Abstract

This paper is focused on the optical properties of nanocomposite plasmonic emitters with core/shell configurations, where a fluorescence emitter is located inside a metal nanoshell. Systematic theoretical investigations are presented for the influence of material type, core radius, shell thickness, and excitation wavelength on the internal optical intensity, radiative quantum yield, and fluorescence enhancement of the nanocomposite emitter. It is our conclusion that: (i) an optimal ratio between the core radius and shell thickness is required to maximize the absorption rate of fluorescence emitters, and (ii) a large core radius is desired to minimize the non-radiative damping and avoid significant quantum yield degradation of light emitters. Several experimental approaches to synthesize these nanocomposite emitters are also discussed. Furthermore, our theoretical results are successfully used to explain several reported experimental observations and should prove useful for designing ultra-bright core/shell nanocomposite emitters.

© 2010 Optical Society of America

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    [CrossRef]

2009 (10)

G. Sun, J. B. Khurgin, and R. A. Soref, “Practical enhancement of photoluminescence by metal nanoparticles,” Appl. Phys. Lett. 94, 101103 (2009).
[CrossRef]

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105, 033107 (2009).
[CrossRef]

X. Li, J. Qian, L. Jiang, and S. He, “Fluorescence quenching of quantum dots by gold nanorods and its application in DNA detection,” Appl. Phys. Lett. 94, 063111 (2009).
[CrossRef]

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avalasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie antenna,” Nature Photon. 3, 654–657 (2009).
[CrossRef]

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

P. Zhang and Y. Guo, “Surface-enhanced Raman scattering inside metal nanoshells,” J. Am. Chem. Soc. 131, 3808–3809 (2009).
[CrossRef] [PubMed]

Y. Jin and X. Gao, “Plasmonic fluorescent quantum-dots,” Nat. Nanotechnol. 4, 571–576 (2009).
[CrossRef] [PubMed]

J. B. Khurgin and G. Sun, “Enhancement of optical properties of nanoscaled objects by metal nanoparticles,” J. Opt. Soc. Am. B 26, 83–95 (2009).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394 (2009).
[CrossRef]

2008 (6)

X. Miao, B. K. Wilson, S. H. Pun, and L. Y. Lin, “Optical manipulation of micron/submicron sized particles and biomolecules through plasmonics,” Opt. Express 16, 13517–13525 (2008).
[CrossRef] [PubMed]

M. Young, D. Willits, M. Uchida, and T. Douglas, “Plant viruses as biotemplates for materials and their use in nanotechnology,” Annu. Rev. Phytopathol. 46, 361–384 (2008).
[CrossRef] [PubMed]

B. E. Brinson, J. B. Lassiter, C. S. Levin, R. Bardhan, N. Mirin, and N. J. Halas, “Nanoshells made easy: improving Au layer growth on nanoparticle surfaces,” Langmuir 24, 14166–14171 (2008).
[CrossRef]

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[CrossRef] [PubMed]

J. R. Lakowicz, K. Ray, M. Chowdhury, H. Szmacinski, Y. Fu, J. Zhang, and K. Nowaczyk, “Plasmon-controlled fluorescence: a new paradigm in fluorescence spectroscopy,” Analyst (Cambridge, U.K.) 133, 1308–1346 (2008).
[CrossRef]

R. Bardhan, N. K. Grady, and N. J. Halas, “Nanoscale control of near-infrared fluorescence enhancement using Au nanoshells,” Small 4, 1716–1722 (2008).
[CrossRef] [PubMed]

2007 (2)

J. Zhang, Y. Fu, and J. R. Lakowicz, “Emission behavior of fluorescently labeled silver nanoshell: enhanced self-quenching by metal nanostructure,” J. Phys. Chem. C 111, 1955–1961 (2007).
[CrossRef]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing and waveguiding,” Nature Photon. 1, 641–648 (2007).
[CrossRef]

2006 (3)

