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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 20 — Sep. 24, 2012
  • pp: 22953–22960
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Gain-assisted Hybrid-superlens Hyperlens for Nano Imaging

Yao Ting Wang, Bo Han Cheng, You Zhe Ho, Yung-Chiang Lan, Pi-Gang Luan, and Din Ping Tsai  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 22953-22960 (2012)
http://dx.doi.org/10.1364/OE.20.022953


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Abstract

We propose an innovative active imaging device named gain-assisted hybrid-superlens hyperlens and examine its resolving power theoretically. This semi-cylindrical device consists of a core of semi-cylindrical super-lens and a half cylindrical outer shell of hyperlens. Both the superlens and hyperlens parts of the device are appropriately designed multi-layered metal-dielectric structures having indefinite eigenvalues of dielectric tensors. The dielectric layers of the hyperlens are doped with Coumarin, which play the role of gain medium. The gain medium is analyzed thoroughly using a generic four-level system model, and the permittivity of the gain medium is extracted from this analysis for simulating the imaging characteristics of the device. According to our simulation at wavelength of 365 nm, an excellent resolution power much better than the diffraction limit value can be achieved.

© 2012 OSA

1. Introduction

On the other hand, usually a superlens or hyperlens cannot be fabricated without using metallic materials. It always accompanies with significant energy loss when operating at terahertz frequency, even at optical frequency. This disadvantage results in that metamaterials are hardly utilized in practical manner since the throughput of the incident waves is extremely low. Consequently, in order to overcome the dissipation problem, some different materials, such as graphenes and superconductors are used [18

18. N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express 18(8), 8353–8359 (2010). [CrossRef] [PubMed]

,19

19. C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett. 107(4), 043901 (2011). [CrossRef] [PubMed]

]. It has, however, other limitations such as the extremely low operating temperature and fragility; all of these restrict their utilities. In 2009, two different types of metamaterials with gain were demonstrated numerically [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

,21

21. 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]

]. It had been proved that gain material is capable of reducing or compensating the loss of the wave medium.

2. Principle

The gain medium in this paper is analyzed by using the same model of a four-level system presented in [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

]. The occupation numbers per unit volume vary according to the following equations:
N4t=rN1N4τ43
(1)
N3t=N4τ43N3τ32+1ωaEPt
(2)
N2t=N3τ32N2τ211ωaEPt
(3)
N1t=N2τ21rN1
(4)
where r is the pumping rate, τij is the relaxation time or lifetime between the ith and jth energy level, ωa is the optical absorption frequency, andEP˙/ωais the induced radiation or excitation rate which relies on its sign. The conservation of the total number of electrons leads to the relation: N(t)=N1(t)+N2(t)+N3(t)+N4(t)=N(0), where N is the total electron density. This relation will be used later.

To solve these coupled equations, a proper approximation will be applied. Note that in the four-level system, in the interest time scale, the occupation numbers at energy level 2 and 4 tend to zero (i.e.,N20,N40) because both of them have extremely short relaxation time: τ2143~10−14 sec. Conversely, the relaxation time τ32 (~10−12 sec.) for the transition from energy level 3 to 2 is much longer than τ21 and τ43, making the population inversion possible. Subtracting Eq. (3) from Eq. (2) and Eq. (1) from Eq. (4), defining ΔN(t)N3(t)N2(t), and then using the conditions N1 >>N4, N3>> N2, N ~N1 + N3 ~N1 + ΔΝ, we get
ΔNt+γaΔN=1ωaEPt+rN
(5)
where γa=(1+rτ32)/τ32. Solving Eq. (5), we get
ΔN=rNγa+eγatωaeγatEPtdt
(6)
In the above expression the integration constant for the integral part is dropped because it is multiplied by a fast decaying factoreγat. Besides, the constant (DC) term of Eq. (6) stands for the expectation value ofΔN, as can be verified by doing time-average of Eq. (5). The time-varying (AC) term gives the first-order perturbation correction to. It is obvious to observe that the magnitude of the non-linear term (the integral part) increases with the applied electric field intensity.

