Abstract

Realistic spasers are numerically modeled within classical electrodynamics scattering framework using intensity-dependent Lorentzian dielectric function. Quantum mechanical effects are accounted for via saturation broadening. Spasers based on silver nano-shells and nanorods with strong field inhomogeneity and retardation are studied in detail. Fields and optical cross-sections are exhaustively analyzed upon variation of three control parameters: the amplitude of the gain Lorentzian, the detuning of the driving frequency from the spaser generation frequency, and the strength of the external E-field. An externally driven spaser demonstrates bistability for E-fields and optical cross-sections, while a freely generating spaser corresponds to the limiting case of vanishing external field. Gain saturation removes singularities and unphysical post-threshold behavior frequently reported with linear simulations. A small shift of the spaser generation frequency with increasing available gain level is observed.

© 2015 Optical Society of America

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2015 (3)

B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photonics 2(7), 993–999 (2015).
[Crossref]

J. B. Khurgin, “Ultimate limit of field confinement by surface plasmon polaritons,” Faraday Discuss. 178, 109–122 (2015).
[Crossref] [PubMed]

J. Song, Y. L. Tian, S. Ye, L. C. Chen, X. Peng, and J. L. Qu, “Characteristic analysis of low-threshold plasmonic lasers using Ag nanoparticles with various shapes using photochemical synthesis,” J. Lightwave Technol. 33(15), 3215–3223 (2015).
[Crossref]

2014 (4)

V. M. Parfenyev and S. S. Vergeles, “Quantum theory of a spaser-based nanolaser,” Opt. Express 22(11), 13671–13679 (2014).
[Crossref] [PubMed]

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
[Crossref]

J. B. Khurgin and G. Sun, “Comparative analysis of spasers, vertical-cavity surface-emitting lasers and surface-plasmon-emitting diodes,” Nat. Photonics 8(6), 468–473 (2014).
[Crossref]

2013 (9)

X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

X. L. Zhong and Z. Y. Li, “All-analytical semiclassical theory of spaser performance in a plasmonic nanocavity,” Phys. Rev. B 88(8), 085101 (2013).
[Crossref]

N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
[Crossref] [PubMed]

X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
[Crossref] [PubMed]

P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

N. Arnold, L. J. Prokopeva, and A. V. Kildishev, “Modeling the local response of gain media in time-domain,” Annual Review of Progress in Applied Computational Electromagnetics 29, 771–776 (2013).

D. G. Baranov, E. S. Andrianov, A. P. Vinogradov, and A. A. Lisyansky, “Exactly solvable toy model for surface plasmon amplification by stimulated emission of radiation,” Opt. Express 21(9), 10779–10791 (2013).
[Crossref] [PubMed]

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21(11), 13467–13478 (2013).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, R. F. Oulton, D. N. Neshev, O. Hess, and Y. S. Kivshar, “Scattering of core-shell nanowires with the interference of electric and magnetic resonances,” Opt. Lett. 38(14), 2621–2624 (2013).
[Crossref] [PubMed]

2012 (7)

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys-Usp 55(10), 1046–1053 (2012).
[Crossref]

H. P. Zhang, J. Zhou, W. B. Zou, and M. He, “Surface plasmon amplification characteristics of an active three-layer nanoshell-based spaser,” J. Appl. Phys. 112, 074309 (2012).

V. M. Parfenyev and S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86(4), 043824 (2012).
[Crossref]

J. B. Khurgin and G. Sun, “How small can “Nano” be in a “Nanolaser”?” Nanophotonics-Berlin 1, 3–8 (2012).

J. Pan, Z. Chen, J. Chen, P. Zhan, C. J. Tang, and Z. L. Wang, “Low-threshold plasmonic lasing based on high-Q dipole void mode in a metallic nanoshell,” Opt. Lett. 37(7), 1181–1183 (2012).
[Crossref] [PubMed]

