Abstract

The electric dyadic Green’s function (dGf) of an eccentrically stratified sphere is built by use of the superposition principle, dyadic algebra, and the addition theorem of vector spherical harmonics. The end result of the analytical formulation is a set of linear equations for the unknown vector wave amplitudes of the dGf. The unknowns are calculated by truncation of the infinite sums and matrix inversion. The theory is exact, as no simplifying assumptions are required in any one of the analytical steps leading to the dGf, and it is general in the sense that any number, position, size, and electrical properties can be considered for the layers of the sphere. The point source can be placed outside of or in any lossless part of the sphere. Energy conservation, reciprocity, and other checks verify that the dGf is correct. A numerical application is made to a stratified sphere made of gold and glass, which operates as a lens.

© 2014 Optical Society of America

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References

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  1. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1993).
  2. L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
    [CrossRef]
  3. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24, 1695–1703 (2007).
    [CrossRef]
  4. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. A 24, 3437–3443 (2007).
    [CrossRef]
  5. K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325–1328 (1995).
  6. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
    [CrossRef]
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    [CrossRef]
  8. S. M. S. Reyhani and R. J. Glover, “Electromagnetic dyadic Green’s function for a multilayered homogeneous lossy dielectric spherical head model for numerical EMC investigation,” Electromagnetics 20, 141–153 (2000).
    [CrossRef]
  9. K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
    [CrossRef]
  10. F. Liu and S. Crozier, “Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods,” Phys. Med. Biol. 49, 1835–1851 (2004).
    [CrossRef]
  11. J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech. 52, 1934–1943 (2004).
    [CrossRef]
  12. H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two–shell antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Propag. 49, 60–69 (2001).
  13. C. T. Tai and R. E. Collin, “Radiation of a Hertzian dipole immersed in a dissipative medium,” IEEE Trans. Antennas Propag. 48, 1501–1506 (2000).
    [CrossRef]
  14. P. M. Morse and H. Feshbach, “Vector fields,” Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II, pp. 1864–1891.
  15. S. Stein, “Addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  16. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  17. Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
    [CrossRef]
  18. Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
    [CrossRef]
  19. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A 433, 599–614 (1991).
  20. A. P. Moneda and D. P. Chrissoulidis, “Focusing of electromagnetic waves by non–spherical, Au–Si nano–particles,” in Proceedings of XIII Mediterranean Conference on Medical and Biological Engineering and Computing 2013 (Springer, 2014), Vol. 41, pp. 837–840.
  21. P. B. Johnson and R. W. Christie, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2007 (2)

2004 (2)

F. Liu and S. Crozier, “Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods,” Phys. Med. Biol. 49, 1835–1851 (2004).
[CrossRef]

J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech. 52, 1934–1943 (2004).
[CrossRef]

2003 (1)

2001 (1)

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two–shell antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Propag. 49, 60–69 (2001).

2000 (3)

C. T. Tai and R. E. Collin, “Radiation of a Hertzian dipole immersed in a dissipative medium,” IEEE Trans. Antennas Propag. 48, 1501–1506 (2000).
[CrossRef]

S. M. S. Reyhani and R. J. Glover, “Electromagnetic dyadic Green’s function for a multilayered homogeneous lossy dielectric spherical head model for numerical EMC investigation,” Electromagnetics 20, 141–153 (2000).
[CrossRef]

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

1998 (1)

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

1996 (2)

Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

1995 (1)

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325–1328 (1995).

1994 (1)

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

1991 (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A 433, 599–614 (1991).

1972 (1)

P. B. Johnson and R. W. Christie, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Andraud, C.

Chrissoulidis, D. P.

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. A 24, 3437–3443 (2007).
[CrossRef]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Focusing of electromagnetic waves by non–spherical, Au–Si nano–particles,” in Proceedings of XIII Mediterranean Conference on Medical and Biological Engineering and Computing 2013 (Springer, 2014), Vol. 41, pp. 837–840.

Christie, R. W.

