Abstract

A semi-analytical method for computing the electric field surrounding a finite linear chain of metal nanospheres and nanospheroids is described. In treating chains or clusters of spheres, a common approach is to use the spherical-harmonic addition theorem to relate the multipole expansion coefficients between different spheres. A method is described here that avoids the use of spherical-harmonic addition theorems, which are not applicable to spheroidal chains. Simulations are given that illustrate the large field enhancements that can occur in the gaps between silver nanoparticles arising from plasmon resonances.

© 2008 Optical Society of America

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  1. T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
    [CrossRef]
  2. T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Analyt. Chem. 17, 557-570 (1998).
    [CrossRef]
  3. M. Wabuyele and T. Vo-Dinh, “Detection of HIV Type 1 DNA sequence using plasmonics nanoprobes,” Anal. Chem. 77, 7810-7815 (2005).
    [CrossRef] [PubMed]
  4. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783-826 (1985).
    [CrossRef]
  5. G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering--Physics and Applications, K.Kneipp, M.Moskovits, and H.Kneipp, eds., Top. Appl. Phys. 103, 19-46 (2006).
    [CrossRef]
  6. H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
    [CrossRef]
  7. Selected Papers on Surface-Enhanced Raman Scattering, M.Kerker, ed., SPIE Milestone Series, Vol. MS 10 (Proc. SPIE, 1990).
  8. J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22, 4950-4959 (1980).
    [CrossRef]
  9. M. Schmeits and L. Dambly, “Fast electron scattering by bispherical surface-plasmon modes,” Phys. Rev. B 44, 12706-12711 (1991).
    [CrossRef]
  10. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331-1333 (1998).
    [CrossRef]
  11. K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
    [CrossRef]
  12. M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
    [CrossRef]
  13. U. Evra and D. J. Bergman, “Lifetime of nano-plasmonic states,” Proc. SPIE 6324, 63240H (2006).
    [CrossRef]
  14. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995), Appendix D.
  15. Translational addition theorems applicable to spheroids have been derived in , but, at least in our opinion, their complexity renders them extremely unwieldy at best and unuseable at worst. They involve double infinite summations with coefficients that themselves require numerical evaluation.
  16. B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. 38, 143-158 (1980).
  17. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
  18. The set of parameters used in our paper is the same set employed in the commercial finite-difference time-domain electromagnetic code Fullwave 6.0 (RSoft Design Group, Inc.), which were derived from the data in . These values are:Δϵk={1759.471,135.344,258.1946,22.90436,1749.06,11756.18},ak={1,1,1,1,1,1},bk={0.243097,19.68071,2.289161,0.329194,4.639097,12.25} andck={0,17.07876,515.022,1718.357,2116.092,10559.42}.
  19. M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23-L25 (1978).
    [CrossRef]
  20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), p. 1284.
  21. P. F. Liao and A. Wokaun, “Lightning rod effect in surface enhanced Raman scattering,” J. Chem. Phys. 76, 751-752 (1982).
    [CrossRef]
  22. K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
    [CrossRef]
  23. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 599.

2006 (1)

U. Evra and D. J. Bergman, “Lifetime of nano-plasmonic states,” Proc. SPIE 6324, 63240H (2006).
[CrossRef]

2005 (1)

M. Wabuyele and T. Vo-Dinh, “Detection of HIV Type 1 DNA sequence using plasmonics nanoprobes,” Anal. Chem. 77, 7810-7815 (2005).
[CrossRef] [PubMed]

2004 (1)

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

2003 (1)

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
[CrossRef]

2000 (1)

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

1998 (2)

1997 (1)

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

1991 (1)

M. Schmeits and L. Dambly, “Fast electron scattering by bispherical surface-plasmon modes,” Phys. Rev. B 44, 12706-12711 (1991).
[CrossRef]

1985 (1)

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

1984 (1)

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

1982 (1)

P. F. Liao and A. Wokaun, “Lightning rod effect in surface enhanced Raman scattering,” J. Chem. Phys. 76, 751-752 (1982).
[CrossRef]

1980 (2)

B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. 38, 143-158 (1980).

J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22, 4950-4959 (1980).
[CrossRef]

1978 (1)

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23-L25 (1978).
[CrossRef]

Aizpurua, J.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Apell, P.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Ausloos, M.

J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22, 4950-4959 (1980).
[CrossRef]

Aussenegg, F. R.

