Abstract

A theoretical analysis of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media is presented. An expression for the noise figure is obtained in terms of the spontaneous emission rate into the amplified surface plasmon-polariton taking into account the different energy decay channels experienced by dipoles in close proximity to the metallic surface. Two amplifier structures are examined: a single-interface between a metal and a gain medium and a thin metal film bounded by identical gain media on both sides. A realistic configuration is considered where the surface plasmon-polariton undergoing amplification has a Gaussian field profile in the plane of the metal and paraxial propagation along the amplifier’s length. The noise figure of these plasmonic amplifiers is studied considering three prototypical gain media with different permittivities. It is shown that the noise figure exhibits a strong dependance on the real part of the permittivities of the metal and gain medium, and that its minimum value is 4/π(3.53dB). The origin of this minimum value is discussed. It is also shown that amplifier configurations supporting strongly confined surface plasmon-polaritons suffer from a large noise figure, which follows from an enhanced spontaneous emission rate due to the Purcell effect.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011 (1)

I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B 83, 081414(R) (2011).
[CrossRef]

2010 (5)

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35, 1197–1199 (2010).
[CrossRef] [PubMed]

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

2009 (7)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

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, 1110–1113 (2009).
[CrossRef] [PubMed]

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009).
[CrossRef]

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

A. Hryciw, Y. Jun, and M. Brongersma, “Plasmon-enhanced emission from optically-doped MOS light sources,” Opt. Express 17, 185–192 (2009).
[CrossRef] [PubMed]

I. De Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express 17, 20191–2020 (2009).
[CrossRef] [PubMed]

2008 (4)

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

2007 (1)

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

2006 (1)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

2005 (2)

J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

2004 (1)

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

2002 (1)

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

2000 (1)

1999 (1)

1986 (1)

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (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]

1978 (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

1964 (1)

H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE 52, 165–172 (1964).
[CrossRef]

Albrektsen, O.

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

Ambati, M.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Atwater, H.

S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Barnes, W. L.

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Berini, P.

I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B 83, 081414(R) (2011).
[CrossRef]

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

I. De Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express 17, 20191–2020 (2009).
[CrossRef] [PubMed]

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009).
[CrossRef]

I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000).
[CrossRef] [PubMed]

Bolger, P. M.

Bouhelier, A.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

Bratkovsky, A.

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

Brongersma, M.

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Chen, C.

Colas des Francs, G.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Danz, N.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

De Leon, I.

I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B 83, 081414(R) (2011).
[CrossRef]

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

I. De Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express 17, 20191–2020 (2009).
[CrossRef] [PubMed]

I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

Dereux, A.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Desurvire, E.

E. Desurvire, Erbium Doped Fiber Amplifiers (Wiley-Interscience, 1994).

Dickson, W.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Eijkemans, T. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Eng, L.

J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Fainman, Y.

Feng, D.

Finot, C.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Ford, G. W.

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

Gather, M. C.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

Geluk, E. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Genov, D. A.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Grafstrom, S.

J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Grandidier, J.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Henry, C. H.

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (1986).
[CrossRef]

Herz, E.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Hickey, S. G.

Hill, M. T.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Holmstrom, P.

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

Hryciw, A.

Inouye, Y.

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

Johansson, M.

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

Jun, Y.

Kambhampati, D.

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

Kawata, S.

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

Knoll, W.

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

Kogelnik, H.

H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE 52, 165–172 (1964).
[CrossRef]

Krasavin, A. V.

Kwon, S.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Lee, Y.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Leosson, K.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

Li, J.

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

Liebscher, L.

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Ma, R.-M.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Maier, S.

S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Markey, L.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Massenot, S.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Meerholz, K.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

Nam, S. H.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Narimanov, E. E.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Neumann, T.

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

Nezhad, M.

Nielsen, M. G.

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

Noginov, M. A.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Notzel, R.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Oei, Y.-S.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Pile, D. F. P.

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Radko, I. P.

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

Seidel, J.

J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Shalaev, V. 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, 1110–1113 (2009).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, California, 1986).

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Skryabin, D. V.

Smalbrugge, B.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Smit, M. K.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Stockman, M. I.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Stout, S.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Suteewong, T.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Tanev, S.

Tetz, K.

Thylen, L.

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

Turkiewicz, J. P.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Tzolov, V. P.

Ulin-Avila, E.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Van Otten, F. W. M.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Veldhoven, P. J. V.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Verma, P.

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

Vries, T. D.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Waardt, H. D.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Wang, S.-Y.

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[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]

Weeber, J.-C.

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Wiesner, U.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Yariv, A.

H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE 52, 165–172 (1964).
[CrossRef]

Zayats, A. V.

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Zhu, G.

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Zhu, Y.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Adv. Funct. Mater. (1)

T. Neumann, M. Johansson, D. Kambhampati, and W. Knoll, “Surface-plasmon fluorescence spectroscopy,” Adv. Funct. Mater. 12, 575–586 (2002).
[CrossRef]

Adv. Opt. Photon. (1)

IEEE J. Quantum Electron. (1)

L. Thylen, P. Holmstrom, A. Bratkovsky, J. Li, and S.-Y. Wang, “Limits on integration as determined by power dissipation and signal-to-noise ratio in loss-compensated photonic integrated circuits based on metal/quantum-dot materials,” IEEE J. Quantum Electron. 46, 518–524 (2010).
[CrossRef]

J. Appl. Phys. (1)

S. Maier and H. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

J. Lightwave Technol. (2)

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4, 288–297 (1986).
[CrossRef]

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

Nano Lett. (2)

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

J. Grandidier, G. Colas des Francs, S. Massenot, A. Bouhelier, L. Markey, J.-C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9, 2935–2939 (2009).
[CrossRef] [PubMed]

