## Overcoming the losses of a split ring resonator array with gain |

Optics Express, Vol. 19, Issue 13, pp. 12688-12699 (2011)

http://dx.doi.org/10.1364/OE.19.012688

Acrobat PDF (890 KB)

### Abstract

We present a computational approach, allowing for a self-consistent treatment of a split ring resonator (SRR) array with a gain layer underneath. We apply three different pumping schemes on the gain layer: (1) homogeneously pumped isotropic gain, (2) homogeneously pumped isotropic gain with a shadow cast by the SRR and (3) anisotropic gain pumped in a selected direction only. We show numerically the magnetic losses of the SRR can be compensated by the gain. The difference on loss compensations among the three pumping schemes is analyzed by the electric field distribution. Studies also show the dielectric background of gain does not affect the loss compensation much for the gain only pumped in the direction parallel to the SRR plane.

© 2011 OSA

## 1. Introduction

1. J. B. Pendry, “Negative refraction,” Contemp. Phys. **45**, 191–202 (2004). [CrossRef]

7. F. Capolino, *Theory and Phenomena of Metamaterials* (CRC Press, Taylor and Francis Group, 2009). [CrossRef]

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

9. 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]

10. J. Zhou, Th. Koschny, and C. M. Soukoulis, “An efficient way to reduce losses of left-handed metamaterials,” Opt. Express **16**, 11147–11152 (2008). [CrossRef] [PubMed]

13. D. O. Guney, Th. Koschny, and C. M. Soukoulis, “Reducing ohmic losses in metamaterials by geometric tailoring,” Phys. Rev. B **80**, 125129 (2009). [CrossRef]

21. A. N. Lagarkov, V. N. Kisel, and A. K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B **27**, 648–659 (2010). [CrossRef]

22. M. Wegener, J. Luis Garca-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express **16**, 19785–19798 (2008). [CrossRef] [PubMed]

26. 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**, 127401 (2010). [CrossRef] [PubMed]

23. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**, 241104 (2009). [CrossRef]

24. A. Fang, Th. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. **12**, 024013 (2010). [CrossRef]

25. A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B **82**, 121102 (2010). [CrossRef]

26. 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**, 127401 (2010). [CrossRef] [PubMed]

*μ*, are compensated. It is demonstrated the gain medium can give an effective gain much larger than its bulk counterpart due to the strong local-field enhancement inside the metamaterial designs [23

23. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**, 241104 (2009). [CrossRef]

27. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics **2**, 351–354 (2008). [CrossRef]

29. M. I. Stockman, “Spasers explained,” Nat. Photonics **2**, 327–329 (2008). [CrossRef]

30. K. Tanaka, E. Plum, J. Y. Ou, T. Uchino, and N. I. Zheludev, “Multifold Enhancement of Quantum Dot Luminescence in Plasmonic Metamaterials,” Phys. Rev. Lett. **105**, 227403 (2010). [CrossRef]

31. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitsky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express **18**, 24140–24151 (2010). [CrossRef] [PubMed]

32. 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**, 735–738 (2010). [CrossRef] [PubMed]

## 2. Theory and model

*N*

_{0}, to the third level,

*N*

_{3}, at a certain pumping rate, Γ

_{pump}, proportional to the optical pumping intensity in an experiment. After a short lifetime,

*τ*

_{32}, electrons transfer non-radiatively into the metastable second level,

*N*

_{2}. The second level (

*N*

_{2}) and the first level (

*N*

_{1}) are called the upper and lower lasing levels. Electrons can be transferred from the upper to the lower lasing level by spontaneous and stimulated emissions. At last, electrons transfer quickly and non-radiatively from the first level (

*N*

_{1}) to the ground state level (

*N*

_{0}). The lifetimes and energies of the upper and lower lasing levels are

*τ*

_{21},

*E*

_{2}and

*τ*

_{10},

*E*

_{1}, respectively. The center frequency of the radiation is

*ω*= (

_{a}*E*

_{2}–

*E*

_{1})/

*h̄*, chosen to equal 2

*π*× 10

^{14}rad/s. The parameters,

*τ*

_{32},

*τ*

_{21}, and

*τ*

_{10}, are chosen 5 × 10

^{−14}, 5 × 10

^{−12}, and 5 × 10

^{−14}s, respectively. The total electron density,

*N*

_{0}(

*t*= 0) =

*N*

_{0}(

*t*) +

*N*

_{1}(

*t*) +

*N*

_{2}(

*t*) +

*N*

_{3}(

*t*) = 5.0 × 10

^{23}/m

^{3}, and the pumping rate, Γ

_{pump}, is an external parameter. These gain parameters are chosen to overlap with the resonance of the split-ring resonator. The time-dependent Maxwell equations are given by ∇ ×