L. R. Hirsch, A. M. Gobin, A. R. Lowery, F. Tam, R. A. Drezek, N. J. Halas, and J. L. West, “Metal nanoshells,” Ann. Biomed. Eng. 34, 15–22 (2006).
[CrossRef] [PubMed]

J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell—bright particle,” J. Phys. Chem. B 110, 8986–8991 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

2004 (2)

J. B. Jackson and N. J. Halas, “Surface-enhanced Raman scattering on tunable plasmonic substrates,” Proc. Natl. Acad. Sci. U.S.A. 101, 17930–17935 (2004).
[CrossRef] [PubMed]

A. M. Derfus, W. C. W. Chan, and S. N. Bhatia, “Probing the cytotoxicity of semiconductor quantum dots,” Nano Lett. 4, 11–18 (2004).
[CrossRef]

2003 (1)

S. A. Maier, P. G. Kik, H. A. Water, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguide,” Nature Mater. 2, 229–232 (2003).
[CrossRef]

2002 (4)

J. Enderlein, “Spectral properties of a fluorescing molecule within a spherical metallic nanocavity,” Phys. Chem. Chem. Phys. 4, 2780–2786 (2002).
[CrossRef]

J. Enderlein, “Theoretical study of single molecule fluorescence in a metallic nanocavity,” Appl. Phys. Lett. 80, 315–317 (2002).
[CrossRef]

E. Dulkeith, A. C. Morteani, T. Niedereichholz, T. A. Klar, J. Friedmann, S. A. Levi, F. C. J. M. van Veggel, D. N. Reinhoudt, M. Möller, and D. I. Gittins, “Fluorescence quenching of dye molecules near gold nanoparticles: radiative and non-radiative effects,” Phys. Rev. Lett. 89, 203002 (2002).
[CrossRef] [PubMed]

N. J. Halas, “The optical properties of nanoshells,” Opt. Photonics News 13, 26–30 (2002).
[CrossRef]

2001 (1)

J. B. Jackson and N. J. Halas, “Silver nanoshells: variations in morphologies and optical properties,” J. Phys. Chem. B 105, 2743–2746 (2001).
[CrossRef]

2000 (2)

A. J. Zarur and J. Y. Ying, “Reverse microemulsion synthesis of nanostructured complex oxides for catalytic combustion,” Nature 403, 65–67 (2000).
[CrossRef] [PubMed]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

1999 (1)

1975 (1)

S. Y. Liao, “Light transmittance and RF shielding effectiveness of a gold film on a glass substrate,” IEEE Trans. Electromagn. Compat. EMC-17, 211–216 (1975).
[CrossRef]

1972 (1)

P. B. Johnson and C. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Alkilany, A. M.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[CrossRef] [PubMed]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Ashley, C.

X. Miao, T. S. Luk, I. Brener, C. Ashley, S. Xiong, D. Peabody, and J. Brinker, “Surface plasmon enhanced fluorescence emission inside metal nanoshells,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest CD (Optical Society of America, 2010), paper JThE20.

Avalasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avalasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie antenna,” Nature Photon. 3, 654–657 (2009).
[CrossRef]

Averitt, R. D.

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bardhan, R.

R. Bardhan, N. K. Grady, and N. J. Halas, “Nanoscale control of near-infrared fluorescence enhancement using Au nanoshells,” Small 4, 1716–1722 (2008).
[CrossRef] [PubMed]

B. E. Brinson, J. B. Lassiter, C. S. Levin, R. Bardhan, N. Mirin, and N. J. Halas, “Nanoshells made easy: improving Au layer growth on nanoparticle surfaces,” Langmuir 24, 14166–14171 (2008).
[CrossRef]

Barnard, E.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

Baxter, S. C.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[CrossRef] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Bhatia, S. N.

A. M. Derfus, W. C. W. Chan, and S. N. Bhatia, “Probing the cytotoxicity of semiconductor quantum dots,” Nano Lett. 4, 11–18 (2004).
[CrossRef]

Brener, I.

