Abstract

We evaluate the spontaneous emission rate (Purcell) enhancement for optically-doped metal–dielectric–semiconductor light-emitting structures by considering the behavior of a semiclassical oscillating point dipole placed within the dielectric layer. For a Ag–SiO2–Si structure containing emitters at the center of a 20-nm-thick SiO2 layer, spontaneous emission rate enhancements of 40 to 60 can be reached in the wavelength range of 600 to 1800 nm, far away from the surface plasmon resonance; similar enhancements are also possible if Al is used instead of Ag. For dipoles contained in the thin oxide layer of a Ag–SiO2–Si–SiO2 structure, the emission exhibits strong preferential coupling to a single well-defined Si waveguide mode. This work suggests a means of designing a new class of power-efficient, high-modulation-speed, CMOS-compatible optical sources that take full advantage of the excellent electrical properties and plasmon-enhanced optical properties afforded by MOS devices.

© 2009 Optical Society of America

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2008 (2)

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

2007 (1)

J. S. Q. Liu and M. Brongersma, "Omnidirectional light emission via surface plasmon polaritons," Appl. Phys. Lett. 90, 091116 (2007).
[CrossRef]

2006 (3)

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Tech. 24, 4600-4615 (2006).
[CrossRef]

A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, "Electrically pumped hybrid AlGaInAs-silicon evanescent laser," Opt. Express 14, 9203-9210 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Lipson, "Guiding, modulating, and emitting light on silicon—Challenges and opportunities," J. Lightwave Tech. 23, 4222-4238 (2005).
[CrossRef]

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

2003 (2)

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

L. Pavesi, "Will silicon be the photonic material of the third millenium?" J. Phys. Condens. Matter 15, R1169- R1196 (2003).
[CrossRef]

2000 (2)

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

J. Vu?kovi?, M. Loncar, and A. Scherer, "Surface plasmon enhanced light-emitting diode," IEEE J. Quantum Electron. 36, 1131-1144 (2000).
[CrossRef]

1999 (1)

W. L. Barnes, "Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices," J. Lightwave Tech. 17, 2170-2182 (1999).
[CrossRef]

1998 (1)

1997 (1)

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

1984 (1)

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

Atwater, H. A.

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

Barnes, W. L.

W. L. Barnes, "Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices," J. Lightwave Tech. 17, 2170-2182 (1999).
[CrossRef]

Bowers, J. E.

Brongersma, M.

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

J. S. Q. Liu and M. Brongersma, "Omnidirectional light emission via surface plasmon polaritons," Appl. Phys. Lett. 90, 091116 (2007).
[CrossRef]

Buchal, C.

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Carius, R.

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

Cho, K. S.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Cohen, O.

Dal Negro, L.

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

de Dood, M. J. A.

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

Di Stefano, G.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Djurisic, A. B.

Eckau, A.

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

Elazar, J. M.

Fallica, P. G.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Fang, A. W.

Fathpour, S.

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Tech. 24, 4600-4615 (2006).
[CrossRef]

Ford, G. W.

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

Franzò, G.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

Iacona, F.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Irrera, A.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Jalali, B.

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Tech. 24, 4600-4615 (2006).
[CrossRef]

Jones, R.

Jun, Y. C.

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

Kalkman, J.

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

Kekatpure, R. D.

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

Kim, K. H.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Kim, T. Y.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Lipson, M.

M. Lipson, "Guiding, modulating, and emitting light on silicon—Challenges and opportunities," J. Lightwave Tech. 23, 4222-4238 (2005).
[CrossRef]

Liu, J. S. Q.

J. S. Q. Liu and M. Brongersma, "Omnidirectional light emission via surface plasmon polaritons," Appl. Phys. Lett. 90, 091116 (2007).
[CrossRef]

Loncar, M.

J. Vu?kovi?, M. Loncar, and A. Scherer, "Surface plasmon enhanced light-emitting diode," IEEE J. Quantum Electron. 36, 1131-1144 (2000).
[CrossRef]

Majewski, M. L.

Mazzoleni, C.

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

Miritello, M.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Neufeld, E.

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

Pacifici, D.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Paniccia, M. J.

Park, H.

Park, N. M.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Pavesi, L.

L. Pavesi, "Will silicon be the photonic material of the third millenium?" J. Phys. Condens. Matter 15, R1169- R1196 (2003).
[CrossRef]

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

Polman, A.

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

Priolo, F.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

Qin, G. G.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Rakic, A. D.

Ran, G. Z.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Sanfillipo, D.

A. Irrera, D. Pacifici, M. Miritello, G. Franzò, F. Priolo, F. Iacona, D. Sanfillipo, G. Di Stefano, and P. G. Fallica, "Electroluminescence properties of light emitting devices based on silicon nanocrystals," Physica E 16, 395-399 (2003).
[CrossRef]

Scherer, A.

J. Vu?kovi?, M. Loncar, and A. Scherer, "Surface plasmon enhanced light-emitting diode," IEEE J. Quantum Electron. 36, 1131-1144 (2000).
[CrossRef]

Shin, J. H.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Sun, K.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Sung, G. Y.

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Vuckovic, J.

