## Toy model for plasmonic metamaterial resonances coupled to two-level system gain

Optics Express, Vol. 16, Issue 24, pp. 19785-19798 (2008)

http://dx.doi.org/10.1364/OE.16.019785

Acrobat PDF (1553 KB)

### 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

## 1. Introduction

*the*major challenges in the emerging field of photonic metamaterials [1

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

3. 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]

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

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]

*α*=3×10

^{4}cm

^{-1}=3 µm

^{-1}— which is even larger than the band-to-band absorption of typical direct-gap semiconductors such as, e.g., GaAs (there,

*α*≈10

^{4}cm

^{-1}). At first sight, this level of absorption looks quite depressing as room-temperature steady-state gain coefficients

*g*=-

*α*of this magnitude are not easily achieved at all.

8. 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]

9. 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]

10. M. I. Stockman, “Spasers explained,” Nature Photon. **2**, 327–329 (2008). [CrossRef]

*effective*gain coefficient of the combined system. Due to pronounced

*local-field*enhancement effects in the spatial vicinity of the periodic metallic nanostructure, the effective gain coefficient can be substantially larger than its bulk counterpart. As an extended planar periodic arrangement of identical plasmonic structures that oscillate in phase due to mutual coupling and that exhibit a sub-wavelength period clearly leads to coherent plane-wave emission of light normal to the plane (in close analogy to phased antenna arrays), this two-dimensional active metamaterial structure has been named the “lasing SPASER” [8

8. 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]

11. 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]

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

13. 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]

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). [CrossRef]

15. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express **16**, 6692–6716 (2008). [CrossRef] [PubMed]

16. 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]

19. 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]

20. 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]

21. C. Manolatou and F. Rana, “Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers,” IEEE J. Quantum Electron. **44**, 435–447 (2008). [CrossRef]

9. 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]

8. 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]

*local-field coupling*analogous to the local-field Lorentz factor. Interestingly, our toy model can be solved analytically for (quasi-) steady-state conditions. Numerical solutions for the time-dependent case are also discussed.

## 2. Definition of the model

*E*

*L*, times the polarization of the bosonic mode,

*P*

_{pl}, i.e.,

*L*times the polarization of the 2LS,

*P*

_{2LS}, i.e.,

*L*is expected to approach zero. Note that we have omitted the self-interactions, which can generally also occur via the local fields. In linear optics, however, self-interaction merely renormalizes the effective resonance center frequency.

*p*

_{2LS}denotes the (dimensionless) complex transition amplitude,

*f*=

*f*

_{2LS}the occupation probability of the upper level,

*d*

_{2LS}the dipole matrix element,

*E*the total electric field,

*Ω*

_{2LS}the transition frequency,

*γ*

_{2LS}the damping (or transverse relaxation) rate, and

*Γ*

_{2LS}the population (or longitudinal) relaxation rate. The individual electric dipole moment is given by the product

*d*

_{2LS}

*p*

_{2LS}. Multiplying with the volume density of two-level systems,

*N*

_{2LS}, leads to the macroscopic 2LS polarization

*P*

_{2LS}, i.e., to

*P*

_{2LS}=

*N*

_{2LS}

*d*

_{2LS}

*p*

_{2LS}+c.c. The equations of motion for a corresponding single bosonic mode are strictly identical to Eq. (4) and Eq. (5), except that the factor (1-2

*f*) in Eq. (4) has to be replaced by unity, i.e.,

*f*

_{pl}of the bosonic mode becomes irrelevant. In analogy to the 2LS, the macroscopic plasmonic polarization is given by

*P*

_{pl}=

*N*

_{pl}

*d*

_{pl}

*p*

_{pl}+c.c.

*local-field effects*. Under these conditions, we simply have to replace the electric fields

*E*in Eq. (4) and Eq. (5) by the term given in Eq. (2) and the electric field

*E*in Eq. (6) by Eq. (3).

*Γ*

_{pump}, of the 2LS on the right-hand side of Eq. (5).

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

3. 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]

*P*) as only that can couple directly to readily available gain media at optical frequencies. Thus, our model is not applicable directly to magnetic-dipole plasmonic resonances or to negative-index metamaterials. Yet, it may provide qualitative trends. Our model is applicable directly to, e.g., the electric-dipole (Mie) resonances of the “V” structures envisioned in Ref. 9, but not to the “dark” modes there.

