Abstract

Spontaneous emission (SE) rate and the fluorescence efficiency of a bare fluorescing nanoparticle and the nanoparticle with a silver nanoshell are analyzed rigorously by using a classical electromagnetic approach with the consideration of the nonlocal effect of the silver nano-shell. The dependences of the SE rate and the fluorescence efficiency on the core-shell structure are carefully studied and the physical interpretations of the results are addressed. The results show that the SE rate of a bare nanoparticle is much slower than that in the infinite medium by almost an order of magnitude and consequently the fluorescence efficiency is usually low. However, by encapsulating the nanoparticle with a silver shell, highly efficient fluorescence can be achieved as a result of a large Purcell enhancement and high out-coupling efficiency for a well-designed core-shell structure. We also show that a higher SE rate may not offer a larger fluorescence efficiency since the fluorescence efficiency not only depends on the internal quantum yield but also the out-coupling efficiency.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  7. L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
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  14. N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
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2006 (4)

A. Burns, H. Ow and U. Wiesner, "Fluorescent core-shell silica nanoparticles: towards "Lab on a Particle" architectures for nanobiotechnology," Chem. Soc. Rev. 35, 1028 (2006).
[CrossRef] [PubMed]

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
[CrossRef] [PubMed]

F. Wang, W. Tan, Y. Zhang, X. Fan and M. Wang, "Luminescent nanomaterials for biological labeling," Nanotechnology 17, R1 (2006)
[CrossRef]

2005 (3)

A. Moroz, "Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell," Chem. Phys. 317, 1(2005).
[CrossRef]

A. Moroz, "A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere," Anna. Phys. 315, 352 (2005).
[CrossRef]

J. Enderlein, "Response to comments on "theoretical study of single molecule fluorescence in a metallic nanocavity,"Appl. Phys. Lett. 87, 066102 (2005).
[CrossRef]

2004 (3)

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, "Surface-plasmon-enhanced light emitters based on InGaN quantum wells," Nat. Mater. 3, 601(2004).
[CrossRef] [PubMed]

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

C. Radloff and N. J. Halas, "Plasmonic Response of Concentric Nanoshells," Nano. Lett. 4, 1323 (2004).
[CrossRef]

2002 (3)

C. Graf and A. van Blaaderen, "Metallodielectric colloidal core-shell particles for photonic applications," Langmuir 18, 524 (2002).
[CrossRef]

J. Enderlein, "Theoretical study of single molecule fluorescence in a metallic nanocavity,"Appl. Phys. Lett. 80,315 (2002).
[CrossRef]

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

2001 (4)

J. R. Lakowicz, "Radiative Decay Engineering: Biophysical and Biomedical Applications," Anal.Biochem. 298, 1 (2001).
[CrossRef] [PubMed]

S. Haubold, M. Haase, A. Kornowski and H. Weller, "Strongly luminescent InP/ZnS core-shell nanoparticles," Chemphyschem 2, 331 (2001).
[CrossRef]

S. R. Sershen, S. L. Westcott, J. L. West and N. J. Halas, "An opto-mechanical nanoshell-polymer composite," Appl. Phys. B-Lasers and Optics 73, 379 (2001).
[CrossRef]

A. Pack, M. Hietschold, and R. Wannemacher, "Failure of local Mie theory: optical spectra of colloidal aggregates," Opt. Commun. 194, 277 (2001).
[CrossRef]

2000 (1)

1998 (3)

K. Neyts, "Simulation of light emission from thin-film microcavities," J. Opt. Soc. Am. A 15, 962 (1998).
[CrossRef]

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, "Nanoengineering of optical resonances," Chem. Phys. Lett. 288, 243 (1998).
[CrossRef]

R. Y. Tsien, "The green fluorescent protein," Annu. Rev. Biochem. 67, 509-544 (1998).
[CrossRef] [PubMed]

1996 (1)

M. A. Hines and P. Guyot-Sionest, "Synthesis and characterization of strong luminescing ZnS-capped CdSe nanocrystals," J. Phys. Chem. 100, 468 (1996).
[CrossRef]

