Abstract

The interaction between metal particles was calculated based on the discrete dipole method. The calculated results revealed that the multiple interactions in a collection of metal particles are dominated by the interaction between the neighboring particles. Thus, a two-particle approximation was deduced and applied to estimate the optical absorption and near-field enhancement property of Ag–Si3N4 composite films. The calculated results indicated that the interaction between Ag particles will induce extra absorption near the surface plasmon resonance peak and improve the near-field enhancement property of composite films. Calculated results were consistent with the experimental results.

© 2006 Optical Society of America

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  1. G. Mie, "Beitrage zur optik trüber medien speziell kolloidaler metallösungen," Ann. Phys. 25, 377-455 (1908).
    [CrossRef]
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  3. M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
    [CrossRef]
  4. A. A. Kokhanovsky and A. Macke, "Integral light-scattering and absorption characteristics of large, nonspherical particles," Appl. Opt. 36, 8785-8790 (1997).
    [CrossRef]
  5. N. C. Skaropoulos and H. W. J. Russchenberg, "Light scattering by arbitrarily oriented rotationally symmetric particles," J. Opt. Soc. Am. A 19, 1583-1591 (2002).
    [CrossRef]
  6. E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
  7. K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
    [CrossRef]
  8. H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
    [CrossRef]
  9. J. I. Gittleman and B. Abeles, "Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals," Phys. Rev. B 15, 3273-3275 (1977).
    [CrossRef]
  10. P. Sheng, "Theory for the dielectric function of granular composite media," Phys. Rev. Lett. 45, 60-63 (1980).
    [CrossRef]
  11. D. Stroud, "The effective medium approximations: some recent developments," Superlatt. Microstruct. 23, 567-573 (1998).
    [CrossRef]
  12. A. C. Levasseur-Regourd and E. Hadamcik, "Light scattering by irregular dust particles in the solar system: observations and interpretation by laboratory measurements," J. Quant. Spectrosc. Radiat. Transfer. 79, 903-910 (2003).
    [CrossRef]
  13. K. Lumme, J. Rahola, and J. W. Hovenier, "Light scattering by dense clusters of spheres," Icarus 126, 455-469 (1997).
    [CrossRef]
  14. L. E. McNeil and R. H. French, "Multiple scattering from rutile TiO2 particles," Acta Mater. 48, 4571-4576 (2000).
    [CrossRef]
  15. L. E. McNeil and R. H. French, "Light scattering from red pigment particles: multiple scattering in a strongly absorbing system," J. Appl. Phys. 89, 283-293 (2001).
    [CrossRef]
  16. A. Giusto, R. Saija, M. A. Latì, P. Denti, F. Borghese, and O. I. Sindoni, "Optical properties of high-density dispersions of particles: application to intralipid solutions," Appl. Opt. 42, 4375-4380 (2003).
    [CrossRef] [PubMed]
  17. V. Twersky, "On propagation in random media of discrete scatterers," Proc. Symp. Appl. Math. 16, 84-116 (1964).
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  19. V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
    [CrossRef]
  20. V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
    [CrossRef]
  21. A. K. Sarychev and V. M. Shalaev, "Electromagnetic field fluctuation and optical nonlinearities in metal-dielectric composites," Phys. Rep. 335, 275-371 (2000).
    [CrossRef]
  22. V. M. Shalaev, Nonlinear Optics of Random Media (Springer, 2000), pp. 24-27.
  23. D. R. Ou, J. Zhu, R. J. Zhu, and J. Wang, "Local near-field enhancement of random Sb-SiN films," Appl. Opt. 43, 3073-3077 (2004).
    [CrossRef] [PubMed]
  24. D. R. Ou, J. Zhu, J. H. Zhao, R. J. Zhu, and J. Wang, "Influence of the interaction between metal particles on optical properties of Ag-Si3N4 composite films. I. Experiment," Appl. Opt. 45, 1244-1248 (2006).
    [CrossRef] [PubMed]
  25. C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1976).
  26. S. Link and M. A. EI-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic in gold and silver nanodots and nanorods," J. Phys. Chem. B 103, 8410-8426 (1999).
    [CrossRef]
  27. L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
    [CrossRef]