P. B. Johnson and R. W. Christie, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Collin, R. E.

C. T. Tai and R. E. Collin, “Radiation of a Hertzian dipole immersed in a dissipative medium,” IEEE Trans. Antennas Propag. 48, 1501–1506 (2000).
[CrossRef]

Crozier, S.

F. Liu and S. Crozier, “Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods,” Phys. Med. Biol. 49, 1835–1851 (2004).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Feshbach, H.

P. M. Morse and H. Feshbach, “Vector fields,” Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II, pp. 1864–1891.

Georgios, S.

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

Glover, R. J.

S. M. S. Reyhani and R. J. Glover, “Electromagnetic dyadic Green’s function for a multilayered homogeneous lossy dielectric spherical head model for numerical EMC investigation,” Electromagnetics 20, 141–153 (2000).
[CrossRef]

Ioannidou, M. P.

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christie, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Karafotias, A.

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

Kim, J.

J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech. 52, 1934–1943 (2004).
[CrossRef]

Kooi, P. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

Lafait, J.

Lee, S. S.

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325–1328 (1995).

Leong, M. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

Li, L. W.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

Lim, K.

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325–1328 (1995).

Liu, F.

F. Liu and S. Crozier, “Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods,” Phys. Med. Biol. 49, 1835–1851 (2004).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A 433, 599–614 (1991).

Moneda, A. P.

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24, 1695–1703 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. A 24, 3437–3443 (2007).
[CrossRef]

A. P. Moneda and D. P. Chrissoulidis, “Focusing of electromagnetic waves by non–spherical, Au–Si nano–particles,” in Proceedings of XIII Mediterranean Conference on Medical and Biological Engineering and Computing 2013 (Springer, 2014), Vol. 41, pp. 837–840.

Morse, P. M.

P. M. Morse and H. Feshbach, “Vector fields,” Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II, pp. 1864–1891.

Mosallaei, H.

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two–shell antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Propag. 49, 60–69 (2001).

Nikita, K. S.

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

Rahmat-Samii, Y.

J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech. 52, 1934–1943 (2004).
[CrossRef]

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two–shell antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Propag. 49, 60–69 (2001).

Reyhani, S. M. S.

S. M. S. Reyhani and R. J. Glover, “Electromagnetic dyadic Green’s function for a multilayered homogeneous lossy dielectric spherical head model for numerical EMC investigation,” Electromagnetics 20, 141–153 (2000).
[CrossRef]

Skaropoulos, N. C.

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

Stamatakos, S.

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

Stein, S.

S. Stein, “Addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Stout, B.

Stout, S.

Tai, C. T.

C. T. Tai and R. E. Collin, “Radiation of a Hertzian dipole immersed in a dissipative medium,” IEEE Trans. Antennas Propag. 48, 1501–1506 (2000).
[CrossRef]

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1993).

Uzunoglu, N. K.

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

Xu, Y. L.

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Yeo, T. S.

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

Electromagnetics (1)

S. M. S. Reyhani and R. J. Glover, “Electromagnetic dyadic Green’s function for a multilayered homogeneous lossy dielectric spherical head model for numerical EMC investigation,” Electromagnetics 20, 141–153 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325–1328 (1995).

H. Mosallaei and Y. Rahmat-Samii, “Nonuniform Luneburg and two–shell antennas: radiation characteristics and design optimization,” IEEE Trans. Antennas Propag. 49, 60–69 (2001).

C. T. Tai and R. E. Collin, “Radiation of a Hertzian dipole immersed in a dissipative medium,” IEEE Trans. Antennas Propag. 48, 1501–1506 (2000).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (4)

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane–wave and localized source exposure,” IEEE Trans. Microw. Theory Tech. 44, 1963–1973 (1996).
[CrossRef]

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech. 42, 2302–2310 (1994).
[CrossRef]

K. S. Nikita, S. Stamatakos, S. Georgios, N. K. Uzunoglu, and A. Karafotias, “Analysis of the interaction between a layered spherical human head model and a finite–length dipole,” IEEE Trans. Microw. Theory Tech. 48, 2003–2013 (2000).
[CrossRef]

J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a human body: simulations, designs, and characterizations,” IEEE Trans. Microw. Theory Tech. 52, 1934–1943 (2004).
[CrossRef]

J. Comput. Phys. (1)

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

Math. Comput. (1)

Y. L. Xu, “Fast evaluation of the Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Phys. Med. Biol. (1)

F. Liu and S. Crozier, “Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods,” Phys. Med. Biol. 49, 1835–1851 (2004).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christie, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Proc. R. Soc. Lond. Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A 433, 599–614 (1991).

