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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 26 — Dec. 25, 2006
  • pp: 12944–12949
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Experimental retrieval of the effective parameters of metamaterials based on a waveguide method

Hongsheng Chen, Jingjing Zhang, Yang Bai, Yu Luo, Lixin Ran, Qin Jiang, and Jin Au Kong  »View Author Affiliations


Optics Express, Vol. 14, Issue 26, pp. 12944-12949 (2006)
http://dx.doi.org/10.1364/OE.14.012944


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Abstract

A waveguide-based retrieval method for measuring complex permittivity and permeability tensors of metamaterials is presented. In the proposed scheme, multiple independent sets of scattering data for the material under test with different orientations are measured in the frequency range corresponding to the dominant TE10 mode. The method is applied to various metamaterials and shows its effectiveness in the effective parameters extraction.

© 2006 Optical Society of America

1. Introduction

Since the first left-handed metamaterial (LHM) was realized based on split-ring resonators (SRR) and rods [1

1. 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 (2000). [CrossRef] [PubMed]

], a variety of designs as candidates to realize the LHMs has been experimentally realized and studied [2–7

2. R. Marques, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and lefthanded metamaterials,” Phys. Rev B. 65, 144440 (2002). [CrossRef]

]. All of them present bulk electromagnetic properties, which can be characterized by constitutive parameter tensors [8–12

8. S. Maslovski, P. Ikonen, I. Kolmakov, S. Tretyakov, and M. Kaunisto, “Artificial magnetic materials based on the new magnetic particle: metasolenoid,” Progress In Electromagnetics Research, PIER 54, 61 (2005). [CrossRef]

].

Because the performance of metamaterial applications [14–18

14. Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. A. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220 (2006). [CrossRef] [PubMed]

] depend much on the materials properties, it is very important that the constitutive parameters of these metamaterials can be well retrieved. In this paper, we aim to present an efficient experimental technique, which can be easily carried out to measure the permittivity and permeability tensors of the metamaterials. The material under test is assumed to be biaxially anisotropic, which, in general, is valid for the metamaterials realized so far. Various kinds of experimental approaches can be used for measuring the electromagnetic parameters of metamaterials, such as the resonator method [19

19. L. Chen, C. K. Ong, and B. T. G. Tan, “Cavity perturbation technique for the measurement of permittivity tensor of uniaxially anisotropic dielectrics,” IEEE Trans. Instrum. Meas. 48, 6 (1999).

], the open-ended coaxial method [20

20. N. B. Tahar and A. F.-Lamer, “Broad-band simultaneous measurement of the complex permittivity tensor for uniaxial materials using a coaxial discontinuity,” IEEE Trans. Microw. Theory Tech. 39, 10 (1991).

], the free-space method [21

21. D. Ghodgaonkar, V. Varadan, and V. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39, 387 (1990). [CrossRef]

], and the rectangular waveguide method [22

22. N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, “The inverse problem for biaxial materials,” IEEE Trans. Microw. Theory Tech. MTT- 32, 4 (1984).

]. All of these methods have both advantages and disadvantages. The resonant method has a quite high accuracy and sensitivity, but it is narrowband, and requires the test specimen having a small electrical size and a specified geometrical shape [19

19. L. Chen, C. K. Ong, and B. T. G. Tan, “Cavity perturbation technique for the measurement of permittivity tensor of uniaxially anisotropic dielectrics,” IEEE Trans. Instrum. Meas. 48, 6 (1999).

], so it is unfit to measure metamaterials with frequency dispersive in nature. The coaxial line based reflection-transmission approach has an advantage in terms of bandwidth, but the test samples should be properly machined in the shape of circular cylinders [20

20. N. B. Tahar and A. F.-Lamer, “Broad-band simultaneous measurement of the complex permittivity tensor for uniaxial materials using a coaxial discontinuity,” IEEE Trans. Microw. Theory Tech. 39, 10 (1991).

], which is a very tough requirement for metamaterials. The free-space method has been reported to measure the parameters of the metamaterials where a plane wave was required to normally incident onto the material [23

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

]. The thickness of the slab has to be very uniform along its transverse cross section because the undesirable reflection and diffraction from the incident wave over a big sample of material will cause the result less accurate. In the rectangular waveguide method [22

22. N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, “The inverse problem for biaxial materials,” IEEE Trans. Microw. Theory Tech. MTT- 32, 4 (1984).

