Abstract

By use of the extinction theorem, the induced electric surface currents involved in electromagnetic scattering and absorption by thin finite wires with arbitrary material parameters are found to be describable by the standard Pocklington integral equation. Unlike for the usual computation, however, when the wire is not highly conducting the scattered wave is described by surface distributions of both electric and magnetic currents, and more than one value of surface impedance may be required. The equation is solved for normal incidence by Galerkin’s method using a single trial function, and results are confirmed by comparison with independent analytical and numerical computations for both coated and uncoated wires.

© 1998 Optical Society of America

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  1. C.-T. Tai, “Electromagnetic back-scattering from cylindrical wires,” J. Appl. Phys. 23, 909–916 (1952).
    [CrossRef]
  2. E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. AP-8, 1–7 (1960).
    [CrossRef]
  3. K. K. Mei, “On the integral equations of thin wire antennas,” IEEE Trans. Antennas Propag. AP-13, 374–378 (1965).
    [CrossRef]
  4. J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE 53, 796–804 (1965).
    [CrossRef]
  5. L. N. Medgyesi-Mitschang, C. Eftimiu, “Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions,” IEEE Trans. Antennas Propag. AP-30, 628–636 (1982).
    [CrossRef]
  6. L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from extended wires and two- and three-dimensional surfaces,” IEEE Trans. Antennas Propag. AP-33, 1090–1100 (1985).
    [CrossRef]
  7. A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
    [CrossRef]
  8. P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
    [CrossRef]
  9. P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
    [CrossRef]
  10. K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
    [CrossRef]
  11. N. K. Uzunoglu, N. G. Alexopoulos, J. G. Fikioris, “Scattering from thin and finite dielectric fibers,” J. Opt. Soc. Am. 68, 194–197 (1978).
    [CrossRef]
  12. E. H. Newman, “A unified theory of thin material wires,” IEEE Trans. Antennas Propag. 39, 1488–1496 (1991).
    [CrossRef]
  13. H. Hönl, A. W. Maue, K. Westpfahl, “Theorie der beugung,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25/1, pp. 224, 240.
  14. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  15. S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
    [CrossRef]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 22.
  17. J. M. Stiles, K. Sarabandi, “A scattering model for thin dielectric cylinders of arbitrary cross section and electrical length,” IEEE Trans. Antennas Propag. 44, 260–266 (1996).
    [CrossRef]
  18. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955); “The long wavelength limit in scattering from a dielectric cylinder at oblique incidence,” Can. J. Phys. 43, 2212–2215 (1965). Some errata are given in R. L. Fante, “Some comments on the scattering of long wavelength waves by dielectric cylinders,” Proc. IEEE 53, 1675 (1965).
    [CrossRef]
  19. J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15, 659–660 (1979).
    [CrossRef]
  20. D. E. Barrick, in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), pp. 205–339.
  21. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 18, 668–670 (1976).
    [CrossRef]

1996

J. M. Stiles, K. Sarabandi, “A scattering model for thin dielectric cylinders of arbitrary cross section and electrical length,” IEEE Trans. Antennas Propag. 44, 260–266 (1996).
[CrossRef]

1995

P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
[CrossRef]

1992

A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
[CrossRef]

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
[CrossRef]

1991

E. H. Newman, “A unified theory of thin material wires,” IEEE Trans. Antennas Propag. 39, 1488–1496 (1991).
[CrossRef]

1985

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from extended wires and two- and three-dimensional surfaces,” IEEE Trans. Antennas Propag. AP-33, 1090–1100 (1985).
[CrossRef]

1982

L. N. Medgyesi-Mitschang, C. Eftimiu, “Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions,” IEEE Trans. Antennas Propag. AP-30, 628–636 (1982).
[CrossRef]

1979

J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15, 659–660 (1979).
[CrossRef]

1978

1976

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 18, 668–670 (1976).
[CrossRef]

K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
[CrossRef]

1975

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1965

K. K. Mei, “On the integral equations of thin wire antennas,” IEEE Trans. Antennas Propag. AP-13, 374–378 (1965).
[CrossRef]

J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE 53, 796–804 (1965).
[CrossRef]

1960

E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. AP-8, 1–7 (1960).
[CrossRef]

1955

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955); “The long wavelength limit in scattering from a dielectric cylinder at oblique incidence,” Can. J. Phys. 43, 2212–2215 (1965). Some errata are given in R. L. Fante, “Some comments on the scattering of long wavelength waves by dielectric cylinders,” Proc. IEEE 53, 1675 (1965).
[CrossRef]

1952

C.-T. Tai, “Electromagnetic back-scattering from cylindrical wires,” J. Appl. Phys. 23, 909–916 (1952).
[CrossRef]

Alexopoulos, N. G.

