диафрагмированные волноводные фильтры / 92cbe61a-fee6-4109-b0e5-a1ed3f5e4df6

Abstract – This paper revisits the topic of transmission in cut-off rectangular waveguides periodically loaded in the E- plane with Split-Ring Resonators (SRR), to present some new results and interpretations: it is shown that along with the known band-pass region where the SRRs behave as a negative magnetic permeability medium (NMPM), another band-pass region with very good insertion loss can exist that could be explored for the design of compact waveguide sharp filters. The analysis is based on method of moment simulations using WIPL-D, and the results are validated with measurements.

Index terms – Split-ring resonators, effective high permittivity media, waveguide filters.

I. Introduction

Split-ring resonators (SRR) were originally proposed by Pendry et al. [1] as electrically small particles with high Q that show strong magnetic polarizability near resonance with high diamagnetic susceptibility above resonance. The motivation was to use them in a periodic lattice to form an artificial effective negative magnetic permeability medium (NMP). This concept was experimentally demonstrated by Smith et al. [2] who associated the SRRs with conducting wire media with effective negative electric permittivity (NEP) to form a double-nega- tive medium or “left-handed” media (LHM), using Veselago terminology [3]. The possibility to artificially synthesize otherwise inexistent in nature double-nega- tive medium opened the possibility to explore new interesting applications some of them already anticipated by Veselago in 1968 [3].

Rectangular metallic waveguides have been used in different ways to study the properties of LHM. For instance Caloz et al. [4] considered a LHM loaded rectangular metallic waveguide carrying a propagating TE10 mode to demonstrate fundamental properties like negative phase velocity and negative angle of refraction at an interface between LHM and right-handed medium; the analysis was based on finite element method, adopting an effective medium approach.

A below cut-off metallic square waveguide has been proposed in [5] as an effective one-dimensional plasma medium, to test and characterize negative magnetic permeability (NMP) particles. It was shown that subwavelength propagation occurs when this waveguide is loaded with an array of Split-Ring Resonator particles

(SRR). It was suggested that such phenomenon happens because the loaded structure behaves as a medium with simultaneously negative permittivity and permeability. The negative permeability is due to a resonance of the magnetic polarizability of the SRRs. An alternative and fully equivalent interpretation that emphasizes the anisotropic characteristics of the SRR array was recently proposed in [2].

Different types of NMP particles were proposed and tested using the described waveguide set-up, and the transmission characteristics of the resulting left-hand- ed medium were extensively studied [5-9]. These studies showed the existence of a very narrow and sharp band-pass region, corresponding to the frequencies where the array of particles presents its NMP characteristic. In all cases however, the insertion loss in the pass band is not better than -20 dB which precludes the use of this type of waveguide structures for practical filtering applications.

In the described set-up, the square waveguide section that contains the array of NMP particles is terminated at both ends by commercial coax-to-waveguide transitions, making a step transition in the transverse waveguide dimensions both in the E- and H-planes. This bandwidth of the coax-to-waveguide transitions imposes a band limitation for the experimental evaluation of the LHM section. Furthermore, the step in the transition plane impacts negatively on the insertion loss and the bulk of the coax-to-waveguide transitions does not favour a useful application.

Decoopman et al. [10] proposed a different approach for a LHM transmission line that eliminates the above limitations. The structure is based on a fin-line topology periodically loaded with SRRs and thin wires that is

Received: November 7, 2005. Revised: January 12, 2006.

1 Dep. of Electrical and Computer Eng., Instituto Superior Técnico/Instituto de Telecomunicações, IST/IT – Torre Norte, Av. Rovisco Pais, 1049-001 Lisboa, Portugal - Tel. +35 121 8418481 - Fax +35 121 8418472 - E-mail: carlos.fernandes@lx.it.pt

2 Dep. de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012 Sevilla, Spain - E-mail: marques@us.es

3 Dep. de Engenharia Electrotécnica, Polo II da Universidade de Coimbra/Instituto de Telecomunicações, 3030 Coimbra, Portugal - E-mail: mario.silveirinha@co.it.pt

enclosed by a metallic waveguide to prevent radiation loss. With appropriate design of the transitions from the hollow waveguide to the periodic structure, reduced transmission loss is obtained in the pass-band. The present paper proposes a modification of the below cut-off waveguide configuration used in [5] that eliminates the above referred band restrictions and mismatch. The new set-up can be made very compact and allows measuring the structure behaviour through a much broader frequency range. Simulations and measurements with this structure through a broad range of frequencies showed that the transmission characteristic generally shows two sharp resonances, one of them presenting very small insertion loss opening the way for very compact filters. An interpretation is given for these two transmission conditions.

