MADSEN AND BOBOWSKI: THE COMPLEX PERMEABILITY OF SPLIT-RING RESONATOR ARRAYS MEASURED AT MICROWAVE FREQUENCIES 1 The Complex Permeability of Split-Ring Resonator Arrays Measured at Microwave Frequencies Sabrina L. Madsen and Jake S. Bobowski Abstract—We have measured the relative permeability of split- other hand, the size of a unit cell is typically greater than the ring resonator (SRR) arrays used in metamaterials designed to SRR outer diameter [1]. 0 have µ < 0 over a narrow range of microwave frequencies. The The second driver of our work has been the fact that extract- SRR arrays were loaded into the bore of a loop-gap resonator (LGR) and reflection coefficient measurements were used to de- ing meaningful effective parameters from metamaterials has termine both the real and imaginary parts of the array’s effective proven to be challenging, both numerically and experimentally. permeability. Data were collected as a function of array size and The most widely used technique is known as the Nicolson- SRR spacing. The results were compared to those obtained from Ross-Weir (NRW) method which involves illuminating the continuous extended split-ring resonators (ESRRs). The arrays metamaterial with plane waves and analyzing the resulting of planar SRRs exhibited enhanced damping and a narrower range of frequencies with µ0 < 0 when compared to the ESRRs. scattering parameters [7], [8]. Using this method, there have The observed differences in damping, however, were diminished been many reports in which either "00 or µ00 have been found to considerably when the array size was expanded from a one- be negative near the resonant frequency of the metamaterial. dimensional array of N SRRs to a 2 × 2 × N array. Our A value of ξ00 < 0, where ξ is either " or µ, is counter to method can also be used to experimentally determine the effective physical intuition because it represents a gain in energy. This permeability of other metamaterial designs. type of behavior has been termed an anti-resonant response Index Terms—Loop-gap resonator (LGR), metamaterials, per- and it has been observed in both numerical and experimental meability, split-ring resonator (SRR). studies [9]–[12]. Koschny et al. argued that ξ00 < 0 does not violate any physical principle so long as there is net dissipation I. INTRODUCTION of the electromagnetic (EM) field by the metamaterial [9]. HIS paper describes experimental measurements of the Furthermore, they argued that ξ00 < 0 is an intrinsic property relative permeability µ0 jµ00 of arrays of planar split- of metamaterials caused by the spatial periodicity of the com- T − ring resonators (SRRs) at microwave frequencies. The SRR posite structures [13]. Alitalo et al. applied the NRW method array was first proposed by Pendry et al. as an artificial to aperiodic metamaterial structures, both experimentally and structure that could be engineered to have magnetic properties numerically, and found that values of ξ00 < 0 persisted. They (µ0 < 0) not found in natural materials [1]. A short time later, interpreted their results as pointing to a deficiency in the NRW a metamaterial with a negative index of refraction was fabri- method when applied to microwave metamaterials [12]. This cated from a lattice of SRRs and conducting rods. Negative view was supported by Woodley and Mojahedi who applied index materials (NIMs) have an effective permeability µ and the NRW method to an array of dielectric spheres for which permittivity " that are simultaneously negative over a narrow there is an analytical solution that shows that "00 is positive band of microwave frequencies [2], [3]. Metamaterials have at all frequencies. In contrast, their numerical studies using generated a lot of interest due to their potential use in exotic the NRW method found "00 < 0 over a narrow frequency applications [4], [5]. band [14]. There are two primary motivators for our present work. Given the questions over the reliability of the NRW method First, Pendry’s initial negative-permeability calculation con- when applied to metamaterials, alternative methods for ex- sidered a pair of concentric conducting sheets arranged in tracting the effective parameters of these structures need to be a split-ring geometry. We will refer to this structure as an developed. Kong and coworkers have developed a method in extended split-ring resonator (ESRR) [6]. Pendry et al. argued which the metamaterial is loaded into a rectangular waveguide arXiv:2006.13861v1 [physics.app-ph] 24 Jun 2020 that arrays of planar SRRs should approximate the behavior operating in the TE10 mode and the complex reflection and of the ESRR and an expression for the permeability of the transmission coefficients are analyzed. As with the NRW array was derived. However, the expression was valid only retrieval method, these authors also report an anti-resonant for a certain set of geometrical constraints that