Barrick, D. E.

D. E. Barrick, in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), pp. 205–339.

Cassedy, E. S.

E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. AP-8, 1–7 (1960).
[CrossRef]

Chatterjee, A.

A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
[CrossRef]

Chen, K.-M.

K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
[CrossRef]

Eftimiu, C.

L. N. Medgyesi-Mitschang, C. Eftimiu, “Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions,” IEEE Trans. Antennas Propag. AP-30, 628–636 (1982).
[CrossRef]

Fainberg, J.

E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. AP-8, 1–7 (1960).
[CrossRef]

Fikioris, J. G.

Guru, B. S.

K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
[CrossRef]

Hönl, H.

H. Hönl, A. W. Maue, K. Westpfahl, “Theorie der beugung,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25/1, pp. 224, 240.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 22.

Kent, W. J.

A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
[CrossRef]

Lentz, W. J.

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 18, 668–670 (1976).
[CrossRef]

Livesay, D. E.

K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
[CrossRef]

Maue, A. W.

H. Hönl, A. W. Maue, K. Westpfahl, “Theorie der beugung,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25/1, pp. 224, 240.

Medgyesi-Mitschang, L. N.

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from extended wires and two- and three-dimensional surfaces,” IEEE Trans. Antennas Propag. AP-33, 1090–1100 (1985).
[CrossRef]

L. N. Medgyesi-Mitschang, C. Eftimiu, “Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions,” IEEE Trans. Antennas Propag. AP-30, 628–636 (1982).
[CrossRef]

Mei, K. K.

K. K. Mei, “On the integral equations of thin wire antennas,” IEEE Trans. Antennas Propag. AP-13, 374–378 (1965).
[CrossRef]

Newman, E. H.

E. H. Newman, “A unified theory of thin material wires,” IEEE Trans. Antennas Propag. 39, 1488–1496 (1991).
[CrossRef]

Pedersen, J. C.

P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
[CrossRef]

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
[CrossRef]

Putnam, J. M.

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from extended wires and two- and three-dimensional surfaces,” IEEE Trans. Antennas Propag. AP-33, 1090–1100 (1985).
[CrossRef]

Richmond, J. H.

J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE 53, 796–804 (1965).
[CrossRef]

Sarabandi, K.

J. M. Stiles, K. Sarabandi, “A scattering model for thin dielectric cylinders of arbitrary cross section and electrical length,” IEEE Trans. Antennas Propag. 44, 260–266 (1996).
[CrossRef]

Stiles, J. M.

J. M. Stiles, K. Sarabandi, “A scattering model for thin dielectric cylinders of arbitrary cross section and electrical length,” IEEE Trans. Antennas Propag. 44, 260–266 (1996).
[CrossRef]

Ström, S.

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

Tai, C.-T.

C.-T. Tai, “Electromagnetic back-scattering from cylindrical wires,” J. Appl. Phys. 23, 909–916 (1952).
[CrossRef]

Uzunoglu, N. K.

Volakis, J. L.

A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
[CrossRef]

Wait, J. R.

J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15, 659–660 (1979).
[CrossRef]

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955); “The long wavelength limit in scattering from a dielectric cylinder at oblique incidence,” Can. J. Phys. 43, 2212–2215 (1965). Some errata are given in R. L. Fante, “Some comments on the scattering of long wavelength waves by dielectric cylinders,” Proc. IEEE 53, 1675 (1965).
[CrossRef]

Waterman, P. C.

P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
[CrossRef]

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Westpfahl, K.

H. Hönl, A. W. Maue, K. Westpfahl, “Theorie der beugung,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25/1, pp. 224, 240.

Am. J. Phys.

S. Ström, “On the integral equations for electromagnetic scattering,” Am. J. Phys. 43, 1060–1069 (1975).
[CrossRef]

Appl. Opt.

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 18, 668–670 (1976).
[CrossRef]

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955); “The long wavelength limit in scattering from a dielectric cylinder at oblique incidence,” Can. J. Phys. 43, 2212–2215 (1965). Some errata are given in R. L. Fante, “Some comments on the scattering of long wavelength waves by dielectric cylinders,” Proc. IEEE 53, 1675 (1965).
[CrossRef]

Electron. Lett.