**E**= –

*∂*

**B**

*/∂t*and ∇ ×

**H**=

*εε*

_{o}∂**E**/

*∂t*+

*∂*

**P**/

*∂t*, where

**B**=

*μμ*

_{o}**H**and

**P**is the dispersive electric polarization density from which the amplification and gain can be obtained. Following the single electron case, we can show [33] the polarization density

**P**(

**r**,

*t*) in the presence of an electric field obeys locally the following equation of motion, where Γ

*is the linewidth of the atomic transition*

_{a}*ω*and is equal to 2

_{a}*π*× 20 × 10

^{12}rad/s. The factor, ∇

*N*(

**r**,

*t*) =

*N*

_{2}(

**r**,

*t*) –

*N*

_{1}(

**r**

*,t*), is the population inversion that drives the polarization, and

*σ*is the coupling strength of

_{a}**P**to the external electric field and its value is taken to be 10

^{−4}C

^{2}/kg. It follows [33] from Eq. 1 that the amplification line shape is Lorentzian and homogeneously broadened. The occupation numbers at each spatial point vary according to where

*t*= 2.0 × 10

^{−18}s and Δ

*x*= 2.5 × 10

^{−9}m. The initial condition is that all the electrons are in the ground state, so there is no field, no polarization, and no spontaneous emission. Then, the electrons are pumped from

*N*

_{0}to

*N*

_{3}(then relaxing to

*N*

_{2}) with a constant pump rate, Γ

_{pump}. The system begins to evolve according to the system of equations above.

## 3. Geometric dimensions of the SRR array

*ω*= 1.37 × 10

_{p}^{16}rad/s and

*γ*= 2.73 × 10

^{13}rad/s. The GaAs layer between the SRR and gain is introduced to avoid the quenching effect. The incident wave propagates along the

*y*direction parallel to the SRR plane and has the magnetic field perpendicular to that plane. The unit cell size along the propagation direction is

*a*. In

*z*direction, the unit cell size is

*h*, which is larger than

*h*

_{1}+

*h*

_{2}+

*h*

_{3}+

*h*, where

_{s}*h*

_{1},

*h*

_{2},

*h*

_{3}and

*h*are the thicknesses of the bottom GaAs layer, the gain layer, the GaAs spacing layer, and the SRR, respectively. Along the unit cell boundaries in

_{s}*x*and

*z*directions, periodic boundary conditions are enforced to simulate the infinite periodic structure. All the dimensions are chosen to have the magnetic resonance overlap with the emission frequency of 100THz of the gain material. For comparison, we also introduce another gain configuration (see Fig. 1(b)), where the gain is embedded in the gap of the SRR instead of a layer underneath. The dimensions are kept the same as Fig. 1(a).

## 4. Numerical simulations and discussions

### 4.1. Isotropic gain

*T*, the reflection

*R*, and the absorption

*A*= 1 –

*T – R*, as a function of frequency in the propagation direction. With the introduction of gain, the absorption near the resonance frequency

*f*= 100THz decreases and the transmission increases. To investigate the loss reduction of the magnetic resonators, we plot the retrieved effective permeabilities,

*μ*, without and with gain by inverting the scattering amplitudes [35, 36

36. Th. Koschny, P. Markoŝ, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B **71**, 245105 (2005). [CrossRef]

*μ*of the SRR becomes much stronger and narrower compared to the case without gain. In Fig. 2(b), we plot the effective permeabilities,

*μ*, without and with gain for the case the gain is in the SRR gap. Similar to the results for 2D SRR in Ref. 23

23. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**, 241104 (2009). [CrossRef]

*μ*becomes strong and narrow with the introduction of gain in the SRR gap. Note that a lower pumping rate (Γ

_{pump}= 7.0×10

^{8}s

^{−1}) leads to a sharper magnetic resonance comparing with the case the gain is underneath the SRR (Γ

