We state and compare four different definitions of magnetic permeability for periodic, artificial media, or metamaterials. The connections among them, and properties in general, are discussed in detail, including causality, passivity, symmetry, asymptotic behavior, and origin dependence. The analysi...

The higher order multipoles above the electric quadrupole are commonly neglected in metamaterial homogenization. We show that they nevertheless can be significant when the second-order spatial dispersive effects, such as the magnetic response, are considered. In this respect, they can be equally imp...