LHM chracterization (Field Average method)
MMM(Metamaterial)construct with an periodical unit cell has an individual local field (microscopic field)generated in response to an incident field. The microscopic field can be transformed into macroscopic field that response to the triggered field, with an used of the field summation method. The macroscopic field description then allow the appropriate effective permittivity and permeability to be assigned.
The method first organize the spatial boundary of an unit cell, say into 2d length side cubic. The unti cell boundary is chosen no to crosse any abrupt field change section (the effect of crossing and explanation is no well-explaned yet), and the Maxwell's curl solution is solved at each cubic faces. (Note : Maxwell's curl equation gives an closed loop line integral, has the curl field in rigth-hand rules direction). Two cubic are constructred with the inner most corner at (-b,-b,-b)for the curl E while (0,0,0) for the curl H cubic, which mean both cubic is shifted by dimension(d,d,d).Then the "avarage field" appear as an point at face centre and at line cross the points.
Next an constitutive relationhsip between the average field is defined. An plane wave now is assumed to propagating and has the field summation macroscopic field defined, thus allow an correction factor to be defined to give the correct permittivity and permeability value, which appear as an sinc2 term. (To be continue)
References:
Homogenization of metamaterials by field averaging
Jorunal of Optics, vol 23, N0 3