], the matching of the test specimen is not so crucial as only slab-shaped samples with small cross sections are required, it has no other tough requirements, which are the advantages of this method. The method has been used to measure non-magnetic materials with uniaxial permittivity tensor [24

24. M. J. Akhtar, L. E. Feher, and M. Thumm, “A waveguide-based two-step approach for measuring complex permittivity tensor of uniaxial composite materials,” IEEE Trans. Microw. Theory Tech. 54, 5 (2006). [CrossRef]

] but, as the best of our knowledge, has not been applied to measure metamaterials with both complex permittivity and permeability tensors so far.

The waveguide-based retrieved algorithm for biaxial metamaterials is developed in this paper. Different with [22

22. N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, “The inverse problem for biaxial materials,” IEEE Trans. Microw. Theory Tech. MTT- 32, 4 (1984).

] where both TE10 and TE20 excitations are needed for dispersive material, here we only need TE10 excitation. The cost is that four independent set of measurements with different orientations of the metamaterials are needed in order to retrieve six unknown parameters. The whole retrieval procedure is firstly verified by a slab of homogenous frequency dispersive material with known parameters, then to an anisotropic metamaterial composed of SRR structures. We confirmed that the effective permeability along the axis of the SRR µ 1 is negative in a frequency band above the resonant frequency while the permeability along other directions is close to 1. The retrieved results using the waveguide-based method are compared with that obtained from normal incident measurement [23

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

,25

25. X. Chen, B.-I Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E. 71, 046610 (2005). [CrossRef]

]. Good agreements are obtained.

2. Retrieve technique

Fig. 1. Experimental scheme for the S parameters measurement in waveguide.

z=±{[(1+S11)2S212][(1S11)2S212]}12
(1)
eink0zd=X±i(1X2)12
(2)

where X=(1-S112+S212)/2S 21. Different with that in [29

29. X. Chen, T. M. Grzegorczyk, B.-I Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E. 70, 016608 (2004). [CrossRef]

,30

30. 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]

], here n and z are:

na=(k02ε3μ1kx2μ1μ2)12(k02kx2)12
(3)
za=μ1(k02kx2)12(k02ε3μ1kx2μ1μ2)12
(4)
nb=(k02ε3μ2kx2μ2μ1)12(k02kx2)12
(5)
zb=μ2(k02kx2)12(k02ε3μ2kx2μ2μ1)12
(6)

where kx =π/a is the transverse wave number in the rectangular waveguide. The subscripts ‘a’ and ‘b’ denote that the results are calculated from the measurements of case (a) and case (b), respectively. Therefore, from Eq. (3–6) we get

μ1=naza,μ2=nbzb
(7)

ε 3 can be calculated from Eqs. (3) and (7), or from Eqs. (5) and (7), respectively,

ε3a=(na2k0z2+kx2μ1μ2)(k02μ1)
(8)
ε3b=(nb2k0z2+kx2μ2μ1)(k02μ2)
(9)

Similarly, we can achieve the other three constitutive parameters µ 3, ε 1, and ε 2 based on two additional measurements: in the third measurement (case c), the principal axes (e 1, e 2, e 3) of the material is along with the coordinate (-, ŷ, ), in this case we can get µ 3 and ε 2; in the fourth measurement (case d), the principal axes (e 1, e 2, e 3) of the material is along with the coordinate (ŷ, , ), then we can get ε 1.

3. Numerical validation

Fig. 2. Comparison of the analytical (markers) and the retrieved results (Solid lines: real part; dashed lines: imaginary part) for a loss homogeneous medium.

For the purpose of assessing the accuracy of the measurement technique, the retrieval technique has been validated based on simulated scattering data measured from a slab of a homogeneous material. The constitutive parameters of the material are set to be: ε 1=1, µ 2=1, µ 3=1, ε 2=1-0.3/(f 2/3.662+i0.00594f-1), ε 3=1-1.2/(f 2/3.182+i0.0785f-1), µ 1=1-1/(f 2/2.012+i0.0792f-1). A WR-430 (a=109.22 mm, b=54.61 mm) waveguide is used in the simulation. The retrieved constitutive parameters obtained from the measured S parameters are shown in Fig. 2. The retrieved ε 1, µ 2, and µ 3 are equal to 1 and haven’t shown here. We see a perfect agreement between the retrieved parameters and the input ones.

