Kramers-Kronig relations for magnetoinductive waves

Kramers-Kronig relations for propagating modes are a fundamental property of many but not all structures supporting wave propagation. While Kramers-Kronig relations for a transfer function of a causal system are always satisfied, it was discovered in the past that Kramers-Kronig relations for individual modes of a system may fail, e.g., in leaky structures. Our aim in this paper is to scrutinize whether magnetoinductive waves propagating in discrete metamaterial structures by virtue of interelement coupling satisfy Kramers-Kronig relations. Starting with the dispersion relations of magnetoinductive waves in the nearest-neighbor approximation we investigated the real and imaginary parts of two complex functions: the propagation coefficient and the transfer function. In order to show that the real and imaginary parts of these functions are related to each other by the Kramers-Kronig relations we had to modify the functions by deducting from them the values of the functions at infinite frequencies. It was shown that these modified functions and their Kramers-Kronig pairs perfectly coincided. The much more complicated case of magnetoinductive wave propagation with long-range coupling assumed was also investigated. The mode structure was derived for interactions up to the third neighbor. It was shown that the Kramers-Kronig relations are not applicable to the individual modes, but are valid for the transfer functions of the waveguide after solving the excitation problem and taking all the modes into account. Our method can be applied to practically relevant cases enabling rapid evaluation of transfer functions, e.g., in metamaterials used for wireless power transfer.