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27 September 2005 Effect of radiation on dispersion of magneto-inductive waves in a metamaterial
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The dispersion equation of a magneto-inductive wave along a line of magnetically coupled resonant elements is investigated under conditions when retardation must be taken into account. It is shown that both the radiation resistance and the imaginary part of the mutual inductance appear in the modified dispersion equation which allows interaction up to the pth neighbour. The problems arising in the solution of the full dispersion equation are discussed and it is concluded that the general solution leading to a large number of high-attenuation branches may not lead to a solution that is easily interpretable physically. It is suggested that the dispersion equation is to be derived from the variation of the current along an array of a finite number of elements excited by a voltage applied to the first element. The solution is obtained for a planar array of capacitively loaded loops in a closed form by inverting the complex mutual inductance matrix. It is shown that retardation and higher order interactions have greater effect upon the attenuation of the arising backward wave than upon the phase change per element. The appearance of a forward wave with a phase velocity close to that of light is also shown.
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O. Zhuromskyy, E. Shamonina, and L. Solymar "Effect of radiation on dispersion of magneto-inductive waves in a metamaterial", Proc. SPIE 5955, Metamaterials, 595506 (27 September 2005);

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