TRANSMISSION PROPERTIES OF TWO SHIFTED MAGNETOINDUCTIVE WAVEGUIDES A. Radkovskaya, 1 O. Sydoruk, 2 M. Shamonin, 3 C. J. Stevens, 4 G. Faulkner, 4 D. J. Edwards, 4 E. Shamonina, 4 and L. Solymar 5 1 Magnetism Division, Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119992, Russia 2 Department of Physics, University of Osnabru ¨ ck, Osnabru ¨ ck D- 49069, Germany 3 Department of Electrical Engineering and Information Technology, University of Applied Sciences, Regensburg D-93025, Germany 4 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom 5 Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BT, United Kingdom Received 5 October 2006 ABSTRACT: Transmission properties of magnetoinductive waves prop- agating in two coupled one-dimensional metamaterial arrays are studied both experimentally and theoretically for the case when one of the ar- rays is shifted relative to the other one. Two different kinds of resonant metamaterial elements, split-pipe and spiral resonators, are investigated in the frequency bands centred at 46.2 and 586 MHz, respectively. It is shown that within a certain frequency range close to the resonant fre- quencies the transmission is strongly dependent on the shift. Theoretical calculations based on the impedance matrix show good agreement with the experimental results. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1054 –1058, 2007; Published online in Wiley Inter- Science (www.interscience.wiley.com). DOI 10.1002/mop.22344 Key words: metamaterials; magnetoinductive waves; magnetic coupling 1. INTRODUCTION The properties of metamaterials have been in the forefront of interest ever since Pendry’s discovery [1] that, under certain cir- cumstances, they are capable of producing not only negative refraction but perfect imaging as well. The conditions under which imaging can be realized have been widely investigated (see e.g. Refs. 2 and 3.) both theoretically and experimentally. A necessary consequence of this quest was the study of the properties of the actual elements [4 –7] followed closely by further research on the interaction between the elements [8 –11]. The most obvious and most effective interaction between the elements relies on magnetic coupling giving rise to magnetoinductive (MI) waves [8, 12]. This magnetic coupling between any two elements can be characterized by a mutual inductance whose value depends on the size and shape of the elements as well as on their relative orientation. In addition to the self-impedance, it plays a decisive role in the propagation properties of magnetoinductive waves, determines the direction of phase and group velocities, and gives the width of the frequency band in which the waves can propagate. Investigations carried out so far have been mainly concerned with arrays of identical ele- ments in one- and two-dimensions [8, 12–14], although dispersion equations are also available for the three-dimensional case [12]. Coupling between arrays has also been considered with the pur- pose of producing waveguide components, filters, and delay lines [10, 11]. It has been shown for example that directional couplers for MI waveguides can be designed by specifying the mutual inductances in the coupling region. The aim of the present article is to investigate both theoretically and experimentally the properties of two coupled arrays consisting of metamaterial elements when one of them is shifted mechani- cally relative to the other one. The theoretical analysis is based on the impedance matrix that relates the currents and applied voltages to each other. For a given excitation, the currents may then be obtained by inverting the impedance matrix. In Section 2, we introduce the elements used and the experi- mental arrangement. The mathematical formulation and theoretical considerations are discussed in Section 3, results are shown in Section 4 and conclusions are drawn in Section 5. 2. EXPERIMENT Two different kinds of metamaterial elements were used in the experiments, capacitively loaded split-pipes and spirals, as shown in Figures 1(a) and 1(b). Their resonant frequencies and guiding properties were studied before [15–18]. The dimensions of the first type (the inner radius, width, height, and the width of the gap) are [Fig. 1(a)] r = 10 mm, w = 1 mm, l = 5 mm, and g = 1 mm. They are loaded by nominally identical capacitors of 330 pF. Their resonant frequencies and quality factors, measured with the aid of a network analyzer of the type HP8753C, were found as f 0 = 42.6  0.2 MHz and Q = 105  5, respectively. The elements of the second type are four-turn spiral resonators produced on a Printed Circuit Board (type FR4) by selective etching as specified by a mask. Their radius, width of the spiral, and the separation between the turns are r = 9 mm, t = 0.9 mm, and s = 1.35 mm as may be seen in Figure 1(b). The resonance frequencies and quality factors of the spirals were measured as f 0 = 0.586  0.004 GHz and the quality factor as Q = 49  4. 2.1. Coupling Between the Elements Next, we investigated magnetic coupling between two elements of the same kind both in the axial (centers of the elements lie on an axis perpendicular to their planes) and in the planar (the elements are in the same plane) configurations. A measure of magnetic coupling is the coupling coefficient  = 2M/ L (1) where L is the self-inductance and M is the mutual inductance between the elements. The measurement is done by placing a transmitting coil near the centre of one element and then measur- ing the currents in both elements within a certain frequency range by moving a receiving coil consecutively to elements 1 and 2. The coupling coefficient may be then derived by determining the frequency variation of the currents. The measurements for the split-pipes were performed at 401 discrete values of frequency in the range of 45– 48 MHz. In the Figure 1 Schematic representation of (a) the split-pipe and (b) the spiral resonators 1054 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 5, May 2007 DOI 10.1002/mop