Quantized Longitudinal and Transverse Shifts Associated with Total Internal Reflection

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Abstract
In additional measurements following their experimental demonstration of the now well-known longitudinal shift of a light beam undergoing total reflection, Goos and Hänchen have observed that an incident beam of natural light is separated into two beams with polarizations normal and parallel to the incidence plane. Although an explanation of this filtering effect of orthogonal modes is easy in terms of either the Artmann or the Renard formulas of the Goos-Hänchen shift, which are deduced from a stationary phase and an energy-conservation argument, respectively, it seems that this explation has not yet been produced. We describe here associated theoretical and experimental work showing that both the longitudinal Goos-Hänchen shift and the new transverse shift are quantized, each having two eigenvalues and two eigenfunctions, which are the principal linear polarization states in the first case and the circularly polarized states in the second case. Thus the longitudinal and the transverse shifts should not be simultaneously observable. This can be justified in terms of the so-called tangential and sagittal focal lines produced by total reflection from a point source. Our theoretical reasoning is based on formal properties of Poynting's energy-flux vector. Our experiments consist of (1) a confirmation, with photographic recording, of the Goos-Hänchen "polarization effect," (2) a similar demonstration of the filtering of the two circular polarization modes by observation of the transverse shift. In the latter case our apparatus is similar to the one we have used for demonstrating the transverse shift, but with the circular polarization analyzer placed after rather than before the totally reflecting prism. Our new measurements also comprise an improved evaluation of the transverse shift.