## Fundamental modal properties of SRR metamaterials and metamaterial based waveguiding structures

Optics Express, Vol. 17, Issue 8, pp. 6101-6117 (2009)

http://dx.doi.org/10.1364/OE.17.006101

Acrobat PDF (364 KB)

### Abstract

A rigorous full wave analysis of bianisotropic split ring resonator (SRR) metamaterials is presented for different electromagnetic field polarization and propagation directions. An alternative physical explanation is gained by revealing the fact that imaginary wave number leads to the SRR resonance. Metamaterial based parallel plate waveguide and rectangular waveguide are then examined to explore the resonance response to transverse magnetic and transverse electric waves. It is shown that different dispersion properties, such as non-cutoff frequency mode propagation and enhanced bandwidth of single mode operation, become into existence under certain circumstances. In addition, salient dispersion properties are imparted to non-radiative dielectric waveguides and H waveguides by uniaxial bianisotropic SRR metamaterials. Both longitudinal-section magnetic and longitudinal-section electric modes are capable of propagating very slowly due to metamaterial bianisotropic effects. Particularly, the abnormal falling behavior of some higher-order modes, eventually leading to the leakage, may appear when metamaterials are double negative. Fortunately, for other modes, leakage can be reduced due to the magnetoelectric coupling. When the metamaterials are of single negative parameters, leakage elimination can be achieved.

© 2009 Optical Society of America

## 1. Introduction

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

6. R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag. **51**, 1516–1529 (2003). [CrossRef]

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B **65**, 144440 (2002). [CrossRef]

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. **100**, 024507 (2006). [CrossRef]

9. V. V. Varadan and A. R. Tellakula, “Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. **100**, 034910 (2006). [CrossRef]

10. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. **84**, 2493–2495 (2004). [CrossRef]

11. 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]

14. P. Markos and C. M. Soukoulis, “Numerical studies of left handed materials and arrays of split ring resonators,” Phys. Rev. E **65**, 036622 (2002). [CrossRef]

15. K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonance for different split ring resonator parameters and designs,” New J. Phys. **7**, 168 (2005). [CrossRef]

17. A. Alú and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers,” IEEE Trans. Microw. Theory Tech. **52**, 199–210 (2004). [CrossRef]

24. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. **48**, 2557–2560 (2006). [CrossRef]

20. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative refractive index waveguides,” Phys. Rev. E **67**, 057602 (2003). [CrossRef]

21. S. Hrabar, J. Bartolic, and Z. Sipus, “Miniaturization of rectangular waveguide using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag. **53**, 110–119 (2005). [CrossRef]

22. A. L. Topa, C. R. Paiva, and A. M. Barbosa, “Novel propagation features of double negative H-guides and H-guide couplers,” Microwave Opt. Technol. Lett. **47**, 185–190 (2005). [CrossRef]

23. P. Yang, D. Lee, and K. Wu, “Nonradiative dielectric waveguide embedded in metamaterial with negative permittivity or permeability,” Microwave Opt. Technol. Lett. **45**, 207–210 (2005). [CrossRef]

24. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. **48**, 2557–2560 (2006). [CrossRef]

## 2. Physical characteristics of the resonance band gap of SRR metamaterials

25. C. Krowne, “Electromagnetic theorems for complex. anisotropic media,” IEEE Trans. Antennas Propag. **32**, 1224–1230 (1984). [CrossRef]

*ε*̄ and

*μ*̄ are the relative electric permittivity and relative magnetic permeability tensors, and

*κ*̄ is the magnetoelectric coupling dimensionless tensor.

*ε*̄,

*μ*̄ and

*κ*̄ tensors are of significance without losses. It should be noted that the aim of Fig. 1 is to illustrate the relative orientation of the SRR unit, and an array of such ring unit will compose the metamaterials. Therefore, the whole theoretical analysis conducted in this paper is based on the macroscopical effects of metamaterials where the coupling of different rings has been counted [7

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B **65**, 144440 (2002). [CrossRef]

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. **100**, 024507 (2006). [CrossRef]

*h*=

*Z*

_{0}

*H*, from Maxwell’s curl equation for source free regions together with Eq.(1) and Eq. (2), one may write

*k*

_{0}.

