Total Reflection of Matter Waves: The Goos–Haenchen Effect for Grazing Incidence*

Abstract

At total reflection, beams are not reflected ideally from their point of incidence at the reflecting boundary but shifted by a length D (Goos–Haenchen effect). Based on the model of wave bundles consisting of a composite of totally reflected partial waves with varying angles of incidence, we treat the case of matter waves for a vanishing potential step at the boundary (the critical angle of total reflection tending to π/2). It is well known from Renard’s formula that the partial wave for grazing incidence has a vanishing shift D, with an angle α0 of incidence midway between the critical angle and π/2. However, the Goos–Haenchen shift of a wave bundle goes to infinity at α0 → π/2. The width W of the bundle also goes to infinity because the angular interval of the partial waves goes to zero. But the quantity D/W remains constant. D/W has a value of 18% if Artmann’s formula for D is used and is 4.5% with Renard’s more refined formula for D.