Abstract

The T-matrix equations are revisited with an eye to computing surface fields. Electromagnetic scattering by cylinders is considered for both surface- and volume-type scatterers. Elliptical and other smooth surfaces are examined, as well as cylinders with edges, and the usefulness of various surface field representations is shown. An absorption matrix is defined, and discarding the skew-symmetric part of the T matrix, i.e., enforcing reciprocity, in the course of numerical computations, is found to better satisfy energy constraints both with and without losses present. For thin penetrable cylinders, extended Rayleigh formulas are found for the case krmax1, |krmax| arbitrary, where k and k are, respectively, the free-space and interior propagation constants. In contrast, existing methods require that both quantities be small compared with unity.

© 1999 Optical Society of America

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References

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  1. V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, New York, 1980).
  2. V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
    [CrossRef]
  3. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
    [CrossRef]
  4. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  5. S. Ström, W. Zheng, “Basic features of the null field method for dielectric scatterers,” Radio Sci. 22, 1273–1281 (1987).
    [CrossRef]
  6. B. Z. Steinberg, Y. Leviatan, “A multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag. 44, 572–579 (1996).
    [CrossRef]
  7. K. K. Mei, J. Van Bladel, “Low-frequency scattering by rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 52–56 (1963);“Scattering by perfectly conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963)
    [CrossRef]
  8. M. G. Andreasen, K. K. Mei, “Comments on scattering by conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-12, 235–236 (1964).
    [CrossRef]
  9. S. Abdelmessih, G. Sinclair, “Treatment of singularities in scattering from perfectly conducting polygonal cylinders—a numerical technique,” Can. J. Phys. 45, 1305–1318 (1967).
    [CrossRef]
  10. L. Shafai, “An improved integral equation for the numerical solution of two-dimensional diffraction problems,” Can. J. Phys. 48, 954–963 (1970).
    [CrossRef]
  11. J. D. Hunter, “Surface current density on perfectly conducting polygonal cylinders,” Can. J. Phys. 50, 139–150 (1972).
    [CrossRef]
  12. R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction. I. General method,” Philos. Trans. R. Soc. London Ser. A 287, 45–78 (1977).
    [CrossRef]
  13. D. T. DiPerna, T. K. Stanton, “Sound scattering by cylinders of noncircular cross section: a conformal mapping approach,” J. Acoust. Soc. Am. 96, 3064–3079 (1994).
    [CrossRef]
  14. Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
    [CrossRef]
  15. K. Iizuka, J. L. Yen, “Surface currents on triangular and square metal cylinders,” IEEE Trans. Antennas Propag. AP-15, 795–801 (1967).
    [CrossRef]
  16. I. Navot, “An extension of the Euler–McLaurin summation formula to functions with a branch singularity,” J. Math. Phys. 40, 271–276 (1961).
  17. P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).
  18. J. Van Bladel, “Low-frequency scattering by cylindrical bodies,” Appl. Sci. Res. Sect. B 10, 195–202 (1963);“Scattering of low-frequency E-waves by dielectric cylinders,” IEEE Trans. Antennas Propag. AP-24, 255–258 (1976).
    [CrossRef]
  19. V. V. Varadan, V. K. Varadan, “Low-frequency expansions for acoustic wave scattering using Waterman’s T-matrix method,” J. Acoust. Soc. Am. 66, 586–589 (1979).
    [CrossRef]

1997

Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
[CrossRef]

1996

B. Z. Steinberg, Y. Leviatan, “A multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag. 44, 572–579 (1996).
[CrossRef]

1994

D. T. DiPerna, T. K. Stanton, “Sound scattering by cylinders of noncircular cross section: a conformal mapping approach,” J. Acoust. Soc. Am. 96, 3064–3079 (1994).
[CrossRef]

1988

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
[CrossRef]

1987

S. Ström, W. Zheng, “Basic features of the null field method for dielectric scatterers,” Radio Sci. 22, 1273–1281 (1987).
[CrossRef]

1979

V. V. Varadan, V. K. Varadan, “Low-frequency expansions for acoustic wave scattering using Waterman’s T-matrix method,” J. Acoust. Soc. Am. 66, 586–589 (1979).
[CrossRef]

1977

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction. I. General method,” Philos. Trans. R. Soc. London Ser. A 287, 45–78 (1977).
[CrossRef]

