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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25912–25921
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Compact electric-LC resonators for metamaterials

Withawat Withayachumnankul, Christophe Fumeaux, and Derek Abbott  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25912-25921 (2010)
http://dx.doi.org/10.1364/OE.18.025912


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Abstract

Alternative designs to an electric-LC (ELC) resonator, which is a type of metamaterial inclusion, are presented in this article. Fitting the resonator with an interdigital capacitor (IDC) helps to increase the total capacitance of the structure. In effect, its resonance frequency is shifted downwards. This implies a decreased overall resonator size with respect to its operating wavelength. As a result, the metamaterial, composed of an array of IDC-loaded ELC resonators with their collective electromagnetic response, possesses improved homogeneity and hence is less influenced by diffraction effects of individual cells. The impact of incorporating an IDC into ELC resonators in terms of the electrical size at resonance and other relevant properties are investigated through both simulation and experiment in the microwave regime. The proposed structures can be applied to the terahertz regime via appropriate lithographic scaling.

© 2010 Optical Society of America

1. Introduction

An electromagnetic metamaterial is a man-made composite material comprising a periodic array of subwavelength inclusions. Typically, a single metallic metamaterial inclusion can be considered as an LC resonant circuit with its inductance and capacitance influenced by its shape and dimensions. These resonators can collectively exhibit macroscopically observed effective values of permittivity and/or permeability that are not found in natural materials. Various forms of resonant inclusions have been introduced to date, e.g., a split-ring resonators (SRR) for a magnetic response [1

1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999). [CrossRef]

] or a pair of cut wires for negative refractive index [2

2. V. Shalaev, W. Cai, U. Chettiar, H. Yuan, A. Sarychev, V. Drachev, and A. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]

]. Because of the possibility of engineering electromagnetic material properties, metamaterials offer immense opportunities in improving existing optical designs along with exploring unprecedented devices such as superlenses [3

3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

, 4

4. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef] [PubMed]

] and invisibility cloaks [5

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

, 6

6. N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. Kivshar, “Ideal and nonideal invisibility cloaks,” Opt. Express 16, 21369–21374 (2008). [CrossRef] [PubMed]

].

For these devices to function properly, the underlying metamaterials must be operated in the effective-medium regime, i.e., the lattice constant or unit cell size should be much smaller than λ0/4, where λ0 is the operating, i.e. resonant, wavelength [7

7. T. Koschny, P. Markoš, 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: Condens. Matter 71, art. no. 245105 (2005). [CrossRef]

]. Under this condition a collection of metamaterial elements appears nearly homogeneous to incident waves and can be characterized by an effective permittivity and permeability. As the unit cell size approaches λ0/4, diffraction effects and poor refraction become significant [8

8. C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys. 7, art. no. 167 (2005). [CrossRef]

]. These parasitic effects are detrimental to the performance of metamaterials in quasi-optical applications. For example, a large unit cell size imposes a limitation on the subwavelength resolution of a metamaterial super-lens [9

9. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003). [CrossRef]

]. When the cell size is comparable to or larger than a quarter-wavelength, the effective material parameters lose their relevance [10

10. M. M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microwaves Antennas Propag. 1, 3–11 (2007). [CrossRef]

]. There are however limits to size reduction, because simply shrinking the volume or area of metamaterial inclusions reduces the capacitance and inductance and hence disturbs other important characteristics of metamaterials. In order to preserve these characteristics, the inductance and capacitance per unit area must be increased accordingly.

In this paper, a practical approach to reducing the electrical footprint of ELC resonators is presented. Section 2 reviews some existing solutions to reduce the size of metamaterial resonators. Section 3 describes the design of IDC-loaded ELC resonators, along with a mathematical analysis based on lumped element theory. In Section 4, simulation and experimental results obtained from the proposed resonators are discussed in terms of the transmission characteristics and the effective medium properties.