are difficult response [15], [16]. to satisfy in practice. For example, their estimate of the SRR Marques´ et al. [17] have developed a method for measuring capacitance required the inner radius of the concentric rings the resonant frequency and magnetic polarizability of a single to be greater than the spacing between SRRs. In practical resonant element used in the construction of metamaterials. realizations of a three-dimensional (3D) metamaterial, on the Their method places the resonator inside of a circular aperture within a rectangular waveguide. They then make use of a Manuscript submitted on June 25, 2020. Lorentz local field theory to deduce the effective parameters S.L. Madsen and J.S. Bobowski are with the Department of Physics, University of British Columbia, Kelowna, BC, V1V 1V7 Canada (e-mail: (permittivity and permeability) of a material made from a [email protected]). regular array of the resonant elements [17]. Our contribution MADSEN AND BOBOWSKI: THE COMPLEX PERMEABILITY OF SPLIT-RING RESONATOR ARRAYS MEASURED AT MICROWAVE FREQUENCIES 2 is distinct from that of Marques´ et al. because our objective from the shift in resonant frequency and µ00 by the change in was to develop and implement an entirely new experimental quality factor [20], [23]. method capable of directly extracting the complex permeabil- In the case of a metamaterial, however, µ0 and µ00 are ity of complete metamaterial structures consisting of numerous expected to be strong functions of frequency near f0 of the resonant elements. LGR. Therefore, to determine the permeability from a fit to A unique feature of our method is that it exposes the an S measurement, a model for the frequency dependence j 11j material under test to an RF magnetic field while keeping its of µ = µ0 jµ00 must be adopted. For our measurements eff − exposure to RF electric fields to a minimum. As pointed out by of SRR arrays, we use the model Pendry et al. originally Koschny et al., µ00 > 0 is required in this situation to ensure calculated for ESRRs net energy dissipation [9]. Therefore, a material found to have 2 1 (fs=fp) µ < 0 when exposed to EM plane waves would have to µeff = 1 − (1) 00 − 1 (f =f)2 j [γ= (2πf)] have the unusual property that its magnetic response changes − s − when exposed to a uniform RF magnetic field. To the best of where fs is the ESRR resonant frequency, fp is called the our knowledge, such a property has not been experimentally magnetic plasma frequency, and γ is a damping constant [1]. demonstrated in any of the existing microwave metamaterials. Note that, provided fp > fs, this model guarantees that µ00 > 0 In this paper we describe the use of loop-gap resonators at all frequencies. Furthermore, it satisfies the Kramers-Kronig (LGRs) to experimentally determine the permeability of both relations for the real and imaginary components of the mag- netic susceptibility, χ = µ 1 [24], [25]. SRR arrays of various sizes and assemblies of ESRRs. The eff eff − paper is organized as follows: Section II provides an overview In the measurements presented in this paper, we first used of the experimental methods. In particular, it describes how the S11 frequency dependence of an LGR with an empty bore j j 2 reflection coefficient measurements of an SRR-loaded LGR to determine f0, Q0, and M0 =R0. Next, the bore of the LGR are used to determine the permeability of the SRR array. In was partially loaded with an SRR array. Assuming a known filling factor η, a second S measurement was made and a fit Section III, we present the design of the LGRs, SRRs, and j 11j ESSRs used in our experiments. Section IV shows reflection to its frequency dependence was used to extract the parameters coefficient measurements and the corresponding extracted per- fs, fp, and γ that characterize the permeability of the SRR meabilities of SRR arrays and ESRRs. In Section V we discuss array. All reflection coefficient measurements were made using the results and Section VI summarizes the main conclusions. an Agilent E5061A vector network analyzer (VNA). In [6], the expected frequency dependence of S11 for an LGR whose bore is partially filled with a negative-permeabilityj j II. EXPERIMENTAL METHODS material was calculated. A µeff given by (1) was assumed for Recently, we have used LGRs to measure electric [18], the material filling the bore and the calculation revealed a [19] and magnetic [20] material properties at microwave reflection coefficient with a double resonance. The frequency frequencies. LGRs are electrically-small resonators that can splitting of the pair of resonances as well as their relative be modeled, approximately, as lumped-element LRC cir- amplitudes and widths were shown to depend on the filling cuits [21], [22]. In their simplest form, LGRs consist of factor η and the SRR permeability parameters fs, fp, and a hollow conducting tube with a narrow slit cut along its γ.