Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum-dielectric interface

Multipole solutions are obtained for the boundary conditions on the macroscopic electromagnetic fields at the surface of a semi-infinite homogeneous medium in a vacuum, which may be anisotropic, magnetic and chiral. The theory is based on the differential forms of Maxwell's equations and is developed within the electric octopole-magnetic quadrupole approximation, since optical effects are known which require this multipole order for their description. To ensure that both spatial invariance and reciprocity (time-reversal symmetry) are satisfied, it is necessary to take the constitutive relations for the D and H fields in relativistically covariant form. These forms are derived to the order of electric octopole and magnetic quadrupole. The boundary conditions are shown to reduce to the standard Maxwell forms when the properties of the medium are described within the electric dipole approximation D = epsilon(0)E + P and H = mu(0)(-1)B. However, when higher multipole contributions are included in the constitutive relations, various components of the electromagnetic fields may be macroscopically discontinuous at the interface due to surface densities of bound current, charge, electric dipole moment, etc. The multipole forms for these discontinuities, which are consistent with the use of relativistically covariant D and H fields, are identified.