I. E. Protsenko, “Quantum theory of dipole nanolasers,” J. Russ. Laser Res. 33(6), 559–577 (2012).
[Crossref]

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
[Crossref] [PubMed]

2011 (4)

2010 (4)

X. Fan, Z. Shen, and B. Luk’yanchuk, “Huge light scattering from active anisotropic spherical particles,” Opt. Express 18(24), 24868–24880 (2010).
[Crossref] [PubMed]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

2009 (1)

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

2006 (2)

A. Y. Smuk and N. M. Lawandy, “Spheroidal particle plasmons in amplifying media,” Appl. Phys. B 84(1-2), 125–129 (2006).
[Crossref]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1106–1115 (2006).
[Crossref]

2005 (1)

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

2004 (1)

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

2003 (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(2), 027402 (2003).
[Crossref] [PubMed]

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83(22), 4625–4627 (2003).
[Crossref]

1998 (1)

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Agrawal, G. P.

W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
[Crossref]

Andrianov, E. S.

Arnold, N.

N. Arnold, L. J. Prokopeva, and A. V. Kildishev, “Modeling the local response of gain media in time-domain,” Annual Review of Progress in Applied Computational Electromagnetics 29, 771–776 (2013).

N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
[Crossref] [PubMed]

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Baranov, D. G.

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(7259), 1110–1112 (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(2), 027402 (2003).
[Crossref] [PubMed]

Cai, G. W.

P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

Chen, J.

Chen, L. C.

Chen, Z.

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Ding, B.

N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
[Crossref] [PubMed]

Ding, P.

P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

Dorofeenko, A. V.

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1106–1115 (2006).
[Crossref]

Esumi, K.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83(22), 4625–4627 (2003).
[Crossref]

Fan, C. Z.

P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

Fan, X.

Fedorov, I. A.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Feldmann, J.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
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X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
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X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
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García-Pomar, J. L.

Gordon, J. A.

Grosse, S.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
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X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
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W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
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A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
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S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
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P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
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He, M.

H. P. Zhang, J. Zhou, W. B. Zou, and M. He, “Surface plasmon amplification characteristics of an active three-layer nanoshell-based spaser,” J. Appl. Phys. 112, 074309 (2012).

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(7259), 1110–1112 (2009).
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S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
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N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
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J. B. Khurgin and G. Sun, “Comparative analysis of spasers, vertical-cavity surface-emitting lasers and surface-plasmon-emitting diodes,” Nat. Photonics 8(6), 468–473 (2014).
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J. B. Khurgin and G. Sun, “How small can “Nano” be in a “Nanolaser”?” Nanophotonics-Berlin 1, 3–8 (2012).

Kildishev, A. V.

X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

N. Arnold, L. J. Prokopeva, and A. V. Kildishev, “Modeling the local response of gain media in time-domain,” Annual Review of Progress in Applied Computational Electromagnetics 29, 771–776 (2013).

X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
[Crossref] [PubMed]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1106–1115 (2006).
[Crossref]

Kivshar, Y. S.

Klar, T.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
[Crossref]

Klar, T. A.

N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1106–1115 (2006).
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Kuwata, H.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83(22), 4625–4627 (2003).
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Kuznetsov, A. I.

B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photonics 2(7), 993–999 (2015).
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A. Y. Smuk and N. M. Lawandy, “Spheroidal particle plasmons in amplifying media,” Appl. Phys. B 84(1-2), 125–129 (2006).
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Li, Z. Y.

X. L. Zhong and Z. Y. Li, “All-analytical semiclassical theory of spaser performance in a plasmonic nanocavity,” Phys. Rev. B 88(8), 085101 (2013).
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S. Y. Liu, J. Li, F. Zhou, L. Gan, and Z. Y. Li, “Efficient surface plasmon amplification from gain-assisted gold nanorods,” Opt. Lett. 36(7), 1296–1298 (2011).
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P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

Linden, S.

Lisyansky, A. A.

Liu, S. Y.

Liu, W.

Luk’yanchuk, B.

Luk’yanchuk, B. S.

