Abstract

For electromagnetic wave incidence on a thin fiber at an angle θi with the fiber axis, induced currents are described by linear combinations of driven and traveling waves exp(±ikzcosθi) and exp(±imkz), where the complex factor m serves to remove singularities in the normalizing constants of the currents. The fiber may be solid, hollow, or coated, with quite general constitutive parameters. Results are given for scattering, absorption, extinction, and radar cross sections, using energy conservation and reciprocity as consistency checks, and compared with independent computations, including the Born approximation for tenuous conductors and dielectrics. In addition, the far field amplitude of a 50 wavelength conductor is obtained and compared with the well-known long-wire approximation.

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References

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  1. O. Einarsson, “The wire,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B.A. Senior, and P. L.E. Uslenghi, eds. (North-Holland, 1969), pp. 472ff.
  2. E. H. Newman, “A unified theory of thin material wires,” IEEE Trans. Antennas Propag. 39, 1488–1496 (1991).
    [CrossRef]
  3. K. W. Whites, “Resistive and conductive tube boundary condition models for material wire-shaped scatterers,” IEEE Trans. Antennas Propag. 46, 1548–1554 (1998).
    [CrossRef]
  4. C.-T. Tai, “Electromagnetic back-scattering from cylindrical wires,” J. Appl. Phys. 23, 909–916 (1952).
    [CrossRef]
  5. E. S. Cassedy, J. Fainberg, “Back scattering cross sections of cylindrical wires of finite conductivity,” IRE Trans. Antennas Propag. 8, 1–7 (1960).
    [CrossRef]
  6. P. C. Waterman, J. C. Pedersen, “Scattering by finite wires,” J. Appl. Phys. 72, 349–359 (1992).
    [CrossRef]
  7. P. C. Waterman, J. C. Pedersen, “Electromagnetic scattering and absorption by finite wires,” J. Appl. Phys. 78, 656–667 (1995).
    [CrossRef]
  8. P. C. Waterman, J. C. Pedersen, “Scattering by finite wires of arbitrary ϵ,μ, and σ ,” J. Opt. Soc. Am. A 15, 174–184 (1998).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.
  10. R. C. Hansen, “Antenna mode and structural mode RCS: dipole,” Microwave Opt. Technol. Lett. 3, 6–10 (1990).
    [CrossRef]
  11. G. J. Burke, “Numerical Electromagnetics Code (NEC)–Method of Moments; Part I: Program Description—Theory, Part II: Program Description—Code, Part III: User’s Guide,” Lawrence Livermore National Lab. Rep. UCID-18834 (January 1981).
  12. R. Schiffer, K. O. Theilheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2474–2483 (1979).
    [CrossRef]
  13. 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]
  14. P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
    [CrossRef]
  15. D. E. Barrick, “Cylinders,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, 1970), pp. 205–339.
    [CrossRef]
  16. A. J. Poggio, E. K. Miller, in “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 159–264.
    [CrossRef]
  17. L. Peters, “End-fire echo area of long, thin bodies,” IRE Trans. Antennas Propag. 6, 133–139 (1958).
    [CrossRef]
  18. A. V. Jelinek, C. W. Bruce, “Extinction spectra of high conductivity fibrous aerosols,” J. Appl. Phys. 78, 2675–2678 (1995).
    [CrossRef]
  19. M. Hart, C. W. Bruce, “Backscatter measurements of thin nickel-coated graphite fibers,” IEEE Trans. Antennas Propag. 48, 842–843 (2000).
    [CrossRef]

2000 (1)

M. Hart, C. W. Bruce, “Backscatter measurements of thin nickel-coated graphite fibers,” IEEE Trans. Antennas Propag. 48, 842–843 (2000).
[CrossRef]

1998 (2)

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires of arbitrary ϵ,μ, and σ ,” J. Opt. Soc. Am. A 15, 174–184 (1998).
[CrossRef]

K. W. Whites, “Resistive and conductive tube boundary condition models for material wire-shaped scatterers,” IEEE Trans. Antennas Propag. 46, 1548–1554 (1998).
[CrossRef]

1996 (1)

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 (2)

