Abstract

We propose, solve, and discuss a simple model for a metamaterial incorporating optical gain: A single bosonic resonance is coupled to a fermionic (inverted) two-level-system resonance via local-field interactions. For given steady-state inversion, this model can be solved analytically, revealing a rich variety of (Fano) absorption/gain lineshapes. We also give an analytic expression for the fixed inversion resulting from gain pinning under steady-state conditions. Furthermore, the dynamic response of the “lasing SPASER”, i.e., its relaxation oscillations, can be obtained by simple numerical calculations within the same model. As a result, this toy model can be viewed as the near-field-optical counterpart of the usual LASER rate equations.

© 2008 Optical Society of America

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  5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
    [CrossRef] [PubMed]
  6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
    [CrossRef]
  7. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
    [CrossRef] [PubMed]
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  14. A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
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    [CrossRef]
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2008 (7)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[CrossRef]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

M. I. Stockman, "Spasers explained," Nature Photon. 2, 327-329 (2008).
[CrossRef]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

C. Manolatou and F. Rana, "Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers," IEEE J. Quantum Electron. 44, 435-447 (2008).
[CrossRef]

J. A. Gordon and R. W. Ziolkowski, "CNP optical metamaterials," Opt. Express 16, 6692-6716 (2008).
[CrossRef] [PubMed]

S.-W. Chang, C.-Y.A. Ni, and S. L. Chuang, "Theory for bowtie plasmonic nanolasers," Opt. Express 16, 10580-10595 (2008).
[CrossRef] [PubMed]

2007 (6)

J. A. Gordon and R. W. Ziolkowski, "The design and simulated performance of a coated nano-particle laser," Opt. Express 15, 2622-2653 (2007).
[CrossRef] [PubMed]

M. I. Stockman, "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality," Phys. Rev. Lett. 98,177404:1-4 (2007).
[CrossRef]

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1,41-48 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

2006 (4)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, "Negative-Index Metamaterials: Going Optical," IEEE J. Sel. Top. Quantum Electron. 12,1106-1115 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, "Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium," Opt. Lett. 31, 3022-3024 (2006).
[CrossRef] [PubMed]

2005 (3)

J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005).
[CrossRef]

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

2004 (2)

I. Avrutsky, "Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain," Phys. Rev. B 70, 155416:1-6 (2004).

P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express 12, 4072-4079 (2004).
[CrossRef] [PubMed]

2003 (1)

D. J. Bergman and M. I. Stockman, "Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems," Phys. Rev. Lett. 90, 027402:1-4 (2003).
[CrossRef]

2000 (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Adegoke, J.

Avrutsky, I.

I. Avrutsky, "Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain," Phys. Rev. B 70, 155416:1-6 (2004).

Bahoura, M.

Bergman, D. J.

D. J. Bergman and M. I. Stockman, "Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems," Phys. Rev. Lett. 90, 027402:1-4 (2003).
[CrossRef]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[CrossRef]

Chang, S.-W.

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[CrossRef]

Chuang, S. L.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

de Vries, T.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

de Waardt, H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Dolling, G.

Drachev, V. P.

Eijkemans, T. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Eng, L.

J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Fainman, Y.

Fedotov, V. A.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

Geluk, E. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Gordon, J. A.

Govyadinov, A. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Grafstroem, S.

J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005).
[CrossRef]

Hill, M. T.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[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. 12,1106-1115 (2006).
[CrossRef]

Klar, T. A.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, "Negative-Index Metamaterials: Going Optical," IEEE J. Sel. Top. Quantum Electron. 12,1106-1115 (2006).
[CrossRef]

Klein, M. W.

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

Koschny, T.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Kwon, S.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Lee, Y.-H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

Manolatou, C.

C. Manolatou and F. Rana, "Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers," IEEE J. Quantum Electron. 44, 435-447 (2008).
[CrossRef]

Mingaleev, S.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Nezhad, P.

Ni, C.-Y.A.

Noginov, M. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, "Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium," Opt. Lett. 31, 3022-3024 (2006).
[CrossRef] [PubMed]

Nötzel, R.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Oei, Y.-S.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Papasimakis, N.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Prosvirnin, S. L.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

Rana, F.

C. Manolatou and F. Rana, "Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers," IEEE J. Quantum Electron. 44, 435-447 (2008).
[CrossRef]

Ritzo, B. A.