1972

J. D. Hunter, “Surface current density on perfectly conducting polygonal cylinders,” Can. J. Phys. 50, 139–150 (1972).
[CrossRef]

1971

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

1970

L. Shafai, “An improved integral equation for the numerical solution of two-dimensional diffraction problems,” Can. J. Phys. 48, 954–963 (1970).
[CrossRef]

1969

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

1967

S. Abdelmessih, G. Sinclair, “Treatment of singularities in scattering from perfectly conducting polygonal cylinders—a numerical technique,” Can. J. Phys. 45, 1305–1318 (1967).
[CrossRef]

K. Iizuka, J. L. Yen, “Surface currents on triangular and square metal cylinders,” IEEE Trans. Antennas Propag. AP-15, 795–801 (1967).
[CrossRef]

1964

P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).

M. G. Andreasen, K. K. Mei, “Comments on scattering by conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-12, 235–236 (1964).
[CrossRef]

1963

K. K. Mei, J. Van Bladel, “Low-frequency scattering by rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 52–56 (1963);“Scattering by perfectly conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963)
[CrossRef]

J. Van Bladel, “Low-frequency scattering by cylindrical bodies,” Appl. Sci. Res. Sect. B 10, 195–202 (1963);“Scattering of low-frequency E-waves by dielectric cylinders,” IEEE Trans. Antennas Propag. AP-24, 255–258 (1976).
[CrossRef]

1961

I. Navot, “An extension of the Euler–McLaurin summation formula to functions with a branch singularity,” J. Math. Phys. 40, 271–276 (1961).

Abdelmessih, S.

S. Abdelmessih, G. Sinclair, “Treatment of singularities in scattering from perfectly conducting polygonal cylinders—a numerical technique,” Can. J. Phys. 45, 1305–1318 (1967).
[CrossRef]

Abodeely, R. J.

P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).

Andreasen, M. G.

M. G. Andreasen, K. K. Mei, “Comments on scattering by conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-12, 235–236 (1964).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction. I. General method,” Philos. Trans. R. Soc. London Ser. A 287, 45–78 (1977).
[CrossRef]

DiPerna, D. T.

D. T. DiPerna, T. K. Stanton, “Sound scattering by cylinders of noncircular cross section: a conformal mapping approach,” J. Acoust. Soc. Am. 96, 3064–3079 (1994).
[CrossRef]

Friedmann, M.

Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
[CrossRef]

Hunter, J. D.

J. D. Hunter, “Surface current density on perfectly conducting polygonal cylinders,” Can. J. Phys. 50, 139–150 (1972).
[CrossRef]

Iizuka, K.

K. Iizuka, J. L. Yen, “Surface currents on triangular and square metal cylinders,” IEEE Trans. Antennas Propag. AP-15, 795–801 (1967).
[CrossRef]

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
[CrossRef]

Leviatan, Y.

Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
[CrossRef]

B. Z. Steinberg, Y. Leviatan, “A multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag. 44, 572–579 (1996).
[CrossRef]

Mei, K. K.

M. G. Andreasen, K. K. Mei, “Comments on scattering by conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-12, 235–236 (1964).
[CrossRef]

K. K. Mei, J. Van Bladel, “Low-frequency scattering by rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 52–56 (1963);“Scattering by perfectly conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963)
[CrossRef]

Navot, I.

I. Navot, “An extension of the Euler–McLaurin summation formula to functions with a branch singularity,” J. Math. Phys. 40, 271–276 (1961).

Shafai, L.

L. Shafai, “An improved integral equation for the numerical solution of two-dimensional diffraction problems,” Can. J. Phys. 48, 954–963 (1970).
[CrossRef]

Shifman, Y.

Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
[CrossRef]

Sinclair, G.

S. Abdelmessih, G. Sinclair, “Treatment of singularities in scattering from perfectly conducting polygonal cylinders—a numerical technique,” Can. J. Phys. 45, 1305–1318 (1967).
[CrossRef]

Stanton, T. K.

D. T. DiPerna, T. K. Stanton, “Sound scattering by cylinders of noncircular cross section: a conformal mapping approach,” J. Acoust. Soc. Am. 96, 3064–3079 (1994).
[CrossRef]

Steinberg, B. Z.

