## Transmission properties and effective electromagnetic parameters of double negative metamaterials

Optics Express, Vol. 11, Issue 7, pp. 649-661 (2003)

http://dx.doi.org/10.1364/OE.11.000649

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### Abstract

We analyze the transmission properties of double negative metamaterials (DNM). Numerical simulations, based on the transfer matrix algorithm, show that some portion of the electromagnetic wave changes its polarization inside the DNM structure. As the transmission properties depend strongly on the polarization, this complicates the interpretation of experimental and numerical data, both inside and outside of the pass band. From the transmission data, the effective permittivity, permeability and refractive index are calculated. In the pass band, we found that the real part of permeability and both the real and the imaginary part of the permittivity are negative. Transmission data for some new structures are also shown. Of particular interest is the structure with cut wires, which possesses two double negative pass bands.

© 2003 Optical Society of America

## 1. Introduction

*et al.*[1

1. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. **76**, 4773 (1996) [CrossRef] [PubMed]

2. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, (1999) “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. on Microwave Theory and Techn. **47**, 2075 (1999) [CrossRef]

3. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz “A Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184 (2000) [CrossRef] [PubMed]

4. D.R. Smith and N. Kroll, “Negative Refractive Index in Left-Handed Materials,” Phys. Rev. Lett. **85**, 2933 (2000) [CrossRef] [PubMed]

5. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed meta material,” Appl. Phys. Lett. **78**, 489 (2001) [CrossRef]

7. P. Markoš and C.M. Soukoulis, “Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators” Phys. Rev. E **65**, 036622 (2002) [CrossRef]

8. P. Markoš, I. Rousochatzakis, and C.M. Soukoulis, “Transmission Losses in Left-handed Materials,” Phys. Rev. E **66**, 045601 (2002) [CrossRef]

9. C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Numerical Studies of Left-handed Metamaterials,” Progress In Electromagnetics Research, PIER **35**, 315 (2002) [CrossRef]

_{eff}and permeability µ

_{eff}

*negative*. They are also called “Left-handed materials” (LHM) after Veselago [10

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

_{eff}given by the formula [1

1. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. **76**, 4773 (1996) [CrossRef] [PubMed]

11. M.M. Sigalas, C.Y. Chan, K.M. Ho, and M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B **52**, 11744 (1999) [CrossRef]

2. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, (1999) “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. on Microwave Theory and Techn. **47**, 2075 (1999) [CrossRef]

*f*

_{e}is the electronic plasma frequency,

*f*

_{m}is the magnetic resonance frequency, γ

*e*(γ

_{m}) represent the losses of the system, and

*F*is a filling factor of the SRR.

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

3. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz “A Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184 (2000) [CrossRef] [PubMed]

*n*was done by measuring the

*negative*Snell’s law [12

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

13. C. G. Parazzoli, R. B. Gregor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,” Phys. Rev. Lett. **90** 107401 (2003) [CrossRef] [PubMed]

_{eff}and µ

_{eff}has been proved by the analysis of numerical data, for the transmission and reflection (magnitude as well as phase) obtained by the transfer matrix method (TMM) [14

14. D.R. Smith, S. Shultz, P. Markoš, and C.M. Soukoulis, “Determination of Effective Permittivity and Permeability of Metamaterials from Reflection and Transmission Coefficient,” Phys. Rev. B **65** 195104 (2002) [CrossRef]

_{zz}<0 and ∊

_{xx}, ∊

_{yy}<0 only [15

15. D. R. Smith and D. Schurig, “Electromagnetic Wave propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. **90** 077405 (2003) [CrossRef] [PubMed]

*z*axis. No wave propagation is possible along the

*x*or the

*y*axis since the product ∊µ is negative for this direction. This might be overcame using more sophisticated structures. For instance, structures analyzed in Refs. [5

5. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed meta material,” Appl. Phys. Lett. **78**, 489 (2001) [CrossRef]

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

_{eff}, but also the

*imaginary*part ∊″

_{eff}is

*negative*. The same was observed in other structures [16

16. S. O’Brien and J.B. Pendry, “Photonic band gap effects and magnetic activity of dielectric composites,” J. Phys.: Condens. Matter **14**4035 (2002) [CrossRef]

## 2. Transfer matrix method (TMM)

7. P. Markoš and C.M. Soukoulis, “Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators” Phys. Rev. E **65**, 036622 (2002) [CrossRef]