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

A. V. Jelinek, C. W. Bruce, “Extinction spectra of high conductivity fibrous aerosols,” J. Appl. Phys. 78, 2675–2678 (1995).
[CrossRef]

1992 (1)

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

1991 (1)

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

1990 (1)

R. C. Hansen, “Antenna mode and structural mode RCS: dipole,” Microwave Opt. Technol. Lett. 3, 6–10 (1990).
[CrossRef]

1982 (1)

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[CrossRef]

1979 (1)

R. Schiffer, K. O. Theilheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2474–2483 (1979).
[CrossRef]

1960 (1)

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

1958 (1)

L. Peters, “End-fire echo area of long, thin bodies,” IRE Trans. Antennas Propag. 6, 133–139 (1958).
[CrossRef]

1952 (1)

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

Barber, P. W.

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[CrossRef]

Barrick, D. E.

D. E. Barrick, “Cylinders,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, 1970), pp. 205–339.
[CrossRef]

Bruce, C. W.

M. Hart, C. W. Bruce, “Backscatter measurements of thin nickel-coated graphite fibers,” IEEE Trans. Antennas Propag. 48, 842–843 (2000).
[CrossRef]

A. V. Jelinek, C. W. Bruce, “Extinction spectra of high conductivity fibrous aerosols,” J. Appl. Phys. 78, 2675–2678 (1995).
[CrossRef]

Burke, G. J.

G. J. Burke, “Numerical Electromagnetics Code (NEC)–Method of Moments; Part I: Program Description—Theory, Part II: Program Description—Code, Part III: User’s Guide,” Lawrence Livermore National Lab. Rep. UCID-18834 (January 1981).

Cassedy, E. S.

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

Chang, R. K.

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[CrossRef]

Einarsson, O.

O. Einarsson, “The wire,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B.A. Senior, and P. L.E. Uslenghi, eds. (North-Holland, 1969), pp. 472ff.

Fainberg, J.

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

Hansen, R. C.

R. C. Hansen, “Antenna mode and structural mode RCS: dipole,” Microwave Opt. Technol. Lett. 3, 6–10 (1990).
[CrossRef]

Hart, M.

M. Hart, C. W. Bruce, “Backscatter measurements of thin nickel-coated graphite fibers,” IEEE Trans. Antennas Propag. 48, 842–843 (2000).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.

Jelinek, A. V.

A. V. Jelinek, C. W. Bruce, “Extinction spectra of high conductivity fibrous aerosols,” J. Appl. Phys. 78, 2675–2678 (1995).
[CrossRef]

Miller, E. K.

A. J. Poggio, E. K. Miller, in “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 159–264.
[CrossRef]

Newman, E. H.

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

Owen, J. F.

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[CrossRef]

Pedersen, J. C.

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires of arbitrary ϵ,μ, and σ ,” J. Opt. Soc. Am. A 15, 174–184 (1998).
[CrossRef]

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]

Peters, L.

L. Peters, “End-fire echo area of long, thin bodies,” IRE Trans. Antennas Propag. 6, 133–139 (1958).
[CrossRef]

Poggio, A. J.

A. J. Poggio, E. K. Miller, in “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 159–264.
[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]

Schiffer, R.

R. Schiffer, K. O. Theilheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2474–2483 (1979).
[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]

Tai, C.-T.

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

Theilheim, K. O.

R. Schiffer, K. O. Theilheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2474–2483 (1979).
[CrossRef]

Waterman, P. C.

P. C. Waterman, J. C. Pedersen, “Scattering by finite wires of arbitrary ϵ,μ, and σ ,” J. Opt. Soc. Am. A 15, 174–184 (1998).
[CrossRef]

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]

Whites, K. W.

