Electromagnetic boundary conditions in multipole theory
Multipole expansions for the macroscopic charge and current densities in a dielectric half-space involve a hierarchy of singular functions comprising the Dirac delta function and its derivatives. For these, Maxwell's differential equations yield corresponding singular expansions of the macroscopic electromagnetic fields E and B, and the response fields D and H, together with their boundary conditions (in terms of macroscopic multipole moment densities) at a dielectric–vacuum (or dielectric–dielectric) interface. Explicit results are obtained up to electric octopole–magnetic quadrupole order. These show that published expressions for boundary conditions are incomplete beyond electric dipole order, due to an invalid assumption concerning two-dimensional behaviour at the interface. The effect of this on studies of certain reflection effects for anisotropic media is detailed. Comparison of the differential theory with the standard integral formulation shows that, beyond electric dipole order, the latter is incomplete and redundant.