4. Experimental results

We use two isotropic materials with known properties: a Teflon (ε r=2.1) and a dielectric FR4 substrate (ε r=4+i0.02) for calibration. A WR-430 waveguide is used in the experiment. The operating frequency range is 1.72~2.61 GHz. The two samples have a thickness of 10 mm and are loaded into the waveguide independently. The S parameters are recorded by an Agilent 8722ES network analyzer. The retrieved constitutive parameters of the two materials are shown in Fig. 3, where we see the relative permittivity of the Teflon sample is around 2.1 and the relative permeability is around 1. For the dielectric FR4 substrate, the relative permittivity is around 4.0, and the relative permeability is around 1. Both of the measured results are in good agreement with the known value, indicating the measurement method works very well.

Fig. 3. Measured ε r and µ r for the Teflon and FR4 substrate.

The SRR structure as shown in Fig. 4(a) is fabricated for measurement. The sizes of the resonator are w=7.6 mm, c=1 mm, g=2 mm, d=2 mm. The metallic SRR strips are printed on the FR4 substrate. The structure has a periodicity of 4 mm along the e 1 direction, 13.6 mm along both e 2 and e 3 direction. It should be noted that enough large number of unit cells should be used in the e 1 directions in order to let the SRR exhibit the same bulk media properties in every measurement [31

31. J. B. Pendry, A. J. Holten, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). [CrossRef]

,32

32. H. Chen, L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915, (2006). [CrossRef]

]. For example, here we focused on µ 1, µ 2, and ε 3, and the two necessary measurements are shown in Fig. 4(b) and Fig. 4(c) corresponding respectively to the measurement of case (a) and case (b) indicated in section 2. In case (b), at least five layers of the ring in the e 1 (or ) direction is needed. While in case (a), the retrieved parameters are not sensitive to the number of layers used in e 2 direction [29

29. X. Chen, T. M. Grzegorczyk, B.-I Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E. 70, 016608 (2004). [CrossRef]

].

Fig. 4. (a) The dimensions of the SRR unit, (b) (c) two measurements corresponding to two different orientations of the SRR in the waveguide.

The retrieved experimental results are shown in Fig. 5. We see that the real part of µ 1 retrieved in the experimental measurement is negative in the frequency range from 1.96 GHz to 2.1 GHz [Fig. 5(a)]. The retrieved µ 1 based on the two numerical simulations with different configurations are also shown. In the first simulation (Sim 1), we use the free-space method, modeling a normal incidence plane wave. In the second simulation (Sim 2), we model the real rectangular waveguide. The results are in good agreement with each other except the experimental result shows a smaller bandwidth. The retrieved µ 2 shown in Fig. 5(a) is close to 1, indicating that the structure is non-magnetic in the e 2 direction.

We present the retrieved ε 3 in Fig. 5(b) from the calculation of Eq. (8). We show the one calculated from Eq. (8) exhibit an anti-resonant behavior around 1.96 GHz, which is accompanied by a negative imaginary part of permittivity. The unphysical phenomenon is very similar to that reported in Ref. [33

33. 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]

], where it was pointed out that around the antiresonant frequency region, the wavelength in the medium is smaller than the periodicity of the SRRs and therefore, can not be characterized as a homogenous medium [33

33. 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]