_{pump}= 1.0 × 10

^{9}s

^{−1}) due to the local electric field concentration in the gap. However, the strong magnetic resonances in Figs. 2(a) and 2(b) are not symmetric due to the periodicity effect [36

36. Th. Koschny, P. Markoŝ, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B **71**, 245105 (2005). [CrossRef]

*L*,

*C*and

*R*are the effective inductance, capacitance and resistance of the SRR, respectively, and

*I*is the current flowing in the SRR and

*ε*

_{emf}is the induced electromotive force. From Faraday’s law,

*A*is the area enclosed by SRR, and

*c*is the speed of light in vacuum.) Then we can obtain the expression with Lorentz resonance shape, where

*η*,

*ω*

_{0}, and

*γ*are

*A/*(

*cL*),

*R/L*, respectively. The detailed results are plotted in Fig. 3(a) for the structure with the gain layer underneath. One can see the current resonances have very nice Lorentz line shapes. As the pumping rate increases, the resonance is getting stronger and narrower. The full width at half maximum (FWHM) reaches 2.5THz when the pumping rate Γ

_{pump}= 2.8 × 10

^{9}s

^{−1}, which is a significant loss reduction compared with the FWHM without gain (FWHM = 6.4THz). So the gain compensates the losses. In addition, we also calculate

*I/*(

*ηω*

^{2}

*E*) vs. frequency for the structure with gain in the SRR gap to compare the efficiency of the loss compensation for these two different gain configurations. The results are shown in Fig. 3(b). One can see the structure with gain in the SRR gap needs less gain (i.e., smaller pumping rate 1.5 × 10

^{9}s

^{−1}) to reach the same FWHM, 2.5THz, of the resonance than the case with gain underneath the SRR with the pumping rate Γ

_{pump}= 2.8 × 10

^{9}s

^{−1}. It is easy to understand the difference in pumping rates in the two designs because of the strong local electric field enhancement in the SRR gap. Though the loss compensation for the structure with a gain layer underneath is not so efficient as the case with the gain in the SRR gap, the results in Fig. 3(a) still show that the magnetic losses can be substantially reduced, especially if we push the pumping rate to a high value.

37. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. **84**, 2943–2945 (2004). [CrossRef]

38. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. **88**, 041109 (2006). [CrossRef]

*h*, which is much smaller than the wavelength

*λ*(

*λ/h*= 37.5), so the resonance is far below the Brillouin zone edge and we can ignore periodicity effect. Figure 4(b) plots the retrieved effective permittivity

*ε*, with and without gain. Both of them have a very nice Lorentz line shape. Without gain, the resonance is broad and weak, and the FWMH is 3THz. With the introduction of gain, the resonance becomes stronger and narrower, and the FWHM reduces to a much smaller value, 0.92THz. So the gain compensates the losses of the SRR for perpendicular incidence.

### 4.2. Isotropic gain with a shadow of the SRR

_{pump}. This is an ideal case. Consider the case in experiments that we incident an external optical pumping wave on the structure (Fig. 1(a)) from the top to optically pump the electrons from level 0 to level 3, there will be a shadow on the gain layer cast by the SRR structure, where the gain is pumped by a much lower rate. As a simplified model, we turn off the gain in the area which lies directly under the SRR to simply emulate the shadow of the SRR structure, while we still keep a homogeneous pumping rate Γ

_{pump}in other gain area (see Fig. 5). In Fig. 6(a), we plot

*I*/(

*ηω*

^{2}

*E*) as a function of frequency in this case. Compared with the case without the shadow on the gain layer (Fig. 3(a)), the resonance gets much weaker and broader (FWHM = 5.7THz and 5.4THz for the pumping rates Γ

_{pump}= 1.0 × 10

^{9}s

^{−1}and 1.5 × 10

^{9}s

^{−1}, respectively). This shows the gain in the shadow area plays an important part in the loss compensation.

### 4.3. Anisotropic gain

*y*direction in Fig. 1(a), i.e., the gap bearing side of the SRR. So, the gain only couples to the electric field in

*y*direction. The corresponding

*I/*(

*ηω*

^{2}

*E*) vs. frequency curves for different pumping rates are plotted in Fig. 6(b). One can see the resonances are also much broader than the case with homogeneously pumped isotropic gain. So the loss compensation is less efficient.

