Abstract

An exact, analytical solution to the problem of point-source radiation in the presence of a sphere with an eccentric spherical inclusion has been obtained by combined use of the dyadic Green’s function formalism and the indirect mode-matching technique. The end result of the analysis is a set of linear equations for the vector wave amplitudes of the electric Green’s dyad. The point source can be anywhere, even within the aforesaid nonspherical body, and there is no restriction with regard to the electrical properties in any part of space. Several checks confirm that this solution obeys the energy conservation and reciprocity principles. Numerical results are presented for an electric Hertz dipole radiating from within an acrylic sphere, which contains an eccentric spherical cavity.

© 2007 Optical Society of America

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  1. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, 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. Microwave Theory Tech. 42, 2302-2310 (1994).
    [Crossref]
  3. K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave Theory Tech. 48, 2003-2013 (2000).
    [Crossref]
  4. 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]
  5. J. Kim and Y. Rahmat-Samii, "Implanted antennas inside a human body: simulations, designs, and characterizations," IEEE Trans. Microwave Theory Tech. 52, 1934-1943 (2004).
    [Crossref]
  6. 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] [PubMed]
  7. H. Mosallaei and Y. Rahmat-Samii, "Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization," IEEE Trans. Antennas Propag. 49, 60-69 (2001).
    [Crossref]
  8. J. G. Fikioris and N. K. Uzunoglu, "Scattering from an eccentrically stratified dielectric sphere," J. Opt. Soc. Am. 69, 1359-1366 (1979).
    [Crossref]
  9. F. Borghese, P. Denti, and R. Saija, "Optical properties of spheres containing a spherical eccentric inclusion," J. Opt. Soc. Am. A 9, 1327-1335 (1992).
    [Crossref]
  10. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, "Indirect mode matching solution to scattering from a dielectric sphere with an eccentric inclusion," J. Opt. Soc. Am. A 11, 1859-1866 (1994).
    [Crossref]
  11. J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
    [Crossref]
  12. M. P. Ioannidou and D. P. Chrissoulidis, "Electromagnetic-wave scattering by a sphere with multiple spherical inclusions," J. Opt. Soc. Am. A 19, 505-512 (2002).
    [Crossref]
  13. 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 excitation," IEEE Trans. Microwave Theory Tech. 44, 1963-1973 (1996).
    [Crossref]
  14. K. Lim and S. S. Lee, "Analysis of electromagnetic scattering from an eccentric multilayered sphere," IEEE Trans. Antennas Propag. 43, 1325-1326 (1995).
  15. B. Stout, C. Andraud, S. Stout, and J. Lafait, "Absorption in multiple-scattering systems of coated spheres," J. Opt. Soc. Am. A 20, 1050-1059 (2003).
    [Crossref]
  16. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II, pp. 1864-1891.
  17. S. Stein, "Addition theorems for spherical vector wave functions," Q. Appl. Math. 19, 15-24 (1961).
  18. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
  19. 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]
  20. J. D. Kanellopoulos and J. G. Fikioris, "Resonant frequencies in an electromagnetic eccentric spherical cavity," Q. Appl. Math. 37, 51-66 (1979).
  21. Y. L. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
    [Crossref]

2004 (2)

J. Kim and Y. Rahmat-Samii, "Implanted antennas inside a human body: simulations, designs, and characterizations," IEEE Trans. Microwave Theory Tech. 52, 1934-1943 (2004).
[Crossref]

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] [PubMed]

2003 (1)

2002 (1)

2001 (1)

H. Mosallaei and Y. Rahmat-Samii, "Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization," IEEE Trans. Antennas Propag. 49, 60-69 (2001).
[Crossref]

2000 (3)

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave Theory Tech. 48, 2003-2013 (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]

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]

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 (1)

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 excitation," IEEE Trans. Microwave Theory Tech. 44, 1963-1973 (1996).
[Crossref]

1995 (2)

K. Lim and S. S. Lee, "Analysis of electromagnetic scattering from an eccentric multilayered sphere," IEEE Trans. Antennas Propag. 43, 1325-1326 (1995).

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
[Crossref]

1994 (2)

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, "Indirect mode matching solution to scattering from a dielectric sphere with an eccentric inclusion," J. Opt. Soc. Am. A 11, 1859-1866 (1994).
[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. Microwave Theory Tech. 42, 2302-2310 (1994).
[Crossref]

1992 (1)

1979 (2)

J. G. Fikioris and N. K. Uzunoglu, "Scattering from an eccentrically stratified dielectric sphere," J. Opt. Soc. Am. 69, 1359-1366 (1979).
[Crossref]

J. D. Kanellopoulos and J. G. Fikioris, "Resonant frequencies in an electromagnetic eccentric spherical cavity," Q. Appl. Math. 37, 51-66 (1979).