]. For the permittivity calculated from Eq. (9), we present the results in Fig. 5(c) for SRR with different layers in the (e 1) direction. An interesting phenomenon can be seen that for SRR with less than five layers, the imaginary part of the permittivity is negative around the plasma frequency. The reason is that in the e 1 direction, the magnetic plasma frequency of SRRs with a few layers, ω′mp , is higher than that of a bulk SRR medium, ωmp . When we use Eq. (9) to retrieve ε 3b, the µ 1 we used is retrieved from measurement of case (a), where the sample is a bulk SRR medium, so µ 1 has a plasma frequency of ωmp . Since the real part of µ 1 is close to zero near ωmp , the second term in the numerator of Eq. (9) has a large negative imaginary part, which leads to a negative imaginary part of permittivity near ωmp , as shown in Fig. 5(c) for the case of SRR with one or three layers. Similarly, the first term in the numerator of Eq. (9), nb2, is retrieved from measurement of case (b), which means µ 1 in case (b) has a plasma frequency of ωmp. From Eq. (5), we can expect that nb2 has a large positive imaginary part near ωmp. As the number of layers increase in the test material, ωmp decreases to ωmp , then the negative imaginary part of permittivity raised by the second term is canceled by the first term, which lead to more reasonable results, as shown in Fig. 5(c) for the case of SRR with five layers. Therefore, the principle backed in this unphysical phenomenon (the negative imaginary part of permittivity retrieved from SRR with less than 5 layers) is different with that in Fig. 5(b). It should also be noted that there is no anti-resonant behavior around the resonant frequency, the reason is that in the orientation of case (b), the wave number in the direction always equals to kx =π/a, so the applied H z field that is perpendicular to the SRR have a spatial variation on a scale significantly larger than the periodicity of SRR in the direction, which means in this orientation, SRR can form an effective medium over the resonant frequency regime.

Fig. 5. Measured constitutive parameters of the SRR structure.

5. Conclusion

We presented a waveguide-based retrieval method for measuring complex permittivity and permeability tensors of metamaterials. Both the retrieval algorithm and experimental realizations are proposed. The whole retrieval procedure is verified both numerically and experimentally, using various kinds of materials with known properties. The successful retrieved results show the effectiveness and robustness of our method. In addition, the experiment measurement is easy to carry out, and there are no tough requirements for the material sample, indicating a good candidate in the homogenizations of metamaterials.

Acknowledgments

We acknowledge the support by the China Postdoctoral Science Foundation under Grant No. 20060390331, and by the Chinese NSF under Grant Nos. 60371010 and 60531020.

References and links

1.

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 (2000). [CrossRef] [PubMed]

2.

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

3.

L. Ran, J. Huangfu, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental study on several left-handed matamaterials,” Progress In Electromagnetics Research, PIER 51, 249 (2005). [CrossRef]

4.

P. Gay-Balmaz and O. J. F. Martin, “Efficient isotropic magnetic resonators,” Appl. Phys. Lett. 81, 5 (2002). [CrossRef]

5.

H. Chen, L. Ran, J. Huangfu, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Magnetic properties of s-shaped split-ring resonators,” Progress In Electromagnetics Research, PIER 51, 231 (2005). [CrossRef]

6.

H. Chen, L. Ran, J. Huangfu, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E. 70, 057605 (2004). [CrossRef]

7.

J. D. Baena, R. Marques, F. Medina, and J. Martel, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B 69, 014402 (2004). [CrossRef]

8.

S. Maslovski, P. Ikonen, I. Kolmakov, S. Tretyakov, and M. Kaunisto, “Artificial magnetic materials based on the new magnetic particle: metasolenoid,” Progress In Electromagnetics Research, PIER 54, 61 (2005). [CrossRef]

9.

A. Sihvola, “Metamaterials and depolarization factors,” Progress In Electromagnetics Research, PIER 51, 65 (2005). [CrossRef]

10.

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]

11.

H. Y. Yao, L. W. Li, Q. Wu, and J. A. Kong, “Macroscopic performance analysis of metamaterials synthesized from micrsocopic 2-D isotropic cross split-ring resonator array,” Progress in Electromagnetics Research-PIER 51, 197 (2005). [CrossRef]

12.

S.-W. Lee, Y. Kuga, and A. Ishimaru, “Quasi-static analysis of materials with small tunable stacked split ring resonators,” Progress in Electromagnetics Research-PIER 51, 219, (2005). [CrossRef]

13.

T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. Nemat-Nasser, D. Schurig, and D. R. Smith, “Freespace microwave focusing by a negative-index gradient lens,” Appl. Phys. Lett. 88, 081101 (2006). [CrossRef]

14.

Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. A. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220 (2006). [CrossRef] [PubMed]

15.

Q. Sui, C. Li, and F. Li, “The experimental study of /4 monopole antennas in meta-material,” Progress In Electromagnetics Research, PIER 51, 281, (2005). [CrossRef]

16.

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas and Wireless Propagation Letters 1, 10 (2002). [CrossRef]

17.

B.-I. Wu, W. Wang, J. Pacheco, X. Chen, T. Grzegorczyk, and J. A. Kong, “A study of using metamaterials as antenna substrate to enhance gain,” Progress In Electromagnetics Research, PIER 51, 395, (2005).

