Abstract

We study a numerical method of calculating the near field of ensembles of arbitrary spheres by extending Mie theory. A recursive method based on the orders of scattering is presented. This method represents a concise way to calculate the near field of aggregates of any number of arbitrary spheres. Numerical examples are given to show its validity.

© 2004 Optical Society of America

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References

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  1. M. Ohtsu, Near-Field Nano-Optics (Kluwer/Plenum, New York, 2000).
  2. R. Hillenbrand, F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000).
    [CrossRef] [PubMed]
  3. J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
    [CrossRef]
  4. For SERS reviews, see M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
    [CrossRef]
  5. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
    [CrossRef]
  6. M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
    [CrossRef]
  7. L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
    [CrossRef]
  8. H. X. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
    [CrossRef] [PubMed]
  9. X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
    [CrossRef] [PubMed]
  10. U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters (Springer, New York, 1995), pp. 155–173.
  11. G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  12. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. 1. Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [CrossRef]
  13. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef] [PubMed]
  14. D. W. Mackowski, M. I. Mishchenko, “Calculation of total cross sections of multiple-shere clusters,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  15. K. A. Fuller, “Optical resonances and 2-sphere system,” Appl. Opt. 30, 4716–4731 (1991).
    [CrossRef] [PubMed]
  16. H. X. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312, 411–419 (2003).
    [CrossRef]
  17. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  18. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  19. O. R. Cruzan, “Translational additional theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  20. H. X. Xu, M. Käll, “Polarization dependent surface-enhanced Raman spectroscopy of isolated silver nanoaggregates,” ChemPhysChem 4, 1001–1005 (2003).
    [CrossRef] [PubMed]
  21. P. B. Johanson, R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2003 (2)

H. X. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312, 411–419 (2003).
[CrossRef]

H. X. Xu, M. Käll, “Polarization dependent surface-enhanced Raman spectroscopy of isolated silver nanoaggregates,” ChemPhysChem 4, 1001–1005 (2003).
[CrossRef] [PubMed]

2002 (4)

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

H. X. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

2000 (1)

R. Hillenbrand, F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000).
[CrossRef] [PubMed]

1999 (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

1997 (1)

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

1996 (1)

1995 (1)

1991 (1)

1985 (1)

For SERS reviews, see M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

1972 (1)

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

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. 1. Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

1962 (1)

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

1961 (1)

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

1908 (1)

G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Bian, R. X.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. 1. Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Christy, R. W.

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

Cruzan, O. R.

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

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

Fuller, K. A.

Gu, B. Y.

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

Gunnarsson, L.

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

Haslett, T.

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

Hillenbrand, R.

R. Hillenbrand, F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000).
[CrossRef] [PubMed]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

Johanson, P. B.

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

Käll, M.

H. X. Xu, M. Käll, “Polarization dependent surface-enhanced Raman spectroscopy of isolated silver nanoaggregates,” ChemPhysChem 4, 1001–1005 (2003).
[CrossRef] [PubMed]

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

H. X. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

Keilmann, F.

R. Hillenbrand, F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000).
[CrossRef] [PubMed]

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

Kreibig, U.

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters (Springer, New York, 1995), pp. 155–173.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. 1. Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Mackowski, D. W.

Mie, G.

G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

Moskovits, M.

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

For SERS reviews, see M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Novotny, L.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

Ohtsu, M.

M. Ohtsu, Near-Field Nano-Optics (Kluwer/Plenum, New York, 2000).

Prikulis, J.

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

Stein, S.

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tay, L. L.

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

Vollmer, M.

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters (Springer, New York, 1995), pp. 155–173.

Wang, R. Z.

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

Wang, X. H.

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

Xie, X. S.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

Xu, H. X.

H. X. Xu, M. Käll, “Polarization dependent surface-enhanced Raman spectroscopy of isolated silver nanoaggregates,” ChemPhysChem 4, 1001–1005 (2003).
[CrossRef] [PubMed]

H. X. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312, 411–419 (2003).
[CrossRef]

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

H. X. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

Xu, Y. L.

Yang, G. Z.

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

Yang, J.