B. Z. Steinberg, Y. Leviatan, “A multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag. 44, 572–579 (1996).
[CrossRef]

Ström, S.

S. Ström, W. Zheng, “Basic features of the null field method for dielectric scatterers,” Radio Sci. 22, 1273–1281 (1987).
[CrossRef]

Van Bladel, J.

J. Van Bladel, “Low-frequency scattering by cylindrical bodies,” Appl. Sci. Res. Sect. B 10, 195–202 (1963);“Scattering of low-frequency E-waves by dielectric cylinders,” IEEE Trans. Antennas Propag. AP-24, 255–258 (1976).
[CrossRef]

K. K. Mei, J. Van Bladel, “Low-frequency scattering by rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 52–56 (1963);“Scattering by perfectly conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963)
[CrossRef]

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
[CrossRef]

V. V. Varadan, V. K. Varadan, “Low-frequency expansions for acoustic wave scattering using Waterman’s T-matrix method,” J. Acoust. Soc. Am. 66, 586–589 (1979).
[CrossRef]

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
[CrossRef]

V. V. Varadan, V. K. Varadan, “Low-frequency expansions for acoustic wave scattering using Waterman’s T-matrix method,” J. Acoust. Soc. Am. 66, 586–589 (1979).
[CrossRef]

Wall, D. J. N.

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction. I. General method,” Philos. Trans. R. Soc. London Ser. A 287, 45–78 (1977).
[CrossRef]

Waterman, P. C.

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

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).

Yen, J. L.

K. Iizuka, J. L. Yen, “Surface currents on triangular and square metal cylinders,” IEEE Trans. Antennas Propag. AP-15, 795–801 (1967).
[CrossRef]

Yos, J. M.

P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).

Zheng, W.

S. Ström, W. Zheng, “Basic features of the null field method for dielectric scatterers,” Radio Sci. 22, 1273–1281 (1987).
[CrossRef]

Appl. Sci. Res. Sect. B

J. Van Bladel, “Low-frequency scattering by cylindrical bodies,” Appl. Sci. Res. Sect. B 10, 195–202 (1963);“Scattering of low-frequency E-waves by dielectric cylinders,” IEEE Trans. Antennas Propag. AP-24, 255–258 (1976).
[CrossRef]

Can. J. Phys.

S. Abdelmessih, G. Sinclair, “Treatment of singularities in scattering from perfectly conducting polygonal cylinders—a numerical technique,” Can. J. Phys. 45, 1305–1318 (1967).
[CrossRef]

L. Shafai, “An improved integral equation for the numerical solution of two-dimensional diffraction problems,” Can. J. Phys. 48, 954–963 (1970).
[CrossRef]

J. D. Hunter, “Surface current density on perfectly conducting polygonal cylinders,” Can. J. Phys. 50, 139–150 (1972).
[CrossRef]

IEE Proc. Microwave Antennas Propag.

Y. Shifman, M. Friedmann, Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” IEE Proc. Microwave Antennas Propag. 144, 235–240 (1997).
[CrossRef]

IEEE Trans. Antennas Propag.

K. Iizuka, J. L. Yen, “Surface currents on triangular and square metal cylinders,” IEEE Trans. Antennas Propag. AP-15, 795–801 (1967).
[CrossRef]

B. Z. Steinberg, Y. Leviatan, “A multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag. 44, 572–579 (1996).
[CrossRef]

K. K. Mei, J. Van Bladel, “Low-frequency scattering by rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 52–56 (1963);“Scattering by perfectly conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-11, 185–192 (1963)
[CrossRef]

M. G. Andreasen, K. K. Mei, “Comments on scattering by conducting rectangular cylinders,” IEEE Trans. Antennas Propag. AP-12, 235–236 (1964).
[CrossRef]

J. Acoust. Soc. Am.

D. T. DiPerna, T. K. Stanton, “Sound scattering by cylinders of noncircular cross section: a conformal mapping approach,” J. Acoust. Soc. Am. 96, 3064–3079 (1994).
[CrossRef]

V. V. Varadan, A. Lakhtakia, V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988).
[CrossRef]

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

V. V. Varadan, V. K. Varadan, “Low-frequency expansions for acoustic wave scattering using Waterman’s T-matrix method,” J. Acoust. Soc. Am. 66, 586–589 (1979).
[CrossRef]

J. Math. Phys.

I. Navot, “An extension of the Euler–McLaurin summation formula to functions with a branch singularity,” J. Math. Phys. 40, 271–276 (1961).