Referring to [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

], the dynamical equation for the induced electric polarization is given by
2Pt2+ΓaPt+ωa2P=σaΔN(t)E
(7)
where Γa is the bandwidth of the atomic transition, and σa is the coupling strength of polarization to the external electric field. In this gain medium we assume that the external sinusoidal electric field coupled to the four-level atomic system has instantaneous amplitude |E|sin(ωαt) and the instantaneous amplitude of the induced polarization is |P|sin(ωat + δ), where δ is the phase delay of the later with respect to the former, and |E| and |P| are the absolute magnitudes of them, respectively. In our model we also assume the condition, ωa >> γa and under this condition Eq. (7) can be approximated as
2Pt2+ΓaPt+ωa2PσarNγaE+σasin(δ)|E||P|2γaE
(8)
As shown in Eq. (8), the non-linear term gives the first-order perturbation correction to the population numbers. Nevertheless, no well-developed method can be readily implemented to solve this differential equation for finding its exact solution. Though we are not able to solve it exactly, approximate solution is still possible to obtain if the electric field is weak enough. Hereafter we assume that the weak field condition is fulfilled, thus the nonlinear term on the right hand side of Eq. (8) can be dropped. The extracted effective permittivity under this assumption is still of Lorentz form, written as

εeff(ω)=1+ωg2/(ω2ωa2+iωΓa)
(9)

Here ωg=rNσa/γaε0 is a characteristic frequency, and the unusual plus sign appearing in the right hand side of the formula between 1 and the remaining term indicates that the gain material indeed provides energy compensation for the absorption of the medium. The validity of the above approximations can be verified numerically using FDTD method [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

]. However, if the electric field intensity increases to 104 or higher orders, the non-linear effect cannot be neglected. This argument had already been mentioned in [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

] and we get a good match for its conclusion.

3. Simulation result

However, hybrid-superlens hyperlens still suffers from the drawback of extremely high energy loss. In fact, when passing through this device, only less than 1% of the incident wave energy remains. In order to resolve this disadvantage, a combined structure consisting of a hybrid-superlens hyperlens and a laser dye thin film is proposed, as shown in Fig. 1. In this letter, the laser dye we choose is Coumarin, which is an excellent candidate of gain material because: 1its absorption wavelength is in the violet-blue region and there is indeed laser operating at the same wavelength [21

21. 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]

], 2it is almost non-toxin to the human body since Coumarin and its derivatives are commonly used as food additive and ingredient in perfume, and 3Coumarin can be easily doped into PMMA to a high concentration and its knowledge for being gain medium is well-established [22

22. F. J. Duarte and L. W. Hillman, “Dye laser principles with applications” (1990), See appendix.

]. To fit our designing parameters, Coumarin 2 (7-(Ethylamino)-4,6-dimethylcoumarin) is chosen to be the gain material because its absorption wavelength (λa = 365 nm) is readily acquirable using commercial laser products. We also assume that the PMMA has been doped with Coumarin 2 to the concentration 1.0 × 10−2 M [23

23. G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin. 90(1-2), 1–5 (2000). [CrossRef]

]. Other experimental parameters of Coumarin 2 are as follows: the total electron density N is 6.0 × 1026 per cubic meter, the lifetime τ32 is 4.2 × 10−9 second [24

24. H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A. 79(6), 2128–2132 (1982). [CrossRef] [PubMed]

], the coupling strength σa is 10−8 C2/kg [20

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

], the bandwidth of the absorption wavelength Δλ is 50 nm, and the pumping rate r is 1.0 × 109. According to these chosen parameters, the real and imaginary parts of the effective permittivity of pure Coumarin 2 are shown in Fig. 2(a)
Fig. 2 According to the parameters, (a) the theoretical prediction to real and imaginary part of permittivity of Coumarin 2 are defined. (b) The isofrequency curve of upper superlens (red) and lower hyperlens (blue). k// (k) represents the effective wave vectors being parallel (perpendicular) to metal/dielectric interface.
. Based on the results shown in Fig. 2, the permittivity of Coumarin 2 at λ = 365 nm is 0.9057 + 94.5550i. As has been mentioned before, the concentration of Coumarin 2 in PMMA is 1.0 × 10−2 M, corresponding to a ratio of weight (denoted as f) about 0.002. This value is derived from the data that the density of PMMA is 1.35 g/cm3 and the molecular weight of Coumarin 2 is 217.26 g/mol. The dielectric constant of the gain medium layer thus becomes 2.222 + 0.1887i. The commercial solver COMSOL MultiphysicsTM 3.5a based on finite element method (FEM) is utilized insimulating the two-dimensional structure, and perfectly matched layers surrounding the simulation region are used. Transverse magnetic (TM) polarized incident light is considered with the incident electric field (x-direction) being perpendicular to the axis of the cylindrical structure (along the y-direction), as shown in Fig. 1. Figure 2(b) plots the isofrequency dispersion relations for the superlens and hyperlens structures of the hybrid-superlens hyperlens at the incident wavelength of 365 nm.