B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photonics 2(7), 993–999 (2015).
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Meinzer, N.

Meng, X.

X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
[Crossref] [PubMed]

Meng, X. G.

X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

Miroshnichenko, A. E.

Miyano, K.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83(22), 4625–4627 (2003).
[Crossref]

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Neshev, D. N.

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

O’Reilly, E. P.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Oulton, R. F.

Pan, J.

Paniagua-Dominguez, R.

B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photonics 2(7), 993–999 (2015).
[Crossref]

Parfenyev, V. M.

V. M. Parfenyev and S. S. Vergeles, “Quantum theory of a spaser-based nanolaser,” Opt. Express 22(11), 13671–13679 (2014).
[Crossref] [PubMed]

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

V. M. Parfenyev and S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86(4), 043824 (2012).
[Crossref]

Peng, X.

Perner, M.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
[Crossref]

Premaratne, M.

W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
[Crossref]

Prokopeva, L. J.

N. Arnold, L. J. Prokopeva, and A. V. Kildishev, “Modeling the local response of gain media in time-domain,” Annual Review of Progress in Applied Computational Electromagnetics 29, 771–776 (2013).

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I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Pukhov, A. A.

Pusch, A.

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Qu, J. L.

Ruther, M.

Samoilov, V. N.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Sarychev, A. K.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Shalaev, V. M.

X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
[Crossref] [PubMed]

X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1106–1115 (2006).
[Crossref]

Shen, Z.

Smuk, A. Y.

A. Y. Smuk and N. M. Lawandy, “Spheroidal particle plasmons in amplifying media,” Appl. Phys. B 84(1-2), 125–129 (2006).
[Crossref]

Song, J.

Soukoulis, C. M.

Spirkl, W.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
[Crossref]

Stockman, M. I.

W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
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M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
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M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
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M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
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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(2), 027402 (2003).
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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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Sun, G.

J. B. Khurgin and G. Sun, “Comparative analysis of spasers, vertical-cavity surface-emitting lasers and surface-plasmon-emitting diodes,” Nat. Photonics 8(6), 468–473 (2014).
[Crossref]

J. B. Khurgin and G. Sun, “How small can “Nano” be in a “Nanolaser”?” Nanophotonics-Berlin 1, 3–8 (2012).

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Tamaru, H.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83(22), 4625–4627 (2003).
[Crossref]

Tanaka, K.

X. G. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

X. Meng, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Wavelength-tunable spasing in the visible,” Nano Lett. 13(9), 4106–4112 (2013).
[Crossref] [PubMed]

Tang, C. J.

Tartakovsky, G. T.

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

Tian, Y. L.

Tsakmakidis, K. L.

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Uskov, A. V.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Vergeles, S. S.

V. M. Parfenyev and S. S. Vergeles, “Quantum theory of a spaser-based nanolaser,” Opt. Express 22(11), 13671–13679 (2014).
[Crossref] [PubMed]

I. A. Fedorov, V. M. Parfenyev, S. S. Vergeles, G. T. Tartakovsky, and A. K. Sarychev, “Allowable Number of Plasmons in Nanoparticle,” JETP Lett. 100(8), 530–534 (2014).
[Crossref]

V. M. Parfenyev and S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86(4), 043824 (2012).
[Crossref]

Vinogradov, A. P.

von Plessen, G.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80(19), 4249–4252 (1998).
[Crossref]

Voshchinnikov, N. V.

B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photonics 2(7), 993–999 (2015).
[Crossref]

Wang, J. Q.

P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
[Crossref]

Wang, Z. L.

Wegener, M.

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Wuestner, S.

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
[Crossref] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Ye, S.

Yuan, H. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Zaimidoroga, O. A.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Zhan, P.

Zhang, H. P.

H. P. Zhang, J. Zhou, W. B. Zou, and M. He, “Surface plasmon amplification characteristics of an active three-layer nanoshell-based spaser,” J. Appl. Phys. 112, 074309 (2012).