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.

Borghese, F.

Chrissoulidis, D. P.

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] [PubMed]

Cruzan, O. R.

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

Denti, P.

Feshbach, H.

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

Fikioris, J. G.

J. D. Kanellopoulos and J. G. Fikioris, "Resonant frequencies in an electromagnetic eccentric spherical cavity," Q. Appl. Math. 37, 51-66 (1979).

J. G. Fikioris and N. K. Uzunoglu, "Scattering from an eccentrically stratified dielectric sphere," J. Opt. Soc. Am. 69, 1359-1366 (1979).
[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.

Kakogiannos, N. B.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
[Crossref]

Kanellopoulos, J. D.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
[Crossref]

J. D. Kanellopoulos and J. G. Fikioris, "Resonant frequencies in an electromagnetic eccentric spherical cavity," Q. Appl. Math. 37, 51-66 (1979).

Karafotias, A.

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave 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. Microwave 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. Microwave 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, 1325-1326 (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. Microwave 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. Microwave 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, 1325-1326 (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] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, 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 lens antennas: radiation characteristics and design optimization," IEEE Trans. Antennas Propag. 49, 60-69 (2001).
[Crossref]

Nikita, K. S.

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave 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. Microwave Theory Tech. 52, 1934-1943 (2004).
[Crossref]

H. Mosallaei and Y. Rahmat-Samii, "Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization," IEEE Trans. Antennas Propag. 49, 60-69 (2001).
[Crossref]

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]

Roumeliotis, J. A.

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
[Crossref]

Saija, R.

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 excitation," IEEE Trans. Microwave Theory Tech. 44, 1963-1973 (1996).
[Crossref]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, "Indirect mode matching solution to scattering from a dielectric sphere with an eccentric inclusion," J. Opt. Soc. Am. A 11, 1859-1866 (1994).
[Crossref]

Stamatakos, G. S.

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave 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 Press, 1993).

Uzunoglu, N. K.

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave Theory Tech. 48, 2003-2013 (2000).
[Crossref]

J. G. Fikioris and N. K. Uzunoglu, "Scattering from an eccentrically stratified dielectric sphere," J. Opt. Soc. Am. 69, 1359-1366 (1979).
[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]

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. Microwave 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)

H. Mosallaei and Y. Rahmat-Samii, "Nonuniform Luneburg and two-shell lens antennas: radiation characteristics and design optimization," IEEE Trans. Antennas Propag. 49, 60-69 (2001).
[Crossref]

K. Lim and S. S. Lee, "Analysis of electromagnetic scattering from an eccentric multilayered sphere," IEEE Trans. Antennas Propag. 43, 1325-1326 (1995).

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. Microwave Theory Tech. (5)

J. A. Roumeliotis, N. B. Kakogiannos, and J. D. Kanellopoulos, "Scattering from a sphere of small radius embedded into a dielectric one," IEEE Trans. Microwave Theory Tech. 43, 155-168 (1995).
[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 excitation," IEEE Trans. Microwave Theory Tech. 44, 1963-1973 (1996).
[Crossref]

J. Kim and Y. Rahmat-Samii, "Implanted antennas inside a human body: simulations, designs, and characterizations," IEEE Trans. Microwave Theory Tech. 52, 1934-1943 (2004).
[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. Microwave Theory Tech. 42, 2302-2310 (1994).
[Crossref]

K. S. Nikita, G. S. Stamatakos, N. K. Uzunoglu, and A. Karafotias, "Analysis of the interaction between a layered spherical human head model and a finite-length dipole," IEEE Trans. Microwave Theory Tech. 48, 2003-2013 (2000).
[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. (1)

J. Opt. Soc. Am. A (4)

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] [PubMed]

Q. Appl. Math. (3)

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).

J. D. Kanellopoulos and J. G. Fikioris, "Resonant frequencies in an electromagnetic eccentric spherical cavity," Q. Appl. Math. 37, 51-66 (1979).

Other (2)

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

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

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Figures (5)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Radiated power versus cavity size.

Fig. 3
Fig. 3

Radiated power versus source-cavity separation.

Fig. 4
Fig. 4

Antenna patterns on the x z and y z planes.

