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

Abstract

Multipole solutions are obtained for the boundary conditions on the macroscopic electromagnetic elds at the surface of a semi-in nite homogeneous medium in a vacuum, which may be anisotropic, magnetic and chiral. The theory is based on the di¬erential forms of Maxwell's equations and is developed within the electric octopole-magnetic quadrupole approximation, since optical e¬ects are known which require this multipole order for their description. To ensure that both spatial invariance and reciprocity (time-reversal symmetry) are satis ed, it is necessary to take the constitutive relations for the D and H elds 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 =°0E + P and H = • ¡1 0 B. However, when higher multipole contributions are included in the constitutive relations, various components of the electromagnetic elds 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 elds, are identi ed.

Transformations of the Multipole Response Fields D and H for Magnetic Media

“…Equations (9)(10)(11)(12)(13)(14) are the constitutive relations, for E and B fields represented by complex harmonic plane waves in magnetic media, given by multipole theory to electric quadrupole-magnetic dipole order. Note that the above results, and ones that follow, reduce to those for a non-magnetic medium if we set the time-odd tensors α ij , a ijk and G ij equal to zero.…”

Optical wave parameters for spatially dispersive and anisotropic nanomaterials

“…To derive the reflection and transmission coefficients, we require that the tangential components E | | and H | | of E and H are continuous across the interface. This boundary condition is valid for most optical materials, excluding however such specific media in which certain electric quadrupole and higher-order multipoles can be excited to generate lower-order multipoles on the material surface [19,20]. We choose the directions of E and H such that for a zero phase shift, the sign of E | | does not change, but the sign of H | | changes upon reflection, as in Fig.…”

The origin dependence of the material constants: the permittivity and the inverse permeability

“…(41) and (42), to electric octopole-magnetic quadrupole order, are defined as [9] A ı = ε 0 δ ı + F ı − 1 12…”

“…Therefore, possible alternative expressions have to be considered. It is shown that a covariant formulation of the auxiliary fields and the constitutive relations leads to origin-independent expressions for the material constants [9,12] that differ from the ones given in Ref. [7].…”

“…Raab and de Lange proposed an origin-independent definition of the frequency-dependent magnetizability in the microscopic case [8] that agrees with the macroscopic expression achieved for the inverse permeability using their transformation theory. The validity of this expression has still to be investigated, since an alternative definition of the inverse permeability can be derived in the macroscopic case [9]. Furthermore, the theory used to derive the magnetizability in the microscopic case relies on the invocation of a number of requirements that the frequency-dependent magnetizability is assumed to possess [8].…”