2006 (1)

2004 (1)

2003 (3)

A. Giusto, R. Saija, M. A. Latì, P. Denti, F. Borghese, and O. I. Sindoni, "Optical properties of high-density dispersions of particles: application to intralipid solutions," Appl. Opt. 42, 4375-4380 (2003).
[CrossRef] [PubMed]

H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
[CrossRef]

A. C. Levasseur-Regourd and E. Hadamcik, "Light scattering by irregular dust particles in the solar system: observations and interpretation by laboratory measurements," J. Quant. Spectrosc. Radiat. Transfer. 79, 903-910 (2003).
[CrossRef]

2002 (1)

2001 (2)

L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
[CrossRef]

L. E. McNeil and R. H. French, "Light scattering from red pigment particles: multiple scattering in a strongly absorbing system," J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

2000 (2)

A. K. Sarychev and V. M. Shalaev, "Electromagnetic field fluctuation and optical nonlinearities in metal-dielectric composites," Phys. Rep. 335, 275-371 (2000).
[CrossRef]

L. E. McNeil and R. H. French, "Multiple scattering from rutile TiO2 particles," Acta Mater. 48, 4571-4576 (2000).
[CrossRef]

1999 (1)

S. Link and M. A. EI-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic in gold and silver nanodots and nanorods," J. Phys. Chem. B 103, 8410-8426 (1999).
[CrossRef]

1998 (1)

D. Stroud, "The effective medium approximations: some recent developments," Superlatt. Microstruct. 23, 567-573 (1998).
[CrossRef]

1997 (2)

1996 (3)

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
[CrossRef]

1994 (1)

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

1980 (1)

P. Sheng, "Theory for the dielectric function of granular composite media," Phys. Rev. Lett. 45, 60-63 (1980).
[CrossRef]

1977 (1)

J. I. Gittleman and B. Abeles, "Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals," Phys. Rev. B 15, 3273-3275 (1977).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

1964 (1)

V. Twersky, "On propagation in random media of discrete scatterers," Proc. Symp. Appl. Math. 16, 84-116 (1964).

1908 (1)

G. Mie, "Beitrage zur optik trüber medien speziell kolloidaler metallösungen," Ann. Phys. 25, 377-455 (1908).
[CrossRef]

Abeles, B.

J. I. Gittleman and B. Abeles, "Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals," Phys. Rev. B 15, 3273-3275 (1977).
[CrossRef]

Armstrong, R. L.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

Bohren, C.

C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1976).

Borghese, F.

Denti, P.

EI-Sayed, M. A.

S. Link and M. A. EI-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic in gold and silver nanodots and nanorods," J. Phys. Chem. B 103, 8410-8426 (1999).
[CrossRef]

French, R. H.

L. E. McNeil and R. H. French, "Light scattering from red pigment particles: multiple scattering in a strongly absorbing system," J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
[CrossRef]

L. E. McNeil and R. H. French, "Multiple scattering from rutile TiO2 particles," Acta Mater. 48, 4571-4576 (2000).
[CrossRef]

Gittleman, J. I.

J. I. Gittleman and B. Abeles, "Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals," Phys. Rev. B 15, 3273-3275 (1977).
[CrossRef]

Giusto, A.

Hadamcik, E.

A. C. Levasseur-Regourd and E. Hadamcik, "Light scattering by irregular dust particles in the solar system: observations and interpretation by laboratory measurements," J. Quant. Spectrosc. Radiat. Transfer. 79, 903-910 (2003).
[CrossRef]

Hanuska, A. R.

L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
[CrossRef]

Hovenier, J. W.

K. Lumme, J. Rahola, and J. W. Hovenier, "Light scattering by dense clusters of spheres," Icarus 126, 455-469 (1997).
[CrossRef]

Huffman, D. R.

C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1976).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Kim, W.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

Kimura, H.

H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
[CrossRef]

Kokhanovsky, A. A.

Kolokolova, L.

H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
[CrossRef]

Latì, M. A.