17. J.B. Pendry and A. MacKinnon, Phys. Rev. Lett.692772 (1992) J.B. Pendry, “Photonic band gap structures,” J. Mod. Opt. 41 (1994) 209 J.B. Pendry and P.M. Bell 1996, “Transfer matrix techniques for electromagnetic waves,” in Photonic Band Gap Materials vol. 315 of NATO ASI Ser. E: Applied Sciences (1996), ed. by C.M. Soukoulis (Plenum, NY) p. 203 [CrossRef] [PubMed]

7. P. Markoš and C.M. Soukoulis, “Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators” Phys. Rev. E **65**, 036622 (2002) [CrossRef]

_{m}=∊′

_{m}+

*i*∊″

_{m}. In our simulations, ∊″

_{m}takes values as large as 10

^{6}. In such strongly inhomogeneous structures there is a danger that the discretization of Maxwell’s equation might not be accurate enough. Therefore, in order to check the ability of the transfer matrix to give us the proper electromagnetic parameters of the LH structure, we analyzed the simpler system, namely the periodic array of thin metallic wires. Thanks to the translational invariance of this system along the wire axis, the system is effectively two-dimensional. This enabled us to use variousdiscretizations of the unit cell. As it is shown in Fig. 2, the obtained effective permittivity is almost independent of the discretization procedure.

*N*=10

^{3}mesh points per unit cell. More detailed discretization will require too long CPU time. As we only use a homogeneous discretization, we are restricted in the sizes of the DNM structures. All the length scales of the DNM are namely given as multiplies of the minimum mesh length which is typically 0.15–0.33 mm.

## 3. Transmission data

*z*direction (Fig. 1). The transfer matrix can obtain not only the transmission for both the

*E⃗*‖

_{y}polarization (

*t*

_{yy}) and the

*E⃗*‖

*x*polarization (

*t*

_{xx}), but also the transmission

*t*

_{xy}and

*t*

_{yx}. The last two transmissions give the change of the polarization of the EM wave inside the structure.

*t*|

^{2}for the DNM as shown in Fig 1 and for an array of SRR. Note that the “off-diagonal” transmissions |

*t*

_{xy}|

^{2}and |

*t*

_{yx}|

^{2}are relatively large also in the band pass of DNM. This means that in the DNM structure, there always exists a possibility for the transmission from one polarization wave into the other one. This effect plays an important role outside the band pass. In Fig. 4 we show the transmission for the DNM structure for the frequency

*f*=9.7 GHz, which is slightly below the lower pass band edge. We see that for a system length shorter than 10 unit cells, the transmission quickly oscillates and decreases exponentially. This indicates that both |

*n*′

_{eff}| and |

*n*″

_{eff}| are large. After passing 10 unit cells, however, the transmission stops to decrease and its length dependence is determined by the index of refraction in vacuum.

*t*

_{yy}consists not only from the “unperturbed” contribution

*y*-polarized wave into

*x*-polarized and back to

*y*-polarized:

*L*) decreases exponentially with the system length

*L*, while the second term

*y*-polarized wave into

*x*-polarized wave and back, remains system-length independent, because

*t*

_{xx}(

*z*,

*z*′) ~ 1 for any distance |

*z*-

*z*′|. For system length larger than certain “critical length”, one observes only the term

*n*″

_{eff}is larger, the “critical length” decreases to 1–2 unit cells. Therefore, in the experiment, one observes only the transmission given by the second term of Eq. (3).

*outside*the sample always contains a small portion of the

*E⃗*‖

*y*wave, which

*inside*the structure behaves as

*E⃗*‖

*x*and knows nothing about the negativeness of effective parameters. In our numerical simulations, the amplitude of this “false wave” is rather small. In Fig 3, the transmission |

*t*

_{yx}|

^{2}is of the order of 10

^{-3}(with sharp maximum |

*t*

_{xy}|

^{2}≈0.036 for

*f*=10.7 GHz). This means that |

*t*

_{xy}is much larger for some other DNM pass band.

## 4. Effective parameters of DNM

*L*are given, in terms of the index of refraction

*n*and impedance

*z*of the slab, by the textbook formulas

*k*is the wave vector in vacuum of the normally incident EM wave. Our aim is to invert Eqs. (5) and (6), i.e., to express

*n*and

*z*as a function of

*t*and

*r*. Once

*n*and

*z*are obtained, permittivity and permeability can be easily calculated from the relations

*Y*|<1). This determines unambiguously both real and imaginary part of the refractive index.