2. Existing solutions

A more sophisticated and specific approach involves redesigning the resonator pattern to accommodate a higher capacitance or inductance. As for example, a multiple SRR (MSRR) is an extension to a conventional SRR, within which smaller split rings are nested to increase the parallel capacitance [14

14. F. Bilotti, A. Toscano, and L. Vegni, “Design of spiral and multiple split-ring resonators for the realization of miniaturized metamaterial samples,” IEEE Trans. Antennas Propag. 55, 2258–2267 (2007). [CrossRef]

]. With comparable dimensions, a spiral resonator (SR) has a larger capacitance than does a typical SRR by at least fourfold [15

15. J. D. Baena, R. Marqués, and F. Medina, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B: Condens. Matter 69, art. no. 014402 (2004). [CrossRef]

]. Essentially, an increment in the structural capacitance results in a realizable unit cell size down to λ0/40 for an MSRR and λ0/250 for an SR [14

14. F. Bilotti, A. Toscano, and L. Vegni, “Design of spiral and multiple split-ring resonators for the realization of miniaturized metamaterial samples,” IEEE Trans. Antennas Propag. 55, 2258–2267 (2007). [CrossRef]

]. Fractal-based metamaterials with magnetic-field coupling similar to double SRR’s have been reported [16

16. E. Lenz and H. Henke, “Homogenization of metamaterials due to fractaloid structures in the microwave regime,” J. Opt. A: Pure Appl. Opt. 11, art. no. 114021 (2009). [CrossRef]

]. The increased perimeter of the structure as a result of the fractal self-similarity leads to larger inductance and capacitance and a reduction in the resonance frequency. S-ring resonators that provide a double-negative response can be shrunken from λ0/6 to λ0/15 by winding parallel strips to increase the capacitance [17

17. H. Chen, L. Ran, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Crankled S-ring resonator with small electrical size,” Progress in Electromagnetics Research 66, 179–190 (2006). [CrossRef]

]. It is interesting to note that all the redesigned structures do not provide a pure electric resonance.

3. IDC-loaded ELC resonators

Fig. 1 Electric-LC resonator. (a) A typical ELC resonator is composed of a center capacitive gap connected to two inductive loops. (b) An equivalent circuit of the resonator constitutes an LC resonator (the resistance is neglected here) [18].

A practical redesign of the structure presented in this article involves replacing the normal capacitive gap in the center of an ELC resonator with an interdigital capacitor (IDC), illustrated in Fig. 2, to increase the capacitance. Essentially, IDC’s are widely used as lumped elements in microwave circuits [20

20. K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines (Artech House, 1996), 2nd ed.

] with an aim to decrease their circuit board footprint. As such, its concept well suits metamaterials, where the unit cell needs to be much smaller than the operating wavelength. The feature size of an IDC does not need to be finer than the gap width of an ordinary ELC resonator to enhance the capacitance. Fabrication of IDC’s can be readily carried out with standard photolithographic techniques, since the structure is fully planar. The single-layered design of IDC-loaded ELC resonators eliminates additional fabrication steps that might be required in other approaches. Owing to this level of simplicity, this design is favorable for implementing electric metamaterials in the terahertz regime.

Fig. 2 Interdigital capacitor (IDC) with 6 fingers. The finger length lIDC extends over the overlapping portion of all fingers.

Earlier, IDC’s are incorporated into microstrip-coupled tunable SRRs aimed at enhancing the local electric field strength at the capacitors’ gaps [21

21. G. Houzet, X. Mélique, and D. Lippens, “Microstrip transmission line loaded by split-ring resonators tuned by ferroelectric thin film,” Progress in Electromagnetics Research 12, 225–236 (2010). [CrossRef]

]. The idea of using IDC’s with metamaterials has also been realized in the transmission-line approach. Composite right/left handed transmission lines (CRLH-TL’s), which exhibit a band of negative phase velocity, are typically fitted with a set of IDC’s that act as series capacitors [22

22. C. Caloz and T. Itoh, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag. 52, 1159–1166 (2004). [CrossRef]

, 23

23. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Compon. Lett. 14, 68–70 (2004). [CrossRef]

]. Even though the effect of IDC’s in transmission-line metamaterials are extensively investigated, no assumption can be made for IDC-loaded resonators because of the radical difference of the two approaches.

Two variants for IDC-loaded ELC resonators are proposed and studied in this article. For the first configuration shown in Fig. 3(a,c), the capacitor’s gaps are oriented perpendicularly to the intended polarization of operation, and the two ports are positioned at the outermost fingers on both sides. Another configuration in Fig. 3(b,d) has the gaps oriented in parallel to the polarization. The ports are located at the terminal strips (the strips that join fingers together). For the first configuration, the local electric field inside the capacitive gap aligns with the electric polarization. Hence, this configuration is expected to perform better in terms of cell size reduction. It is worth noting that the mirror symmetry of the resonator is broken in the first configuration. However, this does not affect the cancellation of the magnetic dipoles, since the area of the two current loops of the resonator are still identical.

