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. 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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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]

Gan, L.

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).
[Crossref]

Guler, U.

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).
[Crossref]

Hamm, J. M.

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).
[Crossref]

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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Hess, O.

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).
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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).
[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).
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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.

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. 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]

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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Lawandy, N. M.

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

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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Liang, E. J.

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).
[Crossref]

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.

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

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

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).
[Crossref] [PubMed]

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).
[Crossref]

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. 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]

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.

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

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, C. Hrelescu, and T. A. Klar, Institute of Applied Physics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria, are preparing a manuscript to be called “Minimal Spaser Threshold within Electrodynamic Framework: Shape, Size and Modes.”

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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 a 1 = 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)∣/E sat in the plane of incidence defined by the k-vector and external field vector e 3, for a post-threshold gain, ε L = 1.5ε thr and an external field amplitude e 3 = 0.006 E sat. 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 e 1 inside the core is practically constant.
Fig. 2
Fig. 2 (a), (c) - core field amplitude |e 1|, and (b), (d) - scattering cross-section σsca of the driven dipolar core-shell spaser (Fig. 1) as a function of ω and external field e 3. (a), (b) refer to a gain exactly at threshold εL = εthr , (c), (d) to the post-threshold gain, ε L = 1.5εthr .
Fig. 3
Fig. 3 (a), (c) - core field amplitude |e 1 |, and (b), (d) - scattering cross-section σ sca of the driven dipolar core-shell spaser (Fig. 1) as a function of ω and external field e 3 . (a), (b) refer to a gain exactly at threshold ε L thr , (c), (d) to the post-threshold gain, ε L =1.5ε thr .
Fig. 4
Fig. 4 Influence of the external field e 3 on the saturated optical cross-sections for the driven quasi-static dipolar core-shell spaser at threshold gain level ε L = ε thr. 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 = |e 1|2/E sat 2 are shown for several external field values e 3 listed in (b). For clarity, not all curves are shown in each panel - (a) and (c) include curves at small e 3, 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 e 3, the extinction cross-section σ ext≤ 0, which corresponds to a loss compensation. (d) shows all three optical cross-sections for the external field value e 3 = 0.006E sat 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 ε thr,QS and ω thr,QS, while numerical ones are normalized to the numerical thresholds ε thr,Num and ω thr,Num. The numerical field value in the core center was used. (a) The dependence of the amplitude of the core field ∣e 1∣ on the external field e 3 for three values of available gain ε L (labeled in the plot) exactly on resonance ω = ω thr. The solid curve for ε L = ε thr is a cut of the surface shown in Fig. 2(a) at ω/ω thr = 1, while the solid bistable curve for ε L = 1.5ε thr is a cut of the Fig. 2(c), also at ω/ω thr = 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 ∣e 1∣ on frequency detuning above threshold (ε L = 1.5ε thr) 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 e 3/E sat values (curve for e 3/E sat = 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 a 1 = 50 nm, Ag shell thickness h 2 = 4.3 nm, outer gain shell h 3 = 50 nm; material parameters are described in the text. Results are normalized to the numerical ε thr, Num and ω thr, Num values given in the text. (a) The dependence of the maximum field in the gain material E max on the external field e 4 for three values of available gain ε L (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 e 4, at an angle 45° with respect to the former. The inset shows the field distribution for a gain level of ε L = 1.5ε thr and an external field e 4/E sat = 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 E max on the frequency detuning above threshold (ε L = 1.5ε thr) for three values of external field e 4 labeled in the plot.
