Abstract

An analytic solution to the scattering by an infinite chiral cylinder, for oblique incidence of an on-axis Gaussian beam, is constructed by expanding the incident Gaussian beam scattered fields as well as internal fields in terms of appropriate cylindrical vector wave functions. The unknown expansion coefficients are determined by a system of linear equations derived from the boundary conditions. For a localized beam model, the scattering characteristics that are different from the case of an infinite dielectric cylinder are described in detail and discussed concisely.

© 2012 Optical Society of America

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References

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  1. D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
    [CrossRef]
  2. C. N. Chiu and C.-I. G. Hsu, “Scattering and shielding properties of a chiral-coated fiber-reinforced plastic composite cylinder,” IEEE Trans. Electromagn. Compat. 47, 123–130 (2005).
    [CrossRef]
  3. T. G. Mackay and A. Lakhtakia, “Simultaneous negative-and-positive-phase-velocity propagation in an isotropic chiral medium,” Microw. Opt. Technol. Lett. 49, 1245–1246 (2007).
    [CrossRef]
  4. J. Dong, “Exotic characteristics of power propagation in the chiral nihility fiber,” Progress Electromagn. Res. 99, 163–178 (2009).
    [CrossRef]
  5. N. Wongkasem and A. Akyurtlu, “Light splitting effects in chiral metamaterials,” J. Opt. 12, 035101 (2010).
    [CrossRef]
  6. M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
    [CrossRef]
  7. V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
    [CrossRef]
  8. D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
    [CrossRef]
  9. A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
    [CrossRef]
  10. M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
    [CrossRef]
  11. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. 26, 225–239 (1995).
    [CrossRef]
  12. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef]
  13. 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]
  14. 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]
  15. K. F. Ren, G. Gréhan, and 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]
  16. L. Mees, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results,” Appl. Opt. 38, 1867–1876 (1999).
    [CrossRef]
  17. H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24, 1383–1391(2007).
    [CrossRef]
  18. H. Y. Zhang and Y. P. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B 25, 131–135 (2008).
    [CrossRef]
  19. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1957), Chap. 4.
  20. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  21. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
    [CrossRef]
  22. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
    [CrossRef]
  23. J. J. Wang and G. Gouesbet, “Note on the use of localized beam models for light scattering theories in spherical coordinates,” Appl. Opt. 51, 3832–3836 (2012).
    [CrossRef]
  24. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Int. Textbook Company, 1971).

2012

2011

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

2010

N. Wongkasem and A. Akyurtlu, “Light splitting effects in chiral metamaterials,” J. Opt. 12, 035101 (2010).
[CrossRef]

2009

J. Dong, “Exotic characteristics of power propagation in the chiral nihility fiber,” Progress Electromagn. Res. 99, 163–178 (2009).
[CrossRef]

2008

2007

H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24, 1383–1391(2007).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Simultaneous negative-and-positive-phase-velocity propagation in an isotropic chiral medium,” Microw. Opt. Technol. Lett. 49, 1245–1246 (2007).
[CrossRef]

2006

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

2005

C. N. Chiu and C.-I. G. Hsu, “Scattering and shielding properties of a chiral-coated fiber-reinforced plastic composite cylinder,” IEEE Trans. Electromagn. Compat. 47, 123–130 (2005).
[CrossRef]

2004

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
[CrossRef]

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

2003

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

2001

M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
[CrossRef]

1999

1998

1997

1995

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

1991

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

1979

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

Akyurtlu, A.

N. Wongkasem and A. Akyurtlu, “Light splitting effects in chiral metamaterials,” J. Opt. 12, 035101 (2010).
[CrossRef]

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Arvas, E.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

Bray, M. G.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Chiu, C. N.

C. N. Chiu and C.-I. G. Hsu, “Scattering and shielding properties of a chiral-coated fiber-reinforced plastic composite cylinder,” IEEE Trans. Electromagn. Compat. 47, 123–130 (2005).
[CrossRef]

Davis, L. W.

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

Demir, V.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

Dong, J.

J. Dong, “Exotic characteristics of power propagation in the chiral nihility fiber,” Progress Electromagn. Res. 99, 163–178 (2009).
[CrossRef]

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1957), Chap. 4.

Elsherbeni, A.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

Gouesbet, G.

Gréhan, G.

Han, G. X.