Zhong, X. L.

X. L. Zhong and Z. Y. Li, “All-analytical semiclassical theory of spaser performance in a plasmonic nanocavity,” Phys. Rev. B 88(8), 085101 (2013).
[Crossref]

Zhou, F.

Zhou, J.

H. P. Zhang, J. Zhou, W. B. Zou, and M. He, “Surface plasmon amplification characteristics of an active three-layer nanoshell-based spaser,” J. Appl. Phys. 112, 074309 (2012).

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(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Zhu, W. R.

W. R. Zhu, M. Premaratne, S. D. Gunapala, G. P. Agrawal, and M. I. Stockman, “Quasi-static analysis of controllable optical cross-sections of a layered nanoparticle with a sandwiched gain layer,” J. Opt. 16(7), 075003 (2014).
[Crossref]

Ziolkowski, R. W.

Zou, W. B.

H. P. Zhang, J. Zhou, W. B. Zou, and M. He, “Surface plasmon amplification characteristics of an active three-layer nanoshell-based spaser,” J. Appl. Phys. 112, 074309 (2012).

ACS Nano (1)

A. Pusch, S. Wuestner, J. M. Hamm, K. L. Tsakmakidis, and O. Hess, “Coherent amplification and noise in gain-enhanced nanoplasmonic metamaterials: a Maxwell-Bloch Langevin approach,” ACS Nano 6(3), 2420–2431 (2012).
[Crossref] [PubMed]

ACS Photonics (1)

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Annual Review of Progress in Applied Computational Electromagnetics (1)

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N. Arnold, B. Ding, C. Hrelescu, and T. A. Klar, “Dye-doped spheres with plasmonic semi-shells: Lasing modes and scattering at realistic gain levels,” Beilstein J. Nanotechnol. 4, 974–987 (2013).
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P. Ding, J. N. He, J. Q. Wang, C. Z. Fan, G. W. Cai, and E. J. Liang, “Low-threshold surface plasmon amplification from a gain-assisted core-shell nanoparticle with broken symmetry,” J. Opt. 15(10), 105001 (2013).
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Figures (14)

Fig. 1
Fig. 1

Geometry (left) and typical field distribution (right) for the (almost) quasi-static dipolar core-shell spaser. Gain core radius a1 = 10 nm, Ag shell thickness h = 2.7 nm, background material is air. The right panel shows the distribution of the normalized field amplitude ∣(E)∣/Esat in the plane of incidence defined by the k-vector and external field vector e3, for a post-threshold gain, εL = 1.5εthr and an external field amplitude e3 = 0.006 Esat. The shape of the field distribution is similar for other parameter values and (in the vicinity of the structure) is similar to that in a freely generating spaser. The field e1 inside the core is practically constant.

Fig. 2
Fig. 2

(a), (c) - core field amplitude |e1|, and (b), (d) - scattering cross-section σsca of the driven dipolar core-shell spaser (Fig. 1) as a function of ω and external field e3. (a), (b) refer to a gain exactly at threshold εL = εthr, (c), (d) to the post-threshold gain, εL = 1.5εthr.

Fig. 3
Fig. 3

(a), (c) - core field amplitude |e1|, and (b), (d) - scattering cross-section σsca of the driven dipolar core-shell spaser (Fig. 1) as a function of ω and external field e3. (a), (b) refer to a gain exactly at threshold εLthr, (c), (d) to the post-threshold gain, εL=1.5εthr.

Fig. 4
Fig. 4

Influence of the external field e3 on the saturated optical cross-sections for the driven quasi-static dipolar core-shell spaser at threshold gain level εL = εthr. Other parameters are as in Fig. 2. Panels (a), (b) and (c) show extinction, scattering and absorption respectively. The curves are labeled in panel (b). Solid curves refer to the linear problem without saturation, s = 0. Curves with nonlinear saturation, s = |e1|2/Esat2 are shown for several external field values e3 listed in (b). For clarity, not all curves are shown in each panel - (a) and (c) include curves at small e3, which better illustrate the transition to the linear case without saturation. The dashed ellipse in (a) marks the region where, for low values of the external field e3, the extinction cross-section σext≤ 0, which corresponds to a loss compensation. (d) shows all three optical cross-sections for the external field value e3 = 0.006Esat without saturation (thick curves) and with saturation (thin curves).