*iβz*'), where

*β*=

*k*/

_{z}*k*

_{0}is the normalized longitudinal wave number, one has

*' stands for ∂/∂*

_{x}*x*' and ∂

*y*' for ∂/∂

*y*',

*x*'=

*k*

_{0}

*x*,

*y*'=

*k*

_{0}

*y*,

*z*'=

*k*

_{0}

*z*. After substituting Eq. (2) into Eq. (4), and apply the condition that

*E*=

_{z}*h*= 0 for TEM waves, one obtains the following relations

_{z}**H**is perpendicular to the SRR plane, and incident

**E**is parallel to the gapbearing sides of SRR. The normalized wave number of these TEM waves is given by

7. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B **65**, 144440 (2002). [CrossRef]

*ω*, only those modes having

*μ*-

_{xx}ε_{yy}*κ*

^{2}>0 will propagate. Those modes with

*μ*-

_{xx}ε_{yy}*κ*

^{2}< 0 will lead to an imaginary

*β*, meaning that all field components will decay exponentially away from the source of excitation. Since

*κ*

^{2}> 0, this SRR orientation will achieve the transmission stop band when the constitutive parameters are single negative, including

*ε*>0 and

_{yy}*μ*<0 case, as well as

_{xx}*ε*<0 and

_{yy}*μ*>0 case. And also when the constitutive parameters are double negative or double positive with the condition |

_{xx}*μ*|<|

_{xx}ε_{yy}*κ*

^{2}|, the transmission stop band will also occur.

**E**is perpendicular to the SRR plane, and magnetic field

**H**is parallel to the gapbearing sides of SRR. The normalized wave number of these TEM waves is given by

*S*

_{21}) curves, let’s see the

*S*parameters for the SRR metamaterial planar slab with complex impedance

*β*determined by the six cases above, and thickness

*z*' as the distance between the calibration planes

## 3. Propagation features of waveguides structures with SRR metamaterials

### 3.1 Parallel Plate Waveguides and Rectangular Waveguides

*W*in Fig. 3(a) is assumed to be much greater than the separation

*l*between the two plates, so that fringing fields and any

*x*-variation can be ignored. And it is standard convention to have the longest side of the rectangular waveguide along the

*x*-axis, so that

*u*>

*v*in Fig. 3(b).

#### 3.1.1 Non-cutoff Frequency Modes

*h*= 0 and a nonzero

_{z}*E*field which satisfies the reduced wave Eq. (10a), with ∂

_{z}*x*'=0,

*n*= 0, 1, 2⋯) is the cutoff wave number constrained to discrete values. /

*n*= 0, the TM

_{0}mode is actually identical to the TEM mode shown in Fig. 2(a), therefore, this TM

_{0}mode has a cutoff phenomenon while SRR resonance. However, from (12) one knows that TM mode has the chance to propagate with no cutoff frequency when

*ε*< 0 and

_{yy}*μ*-

_{xx}ε_{yy}*κ*

^{2}> 0, as shown in Fig. 5. Similar results hold true for the TE modes in Fig. 4(b), and TM modes in Fig. 4(c), which are corresponding to the resonance cases in the Fig. 2.

*κ*≠ 0, decoupling of

*E*and

_{z}*h*occurs only when ∂

_{z}_{x'}≡0 or ∂

_{y'}≡0, therefore we only consider TE

_{mn}modes with

*m*= 0 or

*n*= 0, since neither

*m*nor

*n*can be zero for TM modes in a rectangular waveguide. When ∂

_{y'}≡0, one has the following decoupled equation and boundary condition for TE

_{m0}modes from Eq. (10b)

#### 3.1.2 Enhanced Bandwidth of Single Mode Operation

*E*= 0 and a nonzero

_{z}*h*field which satisfies the reduced wave Eq. (13b), with ∂

_{z}*x*'=0. Through the similar derivation, one can obtain

_{0n}modes in Fig. 4(a) and Fig. 4(c) achieve the cutoff frequency of

_{m0}modes in Fig. 4(b) obtains the cutoff frequency of

### 3.2 Nonradiative Dielectric Waveguides and H Waveguides

*ε*̄,

*μ*̄ and

*κ*̄ tensors in Eq. (1) have the following uniaxial form [26

26. S. A. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML),” J. Electromagn. Wave Applic. **12**, 821–837 (1998). [CrossRef]

*ϛ*,

*ξ*, reflect the different changes of

*ε*and

*μ*components in

*ŷŷ*and

*ẑẑ*direction. Therefore,

*ε*

_{1}and

*μ*

_{1}are always positive, whereas

*ε*

_{2}and

*μ*

_{2}can be negative in certain frequency band.