P. C. Waterman, J. M. Yos, R. J. Abodeely, “Numerical integration of non-analytic functions,” J. Math. Phys. 43, 45–50 (1964).

Philos. Trans. R. Soc. London Ser. A

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction. I. General method,” Philos. Trans. R. Soc. London Ser. A 287, 45–78 (1977).
[CrossRef]

Phys. Rev. D

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

Radio Sci.

S. Ström, W. Zheng, “Basic features of the null field method for dielectric scatterers,” Radio Sci. 22, 1273–1281 (1987).
[CrossRef]

Other

V. K. Varadan, V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, New York, 1980).

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Figures (16)

Fig. 1
Fig. 1

Geometry for scattering by a cylinder.

Fig. 2
Fig. 2

Amplitude (solid curve) and phase (dashed curve) for broadside incidence on an 8:1 Rayleigh cylinder with krmax=0.1.

Fig. 3
Fig. 3

Same as Fig. 2, but for E-perpendicular polarization.

Fig. 4
Fig. 4

Same as Fig. 2, but for krmax=6.14024 (circumference=4λ).

Fig. 5
Fig. 5

Same as Fig. 4, but for E-perpendicular polarization.

Fig. 6
Fig. 6

Amplitude (solid curve) and phase (dashed curve) for broadside incidence on an 8:1 penetrable Rayleigh cylinder with krmax=0.08. 1, normal gradient of the axial E field; 2, axial E field.

Fig. 7
Fig. 7

Same as Fig. 6, but for higher conductivity (no longer a Rayleigh case).

Fig. 8
Fig. 8

Same as Fig. 6, but for krmax=6.14024 (circumference=4λ).

Fig. 9
Fig. 9

Geometry for the inverted elliptical cross section of Eq. (28).

Fig. 10
Fig. 10

Amplitude (solid curve) and phase (dashed curve) for incidence from the right on the inverted elliptical cylinder (E-parallel polarization).

Fig. 11
Fig. 11

Cross-section geometry for three cylinders with edges.

Fig. 12
Fig. 12

Amplitude (solid curve) and phase (dashed curve) for Rayleigh scattering by a square cylinder, with ka=0.001, where 2a is the side length. The dotted curve shows the negligible amplitude variations remaining when the edge factor is removed.

Fig. 13
Fig. 13

Amplitude and phase for a larger square cylinder (ka=10).

Fig. 14
Fig. 14

Amplitude and phase for the equilateral triangular cylinder (ka=2π, where 2a is the length of each face).

Fig. 15
Fig. 15

Amplitude and phase for the semicircular cylinder (ka=2, where a is the cylinder radius).

Fig. 16
Fig. 16

Real and imaginary parts of the dominant T-matrix element T00 plotted against variations in the (complex) index of refraction q of a 2:1 elliptical cylinder (krmax=0.1).

Tables (3)

Tables Icon

Table 1 Dominant Matrix Element T00 versus Truncation (MatrixSize=N × N)

Tables Icon

Table 2 Dominant Matrix Element T00 versus Truncation (Matrix Size N × N)

Tables Icon

Table 3 Dipole–Dipole Matrix Element T11 versus Truncation for a Magnetic Material

Equations (56)

