Abstract

The problem of computing the transition matrix (T matrix) in the framework of the null-field method with discrete sources is treated. Numerical experiments are performed to investigate the symmetry property of the T matrix when localized and distributed vector spherical functions are used for solution construction.

© 1999 Optical Society of America

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References

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  1. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
    [CrossRef]
  2. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  3. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  4. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  5. B. Peterson, S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers,” Phys. Rev. D 8, 3661–3678 (1973).
    [CrossRef]
  6. D. W. Mackowski, “Calculations of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  7. D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
    [CrossRef]
  8. G. Videen, D. Ngo, P. Chylek, R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
    [CrossRef]
  9. Wenxin Zheng, “The null-field approach to electromagnetic scattering from composite objects: the case with three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400 (1988).
    [CrossRef]
  10. T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [CrossRef]
  11. A. Boström, “Scattering of acoustic waves by a layered elastic obstacle immersed in a fluid: an improved null-field approach,” J. Acoust. Soc. Am. 76, 588–593 (1984).
    [CrossRef]
  12. M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
    [CrossRef]
  13. M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–953 (1984).
    [CrossRef] [PubMed]
  14. R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction: I. General method; II. Approximate methods; III. Inverse methods,” Philos. Trans. R. Soc. London A 287, 45–117 (1977).
    [CrossRef]
  15. A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
    [CrossRef]
  16. R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
    [CrossRef]
  17. A. Doicu, T. Wriedt, “EBCM with multipole sources located in the complex plane,” Opt. Commun. 139, 85–98 (1997).
    [CrossRef]
  18. T. Wriedt, A. Doicu, “Formulations of the EBCM for three-dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
    [CrossRef]
  19. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, New York, 1969).
  20. V. V. Varadan, A. Lakhtakia, V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier, Amsterdam, 1991).

1998

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

T. Wriedt, A. Doicu, “Formulations of the EBCM for three-dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

1997

A. Doicu, T. Wriedt, “EBCM with multipole sources located in the complex plane,” Opt. Commun. 139, 85–98 (1997).
[CrossRef]

1996

D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
[CrossRef]

1995

1994

1991

1988

Wenxin Zheng, “The null-field approach to electromagnetic scattering from composite objects: the case with three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400 (1988).
[CrossRef]

1984

A. Boström, “Scattering of acoustic waves by a layered elastic obstacle immersed in a fluid: an improved null-field approach,” J. Acoust. Soc. Am. 76, 588–593 (1984).
[CrossRef]

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–953 (1984).
[CrossRef] [PubMed]

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

1983

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

1977

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction: I. General method; II. Approximate methods; III. Inverse methods,” Philos. Trans. R. Soc. London A 287, 45–117 (1977).
[CrossRef]

1973

B. Peterson, S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

1971

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

1969

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

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Bates, R. H. T.

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction: I. General method; II. Approximate methods; III. Inverse methods,” Philos. Trans. R. Soc. London A 287, 45–117 (1977).
[CrossRef]

Boström, A.

A. Boström, “Scattering of acoustic waves by a layered elastic obstacle immersed in a fluid: an improved null-field approach,” J. Acoust. Soc. Am. 76, 588–593 (1984).
[CrossRef]

Chylek, P.

D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
[CrossRef]

G. Videen, D. Ngo, P. Chylek, R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
[CrossRef]

Doicu, A.

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

T. Wriedt, A. Doicu, “Formulations of the EBCM for three-dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

A. Doicu, T. Wriedt, “EBCM with multipole sources located in the complex plane,” Opt. Commun. 139, 85–98 (1997).
[CrossRef]

Durney, C. H.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Hackman, R. H.

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Iskander, M. F.

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–953 (1984).
[CrossRef] [PubMed]

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Lakhtakia, A.

M. F. Iskander, A. Lakhtakia, “Extension of the iterative EBCM to calculate scattering by low-loss or lossless elongated dielectric objects,” Appl. Opt. 23, 948–953 (1984).
[CrossRef] [PubMed]

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

V. V. Varadan, A. Lakhtakia, V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier, Amsterdam, 1991).

Mackowski, D. W.

Mishchenko, M. I.

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, New York, 1969).

Ngo, D.

D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
[CrossRef]

G. Videen, D. Ngo, P. Chylek, R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
[CrossRef]

Peterson, B.

B. Peterson, S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Pinnick, R. G.

Ström, S.

B. Peterson, S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier, Amsterdam, 1991).

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier, Amsterdam, 1991).

Videen, G.

D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
[CrossRef]

G. Videen, D. Ngo, P. Chylek, R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
[CrossRef]

Wall, D. J. N.

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction: I. General method; II. Approximate methods; III. Inverse methods,” Philos. Trans. R. Soc. London A 287, 45–117 (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]

Wriedt, T.