Fig. 5
Fig. 5

Analytical (lines) and numerical (symbols) results for a small dipolar core-shell spaser in air, which is shown in Fig. 1. Analytical results are normalized to the quasi-static (QS) values εthr,QS and ωthr,QS, while numerical ones are normalized to the numerical thresholds εthr,Num and ωthr,Num. The numerical field value in the core center was used. (a) The dependence of the amplitude of the core field ∣e1∣ on the external field e3 for three values of available gain εL (labeled in the plot) exactly on resonance ω = ωthr. The solid curve for εL = εthr is a cut of the surface shown in Fig. 2(a) at ω/ωthr = 1, while the solid bistable curve for εL = 1.5εthr is a cut of the Fig. 2(c), also at ω/ωthr = 1. The parameters marked by a full circle on the upper curve were used to draw a field distribution in the right panel of Fig. 1. (b) The dependence of the amplitude of the core field ∣e1∣ on frequency detuning above threshold (εL = 1.5εthr) for several values of external field labeled in the plot. Solid curves correspond to the cuts of the surface shown in Fig. 2(c) at corresponding e3/Esat values (curve for e3/Esat = 0.0042 is not shown).

Fig. 6
Fig. 6

Large shell quadrupolar spaser driven by the external field, similar to Fig. 5. Symbols represent numerical results, curves are guides for the eye. The inset in (b) shows the geometry of the structure. Core radius a1 = 50 nm, Ag shell thickness h2 = 4.3 nm, outer gain shell h3 = 50 nm; material parameters are described in the text. Results are normalized to the numerical εthr, Num and ωthr, Num values given in the text. (a) The dependence of the maximum field in the gain material Emax on the external field e4 for three values of available gain εL (labeled in the plot) near the resonance. This maximum is achieved in the outer gain shell (subscript “3”), near the Ag surface, in the plane of incidence defined by the k-vector and external field vector e4, at an angle 45° with respect to the former. The inset shows the field distribution for a gain level of εL = 1.5εthr and an external field e4/Esat = 0.0124, indicated by the filled circle at the upper curve. The shape of the filed distribution is similar for other parameters. (b) The dependence of the maximum field Emax on the frequency detuning above threshold (εL = 1.5εthr) for three values of external field e4 labeled in the plot.

Fig. 7
Fig. 7

Dependence of cross-sections of the large shell quadrupolar spaser on the external driving field e4 in the presence of saturation. Numerical results are given near the generation frequency, ω = 1.0001ωthr. Other parameters are as in Fig. 6. (a) εL = 0.5εthr , (b) εL = εthr, (c) and (d) εL = 1.5εthr , with increasing and decreasing external field respectively, to show different branches of solutions. The dashed rectangle indicates the region where the structure can work as an amplifier, as discussed in the text. Note the difference in ordinate scale between all panels. The behavior can be better understood by comparing with Fig. 2(d) for the scattering and Figs. 13(a), (b) for absorption and extinction.

