Abstract

We present the most general framework, in terms of distributions, for describing a shaped electromagnetic beam in elliptical-cylinder coordinates. This framework is illustrated by investigating the case of a first-order Gaussian beam.

© 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. 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), pp. 339–384.
  3. 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]
  4. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
    [CrossRef]
  5. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  6. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
    [CrossRef]
  7. 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]
  8. E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. (New York) 37, 4705–4710 (1996).
    [CrossRef]
  9. F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, Paris, 1992).
  10. L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1965).
  11. E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass., 1968).
  12. Information on theory of distributions and its application to light scattering can be obtained from G. Gouesbet on request.
  13. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. (New York) 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. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).
  16. T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
    [CrossRef]
  17. F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
    [CrossRef]
  18. Information on electromagnetic scattering of shaped beams (generalized Lorenz–Mie theory) can be obtained from G. Gouesbet on request.
  19. R. Campbell, Théorie générale de l’équation de Mathieu (Masson, Paris, 1955).
  20. N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.
  22. G. Gouesbet, “Exact description of arbitrary-shaped beams for use in light-scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 (1996).
    [CrossRef]
  23. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
    [CrossRef]
  24. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  25. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  26. 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]
  27. 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]
  28. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
    [CrossRef]
  29. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1976).

1997 (2)

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

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

1996 (3)

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

1995 (4)

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations 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, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[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]

1994 (2)

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)

1979 (1)

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

1965 (1)

1963 (1)

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

1939 (1)

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

1919 (1)

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

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]

Arfken, G.

G. Arfken, 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]

Borgnis, F. E.

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

Bromwich, T. J.

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
[CrossRef]

Butkov, E.

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

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. 19, 1177–1179 (1979).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

Gouesbet, G.

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

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

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. (New York) 37, 4705–4710 (1996).
[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, J. A. Lock, G. Gréhan, “Partial-wave representations 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, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[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]

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

Gréhan, G.

Lenglart, E.

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

Lock, J. A.

Maheu, B.

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

McLachlan, N. W.

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

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

Onofri, F.

Roddier, F.

F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, Paris, 1992).

Schwartz, L.

L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1965).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

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. (New York) 4, 65–71 (1963).
[CrossRef]

Ann. Phys. (Leipzig) (1)

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

Appl. Opt. (3)

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. (New York) (2)

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

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

J. Opt. (Paris) (3)

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, “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. (1)

J. Opt. Soc. Am. A (4)

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]

Philos. Mag. (1)

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
[CrossRef]

Phys. Rev. (1)

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

Other (11)

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

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

Information on electromagnetic scattering of shaped beams (generalized Lorenz–Mie theory) can be obtained from G. Gouesbet on request.

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).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

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), pp. 339–384.

F. Roddier, Distributions et transformation de Fourier (McGraw-Hill, Paris, 1992).

L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1965).

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

Information on theory of distributions and its application to light scattering can be obtained from G. Gouesbet on request.

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Figures (1)

Fig. 1
Fig. 1

Relationship between Cartesian coordinates (x, y) and (u, w). Coordinates (x, y) define the ellipse of Eq. (1). The z axis is perpendicular to the figure. The incident beam propagates toward positive w, perpendicularly to the cylinder axis. The ordinate axis u is oriented as indicated in the figure. Elliptical coordinates (z, μ, θ) attached to the ellipse are defined by Eqs. (4).

Equations (147)