Levasseur-Regourd, A. C.

A. C. Levasseur-Regourd and E. Hadamcik, "Light scattering by irregular dust particles in the solar system: observations and interpretation by laboratory measurements," J. Quant. Spectrosc. Radiat. Transfer. 79, 903-910 (2003).
[CrossRef]

Link, S.

S. Link and M. A. EI-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic in gold and silver nanodots and nanorods," J. Phys. Chem. B 103, 8410-8426 (1999).
[CrossRef]

Lumme, K.

K. Lumme, J. Rahola, and J. W. Hovenier, "Light scattering by dense clusters of spheres," Icarus 126, 455-469 (1997).
[CrossRef]

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Macke, A.

Makowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
[CrossRef]

Mann, I.

H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
[CrossRef]

Markel, V. A.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
[CrossRef]

McNeil, L. E.

L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
[CrossRef]

L. E. McNeil and R. H. French, "Light scattering from red pigment particles: multiple scattering in a strongly absorbing system," J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

L. E. McNeil and R. H. French, "Multiple scattering from rutile TiO2 particles," Acta Mater. 48, 4571-4576 (2000).
[CrossRef]

Mie, G.

G. Mie, "Beitrage zur optik trüber medien speziell kolloidaler metallösungen," Ann. Phys. 25, 377-455 (1908).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
[CrossRef]

Ou, D. R.

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Poliakov, E. Y.

V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Rahola, J.

K. Lumme, J. Rahola, and J. W. Hovenier, "Light scattering by dense clusters of spheres," Icarus 126, 455-469 (1997).
[CrossRef]

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Russchenberg, H. W. J.

Saija, R.

Sarychev, A. K.

A. K. Sarychev and V. M. Shalaev, "Electromagnetic field fluctuation and optical nonlinearities in metal-dielectric composites," Phys. Rep. 335, 275-371 (2000).
[CrossRef]

Shalaev, V. M.

A. K. Sarychev and V. M. Shalaev, "Electromagnetic field fluctuation and optical nonlinearities in metal-dielectric composites," Phys. Rep. 335, 275-371 (2000).
[CrossRef]

V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
[CrossRef]

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

V. M. Shalaev, Nonlinear Optics of Random Media (Springer, 2000), pp. 24-27.

Sheng, P.

P. Sheng, "Theory for the dielectric function of granular composite media," Phys. Rev. Lett. 45, 60-63 (1980).
[CrossRef]

Sindoni, O. I.

Skaropoulos, N. C.

Stechel, E. B.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

Stroud, D.

D. Stroud, "The effective medium approximations: some recent developments," Superlatt. Microstruct. 23, 567-573 (1998).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
[CrossRef]

Twersky, V.

V. Twersky, "On propagation in random media of discrete scatterers," Proc. Symp. Appl. Math. 16, 84-116 (1964).

Wang, J.

Zhao, J. H.

Zhu, J.

Zhu, R. J.

Acta Mater. (1)

L. E. McNeil and R. H. French, "Multiple scattering from rutile TiO2 particles," Acta Mater. 48, 4571-4576 (2000).
[CrossRef]

Ann. Phys. (1)

G. Mie, "Beitrage zur optik trüber medien speziell kolloidaler metallösungen," Ann. Phys. 25, 377-455 (1908).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

H. Kimura, L. Kolokolova, and I. Mann, "Optical properties of cometary dust: constraints from numerical studies on light scattering by aggregate particles," Astron. Astrophys. 407, L5-L8 (2003).
[CrossRef]

Astrophys. J. (2)

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Icarus (1)

K. Lumme, J. Rahola, and J. W. Hovenier, "Light scattering by dense clusters of spheres," Icarus 126, 455-469 (1997).
[CrossRef]

J. Appl. Phys. (2)

L. E. McNeil and R. H. French, "Light scattering from red pigment particles: multiple scattering in a strongly absorbing system," J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

L. E. McNeil, A. R. Hanuska, and R. H. French, "Near-field scattering from red pigment particles: absorption and spectral dependence," J. Appl. Phys. 89, 1898-1906 (2001).
[CrossRef]