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Seidel, J.

J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005).
[CrossRef]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[CrossRef]

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1,41-48 (2007).
[CrossRef]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, "Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium," Opt. Lett. 31, 3022-3024 (2006).
[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. 12,1106-1115 (2006).
[CrossRef]

Smalbrugge, B.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Small, C. E.

Smit, M. K.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, "Spasers explained," Nature Photon. 2, 327-329 (2008).
[CrossRef]

M. I. Stockman, "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality," Phys. Rev. Lett. 98,177404:1-4 (2007).
[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, 027402:1-4 (2003).
[CrossRef]

Tetz, K.

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Tritschler, T.

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

Turkiewicz, J. P.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

van Otten, F. W. M.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

van Veldhoven, P. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

Zheludev, N. I.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

Zhou, J.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Zhu, G.

Zhu, Y.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Ziolkowski, R. W.

Zschiedrich, L.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Manolatou and F. Rana, "Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers," IEEE J. Quantum Electron. 44, 435-447 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, "Negative-Index Metamaterials: Going Optical," IEEE J. Sel. Top. Quantum Electron. 12,1106-1115 (2006).
[CrossRef]

Nature Photon. (5)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1,41-48 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008).
[CrossRef]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008).
[CrossRef]

M. I. Stockman, "Spasers explained," Nature Photon. 2, 327-329 (2008).
[CrossRef]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Phys. Rev. B (2)

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).

I. Avrutsky, "Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain," Phys. Rev. B 70, 155416:1-6 (2004).

Phys. Rev. Lett. (5)

J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005).
[CrossRef]

M. I. Stockman, "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality," Phys. Rev. Lett. 98,177404:1-4 (2007).
[CrossRef]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, "Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems," Phys. Rev. Lett. 90, 027402:1-4 (2003).
[CrossRef]

Science (2)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

Other (5)

W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

This software suitable for modern Windows compatible personal computers can be downloaded via www.aph.uni-karlsruhe.de/wegener/de/publikationen and then "Publications 2008".

E. Hecht, Optics (Addison Wesley, 1987)

www.lumerical.com

W. W. Chow, S. W. Koch, and M. SargentIII, Semiconductor-Laser Physics (Springer, New York, 1994).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Illustration of a possible geometry for bringing the optical gain from a thin semiconductor film (bulk, quantum wells, or quantum dots) close to an array of, e.g., plasmonic split-ring resonators. This geometry aims at taking advantage of local-field-enhancement effects. The electric-field vector of the incident light lies in the layer plane. (b) Schematic illustration of the toy model for a plasmonic (bosonic) metamaterial resonance coupled to a (fermionic) two-level-system gain resonance via local-field interactions. This interaction is described by the phenomenological (Lorentz) parameter L.

Fig. 2.
Fig. 2.

Normal-incidence intensity transmittance T (black) and reflectance R (red) spectra. From top to bottom row (see schemes on the left-hand side): split-ring resonators only, gain film for f=0 only, both combined for f=0, and both combined for f=1. (a) Complete numerical finite-difference time-domain solutions of the three-dimensional vector Maxwell equations for the geometry depicted in Fig. 1. (b) Same for the transfer-matrix treatment of the toy model. (c) Same for the Maxwell-Garnett treatment of the toy model. Model parameters are: Ω 2LS=2π×200 THz, Ω pl=2π×200 THz, γ 2LS=7.53×1012 s-1, γ pl=29.5×1012 s-1, d 2LS=6.5×10-29 Cm, d pl=6.2×10-26 Cm, N 2LS=5.05×1023 m-3, N pl=5.56×1020 m-3, l 2LS=50 nm, l pl=20 nm, and L=3.3416×1010 m/F such that V 2LS=7.1×1011 s-1 and V pl=6.449×1014 s-1 result.

Fig. 3.
Fig. 3.

(a) Toy model complex linear susceptibility χ, (b) complex refractive index n, and (c) absorption coefficient α within the Maxwell-Garnett treatment. The solid (dashed) curves are the results with (without) local field coupling L. Real (imaginary) parts of complex quantities are red (green). The two-level system occupation f increases from top to bottom row as indicated. Model parameters are identical to those in Fig. 2.

Fig. 4.
Fig. 4.