Q. Appl. Math. (2)

S. Stein, “Addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Other (3)

P. M. Morse and H. Feshbach, “Vector fields,” Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II, pp. 1864–1891.

A. P. Moneda and D. P. Chrissoulidis, “Focusing of electromagnetic waves by non–spherical, Au–Si nano–particles,” in Proceedings of XIII Mediterranean Conference on Medical and Biological Engineering and Computing 2013 (Springer, 2014), Vol. 41, pp. 837–840.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1993).

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Figures (5)

Fig. 1.
Fig. 1.

Point source and eccentrically stratified sphere.

Fig. 2.
Fig. 2.

Eccentrically stratified sphere for energy conservation checks.

Fig. 3.
Fig. 3.

Eccentrically stratified sphere and source positions for reciprocity checks.

Fig. 4.
Fig. 4.

Stratified nanoparticle made of glass and gold (gray layer); excitation by distant electric Hertz dipole.

Fig. 5.
Fig. 5.

Maps of |E⃗| on E and H planes.

Tables (3)

Tables Icon

Table 1. Energy Conservation Checks—Lossless Stratified Sphere of Normalized Size k0a1=6

Tables Icon

Table 2. Energy Conservation Checks—Stratified Sphere (k0a1=6) with Metallic Core

Tables Icon

Table 3. Reciprocity Formula

Equations (23)