In order to satisfy the requirement in [25

25. B. H. Cheng, Y. Z. Ho, Y. C. Lan, and D. P. Tsai, “Optical hybrid-superlens-hyperlens for superresolution imaging,” IEEE J. Sel. Top. Quantum Electron. (submitted).

], the selection of isofrequency curves should take a specific set of dispersive relations. For the ability of resolution and transferred the signal extracted from the slits, this device should use two different anisotropic media. There are opposite signs for dielectric tensor between the upper planar-superlens and the lower cylindrical-hyperlens resulting from different dielectric formations. This fact would result in different hyperbolic dispersion between the upper and lower anisotropic media. In our work, according to ref [8

8. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]

], the upper planar-superlens with East–west opening hyperbolic form can be regarded as the resolution component of hybrid-superlens hyperlens excited by TM wave. In addition, the lower anisotropic medium needs to be in the form of North–south opening hyperbolic form in order to receive and spread subwavelength signals from the upper planar-superlens [12

12. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008). [CrossRef] [PubMed]

].

4. Conclusion

We have derived an analytic formula for evaluating the permittivity of four-level-system gain medium theoretically. Similar analytical formulas for other kinds of gain medium under weak field condition can also be derived using similar approximation approaches. In addition, we have demonstrated the imaging ability of the active device named gain-assisted hybrid-superlens hyperlens, which consists of metal-dielectric multilayer structures of superlens core and a hyperlens shell, and is assisted with gain. This active device not only has the ability of resolving the sub-wavelength fine features of objects, forming magnified images in the far field, but also enhances the field intensity of the images without distortion. Our numerical simulations further verified these statements. Finally, since the parameters of geometry and the materials we chose are practically realizable by using today’s nano-fabrication technology, we expect this innovative device can be realized in the near future.

Acknowledgment

The authors acknowledge financial support from National Science Council, Taiwan, under grant numbers 100-2923-M-002-007-MY3, NSC 101-3113-P-002-021- and NSC101-2112-M-002-023-. We are also grateful to the National Center for Theoretical Sciences, Taipei Office, Molecular Imaging Center of National Taiwan University and National Center for High-Performance Computing, Taiwan and Research Center for Applied Sciences, Academia Sinica for their kind support.

References and links

1.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

2.

D. P. Tsai and W. C. Lin, “Probing the near fields of the super-resolution near-field optical structure,” Appl. Phys. Lett. 77(10), 1413–1415 (2000). [CrossRef]

3.

D. P. Tsai, C. W. Yang, W. C. Lin, F. H. Ho, H. J. Huang, M. Y. Chen, T. F. Tseng, C. H. Lee, and C. J. Yeh, “Dynamic aperture of near-field super resolution structures,” Jpn. J. Appl. Phys. 39(Part 1, No. 2B), 982–983 (2000). [CrossRef]

4.

W. C. Liu, C. Y. Wen, K. H. Chen, W. C. Lin, and D. P. Tsai, “Near-field images of the AgOx-type super-resolution near-field structure,” Appl. Phys. Lett. 78(6), 685–687 (2001). [CrossRef]

5.

T. C. Chu, D. P. Tsai, and W. C. Liu, “Readout contrast beyond diffraction limit by a slab of random nanostructures,” Opt. Express 15(1), 12–23 (2007). [CrossRef] [PubMed]

6.

K. P. Chiu, K. F. Lai, and D. P. Tsai, “Application of surface polariton coupling between nano recording marks to optical data storage,” Opt. Express 16(18), 13885–13892 (2008). [CrossRef] [PubMed]

7.

D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13(6), 2127–2134 (2005). [CrossRef] [PubMed]

8.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]

9.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

10.

W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72(19), 193101 (2005). [CrossRef]

11.

D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. 7, 162 (2005). [CrossRef]

12.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008). [CrossRef] [PubMed]

13.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

14.

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for Imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007). [CrossRef] [PubMed]

15.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

16.

S. Schwaiger, M. Bröll, A. Krohn, A. Stemmann, C. Heyn, Y. Stark, D. Stickler, D. Heitmann, and S. Mendach, “Rolled-up three-dimensional metamaterials with a tunable plasma frequency in the visible regime,” Phys. Rev. Lett. 102(16), 163903 (2009). [CrossRef] [PubMed]

17.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

18.

N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express 18(8), 8353–8359 (2010). [CrossRef] [PubMed]

19.