18.

A.K. Hamid, “Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials,” Progress In Electromagnetics Research, PIER 51, 329, (2005). [CrossRef]

19.

L. Chen, C. K. Ong, and B. T. G. Tan, “Cavity perturbation technique for the measurement of permittivity tensor of uniaxially anisotropic dielectrics,” IEEE Trans. Instrum. Meas. 48, 6 (1999).

20.

N. B. Tahar and A. F.-Lamer, “Broad-band simultaneous measurement of the complex permittivity tensor for uniaxial materials using a coaxial discontinuity,” IEEE Trans. Microw. Theory Tech. 39, 10 (1991).

21.

D. Ghodgaonkar, V. Varadan, and V. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39, 387 (1990). [CrossRef]

22.

N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, “The inverse problem for biaxial materials,” IEEE Trans. Microw. Theory Tech. MTT- 32, 4 (1984).

23.

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

24.

M. J. Akhtar, L. E. Feher, and M. Thumm, “A waveguide-based two-step approach for measuring complex permittivity tensor of uniaxial composite materials,” IEEE Trans. Microw. Theory Tech. 54, 5 (2006). [CrossRef]

25.

X. Chen, B.-I Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E. 71, 046610 (2005). [CrossRef]

26.

J. A. Kong, Electromagnetic Wave Theory (Wiley and Sons, 1986, 1990, EMW Publishing, 2000, 2005)

27.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Progress In Electromagnetics Research, PIER 51, 83, (2005). [CrossRef]

28.

X. Chen, T. M. Grzegorczyk, and J. A. Kong, “Optimization approach to the retrieval of the constitutive parameters of slab of general bianisotropic medium,” Progress in Electromagnetics Research-PIER 60, 1 (2006). [CrossRef]

29.

X. Chen, T. M. Grzegorczyk, B.-I Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E. 70, 016608 (2004). [CrossRef]

30.

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]

31.

J. B. Pendry, A. J. Holten, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). [CrossRef]

32.

H. Chen, L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Equivalent circuit model for left-handed metamaterials,” J. Appl. Phys. 100, 024915, (2006). [CrossRef]

33.

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]

OCIS Codes
(120.5820) Instrumentation, measurement, and metrology : Scattering measurements
(160.1190) Materials : Anisotropic optical materials

ToC Category:
Metamaterials

History
Original Manuscript: October 10, 2006
Revised Manuscript: December 1, 2006
Manuscript Accepted: December 3, 2006
Published: December 22, 2006

Citation
Hongsheng Chen, Jingjing Zhang, Yang Bai, Yu Luo, Lixin Ran, Qin Jiang, and Jin Au Kong, "Experimental retrieval of the effective parameters of metamaterials based on a waveguide method," Opt. Express 14, 12944-12949 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12944


