Energy flow in a semi-infinite spatially dispersive absorbing dielectric

Abstract

We consider energy flow in a semi-infinite spatially dispersive absorbing dielectric bounded by vacuum, on which light is incident from the vacuum, with the direction of propagation normal to the surface. While the total energy flux in the vacuum is given by its electromagnetic Poynting vector, the total energy flux in the crystal is given by the sum of the electromagnetic Poynting vector and a mechanical Poynting vector, which arises from the energy transported by the excitations in the medium. We consider two models, the first of which is a semi-infinite medium in the dielectric approximation, which consists of assuming that the nonlocal dielectric function of the semi-infinite crystal is that of the infinitely extended medium, so that the surface of the medium enters the theory only through the restriction of the coordinates normal to the surface to the half-space occupied by the medium. We show that this model fails to conserve energy in the sense that the surface acts as a source of energy, because for this model the mechanical Poynting vector is positive on the surface, while the electromagnetic Poynting vector is continuous across the surface. The second model considered includes the effects of a surface in a phenomenological way in the dielectric function. We find that for this model energy is conserved; that is, the surface acts neither as a source nor sink of energy. Finally, we find that for normal incidence, when the additional boundary condition for a given model of a spatially dispersive dielectric medium may be expressed in terms of the polarization $\overrightarrow{P}\mathrm{}\left(z\right)\mathrm{}$ associated with the dipole-active excitation in the medium as ${\left[\alpha \overrightarrow{P}\mathrm{}\right(z)+\beta \frac{d\overrightarrow{P}\mathrm{}\left(z\right)\mathrm{}}{\mathrm{dz}}]}_{z = 0}=0\mathrm{}$, the coefficients $\alpha$ and $\beta$ determine whether or not the surface is a source or sink of energy. The mechanical Poynting vector is proportional to $\mathrm{Im}\left(\frac{\alpha }{\beta }\right)$, so that when this quantity vanishes, the surface is neither a source nor sink of energy, but when it is nonzero, the surface is either a source or sink of energy, depending on the sign of the other constants entering into the expression for the mechanical Poynting vector.