J. R. Wait, “Exact surface impedance for a cylindrical conductor,” Electron. Lett. 15, 659–660 (1979).
[CrossRef]

IEEE Trans. Antennas Propag.

J. M. Stiles, K. Sarabandi, “A scattering model for thin dielectric cylinders of arbitrary cross section and electrical length,” IEEE Trans. Antennas Propag. 44, 260–266 (1996).
[CrossRef]

K.-M. Chen, D. E. Livesay, B. S. Guru, “Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity,” IEEE Trans. Antennas Propag. AP-24, 330–336 (1976).
[CrossRef]

E. H. Newman, “A unified theory of thin material wires,” IEEE Trans. Antennas Propag. 39, 1488–1496 (1991).
[CrossRef]

K. K. Mei, “On the integral equations of thin wire antennas,” IEEE Trans. Antennas Propag. AP-13, 374–378 (1965).
[CrossRef]

L. N. Medgyesi-Mitschang, C. Eftimiu, “Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions,” IEEE Trans. Antennas Propag. AP-30, 628–636 (1982).
[CrossRef]

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from extended wires and two- and three-dimensional surfaces,” IEEE Trans. Antennas Propag. AP-33, 1090–1100 (1985).
[CrossRef]

A. Chatterjee, J. L. Volakis, W. J. Kent, “Scattering by a perfectly conducting and a coated thin wire using a physical basis model,” IEEE Trans. Antennas Propag. 40, 761–769 (1992).
[CrossRef]

IRE Trans. Antennas Propag.

E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. AP-8, 1–7 (1960).
[CrossRef]

J. Appl. Phys.

C.-T. Tai, “Electromagnetic back-scattering from cylindrical wires,” J. Appl. Phys. 23, 909–916 (1952).
[CrossRef]

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
[CrossRef]

P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev. D

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Proc. IEEE

J. H. Richmond, “Digital computer solutions of the rigorous equations for scattering problems,” Proc. IEEE 53, 796–804 (1965).
[CrossRef]

Other

D. E. Barrick, in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, New York, 1970), pp. 205–339.

H. Hönl, A. W. Maue, K. Westpfahl, “Theorie der beugung,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25/1, pp. 224, 240.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 22.

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Figures (8)

Fig. 1
Fig. 1

Geometry for a TM plane wave normally incident on a thin wire.

Fig. 2
Fig. 2

Far-field cross sections for a wire of varying conductivity (kh=2π, ka=0.02π). The dotted curve gives the extinction cross section, and the dashed curves give the limiting forms of relations (33).

Fig. 3
Fig. 3

Far-field cross sections for a wire of varying permittivity (kh=2π, ka=0.02π). The extinction and scattering cross sections are identical in this case; dashed curves are from relations (33).

Fig. 4
Fig. 4

Cross sections for a copper-coated glass fiber versus coating thickness t (δ is the skin depth of copper, ka=0.001, kh =5).

Fig. 5
Fig. 5

Cross sections versus kh for a coated nichrome wire (see text).

Fig. 6
Fig. 6

Magnitude (solid curves) and phase (dashed curves) for the E field along the wire surface. Points give numerical results from Ref. 17. Here kh=π/4, h/a=100.

Fig. 7
Fig. 7

Corresponding surface E fields for wires of several conductivities. Again kh=π/4, h/a=100.

Fig. 8
Fig. 8

Fractional difference in surface impedance for a range of conductivities (solid curves) or dielectric constants (dashed curve).

Tables (2)

Tables Icon

Table 1 Rayleigh Cross Sections for Nichrome Wire [σ=1×106 (ohm m)-1] at 35 GHz, with kh=0.1, kr1=0.001, kr2=0.002

Tables Icon

Table 2 Values of m Obtained from the Limiting Case [Approximations (31)] and General Eq. (21a) Compared for Several Values of r=n2a

Equations (88)