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References

  1. 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 (2000). [CrossRef] [PubMed]
  2. R. Marques, F. Medina, and R. Rafii-El-Idrissi, "Role of bianisotropy in negative permeability and left-handed metamaterials," Phys. Rev B. 65, 144440 (2002). [CrossRef]
  3. L. Ran, J. Huangfu, H. Chen, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Experimental study on several left-handed matamaterials," Progress In Electromagnetics Research, PIER 51, 249 (2005). [CrossRef]
  4. P. Gay-Balmaz, and O. J. F. Martin, "Efficient isotropic magnetic resonators," Appl. Phys. Lett. 81, 5 (2002). [CrossRef]
  5. H. Chen, L. Ran, J. Huangfu, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Magnetic properties of s-shaped split-ring resonators," Progress In Electromagnetics Research, PIER 51, 231 (2005). [CrossRef]
  6. H. Chen, L. Ran, J. Huangfu, X. M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Left-handed materials composed of only S-shaped resonators," Phys. Rev. E. 70, 057605 (2004). [CrossRef]
  7. J. D. Baena, R. Marques, F. Medina, and J. Martel, "Artificial magnetic metamaterial design by using spiral resonators," Phys. Rev. B 69, 014402 (2004). [CrossRef]
  8. S. Maslovski, P. Ikonen, I. Kolmakov, S. Tretyakov, and M. Kaunisto, "Artificial magnetic materials based on the new magnetic particle: metasolenoid," Progress In Electromagnetics Research, PIER 54, 61 (2005). [CrossRef]
  9. A. Sihvola, "Metamaterials and depolarization factors," Progress In Electromagnetics Research, PIER 51, 65 (2005). [CrossRef]
  10. 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]
  11. H. Y. Yao, L. W. Li, Q. Wu, and J. A. Kong, "Macroscopic performance analysis of metamaterials synthesized from micrsocopic 2-D isotropic cross split-ring resonator array," Progress in Electromagnetics Research-PIER 51, 197 (2005). [CrossRef]
  12. S.-W. Lee, Y. Kuga, and A. Ishimaru, "Quasi-static analysis of materials with small tunable stacked split ring resonators," Progress in Electromagnetics Research-PIER 51, 219, (2005). [CrossRef]
  13. T.  Driscoll, D. N.  Basov, A. F.  Starr, P. M.  Rye, S.  Nemat-Nasser, D.  Schurig, and D. R.  Smith, "Free-space microwave focusing by a negative-index gradient lens," Appl. Phys. Lett.  88, 081101 (2006). [CrossRef]
  14. Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. A. Kong, "Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials," Opt. Express  14, 2220 (2006). [CrossRef] [PubMed]
  15. Q. Sui, C. Li, and F. Li, "The experimental study of /4 monopole antennas in meta-material," Progress In Electromagnetics Research, PIER 51, 281, (2005). [CrossRef]
  16. N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas and Wireless Propagation Letters 1, 10 (2002). [CrossRef]
  17. B.-I. Wu, W. Wang, J. Pacheco, X. Chen, T. Grzegorczyk, and J. A. Kong, "A study of using metamaterials as antenna substrate to enhance gain," Progress In Electromagnetics Research, PIER 51, 395, (2005).
  18. A.K. Hamid, "Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials," Progress In Electromagnetics Research, PIER 51, 329, (2005). [CrossRef]
  19. L. Chen, C. K. Ong, and B. T. G. Tan, "Cavity perturbation technique for the measurement of permittivity tensor of uniaxially anisotropic dielectrics," IEEE Trans. Instrum. Meas. 48, 6 (1999).
  20. N. B. Tahar, and A. F.-Lamer, "Broad-band simultaneous measurement of the complex permittivity tensor for uniaxial materials using a coaxial discontinuity," IEEE Trans. Microw. Theory Tech. 39, 10 (1991).
  21. D. Ghodgaonkar, V. Varadan, and V. Varadan, "Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies," IEEE Trans. Instrum. Meas. 39, 387 (1990). [CrossRef]
  22. N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, "The inverse problem for biaxial materials," IEEE Trans. Microw. Theory Tech. MTT- 32, 4 (1984).
  23. A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, "Fabrication and characterization of a negative-refractive-index composite metamaterial," Phys. Rev B. 70, 113102 (2004). [CrossRef]
  24. M. J. Akhtar, L. E. Feher, and M. Thumm, "A waveguide-based two-step approach for measuring complex permittivity tensor of uniaxial composite materials," IEEE Trans. Microw. Theory Tech. 54, 5 (2006). [CrossRef]
  25. X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, "Retrieval of the effective constitutive parameters of bianisotropic metamaterials," Phys. Rev. E. 71, 046610 (2005). [CrossRef]
  26. J. A. Kong, Electromagnetic Wave Theory (Wiley and Sons, 1986, 1990, EMW Publishing, 2000, 2005)
  27. T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B.-I. Wu, and J. A. Kong, "Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials," Progress In Electromagnetics Research, PIER 51, 83, (2005). [CrossRef]
  28. X. Chen, T. M. Grzegorczyk, and J. A. Kong, "Optimization approach to the retrieval of the constitutive parameters of slab of general bianisotropic medium," Progress in Electromagnetics Research-PIER 60, 1 (2006). [CrossRef]
  29. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E. 70, 016608 (2004). [CrossRef]
  30. 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]
  31. J. B. Pendry, A. J. Holten, D. J. Robbins, and W. J. Stewart, "Magnetism from Conductors and Enhanced Non-Linear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). [CrossRef]
  32. H. Chen, L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, "Equivalent circuit model for left-handed metamaterials," J. Appl. Phys. 100, 024915, (2006). [CrossRef]
  33. 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]

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