Fig. 5
Fig. 5

Antenna patterns on the x y plane.

Tables (4)

Tables Icon

Table 1 Point Source Radiating from within an Acrylic Sphere with an Air Bubble a

Tables Icon

Table 2 Point Source Radiating from within an Acrylic Sphere with a Metallic Core a

Tables Icon

Table 3 Points Selected for the Reciprocity Checks

Tables Icon

Table 4 Numerical Results for Both Sides of the Reciprocity Formula (31)

Equations (39)

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

G e ( i s ) = δ s i G e ( s ) + G e , i ( i s ) ,
G e ( 2 s ) = δ s 2 G e ( s ) + G e , 0 ( 2 s ) + G e , 1 ( 2 s ) ,
× × G e ( s ) ( r i r i ) k s 2 G e ( s ) ( r i r i ) = J δ ( r i r i ) ,
G e ( s ) ( r i , r i ) = 1 k s 2 r ̂ i r ̂ i δ ( r i r i ) + j k s 4 π n = 1 m = n n c m n [ M m n ( 1 ̃ ) ( k s r i ) M m n ( 3 ̃ ) ( k s r i ) + N m n ( 1 ̃ ) ( k s r i ) N m n ( 3 ̃ ) ( k s r i ) ] ,
M m n ( ı ) ( k r ) = z n ( ı ) ( k r ) [ j m P n m ( cos θ ) sin θ θ ̂ d P n m ( cos θ ) d θ φ ̂ ] e j m φ ,
N m n ( ı ) ( k r ) = { n ( n + 1 ) k r z n ( ı ) ( k r ) P n m ( cos θ ) r ̂ + η n ( ı ) ( k r ) [ d P n m ( cos θ ) d θ θ ̂ + j m P n m ( cos θ ) sin θ φ ̂ ] } e j m φ ,
V F ( M N ) , m n ( 1 ) ( k r ) F ( N M ) , m n ( 1 ) ( k r ) d V = 0 ,
V F ( M N ) , m n ( 1 ) ( k r ) F ( M N ) , m n ( 1 ) ( k r ) d V = ( 1 ) m π 2 k 2 2 n ( n + 1 ) 2 n + 1 δ ( k k ) δ n n δ m , m ,
G e , i ( i s ) ( r i , r ) = j k s 4 π a , m , n c m n [ δ i 0 F a , m n ( 3 ) ( k i r i ) A a , m n ( i ) ( r ) + δ i 1 F a , m n ( 1 ) ( k i r i ) C a , m n ( i ) ( r ) ] ,
G e , i ( 2 s ) ( r i , r ) = j k s 4 π a , m , n c m n [ δ i 0 F a , m n ( 1 ) ( k 2 r i ) C a , m n ( 2 ) ( r ) + δ i 1 F a , m n ( 3 ) ( k 2 r i ) A a , m n ( 2 ) ( r ) ] ,
G e ( i s ) ( r i , r ) = δ s i 1 k s 2 r ̂ i r ̂ i δ ( r i r i ) + j k s 4 π a , m , n c m n [ δ i 0 F a , m n ( 3 ) ( k i r i ) A a , m n ( i ) ( r ) + δ i 1 F a , m n ( 1 ) ( k i r i ) C a , m n ( i ) ( r ) + δ s i F a , m n ( 1 ̃ ) ( k s r i ) F a , m n ( 3 ̃ ) ( k s r i ) ] ,
G e ( 2 s ) ( r , r ) = δ s 2 1 k s 2 r ̂ i r ̂ i δ ( r i r i ) + j k s 4 π a , m , n c m n [ F a , m n ( 3 ) ( k 2 r 1 ) A a , m n ( 2 ) ( r ) + F a , m n ( 1 ) ( k 2 r 0 ) C a , m n ( 2 ) ( r ) + δ s 2 F a , m n ( 1 ̃ ) ( k s r i ) F a , m n ( 3 ̃ ) ( k s r i ) ] ,
r ̂ i × G e ( i s ) = r ̂ i × G e ( 2 s ) , r i = a i ,
r ̂ i × ( × G e ( i s ) ) = r ̂ i × ( × G e ( 2 s ) ) , r i = a i ,
S F ( M N ) , m n ( ı ) ( k r ) × F ( M N ) , m n ( ı ) ( k r ) r ̂ d S = 0 ,