Fig. 8
Fig. 8

Spheroidal dipolar spaser driven by the external field e3, numerical results. a) The dependence of the maximum field in the gain material (near the tip of the Ag spheroid) on the available gain εL at the resonance ω = ωthr, for the external field e3 = 0.001 Esat. The upper curve corresponds to the spherical gain material with the radius a2 = 4c = 80 nm, while the lower one to a gain shell of constant thickness h2 = 10 nm along all axes of the Ag spheroid. Long semi-axis c = 20 nm in both cases. The results are normalized to the corresponding (somewhat different) numerical εthr and ωthr values. The insets illustrate the geometries of both structures. Further details about the geometries, material parameters and threshold values are given in the text. The two panels above the main plot show the normalized field amplitude (∣(E)∣/Esat) in the plane of incidence defined by the k-vector and external field vector e3, for the gain values εL = 3εthr, indicated by the small filled circle and ellipse on the corresponding curves. The shape of the field distribution is similar for other parameter values. The dashed circle indicates the region studied in further detail in Figs. (b) and (c). (b) The region where (for the a2 = 80 nm case) the high-field branch of solution ceases to exist at ω = ωthr due to a drift in spaser generation frequency with the available gain, ωgen(εL). (c) The numerical cut of the solution surface for εL = 4.7135εthr, near the termination point in Fig. (b). The arrow shows the shift of the spaser generation frequency for this level of gain. The crosses in Figs. (b) and (c) indicate the same solution point.

Fig. 9
Fig. 9

Influence of available gain εL on the driven core-shell spaser at threshold frequency ωthr. Other parameters are the same as in Fig. 2. (a) Core field amplitude |e1| as a function of available gain εL and external field e3. (b) Scattering cross section as a function of the same variables. (c) and (d) are similar to (a) and (b), but for a small detuning ω = 0.999ωthr which removes the singularities for σsca at zero external field.

Fig. 10
Fig. 10

The regions of bistability, where 3 solutions exist for the quasi-static dipolar core-shell spaser. (a) - in the plane of frequency ω and external field e3, above threshold, for εL = 1.5εthr. This picture is a projection of the Figs. 2(c), (d). (b) - in the plane of available gain εL and external field e3, for small detuning ω = 0.999ωthr. This picture is a projection of the Figs. 9(c), (d).

Fig. 11
Fig. 11

Dependence of core field amplitude |e1| and scattering cross-section σsca on frequency ω and available gain εL for the driven core-shell spaser, as in Figs. 3(b) and 3(d), but for the trice smaller external field e3 = 0.002Esat. (a) core field, arrow indicates the limiting case of a freely generating spaser, (b) scattering cross section.

Fig. 12
Fig. 12

Saturated cross-sections, dependence on frequency and external field: absorption (a), (c) and extinction (b), (d). Other parameters as in Fig. 4. (a), (b) - at gain threshold εL = εthr; (c), (d) – above the gain threshold, at εL = 1.5εthr.

Fig. 13
Fig. 13

Saturated cross-sections, dependence on available gain level and external field at fixed frequency: absorption (a), (c) and extinction (b, (d). Other parameters as in Fig. 12 (a), (b) – exactly on resonance, ω = ωthr, (c) (d) - at small blue detuning from threshold frequency ω = 1.001ωthr.

Fig. 14
Fig. 14

Influence of saturation on the absorption and extinction cross-sections for a quasi-static dipolar core-shell spaser driven by the external field e3 = 0.006Esat. Dependence on frequency ω and available gain εL. Parameters are as in Fig. 3. (a), (c) and (e) show absorption σabs, while (b), (d) and (f) show extinction cross section σext. (a) and (b) refer to the linear problem without saturation (s = 0), while (c)-(f) include gain saturation. (e) and (f) illustrate the behavior at higher available gain levels and have reverse orientation of the εL axis, which better illustrates the global shape of the surfaces.

Equations (50)

