Closed-form expression for the Goos-Hänchen lateral displacement

The Artmann formula provides an accurate determination of the Goos-Hänchen 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-Hänchen 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.

Figure 1

Dielectric blocks geometry. Dielectric structure composed by $N$ triangular prisms. The GH shift of the outgoing beam with respect to the path predicted by geometrical optics is proportional to the number of triangular prisms, $N{d}_{\mathrm{GH}}$.

Figure 2

GH shift for Gaussian optical beams. The GH shift of the intensity's maximum and mean value is plotted, for TE and TM waves, as a function of the incidence angle ${\theta }_0$ for Gaussian beams with $\lambda =0.633$ nm and different beam waist $w_0=0.5$, 1.0, 2.0 mm. The analytical expressions for the maximum, given in Eqs. (16) and (21) and represented by solid blue lines, and for the mean value, given in Eqs. (27) and (30) and represented by dashed red lines, are in excellent agreement with the numerical data (gray dots). The difference between these curves clearly shows the breaking of symmetry in the critical region, ${\theta }_0\in [{\theta }_{\mathrm{cri}}-\frac{\lambda }{{\mathrm{w}}_0},{\theta }_{\mathrm{cri}}+\frac{\lambda }{{\mathrm{w}}_0}]$.

Figure 3

Dependence of the GH shift on the angular distribution shape. In the critical region, ${\theta }_0\in [{\theta }_{\mathrm{cri}}-\frac{\lambda }{{\mathrm{w}}_0},{\theta }_{\mathrm{cri}}+\frac{\lambda }{{\mathrm{w}}_0}]$, the GH shift is dependent on the shape of the angular distribution. As an example, we plot the intensity maximum for Gaussian (solid blue lines) and box (dashed green lines) angular distributions. In the Artmann zone, where the plane-wave limit is valid, no difference is found.