Han, Y. P.

He, S.

M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
[CrossRef]

Hsu, C.-I. G.

C. N. Chiu and C.-I. G. Hsu, “Scattering and shielding properties of a chiral-coated fiber-reinforced plastic composite cylinder,” IEEE Trans. Electromagn. Compat. 47, 123–130 (2005).
[CrossRef]

Kern, D.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Kluskens, M. S.

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, “Simultaneous negative-and-positive-phase-velocity propagation in an isotropic chiral medium,” Microw. Opt. Technol. Lett. 49, 1245–1246 (2007).
[CrossRef]

Lock, J. A.

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, “Simultaneous negative-and-positive-phase-velocity propagation in an isotropic chiral medium,” Microw. Opt. Technol. Lett. 49, 1245–1246 (2007).
[CrossRef]

Mautz, J. R.

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

Mees, L.

Newman, E. H.

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Ren, K. F.

Semichaevsky, A.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Int. Textbook Company, 1971).

Takenaka, T.

M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
[CrossRef]

Wang, J. J.

Werner, D. H.

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

Wongkasem, N.

N. Wongkasem and A. Akyurtlu, “Light splitting effects in chiral metamaterials,” J. Opt. 12, 035101 (2010).
[CrossRef]

Worasawate, D.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

Yokota, M.

M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
[CrossRef]

Zhang, H. Y.

Appl. Opt.

IEEE Trans. Antennas Propag.

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic resonances and Q factors of a chiral sphere,” IEEE Trans. Antennas Propag. 52, 213–219 (2004).
[CrossRef]

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

A. Semichaevsky, A. Akyurtlu, D. Kern, D. H. Werner, and M. G. Bray, “Novel BI-FDTD approach for the analysis of chiral cylinders and spheres,” IEEE Trans. Antennas Propag. 54, 925–932 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. Mag.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Trans. Antennas Propag. Mag. 46, 94–99 (2004).
[CrossRef]

IEEE Trans. Electromagn. Compat.

C. N. Chiu and C.-I. G. Hsu, “Scattering and shielding properties of a chiral-coated fiber-reinforced plastic composite cylinder,” IEEE Trans. Electromagn. Compat. 47, 123–130 (2005).
[CrossRef]

J. Opt.

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

N. Wongkasem and A. Akyurtlu, “Light splitting effects in chiral metamaterials,” J. Opt. 12, 035101 (2010).
[CrossRef]

J. Opt. Soc. Am A

M. Yokota, S. He, and T. Takenaka, “Scattering of a Hermite–Gaussian beam field by a chiral sphere,” J. Opt. Soc. Am A 18, 1681–1689 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

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

Microw. Opt. Technol. Lett.

T. G. Mackay and A. Lakhtakia, “Simultaneous negative-and-positive-phase-velocity propagation in an isotropic chiral medium,” Microw. Opt. Technol. Lett. 49, 1245–1246 (2007).
[CrossRef]

Phys. Rev. A

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

Progress Electromagn. Res.

J. Dong, “Exotic characteristics of power propagation in the chiral nihility fiber,” Progress Electromagn. Res. 99, 163–178 (2009).
[CrossRef]

Other

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1957), Chap. 4.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Int. Textbook Company, 1971).

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Figures (3)

Fig. 1.
Fig. 1.

Cartesian coordinate system Oxyz is obtained first by a translation of the Gaussian beam coordinate system Oxyz along the z axis and then by a rotation through a single Euler angle β, and origin O is at (0,0,z0) in Oxyz. An infinite chiral cylinder is natural to Oxyz.

Fig. 2.
Fig. 2.

|T1(ϕ)|2 (solid line) and |T2(ϕ)|2 (dotted line) versus ϕ for an infinite chiral cylinder (εr=4, μr=1, κ=0.15, kr1=18.85, β=π/2), illuminated by the Gaussian beam with s=0.14 (TE mode, z0=0).

Fig. 3.
Fig. 3.

Normalized differential scattering cross section k2σ(ϕ)/(16π) for an infinite chiral cylinder (εr=4, μr=1, κ=0.15, kr1=15.71) and that for an infinite dielectric cylinder (εr=4, μr=1, κ=0, kr1=15.71), both with β=π/2 and illuminated by the Gaussian beam with s=0.14 (z0=0, TE mode).