1990 (1)

P. T. Leung, "Decay of molecules at spherical surfaces: nonlocal effects," Phys. Rev. B 42, 7622 (1990).
[CrossRef]

1988 (1)

H. Chew, "Radiation and lifetime of atoms inside dielectric particles," Phys. Rev. A 38, 3410 (1988).
[CrossRef] [PubMed]

1980 (1)

W. Lukosz, ‘‘Theory of optical-environment-dependent spontaneous emission rates for emitters in thin layers,’’Phys. Rev. B 22, 3030 (1980).
[CrossRef]

1970 (1)

A. R. Melnyk and M. J. Harrison,"Theory of optical excitation of plasmons in metals," Phys. Rev. B 2, 835 (1970).
[CrossRef]

1946 (1)

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Achermann, M.

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

Averitt, R. D.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, "Nanoengineering of optical resonances," Chem. Phys. Lett. 288, 243 (1998).
[CrossRef]

Balet, L. P.

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

Bawendi, M. G.

J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
[CrossRef] [PubMed]

Burns, A.

A. Burns, H. Ow and U. Wiesner, "Fluorescent core-shell silica nanoparticles: towards "Lab on a Particle" architectures for nanobiotechnology," Chem. Soc. Rev. 35, 1028 (2006).
[CrossRef] [PubMed]

Chen, X. W.

X. W. Chen, W. C. H. Choy, S. He and P. C. Chui, "Accurate analysis and optimal design of top-emitting organic light emitting devices," J. Appl. Phys. (in press).
[PubMed]

Chew, H.

H. Chew, "Radiation and lifetime of atoms inside dielectric particles," Phys. Rev. A 38, 3410 (1988).
[CrossRef] [PubMed]

Choy, W. C. H.

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

X. W. Chen, W. C. H. Choy, S. He and P. C. Chui, "Accurate analysis and optimal design of top-emitting organic light emitting devices," J. Appl. Phys. (in press).
[PubMed]

Chui, P. C.

X. W. Chen, W. C. H. Choy, S. He and P. C. Chui, "Accurate analysis and optimal design of top-emitting organic light emitting devices," J. Appl. Phys. (in press).
[PubMed]

Enderlein, J.

J. Enderlein, "Response to comments on "theoretical study of single molecule fluorescence in a metallic nanocavity,"Appl. Phys. Lett. 87, 066102 (2005).
[CrossRef]

J. Enderlein, "Theoretical study of single molecule fluorescence in a metallic nanocavity,"Appl. Phys. Lett. 80,315 (2002).
[CrossRef]

Eychmuller, A.

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

Fan, X.

F. Wang, W. Tan, Y. Zhang, X. Fan and M. Wang, "Luminescent nanomaterials for biological labeling," Nanotechnology 17, R1 (2006)
[CrossRef]

Frangioni, J. V.

J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
[CrossRef] [PubMed]

Gaponik, N.

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

Graf, C.

C. Graf and A. van Blaaderen, "Metallodielectric colloidal core-shell particles for photonic applications," Langmuir 18, 524 (2002).
[CrossRef]

Guyot-Sionest, P.

M. A. Hines and P. Guyot-Sionest, "Synthesis and characterization of strong luminescing ZnS-capped CdSe nanocrystals," J. Phys. Chem. 100, 468 (1996).
[CrossRef]

Haase, M.

S. Haubold, M. Haase, A. Kornowski and H. Weller, "Strongly luminescent InP/ZnS core-shell nanoparticles," Chemphyschem 2, 331 (2001).
[CrossRef]

Halas, N. J.

C. Radloff and N. J. Halas, "Plasmonic Response of Concentric Nanoshells," Nano. Lett. 4, 1323 (2004).
[CrossRef]

S. R. Sershen, S. L. Westcott, J. L. West and N. J. Halas, "An opto-mechanical nanoshell-polymer composite," Appl. Phys. B-Lasers and Optics 73, 379 (2001).
[CrossRef]

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, "Nanoengineering of optical resonances," Chem. Phys. Lett. 288, 243 (1998).
[CrossRef]

Harrison, M. J.

