Closed form expression for the Goos-Haenchen lateral displacement
  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. 
Manoel P. Araujo, Stefano De Leo, Gabriel G. Maia
physics.optics