Fig. 3 Two variants of IDC-loaded ELC resonators. The IDC’s gaps align (a,c) in perpendicular to and (b,d) in parallel with the direction of an incident electric field.

For an ELC resonator, if the inductance loop remains unchanged, it can be estimated from Eq. (4) that the new resonance frequency f0,new after IDC loading equals
f0,new=l0lIDC(N1)f0,
(5)
where f0 is the resonance frequency of a conventional ELC resonator. This simple model gives an impression for the expected change in the resonance frequency of IDC-loaded ELC resonators.

4. Results

4.1. Transmission characteristics

The four designs of IDC-loaded ELC resonators in Fig. 3, along with a corresponding conventional resonator in Fig. 1, are fabricated and characterized in the microwave regime. For the sake of comparison, the five designs share the unit cell size a, gap width g, and cross-sectional length of the loaded capacitor. It is suggested that the finger width should be equal to the gap width, or w = g, to maximize the capacitance density [25

25. G. D. Alley, “Interdigital capacitors and their application to lumped-element microwave integrated circuits,” IEEE Trans. Microwave Theory Tech MTT-18, 1028–1033 (1970). [CrossRef]

]. In details, the structural parameters common to all designs are w = g = 0.4 mm, b = 0.8 mm, d = 12 mm, and a = 14 mm. Other parameters specific to each design are given in Table 1. Each planar metamaterial comprises an array of 9×9 identical resonators, made of copper with a thickness of 35.6 μm (1.4 mil). The planar substrate is an FR4 epoxy board with a thickness of 0.8 mm, a measured dielectric constant of 4.2, and a reported loss tangent of 0.02. Partial views of the fabricated samples are shown in Fig. 4. In the experiment, a metamaterial sample is located between two horn antennas facing each other, from which the transmission through the sample is measured and compared to free-space transmission. The measurement results are given in Fig 5(top).

Fig. 4 ELC resonator arrays forming planar metamaterials used in the experiment. Each sample is fabricated from copper on an FR4 substrate. The photos show 3 by 3 arrays of resonators for convenience, however the actual array size used in the experiment was 9 by 9.
Fig. 5 Transmission profiles of the resonators from (top) the experiment and (bottom) the simulation. The two graphs share the horizontal scale.

Table 1. The structural parameters and resonance frequency of the samples under test.

table-icon
View This Table

In order to verify the experimental results, the simulation for IDC-loaded ELC resonators is performed with a finite-element-based electromagnetic solver, Ansoft HFSS. Periodic boundary conditions are utilized for the transverse boundaries to replicate an infinite planar array of the resonators. Two ports at the open ends allows to determine the response of the sample to a plane wave incident normally to the array. As shown in Fig. 5 the simulation results are in general agreement with the experimental data. The discrepancies are attributed to the finite size of the fabricated array and the nonuniformity among the resonators due to fabrication tolerances. Table 1 indicates a general agreement in the resonance frequencies obtained from the simulation and the experiment. In addition, Eq. 5 can provide a rough estimation in the resonance frequency. Note that in a 3D configuration, the interaction between layers might shift the resonance slightly.

It is clear from Fig. 5 and Table 1 that the resonance frequency of IDC-loaded ELC resonators is remarkably lower than that of the original design. As a consequence, the coupling strength reduces for those structures with a higher capacitance [18

18. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, art. no. 041109 (2006). [CrossRef]

, 13

13. K. Aydin and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” J. Appl. Phys. 101, 024911 (2007). [CrossRef]

]. With a comparable capacitance area and density, sample ELC1a (ELC2a) has a lower resonance frequency compared to sample ELC1b (ELC2b). A reduction in resonance frequency is stronger when the gaps of the IDC are perpendicular to the polarization, as in ELC1a and ELC2a, which can be attributed to a better coupling between the incident electric field and the field in the capacitors. In terms of the effective medium ratio, the maximum improvement can be observed in sample ELC2a, for which λ0/a equals 10.2, in comparison to 6.8 of the conventional ELC design (ELC0). The effective medium ratio of other samples is listed in Table 1. Note that the original ELC design [18

18. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, art. no. 041109 (2006). [CrossRef]

] has a unit cell size of λ0/5.7.