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

x2a2+y2b2=1,a>b.
l=a[1-(b/a)2]1/2,l[0, ],
e=b/a,e[0, 1].
z=z,
x=l cosh μ cos θ,
y=l sinh μ sin θ.
ds2=(hz)2dz2+(hμ)2dμ2+(hθ)2dθ2,
hz=1,
hμ=hθ=l(cosh2 μ-cos2 θ)1/2=lcosh 2µ-cos 2θ21/2
hz=1,
z hμhθ=0.
2Uz2+k2U+1hμhθ μ hθhμ Uμ+θ hμhθ Uθ=0,
2Uz2+k2U+2l2(cosh 2µ-cos 2θ) 2Uμ2+2Uθ2=0,
Hz,TM=0,
Hμ,TM=iωl(cosh2 μ-cos2 θ)1/2 UTMθ,
Hθ,TM=-iωl(cosh2 μ-cos2 θ)1/2 UTMμ,
Ez,TM=2UTMz2+k2UTM,
Eμ,TM=1l(cosh2 μ-cos2 θ)1/2 2UTMzμ,
Eθ,TM=1l(cosh2 μ-cos2 θ)1/2 2UTMzθ,
Ez,TE=0,
Eμ,TE=-iωαl(cosh2 μ-cos2 θ)1/2 UTEθ,
Eθ,TE=iωαl(cosh2 μ-cos2 θ)1/2 UTEμ,
Hz,TE=2UTEz2+k2UTE,
Hμ,TE=1l(cosh2 μ-cos2 θ)1/2 2UTEzμ,
Hθ,TE=1l(cosh2 μ-cos2 θ)1/2 2UTEzθ,
U=Z(z)M(μ)Θ(θ).
d2Z(z)dz2+aZ(z)=0.
Z(z)=exp(ikγz),(kγ)R.
d2M(μ)dμ2-(b-2q2 cosh 2µ)M(μ)=0,bC,
d2Θ(θ)dθ2+(b-2q2 cos 2θ)Θ(θ)=0,bC,
q=kl2(1-γ2)1/2.
UTM=E0k2 n=0[An,TM(γ), cehn(μ, q2)×cen(θ, q2)exp(ikγz)+Bn,TM(γ), sehn(μ, q2)sen(θ, q2)exp(ikγz)],
UTE=H0k2 n=0[An,TE(γ),cehn(μ, q2)×cen(θ, q2)exp(ikγz)+Bn,TE(γ), sehn(μ, q2)sen(θ, q2)exp(ikγz)],
Ez(z, μ, θ)=2UTMz2+k2UTM,
Hz(z, μ, θ)=2UTEz2+k2UTE,
cehn(iμ)=cen(μ),
sehn(iμ)=isen(μ),
Hz(z, iμ, θ)=H0n=0[An,TE(γ), (1-γ2)cen(μ, q2)×cen(θ, q2)exp(ikγz)+iBn,TE(γ), (1-γ2)sen(μ, q2)×sen(θ, q2)exp(ikγz)]
Xzpp=12X0[Xz(z, iμ, θ)+Xz(z, iμ,-θ)],
Xzmm=12iX0[Xz(z, iμ, θ)-Xz(z, iμ,-θ)],
n=0An,TE(γ), (1-γ2)cen(μ, q2)cen(θ, q2)×exp(ikγz)=Hzpp,
n=0Bn,TE(γ), (1-γ2)sen(μ, q2)sen(θ, q2)×exp(ikγz)=Hzmm,
An,TE=j=0An,TEjδ(j)(γ, γ0),
Hzpp=exp(ikγ0z)u=0Hzppu(μ, θ)zu.
δ(j)(γ, γ0), f(γ)=(-1)jdjf(γ)dγjγ=γ0
Hzpp=n=0j=0An,TEj(-1)jdjdγj(1-γ2)×cen(μ, q2)cen(θ, q2)exp(ikγz)γ=γ0,
j=0u=0j=u=0j=u,
Hzpp=exp(ikγ0z)u=0(-ik)uzun=0An,TEu(1-γ02)×cen(μ, q02)cen(θ, q02)+exp(ikγ0z)×u=0(ik)uzuj=u+1(-1)jju×n=0An,TEj×dj-udγj-u(1-γ2)cen(μ, q2)cen(θ, q2)γ=γ0,
Hzppu=(-ik)un=0An,TEu(1-γ02)cen(μ, q02)cen(θ, q02)+(ik)uj=u+1(-1)jjun=0An,TEj×dj-udγj-u(1-γ2)cen(μ, q2)cen(θ, q2)γ=γ0.
02πcen(x, q2)cem(x, q2)dx=πδnm,
