Abstract

We extend the model of Chance, Prock and Silbey[1] and analytically determine the Poynting vector in the direction perpendicular to the plane of a multilayer organic device. The result is used to predict the spatial profile of Förster energy transfer, the radiative output of an organic light emitting device, and to calculate the efficiency of surface plasmon polariton-mediated energy transfer across a thin silver film.

© 2007 Optical Society of America

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  1. R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near metal interfaces," in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, 1978), Vol. 37,pp. 1-65.
  2. V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
    [CrossRef]
  3. M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001).
    [CrossRef]
  4. M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002).
    [CrossRef]
  5. T. Förster, "Transfer mechanisms of electronic excitation," Disc. Faraday Soc. 27, 7-17 (1959).
  6. C.-T. Tai, Dyadic Green's functions in electromagnetic theory (IEEE Press, 1994).
  7. R. L. Hartman, "Green dyadic calculations for inhomogeneous optical media," J. Opt. Soc. Am. A 17, 1067-1076 (2000).
    [CrossRef]
  8. R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
    [CrossRef]
  9. M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
    [CrossRef]
  10. P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004).
    [CrossRef] [PubMed]
  11. P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000).
    [CrossRef] [PubMed]
  12. D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
    [CrossRef]
  13. D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
    [CrossRef]
  14. H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
    [CrossRef]
  15. D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
    [CrossRef] [PubMed]

2004 (1)

P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004).
[CrossRef] [PubMed]

2003 (1)

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

2002 (3)

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
[CrossRef] [PubMed]

M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002).
[CrossRef]

2001 (1)

M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001).
[CrossRef]

2000 (2)

R. L. Hartman, "Green dyadic calculations for inhomogeneous optical media," J. Opt. Soc. Am. A 17, 1067-1076 (2000).
[CrossRef]

P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000).
[CrossRef] [PubMed]

1999 (2)

R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
[CrossRef]

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

1998 (1)

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

1996 (1)

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

1959 (1)

T. Förster, "Transfer mechanisms of electronic excitation," Disc. Faraday Soc. 27, 7-17 (1959).

Agranovich, V. M.

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

Andrew, P.

P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004).
[CrossRef] [PubMed]

P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000).
[CrossRef] [PubMed]

Baldo, M. A.

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

Barnes, W. L.

P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004).
[CrossRef] [PubMed]

P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000).
[CrossRef] [PubMed]

Basko, D. M.

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

Bassani, F.

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

Bulovic, V.

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

Burrows, P. E.

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

Cohen, S. M.

R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
[CrossRef]

Forrest, S. R.

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

Förster, T.

T. Förster, "Transfer mechanisms of electronic excitation," Disc. Faraday Soc. 27, 7-17 (1959).

Garbuzov, D. Z.

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

Gu, G.

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

Hartman, R. L.

R. L. Hartman, "Green dyadic calculations for inhomogeneous optical media," J. Opt. Soc. Am. A 17, 1067-1076 (2000).
[CrossRef]

R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
[CrossRef]

Holmes, R. J.

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

Iizumi, Y.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

Kafafi, Z. H.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

Khalfin, V. B.

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

Kido, J.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

La Rocca, G. C.

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

Leung, P. T.

R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
[CrossRef]

Lu, M. H.

M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002).
[CrossRef]

M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001).
[CrossRef]

Magde, D.

D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
[CrossRef] [PubMed]

Mattoussi, H.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

Merritt, C. D.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

Murata, H.

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

Segal, M.

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

Seybold, P. G.

D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
[CrossRef] [PubMed]

Soos, Z. G.

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

Sturm, J. C.

M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002).
[CrossRef]

M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001).
[CrossRef]

Wong, R.

D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001).
[CrossRef]

Chem. Phys. Lett. (1)

D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996).
[CrossRef]

Disc. Faraday Soc. (1)

T. Förster, "Transfer mechanisms of electronic excitation," Disc. Faraday Soc. 27, 7-17 (1959).

J. Appl. Phys. (2)

M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002).
[CrossRef]

H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999).
[CrossRef]

J. Chem. Phys. (1)

R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999).
[CrossRef]

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

Photochem. Photobiol. (1)

D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002).
[CrossRef] [PubMed]

Phys. Rev. B (2)

M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003).
[CrossRef]

V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998).
[CrossRef]

Physica Status Solidi A (1)

D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002).
[CrossRef]

Science (2)

P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004).
[CrossRef] [PubMed]

P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000).
[CrossRef] [PubMed]

Other (2)

R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near metal interfaces," in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, 1978), Vol. 37,pp. 1-65.