X. Miao, T. S. Luk, I. Brener, C. Ashley, S. Xiong, D. Peabody, and J. Brinker, “Surface plasmon enhanced fluorescence emission inside metal nanoshells,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest CD (Optical Society of America, 2010), paper JThE20.

Brinker, J.

X. Miao, T. S. Luk, I. Brener, C. Ashley, S. Xiong, D. Peabody, and J. Brinker, “Surface plasmon enhanced fluorescence emission inside metal nanoshells,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest CD (Optical Society of America, 2010), paper JThE20.

Brinson, B. E.

B. E. Brinson, J. B. Lassiter, C. S. Levin, R. Bardhan, N. Mirin, and N. J. Halas, “Nanoshells made easy: improving Au layer growth on nanoparticle surfaces,” Langmuir 24, 14166–14171 (2008).
[CrossRef]

Brongersma, M. L.

R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

Chan, W. C. W.

A. M. Derfus, W. C. W. Chan, and S. N. Bhatia, “Probing the cytotoxicity of semiconductor quantum dots,” Nano Lett. 4, 11–18 (2004).
[CrossRef]

Chowdhury, M.

J. R. Lakowicz, K. Ray, M. Chowdhury, H. Szmacinski, Y. Fu, J. Zhang, and K. Nowaczyk, “Plasmon-controlled fluorescence: a new paradigm in fluorescence spectroscopy,” Analyst (Cambridge, U.K.) 133, 1308–1346 (2008).
[CrossRef]

Christy, C.

P. B. Johnson and C. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Derfus, A. M.

A. M. Derfus, W. C. W. Chan, and S. N. Bhatia, “Probing the cytotoxicity of semiconductor quantum dots,” Nano Lett. 4, 11–18 (2004).
[CrossRef]

Douglas, T.

M. Young, D. Willits, M. Uchida, and T. Douglas, “Plant viruses as biotemplates for materials and their use in nanotechnology,” Annu. Rev. Phytopathol. 46, 361–384 (2008).
[CrossRef] [PubMed]

Drezek, R. A.

L. R. Hirsch, A. M. Gobin, A. R. Lowery, F. Tam, R. A. Drezek, N. J. Halas, and J. L. West, “Metal nanoshells,” Ann. Biomed. Eng. 34, 15–22 (2006).
[CrossRef] [PubMed]

Dulkeith, E.

E. Dulkeith, A. C. Morteani, T. Niedereichholz, T. A. Klar, J. Friedmann, S. A. Levi, F. C. J. M. van Veggel, D. N. Reinhoudt, M. Möller, and D. I. Gittins, “Fluorescence quenching of dye molecules near gold nanoparticles: radiative and non-radiative effects,” Phys. Rev. Lett. 89, 203002 (2002).
[CrossRef] [PubMed]

Enderlein, J.

J. Enderlein, “Spectral properties of a fluorescing molecule within a spherical metallic nanocavity,” Phys. Chem. Chem. Phys. 4, 2780–2786 (2002).
[CrossRef]

J. Enderlein, “Theoretical study of single molecule fluorescence in a metallic nanocavity,” Appl. Phys. Lett. 80, 315–317 (2002).
[CrossRef]

Esteban, R.

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105, 033107 (2009).
[CrossRef]

Fan, S.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avalasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie antenna,” Nature Photon. 3, 654–657 (2009).
[CrossRef]

Friedmann, J.

E. Dulkeith, A. C. Morteani, T. Niedereichholz, T. A. Klar, J. Friedmann, S. A. Levi, F. C. J. M. van Veggel, D. N. Reinhoudt, M. Möller, and D. I. Gittins, “Fluorescence quenching of dye molecules near gold nanoparticles: radiative and non-radiative effects,” Phys. Rev. Lett. 89, 203002 (2002).
[CrossRef] [PubMed]

Fu, Y.