Begun, G. M.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Bergman, D. J.

U. Evra and D. J. Bergman, “Lifetime of nano-plasmonic states,” Proc. SPIE 6324, 63240H (2006).
[CrossRef]

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
[CrossRef]

Caola, M. J.

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23-L25 (1978).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995), Appendix D.

Dambly, L.

M. Schmeits and L. Dambly, “Fast electron scattering by bispherical surface-plasmon modes,” Phys. Rev. B 44, 12706-12711 (1991).
[CrossRef]

Dasari, R. R.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Evra, U.

U. Evra and D. J. Bergman, “Lifetime of nano-plasmonic states,” Proc. SPIE 6324, 63240H (2006).
[CrossRef]

Feld, M. S.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 599.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), p. 1284.

Gerardy, J. M.

J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22, 4950-4959 (1980).
[CrossRef]

Hiromoto, M. Y. K.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Itzkan, I.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Kall, M.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Kneipp, H.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Kneipp, K.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Krenn, J. R.

Leitner, A.

Li, K.

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
[CrossRef]

Li, X.

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

Liao, P. F.

P. F. Liao and A. Wokaun, “Lightning rod effect in surface enhanced Raman scattering,” J. Chem. Phys. 76, 751-752 (1982).
[CrossRef]

Macphie, R. H.

B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. 38, 143-158 (1980).

Moody, R. L.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), p. 1284.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 599.

Moskovits, M.

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

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

Perelman, L. T.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Quinten, M.

Schatz, G. C.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering--Physics and Applications, K.Kneipp, M.Moskovits, and H.Kneipp, eds., Top. Appl. Phys. 103, 19-46 (2006).
[CrossRef]

Schmeits, M.

M. Schmeits and L. Dambly, “Fast electron scattering by bispherical surface-plasmon modes,” Phys. Rev. B 44, 12706-12711 (1991).
[CrossRef]

Sinha, B. P.

B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. 38, 143-158 (1980).

Stockman, M. I.

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
[CrossRef]

Van Duyne, R. P.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering--Physics and Applications, K.Kneipp, M.Moskovits, and H.Kneipp, eds., Top. Appl. Phys. 103, 19-46 (2006).
[CrossRef]

Vo-Dinh, T.

M. Wabuyele and T. Vo-Dinh, “Detection of HIV Type 1 DNA sequence using plasmonics nanoprobes,” Anal. Chem. 77, 7810-7815 (2005).
[CrossRef] [PubMed]

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Analyt. Chem. 17, 557-570 (1998).
[CrossRef]

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Wabuyele, M.

M. Wabuyele and T. Vo-Dinh, “Detection of HIV Type 1 DNA sequence using plasmonics nanoprobes,” Anal. Chem. 77, 7810-7815 (2005).
[CrossRef] [PubMed]

Wokaun, A.

P. F. Liao and A. Wokaun, “Lightning rod effect in surface enhanced Raman scattering,” J. Chem. Phys. 76, 751-752 (1982).
[CrossRef]

Xu, H.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Yang, Y.

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Young, M. A.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering--Physics and Applications, K.Kneipp, M.Moskovits, and H.Kneipp, eds., Top. Appl. Phys. 103, 19-46 (2006).
[CrossRef]

Anal. Chem. (2)

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

M. Wabuyele and T. Vo-Dinh, “Detection of HIV Type 1 DNA sequence using plasmonics nanoprobes,” Anal. Chem. 77, 7810-7815 (2005).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

P. F. Liao and A. Wokaun, “Lightning rod effect in surface enhanced Raman scattering,” J. Chem. Phys. 76, 751-752 (1982).
[CrossRef]

J. Phys. A (1)

M. J. Caola, “Solid harmonics and their addition theorems,” J. Phys. A 11, L23-L25 (1978).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22, 4950-4959 (1980).
[CrossRef]

M. Schmeits and L. Dambly, “Fast electron scattering by bispherical surface-plasmon modes,” Phys. Rev. B 44, 12706-12711 (1991).
[CrossRef]

Phys. Rev. E (1)

H. Xu, J. Aizpurua, M. Kall, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402-1-4 (2003).
[CrossRef]

K. Kneipp, Y. Yang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

Proc. SPIE (2)

M. I. Stockman, K. Li, X. Li, and D. J. Bergman, “An efficient nanolens: Self-similar chain of metal nanospheres,” Proc. SPIE 5512, 87-99 (2004).
[CrossRef]

U. Evra and D. J. Bergman, “Lifetime of nano-plasmonic states,” Proc. SPIE 6324, 63240H (2006).
[CrossRef]

Q. Appl. Math. (1)

B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. 38, 143-158 (1980).