Fig. 7
Fig. 7 Dependence of cross-sections of the large shell quadrupolar spaser on the external driving field e 4 in the presence of saturation. Numerical results are given near the generation frequency, ω = 1.0001ω thr. Other parameters are as in Fig. 6. (a) ε L = 0.5ε thr , (b) ε L = ε thr, (c) and (d) ε L = 1.5ε thr , 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 e 3, numerical results. a) The dependence of the maximum field in the gain material (near the tip of the Ag spheroid) on the available gain ε L at the resonance ω = ω thr, for the external field e 3 = 0.001 E sat. The upper curve corresponds to the spherical gain material with the radius a 2 = 4c = 80 nm, while the lower one to a gain shell of constant thickness h 2 = 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 ε thr and ω thr 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 e 3, for the gain values ε L = 3ε thr, 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 a 2 = 80 nm case) the high-field branch of solution ceases to exist at ω = ω thr due to a drift in spaser generation frequency with the available gain, ω gen(ε L). (c) The numerical cut of the solution surface for ε L = 4.7135ε thr, 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 ε L on the driven core-shell spaser at threshold frequency ω thr. Other parameters are the same as in Fig. 2. (a) Core field amplitude |e 1| as a function of available gain ε L and external field e 3. (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 = 0.999ω thr which removes the singularities for σ sca 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 e 3, above threshold, for ε L = 1.5ε thr. This picture is a projection of the Figs. 2(c), (d). (b) - in the plane of available gain ε L and external field e 3, for small detuning ω = 0.999ω thr. This picture is a projection of the Figs. 9(c), (d).
Fig. 11
Fig. 11 Dependence of core field amplitude |e 1| and scattering cross-section σ sca on frequency ω and available gain ε L for the driven core-shell spaser, as in Figs. 3(b) and 3(d), but for the trice smaller external field e 3 = 0.002E sat. (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 ε L = ε thr; (c), (d) – above the gain threshold, at ε L = 1.5ε thr.
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 = ω thr, (c) (d) - at small blue detuning from threshold frequency ω = 1.001ω thr.
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 e 3 = 0.006E sat. Dependence on frequency ω and available gain ε L. Parameters are as in Fig. 3. (a), (c) and (e) show absorption σ abs, while (b), (d) and (f) show extinction cross section σ ext. (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 ε L 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 ) + 2 f ( 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 ) + 2 f ( ε 2 ε 3 )
ω thr QS = 2.4945 eV ( λ thr QS = 497.03 nm ) , 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 = 2.4840 eV ( λ thr Mie = 499.13 nm ) , ε thr Mie = 0.1683
ω thr Num = 2.4839 eV ( λ thr Num = 499.15 nm ) , ε thr Num = 0.1684
ω thr Mie = 1.6369 eV ( λ thr Mie = 757.42 nm) , ε thr Mie = 0.03084
ω thr Num = 1.6368 eV ( λ thr Num = 757.49 nm ) , ε thr Num = 0.03084
e 1 = C ε L / ε thr 1 e 3
ω thr Sph = 1.6067 eV ( λ thr Sph = 771.7 nm) , ε thr Sph = 0.03937
ω thr Num = 1.6124 eV ( λ thr Num = 768.94 nm ) , ε thr Num = 0.03306
ω thr Num = 1.6220 eV ( λ thr Num = 764.41 nm ) , 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 t o t 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 2 π n 2 = 8 π μ 12 2 n k 3 γ L ε CGS = 2 μ 12 2 n k 3 γ L ε ε 0 SI
c ε ε 0 E sat 2 2 n = I sat = γ N ω 12 3 γ L ε ε 0 2 μ 12 2 n k E sat = 3 γ N γ L μ 12 ~ 1.76 10 6 V/m
ε L = 4 π α N 2 ( 0 ) = [ 4 π μ 12 2 2 γ L I p σ 03 ω 03 N tot γ N , W p < < γ N 4 π 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 , 3 x y , 3 x z ) 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 , 3 x y , 3 x z )
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 ) + 2 f ( ε 1 ε 2 ) ( ε 2 ε 3 ) a 2 3 e 3 , f = a 1 3 a 2 3
σ sca = 6 π k 2 | A 1 | 2 , σ ext = 6 π k 2 Re A 1 , σ abs = 6 π k 2 ( | A 1 | 2 Re A 1 ) A 1 = 2 i 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 ) + 2 f ( 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 ) + 2 f ( 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 ) + 2 f ( 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 s a t , δ L = ε L ε thr ε thr , C = 3 2 f 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 = 8 π 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 ) + 2 f ( 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 = 9 i ε 2 ε 3 D ε 1 ε thr = 3 2 f 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
ε 1 ε + 2 = 4 π 3 ( N b χ b + N s χ s ) = 4 π 3 N b χ b + ε b 1 ε b + 2
4 π 3 N b χ b = ε b 1 ε b + 2
ε = 0

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