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

J. Phys. Chem. B (1)

S. Link and M. A. EI-Sayed, "Spectral properties and relaxation dynamics of surface plasmon electronic in gold and silver nanodots and nanorods," J. Phys. Chem. B 103, 8410-8426 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer. (2)

M. I. Mishchenko, L. D. Travis, and D. W. Makowski, "T-Matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transfer. 55, 535-575 (1996).
[CrossRef]

A. C. Levasseur-Regourd and E. Hadamcik, "Light scattering by irregular dust particles in the solar system: observations and interpretation by laboratory measurements," J. Quant. Spectrosc. Radiat. Transfer. 79, 903-910 (2003).
[CrossRef]

Phys. Rep. (1)

A. K. Sarychev and V. M. Shalaev, "Electromagnetic field fluctuation and optical nonlinearities in metal-dielectric composites," Phys. Rep. 335, 275-371 (2000).
[CrossRef]

Phys. Rev. B (3)

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, and R. L. Armstrong, "Small-particle composites. I. Linear optical properties," Phys. Rev. B 53, 2425-2436 (1996).
[CrossRef]

V. M. Shalaev, E. Y. Poliakov, and V. A. Markel, "Small-particle composites. II. Nonlinear optical properties," Phys. Rev. B 53, 2437-2449 (1996).
[CrossRef]

J. I. Gittleman and B. Abeles, "Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals," Phys. Rev. B 15, 3273-3275 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

P. Sheng, "Theory for the dielectric function of granular composite media," Phys. Rev. Lett. 45, 60-63 (1980).
[CrossRef]

Proc. Symp. Appl. Math. (1)

V. Twersky, "On propagation in random media of discrete scatterers," Proc. Symp. Appl. Math. 16, 84-116 (1964).

Superlatt. Microstruct. (1)

D. Stroud, "The effective medium approximations: some recent developments," Superlatt. Microstruct. 23, 567-573 (1998).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

V. M. Shalaev, Nonlinear Optics of Random Media (Springer, 2000), pp. 24-27.

C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1976).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

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Figures (5)

Fig. 1
Fig. 1

Plots of | p 1 | / | p 0 | as a function of distance r 12 / r a , calculated with φ 12 = 0 . The dielectric constants of metal particle and surrounding dielectric are as shown.

Fig. 2
Fig. 2

Average values of equivalent dipole moments ( p aver ) varied as a function of average distance R aver / r a , calculated by using the coupled dipole equation.

Fig. 3
Fig. 3

Relationship between the average distance R aver / r a and the metal fraction f m of composite films, given by computer simulation.

Fig. 4
Fig. 4

Absorption coefficient of Ag– Si 3 N 4 composite films: (a), (b) calculated without regard to the interactions between Ag particles; (c), (d) calculated with the interaction considered by using the two-particle approximation. The average radius and volume fraction of Ag particles are as shown.

Fig. 5
Fig. 5

Simulation of absorption spectrums for silver rods in Si 3 N 4 , with the aspect ratio R varying from 1.4 to 2.

Tables (1)

Tables Icon

Table 1 Maximum Value of |p ave|/|p 0| and the Coupling Distances Calculated from the Collections and from Two Particles with φ12 = 0

Equations (9)

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

p 1 = α 0 ( E 0 + E s 21 ) ,
p 2 = α 0 ( E 0 + E s 12 ) ,
α 0 = 4 π r a 3 ε d ε m ε d ε m + 2 ε d ,
p 1 x = p 2 x = p 0 1 2 ( 1 3 cos 2 φ 12 ) β 1 2 β 2 + β 1 ,
p 1 y = p 2 y = p 0 3 2 sin 2 φ 12 β 2 β 2 + β 1 ,
p 1 z = p 2 z = 0 ,
β = ε m ε d ε m + 2 ε d ( r a r 12 ) 3 ,
p i = α 0 ( E 0 + j i E sij ) .
α = N ( k w × imag { p aver ε d E 0 } + k w 4 6 π | p aver ε d E 0 | 2 ) ,

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