Rev. Mod. Phys. (1)

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

Trends Analyt. Chem. (1)

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Analyt. Chem. 17, 557-570 (1998).
[CrossRef]

Other (8)

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering--Physics and Applications, K.Kneipp, M.Moskovits, and H.Kneipp, eds., Top. Appl. Phys. 103, 19-46 (2006).
[CrossRef]

Selected Papers on Surface-Enhanced Raman Scattering, M.Kerker, ed., SPIE Milestone Series, Vol. MS 10 (Proc. SPIE, 1990).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

The set of parameters used in our paper is the same set employed in the commercial finite-difference time-domain electromagnetic code Fullwave 6.0 (RSoft Design Group, Inc.), which were derived from the data in . These values are:Δϵk={1759.471,135.344,258.1946,22.90436,1749.06,11756.18},ak={1,1,1,1,1,1},bk={0.243097,19.68071,2.289161,0.329194,4.639097,12.25} andck={0,17.07876,515.022,1718.357,2116.092,10559.42}.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1953), p. 1284.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995), Appendix D.

Translational addition theorems applicable to spheroids have been derived in , but, at least in our opinion, their complexity renders them extremely unwieldy at best and unuseable at worst. They involve double infinite summations with coefficients that themselves require numerical evaluation.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), p. 599.

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Figures (6)

Fig. 1
Fig. 1

(a) Linear chain of spheres; (b) the i th and j th pair in the chain. The electric field is directed along the z axis.

Fig. 2
Fig. 2

The i th and j th pair of spheroids in a chain.

Fig. 3
Fig. 3

Cross-sectional view of a sphere (dashed) and two spheroids of volume equal to that of the sphere. The spheroids have aspect ratios of (a) 2 and (b) 4.

Fig. 4
Fig. 4

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. In all three cases, the gap is 10% of the diameter of the sphere (of unit radius). The spheroids are equal in volume to that of the sphere.

Fig. 5
Fig. 5

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. The gap is 5% of the diameter of the sphere. The spheroids are equal in volume to that of the sphere.

Fig. 6
Fig. 6

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. The gap is 2% of the diameter of the sphere. The spheroids are equal in volume to that of the sphere.

Equations (99)