Readers interested in the numerous parameter combinations other than the few selected special cases shown in Figs. 2 and 3 may download a software free of cost [25], allowing to calculate all relevant quantities of our toy model within the Maxwell-Garnett approach. The surface of this software is depicted here. All ten model parameters can be adjusted and the spectra change in real time. Default parameters are those of Figs. 2 and 3 and f=1. All frequencies and dampings are normalized with respect to the fixed 2LS transition frequency Ω 2LS=2π×200 THz (corresponding to about 1.5-µm wavelength).

Fig. 5.
Fig. 5.

(a) The two complex eigenfrequencies ω according to Eq. (11). The occupation factor f runs from 0 (no pumping) to 1 (complete inversion) along the direction indicated by the arrows. Model parameters are chosen as in Figs. 2 and 3. At the crossing with the real frequency axis, the imaginary part of ω becomes zero, corresponding to the only possible non-trivial stationary solution of the lasing SPASER (see open circle). The occupation f is pinned via this condition Eq. (12). The two eigenfrequencies without coupling, i.e., for L=0, are shown by the green filled circles. Note that these uncoupled complex eigenfrequencies do not depend on f at all. It is instructive to compare the complex eigenfrequencies shown here with the linear optical spectra shown in Figs. 2 and 3. (b) Same as (a) but Ω pl→ 0.99×Ω pl, (c) as (b) but LL/2, and (d) as (b) but LL/5. For the latter, stationary SPASER action is obviously no longer possible.

Fig. 6.
Fig. 6.

Switch-on of the lasing SPASER within our toy model leading to pronounced rapid relaxation oscillations of the two-level system occupation f and the effective rate of stimulated emission Γ stim. The pump rate Γ pump=Γ 0(1-f) with constant Γ 0 after time t=0 is parameter. The model parameters are identical to those of Figs. 2 and 3, Γ 2LS=1010 s-1. The lasing SPASER frequency results as ω=Ω 2LS=Ω pl. (a) Γ 0=4×1010 s-1 (just slightly above threshold) and (b) Γ 0=6×1010 s-1.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E ( t ) = E ˜ 0 cos ( ω t ) = E ˜ 0 2 ( exp ( i ω t ) + c . c . )
E E + L P pl .
E E + L P 2 LS .
p ˙ 2 LS + ( i Ω 2 LS + γ 2 LS ) p 2 LS = i 1 d 2 LS E ( 1 2 f ) ,
f ˙ + Γ 2 LS f = i 1 ( p 2 LS * d 2 LS E p 2 LS d 2 LS * E * ) .
p ˙ pl + ( i Ω pl + γ pl ) p pl = i 1 d pl E .
p ˜ 2 LS = ( 1 2 f ) ( 1 d 2 LS E ˜ 0 2 + V LS 1 d pl E ˜ 0 2 ( Ω pl ω ) i γ pl ) ( Ω 2 LS ω ) i γ 2 LS ( 1 2 f ) V pl V 2 LS ( Ω pl ω ) i γ pl ,
p ˜ pl = 1 d pl E ˜ 0 2 + V pl p ˜ 2 LS ( Ω pl ω ) i γ pl .
V 2 LS = 1 d 2 LS L N pl d pl ,
V pl = 1 d pl L N 2 LS d 2 LS ,
P = l 2 LS l 2 LS + l pl P 2 LS + l pl l 2 LS + l pl P pl = ε 0 χ E ˜ 0 2 exp ( i ω t ) + c . c .
ω = Ω 2 LS + Ω pl 2 i γ 2 LS + γ pl 2 ± ( Ω 2 LS Ω pl 2 i γ 2 LS γ pl 2 ) 2 + V 2 LS V pl ( 1 2 f ) .
γ 2 LS + γ pl 2 = ± Im ( ( Ω 2 LS Ω pl 2 i γ 2 LS γ pl 2 ) 2 + V 2 LS V pl ( 1 2 f ) ) .
f = 1 2 ( 1 + γ pl γ 2LS V pl V 2 LS ) 1 2 ( 1 + γ 2 V 2 ) [ 0 , 1 ] .
f = 1 2 ( 1 + γ 2 + ( Δ Ω 2 ) 2 V 2 ) [ 0 , 1 ] .
f ˙ + Γ 2 LS f = i ( p 2 LS * V 2 LS p pl c . c . ) + Γ pump = Γ stim + Γ pump .

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