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ε G = ε h ε L ( ω L ω+i γ L /2) γ L /2 ( ω L ω) 2 + ( γ L /2) 2 (1+s) , s= | E | 2 / E sat 2
e 1 = 9 ε 2 ε 3 D e 3 , D=( ε 1 +2 ε 2 )( ε 2 +2 ε 3 )+2f( ε 1 ε 2 )( ε 2 ε 3 ),f= a 1 3 a 2 3
ε h ε thr ( ω L ω thr +i γ L /2) γ L /2 ( ω L ω thr ) 2 + ( γ L /2) 2 =2 ε 2 ( ε 2 +2 ε 3 )f( ε 2 ε 3 ) ( ε 2 +2 ε 3 )+2f( ε 2 ε 3 )
ω thr QS =2.4945eV( λ thr QS =497.03nm), ε thr QS =0.1549
[ ( ε L ε thr 1 ) | e 1 | 2 E sat 2 ] e 1 E sat =C e 3 E sat
s= | e 1 | 2 E sat 2 =( 1+ ( ω L ω thr γ L /2 ) 2 )( ε L ε thr 1 )
ω thr Mie =2.4840eV( λ thr Mie =499.13nm), ε thr Mie =0.1683
ω thr Num =2.4839eV( λ thr Num =499.15nm), ε thr Num =0.1684
ω thr Mie =1.6369eV( λ thr Mie =757.42nm), ε thr Mie =0.03084
ω thr Num =1.6368eV( λ thr Num =757.49nm), ε thr Num =0.03084
e 1 = C ε L / ε thr 1 e 3
ω thr Sph =1.6067eV( λ thr Sph =771.7nm), ε thr Sph =0.03937
ω thr Num =1.6124eV( λ thr Num =768.94nm), ε thr Num =0.03306
ω thr Num =1.6220eV( λ thr Num =764.41nm), ε thr Num =0.04215
N ˙ 2 = I p σ 03 ω 03 N 0 I s σ 12 ω 12 N 2 γ N N 2 = W p ( N tot N 2 )( W s + γ N ) N 2
I p , I s << γ 32,10 ω σ 03,12 ~1.5 10 9 W/cm 2
N 2 = W p N tot W p + γ N + W s = N 2 (0) 1+ W s W p + γ N
N 2 (0)= N 2 ( W s =0)= W p N tot W p + γ N
W p << γ N , W sat = γ N I sat = γ N ω 12 σ 12 ~6.6 10 5 W/cm 2 W p >> γ N , W sat = W p I sat = I p
σ 12 = σ 12,max 3 = 1 3 γ rad γ L 3 λ 2 2π n 2 = 8π μ 12 2 nk 3 γ L ε CGS = 2 μ 12 2 nk 3 γ L ε ε 0 SI
cε ε 0 E sat 2 2n = I sat = γ N ω 12 3 γ L ε ε 0 2 μ 12 2 nk E sat = 3 γ N γ L μ 12 ~1.76 10 6 V/m
ε L =4πα N 2 (0)= [ 4π μ 12 2 2 γ L I p σ 03 ω 03 N tot γ N , W p << γ N 4π μ 12 2 2 γ L N tot , W p >> γ N
τ inv > ( W p + γ N ) 1
ε L,LL = ε L ( 3 ε h +2 ) 2 I sat,LL = I sat ( 3 ε h +2 ) 2 ,or E sat,LL = E sat 3 ε h +2
n pl ~ ε 0 ε G E sat 2 V ω
φ 1 = e 1 x, E 1 =( e 1 ,0,0) φ 2 = e 2 x+ d 2 x r 3 , E 2 =( e 2 ,0,0)+ d 2 r 5 (3 x 2 r 2 ,3xy,3xz) φ 3 = e 3 x+ d 3 x r 3 , E 3 =( e 3 ,0,0)+ d 3 r 5 (3 x 2 r 2 ,3xy,3xz)