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (2)

Chem. Rev. (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2975 (1999).
[CrossRef]

ChemPhysChem (1)

H. X. Xu, M. Käll, “Polarization dependent surface-enhanced Raman spectroscopy of isolated silver nanoaggregates,” ChemPhysChem 4, 1001–1005 (2003).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. 1. Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. Appl. Phys. (1)

J. Prikulis, H. X. Xu, L. Gunnarsson, M. Käll, “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6213 (2002).
[CrossRef]

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

Phys. Lett. A (1)

H. X. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312, 411–419 (2003).
[CrossRef]

Phys. Rev. B (1)

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

Phys. Rev. Lett. (4)

R. Hillenbrand, F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000).
[CrossRef] [PubMed]

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

H. X. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

X. H. Wang, R. Z. Wang, B. Y. Gu, G. Z. Yang, “Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with pseudogaps,” Phys. Rev. Lett. 88, 093902 (2002).
[CrossRef] [PubMed]

Q. Appl. Math. (2)

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

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

Rev. Mod. Phys. (1)

For SERS reviews, see M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Top. Appl. Phys. (1)

M. Moskovits, L. L. Tay, J. Yang, T. Haslett, “SERS and the single molecule,” Top. Appl. Phys. 82, 215–226 (2002).
[CrossRef]

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Ohtsu, Near-Field Nano-Optics (Kluwer/Plenum, New York, 2000).

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters (Springer, New York, 1995), pp. 155–173.

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Figures (2)

Fig. 1
Fig. 1

Local intensity enhancement distribution I / I 0 in logarithmic scale in the plane of the wave vector k and the electric field E through the centers of three identified Ag spheres with radius R = 35   nm at the incident wavelength 514.5 nm for different incident polarizations as illustrated by the arrows of E and k. The dielectric function of Ag is obtained from Johansson and Christy.21 The number of the multipoles is L = 16 and the orders of scattering N os = 200 , which are enough for the convergence of the calculations.

Fig. 2
Fig. 2

Local intensity distribution I / I 0 in the logarithmic scale in the plane through the centers of five different Ag spheres; 1 ( R = 55   nm ) , 2 ( R = 50   nm ) , 3 ( R = 45   nm ) , 4 ( R = 45   nm ) , and 5 ( R = 50   nm ) at the incident wavelength 514.5 nm, with the wave vector k perpendicular to this plane and the different incident polarization E illustrated by the white arrows. The number of the multipoles is L = 16 and the orders of scattering N os = 200 .

Equations (35)