4.2. Effective medium properties

In order to provide further insight, the samples are characterized for their effective medium parameters. It is worth nothing that the parameters are evaluated on the basis of planar metamaterials. Hence, these parameters would need to be fine-tuned for 3D operation to take into account the weak inter-layer coupling. Here, the effective permittivity and permeability are extracted from the simulated transmission/reflection magnitude and phase using the method of Chen et al. [26

26. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, art. no. 016608 (2004). [CrossRef]

]. For phase de-embedding, it is assumed that the thickness of a metamaterial sample in the direction of wave propagation is equal to the unit cell size or 14 mm in the present case [18

18. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, art. no. 041109 (2006). [CrossRef]

]. Fig. 6 shows the effective permittivity and permeability of the selected resonators. It is obvious that the resonance frequency in the permittivity curve is lowered by the influence of IDC loading. The permittivity further illustrates a narrower bandwidth and a higher effective loss tangent in IDC-loaded ELC resonators. Despite that, the negative value of the permittivity is appropriate for typical applications requiring ε = −1. In addition, further simulation results (not shown here) indicate that with a lower-loss substrate, the effective loss tangent can be significantly reduced.

Fig. 6 The simulated effective medium properties of the samples; (top) the effective permittivity, and (bottom) the effective permeability. The two graphs share the horizontal scale.

As discussed earlier, ELC resonators essentially possess no magnetic response, i.e., the real part of the permeability is close to unity over the frequencies of interest. However, the retrieved parameters do not strictly comply with this principle. As shown in Fig. 6, the real permeability becomes anti-resonant, and the imaginary part is negative. In fact, these anomalies are artifacts introduced during parameter extraction due to the inhomogeneity of metamaterials [7

7. T. Koschny, P. Markoš, 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: Condens. Matter 71, art. no. 245105 (2005). [CrossRef]

, 18

18. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, art. no. 041109 (2006). [CrossRef]

, 27

27. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, art. no. 036617 (2005). [CrossRef]

]. The reduction of these anomalies for ELC1a and ELC2a results from a lower dispersion and the higher homogeneity in these structures.

It is worth noting the characteristics of metamaterials from this and other designs [19

19. 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: Condens. Matter 75, art. no. 041102 (2007). [CrossRef]

, 28

28. J. Zhang and Z. R. Hu, “A novel broadband metamaterial resonator with negative permittivity,” in “PIERS Proceedings, Xi’an, China,” (2010), pp. 1346–1348.

] suggest that there might exist a fundamental limit in the electrical size of the resonator in terms of the operating bandwidth. Such a limit could be in analogy to the well-known trade-off between the size and performance of an antenna [29

29. H. A. Wheeler, “Fundamental limits of small antennas,” Proc. IRE. 35, 1479–1484 (1947). [CrossRef]

, 30

30. L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys. 19, 1163–1175 (1948). [CrossRef]

, 31

31. R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Research National Bureau of Standards—D. Radio Propagation 64D (1960). [PubMed]

], which operates on the basis of a resonance mechanism as well. Therefore, this potential trade-off for metamaterials deserves further theoretical investigation.

5. Conclusion

This article proposes to miniaturize ELC resonators through IDC loading. In the experiment, a set of IDC-loaded ELC samples is fabricated and characterized in the microwave frequency range. The measurement data, in agreement with the simulation results, reveal a significant improvement in the effective medium ratio of these IDC-loaded resonators. The parameter retrieval suggests a tradeoff between the electrical size and absorption in the proposed structure. However, this issue can be partly alleviated by using a substrate with lower loss. An implication of metamaterial size reduction is the structural homogeneity, which leads to lower parasitic effects and hence a higher performance for quasi-optical applications.

The IDC-loading approach can be used in conjunction with other approaches to minimize the electrical size of ELC resonators. Apart from that, the proposed approach can be implemented when other options are not available due to fabrication limits. In addition, the IDC loading can be applied to other resonance-based metamaterials as well. Last but not least, the reduction of the resonator size is not only beneficial for metamaterial homogeneity but also useful in applications of metamaterial-based sensors, where a smaller electrical size of these structures is of a prime importance.

Acknowledgments

The authors acknowledge Pavel Simcik for his technical assistance. This research was supported under the Australian Research Council Discovery Projects funding scheme (project number DP1095151).

References and links

1.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech 47, 2075–2084 (1999). [CrossRef]

2.

V. Shalaev, W. Cai, U. Chettiar, H. Yuan, A. Sarychev, V. Drachev, and A. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]

3.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

4.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef] [PubMed]

5.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

6.

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. Kivshar, “Ideal and nonideal invisibility cloaks,” Opt. Express 16, 21369–21374 (2008). [CrossRef] [PubMed]

7.