J. Vu?kovi?, M. Loncar, and A. Scherer, "Surface plasmon enhanced light-emitting diode," IEEE J. Quantum Electron. 36, 1131-1144 (2000).
[CrossRef]

Walters, R. J.

R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Phys. Rev. B 73, 132302 (2006).
[CrossRef]

Wang, S.

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

Weber, W. H.

G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

White, J. S.

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

Xu, W. J.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

You, L. P.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Zhang, B.

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Appl. Opt. (1)

Appl. Phys. Lett. (3)

S. Wang, A. Eckau, E. Neufeld, R. Carius, and C. Buchal, "Hot electron impact excitation cross-section of Er3+ and electroluminescence from erbium-implanted silicon metal-oxide-semiconductor tunnel diodes," Appl. Phys. Lett. 71, 2824-2826 (1997).
[CrossRef]

J. S. Q. Liu and M. Brongersma, "Omnidirectional light emission via surface plasmon polaritons," Appl. Phys. Lett. 90, 091116 (2007).
[CrossRef]

K. S. Cho, N. M. Park, T. Y. Kim, K. H. Kim, G. Y. Sung, and J. H. Shin, "High efficiency visible electroluminescence from silicon nanocrystals embedded in silicon nitride using a transparent doping layer", Appl. Phys. Lett. 86, 071909 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Vu?kovi?, M. Loncar, and A. Scherer, "Surface plasmon enhanced light-emitting diode," IEEE J. Quantum Electron. 36, 1131-1144 (2000).
[CrossRef]

J. Lightwave Tech. (3)

W. L. Barnes, "Electromagnetic crystals for surface plasmon polaritons and the extraction of light from emissive devices," J. Lightwave Tech. 17, 2170-2182 (1999).
[CrossRef]

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Tech. 24, 4600-4615 (2006).
[CrossRef]

M. Lipson, "Guiding, modulating, and emitting light on silicon—Challenges and opportunities," J. Lightwave Tech. 23, 4222-4238 (2005).
[CrossRef]

J. Phys. Condens. Matter (1)

L. Pavesi, "Will silicon be the photonic material of the third millenium?" J. Phys. Condens. Matter 15, R1169- R1196 (2003).
[CrossRef]

Nanotechnology (1)

K. Sun, W. J. Xu, B. Zhang, L. P. You, G. Z. Ran, and G. G. Qin, "Strong enhancement of Er3+ 1.54 ?m electroluminescence through amorphous Si nanoparticles," Nanotechnology 19, 708 (2008).

Nature (1)

L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzò, and F. Priolo, "Optical gain in silicon nanocrystals," Nature 408, 440-444 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rep. (1)

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Phys. Rev. B (3)

Y. C. Jun, R. D. Kekatpure, J. S. White, and M. Brongersma, "Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures," Phys. Rev. B 78, 153111 (2008).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of a prototype CMOS-compatible light-emitting device architecture exploiting plasmonic decay rate enhancement of emitters in a planar metal–dielectric–semiconductor–dielectric (MDSD) structure and the possibility of coupling the emission into a SOI waveguide. Typical electric field profiles for supported transverse magnetic (TM) modes are shown in red.

Fig. 2.
Fig. 2.

Normalized decay rate density for a perpendicular dipole located at the center of the 20-nm-thick dielectric layer in a Ag–SiO2–Si MDS structure. E-field profiles, normalized to unit H-field magnitude at the M–D interface, are shown in the insets.

Fig. 3.
Fig. 3.

Decay rate density vs. k at λ 0 = 1535 nm and simulation geometries for (a) the semi-infinite MDS and MDM structures, and (b) the MDSD structure; in both cases, the dipole is situated at the center of the dielectric layer. The inset to (b) illustrates the effect of dipole position on the relative contributions of the surface plasmon polariton (SPP), lossy surface wave (LSW), and far-field radiation (rad.) pathways to the overall decay rate in the MDSD structure.

Fig. 4.
Fig. 4.

(a) Decay rate enhancements for DM, MDS, and MDM structures; for these three cases, the dipole is 10 nm from the metal surface. Two plots for the MDSD case of Fig. 3b are included for comparison: (red dashed line) perpendicular dipole at the center of the dielectric (d + = 10 nm); (black dashed line) emitters uniformly distributed such that d + ∊ [1.5,18.5] nm in the dielectric layer, averaged over all dipole orientations. Insets: E-field profiles for λ 0 = 850 nm. (b) The effect of internal quantum efficiency (η) on the decay rate enhancement for the MDSD structure shown in the inset (perpendicular dipole, d + = 10 nm).

Equations (5)

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γ̂γγfree=1η+η01freeddkdk.
1freeddk=321kD3e{kqD[cos2θ(k2r̃P)+12sin2θ(kD2r̃S+qD2r̃P)]}
r̃P=[1+rDPexp(2iqDd)][1+rD+Pexp(2iqDd+)]1rDPrD+Pexp(2iqDLD)
r̃S=[1+rDSexp(2iqDd)][1+rD+Sexp(2iqDd+)]1rDSrD+Sexp(2iqDLD)
r̃P=[1rDPexp(2iqDd)][1rD+Pexp(2iqDd+)]1rDPrD+Pexp(2iqDLD)

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