T. Wriedt, A. Doicu, “Formulations of the EBCM for three-dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

A. Doicu, T. Wriedt, “EBCM with multipole sources located in the complex plane,” Opt. Commun. 139, 85–98 (1997).
[CrossRef]

Zheng, Wenxin

Wenxin Zheng, “The null-field approach to electromagnetic scattering from composite objects: the case with three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400 (1988).
[CrossRef]

Appl. Opt.

Comput. Phys. Commun.

D. Ngo, G. Videen, P. Chylek, “A Fortran code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112 (1996).
[CrossRef]

IEEE Trans. Antennas Propag.

Wenxin Zheng, “The null-field approach to electromagnetic scattering from composite objects: the case with three or more constituents,” IEEE Trans. Antennas Propag. 36, 1396–1400 (1988).
[CrossRef]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative EBCM for solving the absorbtion characteristics of lossy dielectric objects of large aspect ratios,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

J. Acoust. Soc. Am.

R. H. Hackman, “The transition matrix for acoustic and elastic wave scattering in prolate spheroidal coordinates,” J. Acoust. Soc. Am. 75, 35–45 (1984).
[CrossRef]

A. Boström, “Scattering of acoustic waves by a layered elastic obstacle immersed in a fluid: an improved null-field approach,” J. Acoust. Soc. Am. 76, 588–593 (1984).
[CrossRef]

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

J. Mod. Opt.

T. Wriedt, A. Doicu, “Formulations of the EBCM for three-dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. Doicu, T. Wriedt, “EBCM with multipole sources located in the complex plane,” Opt. Commun. 139, 85–98 (1997).
[CrossRef]

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Philos. Trans. R. Soc. London A

R. H. T. Bates, D. J. N. Wall, “Null field approach to scalar diffraction: I. General method; II. Approximate methods; III. Inverse methods,” Philos. Trans. R. Soc. London A 287, 45–117 (1977).
[CrossRef]

Phys. Rev. D

B. Peterson, S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers,” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

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

Other

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, New York, 1969).

V. V. Varadan, A. Lakhtakia, V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier, Amsterdam, 1991).

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Figures (7)

Fig. 1
Fig. 1

Relative symmetry error of the transition matrix for the azimuthal mode m=1 for prolate spheroids with large semiaxis ksa=10 and small semiaxis varying between ksb=0.5 and ksb=5. The refractive index of the particle is M=1.5. The incident field is a p-polarized plane wave traveling along the z axis, which corresponds to the particle symmetry axis. The truncation index is N=17 for all cases.

Fig. 2
Fig. 2

Relative error between the transition matrices computed by the null-field method with localized and distributed vector spherical functions and by the conventional method for prolate spheroids with large semiaxis ksa=10 and small semiaxis varying between ksb=2 and ksb=5. The refractive index of the particle is M=1.5, and the truncation index is N=17.

Fig. 3
Fig. 3

Normalized differential scattering cross section at a scattering angle of 180° versus truncation index. The particle is a prolate spheroid with ksa=10, ksb=1, and M=1.5. The incident field is a p-polarized plane wave traveling along the z axis, which corresponds to the particle symmetry axis. The curves correspond to the azimuthal mode m=1.

Fig. 4
Fig. 4

Relative symmetry error of the transition matrix for the azimuthal mode m=1 for an oblate spheroid with large semiaxis ksb=10 and small semiaxis varying between ksa=1 and ksa=5. The refractive index of the particle is M=1.5. The incident field is a p-polarized plane wave traveling along the z axis, which corresponds to the particle symmetry axis. The truncation index is N=17 for all cases.

Fig. 5
Fig. 5

Relative error between the transition matrices computed by the null-field method with localized and distributed vector spherical functions for oblate spheroids with large semiaxis ksb=10 and small semiaxis varying between ksa=3 and ksa=5. The refractive index of the particle is M=1.5, and the truncation index is N=17.

Fig. 6
Fig. 6

Relative symmetry error of the transition matrix for the azimuthal mode m=1 for a prolate spheroid with ksa=10 and ksb=2 when the refractive index varies from M=1.5 to M=4. The incident field is a p-polarized plane wave traveling along the z axis, which corresponds to the particle symmetry axis. The values of the truncation index are N=17, 18, 20, 24, 26, 28, and 30 and correspond to values of the refractive index M=1.5, 1.75, 2.0, 2.5, 3.0, 3.5, and 4.0, respectively.

Fig. 7
Fig. 7

Normalized differential scattering cross section at a scattering angle of 180° versus truncation index for a prolate spheroid with ksa=10, ksb=2, and M=4. The incident field is a p-polarized plane wave traveling along the z axis, which corresponds to the particle symmetry axis. The curves correspond to the azimuthal mode m=1.