T. Koschny, P. Markoš, 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: Condens. Matter 71, art. no. 245105 (2005). [CrossRef]

8.

C. Caloz, A. Lai, and T. Itoh, “The challenge of homogenization in metamaterials,” New J. Phys. 7, art. no. 167 (2005). [CrossRef]

9.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506–1508 (2003). [CrossRef]

10.

M. M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microwaves Antennas Propag. 1, 3–11 (2007). [CrossRef]

11.

D. Rialet, A. Sharaiha, A.-C. Tarot, and C. Delaveaud, “Characterization of antennas on dielectric and magnetic substrates effective medium approximation,” in “Third European Conference on Antennas and Propagation (EuCAP),” (2009), pp. 3163–3166.

12.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” New J. Phys. 7, art. no. 168 (2005). [CrossRef]

13.

K. Aydin and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” J. Appl. Phys. 101, 024911 (2007). [CrossRef]

14.

F. Bilotti, A. Toscano, and L. Vegni, “Design of spiral and multiple split-ring resonators for the realization of miniaturized metamaterial samples,” IEEE Trans. Antennas Propag. 55, 2258–2267 (2007). [CrossRef]

15.

J. D. Baena, R. Marqués, and F. Medina, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B: Condens. Matter 69, art. no. 014402 (2004). [CrossRef]

16.

E. Lenz and H. Henke, “Homogenization of metamaterials due to fractaloid structures in the microwave regime,” J. Opt. A: Pure Appl. Opt. 11, art. no. 114021 (2009). [CrossRef]

17.

H. Chen, L. Ran, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Crankled S-ring resonator with small electrical size,” Progress in Electromagnetics Research 66, 179–190 (2006). [CrossRef]

18.

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, art. no. 041109 (2006). [CrossRef]

19.

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: Condens. Matter 75, art. no. 041102 (2007). [CrossRef]

20.

K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines (Artech House, 1996), 2nd ed.

21.

G. Houzet, X. Mélique, and D. Lippens, “Microstrip transmission line loaded by split-ring resonators tuned by ferroelectric thin film,” Progress in Electromagnetics Research 12, 225–236 (2010). [CrossRef]

22.

C. Caloz and T. Itoh, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag. 52, 1159–1166 (2004). [CrossRef]

23.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Compon. Lett. 14, 68–70 (2004). [CrossRef]

24.

I. J. Bahl, Lumped Element for RF and Microwave Circuits (Artech House, 2003).

25.

G. D. Alley, “Interdigital capacitors and their application to lumped-element microwave integrated circuits,” IEEE Trans. Microwave Theory Tech MTT-18, 1028–1033 (1970). [CrossRef]

26.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, art. no. 016608 (2004). [CrossRef]

27.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, art. no. 036617 (2005). [CrossRef]

28.

J. Zhang and Z. R. Hu, “A novel broadband metamaterial resonator with negative permittivity,” in “PIERS Proceedings, Xi’an, China,” (2010), pp. 1346–1348.

29.

H. A. Wheeler, “Fundamental limits of small antennas,” Proc. IRE. 35, 1479–1484 (1947). [CrossRef]

30.

L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys. 19, 1163–1175 (1948). [CrossRef]

31.

R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Research National Bureau of Standards—D. Radio Propagation 64D (1960). [PubMed]

32.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22, 1099–1119 (1983). [CrossRef] [PubMed]

33.

J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21, 1379–1386 (2004). [CrossRef]

34.

K. Takano, T. Kawabata, C.-F. Hsieh, K. Akiyama, F. Miyamaru, Y. Abe, Y. Tokuda, R.-P. Pan, C.-L. Pan, and M. Hangyo, “Fabrication of terahertz planar metamaterials using asuper-fine ink-jet printer,” Appl. Phys. Express , art. no. 016701 (2010). [CrossRef]

35.

W. Withayachumnankul and D. Abbott, “Metamaterials in the terahertz regime,” IEEE Photonics J. 1, 99–118 (2009). [CrossRef]

OCIS Codes
(350.4010) Other areas of optics : Microwaves
(160.1245) Materials : Artificially engineered materials
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: September 23, 2010
Revised Manuscript: November 3, 2010
Manuscript Accepted: November 23, 2010
Published: November 26, 2010

Citation
Withawat Withayachumnankul, Christophe Fumeaux, and Derek Abbott, "Compact electric-LC resonators for metamaterials," Opt. Express 18, 25912-25921 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25912


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References

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