II. Simulation results

The proposed modified waveguide structure is schematically represented in Figure 1. It is formed by a waveguide with square cross-section that is short-cir- cuited at both ends. Cross section dimensions are uniform throughout the structure, and such that cut-off conditions are met for the whole band of interest. An array of SRRs is inserted into the waveguide E-plane as shown. Power coupling is made through coaxial connectors with central conductor extending from the H-plane top wall into the waveguide, co-planar with the SRRs plane. Depth, diameter and positioning of these probes is optimized for maximum power coupling. Because this new structure does not require a TE10 propagating section, it eliminates the low frequency bound for structure excitation, thus enabling its use for arbitrarily low frequencies.

Simulations were made with a commercially available EM solver based on the method of moments – WIPL- D [11]. In the simulation model the cross-section dimensions are set as 10.4 mm × 10 mm (a × b), which correspond to 14.4 GHz cut-off frequency. Total length of the structure is L = 80 mm. The monopole probes are

implanted d = 13 mm off from the end short-circuit walls of the waveguide at (x = 0, y = b, z = d) and (x = 0, y = b, z = L - d). Monopole length is h = 7 mm and its diameter is φ = 0.9 mm. The mid plane of waveguide is loaded with five SRRs with the shape that is indicated in Figure 1, with mean radius r1 = 2.65 mm and r2 = 3.85 mm. The centres of successive SRR cells are longitudinally separated by 10 mm. The centres of the end SRRs are 7 mm away from the adjacent probes, in z-direction. In the WIPL-D simulation model, the waveguide is filled with air and no dielectric support is considered for the SRRs. Metal losses are considered, assuming conductivity σ = 6 10-7 Ω-1m-1. For brevity, this will be onwards referenced as “structure S1”.

Figure 2 shows the corresponding s21 results in the 1- 14 GHz frequency interval. Two transmission bands appear: one near 5.6 GHz with insertion loss of the order of -20 dB and a broader one, near 9.5 GHz, with peak -0.65 dB insertion loss. The first transmission band corresponds to the previously identified resonance [1-9] where the SRRs present effective negative permeability behaviour, as will be discussed ahead. In the second band-pass region, the inner and outer ring perimeters of the SRRs are comparable to λ; this resonance is quite different from the first one and its behaviour can no longer be interpreted as an “effective medium with negative permeability”. For brevity, this transmission band will be onwards referred as the “dynamic resonance”, and the first resonance as the “quasistatic resonance”.

In order to identify the nature of these two resonances, they are compared with the intrinsic resonances of a single SRR particle of the same dimensions when inserted into a waveguide under full propagation conditions. The waveguide cross-section dimensions used for this test are 28 mm × 16 mm (a × b). The first resonance appears at f = 5.4 GHz, which is lower than the previously obtained quasi-static resonance of structure

S1. As discussed in [1], the SRR array exhibits effective negative permeability properties just above the resonance of a single SRR. Therefore, the proposed interpretation of the quasi-static resonance of Figure 2 is consistent with this result. The frequency of the second intrinsic resonance of the single SRR particle, f = 10.5 GHz, is higher than the dynamic resonance of structure S1, which indicates that the transmission in the second band is not related with propagation in lefthanded media.

An interpretation of the underlying physical phenomena is possible based on the analysis of [7]. In that work higher order resonances of the SRR are analyzed. It is shown that the second resonance is an electric resonance, which can be excited by an uniform external electric field and produces a very high electric dipolar moment in the SRR. Thus, the dynamic pass-band of structure S1 can be interpreted in terms of a positive effective dielectric constant for the composite waveguide + SRR array. In fact, just below the second resonance of the individual SRRs, the aforementioned electric dipoles must be very high and parallel to the excitation field. These electric dipoles result in a very high and positive effective dielectric constant for the SRR array, which will cancel the negative effective dielectric constant provided by the waveguide in the dynamic pass-band shown in Figure 2. The fact of that the dynamic resonance of structure S1 is very effectively excited by the electric

a

monopoles shown in Figure 1 is also consistent with this interpretation.

III. Experimental results

A prototype was constructed based on the previously defined S1 structure (Figure 1). The waveguide, with inner waveguide dimensions of 10.4 mm × 10 mm × 800 mm (a × b × L) was constructed in brass material. The SRRs were produced with printed circuit technology using RT-Duroid 6002 substrate (10 mills thickness, εr = 2.9, tanδ = 0.04). Corresponding measured s21 results are presented in Figure 3.