C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett. 107(4), 043901 (2011). [CrossRef] [PubMed]

20.

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

21.

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]

22.

F. J. Duarte and L. W. Hillman, “Dye laser principles with applications” (1990), See appendix.

23.

G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin. 90(1-2), 1–5 (2000). [CrossRef]

24.

H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A. 79(6), 2128–2132 (1982). [CrossRef] [PubMed]

25.

B. H. Cheng, Y. Z. Ho, Y. C. Lan, and D. P. Tsai, “Optical hybrid-superlens-hyperlens for superresolution imaging,” IEEE J. Sel. Top. Quantum Electron. (submitted).

OCIS Codes
(100.6640) Image processing : Superresolution
(110.0180) Imaging systems : Microscopy
(160.1190) Materials : Anisotropic optical materials
(160.3380) Materials : Laser materials
(160.3918) Materials : Metamaterials

ToC Category:
Microscopy

History
Original Manuscript: July 30, 2012
Revised Manuscript: September 13, 2012
Manuscript Accepted: September 15, 2012
Published: September 21, 2012

Virtual Issues
Vol. 7, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Yao Ting Wang, Bo Han Cheng, You Zhe Ho, Yung-Chiang Lan, Pi-Gang Luan, and Din Ping Tsai, "Gain-assisted Hybrid-superlens Hyperlens for Nano Imaging," Opt. Express 20, 22953-22960 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22953


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References

  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  2. D. P. Tsai and W. C. Lin, “Probing the near fields of the super-resolution near-field optical structure,” Appl. Phys. Lett.77(10), 1413–1415 (2000). [CrossRef]
  3. D. P. Tsai, C. W. Yang, W. C. Lin, F. H. Ho, H. J. Huang, M. Y. Chen, T. F. Tseng, C. H. Lee, and C. J. Yeh, “Dynamic aperture of near-field super resolution structures,” Jpn. J. Appl. Phys.39(Part 1, No. 2B), 982–983 (2000). [CrossRef]
  4. W. C. Liu, C. Y. Wen, K. H. Chen, W. C. Lin, and D. P. Tsai, “Near-field images of the AgOx-type super-resolution near-field structure,” Appl. Phys. Lett.78(6), 685–687 (2001). [CrossRef]
  5. T. C. Chu, D. P. Tsai, and W. C. Liu, “Readout contrast beyond diffraction limit by a slab of random nanostructures,” Opt. Express15(1), 12–23 (2007). [CrossRef] [PubMed]
  6. K. P. Chiu, K. F. Lai, and D. P. Tsai, “Application of surface polariton coupling between nano recording marks to optical data storage,” Opt. Express16(18), 13885–13892 (2008). [CrossRef] [PubMed]
  7. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005). [CrossRef] [PubMed]
  8. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74(11), 115116 (2006). [CrossRef]
  9. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006). [CrossRef]
  10. W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B72(19), 193101 (2005). [CrossRef]
  11. D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys.7, 162 (2005). [CrossRef]
  12. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008). [CrossRef] [PubMed]
  13. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006). [CrossRef] [PubMed]
  14. H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for Imaging below the diffraction limit,” Opt. Express15(24), 15886–15891 (2007). [CrossRef] [PubMed]
  15. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007). [CrossRef] [PubMed]
  16. S. Schwaiger, M. Bröll, A. Krohn, A. Stemmann, C. Heyn, Y. Stark, D. Stickler, D. Heitmann, and S. Mendach, “Rolled-up three-dimensional metamaterials with a tunable plasma frequency in the visible regime,” Phys. Rev. Lett.102(16), 163903 (2009). [CrossRef] [PubMed]
  17. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science315(5819), 1699–1701 (2007). [CrossRef] [PubMed]
  18. N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express18(8), 8353–8359 (2010). [CrossRef] [PubMed]
  19. C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett.107(4), 043901 (2011). [CrossRef] [PubMed]
  20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79(24), 241104 (2009). [CrossRef]
  21. 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]
  22. F. J. Duarte and L. W. Hillman, “Dye laser principles with applications” (1990), See appendix.
  23. G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin.90(1-2), 1–5 (2000). [CrossRef]
  24. H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A.79(6), 2128–2132 (1982). [CrossRef] [PubMed]
  25. B. H. Cheng, Y. Z. Ho, Y. C. Lan, and D. P. Tsai, “Optical hybrid-superlens-hyperlens for superresolution imaging,” IEEE J. Sel. Top. Quantum Electron. (submitted).

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