*L*, relation (11) enables us to find unambiguously both the real and the imaginary part of the index of refraction. We collect the data for various system lengths

*L*, and calculate

*n*′ and

*n*″ from the linear fits of

*n*″

*kL*and

*n*′

*kL*

*versus*the system length

*L*.

_{eff}and µ′

_{eff}have been found to be negative.

*t*

_{xy}| is much larger than |

*t*

_{yy}| already for a system length of 2–3 unit cells. As we will discuss later, for more complicated structures the condition (13) is not satisfied even in the pass band. Then, the method used by Smith

*et al.*must be applied [14

14. D.R. Smith, S. Shultz, P. Markoš, and C.M. Soukoulis, “Determination of Effective Permittivity and Permeability of Metamaterials from Reflection and Transmission Coefficient,” Phys. Rev. B **65** 195104 (2002) [CrossRef]

*n*′ is negative, but |

*n*′|<1. We can also improve the ratio wavelength/(size of the unit cell), by filling the structure with a dielectric with a high permittivity ∊

_{d}. This reduces the resonance frequency of the SRR by factor

_{d}=10 (data not shown here) and observed the same qualitative behavior of the effective parameters as in this paper.

*n*′

_{eff}| becomes so large that λ is comparable with the size of the unit cell. In this frequency region Eqs. (5) and (6) are not sufficient to estimate

*n*.

_{eff}follows the analytical formula (2). The same was observed in Ref. [14

14. D.R. Smith, S. Shultz, P. Markoš, and C.M. Soukoulis, “Determination of Effective Permittivity and Permeability of Metamaterials from Reflection and Transmission Coefficient,” Phys. Rev. B **65** 195104 (2002) [CrossRef]

_{eff}is more complicated. It is evident that it can not be described by simple relation (1. We found the non-monotonic frequency-dependence of ∊′

_{eff}, again in agreement with previous results [14

**65** 195104 (2002) [CrossRef]

_{eff}exhibits peaks at frequencies where

*n*′

_{eff}or µ′

_{eff}change their sign.

## 5. Negative ∊″_{eff} materials?

_{eff}and ∊″

_{eff}exhibits resonant behavior in the neighborhood of the frequency

*f*′ where µ′

_{eff}(

*f*′)=0. This is easy to understand mathematically. Relation

*n*

^{2}=∊µ gives for the real and the imaginary part of the refractive index

*n*′

*n*″-∊′µ″)/µ′ indeed has a poles at frequency

*f*′ and changes its sign from negative (for

*f*<

*f*′ to positive (for

*f*>

*f*′).

*n*′|≫

*n*″ for frequencies

*f*≤

*f*′. Then the r.h.s. of Eq. (14) must be positive in the limit

*f*→

*f*′. As the product ∊′µ′→0 for

*f*→

*f*′, Eq. (14) implies that the product ∊″µ″ is

*negative*for

*f*≤

*f*′. Hence the small value of

*n*″

_{eff}implies not only very good transmission transmission properties of DNM, but also negativeness of ∊″.

_{eff}is not a result of oversimplified model Eqs. (5) and (6). The generalization of the relations Eqs. (5) and (6) to anisotropic systems would be desired, which calculate tensors of the permittivity and permeability from all transmission data

*t*

_{yy},

*t*

_{xx},

*t*

_{xy}and

*t*

_{yx}. However, the transmission

*t*

_{xy}is small: |

*t*

_{xy}|

^{2}<10

^{-3}for

*f*≤11 GHz, while |

^{-2}for the system lengths 200–300 unit cells. Therefore the correction to transmission, given by Eq. (4), is |

^{-3}≪|

*t*

^{(0)}|. We assume therefore that the “off diagonal” transmissions would only smooth the “resonance” behavior of ∊″. The qualitative frequency dependence would be unchanged. We note also that the negative ∊″

_{eff}was observed in the system of dielectric wires, [16

16. S. O’Brien and J.B. Pendry, “Photonic band gap effects and magnetic activity of dielectric composites,” J. Phys.: Condens. Matter **14**4035 (2002) [CrossRef]

*diagonal*so that

_{eff}by a more detailed theoretical analysis. To the best of our knowledge, there is no theoretical work which determine the energy and the losses of the EM wave propagated in the medium with arbitrary values of (complex) ∊ and µ. Nevertheless, we note that the negative sign of the ∊″ is not in contradiction with any physical law. In particular, the transmission losses, which are usually expressed in terms of ∊″ and µ″ [20] could be re-written, at least in the special case of the normally incident plane wave, in terms of

*n*″ and

*z*′ as

*n*″ and

*z*′ are positive (see Eqs. (12) and (9)). We did not find any general analysis of the transmission losses in systems with arbitrary sign and values of the (complex) permittivity and permeability.

