Lateral Displacement of a Light Beam at a Dielectric Interface*

TLDR

The study investigates the Goos–Hänchen shift of a Gaussian light beam incident from a denser to a rarer medium using a rigorous integral representation of its continuous plane-wave spectrum. By applying a Fresnel approximation to this integral, the authors derive a lateral displacement expression valid for incidence angles arbitrarily close to the critical angle of total reflection. The results show that the shift depends on beam width and incidence angle, deviating from the classical expression near the critical angle where it approaches a beam‑width–dependent constant and attains a maximum slightly above the critical angle, with numerical curves illustrating these behaviors.

Abstract

The lateral (Goos–Hänchen) shift of a gaussian light beam incident from a denser medium upon the interface to a rarer medium is investigated by means of a rigorous integral representation comprising a continuous plane-wave spectrum. By applying a Fresnel approximation to that integral, we derive the lateral displacement for angles of incidence that are arbitrarily close to the critical angle of total reflection. Our results show that, in general, the lateral displacement is a function of the beam width, as well as the incidence angle; the classical expression appears as a limit case which holds only for large beam widths and for incidence angles that are not too close to the critical angle. An analysis of our expression for the beam shift reveals that, as the incidence angle approaches the critical angle of total reflection, the beam shift approaches a constant value that is strongly dependent on the beam width, in contrast to the classical expression, which predicts an infinitely large displacement; we also find that the maximum lateral displacement occurs at an angle that is slightly larger than the critical angle. Numerical results are presented in terms of normalized curves that are applicable to a wide range of realistic beams.