## Evaluation of effective electric permittivity and magnetic permeability in metamaterial slabs by terahertz time-domain spectroscopy

Optics Express, Vol. 16, Issue 7, pp. 4785-4796 (2008)

http://dx.doi.org/10.1364/OE.16.004785

Acrobat PDF (429 KB)

### Abstract

We established a novel method to evaluate effective optical constants by terahertz (THz) time domain spectroscopy and suggested a strict definition of optical constants and an expression for electromagnetic energy loss following the second law of thermodynamics. We deduced the effective optical constants of phosphor bronze wire grids in the THz region experimentally and theoretically. The results depend strongly on the polarization of the THz waves. When the electric field is parallel to the wires, we observed Drude-like electric permittivities with a plasma frequency reduced by a factor of 10^{-3}, whereas when the field is perpendicular, the sample behaved as a simple dielectric film. We also observed unexpected magnetic permeabilities, which originate from the non-resonant real magnetic response of finite size-conductors.

© 2008 Optical Society of America

## 1. Introduction

1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

2. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. **32**, 53–55 (2007). [CrossRef]

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

4. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter **10**4785–4809 (1998). [CrossRef]

*ε*and magnetic permeability

*μ*. There are several ways [5–7

5. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of material by time-domain techniques,” IEEE Trans. Instrum. Meas. **IM-19**, 377–382 (1970). [CrossRef]

5. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of material by time-domain techniques,” IEEE Trans. Instrum. Meas. **IM-19**, 377–382 (1970). [CrossRef]

6. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

8. K. Cho, “Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the “*ε* and *μ*” description,” arXiv:cond-mat/0611235v4 [cond-mat.mtrl-sci] http://arxiv.org/abs/condmat/0611235v4.

8. K. Cho, “Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the “*ε* and *μ*” description,” arXiv:cond-mat/0611235v4 [cond-mat.mtrl-sci] http://arxiv.org/abs/condmat/0611235v4.

15. W. B. Weir, “Automatic Measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE **62**, 33–36 (1974). [CrossRef]

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

17. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science **303**, 1494–1496 (2004). [CrossRef] [PubMed]

23. K. Ohataka, T. Ueta, and K. Amemiya “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys Rev. B **57**, 2550–2568 (1998). [CrossRef]

## 2. Theory

5. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of material by time-domain techniques,” IEEE Trans. Instrum. Meas. **IM-19**, 377–382 (1970). [CrossRef]

6. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

13. P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express **11**, 649–661 (2003). [CrossRef] [PubMed]

14. E. Saenzz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. **101**, 114910 (2007). [CrossRef]

6. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

13. P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express **11**, 649–661 (2003). [CrossRef] [PubMed]

24. A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, “Fabrication and characterization o a negative-refractive-index composite metamaterial,” Phys. Rev. B **70**, 113102 (2004). [CrossRef]

*z*>0 and the Kramers–Kronig relation, where

*z*is the impedance. Note that the thermodynamical demand about the refractive index

*n*, Im

*n*>0, is not used during the entire extraction process. Thus, we can verify the resultant effective optical constants using this demand. This examination is essentially important when we apply the extraction method to the composite metamaterials, in which we have to examine the validity of the effective optical constants concepts. This is also helpful for the correction of the experimental errors. Especially in reflection type THz-TDS, the phase data of the reflected THz waves may include some errors because of the sample-misplacement problem. In this case, the examination enables us to correct the phase errors.

*n*

^{2}=

*k*

^{2}

*c*

^{2}/

*ω*

^{2}is confusing about their signs. Definitions of

*n*and

*z*that are meaningful even in negative index materials are

**ŝ**is the unit vector in the direction of the energy flux,

*z*

_{0}denotes the vacuum impedance (

*μ*

_{0}/

*ε*

_{0})

^{1/2}, and e and h express the complex electromagnetic field [25].

