
    [CrossRef]
  24. D. Guo, Methods of Mathematical Physics (People’s Education Press, Beijing, China, 1965), in Chinese.

1998 (1)

1997 (6)

1996 (2)

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]

1995 (3)

1994 (2)

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 (1)

1986 (1)

1979 (1)

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

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” J. Phys. 33, 189–195 (1955).

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]

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]

Gauchet, N.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).

Girasole, T.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).

Gouesbet, G.

G. Gouesbet, K. F. Ren, G. Gréhan, “Rigorous justification of the cylindrical approximation to speed up computations in generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: 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]

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).

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

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]

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]

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]

J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. 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, 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]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

Gréhan, G.

G. Gouesbet, K. F. Ren, G. Gréhan, “Rigorous justification of the cylindrical approximation to speed up computations in generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

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

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).

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]

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]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

Guo, D.

D. Guo, Methods of Mathematical Physics (People’s Education Press, Beijing, China, 1965), in Chinese.

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]

Lock, J. A.

Maheu, B.

Onofri, F.

Ren, K. F.

Roddier, F.

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

Schwartz, L.

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

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” J. Phys. 33, 189–195 (1955).

Appl. Opt. (4)

J of Opt. (Paris) (1)

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

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) (1)

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

J. Opt. Soc. Am. A (8)

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]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
[CrossRef]

J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–661 (1997).
[CrossRef]

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

J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. 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, K. F. Ren, G. Gréhan, “Rigorous justification of the cylindrical approximation to speed up computations in generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” J. Phys. 33, 189–195 (1955).

Opt. Diagnost. Eng. (1)

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diagnost. Eng. 2, 1–10 (1997).

Part. Part. Syst. Charact. (1)

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)

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

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

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

D. Guo, Methods of Mathematical Physics (People’s Education Press, Beijing, China, 1965), in Chinese.

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Figures (14)

Fig. 1
Fig. 1

Geometry under study.

Fig. 2
Fig. 2

Cartesian coordinate systems for beam description.

Fig. 3
Fig. 3

Coordinate system for a rotated off-axis description.

Fig. 4
Fig. 4

BSD’s versus γ with s = 10-2, Γ = π/4, and m = 0.

Fig. 5
Fig. 5

BSD’s versus γ with the same parameters as in Fig. 4 but for s = 5 × 10-2.

Fig. 6
Fig. 6

BSD’s versus γ with the same parameters as in Fig. 4 but for m = 100.

Fig. 7
Fig. 7

BSD’s versus γ with the same parameters as in Fig. 4 but for Γ = 3π/4.

Fig. 8
Fig. 8

Comparison of |S z | computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.01.

Fig. 9
Fig. 9

Comparison of |S z | computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.1.

Fig. 10
Fig. 10

Modulus of H z versus kz computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.01.

Fig. 11
Fig. 11

Phase of H z versus kz computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.01.

Fig. 12
Fig. 12

Modulus of H z versus kz computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.1.

Fig. 13
Fig. 13

Phase of H z versus kz computed by summation (and with the localized approximation) and from a first Davis beam with s = 0.1.

Fig. 14
Fig. 14

Integration contour.

Equations (196)