J. R. Lakowicz, K. Ray, M. Chowdhury, H. Szmacinski, Y. Fu, J. Zhang, and K. Nowaczyk, “Plasmon-controlled fluorescence: a new paradigm in fluorescence spectroscopy,” Analyst (Cambridge, U.K.) 133, 1308–1346 (2008).
[CrossRef]

J. Zhang, Y. Fu, and J. R. Lakowicz, “Emission behavior of fluorescently labeled silver nanoshell: enhanced self-quenching by metal nanostructure,” J. Phys. Chem. C 111, 1955–1961 (2007).
[CrossRef]

Gao, X.

Y. Jin and X. Gao, “Plasmonic fluorescent quantum-dots,” Nat. Nanotechnol. 4, 571–576 (2009).
[CrossRef] [PubMed]

Gittins, D. I.

E. Dulkeith, A. C. Morteani, T. Niedereichholz, T. A. Klar, J. Friedmann, S. A. Levi, F. C. J. M. van Veggel, D. N. Reinhoudt, M. Möller, and D. I. Gittins, “Fluorescence quenching of dye molecules near gold nanoparticles: radiative and non-radiative effects,” Phys. Rev. Lett. 89, 203002 (2002).
[CrossRef] [PubMed]

Gobin, A. M.

L. R. Hirsch, A. M. Gobin, A. R. Lowery, F. Tam, R. A. Drezek, N. J. Halas, and J. L. West, “Metal nanoshells,” Ann. Biomed. Eng. 34, 15–22 (2006).
[CrossRef] [PubMed]

Goldsmith, E. C.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[CrossRef] [PubMed]

Gole, A. M.

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
[CrossRef] [PubMed]

Grady, N. K.

R. Bardhan, N. K. Grady, and N. J. Halas, “Nanoscale control of near-infrared fluorescence enhancement using Au nanoshells,” Small 4, 1716–1722 (2008).
[CrossRef] [PubMed]

Greffet, J. -J.

R. Esteban, M. Laroche, and J.-J. Greffet, “Influence of metallic nanoparticles on upconversion processes,” J. Appl. Phys. 105, 033107 (2009).
[CrossRef]

Gryczynski, I.

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Acc. Chem. Res. (1)

C. J. Murphy, A. M. Gole, J. W. Stone, P. N. Sisco, A. M. Alkilany, E. C. Goldsmith, and S. C. Baxter, “Gold nanoparticles in biology: beyond toxicity to cellular imaging,” Acc. Chem. Res. 41, 1721–1730 (2008).
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Figures (11)

Fig. 1
Fig. 1

Core/shell nanoparticle geometry: ε i ( i = 1 , 2 , 3 ) are the dielectric functions of the core, shell, and surrounding medium, respectively; r 1 is the core radius and r 2 is the total particle radius.

Fig. 2
Fig. 2

Dielectric functions of (a) Au and (b) Ag. The blue and red curves show the real and imaginary parts of the dielectric function, respectively. The solid and dashed curves show the bulk value and modified value after incorporating the size effect, respectively. Here the core radius and shell thickness of the nanoparticle are assumed to be 5 and 2.5 nm, respectively.

Fig. 3
Fig. 3

(a) Extinction (blue solid curve) and internal optical intensity enhancement (red dashed curve) of a SiO 2 -core Au-shell nanoparticle with the core radius and shell thickness of 20 and 6 nm, respectively. (b) A cross section of the optical intensity distribution around the core/shell nanoparticle when it is excited at 633 nm.

Fig. 4
Fig. 4

Internal optical intensity enhancement as a function of core radius and thickness when the size-dependent values of the metal’s dielectric functions are used. The core material is assumed to be SiO 2 . The shell is assumed to be Au for (a)–(d) and Ag for (e)–(h). The excitation wavelengths in (a) and (e), (b) and (f), (c) and (g), (d) and (h) are the absorption peaks of Alexa Fluora 488 (495 nm), Alexa Fluora 532 (532 nm), Alexa Fluora 633 (632 nm), and Alexa Fluora 750 (749 nm), respectively.