26. 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**=

*ε*

_{0}

*ε*

**E**and

**B**=

*μ*

_{0}

*μ*

**H**. First, we derive mathematically Re

*z*>0 from the time averaged energy flux

**S̅**by comparing both sides of the following equation:

**S**is the real Poynting vector. This also ensures that the reflection coefficients do not exceed 1. Second, we can deduce Im

*n*>0 from the physical fact that the electromagnetic energy should not become greater in passive media. Energy absorption

*Q*is

*U*is the electromagnetic energy density [25]. By taking the time average, the last term on the right-hand side goes to zero. We can transform the time-averaged energy absorption using Eq. (1) for the plane electromagnetic waves:

*e*and h are complex amplitudes of the electromagnetic field vectors, and the asterisk represents the complex conjugate [13

13. P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express **11**, 649–661 (2003). [CrossRef] [PubMed]

*Q̅*>0 requires the condition Im

*n*>0, because of Re

*z*>0. Therefore we have the two physical demands that Im

*n*>0 and Re

*z*>0. Furthermore, we can rewrite Eq. (4) in terms of

*ε*and

*μ*:

*ε*>0 [25], when we consider that for non-magnetic and passive materials,

*μ*=1. However, if we consider the response of magnetic materials to the plane wave, Im

*ε*>0 is not necessary.

*T*) and reflection coefficients (

*R*) of the homogeneous slab as

*d*is the electromagnetic effective thickness[14

14. E. Saenzz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. **101**, 114910 (2007). [CrossRef]

27. T. Driscoll, D. N. Basov, E. J. Padilla, J. J. Mock, and D. R. Smith, “Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements,” Phys. Rev. B **75**, 115114 (2007). [CrossRef]

*T*′=

*T*exp(

*iωd*/

*c*),

*R*′=-

*R*. These expressions are mathematically the same as those obtained in the transfer matrix method for the microwave region [15

15. W. B. Weir, “Automatic Measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE **62**, 33–36 (1974). [CrossRef]

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

*ε*and

*μ*experimentally:

*z*>0. Therefore, we have an infinite number of solution sets, (

*z*,

*n*)=(

*z*

_{0},

*n*

_{0}+

*cN*/

*fd*), where

*f*=

*ω*/2

*π*, and

*N*denotes an arbitrary integer. Second, we choose the value of the complex logarithm from the Kramers–Kronig relation of

*n*[30

30. E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, “Efficient dispersion relations for terahertz spectroscopy,” Appl. Phys. Lett. **89**, 142903 (2006). [CrossRef]

*n*>0. Therefore, we can verify and correct the experimental data applying this themodynamical demand.

## 3. Experimental setup

*ν*is the frequency at which the vacuum wavelength is the same as the period of the wires.

_{a}## 4. Results

31. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

**among the four samples are almost equal to those of**

*ν*_{peak}*ν*.

_{a}**65**, 195104 (2002). [CrossRef]

14. E. Saenzz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. **101**, 114910 (2007). [CrossRef]

27. T. Driscoll, D. N. Basov, E. J. Padilla, J. J. Mock, and D. R. Smith, “Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements,” Phys. Rev. B **75**, 115114 (2007). [CrossRef]

_{peak}. This suggests that the extraordinary transmissions are caused by an impedance match. When the impedance of the material is equal to that of the vacuum (1), the reflectivity of this material goes to 0 so that almost all electromagnetic energy transfers into it. The electric permittivity

*ε*is equal to 0 near ν

_{peak}.

*ε*exhibits Drude-like behavior with plasma frequency of 0.61 THz and extremely low loss. This value of the plasma frequency is much smaller than that of ordinal bulk metals by a factor of 10

^{-3}. The effective optical constants of sample A in the perpendicular configuration are plotted in Fig. 6. All the optical constants show a little dispersion, and the values are almost equal to 1. The sample behaves as a dielectric film with no loss.

*μ*are unexpected: The phosphor bronze is not a magnetic material, but the values of

*μ*=0.73, 0.8 in the parallel and perpendicular configuration are smaller than 1.

## 5. Discussion

4. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter **10**4785–4809 (1998). [CrossRef]

*c*is the speed of light in vacuum, and

*σ*is the bulk conductivity of the wires. The theoretical plasma frequency in sample A is 0.64 THz and is consistent with the experimental value. We also deduce the effective optical constants of sample B in the parallel configuration by THz-TDS and using Eqs. (7)–(9). We can see qualitatively same behavior as that of sample A: Drude-like electric permittivity. The plasma frequency is 1.08 THz, and the loss is extremely low. This behavior is also consistent with the theoretical result (the theoretical plasma frequency is 1.08 THz).

*C*is the capacitance between the wires.

*C*is very small in the present settings; therefore this model accounts for the no dispersion of the extracted permittivity.