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UTMi=E0k2m=-+-im expimφ×Im,TMγ, JmR1-γ2expiγZ,
UTEi=H0k2m=-+-im expimφ×Im,TEγ, JmR1-γ2expiγZ,
Z=kz,  R=kρ,
Im,TMγ, 1-γ2JmR1-γ2expiγZ=12π-im02πEziE0exp-imφdφ,
Im,TEγ, 1-γ2JmR1-γ2expiγZ=12π-im02πHziH0exp-imφdφ.
Im,Xγ=-1+1 Im,Xγδγ-γdγ,
UTMi=E0k2m=-+-im expimφ×-1+1 Im,TMγJmR1-γ2expiγZdγ,
UTEi=H0k2m=-+-im expimφ×-1+1 Im,TEγJmR1-γ2expiγZdγ,
Im,TMγ=im4π21-γ2JmR1-γ202π×exp-imφdφ -+EziE0exp-iγZdZ,
Im,TEγ=im4π21-γ2JmR1-γ202π×exp-imφdφ -+HziH0exp-iγZdZ.
Γ=OPz, -eˆi.
eˆi=-sin Γeˆx-cos Γeˆz.
Ezi=2E0s2CΨ0ΨEQR sin φ-U0,
Hzi=H0Ψ0ΨE-S+2Qs2CRC cos φ-ZS-V0,
ΨE=expiW0expiRS cos φ+ZC,
Ψ0=iQ exp-iQs2RC cos φ-ZS-V02+R sin φ-U02,
Q=1i-2s2RS cos φ+ZC+W0,
U0, V0, W0=ku0, v0, w0,
S=sin Γ,
C=cos Γ.
s=1kw0,
Ezi=-2E0iC expiW0expiZCR sin φ-U0s2×expiRS cos φ,
Hzi=H0 expiW0expiZCexpiRS cos φa0+a2s2,
a0=-S,
a2=b0+b1Z+b2Z2,
b0=SR2C2 cos2 φ+2iRS2-2SRCV0-2iRC2cos φ+SU02+V02+2iCV0+2iSW0+SR2 sin2 φ-2RS sin φU0,
b1=2S2V0+4iCS-2RCS2 cos φ,
b2=S3.
Im,TM¯=-1m2π1-γ2-+Ez0i¯E0exp-iγZdZ,
Im,TE¯=-1m2π1-γ2-+Hz0i¯H0exp-iγZdZ,
Ezi, Hzi=Ez0i, Hz0iexpiR cos φ,
Ez0i¯, Hz0i¯=GˆEz0i, Hz0i,
Gˆ:Rm,  φπ/2.
Im,TMγ¯=K1-γ2-+Ez0i¯E0exp-iγZdZ,
Im,TEγ¯=K1-γ2-+Hz0i¯H0exp-iγZdZ,
Ezi, Hzi=Ez0i, Hz0iexpiRS cos φ,
Gˆ:RR0,  φφ0.
Hz0i=H0 expiW0expiZCa0+a2s2.
Im,TE0γ¯=K1-γ2-+Hz0|s0i¯H0exp-iγZdZ,
Hz0|s0i¯H0=Hz0|s0iH0=-S expiW0expiZC.
-+expiZx-xdZ=2πδx-x,
Im,TE0γ¯=-2πKS expiW0δγ-C1-γ2.
Im,TE0γ=--1mSexpiW0δγ-C.
δγ-C1-γ2=δγ-C1-C2=δγ-CS2,
K=-1m2π,
Im,TE2γ¯=-1m2π1-γ2expiW0×b0¯-+expiZC-γdZ+b1¯-+expiZC-γZdZ+b2¯-+expiZC-γZ2dZ,
Im,TEγ=Im,TE0γ+s2Im,TE2γ
Im,TE2γ¯=-1m expiW0b0¯δγ-C1-γ2+ib1¯δγ-C1-γ2-b2¯δ"γ-C1-γ2.
δγ-C1-γ2=δγ-CS2-2CS4 δγ-C,
δ"γ-C1-γ2=δ"γ-CS2-4CS4 δγ-C+2S41+4C2S2δγ-C
Im,TE2γ¯=-1m expiW0b0¯S2-ib1¯2CS4-2 b2¯S4-8 C2S6b2¯δγ-C+ib1¯S2+4CS4b2¯δγ-C-b2¯S2 δ"γ-C.
Im,TE2γ=i=02 Im,TE2iδiγ-C,
Im,TE20=-1mS2expiW0SU02+V02+2iSW0-2iCV0-2S+2mU0+m2S,
Im,TE21=2i-1mV0 expiW0,
Im,TE22=--1mS expiW0.
b1¯=2S2V0+4iCS-2R0CS2 cos φ0.
2iV0-R0 cos φ0=2iV0,
φ0=π/2,
1SU02+V02+2i W0S-2iC V0S2-2S-2R0U0S+R02S=1SU02+V02+2i W0S-2iC V0S2-2S+2m U0S2+m2S3,