K. W. Whites, “Resistive and conductive tube boundary condition models for material wire-shaped scatterers,” IEEE Trans. Antennas Propag. 46, 1548–1554 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

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

K. W. Whites, “Resistive and conductive tube boundary condition models for material wire-shaped scatterers,” IEEE Trans. Antennas Propag. 46, 1548–1554 (1998).
[CrossRef]

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]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[CrossRef]

M. Hart, C. W. Bruce, “Backscatter measurements of thin nickel-coated graphite fibers,” IEEE Trans. Antennas Propag. 48, 842–843 (2000).
[CrossRef]

IRE Trans. Antennas Propag. (2)

L. Peters, “End-fire echo area of long, thin bodies,” IRE Trans. Antennas Propag. 6, 133–139 (1958).
[CrossRef]

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

J. Appl. Phys. (5)

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]

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

A. V. Jelinek, C. W. Bruce, “Extinction spectra of high conductivity fibrous aerosols,” J. Appl. Phys. 78, 2675–2678 (1995).
[CrossRef]

R. Schiffer, K. O. Theilheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2474–2483 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

Microwave Opt. Technol. Lett. (1)

R. C. Hansen, “Antenna mode and structural mode RCS: dipole,” Microwave Opt. Technol. Lett. 3, 6–10 (1990).
[CrossRef]

Other (5)

G. J. Burke, “Numerical Electromagnetics Code (NEC)–Method of Moments; Part I: Program Description—Theory, Part II: Program Description—Code, Part III: User’s Guide,” Lawrence Livermore National Lab. Rep. UCID-18834 (January 1981).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.

O. Einarsson, “The wire,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B.A. Senior, and P. L.E. Uslenghi, eds. (North-Holland, 1969), pp. 472ff.

D. E. Barrick, “Cylinders,” in Radar Cross Section Handbook, G. T. Ruck, ed. (Plenum, 1970), pp. 205–339.
[CrossRef]

A. J. Poggio, E. K. Miller, in “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 159–264.
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Geometry of the thin fiber, which may be solid, hollow, or coated.

Fig. 2
Fig. 2

The extinction cross section (solid curve) and cross section defect (dotted curve) are shown versus incident angle, including erroneous fluctuations, for a perfectly conducting fiber 1 1 4 wavelengths long ( k a x = 0.001 ) .

Fig. 3
Fig. 3

Loci of the roots of D e (solid curve) and D o (dashed curve) versus fiber length when m = 1 .

Fig. 4
Fig. 4

Backscattering cross sections plotted versus length for a perfectly conducting fiber for both normal incidence (solid curve) and random orientation (dashed curve). Resonance frequency values (points) were computed independently by Hansen.[10]

Fig. 5
Fig. 5

Same as Fig. 4 but for extinction cross sections.

Fig. 6
Fig. 6

Extinction cross section (solid curve) and cross section defect (dotted curve) for the example of Fig. 2 after correcting m. Points were computed from the Lawrence Livermore NEC code.[11]

Fig. 7
Fig. 7

Radar cross section for the example of Fig. 6, again compared with the NEC code.

Fig. 8
Fig. 8

Far field amplitude (solid curve) and defect (dotted curve) versus observation angle for normal incidence (example of Fig. 6).

Fig. 9
Fig. 9

Same as Fig. 8 but for incidence at 30 deg from the fiber axis.

Fig. 10
Fig. 10

The even and odd normalizing constants K e (solid curve) and K o (dashed curve) are shown versus incident angle for the case of Fig. 6.

Fig. 11
Fig. 11

Extinction (solid curve), scattering (dashed curve), and absorption (dotted–dashed curve) cross sections, cross section defect (dotted curve) and NEC code results (points) for the example of Fig. 6 but now with conductivity 10 6 ( Ω m ) 1 (frequency is 35 GHz).

Fig. 12
Fig. 12

Same as Fig. 11 but now for σ = 10 5 ( Ω m ) 1 . The fiber is now primarily absorbing (see the text).

Fig. 13
Fig. 13

The extinction (solid curve) and enhanced scattering (dashed curve) cross sections are given for σ = 10 3 ( Ω m ) 1 , as compared with the first Born approximation (dotted curves). Absorption and extinction are identical.

Fig. 14
Fig. 14

Same as Fig. 13 but for the backscattering cross section.

Fig. 15
Fig. 15

Radar cross section for a long, perfectly conducting wire ( k h = 14.57 π , k a = 0.07 π ) . Peak values (points) were computed independently by Poggio and Miller.[16]

Fig. 16
Fig. 16

Extinction cross section (solid curve) and cross section defect (dotted curve) for a 50 wavelength perfectly conducting wire ( k a = 0.05 π ) .

Fig. 17
Fig. 17

Radar cross section for the wire of Fig. 16.