*μ*is not equal to 1 in the parallel and perpendicular configuration, which seems to be unreasonable, because the constituent phosphor bronze is a non-magnetic material. This behavior cannot be explained within the framework of the homogeneous electric current density in one wire. We must take into consideration the loop current in one wire. Non-magnetic conductors of finite size have non-zero magnetic polarizability because of the loop current [25]. A perfect or good conducting circular wire has non-dispersive magnetic polarizability per unit length under the magnetic field perpendicular to the wire axis of

*r*is the radius. In the case of the magnetic field parallel to the wire axis, the magnetic polarizability is half that in the perpendicular case. Therefore, within the homogenization approximation, we derive

*μ*=0.60 and 0.80 in the parallel and perpendicular configuration. This magnetic polariizability is derived in [25] under the assumption that there is no electromagnetic interaction between the wires, and therefore this interaction effect may explain the small deviation between the theoretical and experimental values.

*Q̅*that is calculated from Eq. (5) using the extracted optical constants (Fig. (7)). Below

*ν*,

_{a}*Q̅*is almost equal to 0 in the parallel and perpendicular configuration. This small energy absorption behavior can be described using the present models. In both configurations, the magnetic energy loss |

*ε*| Im

*μ*is 0, because the magnetic permeability is real. In the parallel case,

*Q̅*∝

*γω*

_{p}^{2}/

*ω*

^{2}is almost equal to 0 in the frequency region near and over the plasma frequency. In contrast, in the perpendicular case, the total energy loss is 0 because

*ε*is also real.

## 6. Conclusion

^{-3}and very low losses. The observed extraordinary transmission results from the impedance-matching effect. In the perpendicular configuration, we can regard the sample as a simple dielectric film with very low losses. In the parallel and perpendicular configuration, the magnetic permeabilities are lower than 1,which can be explained by the inhomogeneity of the current flows in one wire. This non-resonant magnetic response may be helpful for the realization of negative-index metamaterials. Further studies into metamaterials in the terahertz frequency region are promising in the application of various optical devices, and the THz-TDS technique is suitable for metamaterial exploration.

## Acknowledgment

## References and links

1. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

2. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. |

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

4. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter |

5. | A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of material by time-domain techniques,” IEEE Trans. Instrum. Meas. |

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

7. | M. Iwanaga, “Effective optical constants in stratified metal-dielectric metamaterial,” Opt. Lett. |

8. | K. Cho, “Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the “ |

9. | T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

10. | Ricardo A. Depine and Akhlesh Lakhtakia, “Comment I on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E |

11. | A. L. Efros, “Comment II on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

12. | T. Koschny, P. Markos, D. R. Smith, and C. Soukoulis, “Reply to Comments on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E |

13. | P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express |

14. | E. Saenzz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, “On the definition of effective permittivity and permeability for thin composite layers,” J. Appl. Phys. |

15. | W. B. Weir, “Automatic Measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE |

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

17. | T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science |

18. | W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. |

19. | H.-Tong Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature |

20. | W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Phys. Rev. B |

21. | H.-Tong Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M. Lee, and W. J. Padilla, “Complementary planar terahertz metamaterials,” Opt. Express |

22. | H.-Tong Chen, W. J. Padilla, J. M. O. Zide, S. R. Bank, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Ultrafast optical switching of terahertz metamaterials fabricated on ErAs/GaAs nanoisland superlattices,” Opt. lett. |

23. | K. Ohataka, T. Ueta, and K. Amemiya “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys Rev. B |

24. | A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, “Fabrication and characterization o a negative-refractive-index composite metamaterial,” Phys. Rev. B |

25. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

26. | 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 |

27. | T. Driscoll, D. N. Basov, E. J. Padilla, J. J. Mock, and D. R. Smith, “Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements,” Phys. Rev. B |

28. | T. Driscoll, G. O. Andreev, D. N. Basov, S. Palit, T. Ren, J. Mock, S.-Y. Cho, N. M. Jokerst, and D. R. Smith, “Quantitative investigation of terahertz artificial magnetic resonance using oblique angle spectroscopy,” Appl. Phys. Lett. |

29. | B.-I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B |

30. | E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, “Efficient dispersion relations for terahertz spectroscopy,” Appl. Phys. Lett. |

31. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

**OCIS Codes**

(260.3910) Physical optics : Metal optics

(300.6270) Spectroscopy : Spectroscopy, far infrared

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: February 10, 2008

Manuscript Accepted: February 25, 2008

Published: March 24, 2008

**Citation**

Yosuke Minowa, Takashi Fujii, Masaya Nagai, Tetsuyuki Ochiai, Kazuaki Sakoda, Kazuyuki Hirao, and Koichiro Tanaka, "Evaluation of effective electric permittivity and magnetic permeability in metamaterial slabs by terahertz time-domain spectroscopy," Opt. Express **16**, 4785-4796 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4785

