Modifications of the Kramers-Kronig Relations for the Magnetic Permeability
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The Kramers-Kronig (KK) relations provide a mathematical relation between the real and imaginary part of a complex function that is analytic in the upper half plane and tends to zero as the argument tends to infinity. The magnetic susceptibility may satisfy the KK relations, but not in all cases. In cases where the susceptibility does not satisfy the usual KK relations, the analogous Modified Kramers-Kronig (MKK) relations were defined. The MKK relations provided a connection between the real and imaginary part of the magnetic permeability. It is the behavior of the permeability for low and high frequencies that determines if the MKK relations are satisfied.The behavior of the magnetic permeability as a function of the frequency was studied, for a general metamaterial. From causality, the permeability was found to behave only as unity, 1/frequency or 1/frequency^2 when the frequency approached zero. The same result were obtained when the frequency approached infinity. This lead to 3^2=9 different possibilities of how the permeability could behave. The nine different possibilities show that the permeability may behave in ways not characterized by the usual KK relations. Thus, causality was found to be less restrictive than the KK relations.Some of the nine possibilities would satisfy the MKK relations directly. However, for other possibilities, a slight modification of the input function was needed. These modifications were found, and based on the nine different possibilities, six different functions of the permeability were found to satisfy the MKK relations.Passive transmission line metamaterials were also studied specifically. Their magnetic permeability permeability were found to generally satisfy the MKK relations, given that their circuit topology consisted of solely serial and parallel circuits with loss.