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A theoretical and experimental study of magnetic metamaterials with unit cells containing two resonant elements is presented. The properties of these structures, consisting of split rings, are governed by strongly anisotropic magnetic coupling between individual elements. This coupling leads to propagation of slow magnetoinductive waves in the vicinity of the resonant frequency. The wavelength of magnetoinductive waves is much smaller than the free-space wavelength of the electromagnetic radiation. This opens up the possibility of manipulating the near field on a subwavelength scale. We develop a theoretical formulation for coupled chains of metamaterial elements allowing the tailoring of their guiding properties in the near field. In a comprehensive analysis modes of coupled waveguides supporting forward and/or backward waves are identified and the corresponding hybridization mechanisms for dispersion equations of magnetoinductive waves are determined. Analytical predictions are verified both experimentally and numerically on a variety of coupled waveguides. The approach can be employed for the design of near-field manipulating devices.
An experimental study of the properties of magnetoinductive waves in the presence of retardation
(2006)
Magnetoinductive (MI) waves owe their existence to the magnetic coupling between metamaterial elements. First experiments confirming the existence of MI waves were carried out on capacitively loaded loops and Swiss Rolls about three orders of magnitude smaller than the operating wavelengths (5–15 m) so that the radiation effects did not play any significant role. In the present paper MI waves are studied experimentally on various types of split ring resonators of about 1 cm diameter operating in the microwave region between 1 and 2 GHz. Our results prove that retardation has a significant effect upon the propagation of MI waves.
Resonant frequencies of a combination of split rings: Experimental, analytical and numerical study
(2005)
The resonant frequencies of five different ring resonators are measured with the aid of a network analyser within the frequency range of about 1.5 to 2.8 GHz. The resonant frequencies for those configurations are also determined from numerical calculations using the commercially available MICRO-STRIPES package. The experimental and numerical results are shown to be very close to each other. Analytical results from various authors, available for three of the configurations, are also compared with the experimental results; one of them leads to a large discrepancy, but the other analytical approximations are shown to be not too far off.
Resonant frequencies of a split-ring resonator: Analytical solutions and numerical simulations
(2004)
A set of differential equations describing the current andvoltage distribution in a split-ring resonator is derived and solved ana-lytically. The resonant frequencies may be obtained from the solution of a characteristic equation. An approximate solution for the lowest reso-nant frequency agrees with that obtained by heuristic arguments. The analytical results are supported by numerical simulations
An equivalent circuit, consisting of bulk and distributed elements, is derived for describing the properties of a potential metamaterial element capable of providing negative effective permeability. It is the singly split double ring (SSDR), a special case of the split ring resonator (J. B. Pendry et al., IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)), obtained when the gap capacitance in the inner ring is infinitely large. The variables are the inter-ring voltage and the currents flowing in the inner and outer rings. The excitation is assumed in the form of a spatially constant temporally varying magnetic field. The functions, showing the angular variation of the variables, are found by solving a set of differential equations with boundary conditions imposed at the position of the split. It is shown from the analytical solution that the SSDR can have resonant frequencies in the full spectrum from very low to very high frequencies. It is pointed out in particular that whenever the mean diameter of the ring is equal to an odd multiple of the half wavelength it is always possible to find a set of parameters which will give rise to resonance. As examples the resonant frequencies are determined for eight sets of parameters. Results are also derived by replacing the distributed circuit with a number of discrete circuits. It is finally shown that the results obtained from the equivalent circuit model are in excellent agreement with those derived from the MICRO-STRIPES numerical package which solves Maxwell’s equations in the time domain.
Mechanism of subwavelength imaging with bilayered magnetic metamaterials: Theory and experiment
(2007)
We present a theoretical and experimental study of a bilayered metamaterial structure for subwavelength imaging of magnetic field. The simplest version of such a structure consists of one or two linear arrays of capacitively loaded split pipe resonators. Its subwavelength physics is governed by strongly anisotropic magnetic coupling between individual resonators and by propagation of magnetoinductive waves with wavelength much shorter than the wavelength of the electromagnetic radiation in free space. It is shown that magnetoinductive waves propagating in the lateral direction are undesirable because they spread the image. Good subwavelength imaging is achieved when, due to the strong interlayer coupling, a stop band in the vicinity of the resonant frequency appears in the dispersion characteristics. The imaging properties of the single and double lens are compared and it is shown that the double lens has a superior performance. Excellent agreement is obtained between experimental and theoretical results for the magnetic field in the image plane in the operation frequency range of 30–60 MHz. It is shown that the same mechanism is responsible for image formation using bilayered planar metamaterial structures and a design of such a lens comprising two planar layers with a total of 542 elements is provided. The conclusions are not restricted to the radio frequency region because the elements can be scaled down.
Magnetoinductive waves propagating along a line consisting of two kinds of metamaterial elements are studied. Both elements are made up by the same metallic loop but are loaded by different capacitors resulting in different resonant frequencies. The dispersion characteristics are derived from phase and amplitude measurements for the cases when the line consists of (i) identical elements and (ii) of alternating elements. Both planar (elements in the same plane as the axis of the line) and axial (elements perpendicular to the axis of the line) configurations are investigated. It is shown that in the bi-periodic arrangement of the elements, the dispersion curves have a forward wave in the lower frequency branch and a backward wave in the upper frequency branch independent of the configuration whether it is planar or axial. Comparisons between theoretical and experimental results show good agreement.
Transmission properties of magnetoinductive waves propagating in two coupled one-dimensional metamaterial arrays are studied both experimentally and theoretically for the case when one of the arrays 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 frequencies the transmission is strongly dependent on the shift. Theoretical calculations based on the impedance matrix show good agreement with the experimental results