## Planar designs for electromagnetically induced transparency in metamaterials

Optics Express, Vol. 17, Issue 7, pp. 5595-5605 (2009)

http://dx.doi.org/10.1364/OE.17.005595

Acrobat PDF (492 KB)

### Abstract

We present a planar design of a metamaterial exhibiting electromagnetically induced transparency that is amenable to experimental verification in the microwave frequency band. The design is based on the coupling of a split-ring resonator with a cut-wire in the same plane. We investigate the sensitivity of the parameters of the transmission window on the coupling strength and on the circuit elements of the individual resonators, and we interpret the results in terms of two linearly coupled Lorentzian resonators. Our metamaterial designs combine low losses with the extremely small group velocity associated with the resonant response in the transmission window, rendering them suitable for slow light applications at room temperature.

© 2009 Optical Society of America

## 1. Introduction

1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science **305**, 788–792 (2004). [CrossRef] [PubMed]

2. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B **23**, 391–403 (2006). [CrossRef]

3. T. Koschny, P. Markos, 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]

4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

5. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: New Frontiers in Optics,” Adv. Mater. **18**, 1941–1952 (2006). [CrossRef]

6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* nand *μ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

7. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

8. N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Ant. Wireless Prop. Lett. **1**, 10–13 (2002). [CrossRef]

12. U. Leonhardt and T. G. Philbin, “Transformation Optics and the Geometry of Light,” Prog. Opt., in press (2008). http://arxiv.org/abs/0805.4778v2.

13. U. Leonhardt, “Optical Conformal Mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

16. S. Guenneau, A. Movchan, G. Pétursson, and S. A. Ramakrishna, “Acoustic metamaterials for sound focusing and confinement,” New J. Phys. **9**, 399 (2007). [CrossRef]

27. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-Induced Transparency in Metamaterials,” Phys. Rev. Lett. **101**, 047401 (2008). [CrossRef] [PubMed]

29. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. **102**, 053901 (2009). [CrossRef] [PubMed]

30. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagneti-cally induced transparency,” Phys. Rev. Lett. **101**, 253903 (2008). [CrossRef] [PubMed]

31. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005). [CrossRef]

31. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. **77**, 633–673 (2005). [CrossRef]

## 2. Planar EIT Metamaterials

29. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. **102**, 053901 (2009). [CrossRef] [PubMed]

29. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. **102**, 053901 (2009). [CrossRef] [PubMed]

*ω*

_{0}by the full width half maximum bandwidth,

*Q*=

*ω*

_{0}/Δ

*ω*

_{FWHM}. The same procedure but with the SRR removed is repeated in order to calculate the quality factor of the cut-wire. We find that the quality factor of the dark resonator (

*Q*

_{d}≈ 10) is larger than the quality factor of the radiative circuit (

*Q*

_{r}≈ 3.5), as is necessary in order to observe EIT. The SRR and the cut-wire are also designed such that they have the same resonant frequency at

*ω*

_{0}= 9.35 GHz.

*f*= 9.85 GHz), the highest current can be observed in the wire and the current in the split-ring resonator is significantly smaller; there is only a slight coupling to an electric resonance of the SRR. At the transparency frequency (

*f*= 9.72 GHz), the high current in the split-ring resonator shows that the now resonant split-ring is strongly excited; on the other hand, the current in the dipole-coupled wire is small. This behavior is in agreement with our analytical model presented in Sec. 4 and also supports our claim that this metamaterial exhibits classical electromagnetically induced transparency.

*A*= 1–∣

*S*

_{11}∣

^{2}– ∣

*S*

_{12}∣

^{2}) and the permittivity using the parameter retrieval procedure developed by Smith

*et al*. [3

3. T. Koschny, P. Markos, 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]

32. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E **71**, 036617 (2005). [CrossRef]

*ε*). Inside the transparency window, we observe steep dispersion, which will lead to a significantly increased group index and could be useful for slow-light applications. In the remainder of this paper, we will focus on how we can increase the group index of the proposed metamaterial structure.

## 3. Analytical Model for the Metamaterial

*i*

_{1}and

*i*

_{2}circulating in the radiative and dark circuit loops, respectively, can be calculated from standard loop current analysis [33]:

*β*is proportional to the density of cut-wires in the metamaterial and depends also on a number of unknown proportionality constants that are only function of the exact geometry of the wire. Eq. (3) allows to assess quickly how the circuit parameters—and, ultimately, changes in the geometry of the design—affect the properties of the EIT transmission window, such as absorption and group index. Furthermore, since the coupled SRR-wire structure has no magnetic resonances in the EIT transmission window (this is confirmed by the retrieved permeability which is not shown here), the index of refraction can be estimated from

*n*≈ √

*ε*and the group index from

*n*=

_{g}*n*+

*ω*d

*n*/d

*ω*.