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### References

- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007). [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 104785-4809 (1998). [CrossRef]
- A. M. Nicolson and G. F. Ross, "Measurement of the intrinsic properties of material by time-domain techniques," IEEE Trans. Instrum. Meas. IM-19, 377-382 (1970). [CrossRef]
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002). [CrossRef]
- M. Iwanaga, "Effective optical constants in stratified metal-dielectric metamaterial," Opt. Lett. 32, 1314-1316 (2007). [CrossRef] [PubMed]
- K. Cho, "Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the "ε and μ" description," arXiv:cond-mat/0611235v4 [cond-mat.mtrl-sci] http://arxiv.org/abs/cond-mat/0611235v4.
- T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 68, 065602 (2003). [CrossRef]
- R. A. Depine and A. Lakhtakia, "Comment I on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials", Phys. Rev. E 70, 048601 (2004). [CrossRef]
- A. L. Efros, "Comment II on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 70, 048602 (2004). [CrossRef]
- T. Koschny, P. Markos, D. R. Smith, and C. Soukoulis, "Reply to Comments on Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 70, 048603 (2004). [CrossRef]
- P. Markos and C. M. Soukoulis, "Transmission properties and effective electromagnetic parameters of double negative metamaterials," Opt. Express 11, 649-661 (2003). [CrossRef] [PubMed]
- E. Saenzz, P. M. T. Ikonen, R. Gonzalo, and S. A. Tretyakov, "On the definition of effective permittivity and permeability for thin composite layers," J. Appl. Phys. 101, 114910 (2007). [CrossRef]
- W. B. Weir, "Automatic Measurement of complex dielectric constant and permeability at microwave frequencies," Proc. IEEE 62, 33-36 (1974). [CrossRef]
- D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005). [CrossRef]
- T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
- W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, "Dynamical electric and magnetic metamaterial response at terahertz frequencies," Phys. Rev. Lett. 96, 107401 (2006). [CrossRef] [PubMed]
- H.-Tong Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, "Active terahertz metamaterial devices," Nature 444, 597-600 (2006). [CrossRef] [PubMed]
- W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, "Electrically resonant terahertz metamaterials: Theoretical and experimental investigations," Phys. Rev. B 75, 041102 (2007). [CrossRef]
- H.-Tong Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M. Lee, and W. J. Padilla, "Complementary planar terahertz metamaterials," Opt. Express 15, 1084-1095 (2007). [CrossRef] [PubMed]
- H.-Tong Chen, W. J. Padilla, J. M. O. Zide, S. R. Bank, A. C. Gossard, A. J. Taylor, and R. D. Averitt, "Ultrafast optical switching of terahertz metamaterials fabricated on ErAs/GaAs nanoisland superlattices," Opt. lett. 32, 1620-1622 (2007). [CrossRef] [PubMed]
- K. Ohataka, T. Ueta, and K. Amemiya "Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods," Phys Rev. B 57, 2550-2568 (1998). [CrossRef]
- A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, "Fabrication and characterization o a negative-refractive-index composite metamaterial," Phys. Rev. B 70, 113102 (2004). [CrossRef]
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 2nd ed. (Elsevier, New York, 1984).
- 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]
- T. Driscoll, D. N. Basov, E. J. Padilla, J. J. Mock, and D. R. Smith, "Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements," Phys. Rev. B 75, 115114 (2007). [CrossRef]
- T. Driscoll, G. O. Andreev, D. N. Basov, S. Palit, T. Ren, J. Mock, S.-Y. Cho, N. M. Jokerst, and D. R. Smith, "Quantitative investigation of terahertz artificial magnetic resonance using oblique angle spectroscopy," Appl. Phys. Lett. 90, 092508 (2007). [CrossRef]
- B.-I. Popa and S. A. Cummer, "Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields," Phys. Rev. B 72, 165102 (2005). [CrossRef]
- E. Gornov, K.-E. Peiponen, Y. Svirko, Y. Ino, and M. Kuwata-Gonokami, "Efficient dispersion relations for terahertz spectroscopy," Appl. Phys. Lett. 89, 142903 (2006). [CrossRef]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]

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