02πsen(x, q2)sem(x, q2)dx=πδnm,
02πcen(x, q2)sem(x, q2)dx=0.
02π02πcem(μ, q02)cem(θ, q02)dμdθ
02π02πHzppucem(μ, q02)cem(θ, q02)dμdθ=(-ik)u(1-γ02)π2Am,TEu+(ik)uj=u+1(-1)jjun=0An,TEjdj-udγj-u(1-γ2)×02πcen(x, q2)cem(x, q02)dx2γ=γ0.
02π02πHzppucem(μ, q02)cem(θ, q02)dμdθ=(-ik)u(1-γ02)π2Am,TEu+(ik)uj=u+1Nj(-1)jjun=0An,TEjdj-udγj-u(1-γ2)×02πcen(x, q2)cem(x, q02)dx2γ=γ0, foru=0,, Nj
02π02πHzmmusem(μ, q02)sem(θ, q02)dμdθ=(-ik)u(1-γ02)π2Bm,TEu+(ik)uj=u+1Nj(-1)jjun=0Bn,TEjdj-udγj-u(1-γ2)×02πsen(x, q2)sem(x, q02)dx2γ=γ0,
Bn,TE=j=0NjBn,TEjδ(j)(γ, γ0),
Hzmm=exp(ikγ0z)u=0NjHzmmu(μ, θ)zu.
Eu=E0 exp(-iW)[1+s2(2iW-U2-V2)],
Ew=2is2E0U exp(-iW),
Hv=H0 exp(-iW)[1+s2(2iW-U2-V2)],
Hw=2is2H0V exp(-iW),
Ev=Hu=0,
(U, V, W)=(ku, kv, kw)
s=1kw0,
Xx=-sin β Xu+cos β Xw,
Xy=cos β Xu+sin β Xw,
Xz=Xv,
u=-x sin β+y cos β,
v=z,
w=x cos β+y sin β,
Xi=ΛijXj,
Λij=1hi xjxi,
Hz=H0E{1+s2[k2l2(cosh2 μ cos2 θ cos2 β-cosh2 μ cos2 θ-sinh2 μ sin2 θ cos2 β+2 cosh μ sinh μ cos θ sin θ cos β sin β)+2ikl(cosh μ cos θ cos β+sinh μ sin θ sin β)-k2z2]},
Hμ=2H0ikzs2(cosh2 μ-cos2 θ)1/2 E(sinh μ cos θ cos β+cosh μ sin θ sin β),
Hθ=2H0ikzs2(cosh2 μ-cos2 θ)1/2 E(sinh μ cos θ sin β-cosh μ sin θ cos β),
Eμ=E0E(cosh2 μ-cos2 θ)1/2 ((cosh μ sin θ cos β-sinh μ cos θ sin β)×{1+s2[2ikl(cosh μ cos θ cos β+sinh μ sin θ sin β)-k2l2(sinh μ sin θ cos β-cosh μ cos θ sin β)2-k2z2]}+2is2kl(sinh μ cos θ cos β+cosh μ sin θ sin β)(sinh μ sin θ cos β-cosh μ cos θ sin β)),
Eθ=E0E(cosh2 μ-cos2 θ)1/2 ((cosh μ sin θ sin β+sinh μ cos θ cos β)×{1+s2[2ikl(cosh μ cos θ cos β+sinh μ sin θ sin β)-k2l2(sinh μ sin θ cos β-cosh μ cos θ sin β)2-k2z2]}+2is2kl(sinh μ cos θ sin β-cosh μ sin θ cos β)(sinh μ sin θ cos β-cosh μ cos θ sin β)),
E=exp[-ikl(cosh μ cos θ cos β+sinh μ sin θ sin β)].
UTM=0.
Hzpp=12 E1[1+s2(a-k2z2)]+12 E˜1[1+s2(a˜-k2z2)],
Hzmm=-i2 E1[1+s2(a-k2z2)]+i2 E˜1[1+s2(a˜-k2z2)],
E1=exp[-ikl(cos μ cos θ cos β+i sin μ sin θ sin β)],
