Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium

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Abstract
We study the temporal and spatial behavior of the real, instantaneous Poynting vector in an absorptive and/or conductive medium, which is characterized by complex permittivity and permeability and bounded by vacuum. For light polarized either parallel or perpendicular to the plane of incidence, the direction of the Poynting vector changes with time. The tip of this vector describes an ellipse—the Poynting ellipse—for any point in the medium. The plane of this ellipse is parallel to the plane of incidence, its center is given by the usual time-averaged Poynting vector, and it is tilted with respect to the surface normal. The size of the Poynting ellipse decreases exponentially with depth into the medium. In general the time-averaged Poynting vectors for p (TM) and s (TE) polarized light are not parallel, and neither vector is perpendicular to the planes of constant phase. Therefore a beam of arbitrary polarization must split, i.e., double refraction takes place as a consequence of attenuation. The effect is small for transparant media; however, the effect should be significant as absorption lines are approached.