Nature (3)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

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, 1110–1113 (2009).
[CrossRef] [PubMed]

Nature Mater. (1)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Mater. 7, 442–453 (2008).
[CrossRef]

Nature Photon. (4)

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

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer.” Nature Photon. 4, 457–461 (2010).
[CrossRef]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. V. Veldhoven, F. W. M. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007).
[CrossRef]

New J. Phys. (1)

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” New J. Phys. 10, 105018 (2008).
[CrossRef]

Opt. Express (4)

Opt. Express. (1)

I. P. Radko, M. G. Nielsen, O. Albrektsen, and S. I. Bozhevolnyi, “Stimulated emission of surface plasmon polaritons by lead-sulphide quantum dots at near infra-red wavelengths.” Opt. Express. 18, 18633–18641 (2010).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. B (2)

I. De Leon and P. Berini, “Spontaneous emission in long-range surface plasmon-polariton amplifiers,” Phys. Rev. B 83, 081414(R) (2011).
[CrossRef]

I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401(R) (2008).
[CrossRef]

Phys. Rev. Lett. (2)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution.” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

Proc. IEEE (1)

H. Kogelnik and A. Yariv, “Considerations of noise and schemes for its reduction in laser amplifiers,” Proc. IEEE 52, 165–172 (1964).
[CrossRef]

Science (1)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef] [PubMed]

Other (3)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

E. Desurvire, Erbium Doped Fiber Amplifiers (Wiley-Interscience, 1994).

A. E. Siegman, Lasers (University Science Books, California, 1986).

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Figures (3)

Fig. 1
Fig. 1

Amplifier geometries considered: cross-sectional view of (a) a single interface between a metal and a gain medium, and (b) a metal film of thickness tm bounded by identical gain media on both sides; (c) top view of either (a) or (b). The metal and the gain media are characterized by the complex permittivities ɛm and ɛg, respectively. The active structure consists of an amplifier region of length la, limited by passive regions for z < –la and z > 0. A Gaussian SPP propagates through the amplifier in the +z direction having its beam waist, w0, located at z = 0.

Fig. 2
Fig. 2

High gain noise figure (a,c) and normalized spontaneous emission rates in logarithmic scale (b,d) as a function of the wavelength, calculated for the two SPP amplifiers in Fig. 1 using silver (Drude model) as the metal and three gain media exhibiting different ɛg. The vertical dashed lines in (a,c) indicate the location of the SPP energy asymptote for the case ɛg = 13.8. The horizontal dashed lines in (a,c) mark the noise figure value of 4 / π. ηι = 1 is assumed for all calculations. NF1, 〈Γ1〉: single-interface SPP. NF2, 〈Γ2〉: LRSPP.

Fig. 3
Fig. 3

(a) High gain noise figure of the LRSPP amplifier as a function of the wavelength and for several values of tm. NF1 is also shown for reference. (b) The values of fa and − fb in Eq. (13) plotted as a function of the wavelength and for several values of tm. ηι = 1 and ɛg = 2.25 are assumed for all calculations.

Equations (19)

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E y , ι ( x , y , z ) = E y , ι ( y ) q ˜ ι ( z ) exp { i κ ι [ z + x 2 2 q ˜ ι ( z ) ] } .
NF = 2 n N + 1 G + n N ( n N + 1 ) G 2 n 0 ,
NF 2 ξ γ A B ν .
A ι = [ I ι ( x , y ) dxdy ] 2 [ I ι ( x , y ) ] 2 dxdy ,
I ι ( x , y ) | E y , ι ( x , y , z = 0 ) | 2 = | E y , ι ( y ) | 2 exp ( β ι z R , ι x 2 )
A ι = ι δ ι π w 0 2 ,
Γ ι ( y 0 ) = Γ 0 Re u ι du S ( u , y 0 ) .
Γ ι = Γ 0 ι δ ι g dy Re u ι du S ( u , y 0 ) ,
N ι = γ ι + α ι σ e ( ν 0 ) = 2 β g 2 g ( ν 0 ) Γ 0 π ( γ ι + α ι ) ,
ξ ι = g ( ν 0 ) B ν N ι Γ ι F ι ,
N F ι = 2 K ι η ι g dy Re u ι du S ( u , y ) ,
N F ι = 4 η 1 π ( ɛ m ɛ g + ɛ m ) 3 / 2 .
( N F ι N F 0 ) dB = ( Γ ι Γ 0 / 2 ) dB f a + ( F ι A ι F 0 A 0 ) dB f b ,
( y 0 ) = Γ 0 h ν 0 Re 0 du S ( u , y 0 ) .
S ( u , y 0 ) = u 2 1 u 2 [ u 2 R TM + ( u , y 0 ) + ( 1 u 2 ) R TM ( u , y 0 ) + R TE + ( u , y 0 ) ]
Γ ι ( y 0 ) = ι ( y 0 ) h ν 0 = Γ 0 Re u ι du S ( u , y 0 ) .
1 ( y 0 ) = Γ 0 h ν 0 Re [ π ( κ 1 / β 0 ) 5 ( ɛ g ) 2 ( ɛ m ) 1 / 2 exp ( 2 κ 1 y 0 ɛ g / ɛ m ) ] ,
Re u 1 du S ( u , y 0 ) = π ( β 1 / β 0 ) 5 ( ɛ g ) 2 ( ɛ m ) 1 / 2 exp ( 2 β 1 y 0 ɛ g / ɛ m ) + O ( ɛ m ) ,
0 dy Re u 1 du S ( u , y ) λ 0 4 ɛ g ɛ m 2 ( ɛ g + ɛ m ) 2 .

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