Completion of multipole theory for the electromagnetic response fields D and H

Multipole theory of linear constitutive relations for the response fields D and H in dielectrics is incomplete in the sense that it yields unphysical (for example, origin-dependent) results when taken beyond electric dipole order. Because of the inherent non-uniqueness of D and H in Maxwell's equations, we consider the role of transformations of these fields in multipole theory. Specifically, we construct (for a non-dissipative magnetic medium to electric quadrupole-magnetic dipole order, and for E and B fields represented by complex harmonic plane waves) the most general transformation which (i) leaves the inhomogeneous Maxwell equations for D and H unchanged;, (ii) preserves the linear, homogeneous dependence of D and H on the macroscopic polarizability tensors and fields E and B; (iii) is consistent with the requirements of space inversion and time reversal; (iv) maintains the order of the multipole expansion; (v) imposes origin independence on the tensors representing material constants, (vi) preserves necessary symmetries of these tensors. The resulting transformed constitutive relations are unique. They satisfy the Post constraint (equality of traces of the magnetoelectric tensors), yield a Fresnel reflection matrix which satisfies reciprocity, and imply multipole moment densities which are origin independent. The transformation theory can be applied to higher multipole orders, and the results adapted to dissipative media.