*h*= 0 with nonzero

_{x}*h*as the supporting field which satisfies

_{y}*k*' =

_{z}*k*/

_{z}*k*

_{0}=

*β*

_{z}-

*iα*is the complex normalized longitudinal wave number. All other field components can be expressed as

_{z}*h*as a product of two separate variable functions in the form

_{y}*k*'=

_{x}*k*/

_{x}*k*

_{0}and

*k*' =

_{y}*k*/

_{y}*k*

_{0}are the complex normalized transverse wave numbers. Substituting back into Eq. (18), the normalized wave numbers can be expressed

*k*' =

_{x}*k*

_{x1}' =

*β*

_{x1}-

*iα*

_{x1}for |

*x*'| <

*q*', while for |

*x*'| >

*q*', one should take

*k*' =

_{x}*k*

_{x0}' =

*β*

_{x0}-

*iα*

_{x0}. Apply the boundary conditions on the perfectly electric conductor planes to other field components, one can write

*G*is the amplitude constant, and

*s*'=

*k*

_{0}

*s*. The

*n*index gives the number of half waves along

*y*. And

*x*'=±

*q*', the modal equation for the LSM modes can be finally derived

*m*index (

*m*= 0, 1, 2, ⋯) appearing in LSM

_{mn}From the similar derivation, the LSE modes can be defined as

### 3.2.1 Slow Wave Propagation

*β*| decreases gradually as the magnetoelectric coupling turns larger in the case that

_{z}*ε*

_{2}and

*μ*

_{2}are of positive values. Maximum

*κ*is achieved under the cutoff condition

*k*' = 0.

_{z}*v*=

*Tλ*of the modes will be much slower. From Eq. (17), one can see that in the frequency that

*ω*is far larger than the resonance frequency

*ω*

_{0}, both positive

*ε*

_{2}and

*μ*

_{2}as well as smaller absolute value of

*κ*can be obtained, thus slow wave propagation will appear.

*β*| shows a general upward trend when the magnetoelectric coupling becomes significant in the case that

_{z}*ε*

_{2}and

*μ*

_{2}are both negative. Minimum value for

*κ*with the cutoff condition

*k*' = 0 can be obtained,

_{z}*ε*

_{2}and

*μ*

_{2}have the chance to become negative when

*ω*is little larger than the resonance frequency

*ω*

_{0}. Meanwhile, the magnetoelectric coupling

*κ*has the absolute value which can be infinitely large within this frequency band. Therefore, the guided waves are able to propagate very slowly, and even approach zero velocity.

_{01}modes in a conventional nonradiative dielectric waveguide and a nonradiative dielectric waveguide with SRR metamaterials. Since there exists minimum value for magnetoelectric coupling

*κ*in the double negative case, we choose

*κ*= 5 and 10 to make sure that LSM

_{01}propagates. As can be seen within [30 GHz, 31 GHz], the longitudinal wave number of LSM

_{01}mode in nonradiative dielectric with SRR metamaterials is always larger than that of the conventional one, thus traveling more slowly, which indicates that nonradiative dielectric waveguide with SRR metamaterials allows more number of wavelength to propagate within the same length, providing feasibility of miniaturization for nonradiative dielectric waveguide.