R0=-mS.
Ez0i¯E0=-2iC expiW0expiZC-mS-U0s2.
Im,TMγ¯=2i-1mCS2U0+mSexpiW0s2δγ-C,
Gˆ:R-mS,  φπ/2.
A=Hz0i¯H0=-i expiW0expiZCQ¯×exp-iQ¯s2Z2S2+V02+2V0ZS+m2S2+U02+2m U0SS+2Q¯s2CZS+V0,
Q¯=1i-2s2ZC+W0.
A=expiW0expiZCj=05 b2js2j,
b2j=i=02j αi2jZi.
Im,TEγ¯=-1m expiW02π1-γ2-+j=05 b2js2j×expiZC-γdZ.
Im,TEγ¯=-1m expiW02π1-γ2j=010 Aj-+ Zj×expiZC-γdZ,
Aj=l=intj+1/25 αj2ls2l.
-+ Zj expiZC-γdZ=2πijδjγ-C,
Im,TEγ¯=-1m expiW0j=010 Ajijδjγ-C1-γ2.
δjγ-C1-γ2=l=0j-1lCjlδlγ-C.
Im,TEγ¯=--1m expiW0j=05a2j¯s2j,
Im,TEγ¯=j=05Im,TE2jγ¯s2j.
a2j¯=l=02ja2j¯lδlγ-C.
La4¯=12S5 m4δγ-C+2S2iV0-CSm2δγ-C-m2S δ"γ-C+2SC-iSV0δ3γ-C+12 S3δ4γ-C,
La6¯=-16S7 m6δγ-C+1S42CS-iV0m4δγ-C+12S3 m4δ"γ-C+2iV0-2 CSm2δ3γ-C-12 Sm2δ4γ-C+S32C-iSV0δ5γ-C+16 S5δ6γ-C,
La8¯=124S9 m8δγ-C+1S613 iV0-CSm6δγ-C-16S5 m6δ"γ-C+1S23 CS-iV0m4δ3γ-C+14S m4δ4γ-C+SiSV0-3Cm2δ5γ-C-16 S3m2δ6γ-C+S5C-13 iSV0δ7γ-C+124 S7δ8γ-C,
La10¯=-1120S11 m10δγ-C+13S8CS-14 iV0m8δγ-C+124S7 m8δ"γ-C+13S4iV0-4 CSm6δ3γ-C-112S3 m6δ4γ-C+2 CS-12 iV0m4δ5γ-C+112 Sm4δ6γ-C+S33iSV0-4Cm2δ7γ-C-124 S5m2δ8γ-C+13 S7C-14 iSV0δ9γ-C+1120 S9δ10γ-C.
Im,TEγ=--1m expiW0j=05 a2js2j.
La4=12S5 m4δγ-C+2S2iV0+CS*m2δγ-C-m2S δ"γ-C+2SC-iSV0δ3γ-C+12 S3δ4γ-C,
La6=-16S7 m6δγ-C+1S4-iV0-2 CS*m4δγ-C+12S3 m4δ"γ-C+2iV0+0*m2δ3γ-C-12 Sm2δ4γ-C+S32C-iSV0δ5γ-C+16 S5δ6γ-C,
La8=124S9 m8δγ-C+1S613 iV0+CS*m6δγ-C-16S5 m6δ"γ-C+1S2-CS*-iV0m4δ3γ-C+14S m4δ4γ-C+SiSV0-C*m2δ5γ-C-16 S3m2δ6γ-C+S5C-13 iSV0δ7γ-C+124 S7δ8γ-C,
La10=-1120S11 m10δγ-C+13S8-CS*-14 iV0m8δγ-C+124S7 m8δ"γ-C+13S4iV0+2 CS*m6δ3γ-C-112S3 m6δ4γ-C+0*-12 iV0m4δ5γ-C+112 Sm4δ6γ-C+S33iSV0-2C*m2δ7γ-C-124 S5m2δ8γ-C+13 S7C-14 iSV0δ9γ-C+1120 S9δ10γ-C.
LLa2j=LLa2j¯.
Ez0i¯E0=-2is2C expiW0expiZCQ¯2×exp-iQ¯s2Z2S2+V02+2V0ZS+m2S2+U02+2m U0SmS+U0.
Im,TMγ¯=--1m expiW0j=15c2j¯s2j,
Lc4¯=2i CS5 m3δγ-C-4 CS2 V0mδγ-C-2i CS mδ"γ-C,
Lc6¯=-i CS7 m5δγ-C+4 CS4i CS+V0m3δγ-C+2i CS3 m3δ"γ-C-4Ci CS+V0mδ3γ-C-iCSmδ4γ-C,
Lc8¯=13 i CS9 m7δγ-C-2 CS6V0+2i CSm5δγ-C-i CS5 m5δ"γ-C+4CS2V0+2i CSm3δ3γ-C+i CS m3δ4γ-C-2CS2iC+SV0mδ5γ-C-13 iCS3mδ6γ-C,