S F ( M N ) , m n ( ı ) ( k r ) × F ( N M ) , m n ( ı ) ( k r ) r ̂ d S = ± 4 π a 2 ( 1 ) m n ( n + 1 ) 2 n + 1 z n ( ı ) ( k a ) η n ( ı ) ( k a ) δ m , m δ n n .
A ( M N ) , k l ( 0 ) ( r ) = I ( M N ) , l ( 1 , 3 , 3 ) ( k 2 , k 0 , k 2 , a 0 ) C ( M N ) , k l ( 2 ) ( r ) δ s 0 I ( M N ) , l ( 1 , 3 , 3 ) ( k 0 , k 0 , k 2 , a 0 ) F ( M N ) , k l ( 3 ) ( k s r 0 ) ,
C ( M N ) , k l ( 1 ) ( r ) = I ( M N ) , l ( 3 , 1 , 1 ) ( k 2 , k 1 , k 2 , a 1 ) A ( M N ) , k l ( 2 ) ( r ) δ s 1 I ( M N ) , l ( 3 , 1 , 1 ) ( k 1 , k 1 , k 2 , a 1 ) F ( M N ) , k l ( 1 ) ( k s r 1 ) ,
m , n [ A m n , 1 k l ( k 2 d 01 ) A ( M N ) , m n ( 2 ) ( r ) + B m n , 1 k l ( k 2 d 01 ) A ( N M ) , m n ( 2 ) ( r ) ] + I ( M N ) , l ( 1 , 3 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) C ( M N ) , k l ( 2 ) ( r ) = δ s 0 I ( M N ) , l ( 1 , 3 , 3 ) ( k 0 , k 2 , k 0 , a 0 ) F ( M N ) , k l ( 3 ) ( k s r 0 ) δ s 2 F ( M N ) , k l ( 1 ) ( k s r 0 ) ,
I ( M N ) , l ( 3 , 1 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) A ( M N ) , k l ( 2 ) ( r ) + m , n [ A m n , 1 k l ( k 2 d 01 ) C ( M N ) , m n ( 2 ) ( r ) + B m n , 1 k l ( k 2 d 01 ) C ( N M ) , m n ( 2 ) ( r ) ] = δ s 1 I ( M N ) , l ( 3 , 1 , 1 ) ( k 1 , k 2 , k 1 , a 1 ) F ( M N ) , k l ( 1 ) ( k s r 1 ) δ s 2 F ( M N ) , k l ( 3 ) ( k s r 1 ) ,
I M , n ( ı 1 , ı 2 , ı 3 ) ( k 1 , k 2 , k 3 , a ) = k 3 z n ( ı 1 ) ( k 1 a ) η n ( ı 3 ) ( k 3 a ) k 1 η n ( ı 1 ) ( k 1 a ) z n ( ı 3 ) ( k 3 a ) k 3 z n ( ı 2 ) ( k 2 a ) η n ( ı 3 ) ( k 3 a ) k 2 η n ( ı 2 ) ( k 2 a ) z n ( ı 3 ) ( k 3 a ) ,
I N , n ( ı 1 , ı 2 , ı 3 ) ( k 1 , k 2 , k 3 , a ) = k 1 z n ( ı 1 ) ( k 1 a ) η n ( ı 3 ) ( k 3 a ) k 3 η n ( ı 1 ) ( k 1 a ) z n ( ı 3 ) ( k 3 a ) k 2 z n ( ı 2 ) ( k 2 a ) η n ( ı 3 ) ( k 3 a ) k 3 η n ( ı 2 ) ( k 2 a ) z n ( ı 3 ) ( k 3 a ) ,
E ( r ) = j ω μ 0 V G e ( f s ) ( r , r ) J ( r ) d V ,
H ( r ) = V [ × G e ( f s ) ( r , r ) ] J ( r ) d V .
G e ( 0 s ) ( r 1 , r 1 ) = δ s 0 1 k s 2 r ̂ 1 r ̂ 1 δ ( r 1 r 1 ) + j k s 4 π a , m , n c m n { δ s 1 I a , n ( 3 , 3 , 1 ) ( k 1 , k 0 , k 1 , a 1 ) F a , m n ( 3 ) ( k 0 r 1 ) F a , m n ( 1 ) ( k 1 r 1 ) δ s 0 [ I a , n ( 1 , 3 , 1 ) ( k 0 , k 0 , k 1 , a 1 ) F a , m n ( 3 ) ( k 0 r 1 ) F a , m n ( 3 ) ( k 0 r 1 ) + F a , m n ( 1 ̃ ) ( k s r 1 ) F a , m n ( 3 ̃ ) ( k s r 1 ) ] } ,
G e ( 1 s ) ( r 1 , r 1 ) = δ s 1 1 k s 2 r ̂ 1 r ̂ 1 δ ( r 1 r 1 ) + j k s 4 π a , m , n c m n { δ s 0 I a , n ( 1 , 1 , 3 ) ( k 0 , k 1 , k 0 , a 1 ) F a , m n ( 1 ) ( k 1 r 1 ) F a , m n ( 3 ) ( k 0 r 1 ) δ s 1 [ I a , n ( 3 , 1 , 3 ) ( k 1 , k 1 , k 0 , a 1 ) F a , m n ( 1 ) ( k 1 r 1 ) F a , m n ( 1 ) ( k 1 r 1 ) + F a , m n ( 1 ̃ ) ( k s r 1 ) F a , m n ( 3 ̃ ) ( k s r 1 ) ] } ,