### 4.4. Explanation of the differences among the loss compensations by the three pumping schemes

*xy*plane in Fig. 1(a)). The detailed results are plotted in Figs. 7(a)–7(c). One can see the

*x*component of electric field,

*E*, is very weak while the other two components,

_{x}*E*and

_{y}*E*, are relatively strong. So we can ignore the gain contribution by

_{z}*E*and focus on the gain from the coupling with

_{x}*E*and

_{y}*E*. Notice that

_{z}*E*is bounded in the area right below the SRR gap (Fig. 7(b)) while

_{y}*E*mainly has a significant value in the projection of the SRR on the gain layer (Fig. 7(c)). This characteristic of the field amplitude distribution leads to almost no contribution by

_{z}*E*when we have a shadow in the gain layer since there is no gain in that area. Similarly, the gain contribution by

_{z}*E*goes away for the anisotropic gain because the gain only couples with the

_{z}*y*component of the electric field,

*E*. This fact explains the big difference between the homogeneously pumped isotropic gain and the other two gain pumping schemes.

_{y}### 4.5. The effect of the dielectric background of gain

*ε*= 11) and gain (

*ε*= 2) layers, the electromagnetic fields may be bounded in the high dielectric layer. In this section, we will discuss the effect of the dielectric background of gain on the loss compensation. In Fig. 8, we plot the detailed results for the imaginary parts of

*I*/(

*ηω*

^{2}

*E*) as a function of frequency, with and without gain, for the background dielectric constants of the gain layer

*ε*= 2, 5 and 11. The gain is anisotropic and only couples to the electric field in

_{g}*y*direction. We can see the resonance frequency shifts down as the dielectric constant increases. This is expected since the effective capacitance increases with the increment of the dielectric constant. To effectively compensate the losses, we scale the emission frequency to overlap with the corresponding resonance frequencies and then pump with the same rate Γ

_{pump}= 1.5 × 10

^{9}s

^{−1}. We can see from Fig. 8 the resonance enhancements are almost the same for different background dielectric constants of the gain.

*ε*= 2, 5 and 11, respectively. The

_{g}*E*component is ignored since it is very weak as shown in Fig. 7(a). From Fig. 9(a), we can see the field amplitude distribution of

_{x}*E*, the only component which couples to the gain, does not change much in the gain layer as the gain background dielectric constant changes. Although there is a bounding effect on the fields, the

_{y}*y*component of the electric field,

*E*, does not substantially decay in such a very narrow gain layer (10nm) neighboring to the high dielectric GaAs layer. The main change in the electric field is the

_{y}*z*component of the electric field,

*E*, decreases in the gain layer as the gain background dielectric constant,

_{z}*ε*, increases, as shown in Fig. 9(b). This is due to the continuity of the normal component of the electric displacement across the interface since there is no free charge accumulation. Hence the normal component of the electric field is inversely proportional to the dielectric constant. The change of

_{g}*E*does not affect the loss compensation due to no coupling between the gain and

_{z}*E*. If the gain can couple to

_{z}*E*, such as the isotropic gain, the background dielectric constant of the gain will significantly affect the loss compensation.

_{z}## 5. Conclusions

*E*, the electric field component parallel to the gap bearing side of the SRR, are much less efficient in the loss compensation compared to the isotropic gain case, due to no interactions between the electric field perpendicular to the SRR plane,

_{y}*E*, and the gain in these two schemes. We have also studied the effect of the background dielectric of gain. In a very narrow gain layer, the gain dielectric background mainly affects the electric field perpendicular to the GaAs-gain interface due to the continuity of the normal component of the electric displacement across the interface. So, the dielectric background of gain does not make much difference for the gain pumped in the parallel direction only.

_{z}## Acknowledgments

## References and links

1. | J. B. Pendry, “Negative refraction,” Contemp. Phys. |

2. | S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. |

3. | C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: New frontiers in optics,” Adv. Mater. |

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

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

6. | C. M. Soukoulis and M. Wegener, “Optical metamaterials–more bulky and less lossy,” Science |

7. | F. Capolino, |

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

9. | 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 |

10. | J. Zhou, Th. Koschny, and C. M. Soukoulis, “An efficient way to reduce losses of left-handed metamaterials,” Opt. Express |

11. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

12. | J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B |

13. | D. O. Guney, Th. Koschny, and C. M. Soukoulis, “Reducing ohmic losses in metamaterials by geometric tailoring,” Phys. Rev. B |