Lc10¯=-112 i CS11 m9δγ-C+2 CS813 V0+i CSm7δγ-C+13 i CS7 m7δ"γ-C-2CS43i CS+V0m5δ3γ-C-12 i CS3 m5δ4γ-C+2C3i CS+V0m3δ5γ-C+13 iCSm3δ6γ-C-2CS313 SV0+iCmδ7γ-C-112 iCS5mδ8γ-C.
Im,TMγ=--1m expiW0j=15 c2js2j.
Lc4=2i CS5 m3δγ-C+-4 CS2 V0+2iS1+2C2S2*×mδγ-C-2i CS mδ"γ-C,
Lc6=-i CS7 m5δγ-C+4CS4V0+12C3iS-4iS*×m3δγ-C+2i CS3 m3δ"γ-C-4CV0-iS2C*×mδ3γ-C-iCSmδ4γ-C,
Lc8=13 i CS9 m7δγ-C-2CS6V0-iS2C+3 CS*×m5δγ-C-i CS5 m5δ"γ-C+4CS2V0-iCS+S2C*m3δ3γ-C+i CS m3δ4γ-C-2CSSV0+iC-12S2C*×mδ5γ-C-13 iCS3mδ6γ-C,
Lc10=-112 i CS11 m9δγ-C+2CS813 V0-13 i12SC+4 CS*×m7δγ-C+13 i CS7 m7δ"γ-C-2 CS4V0-i12SC+2 CS*×m5δ3γ-C-12 i CS3 m5δ4γ-C+2CV0-i S2C*m3δ5γ-C+13 iCSm3δ6×γ-C-2CS313 SV0+13 i2C-12S2C*×mδ7γ-C-112 iCS5mδ8γ-C.
LLc2j=LLc2j¯.
C=0,  S=1.
Hz0i¯H0=-iQ¯ expiW0exp-iQ¯s2Z2+2V0Z+V02+m2+U02+2mU0,
Q¯=1i-2s2W0,
Im,TEγ¯=-i-1m2π1-γ2expiW0i-2s2W0exp-aTI,
a=s2-2is4W01+4s4W02,
T=V02+m2+U02+2mU0,
I=-+exp-aZ2expibZdZ,
b=2iaV0-γ.
-+exp-aZ2expibZdZ=πa exp-b24a,  Rea>0,
Im,TEγ¯=-i-1m2π1-γ2πaexpiW0i-2s2W0×exp-aTexp-b24a,
0+exp-ax2cosbxdx=12πa exp-b24a,  a>0.
I=-+expiαZ2exp-βZexp-μZ2expiνZdZ,
α=-Ima,
β=Imb,
μ=Rea,
ν=Reb.
I=n=0m=0iαnn!-βmm! Jnm,
Jnm=-+ Z2n+m exp-μZ2expiνZdZ.
-+exp-μZ2expiνZdZ=πμ exp-ν24μ,  μ>0, ν,
-+ Zj exp-μZ2expiνZdZ=1ijπμjνjexp-ν24μ.
I=πμn=0-iαnn!2nν2nm=0iβmm!mνmexp-ν24μ,
F=1+4s4W02,
α=2W0s4F,
β=2V0s2F.
I=πμm=02iV0s2Fm1m!mνm×n=0-2iW0s4Fn1n!2nν2nexp-ν24μ.
I=πμ exp-ν24μj=05 i2js2j,
i0=1,
i2=-iV0Fμν,
i4=1FμV02F+iW01-ν22μ,
i6=V0νF2μ23W0-i V02F1-ν26μ,
i8=1F2μ2V042F2+3i V02W0F-32 W02×1-ν2μ+112ν4μ2,
i10=νF3μ3152 iV0W02+5 V03W0F-12 i V05F2×1-13ν2μ+160ν4μ2.
m=0iβmm!mνmexp-ν24μ=m=0iβmm!mν+iβmexp-ν+iβ24μiβ=0=exp-ν+iβ24μ
I=πμn=0-iαnn!2nν2nexp-ν+iβ24μ=πμ-iα exp-ν+iβ24μ-iα,
Im,TMγ¯=0.
k=1l=0k-1=l=0k=l+1
Q¯=-i l=0 AlZlCl,
Al=k=l+1-2is2k-1k-1lW0k-1-l=-2is2l1+2is2W0-l-1.
Q¯=l=0 qlZl,
ql=-i -2is2Cl1+2is2W0l+1,
Im,TEγ¯=-i-1m2π1-γ2expiW0×SJ1+2s2CSJ2+V0J3,
J1=-+l=0 qlZlexp-is2l=0 qlZl×Z2S2+2V0ZS+TexpiZC-γdZ,
J2=-+l=0 qlZl2Z exp-is2l=0 qlZl×Z2S2+2V0ZS+TexpiZC-γdZ,
J3=-+l=0 qlZl2 exp-is2l=0 qlZl×Z2S2+2V0ZS+TexpiZC-γdZ
T=V02+m2S2+U02+2m U0S.
