Reflection and Refraction of a Beam of Light at a Plane Interface

Abstract

The reflection and refraction of a beam of light is treated. Approximate solutions of Maxwell’s equations are used to describe the electromagnetic field of the beam being limited in the transverse direction. The point of departure is the classical paper by Schaefer and Pich. The laws of reflection and refraction are derived. Fresnel’s formulas and their corrections are presented for both polarizations. The case of total reflection is investigated for E polarization in greater detail. The electromagnetic fields and the time-average Poynting vectors are explicitly derived for both the optically dense and less-dense media. The flow of energy at total reflection is studied extensively. It is shown that, due to the flow of energy in the less-dense medium, the center of gravity of the reflected beam is displaced, as was suggested by v. Fragstein. This leads to a shift of the totally reflected beam with respect to a geometrically reflected beam, as was experimentally demonstrated by Goos and (Lindberg-) Haenchen. New expressions for this “Goos–Haenchen shift” are derived. These expressions reduce to the classical formulas deduced by Artmann, v. Fragstein, Wolter, and Maecker if the angle of incidence is only slightly larger than the critical angle of total reflection. In consistency with Renard’s viewpoint they, however, predict vanishing shifts in the limit of grazing incidence.