Fig. 5
Fig. 5

The resonant wavelength for internal optical intensity enhancement as a function of ratio between shell thickness and core radius of (a) Au-shell and (b) silver-shell nanoparticles. The core materials are assumed to be SiO 2 (blue solid curve) and TiO 2 (red dashed curve). The refractive index of TiO 2 (rutile phase) is 2.609.

Fig. 6
Fig. 6

Internal optical intensity enhancement as a function of core radius and thickness when the bulk values of the metal’s dielectric functions are used. The excitation wavelength is at 633 nm. The core material is assumed to be SiO 2 . The shell is assumed to be Au in (a) and Ag in (b).

Fig. 7
Fig. 7

(a) Enhancement of radiative (blue solid curve) and non-radiative (red dashed curve) decay rates, and (b) radiative quantum yield of the nanocomposite emitter. Here an Alexa Fluora 633 dye molecule is assumed to be at the center of a SiO 2 -core Au-shell nanoparticle with the core radius and shell thickness of 20 and 6 nm, respectively.

Fig. 8
Fig. 8

(a) Radiative quantum yield of the nanocomposite emitter versus core radius while the shell thickness is kept at 2 nm. (b) Quantum yield of the nanocomposite emitter versus shell thickness while the core radius is kept at 20 nm. In both (a) and (b), the emitter locates at the center of the particle and the emission wavelength is assumed to be 647 nm.

Fig. 9
Fig. 9

Radiative quantum yield as a function of the distance from the emitter to the inner metal surface when the core radius is 20 nm and shell thickness is 6 nm. The core and shell material are assumed to be SiO 2 and Au, respectively.

Fig. 10
Fig. 10

An Alexa Fluora 633 dye molecule located at the center of an Au-shell SiO 2 -core nanoparticle with a 20 nm core radius and 6 nm shell thickness. (a) Absorption spectrum of the dye (blue solid curve [25]) and the nanocomposite emitter (red dashed curve). (b) Emission spectrum of the dye (blue solid curve [25]) and the nanocomposite emitter (red dashed curve).

Fig. 11
Fig. 11

Fluorescence enhancement as a function of excitation and detection wavelength for an Alexa Fluora 633 dye molecule encapsulated at the center of an Au-shell SiO 2 -core nanoparticle with a 20 nm core radius and 6 nm shell thickness.

Equations (8)

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N 0 + N i = N all ,
N 0 t = N 0 σ 01 , abs I 01 ω 01 + N i Γ i 0 , total ,
Δ = P r a d P 0 = | K ( ω 01 ) | 2 [ η ( ω i 0 ) / η 0 ( ω i 0 ) ] ,
E 1 = 9 ε 2 ε 3 ε 2 ε a + 2 ε 3 ε b E 0 ( cos   θ r ̂ sin   θ θ ̂ ) ,
E 2 = 3 ε 3 ε 2 ε a + 2 ε 3 ε b { [ ( ε 1 + 2 ε 2 ) + 2 ( ε 1 ε 2 ) ( r 1 r ) 3 ] E 0   cos   θ r ̂ [ ( ε 1 + 2 ε 2 ) ( ε 1 ε 2 ) ( r 1 r ) 3 ] E 0   sin   θ θ ̂ } ,
E 3 = ( 2 ε 2 ε a ε 3 ε b ε 2 ε a + 2 ε 3 ε b ( r 2 r ) 3 1 ) E 0   cos   θ r ̂ + ( ε 2 ε a ε 3 ε b ε 2 ε a + 2 ε 3 ε b ( r 2 r ) 3 1 ) E 0   sin   θ ̂ r ,
ε ( r 1 , r 2 , ω ) = ε ( ω ) b u l k + ω p 2 ω 2 + i ω γ 0 ω p 2 ω 2 + i ω [ γ 0 + Δ γ ( r 1 , r 2 ) ] ,
Γ total Γ rad , 0 = P total P 0 ,

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