C.-T. Tai, Dyadic Green's functions in electromagnetic theory (IEEE Press, 1994).

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Figures (8)

Fig. 1.
Fig. 1.

Coefficients and indexing of the general multilayer structure used for the modeling.

Fig. 2.
Fig. 2.

(a) Förster energy transfer as a function of z position and normalized to the surface-parallel wavevector. The excited molecule is embedded within an infinite film of 1% CuPC in Alq3. The dipole is located at r = z = 0 and the emission wavelength is λ= 535 nm. Bright features correspond to a faster energy transfer. (b) The z dependence shows a 1/z 4 power law consistent with Eq. (22) and a Förster radius of R 0 = 38Å. At λ= 535 nm, the dielectric constants for CuPC and Alq3 are: ε= 1.908 + 0.265i and ε= 2.962, respectively.

Fig. 3.
Fig. 3.

(a). The structure and experimental setup of the OLED from Segal et al. (b) The detailed measurement setup for the outcoupling measurement.[9]

Fig. 4.
Fig. 4.

(a). Angular emission profile of the perpendicular dipoles of the OLED from Segal et al.[9] Each red or blue curve corresponds to the angular emission profile into air and into glass layer, respectively, for 10 different dipole positions spaced 20Å apart in the Alq3 layer. The curves with maxima at larger angles correspond to the dipoles nearer to the metal cathode. The green circle represents the ideal Lambertian emission profile. (b) Angular emission profile for the parallel dipoles. (c) Angular emission profile for the isotropic dipoles. Emission into air closely approximates the expected Lambertian profile.

Fig. 5.
Fig. 5.

(a). Absorption of the parallel dipole energy as a function of the position and normalized surface-parallel wavevector. The dipole is located at the middle of the Alq3 layer and the emission wavelength is λ = 535 nm. Bright features correspond to a higher absorption. The green curve shows the outcoupled energy flux. (b) Same as part (a) but for perpendicular dipole. Perpendicular dashed lines divide this flux into air-outcoupled, glass-waveguided, organics-waveguided and surface plasmon polariton (SPP) portions. At λ = 535 nm, the dielectric constants for Mg, BCP, Alq3, TPD, PEDOT and ITO are: ε = 1.908 + 0.265i, ε = 2.985 + (4.11 × 10-5)i, ε = 2.962, ε = 2.985 + (4.11 × 10-5)i, ε = 2.304 + (3.33 × 10-2)i and ε = 3.295 + (3.63 × 10-2)i, respectively. Note that the dielectric constant of TPD was assumed to be equal that of BCP.

Fig. 6.
Fig. 6.

(a). The calculated distribution of the Alq3 dipole energy versus the dipole distance from the BCP layer. (b) The basic structure and the emitting pathways of the OLED of Segal et al. [9]

Fig. 7.
Fig. 7.

(a). Absorption of the parallel dipole energy as a function of the position and normalized surface-parallel wavevector. The dipole is located at the middle of the PMMA:Alq3 layer and the emission wavelength is λ = 535 nm. The green and blue curves show the outcoupled energy flux from the PMMA:R6G-air and PMMA:Alq3-glass interfaces, respectively. The blue curve is rescaled by 1/2000 to share the same y-axis with the green curve. (b) Same as part (a) but for perpendicular dipole. Perpendicular dashed lines divide the flux into air-outcoupled and glass-waveguided portions. Dielectric constants for PMMA and R6G were extracted from Ref. [10].

Fig. 8.
Fig. 8.

(a). The calculated ratio of emitted power to the total dipole energy for the structure of Andrew and Barnes.[10] The data is shown for samples without the R6G acceptor (PMMA:Alq3/Ag/PMMA) (blue curve), samples without the Alq3 donor (PMMA/Ag/PMMA:R6G) (green curve) and samples containing both Alq3 and R6G (PMMA: Alq3/Ag/PMMA:R6G) (red curve). The silver layer thickness is 60 nm. The R6G absorption and PL spectra of R6G and Alq3 are extracted from Fig. 1(d) of Andrew and Barnes.[10] (b) The energy transfer efficiency, which is the ratio of the energy absorbed by the R6G-doped PMMA layer to the total Alq3 dipole energy, versus silver thickness.