l=0 qlZl2=l=0 ql3Zl,
ql3=i=0l qiql-i,
l=0 qlZl2Z=l=0 ql2Zl,
q02=0
ql2=i=0l-1 qiql-1-i,  l1.
l=0 qlZlZ2S2+2V0ZS+T=l=0 ElZl,
El=ε1lS2ql-2+2ε0lV0Sql-1+Tql,
ε0l=0l=01otherwise,
ε1l=0l=0, 11otherwise.
Im,TEγ¯=-i-1m2π1-γ2expiW0l=0 klKl,
kl=Sql+2s2CSql2+V0ql3
Kl=-+ Zl exp-is2l=0 ElZl×expiZC-γdZ.
ql=-i-2is2ClGl+1
G=1+2is2W0.
q0=-1GiS+2s2CV0G,
qj=4αjs2CiGjq0,  j=1,, 5,
K0=-+exp-is2l=0 ElZlexpiZC-γdZ.
K0=exp-is2E0-+exps2E1iZexp-is2E2riZ2×exp-is2E3Z3exp-is2E4Z4exp-is2E5Z5×exp-is2E6Z6exp-μS2Z2expiZθdZ,
E1i=ImE1,
E2ri=ReE2+iImE2+S2F,
θ=C-γ-s2 ReE1,
K0=exp-is2E0n1=0s2E1in1n1!n2=0-is2E2rin2n2!×n3=0-is2E3n3n3!n4=0-is2E4n4n4!n5=0-is2E5n5n5!×n6=0-is2E6n6n6!-+ Zn1+2n2+3n3+4n4+5n5+6n6×exp-μS2Z2expiZθdZ.
K0=πμS2 exp-is2E0n1=0-s2E1in1n1!n1θn1×n2=0is2E2rin2n2!2n2θ2n2n3=0s2E3n3n3!3n3θ3n3×n4=0-is2E4n4n4!4n4θ4n4n5=0-s2E5n5n5!5n5θ5n5×n6=0is2E6n6n6!6n6θ6n6exp-θ24μS2,
Kl=-illK0θl.
Im,TEγ¯=-i-1m2π1-γ2expiW0πμS2×exp-is2E0exp-θ24μS2j=05 e2js2j.
e0=-i SG,
e2=-1G22CV0+12 i 4C+Ei1GSθμ,
e4=14G3F2μS E1-θ22S2μ+12iCθG3μ2S2×2iV0Ei1μG+8iCV0μ+3S-12θ2μS,
E=4S2W0G2F+iEi12G2F2+8iCEi1GF2+8CV0G2S+24iC2F2.
Im,TMγ¯=-2i -1m2π1-γ2 s2CmS+U0×expiW0πμS2 exp-is2E0×exp-θ24μS2j=04 f2js2j,
f0=-1G2
f2=-12Ei1G+4CG3μS2 θ,
f4=-i4G4S2F2μ E1-θ22S2μ+12CθG4μ2S23-12θ2μS2,
δγ-C1-γ2, Fγ=δγ-C, Fγ1-γ2=Fγ1-γ2γ=C=FC1-C2,
δγ-C1-C2, Fγ=11-C2 δγ-C, Fγ=11-C2Fγγ=C=FC1-C2,
δγ-C1-γ2=δγ-C1-C2,
δnγ-C1-γ2, Fγ=-1nnγnFγ1-γ2γ=C.
δγ-C1-γ2, Fγ=-γFγ1-γ2γ=C=-FCS2-FC2CS4,
I=-+exp-αt2+iβtdt=πα exp-β24α
I1=-+exp-αt2dt=-+exp-at2-ibt2dt=-+exp-at2cosbt2-i sinbt2dt,
I1=πa2+b21/2cos12arctanba-i sin12arctanba.
I1=πa+ib1/2=πα.
I=exp-β24α-+exp-a+ibt-ic+id2a+ib2dt.
ic+id2a+ib=A+iB,
x=t-A,
I2=-+exp-a+ibx-iB2dx.
C exp-a+ibz2dz=C1+C2+C3+C4×exp-a+ibz2dz=0,
C2 exp-a+ibz2dz =0Bexp-a+ibR+iy2dy =exp-a+ibR20Bexp2b-iaRy×expa+iby2dy |exp-a+ibR2B exp2|b|RB+aB2| =B exp-aR2+2|b|RB+aB2,
limRC2 exp-a+ibz2dz=0.
limRC4 exp-a+ibz2dz=0.
limRC1 exp-a+ibz2dz=-+exp-a+ibx2dx=π/α.
I2=-limRC3 exp-a+ibz2dz=limRC1 exp-a+ibz2dz=π/α,
I=πα exp-β24α.

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