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

ϵ ( ω ) = 1 + k = 1 6 Δ ϵ k a k ω 2 i b k ω + c k ,
r i = L i j 2 + r j 2 + 2 L i j r j u j ,
u i = r j u j + L i j L i j 2 + r j 2 + 2 L i j r j u j ,
r j = L i j 2 + r i 2 2 L i j r i u i ,
u j = r i u i L i j L i j 2 + r i 2 2 L i j r i u i ,
ψ p ( i ) = E 0 r i P 1 ( u i ) ,
ψ s ( i ) = n = 1 a n ( i ) 1 r i n + 1 P n ( u i ) ,
ψ i n ( i ) = n = 1 b n ( i ) r i n P n ( u i ) ,
ψ ( r ) = ψ p ( r ) + j = 1 M ψ s ( j ) ( r j ) .
[ ψ p ( i ) + ψ s ( i ) + j = 1 j i M ψ s ( j ) ] r i = R i = ψ i n ( i ) r i = R i ,
ϵ 0 r i [ ψ p ( i ) + ψ s ( i ) + j = 1 j i M ψ s ( j ) ] r i = R i = ϵ i r i ψ i n ( i ) r i = R i .
1 1 P m ( u ) P n ( u ) d u = 2 2 m + 1 δ m n
E 0 R i δ 1 m + 1 R i m + 1 a m ( i ) + j = 1 j i M n = 1 Q m n ( i j ) a n ( j ) = R i m b m ( i ) ,
E 0 R i δ 1 m ( m + 1 ) R i m + 1 a m ( i ) + j = 1 j i M n = 1 R m n ( i j ) a n ( j ) = m γ i R i m b m ( i ) ,
Q m n ( i j ) c m 1 1 [ 1 r j n + 1 P n ( u j ) ] r i = R i P m ( u i ) d u i ,
R m n ( i j ) c m R i 1 1 r i [ 1 r j n + 1 P n ( u j ) ] r i = R i P m ( u i ) d u i ,
R m n ( i j ) = m Q m n ( i j ) .
λ m ( i ) a m ( i ) + j = 1 j i M n = 1 N X m n ( i j ) a n ( j ) = p m ( i ) ,
p m ( i ) E 0 R i ( γ i 1 ) δ 1 m ,
λ m ( i ) m ( 1 + γ i ) + 1 R i 2 ,
X m n ( i j ) m ( γ i 1 ) R i m 1 Q m n ( i j ) .
ψ s ( r ) = j = 1 M n = 1 N a n ( j ) 1 r j n + 1 P n ( u j ) .
E s ( r ) = j = 1 M E s ( j ) ( r ) = j = 1 M j ψ s ( j ) ( r ) ,
ψ s ( j ) ( r ) n = 1 N a n ( j ) 1 r j n + 1 P n ( u j ) .
E s ( j ) ( r ) = j ψ s ( j ) ( r ) = [ r ̂ j ψ s ( j ) r j + θ ̂ j 1 r j ψ s ( j ) θ j ] ,
E s ( j ) ( r ) = n = 1 N a n ( j ) [ r ̂ j ( n + 1 ) r j n + 2 P n ( u j ) + θ ̂ j 1 u j 2 r j n + 2 P n ( u j ) ] ,
E r ( j ) ( r ) n = 1 N a n ( j ) ( n + 1 ) r j n + 2 P n ( u j ) ,
E θ ( j ) ( r ) n = 1 N a n ( j ) 1 u j 2 r j n + 2 P n ( u j ) ,
E s ( j ) ( r ) = r ̂ j E r ( j ) ( r ) + θ ̂ j E θ ( j ) ( r ) .
E s ( r ) = x ̂ E x ( r ) + z ̂ E z ( r ) = j = 1 M [ r ̂ j E r ( j ) ( r ) + θ ̂ j E θ ( j ) ( r ) ] .
E x ( r ) = j = 1 M [ ( x ̂ r ̂ j ) E r ( j ) ( r ) + ( x ̂ θ ̂ j ) E θ ( j ) ( r ) ] ,
E z ( r ) = j = 1 M [ ( z ̂ r ̂ j ) E r ( j ) ( r ) + ( z ̂ θ ̂ j ) E θ ( j ) ( r ) ] ,
E 0 R i ( γ i 1 ) + ( 2 + γ i ) R i 2 a 1 ( i ) + ( γ i 1 ) j = 1 j i N Q 11 ( i j ) a 1 ( j ) = 0 ,