e 1 = e 2 + d 2 a 1 3 , ε 1 e 1 = ε 2 e 2 + 2 ε 2 d 2 a 1 3 e 2 + d 2 a 2 3 = e 3 + d 3 a 2 3 , ε 2 e 2 + 2 ε 2 d 2 a 2 3 = ε 3 e 3 + 2 ε 3 d 3 a 2 3
d 3 = ( ε 1 +2 ε 2 )( ε 2 ε 3 )+f( ε 1 ε 2 )(2 ε 2 + ε 3 ) ( ε 1 +2 ε 2 )( ε 2 +2 ε 3 )+2f( ε 1 ε 2 )( ε 2 ε 3 ) a 2 3 e 3 ,f= a 1 3 a 2 3
σ sca = 6π k 2 | A 1 | 2 , σ ext = 6π k 2 Re A 1 , σ abs = 6π k 2 ( | A 1 | 2 Re A 1 ) A 1 = 2i 3 k 3 a 2 3 ε 3 3/2 ( ε 2 ε 3 )( ε 1 +2 ε 2 )+f( ε 1 ε 2 )(2 ε 2 + ε 3 ) ( ε 1 +2 ε 2 )( ε 2 +2 ε 3 )+2f( ε 1 ε 2 )( ε 2 ε 3 )
ε G (ω, ε L ,s) ε 1 =2 ε 2 ( ε 2 +2 ε 3 )f( ε 2 ε 3 ) ( ε 2 +2 ε 3 )+2f( ε 2 ε 3 ) F(ω,geometry)
ε G ( ω thr , ε thr ,0)=F( ω thr )
ε G ( ω gen , ε L ,s)=F( ω gen )
ε G ( ω gen = ω thr , ε L ,s)= ε G ( ω thr , ε thr ,0)
s= | e 1 | 2 E sat 2 =( 1+ ( ω L ω thr γ L /2 ) 2 )( ε L ε thr 1 )
D=( ε 1 ( e 1 )+2 ε 2 )( ε 2 +2 ε 3 )+2f( ε 1 ( e 1 ) ε 2 )( ε 2 ε 3 )=9 ε 2 ε 3 e 3 / e 1
D D ω (ω ω thr )+ D ε 1 ε 1,thr ε h ε thr ( ε L ε thr + ε 1,thr ε h ω L ω thr γ L /2 +i | e 1 | 2 E sat 2 )=9 ε 2 ε 3 e 3 e 1
[ δ L | e 1 * | 2 ] e 1 * =C e 3 *
e 1,3 * = e 1,3 / E sat , δ L = ε L ε thr ε thr ,C= 3 2f ε 3 ε 2 ε 3 ( 1+i ε h +2 ε 2 ε thr )
| e 1 * |= (| C | e 3 * ) 1/3 ,or| e 1 |= (| C | e 3 E sat 2 ) 1/3
C 1 =( ε 2 ε 3 )( ε 1 +2 ε 2 )+f( ε 1 ε 2 )(2 ε 2 + ε 3 ) σ sca = 8π 3 k 4 a 2 6 ε 3 | C 1 C 1/3 9 ε 2 | 2 ( E sat e 3 ) 4/3
D D thr + D ω (ω ω thr )+ D ε 1 [ ε 1 ε L ( ε L ε thr )+ ε 1 s s ]
D ε 1 =( ε 2 +2 ε 3 )+2f( ε 2 ε 3 )
ε 1,thr = ε h ε thr ( ω L ω thr +i γ L /2) γ L /2 ( ω L ω thr ) 2 + ( γ L /2) 2
ε 1 ε L = ( ω L ω thr +i γ L /2) γ L /2 ( ω L ω thr ) 2 + ( γ L /2) 2 = ε 1,thr ε h ε thr ε 1 s = ε thr ( ω L ω thr +i γ L /2) ( γ L /2) 3 ( ( ω L ω thr ) 2 + ( γ L /2) 2 ) 2 = ε thr ( ε 1,thr ε h ε thr ) 2 1 ω L ω thr γ L /2 +i
( ε L ε thr ε thr | e 1 | 2 E sat 2 ) e 1 e 3 = 9i ε 2 ε 3 D ε 1 ε thr = 3 2f ε 3 ε 2 ε 3 ( 1+i 2 ε 2 + ε h ε thr )
e 1 = C δ L e 3 ,or e 1 E sat = C δ L e 3 E sat
χ s [c m 3 ]= d 3 / e 3
ε1 ε+2 = 4π 3 ( N b χ b + N s χ s )= 4π 3 N b χ b + ε b 1 ε b +2
4π 3 N b χ b = ε b 1 ε b +2
ε =0

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