Equations (34)

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

Ei=E0m=0π[Im,TE(ζ)mmλ(1)+Im,TM(ζ)nmλ(1)]exp(ihz)dζ,
Hi=iE01η0m=0π[Im,TE(ζ)nmλ(1)+Im,TM(ζ)mmλ(1)]exp(ihz)dζ,
Im,TE=(i)m+1ksinζLm,TE,
Im,TM=(i)m+1ksinζLm,TM,
Lm,TE=n=|m|(nm)!(n+m)!2n+12n(n+1)gn[m2Pnm(cosβ)sinβPnm(cosζ)+dPnm(cosβ)dβdPnm(cosζ)dζsinζ],
Lm,TM=mn=|m|(nm)!(n+m)!2n+12n(n+1)gn[Pnm(cosβ)sinβdPnm(cosζ)dζsinζ+dPnm(cosβ)dβPnm(cosζ)],
gn=11+2isz0/w0exp(ikz0)exp[s2(n+1/2)21+2isz0/w0],
Es=E0m=0π[αm(ζ)mmλ(3)+βm(ζ)nmλ(3)]eihzdζ,
Hs=iE01η0m=0π[αm(ζ)nmλ(3)+βm(ζ)mmλ(3)]eihzdζ,
D=ε0εrE+iκμ0ε0H,
B=μ0μrHiκμ0ε0E,
[EH]=[E+H+]+[EH],
E±=±iη0μrεrH±=±iηH±.
2[E+E]+[k+2E+k2E]=[00],
k±=k(μrεr±κ).
×[E+E]=[k+E+kE],
·[E+E]=[00].
Ew=E0m=0π{χm(ζ)[mmλ+(1)+nmλ+(1)]+τm(ζ)[mmλ(1)nmλ(1)]}eihzdζ,
Hw=iE0ηm=0π{χm(ζ)[mmλ+(1)+nmλ+(1)]τm(ζ)[mmλ(1)nmλ(1)]}eihzdζ.
Eϕi+Eϕs=Eϕw,Ezi+Ezs=EzwHϕi+Hϕs=Hϕw,Hzi+Hzs=Hzw}atr=r1.
ξddξJm(ξ)Im,TE(ζ)+mcosζJm(ξ)Im,TM(ζ)+ξddξHm(1)(ξ)αm(ζ)+mcosζHm(1)(ξ)βm(ζ)=[ξ+ddξ+Jm(ξ+)+mcosζn˜+Jm(ξ+)]χm(ζ)+[ξddξJm(ξ)mcosζn˜Jm(ξ)]τm(ζ),
ξ2[Jm(ξ)Im,TM+Hm(1)(ξ)βm(ζ)]=1n˜+ξ+2Jm(ξ+)χm1n˜ξ2Jm(ξ)τm,
mcosζJm(ξ)Im,TE(ζ)+ξddξJm(ξ)Im,TM(ζ)+mcosζHm(1)(ξ)αm(ζ)+ξddξHm(1)(ξ)βm(ζ)=η0η[ξ+ddξ+Jm(ξ+)+mcosζn˜+Jm(ξ+)]χm(ζ)η0η[ξddξJm(ξ)mcosζn˜Jm(ξ)]τm(ζ),
ξ2[Jm(ξ)Im,TE+Hm(1)(ξ)αm(ζ)]=η0η1n˜+ξ+2Jm(ξ+)χm+η0η1n˜ξ2Jm(ξ)τm,
Es=E02πkreiπ4m=(1)meimϕ[amr^+bmϕ^+cmz^],
am=i0πβm(ζ)cosζsinζeik0(rsinζ+zcosζ)dζ,
bm=0παm(ζ)1sinζeik0(rsinζ+zcosζ)dζ,
cm=i0πβm(ζ)sinζeik0(rsinζ+zcosζ)dζ,
αm(ζ)=(i)m+1ksinζαm(ζ),
βm(ζ)=(i)m+1ksinζβm(ζ).
Es=iE02kreikrm=(1)meimϕ[αm(π2)ϕ^+iβm(π2)z^].
σ(ϕ)=4πr2|EsE0|2=16πk2(|T1(ϕ)|2+|T2(ϕ)|2)2,
T1(ϕ)=m=(1)meimϕαm(π2),
T2(ϕ)=m=(1)meimϕiβm(π2).

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