## 3. Linear optical response for fixed occupation

*f*of the 2LS and calculate the linear optical response with respect to some probe light with electric field

*E*. As long as the system does not exhibit effective stimulated emission, this assumption of fixed

*f*is completely unproblematic. This situation can occur under steady-state pumping conditions indeed. In contrast, as soon as

*f*is such that the system does reveal effective stimulated emission, the situation becomes unstable, hence, it cannot occur under steady-state conditions (also see sections 4 and 5). Still, such values of

*f*can be meaningful under transient conditions, for example, for a probe pulse with electric field

*E*that follows, with some delay, the excitation by a pump pulse at larger photon energies (via the pump rate

*Γ*

_{pump}described above).

*f*, it is straightforward to solve the above coupled equations. Using the usual rotating-wave approximation (RWA) [17], we derive the transition amplitudes

*p*=

*p*̃ exp(-i

*ωt*)

*p*

_{2LS}according to Eq. (7) also enters in the numerator of Eq. (8), the spectral dependence of

*p*

_{pl}is quite complex as well. From this point on, one has two different options to arrive at optical spectra: via a transfer-matrix approach or via a Maxwell-Garnett effective-medium approach.

## 3.1. Transfer-Matrix approach

*l*

_{2LS}and

*l*

_{pl}for the 2LS gain and the plasmonic layer, respectively. The two corresponding dielectric functions and refractive indices result immediately from the two macroscopic polarizations

*P*

_{2LS}and

*P*

_{pl}given above. Normal-incidence intensity transmittance

*T*and reflectance

*R*spectra can be computed from the well known transfer-matrix approach [26] for layered films. In order to test the relevance of our toy model and in order to adjust the toy model parameters, especially the local-field parameter

*L*, we compare such calculations with microscopic numerical calculations on the basis of the complete three-dimensional vector Maxwell equations. These calculations based on the commercially available finite-difference time-domain software package lumerical [27] are rather similar to those in Ref. 8, however, with a frequency-dependent two-level-system gain resonance (same parameters as for the toy model) and for plasmonic split-ring resonators (SRR) roughly similar to the ones discussed in Ref. 24. The SRR are

*l*

_{pl}=20-nm thick and are arranged on a quadratic lattice with lattice constant

*a*=300 nm. The SRR lateral dimensions can be seen from Fig. 1(a). The silver, the SRR are assumed to be composed of, is described by the Drude free-electron model with standard parameters, i.e., plasma frequency

*ω*

_{pl}=1.32×10

^{16}s

^{-1}and collision frequency

*ω*

_{col}=1.2×10

^{14}s

^{-1}.

## 3.2. Maxwell-Garnett effective-medium approach

*single*effective film with thickness

*l*=

*l*

_{2LS}+

*l*

_{pl}. The advantage as compared to the two-layer approach of the previous section is that a single set of optical parameters follows the spirit of a metamaterial and that it allows for obtaining a more intuitive understanding. The corresponding effective parameters can immediately be derived from Maxwell-Garnett effective-medium theory [recall that the electric-field vector is parallel to the plane of the layers, see Fig. 1(a)]. In this case, the total effective macroscopic polarization

*P*results as

*χ*, the effective electric permittivity

*ε*=1+

*χ*, and the effective complex refractive index

*n*=±

*ε*

^{1/2}. The sign of the root has to be chosen such that Re(

*n*)≥0. As

*µ*=1 here, we can equivalently say that the sign of the root has to be chosen such that the sign of Im(

*n*) is the same as the sign of Im(

*χ*). The formulas for the normal-incidence intensity transmittance

*T*and reflectance

*R*of a homogeneous slab with complex refractive index

*n*(hence, as

*µ*=1, complex impedance Z=Z

_{0}/

*n*, with the vacuum impedance Z

_{0}) and thickness

*l*=

*l*

_{2LS}+

*l*

_{pl}can, e.g., easily be obtained from Eq. (5.119) and Eq. (5.120) of the review article [3

3. 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]

*L*=0. The comparison of solid and dashed curves, respectively, allows for assessing the influence of the local-field coupling

*L*. Generally, the absorption lineshape is Fano-like — an aspect that we have previously pointed out for coupled classical harmonic oscillators [23]. The Fano character becomes especially obvious for

*f*=0 and if one of the two resonances exhibits a much larger damping than the other one (not shown here, see Fig. 4). More importantly, Fig. 3 also shows the evolution of the absorption spectrum for increasing two-level-system upper-state occupation