A. R. Melnyk and M. J. Harrison,"Theory of optical excitation of plasmons in metals," Phys. Rev. B 2, 835 (1970).
[CrossRef]

Haubold, S.

S. Haubold, M. Haase, A. Kornowski and H. Weller, "Strongly luminescent InP/ZnS core-shell nanoparticles," Chemphyschem 2, 331 (2001).
[CrossRef]

He, S.

X. W. Chen, W. C. H. Choy, S. He and P. C. Chui, "Accurate analysis and optimal design of top-emitting organic light emitting devices," J. Appl. Phys. (in press).
[PubMed]

Hietschold, M.

A. Pack, M. Hietschold, and R. Wannemacher, "Failure of local Mie theory: optical spectra of colloidal aggregates," Opt. Commun. 194, 277 (2001).
[CrossRef]

Hines, M. A.

M. A. Hines and P. Guyot-Sionest, "Synthesis and characterization of strong luminescing ZnS-capped CdSe nanocrystals," J. Phys. Chem. 100, 468 (1996).
[CrossRef]

Hoppe, K.

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

Ivanov, S. A.

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

Kim, S. W.

J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
[CrossRef] [PubMed]

Klimov, V. I.

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

Kornowski, A.

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

S. Haubold, M. Haase, A. Kornowski and H. Weller, "Strongly luminescent InP/ZnS core-shell nanoparticles," Chemphyschem 2, 331 (2001).
[CrossRef]

Lakowicz, J. R.

J. R. Lakowicz, "Radiative Decay Engineering: Biophysical and Biomedical Applications," Anal.Biochem. 298, 1 (2001).
[CrossRef] [PubMed]

Lee, R. K.

Leung, P. T.

P. T. Leung, "Decay of molecules at spherical surfaces: nonlocal effects," Phys. Rev. B 42, 7622 (1990).
[CrossRef]

Leung, Y. P.

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

Lukosz, W.

W. Lukosz, ‘‘Theory of optical-environment-dependent spontaneous emission rates for emitters in thin layers,’’Phys. Rev. B 22, 3030 (1980).
[CrossRef]

Markov, I.

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

Melnyk, A. R.

A. R. Melnyk and M. J. Harrison,"Theory of optical excitation of plasmons in metals," Phys. Rev. B 2, 835 (1970).
[CrossRef]

Moroz, A.

A. Moroz, "Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell," Chem. Phys. 317, 1(2005).
[CrossRef]

A. Moroz, "A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere," Anna. Phys. 315, 352 (2005).
[CrossRef]

Mukai, T.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, "Surface-plasmon-enhanced light emitters based on InGaN quantum wells," Nat. Mater. 3, 601(2004).
[CrossRef] [PubMed]

Narukawa, Y.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, "Surface-plasmon-enhanced light emitters based on InGaN quantum wells," Nat. Mater. 3, 601(2004).
[CrossRef] [PubMed]

Neyts, K.

Niki, I.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, "Surface-plasmon-enhanced light emitters based on InGaN quantum wells," Nat. Mater. 3, 601(2004).
[CrossRef] [PubMed]

Ohnishi, S.

J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
[CrossRef] [PubMed]

Okamoto, K.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, "Surface-plasmon-enhanced light emitters based on InGaN quantum wells," Nat. Mater. 3, 601(2004).
[CrossRef] [PubMed]

Oldenburg, S. J.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, "Nanoengineering of optical resonances," Chem. Phys. Lett. 288, 243 (1998).
[CrossRef]

Ong, H. C.

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

Ow, H.

A. Burns, H. Ow and U. Wiesner, "Fluorescent core-shell silica nanoparticles: towards "Lab on a Particle" architectures for nanobiotechnology," Chem. Soc. Rev. 35, 1028 (2006).
[CrossRef] [PubMed]

Pack, A.