Fig. 18
Fig. 18

Superimposed far field amplitudes for 30, 60, and 90 deg incidence for the 50 wavelength wire of Fig. 16. Small peaks near the two grazing directions stem from the 30 deg incidence case.

Fig. 19
Fig. 19

Extinction (solid curve), scattering (dashed curve), absorption (dotted–dashed curve) cross sections along with the cross section defect (dotted curve) for a hollow tube, obtained by removing about 80% of the inner material from the fiber of Fig. 11 (see the text).

Equations (61)

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E i ( s ) = E ( s ) x x d s u ( s ) ( 1 + 2 s 2 ) g ( k R ) ,
E i e ( s ) = E 0 sin θ i cos ( s cos θ i ) ,
E i o ( s ) = E 0 sin θ i sin ( s cos θ i )
E i ( s ) = E i e ( s ) + i E i o ( s ) .
u e ( s ) = ( 1 K e cos m x ) ( cos m x cos q s cos q x cos m s ) ,
u o ( s ) = ( 1 K o sin m x ) ( sin m x sin q s sin q x sin m s ) ,
u ( s ) = u e ( s ) + i u o ( s ) .
η ( q ) = 2 μ r ( κ 1 k 1 ) 2 J 0 ( κ 1 a ) κ 1 a J 1 ( κ 1 a ) ,
E e = ( 1 K e cos m x ) [ η ( q ) cos m x cos q s η ( m ) cos q x cos m s ] ,
E ( s ) = η ( 0 ) u ( s ) ,
g σ = λ σ γ σ , σ = e , o ( even , odd ) ,
g σ = x x d s u σ ( s ) E i σ ( s ) ,
λ σ = x x d s u σ ( s ) E σ ( s ) ,
γ σ = x x d s u σ ( s ) x x d s u σ ( s ) ( 1 + 2 s 2 ) g ( k R ) .
γ σ = x x d s x x d s [ u σ ( s ) u σ ( s ) u σ ( s ) u σ ( s ) ] g ( k R ) .
K ( q ) = sin θ i [ 2 ln ( 4 x k a ) + L + ( q , x ) ] η ( q ) sin θ i
sin θ i [ 2 ln ( 2 k a sin θ i ) 2 γ + i π ] η ( q ) sin θ i for x 1 ,
K e ( m , q ) = K ( q ) + ( 1 x sin θ i cos m x ) N e ( m , q ) D e ( m , q ) ,
K o ( m , q ) = K ( q ) + ( 1 x sin θ i sin m x ) N o ( m , q ) D o ( m , q ) ,
E θ ( r , θ ) E 0 f ( q , p ) ( 1 i k r ) exp ( i k r )
f ( q , p ) = sin θ x x d s [ u ( s ) ( k a 2 ) 2 E ( s ) ] exp ( i p s ) ,
f ( q , p ) = [ 1 ( k a 2 ) 2 η ( 0 ) ] sin θ x x d s u ( s ) exp ( i p s ) .
f ( p , q ) = f ( q , p ) .
σ e ( q ) λ 2 = ( 1 π ) Im f ( q , q ) ,
σ s ( q ) λ 2 = ( 1 2 π ) 1 1 d p f ( q , p ) 2 ,
σ a ( q ) λ 2 = ( 1 π ) Im x x d s E ( s ) u * ( s ) ,
σ b ( q ) λ 2 = ( 1 π ) f ( q , q ) 2 .
σ a ( q ) λ 2 = ( 1 π ) Im η ( 0 ) x x d s u ( s ) 2 .
σ = 1 2 0 1 d q σ ( q ) .
N e ( m , 0 ) = 0 .
m n + 1 = m n n e ( m n , 0 ) N e , m ( m n , 0 ) , n = 0 , 1 , ,
m = 1 i L ( 4 x ) + x η ( 0 ) 4 x [ 1 n ( 4 x k a ) + 0.5 L ( 4 x ) 1 ]