*f*=0, 0.50, 0.7426, and 1. As transparency of the 2LS at

*f*=0.5 is approached, the avoided crossing obviously collapses. Increasing f further, a sharp feature develops that becomes infinitely sharp if the gain exactly compensates the loss. At this critical point (

*f*=0.7426 for the parameters chosen in Figs. 2 and 3), we expect lasing (or “spasing” [9

9. 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]

**2**, 351–354 (2008). [CrossRef]

*f*in the following section. Further increase of

*f*again broadens the spectral features because the gain over-compensates the loss. It is important to note that no gain is observed at all in Fig. 3 in case that the local-field coupling is artificially switched off (dashed curves for

*L*=0). This behavior is expected on the basis of the intuitive reasoning in section 1 in terms of the

*effective*gain rather than the 2LS gain itself that matters. The discontinuities of Im(

*n*) occur at zeroes of Re(

*n*), where the sign of the root of the electric permittivity has to change in order to fulfill the condition Re(

*n*)≥0.

*T*will exceed unity as soon as effective gain occurs (see above). This can be the case, but Figs. 2 and 3 show that this is not necessarily the case. At and around the critical point, the real part of the refractive index becomes very large compared to unity or close to zero, either of which leads to a large impedance mismatch with respect to vacuum that gives rise to a large intensity reflectance

*R*. As a result, the transmittance

*T*is well below unity and shows no obvious signs of gain at all. We can conclude that transmittance alone is not necessarily a good experimental observable at this point.

*n*for the critical value of the occupation

*f*in Fig. 3 implies that the effective group velocity is getting extremely small or even zero in a very narrow spectral range. Notably, one could furthermore envision dynamic tailoring of the effective group velocity via reducing

*f*with respect to the critical value, which is possible by reducing the pump rate

*Γ*

_{pump}. (Steady-state increase of

*f*with respect to the critical value is not possible, see next section). These small effective group velocities might be interesting in the context of “slow light”.

25.
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”.

*Ω*, damping

*γ*, dipole moment

*d*, and density

*N*). In addition, the important phenomenological local-field coupling parameter

*L*and the two-level-system occupation

*f*can be freely adjusted — altogether 10 parameters.

## 4. Steady-state gain and occupation pinning

*f∈*[0,1

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

*f*until some steady-state value is reached — a phenomenon, which is well known as gain pinning in the context of a LASER [22,28

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

*f*above that steady-state value must be treated with a grain of salt. Also see Ref. 29 for a corresponding discussion regarding constraints for negative-index metamaterials due to causality.

*ω*of the coupled system has to be strictly zero. Without external light field, i.e., for

*E*=0 on the RHS of Eq. (2) and Eq. (3), and for constant pump rate

*Γ*

_{pump}, the two complex-valued eigenfrequencies of the above coupled Eqs. (2)–(6) are given by

*ω*in the complex frequency plane versus

*f*(running from 0 to 1) according to Eq. (11) is illustrated in Fig. 5. Figure 5(a) corresponds to the degenerate case, i.e., to

*Ω*

_{2LS}=

*Ω*

_{pl}. The other parameters are as in Figs. 2 and 3. In Fig. 5(b), the degeneracy is slightly lifted. In Figs. 5(c) and (d), the local-field coupling

*L*is successively decreased. As a result, an eigenfrequency with zero imaginary part no longer occurs in (d) [see open circles in (a)–(c)]. Let us explicitly evaluate the occupation

*f*corresponding to Im(

*ω*)=0.

*f*

*ω*resulting from Eq. (11), hence the linear optical response according to Eq. (10) diverges at this point.

*Ω*

_{2LS}=

*Ω*

_{pl}. We obtain

*f*=0.7426 — which is why we have depicted this particular value in the third row of Fig. 3.

*f*in Eq. (13) needs to be in the interval [0,1

**1**, 41–48 (2007). [CrossRef]

*V*(= geometric mean of the two couplings) smaller than the effective damping

*γ*(= geometric mean of the two dampings) leads to values of

*f*exceeding unity. In other words: We have found a critical threshold value for the strength of the effective local-field interaction

*V*. For values below that critical value, no lasing (spasing) action can occur under steady-state conditions. From Eq. (11), the (real-valued) lasing SPASER frequency results as

*ω*=

*Ω*

_{2LS}=

*Ω*

_{pl}.

*V*≥

*γ*clearly also implies that an avoided crossing of the two-level-system resonance and the plasmonic resonance can be seen in the linear optical spectra without inversion, i.e., for

*f*=0 (see previous section). Hence, within our model, the avoided crossing can be viewed as a prerequisite for obtaining lasing SPASER action (also compare Fig. 5).