*f*→∞, but also at the left boundary of the pass band, the permittivity and permeability is not achievable by present methods.

## 6. Other one-dimensional DNM structures

**65**, 036622 (2002) [CrossRef]

*x*direction) decreases

19. P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires”, *e-print* cond-mat/0212343 to appear in Opt. Lett. (2003) [CrossRef]

5. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed meta material,” Appl. Phys. Lett. **78**, 489 (2001) [CrossRef]

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

21. E.V. Ponizovskaya, M. Nieto-Vesperinas, and N. Garcia, “Losses for microwave transmission metamaterials for producing left-handed materials: The strip wires,” Appl. Phys. Lett. **81**, 4470 (2002) [CrossRef]

19. P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires”, *e-print* cond-mat/0212343 to appear in Opt. Lett. (2003) [CrossRef]

3. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz “A Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184 (2000) [CrossRef] [PubMed]

**78**, 489 (2001) [CrossRef]

8. P. Markoš, I. Rousochatzakis, and C.M. Soukoulis, “Transmission Losses in Left-handed Materials,” Phys. Rev. E **66**, 045601 (2002) [CrossRef]

23. M. Bayindir, K. Aydin, E. Ozbay, P. Markoš, and C.M. Soukoulis, “Transmission Properties of Composite Metamaterials in Free Space,” Appl. Phys. Lett. **81**, 120 (2002) [CrossRef]

*versus*frequency for a DNM used in the experiment of Ref. [23

23. M. Bayindir, K. Aydin, E. Ozbay, P. Markoš, and C.M. Soukoulis, “Transmission Properties of Composite Metamaterials in Free Space,” Appl. Phys. Lett. **81**, 120 (2002) [CrossRef]

23. M. Bayindir, K. Aydin, E. Ozbay, P. Markoš, and C.M. Soukoulis, “Transmission Properties of Composite Metamaterials in Free Space,” Appl. Phys. Lett. **81**, 120 (2002) [CrossRef]

*versus*frequency for a lattice of cut wires is shown by a dashed line on the top panel of Fig. 8. The transmission is close to one for frequencies up to 10 GHz and drops to very small values for higher frequencies. So that the behavior of the cut wires is completely different from that of continuous wires. While the continuous wires behave as high pass filters, the cut wires behave as photonic crystals [25]. As can be seen from the bottom panel of Fig. 8, the first transmission peak of the DNM gives a positive index of refraction, and the two higher peaks give negative index of refraction.

## 7. Conclusion

**65**, 036622 (2002) [CrossRef]

8. P. Markoš, I. Rousochatzakis, and C.M. Soukoulis, “Transmission Losses in Left-handed Materials,” Phys. Rev. E **66**, 045601 (2002) [CrossRef]

*E⃗*‖

*y*wave as input we observe that some small part of the outgoing

*E⃗*‖

*y*wave does not experienced the left-handedness, because it propagates through the sample as a

*E⃗*‖

*x*wave. Outside the pass band, this “false wave” represents the only output, if the sample is big enough to absorb the “true wave” completely.

*effective*permeability to be negative [25]. Thus we believe that systems with negative sign of imaginary part of either

*effective*permittivity or

*effective*permeability might be quite often created in today’s laboratories, although it may not be present in nature.

26. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**309 (2002) [CrossRef]

## Acknowledgments

## References and links

1. | J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. |

2. | J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, (1999) “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. on Microwave Theory and Techn. |

3. | D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz “A Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

4. | D.R. Smith and N. Kroll, “Negative Refractive Index in Left-Handed Materials,” Phys. Rev. Lett. |

5. | R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed meta material,” Appl. Phys. Lett. |

6. | P. Markoš and C.M. Soukoulis, “Transmission Studies of the Left-handed Materials,” Phys. Rev. B |

7. | P. Markoš and C.M. Soukoulis, “Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators” Phys. Rev. E |

8. | P. Markoš, I. Rousochatzakis, and C.M. Soukoulis, “Transmission Losses in Left-handed Materials,” Phys. Rev. E |

9. | C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Numerical Studies of Left-handed Metamaterials,” Progress In Electromagnetics Research, PIER |

10. | V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. |

11. | M.M. Sigalas, C.Y. Chan, K.M. Ho, and M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B |