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i E l = n = 1 m = - n n p = 1 2   i C mnp l | mn 1 p ,
s E = l = 1 L   s E l = l = 1 L n = 1 m = - n n p = 1 2   s C mnp l | mn 3 p ,
s C mnp l = L T l ( i C μ υ q h ,   a ν h ,   b υ h ,   lh A mn μ ν ,   lh B mn μ ν ) .
H i l = k i ω μ n = 1 m = - n n p = 1 p 2   i C mnp l | mn 1 p ,
H s = k i ω μ l = 1 L H s l = k i ω μ l = 1 L n = 1 m = - n n p = 1 p 2   s C mnp l | mn 3 p ,
X p l = [ i C - 11 p l i C 01 p l i C 11 p l i C - 22 p l i C - 12 p l i C 021 l i C 12 p l i C 22 p l i C - MNp l i C - ( M - 1 ) Np l i C MNp l ] ,
Y jp l = [ | - 11 jp l | 01 jp l | 11 jp l | - 22 jp l | 22 jp l | - MNjp l | MNjp l ] ,
Z 1 l = [ b 1 l b 1 l b 1 l b 2 l b 2 l b 2 l b 2 l b 2 l b N l b N l ] D ,
Z 2 l = [ a 1 l a 1 l a 1 l a 2 l a 2 l a 2 l a 2 l a 2 l a N l a N l ] D ,
U 1 l = [ d 1 l d 1 l d 1 l d 2 l d 2 l d 2 l d 2 l d 2 l d N l d N l ] D ,
U 2 l = [ c 1 l c 1 l c 1 l c 2 l c 2 l c 2 l c 2 l c 2 l c N l c N l ] D ,
ih A ¯ = lh A - 11 - 11 lh A 01 - 11 lh A 11 - 11 lh A - 22 - 11 lh A 22 - 11 lh A - MN - 11 lh A MN - 11 lh A - 11 01 lh A 01 01 lh A 11 01 lh A - 22 01 . . . lh A - 11 11 lh A 01 11 lh A 11 11 lh A - 22 11 . . . lh A - 11 - 22 lh A 01 - 22 lh A 11 - 22 lh A - 22 - 22 . lh A - MN - 22 lh A MN - 22 lh A - · 11 22 . . . . lh A 22 22 . . lh A - · 11 - MN . . lh A - 22 - MN . . . lh A - · 11 MN . . lh A - 22 MN . . lh A MN MN ,
lh B ¯ = lh B - 11 - 11 lh B 01 - 11 lh B 11 - 11 lh B - 22 - 11 lh B 22 - 11 lh B - MN - 11 lh B MN - 11 lh B - 11 01 lh B 01 01 lh B 11 01 lh B - 22 01 . . . lh B - 11 11 lh B 01 11 lh B 11 11 lh B - 22 11 . . . lh B - 11 - 22 lh B 01 - 22 lh B 11 - 22 lh B - 22 - 22 . lh B - MN - 22 lh B MN - 22 lh B - · 11 22 . . . . lh B 22 22 . . lh B - · 11 - MN . . lh B - 22 - MN . . . lh B - · 11 MN . . lh B - 22 MN . . lh B MN MN ,
G l = [ X 1 l X 2 l ] ,
l W 1 E = [ Y 11 l Y 12 l ] T ,
l W 1 H = [ Y 12 l Y 11 l ] T ,
l W 3 E = [ Y 31 l Y 32 l ] T ,
l W 3 H = [ Y 32 l Y 31 l ] T ,
S l = Z 1 l 0 0 Z 2 l ,
P l = U 1 l 0 0 U 2 l ,
Ω lh = lh A ¯ lh B ¯ lh B ¯ lh A ¯ .
1 T = GS ,
E s = 1 TW 3 E ,
H s = k i ω μ   1 TW 3 H .
2 T 1 = ( 1 T 1 +   1 T 2 Ω 21 S 1 ) i = 0 N os ( Ω 12 S 2 Ω 21 S 1 ) i ,
2 T 2 = ( 1 T 2 +   1 T 1 Ω 12 S 2 ) i = 0 N os ( Ω 21 S 1 Ω 12 S 2 ) i ,
3 T 3 = ( 1 T 3 +   2 T 1 Ω 13 S 3 ) i = 0 N os ( Ω 31 S 1 Ω 13 S 3 ) i + ( 1 T 3 +   2 T 2 Ω 23 S 3 ) i = 0 N os ( Ω 32 S 2 Ω 23 S 3 ) i - 1 T 3 .
3 T 1 = ( 1 T 1 +   2 T 3 Ω 31 S 1 ) i = 0 N os ( Ω 13 S 3 Ω 31 S 1 ) i + ( 1 T 1 +   2 T 2 Ω 21 S 1 ) i = 0 N os ( Ω 12 S 1 Ω 21 S 1 ) i - 1 T 1 ,
3 T 2 = ( 1 T 2 +   2 T 1 Ω 12 S 2 ) i = 0 N os ( Ω 21 S 1 Ω 12 S 2 ) i + ( 1 T 3 +   2 T 3 Ω 32 S 2 ) i = 0 N os ( Ω 23 S 3 Ω 32 S 2 ) i - 1 T 2 .
L T L = j = 1 L - 1 ( 1 T L +   L - 1 T j Ω jL S L ) i = 0 N os ( Ω Lj S j Ω jL S L ) i - ( L - 2 )   1 T L .
L T l = j = 1 l L ( 1 T l +   L - 1 T j Ω jl S l ) i = 0 N os ( Ω lj S j Ω jl S l ) i - ( L - 2 )   1 T l .
E s = l = 1 L E s l = l = 1 L   L T l   l W 3 E ,
H s = l = 1 L H s l = k i ω μ l = 1 L   L T l   l W 3 H .
E p l = L T l S l   P l   l W 1 E ,
H p l = k i ω μ L T l S l   P l   l W 1 H .

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