a=k2l2(2 cos2 μ cos2 θ cos2 β+2i cos μ sin μ cos θ sin θ cos β sin β-cos2 μ cos2 θ+cos2 β-cos2 θ cos2 β-cos2 μ cos2 β)+2kl(i cos μ cos θ cos β-sin μ sin θ sin β)=a1kl+a2(kl)2
An,TE(γ)=j=02An,TEjδ(j)(γ),
Hzpp=u=02Hzppu(μ, θ)zu,
Hzpp2=-k2s22(E1+E˜1),
Hzpp1=0,
Hzpp0=12(E1+E˜1)+s22(aE1+a˜E˜1).
Am,TE2=s22π2Im0,
Im0=02π02π(E1+E˜1)cem(μ, q02)cem(θ, q02)dμdθ,
q0=[q]γ=0=kl/2.
Im0=202π02πE1cem(μ, q02)cem(θ, q02)dμdθ.
E1=2n=0(-i)ppncen(μ, q02)cen(θ, q02)cen(β, q02)+i (-i)psnsen(μ, q02)sen(θ, q02)sen(β, q02),
Im0=4π2(-i)p cem(β, q02)pm,
Am,TE2=2s2(-i)p cem(β, q02)pm.
Am,TE1=4πAm,TE2Cmm1,
Cnmj=djdγj 02πcen(θ, q2)cem(θ, q02)dθγ=0.
ddγ=q2γ q2=-k2l22γ q2,
Am,TE1=Cmm1=0.
lhs=12 02π02π(E1+E˜1)cem(μ, q02)×cem(θ, q02)dμdθ+s22 02π02π(aE1+a˜E˜1)×cem(μ, q02)cem(θ, q02)dμdθ.
2E1β2=iklηE1-k2l2ξ2E1,
η=-i2a1=cos μ cos θ cos β+i sin μ sin θ sin β,
ξ2=-a2,
lhs=12Im0+s22 d2dβ2Im0+ikls2ηc,m,
ηc,m=02π02πηE1cem(μ, q02)cem(θ, q02)dμdθ,
E1=exp(-iklη)=exp(-2iq0η)
E¯1=exp(-2iqη)
ηE1=ikl2 E¯1q2γ=0.
ηE1=-kln=0(-i)p-i cen(β, q02)pn q2×cen(μ, q2)cen(θ, q2)+sen(β, q02)sn q2sen(μ, q2)sen(θ, q2)-ipncen(μ, q02)cen(θ, q02) q2cen(β, q2)+1snsen(μ, q02)sen(θ, q02) q2sen(β, q2)γ=0.
ηc,m=ikl (-i)ppmπ2q2cem(β, q2)γ=0+ikln=0(-i)p cen(β, q02)pn×02π02πq2cen(μ, q2)cen(θ, q2)γ=0×cem(μ, q02)cem(θ, q02)dμdθ.
I=2πδnm02πq2cen(μ, q2)γ=0cem(μ, q02)dμ.
J=02πq2cen(μ, q2)γ=0cen(μ, q02)dμ.
ddq2 02πcen(μ, q2)cen(μ, q2)dμγ=0=dπdq2γ=0=0=202πq2cen(μ, q2)γ=0×cen(μ, q02)dμ.
ηc,m=ikl (-i)ppmπ2q2cem(β, q2)γ=0,
lhs=12Im0+s22 d2dβ2Im0-k2l2s2 (-i)ppm×π2q2cem(β, q2)γ=0.
Cmm2=d2dγ2 02πcem(θ, q2)cem(θ, q02)dθγ=0=-k2l22 02πq2cem(θ, q2)γ=0cem(θ, q02)dθ=0.
rhs=π2(Am,TE0-2Am,TE2).
Am,TE0=(-i)ppm 2cem(β, q02)+s24cem(β, q02)+2 d2dβ2cem(β, q02)-k2l2q2cem(β, q2)γ=0.
Bn,TE(γ)=j=02Bn,TEjδ(j)(γ)
Bm,TE2=2s2(-i)p sem(β, q02)sm,
Bm,TE1=0,
Bm,TE0=(-i)psm 2sem(β, q02)+s24sem(β, q02)+2 d2dβ2sem(β, q02)-k2l2q2sem(β, q2)γ=0.
UTM=0,