Q 11 ( i j ) = 3 2 1 1 [ u j r j 2 ] r i = R i u i d u i = 3 2 1 1 ( R i u i L i j ) u i d u i ( L i j 2 + R i 2 2 L i j R i u i ) 3 2 ,
Q 11 ( i j ) = 2 R i L i j 3 .
E 0 R i ( γ i 1 ) + ( 2 + γ i ) R i 2 a 1 ( i ) 2 R i ( γ i 1 ) j = 1 j i M 1 L i j 3 a 1 ( j ) = 0 .
E 0 R 1 ( γ 1 1 ) + ( 2 + γ 1 ) R 1 2 a 1 ( 1 ) 2 R 1 ( γ 1 1 ) L 3 a 1 ( 2 ) = 0 ,
E 0 R 2 ( γ 2 1 ) + ( 2 + γ 2 ) R 2 2 a 1 ( 2 ) 2 R 2 ( γ 2 1 ) L 3 a 1 ( 1 ) = 0 .
a 1 ( 1 ) = E 0 R 1 3 [ ( γ 2 + 2 ) ( γ 1 1 ) + 2 ( γ 1 1 ) ( γ 2 1 ) R 2 3 L 3 ] ( γ 2 + 2 ) ( γ 1 + 2 ) 4 ( γ 1 1 ) ( γ 2 1 ) R 1 3 R 2 3 L 6 ,
a 1 ( 2 ) = E 0 R 2 3 [ ( γ 1 + 2 ) ( γ 2 1 ) + 2 ( γ 2 1 ) ( γ 1 1 ) R 1 3 L 3 ] ( γ 2 + 2 ) ( γ 1 + 2 ) 4 ( γ 1 1 ) ( γ 2 1 ) R 1 3 R 2 3 L 6 .
a 1 ( 1 ) = E 0 R 1 3 ( γ 1 1 ) γ 1 + 2 [ 1 + 2 ( R 2 L ) 3 ( γ 2 1 ) ( γ 2 + 2 ) ] ,
a 1 ( 2 ) = E 0 R 2 3 ( γ 2 1 ) γ 2 + 2 [ 1 + 2 ( R 1 L ) 3 ( γ 1 1 ) ( γ 1 + 2 ) ] .
a 1 ( 1 ) = E 0 R 1 3 ( γ 1 1 ) γ 1 + 2 = E 0 R 1 3 ( ϵ 1 ϵ 0 ) ϵ 1 + 2 ϵ 0 .
E s ( 1 ) = ψ s ( 1 ) ( r 1 , u 1 ) = a 1 ( 1 ) r 1 3 ( 2 r ̂ 1 cos θ 1 + θ ̂ 1 sin θ 1 ) ,
a 1 ( 1 ) = E 0 R 1 3 ( γ 1 1 ) γ 1 + 2 .
E z ( 1 ) = 2 a 1 ( 1 ) r 1 3 .
E ̃ 0 = E 0 + 2 a 1 ( 2 ) L 3 ,
a 1 ( 2 ) = E 0 R 2 3 ( γ 2 1 ) γ 2 + 2 .
ξ i = ( r 1 ( i ) + r 2 ( i ) ) 2 d i , η i = ( r 1 ( i ) r 2 ( i ) ) 2 d i ,
r 1 ( i ) = ( z L i + d i ) 2 + x 2 ,
r 2 ( i ) = ( z L i d i ) 2 + x 2 ,
x = d i ( ξ i 2 1 ) ( 1 η i 2 ) ,
z = d i ξ i η i + L i .
ψ = ξ ̂ 1 h ξ ψ ξ + η ̂ 1 h η ψ η + ϕ ̂ 1 h ϕ ψ ϕ ,
h ξ = d ( ξ 2 η 2 ) ( ξ 2 1 ) ,
h η = d ( ξ 2 η 2 ) ( 1 η 2 ) ,
h ϕ = d ( ξ 2 1 ) ( 1 η 2 ) .
ψ p ( i ) = E 0 d i P 1 ( ξ i ) P 1 ( η i ) ,
ψ s ( i ) = n = 1 a n ( i ) Q n ( ξ i ) P n ( η i ) ,
ψ i n ( i ) = n = 1 b n ( i ) P n ( ξ i ) P n ( η i ) .
[ ψ p ( i ) + ψ s ( i ) + j = 1 j i M ψ s ( j ) ] ξ i = ξ ¯ i = ψ i n ( i ) ξ i = ξ ¯ i ,
ϵ 0 ξ i [ ψ p ( i ) + ψ s ( i ) + j = 1 j i M ψ s ( j ) ] ξ i = ξ ¯ i = ϵ i ψ i n ( i ) ξ i ξ i = ξ ¯ i .
E 0 d i ξ ¯ i δ 1 m + Q m ( ξ ¯ i ) a m ( i ) + n = 1 Q m n ( i j ) a n ( j ) = P m ( ξ ¯ i ) b m ( i ) ,
E 0 d 1 ξ ¯ i δ 1 m + ξ ¯ i Q m ( ξ ¯ i ) a m ( i ) + n = 1 R m n ( i j ) a n ( j ) = γ i ξ ¯ i P m ( ξ ¯ i ) b m ( i ) ,
Q m n ( i j ) c m 1 1 [ Q n ( ξ j ) P n ( η j ) ] ξ i = ξ ¯ i P m ( η i ) d η i ,