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

G¯¯e(fs)=δsfG¯¯e(s)+(1δf0)G¯¯e,f(fs)+(1δfN)G¯¯e,f+1(fs).
G¯¯e(s)(r⃗i,r⃗i)=1ks2r^ir^iδ(r⃗ir⃗i)+jks4πv,m,ncmnF⃗v,mn(1˜)(ksr⃗i)F⃗v,mn(3˜)(ksr⃗i),
G¯¯e,f(fs)(r⃗f,r⃗)=jks4πv,m,ncmnF⃗v,mn(1)(kfr⃗f)C⃗v,mn(f)(r⃗),
G¯¯e,f+1(fs)(r⃗f+1,r⃗)=jks4πv,m,ncmnF⃗v,mn(3)(kfr⃗f+1)A⃗v,mn(f+1)(r⃗).
G¯¯e(fs)(r⃗,r⃗)=1ks2δsfr^ir^iδ(r⃗ir⃗i)+jks4πv,m,ncmn[δsfF⃗v,mn(1˜)(ksr⃗i)F⃗v,mn(3˜)(ksr⃗i)+(1δf0)F⃗v,mn(1)(kfr⃗f)C⃗v,mn(f)(r⃗)+(1δfN)F⃗v,mn(3)(kfr⃗f+1)A⃗v,mn(f+1)(r⃗)].
SF⃗v,mn(ι)(kr⃗)×F⃗v,mn(ι)(kr⃗)·r^ds=0,
SF⃗M,mn(ι)(kr⃗)×F⃗N,mn(ι)(kr⃗)·r^ds=δm,mδnn4πa2cmnzM,n(ι)(ka)zN,n(ι)(ka),
IMN,l(1,1,3)(ki,ki1,ki,ai)C⃗MN,kl(i)(r⃗)=δi1,s(1δsN)F⃗MN,kl(3)(ksr⃗s+1)+IMN,l(3,1,3)(ki1,ki1,ki,ai)A⃗MN,kl(i)(r⃗)+(1δi1)mn[Amn,1kl(ki1d⃗i,i1)C⃗MN,mn(i1)(r⃗)+Bmn,1kl(ki1d⃗i,i1)C⃗NM,mn(i1)(r⃗)],
δis(1δs0)F⃗MN,kl(1)(ksr⃗s)+IMN,l(1,3,1)(ki,ki,ki1,ai)C⃗MN,kl(i)(r⃗)+(1δiN)mn[Amn,1kl(kid⃗i,i+1)A⃗MN,mn(i+1)(r⃗)+Bmn,1kl(kid⃗i,i+1)A⃗NM,mn(i+1)(r⃗)]=IMN,l(3,3,1)(ki1,ki,ki1,ai)A⃗MN,kl(i)(r⃗),
IMN,n(ι1,ι2,ι3)(k1,k2,k3,a)=k2zMN,n(ι1)(k1a)zNM,n(ι3)(k2a)k1zNM,n(ι1)(k1a)zMN,n(ι3)(k2a)k2zMN,n(ι2)(k1a)zNM,n(ι3)(k2a)k1zNM,n(ι2)(k1a)zMN,n(ι3)(k2a),
[(1δiN)A⃗i+1C⃗i]=[Ai00I][Ii(3,3,1)Ii(1,3,1)Ii(3,1,3)Ii(1,1,3)][I00Ai1][A⃗i(1δi1)C⃗i1]+(1δsN)[Ai00I][Ii(3,3,1)Ii(1,3,1)Ii(3,1,3)Ii(1,1,3)][0F⃗(3)](1δs0)[Ai00I][F⃗(1)0],
[0C⃗N]=[T111T112T121T122][A⃗10]δsN[F⃗(1)0]+(1δsN)[Ts+111Ts+112Ts+121Ts+122][(δs01)AsF⃗(1)F⃗(3)],
[Ti11Ti12Ti21Ti22]=[IN(3,3,1)IN(1,3,1)IN(3,1,3)IN(1,1,3)][AN100AN1][IN1(3,3,1)IN1(1,3,1)IN1(3,1,3)IN1(1,1,3)][Ai00Ai][Ii(3,3,1)Ii(1,3,1)Ii(3,1,3)Ii(1,1,3)]
[TN11TN12TN21TN22]=[IN(3,3,1)IN(1,3,1)IN(3,1,3)IN(1,1,3)].
T111A⃗1=δsNF⃗(1)+(1δsN)[(1δs0)Ts+111AsF⃗(1)Ts+112F⃗(3)],
C⃗N=T121A⃗1+(1δsN)[Ts+122F⃗(3)(1δs0)Ts+121AsF⃗(1)],
J⃗(f)(r⃗)·E⃗(fs)(r⃗)d3r⃗=J⃗(s)(r⃗)·E⃗(sf)(r⃗)d3r⃗,
A⃗i=[A⃗M,11(i)A⃗M,nmax,nmax(i)A⃗N,11(i)A⃗N,nmax,nmax(i)]T,
C⃗i=[C⃗M,11(i)C⃗M,nmax,nmax(i)C⃗N,11(i)C⃗N,nmax,nmax(i)]T
F⃗(1)=[F⃗M,11(1)F⃗M,nmax,nmax(1)F⃗N,11(1)F⃗N,nmax,nmax(1)]T,
F⃗(3)=[F⃗M,11(3)F⃗M,nmax,nmax(3)F⃗N,11(3)F⃗N,nmax,nmax(3)]T,
Ai=[A11,111All,111B11,111BLL,111A11,1LLALL,1LLB11,1LLBLL,1LLB11,111BLL,111A11,111All,111B11,1LLBLL,1LLA11,1LLALL,1LL],
Ii(i1,i2,i3)=[IM,1(ι1,ι2,ι3)00000000IM,1(ι1,ι2,ι3)00000000IM,1(ι1,ι2,ι3)00000000IM,L(ι1,ι2,ι3)00000000IN,1(ι1,ι2,ι3)00000000IN,1(ι1,ι2,ι3)00000000IN,1(ι1,ι2,ι3)000000000IN,L(ι1,ι2,ι3)];

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