A. Pack, M. Hietschold, and R. Wannemacher, "Failure of local Mie theory: optical spectra of colloidal aggregates," Opt. Commun. 194, 277 (2001).
[CrossRef]

Pang, G. K. H.

Y. P. Leung, W. C. H. Choy, I. Markov, G. K. H. Pang, H. C. Ong, and T.I. Yuk, "Synthesis of wurtzite ZnSe nanorings by thermal evaporation," Appl. Phys. Lett 88, 183110 (2006).
[CrossRef]

Piryantinski, A.

L. P. Balet, S. A. Ivanov, A. Piryantinski, M. Achermann, and V. I. Klimov, "Inverted core/shell nanocrystals continuously tunable between type-I and type-II localization regimes," Nano. Lett. 4, 1485 (2004).
[CrossRef]

Purcell, E. M.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Radloff, C.

C. Radloff and N. J. Halas, "Plasmonic Response of Concentric Nanoshells," Nano. Lett. 4, 1323 (2004).
[CrossRef]

Rogach, A. L.

N. Gaponik, D. V. Talapin, A. L. Rogach, K. Hoppe, E. V. Shevchenko, A. Kornowski, A. Eychmuller and H. Weller, "Thiol-Capping of CdTe nanocrystals:an alternative to organometallic synthetic routes," J. Phys. Chem. B 106, 7177 (2002).
[CrossRef]

Scherer, A.

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J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
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J. P. Zimmer, S. W. Kim, S. Ohnishi, E. Tanaka, J. V. Frangioni, and M. G. Bawendi, "Size series of small indium. arsenide-zinc selenide core-shell nanocrystals and their application to in vivo imaging," J. Am. Chem. Soc. 128, 2526 (2006).
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Figures (3)

Fig. 1
Fig. 1

Geometry and parameters of a spherically multilayered structure with Q layers below and P layers above the 0th shell where an electric dipole is located. ri (i=-Q+1,…,P) denotes the position of the ith boundary; εi, Ai and Bi (i=-Q,…,P) denote the relative permittivity, decomposition coefficients of the inward and outward waves, respectively; R E/M out,l , R E/M in,l and T E/M out,l are the total reflection coefficient from the outer shells, the total reflection coefficient from the inner shells and the total transmission coefficient of TE/TM polarization, respectively.

Fig. 2(a).
Fig. 2(a).

Wavelength dependence of the normalized SE rate for a fixed nanoparticle r = 20nm (b) Variance of the normalized SE rate and the fluorescence efficiency for ηq 0 = 0.25 and ηq 0 = 0.75 with the increase of the size of the nanoparticle at the wavelength of 500nm.

Fig. 3.
Fig. 3.

(a). Wavelength dependence of the normalized SE rate in silver encapsulated nanoparticles of various Ag shell thicknesses s (nm) and fixed core radius; inset shows the variance of the effective radius with the increase of the shell thickness. (b) Variance of the fluorescence efficiency with the increase of Ag shell thickness and fixed core radius. (c) Wavelength dependence of the normalized SE rate in silver encapsulated nanoparticles of various core radii r (nm) and fixed Ag shell thickness; inset shows the variance of the effective radius with the increase of the core radius (d) Variance of the fluorescence efficiency with the increase of core radius and fixed Ag shell thickness.

Equations (41)