R m n ( i j ) c m ξ ¯ i 1 1 ξ i [ Q n ( ξ j ) P n ( u j ) ] ξ i = ξ ¯ i P m ( η i ) d η i ,
R m n ( i j ) = ξ ¯ i P m ( ξ ¯ i ) P m ( ξ ¯ i ) Q m n ( i j ) .
λ m ( i ) a m ( i ) + j = 1 j i M n = 1 N X m n ( i j ) a n ( j ) = p m ( i ) ,
X m n ( i j ) ( γ i 1 ) P m ( ξ ¯ i ) Q m n ( i j ) ,
λ m ( i ) γ i Q m ( ξ ¯ i ) P m ( ξ ¯ i ) Q m ( ξ ¯ i ) P m ( ξ ¯ i ) ,
p m ( i ) E 0 d i ξ ¯ i ( γ i 1 ) δ 1 m .
λ m ( i ) = ( γ i 1 ) Q m ( ξ ¯ i ) P m ( ξ ¯ i ) + 1 ( ξ ¯ i 2 1 ) .
E s ( r ) = j = 1 M j ψ s ( j ) ( r ) ,
ψ s ( j ) ( r ) = n = 1 N a n ( j ) Q n ( ξ j ) P n ( η j ) .
E s ( j ) = j ψ s ( j ) = [ ξ ̂ j 1 h ξ j ψ s ( j ) ξ j + η ̂ j 1 h η j ψ s ( j ) η j ] ,
E s ( j ) = n = 1 N a n ( j ) [ ξ ̂ j 1 h ξ j Q n ( ξ j ) P n ( η j ) + η ̂ j 1 h η j Q n ( ξ j ) P n ( η j ) ] .
E s ( r ) = ξ ̂ j E ξ ( j ) + η ̂ j E η ( j ) ,
E ξ ( j ) = n = 1 N a n ( j ) 1 h ξ j Q n ( ξ j ) P n ( η j ) ,
E η ( j ) = n = 1 N a n ( j ) 1 h η j Q n ( ξ j ) P n ( η j ) .
E s ( r ) = x ̂ E x ( r ) + z ̂ E z ( r ) = j = 1 N [ ξ ̂ j E ξ ( j ) + η ̂ j E η ( j ) ] ,
E x ( r ) = n = 1 N [ ( x ̂ ξ ̂ j ) E ξ ( j ) + ( x ̂ η ̂ j ) E η ( j ) ] ,
E z ( r ) = n = 1 N [ ( z ̂ ξ ̂ j ) E ξ ( j ) + ( z ̂ η ̂ j ) E η ( j ) ] .
x ̂ ξ ̂ j = d j ξ j h η j ,
x ̂ η ̂ j = d j η j h ξ j ,
z ̂ ξ ̂ j = d j η j h ξ j ,
z ̂ η ̂ j = d j ξ j h η j ,
[ ψ j ψ i r ψ i ψ j r ] r = R i d 2 r = 0 ,
[ ψ j ψ i ψ i ψ j ] ξ ̂ i ξ i = ξ ¯ i d 2 r = 0 ,
ψ ξ ̂ = 1 h ξ ψ ξ .
2 π 1 1 [ ψ j ψ i ξ ψ i ψ j ξ ] ξ = ξ ¯ h η h ϕ h ξ d η = π d ( ξ ¯ 2 1 ) 1 1 [ ψ j ψ i ξ ψ i ψ j ξ ] ξ = ξ ¯ d η = 0 ,
P m ( ξ ¯ i ) 1 1 [ Q n ( ξ j ) P n ( η j ) ] ξ i = ξ ¯ i P m ( η i ) d η i = P m ( ξ ¯ i ) 1 1 ξ i [ Q n ( ξ j ) P n ( η j ) ] ξ i = ξ ¯ i P m ( η i ) d η i ,
ψ = x ̂ ψ x + z ̂ ψ z = ξ ̂ 1 h ξ ψ ξ + η ̂ 1 h η ψ η .
Q n ( ξ ) α n ξ n + 1 = α n d n + 1 ( ξ d ) n + 1 α n d n + 1 r n + 1
P n ( ξ ) β n ξ n = β n d n ( ξ d ) n β n d n r n ,
a ̃ m ( i ) = α m d i m + 1 a m ( i ) ,
X ̃ m n ( i j ) = d i m 1 α n β m d j m + 1 X m n ( i j ) ,
λ ̃ m ( i ) = 1 d i 2 α m β m λ m ( i ) = 2 m + 1 d i 2 λ m ( i ) ,
λ ̃ m ( i ) a ̃ m ( i ) + j = 1 j i M n = 1 N X ̃ m n ( i j ) a ̃ n ( j ) = p m ( i ) .
ψ s ( i ) ( ξ i , η i ) = n = 1 N a n ( i ) Q n ( ξ i ) P n ( η i ) = n = 1 N a ̃ n ( i ) Q n ( ξ i ) α n d i n + 1 P n ( η i ) .

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