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

Ei(r)=n=0an Reg ψn(kr),
Es(r)=n=0fnψn(kr)
ψn(kr)=n0.5Hn(kr)cos nθ,n=0, 1,,
kr(θ)=kaρ(θ),
fm=n=0Tmnan,
f=Ta,
an=-(i/4)ds[(nˆψn)E+-ψn(nˆ+E)],
fn=(i/4)ds[(nˆ Reg ψn)E+-Reg(ψn)(nˆ+E)],
n=0, 1,,
s(θ)=0θdθ s(θ)
s(θ)=a{ρ2(θ)+[ρ(θ)]2}1/2.
nˆ+E=n=0annˆ Reg ψn(kr),
iQα=a,f=-i Reg(Q)α,
QmnD=(1/4)ds ψm(nˆ Reg ψn),m, n=0, 1,,
a=-iQ-1a,
T=-Reg(Q)Q-1.
E+=E-,nˆ+E=μrel-1nˆ-E
E-(r)=n=0αn Reg ψn(kr),
nˆ-E(r)=n=0αnnˆ Reg ψn(kr),
Qmn=-(1/4)ds[nˆψm(kr)Reg ψn(kr)-μrel-1ψm(kr)nˆ Reg ψn(kr)],
m, n=0, 1, .
k2=ω2μ+iωμσ
H+=n=0an Reg ψn(kr),
QmnN=(1/4)ds(nˆψm)Reg ψn,m, n=0, 1, .
S=S(orT=T),
S*S=1-A[orT*T+(1/4)A=-Re T],
(1/2)Re(α*Wα)=(1/4)α*(W+W*)α=(1/4)a*(Q*)-1(W+W*)Q-1a,
Wmn=ds μrel Reg[ψm(k*r)]inˆ Reg ψn(kr).
A=(1/4)(Q*)-1(W+W*)Q-1.
QmnD=(1/4)(mn)0.502πdθ Hm(kaρ)cos mθ×[kaρJn(kaρ)cos nθ+n(ρ/ρ)Jn(kaρ)×sin nθ],m, n=0, 1,,
ρ(θ)=[cos2 θ+(a/b)2 sin2 θ]-0.5
ρ2(θ)/s(θ),
Qmn=(1/4)(mn)0.502πdθ{[kaρHm(kaρ)cos mθ+m(ρ/ρ)Hm(kaρ)sin mθ)]×Jn(kaρ)cos nθ-μrel-1Hm(kaρ)×cos(mθ)[kaρJn(kaρ)cos nθ+n(ρ/ρ)Jn(kaρ)sin nθ]}.
Qmn=(1/4)(mn)0.502πdθ{[kaρHm-1(kaρ)cos mθ-(1/2)mρ2Hm(kaρ)c1m(θ)]Jn(kaρ)cos nθ+μrel-1Hm(kaρ)cos(mθ)[kaρJn+1(kaρ)×cos nθ-(1/2)nρ2Jn(kaρ)c2m(θ)]}.
c1m(θ)=[1-(a/b)2]cos(m+2)θ+[1+(a/b)2]cos mθ,
c2m(θ)=[1+(a/b)2]cos mθ+[1-(a/b)2]cos (m-2)θ
ρ2(θ)x(θ)/s(θ),
ρ(θ)=[cos2 θ+(a/b)2 sin2 θ]0.5.
n0.5 cos nθ,n=0, 1,,
QmnD=(1/4)(mn)0.502πdθ s(θ)Hm(kaρ)×cos mθ cos nθ,m, n=0, 1, .
nˆ+E=edge(σ)n=0αn cos(2πnσ),
edge(σ)=m|δm+δm-|-β.
edge(σ)=|(σ2-1/64)(σ2-9/64)(σ2-25/64)(σ2-49/64)|-β,0θπ/4|(σ2-1/64)(σ2-9/64)(σ2-25/64)(σ-7/8)(σ-9/8)|-β,π/4θπ/2.
T00=i(q2/μrel-1)k2Area/4.
Re T00(1/4)A00-Im(q2/μrel)k2Area/4,
Q00=Q00D=(1/4)02πdθ H0(kr)krJ0(kr)02πdθ(kr)2H0(kr),
Q00=(1/4)02πdθ[krH0(kr)J0(kr)-μrel-1H0(kr)krJ0(kr)]02πdθ krJ1(kr)[H0(kr)-(1/2)η0(kr)krH1(kr)]02πdθ(kr)2[H0(kr)-(1/2)η0(kr)krH1(kr)],
η0(kr)=2μrelJ0(kr)/krJ1(kr).
T00-Reg(Q0)/Q0,
Q0=02πdθ(kr)2[H0(kr)-(1/2)η0(kr)krH1(kr)].
nˆψ1(kr)[1/s(θ)][kr/(kr)-(ρ/ρ)/θ]2H1(kr)cos θ,
η1(kr)=2μrel Reg[ψ1(kr)]/nˆ Reg ψ1(kr),
T11-Reg(Q1)/Q1,
Q1=02πdθ[ρ(θ)sin θ][ψ1(kr)-(1/2)η1(kr)nˆψ1(kr)].
Q1=δk2Area/4+i[1+(δ/π)02πdθ cos 2θ ln ρ(θ)],
|krmax|1,

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