These results are similar to those obtained from WIPL- D simulation in Figure 2. They clearly confirm the previously found “quasi-static” resonance, the “dynamic” resonance, and the corresponding insertion loss values. The VNA noise floor makes some difference at the rejection level for very low frequencies.

Despite the general agreement, there is a significant frequency shift of both resonances due to the dielectric substrate that was not accounted for in the WIPL-D model. Simulations predict 5.6 GHz and 9.62 GHz for “quasi-static” and “dynamic” resonances, while measurements give 4.9 GHz and 8.28 GHz respectively.

IV. Discussion and conclusions

The results show that the waveguide structure with SRRs [1] can be reduced to a simpler one, with constant cross-section over the entire length, fed by coaxial ports. This structure can be used over an arbitrarily large frequency band, thus bypassing the normal limitations of waveguide operation bands.

Generally the transmission curves show two resonance bands, the first one being very narrow with

high insertion loss, and the second one being much wider with very small insertion loss. Both these frequency bands are below the waveguide cut-off frequency. The first resonance occurs due to a resonance of the magnetic polarizability of the SRRs, as studied in [1]. In this case, transmission is explained by the fact that the association of the SRRs and waveguide below cut-off behaves effectively as a medium with negative permittivity and permeability. This was referred in this paper as the “quasi-static resonance”. The transmission at the second band – which was referred along the text as “dynamic resonance” – can be interpreted as resulting from the reduction of the waveguide cut-off frequency associated to the high and positive effective permittivity associated to the second SRR resonance. Work is being carried out to further confirm the proposed interpretation of the two resonances.

Applications of this second pass-band for miniaturized waveguide filter design are envisaged. It is stressed that uniform loading of the inner part of the waveguide with dielectric can be used to further miniaturize the whole structure without affecting the discussed physical mechanisms. Work is being carried out to identify strategies to isolate the second resonance and control its width.

Acknowledgements

The authors are indebted to prof. Jesus Rebollar for fruitful discussions on the waveguide set-up that was proposed in this work to test NMP particles. This work was partially supported by Portuguese Fundação para a Ciência e a Tecnologia under project POSI 34860/99, and by the Spanish Ministry of Education and Science under project contract TEC2004-04249-C02-02.

Carlos A. Fernandes (M’89) received the “Licenciado”, MSc, and PhD degrees in Electrical and Computer Engineering from Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 1980, 1985, and 1990, respectively. He joined the IST in 1980, and since 1993 is Associate Professor at the Department of Electrical and Computer Engineering in the areas of

microwaves, radiowave propagation and antennas. Also from 1993 he is senior researcher at the Instituto de Telecomunicações, where he currently is the coordinator of the Wireless Communications scientific area. His current research

References

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[2]Smith, D.R.; Padilla, W.J.; Vier, D.C.; Nemat-Nasser, S.C.; Schultz, S.: Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000), 4184-4187.

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

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[5]Marqués, R.; Martel, J.; Mesa, F.; Medina, F.: Left-handed Media Simulation and Transmission of EM Waves in Subwavelength Split-Ring-Resonator-Loaded Metallic Waveguides, Phys. Rev. Lett. 89 (2002), paper num. 13901.

[6]Hrabar, S.; Bartolic, J.; Sipus, Z.: Waveguide miniaturization using uniaxial negative permeability metamaterial, IEEE Trans. Antennas and Prop. 53 (2005), 110-119.

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interests include artificial dielectrics, dielectric antennas for millimetre wave applications with focus on lens antennas, and radiowave propagation modeling for mobile communication systems. He is a member of IEEE.

Ricardo Marqués (M’95) was born in San Fernando (Cádiz), Spain, in 1954. He received the Ph.D. from the Universidad de Sevilla, Sevilla, Spain. He is currently Associate Professor at the Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla. Since 1984 he has been with the Microwave Group of Research

at such University, where he developed research on computer aided design of microwave circuits, wave propagation in ferrites and complex media, and transmission line theory. More recently, his interest also focused on the analysis and design of artificial microstructured media with exotic electromagnetic properties (metamaterials), including negative refraction, subdiffraction imaging and its applications for microwave technology.

Mário G. Silveirinha (S’99-M’03) received the “Licenciado” degree in Electrical Engineering from the University of Coimbra, Portugal, in 1998, and the PhD degree in Electrical and Computer Engineering from Instituto Superior Técnico (IST), Technical University of Lisbon, Portugal, in 2003. His research interests include propagation in photonic crystals and homogenization and modeling of metamaterials.