## 4. Influence of Circuit Parameters on the EIT Transmission Window

34. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” vol. 16 of *Waves in Random and Complex Media* (Taylor and Francis, London, 2006). [CrossRef]

*i*

_{1}) is small, so that we can approximate the permittivity by

*ω*

_{0}= 1/(

*L*

_{1}

*C*

_{1})

^{1/2}= 1/(

*L*

_{2}

*C*

_{2})

^{1/2}the center frequency of the transparency window.

- In first approximation, the group index does not depend on the resistances
*R*_{1}and*R*_{2}and, hence, not on the ohmic losses in the metamaterial. - The group index is proportional to
*C*^{2}_{c}. This means that weaker coupling (i.e., larger spatial distance between both resonators) leads to smaller group velocities. - We have an additional degree of freedom to control the group velocity: the inductance of the dark circuit.

### 4.1. Resistance of the radiative resonator

*R*

_{1}(blue curves). We observe high group index in the frequency range where Fig. 1(c) has the steepest dispersion. The resistance of the wire was increased by lowering the conductivity of the metal from 5.80×10

^{7}S m

^{-1}to 1.16×10

^{7}S m

^{-1}. We observe that the group index inside the transparency window is almost independent of

*R*

_{1}. The absorption is reduced at the peaks, but is more or less unchanged inside the frequency region of interest. Consequently, the resistance

*R*

_{1}is a parameter that hardly influences the properties of the transparency window.

*R*

_{1}, and therefore the group velocity and absorption in the transparency window are largely insensitive on the value of

*R*

_{1}or on the losses in the wire.

### 4.2. Resistance of the dark resonator

*R*

_{2}(blue lines). This was achieved by increasing the conductivity of the metal of the SRR with one order of magnitude to 5.80×10

^{8}S m

^{-1}. We conclude that the group velocity is only slightly influenced by the resistance of the SRR. However, from the inset of Fig. 6(b) we can see that this time the absorption inside the transparency window has also decreased. It will therefore be advantageous to minimize the losses in the dark resonator in order to achieve low absorption, which is also an important parameter for slow light applications.

### 4.3. Coupling strength

*d*= 2.0 mm (black dashed lines) to

*d*= 2.5 mm (blue lines). This behavior can be understood by considering the EIT effect as a frequency splitting of the originally degenerate resonances of wire and SRR. The farther away these two elements, the weaker the coupling, and the narrower the transparency window; this in turn leads to stronger dispersion and higher group index. Two remarks must be made here. (i) The group velocity will be ultimately limited by the size of the unit cell. If the distance between the SRR and the wire is approximately half a lattice constant, the interaction between the wire of one unit cell and the SRR of its nearest neighbor will contribute significantly to the response. Increasing the lattice constant will not help, since the parameter

*β*is proportional to the density of the unit cells. (ii) The increased group velocity goes together with a smaller bandwidth.

### 4.4. Inductances of dark/radiative resonator

*L*

_{1}. Our analytical model predicts that a higher inductance of the SRR will lead to a higher group index. This case is somewhat more difficult to check, because a change in the geometry of the SRR will automatically go together with a change in coupling strength. We have therefore first decreased the enclosed surface of the SRR; this will make

*L*

_{2}smaller; at the same time we have also adjusted the thickness of the gaps of the SRRs in order to keep the resonance frequency constant. As to leave the coupling strength unchanged, we have increased the distance between the wire and the SRR until the electric field integrated along the inner loop of the SRR is the same as in the reference case. We can indeed take the integrated electric field as a measure for the coupling strength, since the electric fields in the SRR will increase if the coupling would have been stronger and vice versa.

*L*

_{2}, indeed leads to a somewhat smaller group velocity in the transparency window, although the effect is not very strong. The influence on the absorption is negligible. The first remark made in Sec. 4.3 is also appropriate here, namely that the increase in inductance will finally be limited by the size of the unit cell. On the other hand, it is important to observe that a smaller inductance

*L*

_{2}allows for a significantly larger bandwidth and, hence, delay-bandwidth product, which is another important figure of merit for slow light applications.

## 5. Summary

## Acknowledgments

## References and links

1. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science |

2. | D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B |

3. | T. Koschny, P. Markos, 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 |

4. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. |

5. | C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-Index Materials: New Frontiers in Optics,” Adv. Mater. |

6. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

7. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

8. | N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Ant. Wireless Prop. Lett. |

9. | P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A |

10. | A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index meta-materials and their electromagnetic applications,” IEEE Trans. Antennas Propag. |

11. | P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. |

12. | U. Leonhardt and T. G. Philbin, “Transformation Optics and the Geometry of Light,” Prog. Opt., in press (2008). http://arxiv.org/abs/0805.4778v2. |

13. | U. Leonhardt, “Optical Conformal Mapping,” Science |

14. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

15. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwells equations,” Photon. Nanostruct.: Fundam. Applic. |

16. | S. Guenneau, A. Movchan, G. Pétursson, and S. A. Ramakrishna, “Acoustic metamaterials for sound focusing and confinement,” New J. Phys. |

17. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

18. | N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric Coupling to the Electric Resonance of Split-Ring Resonators,” Appl. Phys. Lett. |