*γ*=

*γ*

_{2LS}=

*γ*

_{pl}, but finite detuning

*ΔΩ*=

*Ω*

_{2LS}-

*Ω*

_{pl}≠0. We derive

*ΔΩ*increases the necessary 2LS occupation

*f*, until eventually no physical solution

*f*∈[0,1

**1**, 41–48 (2007). [CrossRef]

*γ*

^{2}+(

*ΔΩ*/2)

^{2}>

*V*

^{2}. The detuning simply reduces the optical gain that is accessible for the plasmonic resonance from the two-level system (and so does the damping). From Eq. (11), the (real-valued) lasing SPASER frequency results as

*ω*=(

*Ω*

_{2LS}+

*Ω*

_{pl})/2.

*propagating*electromagnetic waves (i.e., we have set

*E*=0 on the RHS of Eq. (2) and Eq. (3)) because our aim has been to discuss the case of pure

*evanescent-field*(or local-field) optical feedback. The same holds true for the following section. Experimentally, this ideal case could, in principle, be realized by appropriate anti-reflection coatings of the structure at the lasing wavelength. It should be clear, however, that there will be a continuous transition from an ideal SPASER to a usual LASER if the relative strength of the far-field optical feedback with respect to the near-field (local-field) optical feedback is increased continuously. Studying this transition is not subject of the present letter.

## 5. Lasing SPASER relaxation oscillations

*Γ*

_{pump}=

*Γ*

_{0}(1-

*f*). The Pauli blocking factor (1-

*f*) acknowledges that the upper state of each two-level system cannot be occupied with more than one electron, i.e., it guarantees

*f∈*[0,1

**1**, 41–48 (2007). [CrossRef]

*Γ*

_{0}shall be zero until time

*t*=0. Then it is switched on to a constant value. Also, we seed the polarizations with a tiny but finite value. This is necessary because our toy model does not contain any spontaneous emission whatsoever. As a result, the transition amplitudes would otherwise be strictly zero — no matter how much gain the system develops. Alternatively, one could use a weak external seed pulse of light

*E*(

*t*). None of these details is really important, the initial conditions just need to be non-zero.

*E*=0 on the RHS of Eq. (2), Eq. (5) in RWA becomes

*Γ*

_{stim}. For example, for the degenerate case, the steady-state lasing threshold (where

*Γ*

_{stim}=0) is reached for the pump rate

*Γ*

_{pump}=

*Γ*

_{2LS}

*f*, with the pinned 2LS occupation

*f*according to Eq. (12). As usual, above this threshold pump rate, the effective rate of stimulated emission increases linearly with

*Γ*

_{pump}, i.e.,

*Γ*

_{stim}=

*Γ*

_{pump}-

*Γ*

_{2LS}

*f*.

## 6. Conclusions

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

## Acknowledgments

## References and links

1. | V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon. |

2. | C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science |

3. | K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. |

4. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

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 |

6. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. |

7. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “A low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

8. | N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. |

9. | 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. |

10. | M. I. Stockman, “Spasers explained,” Nature Photon. |

11. | J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express |

12. | S.-W. Chang, C.-Y.A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express |

13. | T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-Index Metamaterials: Going Optical,” IEEE J. Sel. Top. Quantum Electron. |

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. |

15. | J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express |

16. | P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express |

17. | I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. |

18. | 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. |

19. | 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. |

20. | 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. |

21. | C. Manolatou and F. Rana, “Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers,” IEEE J. Quantum Electron. |

22. | W. Schäfer and M. Wegener, |

23. | M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, “Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs,” Phys. Rev. |

24. | 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. |

25. |
This software suitable for modern |

26. | E. Hecht, |

27. | |

28. | W. W. Chow, S. W. Koch, and M. Sargent III, |

29. | M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(260.5740) Physical optics : Resonance

**ToC Category:**

Metamaterials

**History**

Original Manuscript: September 2, 2008

Revised Manuscript: October 28, 2008

Manuscript Accepted: October 28, 2008

Published: November 14, 2008

**Citation**

Martin Wegener, Juan Luis García-Pomar, Costas M. Soukoulis, Nina Meinzer, Matthias Ruther, and Stefan Linden, "Toy model for plasmonic metamaterial resonances coupled to two-level system gain," Opt. Express **16**, 19785-19798 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19785

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### References

- 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]
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