_{01}modes in the nonradiative dielectric waveguide with SRR metamaterials. The time-average power passing a transverse cross-section of the nonradiative dielectric waveguide is

*ε*

_{2}and

*μ*

_{2}are both positive, we choose ‘+’, and for the double negative metamaterial case, we choose ‘-’. As we all know, the double negative metamaterials have the negative wave number, which leads to the positive Poynting vector in Eq. (30). With the choice of

*s*= 0.4

*λ*

_{0}, Fig. 10 shows general trend of power flow varied with absolute value of longitudinal wave number by not taking the constant coefficient into account. It can be seen that

*P*

_{01}will increase as |

*β*| becomes larger than

_{z}*β*| will enhance the power flow. From the above analysis, we can discern that metamaterial waveguide with double negative parameters has the chance to achieve infinite large |

_{z}*β*| around the metamaterial resonance frequency. Therefore, inserting metamaterials will strengthen the power flow of the conventional NRD waveguide. It is worth noting that such strengthen trend of power flow is based on the variation of |

_{z}*β*|, and the total energy will still be conserved since the electromagnetic wave propagates much more slowly.

_{z}*s*< 0.5

*λ*

_{0}is assumed throughout which means that the proposed waveguiding structure functions as nonradiative dielectric waveguide. When

*s*increases to

*s*> 0.5

*λ*

_{0}, the waveguiding structure in Fig. 6 becomes H waveguide, and the results such as slow wave propagation and enhanced energy flow shown in Fig. 8–10 still work.

### 3.2.2 Abnormal Guidance and Leakage Suppression

*n*= 1 dependence may be assumed. Therefore, if the modes can leak power, they must do so in the form of TM

_{1}or TE

_{1}mode in the air-filled parallel plate region. Consequently, the condition for leakage can be written as

*β*<

_{z}*k*, where

_{p}*k*is the normalized longitudinal wave number of the TM

_{p}_{1}or TE

_{1}satisfying

*s*= 5 mm, the waveguiding structure in Fig. 2 works as a nonradiative dielectric waveguide when frequency

*f*< 30 GHz. No electromagnetic wave can propagate between the parallel plates because of the cutoff property, thus no leaky wave exits. However, when

*f*> 30 GHz, it functions as an H waveguide, therefore, leakage may come into being.

*β*| of most modes become larger when

_{z}*f*increases like the modes in conventional waveguides. However, some higher-order LSM modes in the proposed waveguiding structure may operate dramatically differently. From Eq. (22a),

*k*| will demonstrate a general upward trend as

_{z}*f*increase, and may have the chance to get a fall while

*β*|. Fig. 11 presents the abnormal falling behavior of LSM higher-order modes in the proposed nonradiative dielectric waveguide and H waveguide with double negative parameters, and TM

_{z}_{1}mode in air filled parallel plate guide. In the region where the dispersion curve of LSM mode is located below the curve of TM

_{1}mode, one expects the leakage. As can be seen, if the falling behavior continues in the H-guide, leakage will eventually happens. Fortunately, such higher order modes can propagate only when the parameters are both negative. Moreover, lots of them cannot exit in the H-guide region since the cutoff of |

*β*| = 0 while

_{z}*f*becomes larger.

*β*| versus frequency

_{z}*f*for LSM modes in proposed H waveguide under the condition

*β*| presents an increase when

_{z}*f*becomes lager, but experience a decrease as

*κ*turn larger at the same frequency while

*ε*

_{2}and

*μ*

_{2}are positive. Therefore, smaller magnetoelectric coupling

*κ*will reduce the leakage of the propagating modes. To examine further, from Eq. (17) one knows that positive

*ε*

_{2}and

*μ*

_{2}as well as minimum

*κ*can exit together in the frequency bands that are far larger than the SRR resonance frequency

*ω*

_{0}, in which the proposed waveguiding structure may exactly works as a H waveguide, thus such reduced leakage scheme can be fulfilled. On the contrary, Fig. 12(b) shows that under the same frequency, |

*β*| increases when

_{z}*κ*becomes more significant in the case that

*ε*

_{2}and

*μ*

_{2}are both negative. From Eq. (17), one can see that

*ε*

_{2}and

*μ*

_{2}have the chance to become both negative when the frequency is little larger than the SRR resonance frequency

*ω*

_{0}. Meanwhile, the magnetoelectric coupling

*κ*has the absolute value which can be infinitely large within this frequency band. Therefore, the leakage of guided waves is able to be greatly reduced.