19. | T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective Medium Theory of Left-handed Materials,” Phys. Rev. Lett. |

20. | J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, “On the resonances and polarizabilities of split-ring resonators,” J. Appl. Phys. |

21. | F. Bilotti, A. Toscano, and L. Vegni, “Design of Spiral and Multiple Split-Ring Resonators for the Realization of Miniaturized Metamaterial Samples,” IEEE Trans. Antennas Propag. |

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

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

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

25. | P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic Resonances in Individual and Coupled Split-ring Resonators,” J. Appl. Phys. |

26. | R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express |

27. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-Induced Transparency in Metamaterials,” Phys. Rev. Lett. |

28. | N. Liu, S. Kaiser, T. Pfau, and H. Giessen, “Electromagnetically Induced Transparency in Optical Metamaterials,” presented at the QELS Postdeadline Session II of CLEO/QELS, San Jose, California, USA, 2008. |

29. | P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency,” Phys. Rev. Lett. |

30. | N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “A metamaterial analog of electromagneti-cally induced transparency,” Phys. Rev. Lett. |

31. | M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. |

32. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E |

33. | C. Caloz and T. Itoh, “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications” (Wiley, New Jersey, 2005). |

34. | A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” vol. 16 of |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: March 5, 2009

Revised Manuscript: March 20, 2009

Manuscript Accepted: March 21, 2009

Published: March 24, 2009

**Citation**

Philippe Tassin, Lei Zhang, Thomas Koschny, E. N. Economou, and C. M. Soukoulis, "Planar designs for electromagnetically
induced transparency in metamaterials," Opt. Express **17**, 5595-5605 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5595

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### References

- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- D. R. Smith and J. B. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B 23, 391-403 (2006). [CrossRef]
- T. Koschny, P. Markos, 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]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite Medium with Simultaneously Negative Permeability and Permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006). [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ∑ and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002). [CrossRef]
- P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, "Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials," Phys. Rev. A 74, 033822 (2006). [CrossRef]
- A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, "Single-negative, double-negative and low index metamaterials and their electromagnetic applications," IEEE Trans. Antennas Propag. 49, 23-36 (2007).
- P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008). [CrossRef]
- U. Leonhardt and T. G. Philbin, "Transformation Optics and the Geometry of Light," Prog. Opt., in press (2008). http://arxiv.org/abs/0805.4778v2
- U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwells equations," Photon. Nanostruct.: Fundam. Applic. 6, 87 (2008). [CrossRef]
- S. Guenneau, A. Movchan, G. P’etursson, and S. A. Ramakrishna, "Acoustic metamaterials for sound focusing and confinement," New J. Phys. 9, 399 (2007). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Electric Coupling to the Electric Resonance of Split-Ring Resonators," Appl. Phys. Lett. 84, 2943-2945 (2004). [CrossRef]
- T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective Medium Theory of Left-handed Materials," Phys. Rev. Lett. 93, 107402 (2004). [CrossRef] [PubMed]
- J. Garcia-Garcia, F. Martin, J. D. Baena, R. Marques, and L. Jelink, "On the resonances and polarizabilities of split-ring resonators," J. Appl. Phys. 98, 033103 (2005). [CrossRef]
- F. Bilotti, A. Toscano, and L. Vegni, "Design of Spiral and Multiple Split-Ring Resonators for the Realization of Miniaturized Metamaterial Samples," IEEE Trans. Antennas Propag. 55, 2258-2267 (2007). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006). [CrossRef] [PubMed]
- V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007). [CrossRef]
- C. M. Soukoulis, S. Linden, and M. Wegener, "Negative index metamaterials at optical wavelengths," Science 315, 47-49 (2007). [CrossRef] [PubMed]
- P. Gay-Balmaz and O. J. F. Martin, "Electromagnetic Resonances in Individual and Coupled Split-ring Resonators," J. Appl. Phys. 92, 2929-2936 (2002). [CrossRef]
- R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, "Multi-gap individual and coupled split-ring resonator structures," Opt. Express 16, 18131-18144 (2008). [CrossRef] [PubMed]
- S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, "Plasmon-Induced Transparency in Metamaterials," Phys. Rev. Lett. 101, 047401 (2008). [CrossRef] [PubMed]
- N. Liu, S. Kaiser, T. Pfau, and H. Giessen, "Electromagnetically Induced Transparency in Optical Metamaterials," presented at the QELS Postdeadline Session II of CLEO/QELS, San Jose, California, USA, 2008.
- P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009). [CrossRef] [PubMed]
- N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, "A metamaterial analog of electromagnetically induced transparency," Phys. Rev. Lett. 101, 253903 (2008). [CrossRef] [PubMed]
- M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005). [CrossRef]
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005). [CrossRef]
- C. Caloz and T. Itoh, "Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications" (Wiley, New Jersey, 2005).
- A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," in Waves in Random and Complex Media (Taylor and Francis, London, 2006) Vol. 16. [CrossRef]

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