Equations (51)

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

×Et=jkμtHt,
×Ht=-jkεtEt,
n×Ei-n×Es=n×E0,
n×Hi-n×Hs=n×H0,
S(e-e0)Ψν3+jμsεs (h-h0)Φν3dS=0
S(e-e0)Φν3+jμsεs (h-h0)Ψν3dS=0,
ν=1, 2,,
Mmn1,3(kx)=Dmnzn(kr)×jm Pn|m|(cos θ)sin θ eθ-dPn|m|(cos θ)dθ eϕexp(jmϕ)
Nmn1,3(kx)=Dmnn(n+1)×zn(kr)kr Pn|m|(cos θ)exp(jmϕ)er+[krzn(kr)]kr dPn|m|(cos θ)dθ eθ+jm Pn|m|(cos θ)sin θ eϕexp(jmϕ),
Dmn=2n+14n(n+1)  (n-|m|)!(n+|m|)!;
Mmn1,3(kx)=Mm,|m|+l1,3[k(x-zne3)],
xR3-{zne3}n=1,
Nmn1,3(kx)=Nm,|m|+l1,3[k(x-zne3)],
xR3-{zne3}n=1,
Mni1,3(kx)=m(xn±, x, τni±),xR3-{xn±}n=1,
Nni1,3(kx)=n(xn±, x, τni±),xR3-{xn±}n=1,
m(x, y, a)=1k2 a(x)×yg(x, y, k),
n(x, y, a)=1k y×m(x, y, a),xy,
Mn1,3(kx)=1k ×[ϕn±(x)x],xR3-{xn±}n=1,
Nn1,3(kx)=1k ×Mn1,3(kx),xR3-{xn±}n=1,
ϕn±(x)=g(xn±, x, k),n=1, 2,,
e(y)=μ=1aμn×Ψμ1(kiy)+bμn×Φμ1(kiy),yS,
h(y)=-jεiμi μ=1aμn×Φμ1(kiy)+bμn×Ψμ1(kiy),
yS.
Es(x)=ν=1fνMν3(ksx)+gνNν3(ksx),
fν=jks2π Se(y)Nν¯1(ksy)+jμsεs h(y)Mν¯1(ksy)dS(y)
gν=jks2π Se(y)Mν¯1(ksy)+jμsεs h(y)Nν¯1(ksy)dS(y).
E0(x)=ν=1aν0Mν1(ksx)+bν0Nν1(ksx),
H0(x)=-jεsμs ν=1aν0Nν1(ksx)+bν0Mν1(ksx).
fνgν=Taν0bν0.
T=BA-1A0,
X=Xνμ11Xνμ12Xνμ21Xνμ22,ν, μ=1, 2,,
TN=BNAN-1A0N,
T-mn-mnij=Tmnmnji.
Tmnmnij=δmmTmnmnij,
Tmnmnij=(-1)i+jT-mn-mnij.
Tmnmnij=(-1)i+jTmnmnji.
σm=i=12n=1Nn>nN|Tmnmnii-Tmnmnii|/|Tmnmnii|+n,n=1N|Tmnmn12+Tmnmn21|/|Tmnmn12|[%],
εm=i,j=12n,n=1N(Tmnmnij,A-Tmnmnij,B)/Tmnmnij,A[%].
Aνμ11=S[(n×Ψμ1)Ψν3+M(n×Φμ1)Φν3]dS,
Aνμ12=S[(n×Φμ1)Ψν3+M(n×Ψμ1)Φν3]dS,
Aνμ21=S[(n×Ψμ1)Φν3+M(n×Φμ1)Ψν3]dS,
Aνμ22=S[(n×Φμ1)Φν3+M(n×Ψμ1)Ψν3]dS,
Bνμ11=jks2π S[(n×Ψμ1)Nν¯1+M(n×Φμ1)Mν¯1]dS,
Bνμ12=jks2π S[(n×Φμ1)Nν¯1+M(n×Ψμ1)Mν¯1]dS,
Bνμ21=jks2π S[(n×Ψμ1)Mν¯1+M(n×Φμ1)Nν¯1]dS,
Bνμ22=jks2π S[(n×Φμ1)Mν¯1+M(n×Ψμ1)Nν¯1]dS,
A0νμ11=S[(n×Mμ1)Ψν3+(n×Nμ1)Φν3]dS,
A0νμ12=S[(n×Nμ1)Ψν3+(n×Mμ1)Φν3]dS,
A0νμ21=S[(n×Mμ1)Φν3+(n×Nμ1)Ψν3]dS,
A0νμ22=S[(n×Nμ1)Φν3+(n×Mμ1)Ψν3]dS,

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