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

13. | C. G. Parazzoli, R. B. Gregor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,” Phys. Rev. Lett. |

14. | D.R. Smith, S. Shultz, P. Markoš, and C.M. Soukoulis, “Determination of Effective Permittivity and Permeability of Metamaterials from Reflection and Transmission Coefficient,” Phys. Rev. B |

15. | D. R. Smith and D. Schurig, “Electromagnetic Wave propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. |

16. | S. O’Brien and J.B. Pendry, “Photonic band gap effects and magnetic activity of dielectric composites,” J. Phys.: Condens. Matter |

17. | J.B. Pendry and A. MacKinnon, Phys. Rev. Lett.692772 (1992) J.B. Pendry, “Photonic band gap structures,” J. Mod. Opt. 41 (1994) 209 J.B. Pendry and P.M. Bell 1996, “Transfer matrix techniques for electromagnetic waves,” in Photonic Band Gap Materials vol. 315 of NATO ASI Ser. E: Applied Sciences (1996), ed. by C.M. Soukoulis (Plenum, NY) p. 203 [CrossRef] [PubMed] |

18. | Photonic Band GapMaterials, ed. by C.M. Soukoulis (Kluwer, Dordrecht, 1996); Photonic Crystals and Light Localization in the 21st Century, ed. by C.M. Soukoulis (Kluwer, Dordrecht, 2001) |

19. | P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires”, |

20. | L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskĭ, “Electrodynamics of Continuous Media,” Pergamon Press 1984 |

21. | E.V. Ponizovskaya, M. Nieto-Vesperinas, and N. Garcia, “Losses for microwave transmission metamaterials for producing left-handed materials: The strip wires,” Appl. Phys. Lett. |

22. | J.D. Jackson, “Classical Electrodynamic,” (3rd edition), J.Willey and Sons, 1999, p. 312 |

23. | M. Bayindir, K. Aydin, E. Ozbay, P. Markoš, and C.M. Soukoulis, “Transmission Properties of Composite Metamaterials in Free Space,” Appl. Phys. Lett. |

24. | E. Ozbay, K. Aydin, E. Cubukcu, and M. Bayindir, “Transmission and reflection properties of composite double negative metamaterials in free-space,” to be published (2003) |

25. | P Markoš, D.R. Smith, and C.M. Soukoulis, unpublished. |

26. | R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A |

**OCIS Codes**

(000.2690) General : General physics

(000.4430) General : Numerical approximation and analysis

**ToC Category:**

Focus Issue: Negative refraction and metamaterials

**History**

Original Manuscript: January 21, 2003

Revised Manuscript: March 17, 2003

Published: April 7, 2003

**Citation**

Peter Markos and Costas Soukoulis, "Transmission properties and effective electromagnetic parameters of double negative metamaterials," Opt. Express **11**, 649-661 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-649

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### References

- J.B. Pendry, A.J. Holden, W.J. Stewart and I. Youngs, �??Extremely Low Frequency Plasmons in Metallic Mesostructures,�?? Phys. Rev. Lett. 76, 4773 (1996) [CrossRef] [PubMed]
- J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart, (1999) �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. on Microwave Theory and Technol. 47, 2075 (1999) [CrossRef]
- D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz �??A Composite medium with simultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000) [CrossRef] [PubMed]
- D.R. Smith and N. Kroll, �??Negative Refractive Index in Left-Handed Materials,�?? Phys. Rev. Lett. 85, 2933 (2000) [CrossRef] [PubMed]
- R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic, left-handed meta material,�?? Appl. Phys. Lett. 78, 489 (2001) [CrossRef]
- P. Marko¡s and C.M. Soukoulis, �??Transmission Studies of the Left-handed Materials,�?? Phys. Rev. B 65 033401 (2002)
- P. Marko¡s and C.M. Soukoulis, �??Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators,�?? Phys. Rev. E 65, 036622 (2002) [CrossRef]
- P. Marko¡s, I. Rousochatzakis, and C.M. Soukoulis, �??Transmission Losses in Left-handed Materials,�?? Phys. Rev. E 66, 045601 (2002) [CrossRef]
- C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, �??Numerical Studies of Left-handed Metamaterials,�?? Prog. Electromagnetics Res. PIER 35, 315 (2002) [CrossRef]
- V.G. Veselago, �??The electrodynamics of substances with simultaneously negative values of permittivity and permeability,�?? Sov. Phys. Usp. 10, 509 (1968) [CrossRef]
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