UTE=H0k2 n=0[An,TE(γ), cehn(μ, q2)cen(θ, q2)×exp(ikγz)+Bn,TEi(γ), sehn(μ, q2)×sen(θ, q2)exp(ikγz)],
UTE=H0k2 n=0(An,TE0-k2z2An,TE2)×cehn(μ, q02)cen(θ, q02)+(Bn,TE0-k2z2Bn,TE2)×sehn(μ, q02)sen(θ, q02)-k2l22 An,TE2q2cehn(μ, q2)cen(θ, q2)γ=0+Bn,TE2q2sehn(μ, q2)sen(θ, q2)γ=0.
Hz=H0n=0[An,TE0-k2z2An,TE2-2An,TE2]×cehn(μ, q02)cen(θ, q02)+(Bn,TE0-k2z2Bn,TE2-2Bn,TE2)sehn(μ, q02)sen(θ, q02)-k2l22 An,TE2q2cehn(μ, q2)cen(θ, q2)γ=0+Bn,TE2q2sehn(μ, q2)sen(θ, q2)γ=0,
s2E=n=0[An,TE2cehn(μ, q02)cen(θ, q02)+Bn,TE2sehn(μ, q02)sen(θ, q02)].
E=2n=0(-i)ppncehn(μ, q02)cen(θ, q02)cen(β, q02)+(-i)psnsehn(μ, q02)sen(θ, q02)sen(β, q02).
n=0(-i)ppn 2cehn(μ, q02)cen(θ, q02) d2dβ2cen(β, q02)-k2l2cehn(μ, q02)cen(θ, q02)q2cen(β, q2)γ=0+cen(β, q02)q2cehn(μ, q2)cen(θ, q2)γ=0+(-i)psn 2sehn(μ, q02)sen(θ, q02) d2dβ2sen(β, q02)-k2l2sehn(μ, q02)sen(θ, q02)q2sen(β, q2)γ=0+sen(β, q02)q2sehn(μ, q2)sen(θ, q2)γ=0=E[k2l2(cosh2 μ cos2 θ cos2 β-cosh2 μ cos2 θ-sinh2 μ sin2 θ cos2 β+2 cosh μ sinh μ cos θ sin θ cos β sin β)+2ikl(cosh μ cos θ cos β+sinh μ sin θ sin β)].
2n=0(-i)ppncen(μ, q02)cen(θ, q02) d2dβ2cen(β, q02)+i(-i)psnsen(μ, q02)sen(θ, q02) d2dβ2sen(β, q02)-k2l2(-i)ppn q2cen(μ, q2)cen(θ, q2)cen(β, q2)+i(-i)psnsen(μ, q2)sen(θ, q2)sen(β, q2)γ=0=klE1(2iη-klξ2),
UTM=0,
UTE=H0k22n=0(-i)ppncen(β, q02)δ(γ), cehn(μ, q2)×cen(θ, q2)exp(ikγz)+(-i)psnsen(β, q02)×δ(γ), sehn(μ, q2)sen(θ, q2)exp(ikγz),
UTE=H0k22n=0(-i)ppncehn(μ, q02)cen(θ, q02)cen(β, q02)+(-i)psnsehn(μ, q02)sen(θ, q02)sen(β, q02),
cehn(μ, q02)=pn(-1)sMcn(1)(μ, q02),
sehn(μ, q02)=sn(-1)sMsn(1)(μ, q02),
n=2s+p.
Mc2n(1)(μ, q02)=1ce2n(0, q02) m=0(-1)n+mA2m(2n)×J2m(klchμ),
Ms2n(1)(μ, q02)=2 tanh(μ)se2n(0, q02) m=1(-1)n+mB2m(2n)×J2m(klchμ),
liml0 A2m(2n)=δmn,n1,
liml0 ce2n(μ, q02)=cos(2nμ),n1,
liml0 A0(2n)=22,
liml0 ce0(μ, q2)=22.
UTE=H0k2 J0(kr)+2n=1(-i)n cos[n(θ-β)]Jn(kr).
ϕ=π+θ.
J-n(x)=(-1)nJn(x)
UTE=H0k2 n=-+in exp(inϕ)Jn(kr),

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