Equations (26)

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

E ( R ) = i ω μ 0 G ( R R ) · J ( R ) d 3 R′
M n κ o e ( h ) = e ihz [ n J n ( κ r ) r sin cos n ϕ r ̂ J n ( κ r ) r cos sin n ϕ ϕ ̂ ]
N n κ o e ( h ) = e ihz k j [ ih J n ( κ r ) r cos sin n ϕ r ̂ inh J n ( κ r ) r sin cos n ϕ ϕ ̂ + κ 2 J n ( κ r ) cos sin n ϕ z ̂ ]
G 0 ( R R′ ) = i 4 π 0 d κ n = 0 t = e , o 2 δ n 0 κ h s [ M tnκ ( h s ) M tnκ ( h s ) + N tnκ ( h s ) N tnκ ( h s ) M tnκ ( h s ) M tnκ ( h s ) + N tnκ ( h s ) N tnκ ( h s ) ] z 0 z 0
G j ( R R ) = i 4 π 0 d κ n = 0 t = e , o 2 δ n 0 κ h s [ c j M tn κ ( h j ) M tn κ ( h s ) + f j N tn κ ( h j ) N tn κ ( h s )
+ c j M tn κ ( h j ) M tn κ ( h s ) + f j N t n κ ( h j ) N tn κ ( h s ) ]
c j e i h j z j + c j e i h j z j = c j + 1 e i h j + 1 z j + c j + 1 e i h j + 1 z j
h j k j ( f j e i h j z j + f j e i h j z j ) = h j + 1 k j + 1 ( f j + 1 e i h j + 1 z j + f j + 1 e i h j + 1 z j )
c j h j e i h j z j + c j h j e i h j z j = h j + 1 c j + 1 e i h j + 1 z j + h j + 1 c j + 1 e i h j + 1 z j
k j f j e i h j z j + k j f j e i h j z j = k j + 1 f j + 1 e i h j + 1 z j + k j + 1 f j + 1 e i h j + 1 z j
b = e 2 k s 3 6 πmωε [ 1 + 3 q ε 2 p 0 k s 3 Im ( E 0 ) ]
b b 0 = 1 q + q { 1 + 3 2 Re [ 0 d κ κ 3 h s k s 3 ( f s + f s ) ] }
b b 0 = 1 q + q { 1 + 3 4 Re [ 0 d κ κ h s k s ( c s + c s + h s 2 k s 2 ( f s f s ) ) ] }
. S d V = S . d A S z d A .
S z = i 2 μ 0 ω [ E r ( E r z E z r ) * + E ϕ ( E ϕ z 1 r E z ϕ ) * ]
E jr = ω 2 p 0 4 π d κ κ h j k s k j h s J 0 ( κ r ) r ( f j e i h j z f j e i h j z )
E jz = i ω 2 p 0 4 π d κ κ 3 h s k s k j J 0 ( κ r ) ( f j e i h j z + f j e i h j z )
E j r = i ω 2 p 0 4 π d κ [ 1 h s J 1 ( κ r ) r ( cos ϕ sin ϕ ) ( c j e i h j z + c j e i h j z )
+ h j k j k s J 1 ( κ r ) r ( cos ϕ + sin ϕ ) ( f j e i h j z f j e i h j z )
E j ϕ = i ω 2 p 0 4 π d κ [ 1 h s J 1 ( κ r ) r ( cos ϕ + sin ϕ ) ( c j e i h j z + c j e i h j z )
+ h j k j k s J 1 ( κ r ) r ( cos ϕ sin ϕ ) ( f j e i h j z f j e i h j z ) ]
E jz = i ω 2 p 0 4 π d κ κ 2 2 k s k j J 1 ( κ r ) ( cos ϕ + sin ϕ ) ( f j e i h j z + f j e i h j z )
Re ( S z , j * dA ) = 3 q 4 Re [ 0 du u 3 ( ε j ) * 1 u 2 ε j ( ε j ε s u 2 ) 1 2 ( f j e i h j z f j e i h j z ) ( f j e i h j z + f j e i h j z ) * ]
Re ( S z , j dA * ) = 3 q 8 Re [ 0 d u u ( ε j ) * ε j ( ε j ε s u 2 ) 1 2 ( f j e i h j z f j e i h j z ) ( f j e i h j z + f j e i h j z ) *
+ 0 d u u ( ( ε j ε s u 2 ) 1 2 ) * 1 u 2 ( c j e i h j z + c j e i h j z ) ( c j e i h j z c j e i h j z ) * ]
1 b 0 d u Re ( dA d S z , j , * dz ) = b ET ( z ) b 0 = 0 rdr 0 2 π d ϕ ρ R 0 6 ( r 2 + z 2 ) 3 = ρ π 2 R 0 6 z 4

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