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

η=2μrJ0(κa)/κaJ1(κa)
ρ=1-(ka/2)2η.
Ei(r)+××dσiωμ0nˆ×H(r)g(kR)+×dσknˆ×E(r)g(kR)=E(r),r outsidethebody0,r insidethebody.
g(kR)=exp(ikR)/4πkR,R=|r-r|,
zˆ·××dσiωμ0nˆ×H(r)g(kR)+zˆ·×dσknˆ×E(r)g(kR)=-zˆ·Ei(r),ronaxis,insidethebody.
nˆ×H(r)=zˆHϕ(z),
nˆ×E(r)=nˆ×zˆEz(z).
2πωμ0ia-h+hdzHϕ(z)(2/z2+k2)g(kR)+2π(ka)2-h+hdzEz(z)g(kR)/R=-E0,
2π(ka)2-h+hdzEz(z)g(kR)/R
-(1/2)a2Ez(z)-+dz[(z-z)2+a2]-3/2
=-Ez(z).
u(z)=2πiωμ0kaHϕ(z)
-h+hkdzu(z)(2/k2z2+1)g(kR)-Ez(z)=-E0.
u(z)=sus(z),
Ez(z)=sηsus(z).
Es(r)-θˆE0f(p)exp(ikr)/ikr,
f(p)=sρsk sin θ-h+hdzus(z)exp(-ikzp),
σe/λ2=-(1/π)Im f(0),
σs/λ2=(1/2π)-1+1|f(p)|2dp,
σa/λ2=-(1/π)Im -h+hkdzEz*(z)u(z),
σb/λ2=(1/π)|f(0)|2.
u(z)=(1/K)(1-cos mkz/cos mkh),-hzh,
u(z)-(m2/2K)(kh)2(1-z2/h2).
u(z)1/K
Ez(z)=(1/K)[η(0)-η(m)cos mkz/cos mkh],
K=2[L(2x)+ln(4x/ka)-η(0)/2]+N(m)/D(m),
L(x)=0x[exp(iu)-1]du/u=Ci(x)+iSi(x)-ln x-γ
N(m)=i(sin 2mx-2mx)[2(m2-1)ln(4h/a)+η(m)]+{(m2-1)[exp(2imx)-2imx]-2}L[2(1-m)x]-{(m2-1)[exp(-2imx)+2imx]-2}L[2(1+m)x]+2im×exp(ix)[(m-1)sin(m+1)x exp(imx)+(m+1)sin(m-1)x exp(-imx)-4 sin x cos2 mx],
D(m)=4i cos2 mx(mx-tan mx).
N(m)/D(m)=0.
K=2[L(2x)+ln(4x/ka)-η(0)/2],
KK=2 ln(2/ka)-2γ+iπ-η(0),x1,
m2/K0.5{Ω+(1/2m2)[η(m)-η(0)]+[3η(0)-η(m)]x2/10-2ix3/9}-1,
K2[Ω+7/3+2ix-x2-4ix3/9-η(0)/2].
mΩ+7/3+2ix-x2-4ix3/9-η(0)/2Ω+(1/2m2)[η(m)-η(0)]+[3η(0)-η(m)]x2/10-2ix3/90.5.
ηc(m)η(m)-2(μ2/μ0)(1-m2/n22)ln(r2/r1),
f(p)-(2x3 sin θ/3)(m2/K);
σe/λ2(2x3/3π)Im(m2/K),
σs/λ2(8x6/27π)|m2/K|2,
σb/λ2(4x6/9π)|m2/K|2,
σa/λ2-(2x3/3π)A/|K|2,
AIm{[η(m)-η(0)]m*2}+Re[η(m)-η(0)]Im(m4)x2/10-2 Im[η(m)-η(0)](m14+m24)x2/5+Im[η(m)+η(0)]|m|4x2/5
m=1-iL(4x)+xη(m)2x[2 ln(4h/a)+L(4x)-2].
(m2-1)ln2(m+1)ka-γ+iπ2-m+η(m)2=0.
mib/ka,
η(m)[2μrb/(nka)2]J0(b)/J1(b),
b[ln(2/b)-γ](μr/n2)J0(b)/J1(b)
η(0)4μr/(nka)2.
f(p)-(ka)2khΔ sin θ sin pkh/2pkh,
σs/λ2(1/8π)(ka)4(kh)2|Δ|2-1+1dp(1-p2)×(sin pkh/pkh)2,
σb/λ2(1/4π)(ka)4(kh)2|Δ|2,
σa/λ2(1/2π)(ka)2kh Im Δ,