b [ ln ( 2 b ) γ ] = ( μ r μ 2 ) J 0 ( b ) J 1 ( b ) .
m = [ 1 i L ( 4 x ) + x η ( 0 ) 2 x [ ln ( 4 x k a ) + 0.5 L ( 4 x ) 1 ] ] 1 2 .
Δ σ ( m , q ) = σ e ( m , q ) σ s ( m , q ) σ a ( m , q ) λ 2 ,
Δ σ ( m e 1 , 0 ) = 0 .
N e ( m e 2 , q e 2 ) = 0 ,
D e ( m e 2 , q e 2 ) = 0 .
( m n + 1 q n + 1 ) = ( m n N e ( m n , q n ) N e , m ( m n , q n ) q n D e ( m n , q n ) D e , q ( m n , q n ) ) , n = 0 , 1 , .
m e ( q ) = norm e ( q ) 1 [ m e 1 ( q 2 q e 2 2 ) 2 + act e m e 2 q 2 ] ,
norm e ( q ) = ( q 2 q e 2 2 ) 2 + act e q 2 ,
Δ f ( q , p ) = f ( q , p ) f ( p , q ) ,
E I n t e r i o r ( r ) = A E i ( r ) ,
A = [ 2 ( n 2 + 1 ) 0 0 0 2 ( n 2 + 1 ) 0 0 0 1 ] ,
σ e B o r n ( q ) λ 2 = ( 1 π ) Im f B o r n ( q , q ) + σ s B o r n ( q ) λ 2 .
f B o r n ( q , p ) = 1 2 ( k a ) 2 x ( n 2 1 ) sin θ i sin θ sinc [ ( q p ) x ] .
f ( cos θ , cos θ ) x [ ln ( 2 k a sin θ ) γ + 1 2 π i ] .
N 1 ( q ) = ( n 1 2 q 2 ) 1 2 , N 2 ( q ) = ( n 2 2 q 2 ) 1 2 ,
u ( q ) = μ r 2 n 1 2 N 2 μ r 1 n 2 2 N 1
w n ( q ) = J 1 ( N 2 k a 1 ) J 0 ( N 1 k a 1 ) u ( q ) J 0 ( N 2 k a 1 ) J 1 ( N 1 k a 1 ) ,
w d ( q ) = H 1 ( N 2 k a 1 ) J 0 ( N 1 k a 1 ) u ( q ) H 0 ( N 2 k a 1 ) J 1 ( N 1 k a 1 ) ,
w ( q ) = { J 0 ( N 2 k a 1 ) H 0 ( N 2 k a 1 ) , perfectly conducting core w n ( q ) w d ( q ) , otherwise } .
η ( q ) = 2 μ r 2 N 2 n 2 2 k a 2 J 0 ( N 2 k a 2 ) w ( q ) H 0 ( N 2 k a 2 ) J 1 ( N 2 k a 2 ) w ( q ) H 1 ( N 2 k a 2 ) .
L ( x ) = Ci ( x ) + i Si ( x ) ln x γ
ln x γ + 1 2 i π ( i x ) exp ( i x ) for x 1 , arg ( x ) < π ,
L + ( q , x ) = L [ 2 ( 1 q ) x ] + L [ 2 ( 1 + q ) x ] ,
i x ( m 2 q 2 ) D e ( m , q ) = 2 ( m 2 q 2 ) x cos m x ( 1 q ) ( m 2 + q 2 ) cos m x sin 2 q x + 4 m sin m x cos 2 q x ,
i x ( m 2 q 2 ) D o ( m , q ) = 2 ( m 2 q 2 ) x sin m x ( 1 q ) ( m 2 + q 2 ) sin m x sin 2 q x + 4 m cos m x sin 2 q x .
exp ( i z ) cos z 2 [ 1 + exp ( 2 i z ) ] .
σ e x p ( q ) = [ 2 σ ( q c ) + ( 1 2 q c ) sin 2 θ c σ ( q c ) ] ( sin θ sin θ c ) 2 [ σ ( q c ) + ( 1 2 q c ) sin 2 θ c σ ( q c ) ] ( sin θ sin θ c ) 4 , θ θ c ,
σ b e x p ( q ) = [ 3 σ ( q c ) + ( 1 2 q c ) sin 2 θ c σ ( q c ) ] ( sin θ sin θ c ) 4 [ 2 σ ( q c ) + ( 1 2 q c ) sin 2 θ c σ ( q c ) ] ( sin θ sin θ c ) 6 , θ θ c .

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