Closed form expression for the Goos-Haenchen lateral displacement

TL;DR

This paper derives a closed-form expression for the Goos-Haenchen lateral displacement that overcomes divergence issues at the critical angle, aligning well with numerical and experimental data.

Contribution

It introduces a novel closed-form formula for the Goos-Haenchen shift for Gaussian beams, resolving divergence at the critical angle and clarifying the impact of angular distribution shape.

Findings

The new formula matches numerical calculations accurately.

It reproduces the Artmann formula for angles beyond the critical angle.

The study highlights the influence of beam angular distribution on the displacement.

Abstract

The Artmann formula provides an accurate determination of the Goos-Haenchen lateral displacement in terms of the light wavelength, refractive index and incidence angle. In the total reflection region, this formula is widely used in the literature and confirmed by experiments. Nevertheless, for incidence at critical angle, it tends to infinity and numerical calculations are needed to reproduce the experimental data. In this paper, we overcome the divergence problem at critical angle and find, for Gaussian beams, a closed formula in terms of modified Bessel functions of the first kind. The formula is in excellent agreement with numerical calculations and reproduces, for incidence angles greater than critical ones, the Artmann formula. The closed form also allows one to understand how the breaking of symmetry in the angular distribution is responsible for the difference between measurements done by considering the maximum and the mean value of the beam intensity. The results obtained in this study clearly show the Goos-Haenchen lateral displacement dependence on the angular distribution shape of the incoming beam. Finally, we also present a brief comparison with experimental data and other analytical formulas found in the literature.