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Γ r s = F Γ r 0
η q s Γ r s Γ r s + Γ nr = F F η q 0 + ( 1 η q 0 ) η q 0
η ext η q s U F = U F η q 0 + ( 1 η q 0 ) η q 0
E 0 ( r , r' ) = ( I + k 0 2 ρ Θ e i k 0 r r' 4 π r r' )
E s = E 0 + E ρ r
F = 1 + 6 π k 0 Θ ρ ( E ρ r ( r' , r' ) )
F = 1 + 2 π k 0 Θ [ γ ( E γ r ( r' , r' ) ) + θ ( E θ r ( r' , r' ) ) + ϕ ( E ϕ r ( r' , r' ) ) ]
E 0 ( r' , r' ) = l = 1 m = l l ( i α lm E l ( l + 1 ) α lm M ωε ε 0 l ( l + 1 ) × ) × ( r h l ( 1 ) ( k 0 r ) Y lm ( θ , ϕ ) ) for ( r > r' )
E 0 ( r' , r' ) = l = 1 m = l l ( i β lm E l ( l + 1 ) β lm M ωε ε 0 l ( l + 1 ) × ) × ( r j l ( k 0 r ) Y lm ( θ , ϕ ) ) for ( r > r' )
α lm E = i k 0 Θ ρ ( ∇' × r' j l ( k 0 r' ) ( i Y lm ( θ' , ϕ' ) l ( l + 1 ) ) * ) ,
α lm E = k 0 ωμ Θ ρ ( ∇' × ∇' × r' j l ( k 0 r' ) ( i Y lm ( θ' , ϕ' ) l ( l + 1 ) ) * )
β lm E = i k 0 Θ ρ ( ∇' × r' h l ( 1 ) ( k 0 r' ) ( i Y lm ( θ' , ϕ' ) l ( l + 1 ) ) * ) ,
β lm E = k 0 ωμ Θ ρ ( ∇' × ∇' × r' h l ( 1 ) ( k 0 r' ) ( i Y lm ( θ' , ϕ' ) l ( l + 1 ) ) * )
E r ( r , r' ) = l = 1 m = l l ( i R in , l E ( α lm E R out , l E + β lm E ) ( 1 R out , l E R in , l E ) R in , l M ( α lm M R out , l M + β lm M ) ωε ε 0 ( 1 R out , l M R in , l M ) × ) × ( r h l ( 1 ) ( k 0 r ) Y lm ( θ , ϕ ) ) l ( l + 1 )
+ l = 1 m = l l ( i R out , l E ( α lm E + R in , l E β lm E ) ( 1 R out , l E R in , l E ) R out , l M ( α lm M + R in , l M β lm M ) ωε ε 0 ( 1 R out , l M R in , l M ) × ) × ( r j l ( k 0 r ) Y lm ( θ , ϕ ) ) l ( l + 1 )
U = l = 1 m = 1 l ε 0 ε p ( ( α lm E + β lm E R in , l E ) T out , l E ( 1 R out , l E R in , l E ) 2 + μ ε ε p ( α lm M + β lm E R in , l M ) T out , l M ( 1 R out , l M R in , l M ) 2 ) / ( α lm E 2 + μ ε 0 ε α lm M 2 )
E TE ( r ) = l = 1 l l [ i × r ( A lm E ( n ) j l ( k n r ) + B lm E ( n ) h l ( 1 ) ( k n r ) ) Y lm ( θ , ϕ ) l ( l + 1 ) ]
E TM ( r ) = l = 1 l l [ × × r ( A lm M ( n ) j l ( k n r ) + B lm M ( n ) h l ( 1 ) ( k n r ) ) Y lm ( θ , ϕ ) ωε ε n l ( l + 1 ) ]
u l ( k n 1 r n ) A lm E ( n 1 ) + w l ( k n 1 r n ) B lm E ( n 1 ) = u l ( k n r n ) A lm E ( n ) + w l ( k n r n ) B lm E ( n )
u' l ( k n 1 r n ) A lm E ( n 1 ) + w' l ( k n 1 r n ) B lm E ( n 1 ) = u' l ( k n r n ) A lm E ( n ) + w' l ( k n r n ) B lm E ( n )
u l ( k n 1 r n ) A lm M ( n 1 ) + w l ( k n 1 r n ) B lm M ( n 1 ) = u l ( k n r n ) A lm M ( n ) + w l ( k n r n ) B lm M ( n )