A a , k l ( 2 ) ( r ) = δ s 0 I a , l ( 1 , 1 , 3 ) ( k 0 , k 2 , k 0 , a 0 ) F a , k l ( 3 ) ( k s r ) δ s 1 I a , l ( 3 , 1 , 1 ) ( k 1 , k 2 , k 1 , a 1 ) F a , k l ( 1 ) ( k s r ) I a , l ( 3 , 1 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) I a , l ( 3 , 1 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) + δ s 2 [ F a , k l ( 3 ) ( k s r ) I a , l ( 3 , 1 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) F a , k l ( 1 ) ( k s r ) ] I a , l ( 3 , 1 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) I a , l ( 3 , 1 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) ,
C a , k l ( 2 ) ( r ) = δ s 0 I a , l ( 1 , 3 , 3 ) ( k 0 , k 2 , k 0 , a 0 ) F a , k l ( 3 ) ( k s r 0 ) δ s 1 I a , l ( 3 , 3 , 1 ) ( k 1 , k 2 , k 1 , a 1 ) F a , k l ( 1 ) ( k s r 1 ) I a , l ( 1 , 3 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) I a , l ( 1 , 3 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) + δ s 2 [ I a , l ( 1 , 3 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) F a , k l ( 3 ) ( k s r 1 ) F a , k l ( 1 ) ( k s r 0 ) ] I a , l ( 1 , 3 , 3 ) ( k 2 , k 2 , k 0 , a 0 ) I a , l ( 1 , 3 , 1 ) ( k 2 , k 2 , k 1 , a 1 ) ,
J ( r 0 ) = I 0 l δ ( r 0 r 0 ) δ ( θ 0 θ 0 ) δ ( φ 0 φ 0 ) r 0 2 sin θ 0 e ̂ ,
P del = 1 2 Re { S 0 ( E 2 × H 2 * ) n ̂ d S } ,
P abs = 1 2 Re { S 1 ( E 1 × H 1 * ) n ̂ 1 d S 1 } ,
P rad = 1 2 Re { S ( E 0 × H 0 * ) n ̂ d S } ,
I 1 = V a J i a E i b d V = V b J j b E j a d V = I 2 ,
F ( M N ) , m n ( ı ) ( k r 1 ) = l = 1 k = l l [ A k l , ı ̃ m n ( k d 12 ) F ( M N ) , k l ( 1 ̃ ) + B k l , ı ̃ m n ( k d 12 ) F ( N M ) , k l ( 1 ̃ ) ] ,
[ A k l , i m n ( k d 12 ) B k l , i m n ( k d 12 ) ] = ( 1 ) k p [ a ( m , n k , l p ) a ( n , l , p ) z p ( ı ) ( k d 12 ) P p m k ( cos θ 12 ) a ( m , n k , l p + 1 , p ) b ( n , l , p ) z p + 1 ( ı ) ( k d 12 ) P p + 1 m k ( cos θ 12 ) ] e j ( m k ) φ 12 ,
[ a ( n , l , p ) b ( n , l , p ) ] = j p + l n 2 l + 1 2 l ( l + 1 ) [ l ( l + 1 ) + n ( n + 1 ) p ( p + 1 ) p 2 ( l n ) 2 ( l + n + 1 ) 2 p 2 ] .
[ A k l , ı m n ( k d ) B k l , ı m n ( k d ) ] = ( 1 ) n + l [ A k l , ı m n ( k d ) B k l , ı m n ( k d ) ] ,
c m n [ A k l , ı m n ( k d ) B k l , ı m n ( k d ) ] = c k l [ A m n , ı k l ( k d ) B m n , ı k l ( k d ) ] .
( n l p m k m k )

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