*ε*

_{2}and

*μ*

_{2}are of single negative value, Fig. 13 illustrates that when

*μ*

_{2}> 0 and

*ε*

_{2}< 0, |

*β*| of LSM modes become larger as

_{z}*κ*increases. Furthermore, the leakage can hardly happen in this case even when

*κ*= 0, since

*β*>

_{z}*k*can be easily satisfied. However, LSE modes will no longer exist under such condition. On the other hand, when

_{p}*μ*

_{2}< 0 and

*ε*

_{2}> 0, only LSE modes can propagate in the proposed nonradiative dielectric waveguide amd H waveguide, and by choosing proper parameters, the leakage can also be eliminated.

## 4. Conclusion

8. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. **100**, 024507 (2006). [CrossRef]

## Acknowledgments

## References and links

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

2. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenonmena,” IEEE Trans. Microwave Theory Tech. |

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

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

6. | R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag. |

7. | R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B |

8. | D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig “Calculation and measurement of bianisotropy in a split ring resonator,” J. Appl. Phys. |

9. | V. V. Varadan and A. R. Tellakula, “Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation,” J. Appl. Phys. |

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

11. | P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split ring resonators,” J. Appl. Phys. |

12. | T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. |

13. | T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, “Ab initio numerical simulation of left handed metamaterials: comparison of calculation and experiments,” J. Appl. Phys. |

14. | P. Markos and C. M. Soukoulis, “Numerical studies of left handed materials and arrays of split ring resonators,” Phys. Rev. E |

15. | K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonance for different split ring resonator parameters and designs,” New J. Phys. |

16. | R. Marqués, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside- couple split ring resonators for metamaterials design-theory and experiments,” IEEE Trans. Microwave Theory Tech. |

17. | A. Alú and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers,” IEEE Trans. Microw. Theory Tech. |

18. | B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. |

19. | Y. S. Xu, “A study of waveguides field with anisotropic metamaterials,” Microwave Opt. Technol. Lett. |

20. | I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative refractive index waveguides,” Phys. Rev. E |

21. | S. Hrabar, J. Bartolic, and Z. Sipus, “Miniaturization of rectangular waveguide using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag. |

22. | A. L. Topa, C. R. Paiva, and A. M. Barbosa, “Novel propagation features of double negative H-guides and H-guide couplers,” Microwave Opt. Technol. Lett. |

23. | P. Yang, D. Lee, and K. Wu, “Nonradiative dielectric waveguide embedded in metamaterial with negative permittivity or permeability,” Microwave Opt. Technol. Lett. |

24. | P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, and S. Paulotto, “Unimodal surface wave propagation in metamaterial nonradiative dielectric waveguides,” Microwave Opt. Technol. Lett. |

25. | C. Krowne, “Electromagnetic theorems for complex. anisotropic media,” IEEE Trans. Antennas Propag. |

26. | S. A. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (PML),” J. Electromagn. Wave Applic. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 29, 2009

Revised Manuscript: March 5, 2009

Manuscript Accepted: March 5, 2009

Published: March 31, 2009

**Citation**

Rui Yang, Yongjun Xie, Xiaodong Yang, Rui Wang, and Botao Chen, "Fundamental modal properties of SRR
metamaterials and metamaterial based
waveguiding structures," Opt. Express **17**, 6101-6117 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6101

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and μ," Soviet Phys. Uspekhi. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenonmena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability an permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- R. W. Ziolkowski, "Design, fabrication and testing of double negative metamaterials," IEEE Trans. Antennas Propag. 51, 1516-1529 (2003). [CrossRef]
- R. Marqués, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and lefthanded metamaterials," Phys. Rev. B 65, 144440 (2002). [CrossRef]
- D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig "Calculation and measurement of bianisotropy in a split ring resonator," J. Appl. Phys. 100, 024507 (2006). [CrossRef]
- V. V. Varadan, A. R. Tellakula, "Effective properties of split ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100, 034910 (2006). [CrossRef]
- N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, C. M. Soukoulis, "Electric coupling to the magnetic resonance of split ring resonators," Appl. Phys. Lett. 84, 2493-2495 (2004). [CrossRef]
- 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]
- T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402(1-4), 2004. [CrossRef] [PubMed]
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