14. | S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B |

15. | N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. |

16. | 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. |

17. | T. A. Klar, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. |

18. | A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B |

19. | Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express |

20. | A. D. Boardman, Yu. G. Rapoport, N. King, and V. N. Malnev, “Creating stable gain in active metamaterials,” J. Opt. Soc. Am. B |

21. | A. N. Lagarkov, V. N. Kisel, and A. K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B |

22. | M. Wegener, J. Luis Garca-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express |

23. | A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B |

24. | A. Fang, Th. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. |

25. | A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B |

26. | 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. |

27. | N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics |

28. | 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. |

29. | M. I. Stockman, “Spasers explained,” Nat. Photonics |

30. | K. Tanaka, E. Plum, J. Y. Ou, T. Uchino, and N. I. Zheludev, “Multifold Enhancement of Quantum Dot Luminescence in Plasmonic Metamaterials,” Phys. Rev. Lett. |

31. | N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitsky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express |

32. | 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 |

33. | A. E. Siegman, |

34. | A. Taflove, |

35. | D. R. Smith, S. Schultz, P. Markoŝ, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

36. | Th. Koschny, P. Markoŝ, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B |

37. | N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. |

38. | D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.5740) Physical optics : Resonance

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: May 24, 2011

Manuscript Accepted: June 2, 2011

Published: June 16, 2011

**Citation**

Anan Fang, Zhixiang Huang, Thomas Koschny, and Costas M. Soukoulis, "Overcoming the losses of a split ring resonator array with gain," Opt. Express **19**, 12688-12699 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12688

Sort: Year | Journal | Reset

### References

- J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004). [CrossRef]
- S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005). [CrossRef]
- C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: New frontiers in optics,” Adv. Mater. 18, 1941–1952 (2006). [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]
- C. M. Soukoulis and M. Wegener, “Optical metamaterials–more bulky and less lossy,” Science 330, 1633–1634 (2010). [CrossRef] [PubMed]
- F. Capolino, Theory and Phenomena of Metamaterials (CRC Press, Taylor and Francis Group, 2009). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]
- 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. Zhou, Th. Koschny, and C. M. Soukoulis, “An efficient way to reduce losses of left-handed metamaterials,” Opt. Express 16, 11147–11152 (2008). [CrossRef] [PubMed]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]
- J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B 80, 035109 (2009). [CrossRef]
- D. O. Guney, Th. Koschny, and C. M. Soukoulis, “Reducing ohmic losses in metamaterials by geometric tailoring,” Phys. Rev. B 80, 125129 (2009). [CrossRef]
- S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
- N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85, 5040–5042 (2004). [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, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006). [CrossRef]
- A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75, 085436 (2007). [CrossRef]
- Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express 17, 24060–24074 (2009). [CrossRef]
- A. D. Boardman, Yu. G. Rapoport, N. King, and V. N. Malnev, “Creating stable gain in active metamaterials,” J. Opt. Soc. Am. B 24, A53–A61 (2007). [CrossRef]
- A. N. Lagarkov, V. N. Kisel, and A. K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B 27, 648–659 (2010). [CrossRef]
- M. Wegener, J. Luis Garca-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16, 19785–19798 (2008). [CrossRef] [PubMed]
- A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009). [CrossRef]
- A. Fang, Th. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12, 024013 (2010). [CrossRef]
- A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102 (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, 127401 (2010). [CrossRef] [PubMed]
- N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2, 351–354 (2008). [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 (2003). [CrossRef] [PubMed]
- M. I. Stockman, “Spasers explained,” Nat. Photonics 2, 327–329 (2008). [CrossRef]
- K. Tanaka, E. Plum, J. Y. Ou, T. Uchino, and N. I. Zheludev, “Multifold Enhancement of Quantum Dot Luminescence in Plasmonic Metamaterials,” Phys. Rev. Lett. 105, 227403 (2010). [CrossRef]
- N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitsky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18, 24140–24151 (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, 735–738 (2010). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (Hill Valley, 1986), Chaps. 2, 3, 6, and 13.
- A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech House, London, 1995). See Chaps. 3, 6, and 7.
- D. R. Smith, S. Schultz, P. Markoŝ, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
- Th. Koschny, P. Markoŝ, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]
- N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84, 2943–2945 (2004). [CrossRef]
- D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.