fB(p)(1/4π)k3 sin θ(r-1)exp(-ipkz)dV.
ηc(μ2/μ0)(2/k2a)2×11+[(μ2/μ1)(n1/n2)2-1](r1/a)2.
ρ/ηc=1/ηc-(k0a/2)2(k2a/2)2{1+[(n1/n2)2-1](r1/a)2}-(k0a/2)2(k0a/2)2[2(n22-n12)(t/a)+n12-1].
Im(1/ηc)2(k0a/2)2 Im(n22)(t/a),
K=2L(2x)+2 ln(4h/r2)-ηc(0)=2L(2x)+2 ln(4h/r2)-[η(0)-2 ln(r2/r1)]=2L(2x)+2 ln(4h/r1)-η(0),
2(m2-1)ln(4h/r2)+ηc(m)=2(m2-1)ln(4h/r2)+[η(m)-2(1-m2)ln(r2/r1)]=2(m2-1)ln(4h/r1)+η(m).
σe/λ2-(2x/π)Im(1/K),
σs/λ22x/|K|2,
σb/λ24x2/π|K|2,
σa/λ2-2x Im[η(0)]/π|K|2.
-1+1dp(1-p2)(sin px/px)2(1/x)-+ds(sin s/s)2=π/x,x1.
Ez(±h)[η(0)-η(m)]/η(0),
k1=(ω2μ11+iωμ1σ1)1/2,
Ez=E0J0(κ1r)raE1[J0(κ0r)+c1H0(κ0r)]ra.
κnkn[1-(k0 cos θi/kn)2]1/2,n=0, 1
μ0η(q)=2μ1(κ1/k1)2J0(κ1a)/κ1aJ1(κ1a),
μ0η(q)=2μ0(κ0/k0)2κ0aJ0(κ0a)+c1H0(κ0a)J1(κ0a)+c1H1(κ0a),
c1=-J1(κ0a)J0(κ1a)-(μ0k12κ0/μ1k02κ1)J0(κ0a)J1(κ1a)H1(κ0a)J0(κ1a)-(μ0k12κ0/μ1k02κ1)H0(κ0a)J1(κ1a),
η(q)(2μ1/μ0k1a)cot(k1a-π/4).
η(q)(2/k1a)2[1+(1/8)(k0a)2q2]
δη/η|[η(1)-η(0)]/η(0)|(1/8)(k0a)2,
δη/η|1/2(n2-1)|.
Ez=E0J0(κ1r)0rr1E1[J0(κ2r)+c2H0(κ2r)]r1ra,
μ0ηc(q)=2μ2(κ2/k2)2κ2aJ0(κ2a)+c2H0(κ2a)J1(κ2a)+c2H1(κ2a),
c2=-J1(κ2r1)J0(κ1r1)-(μ2k12κ2/μ1k22κ1)J0(κ2r1)J1(κ1r1)H1(κ2r1)J0(κ1r1)-(μ2k12κ2/μ1k22κ1)H0(κ2r1)J1(κ1r1).
R=[(x-x)2+(y-y)2+(z-z)2]0.5R0[1-a(x cos φ+y sin φ)/R02],
R0=[x2+y2+(z-z)2]0.5.
A××[nˆ×H(r)]g(kR)×(xˆy-yˆx)k2Hφ(z)g(kR0)/kR0=k2Hφ(z){[xˆx(z-z)+yˆy(z-z)-zˆ(x2+y2)]
×[g(kR0)/kR0]/kR0-2zˆg(kR0)/kR0}.
02πadφA=2πaA.
B×[nˆ×E(r)]g(kR)=g(kR)×[nˆ×E(r)]=[xˆ(x-x)+yˆ(y-y)+zˆ(z-z)]×(1/R)(d/dR)g(kR)(xˆ sin φ-yˆ cos φ)Ez(z)=k2Ez(z){[xˆ cos φ+yˆ sin φ](z-z)-zˆ[(x-x)cos φ+(y-y)sin φ]}g(kR)/kR.
g(kR)/kRg(kR0)/kR0+k2a(x cos φ+y sin φ)×[(kR0)-3+3i(kR0)-4-3(kR0)-5]exp(ikR0).
02πadφB=π(ka)2Ez(z)×{k2[xˆx(z-z)+yˆy(z-z)-zˆ(x2+y2)][(kR0)-3+3i(kR0)-4-3(kR0)-5]exp(ikR0)+2zˆ[i(kR0)-2-(kR0)-3]exp(ikR0)}.
Es(r)=ρsk-h+hdz{k2[xˆx(z-z)+yˆy(z-z)-zˆ(x2+y2)][g(kR0)/kR0]/kR0-2zˆg(kR0)/kR0}us(z),(R0)mina
xˆxz+yˆyz-zˆ(x2+y2)=θˆR02 sin θ,
R0r-z cos θ,kr.

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