u' l ( k n 1 r n ) A lm M ( n 1 ) + w' l ( k n 1 r n ) B lm M ( n 1 ) = ε n 1 ε n [ u ' l ( k n r n ) A lm M ( n ) + w' l ( k n r n ) B lm M ( n ) ]
( A lm E / M ( n 1 ) B lm E / M ( n 1 ) ) = [ M l , 11 E / M ( n ) M l , 12 E / M ( n ) M l , 21 E / M ( n ) M l , 22 E / M ( n ) ] ( A lm E / M ( n ) A lm E / M ( n ) )
( A lm E / M ( 0 ) B lm E / M ( 0 ) ) = [ H l , 11 E / M H l , 12 E / M H l , 21 E / M ( n ) H l , 22 E / M ] ( A lm E / M ( P ) B lm E / M ( P ) )
R out , l E / M = H l , 12 E / M H l , 22 E / M
T out , l E / M = 1 H l , 22 E / M
E TM ( r ) = l = 1 m = l l [ × × r ( A lm M ( n ) j l ( k n r ) + B lm M ( n ) h l ( 1 ) ( k n r ) ) Y lm ( θ , ϕ ) ωε ε n l ( l + 1 ) ]
+ l = 1 m = 1 l 1 ωεε n [ ( C lm j l ( k L r ) + D lm h l ( 1 ) ( k L r ) ) Y lm ( θ , ϕ ) ]
ε ( ω ) = 1 ω p 2 ω 2 + i ω γ
ε L ( k L , ω ) = 1 ω p 2 ω 2 0.6 v f 2 k L 2 + i ω γ
u l ( k n 1 r n ) A lm M ( n 1 ) + w l ( k n 1 r n ) B lm M ( n 1 ) = u l ( k n r n ) A lm M ( n ) + w l ( k n r n ) B lm M ( n )
1 ε n 1 [ u' l ( k n 1 r n ) A lm M ( n 1 ) + w' l ( k n 1 r n ) B lm M ( n 1 ) ] = 1 ε n [ u' l ( k n r n ) A lm M ( n ) + w' l ( k n r n ) B lm M ( n ) ]
i l ( l + 1 ) ε n [ j l ( k L r n ) C lm + h l ( 1 ) ( k L r n ) D lm ]
l ( l + 1 ) ε n 1 r n [ A lm M ( n 1 ) j l ( k n 1 r n ) + B lm M ( n 1 ) h l ( 1 ) ( k n 1 r n ) ] = l ( l + 1 ) ε n r n [ A lm M ( n ) j l ( k n r n ) + B lm M ( n ) h l ( 1 ) ( k n r n ) ]
i ε n [ j' l ( k L r n ) C lm + h' l ( 1 ) ( k L r n ) D lm ]
u l ( k n + 1 r n + 1 ) A lm M ( n + 1 ) + w l ( k n + 1 r n + 1 ) B lm M ( n + 1 ) = u l ( k n r n + 1 ) A lm M ( n ) + w l ( k n r n + 1 ) B lm M ( n )
1 ε n + 1 [ u' l ( k n + 1 r n + 1 ) A lm M ( n + 1 ) + w' l ( k n + 1 r n + 1 ) B lm M ( n + 1 ) ] = 1 ε n [ u' l ( k n r n + 1 ) A lm M ( n ) + w' l ( k n r n + 1 ) B lm M ( n ) ]
i l ( l + 1 ) ε n [ j l ( k L r n + 1 ) C lm + h l ( 1 ) ( k L r n + 1 ) D lm ]
l ( l + 1 ) ε n + 1 r n + 1 [ A lm M ( n + 1 ) j l ( k n + 1 r n + 1 ) + B lm M ( n + 1 ) h l ( 1 ) ( k n + 1 r n + 1 ) ] = l ( l + 1 ) ε n r n + 1 [ A lm M ( n ) j l ( k n r n + 1 ) + B lm M ( n ) h l ( 1 ) ( k n r n + 1 ) ]
i ε n [ j' l ( k L r n + 1 ) C lm + h' l ( 1 ) ( k L r n + 1 ) D lm ]
( A lm M ( n 1 ) B lm M ( n 1 ) ) = [ M l , 11 M ( n ) M l , 12 M ( n ) M l , 21 M ( n ) M l , 22 M ( n ) ] ( A lm M ( n + 1 ) B lm M ( n + 1 ) )

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