Manoel P. Araújo

Closed form expression for the Goos-Haenchen lateral displacement

The Artmann formula provides an accurate determination of the Goos-Hanchen lateral displacement in terms of the light wavelength, refractive index and in- cidence angle. In the total reflection region, this formula is widely used in the literature and confirmed by exper- iments. 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 for- mula is in excellent agreement with numerical calcula- tions and reproduces, for incidence angles greater than critical ones, the Artmann formula. The closed form also allows one to understand how the breaking of sym- metry in the angular distribution is responsible for the difference between measurements done by considering the maximum and the mean value of the beam inten- sity. The results obtained in this study clearly show the Goos-Hanchen lateral displacement dependence on the angular distribution shape of the incoming beam. Finally, we also present a brief comparison with experi- mental data and other analytical formulas found in the literature.

The asymmetric Goos–Hänchen effect

We show under which conditions optical Gaussian beams, propagating throughout an homogeneous dielectric right angle prism, present an asymmetric Goos–Hanchen (GH) effect. This asymmetric behavior is seen for incidence at critical angles and happens in the propagation direction of the outgoing beam. The asymmetric GH effect can also be seen as an amplification of the standard GH shift. Due to the fact that it only depends on the ratio between the wavelength and the minimal waist size of the incoming Gaussian beam, it can also be used to determine one of these parameters. Multiple-peak interference is an additional phenomenon seen in the presence of such asymmetric effects.

AXIAL DEPENDENCE OF OPTICAL WEAK MEASUREMENTS IN THE CRITICAL REGION

The interference between optical beams of different polarizations plays a fundamental role in reproducing the optical analog of the electron spin weak measurement. The extraordinary point in optical weak measurements is represented by the possibility to estimate with great accuracy the Goos-Haenchen (GH) shift by measuring the distance between the peak of the outgoing beams for two opposite rotation angles of the polarizers located before and after the dielectric block. Starting from the numerical calculation of the GH shift, which clearly shows a frequency crossover for incidence near to the critical angle, we present a detailed study of the interference between s and p polarized waves in the critical region. This allows to determine in which conditions it is possible to avoid axial deformations and reproduce the GH curves. In view of a possible experimental implementation, we give the expected weak measurement curves for Gaussian lasers of different beam waist sizes propagating through borosilicate (BK7) and fused silica dielectric blocks.

The frequency crossover for the Goos–Hänchen shift

For total reflection, the Goos–Hänchen (GH) shift is proportional to the wavelength of the laser beam. At critical angles, such a shift is instead proportional to the square root of the product of the beam waist and wavelength. By using the stationary phase method (SPM) and, when necessary, numerical calculations, we present a detailed analysis of the frequency crossover for the GH shift. The study, done in different incidence regions, sheds new light on the validity of the analytic formulas found in literature.

Transversal symmetry breaking and axial spreading modification for gaussian optical beams

For a long time, it was believed there was no reason to include the geometrical phase in studying the propagation of gaussian optical beams through dielectric blocks. This can be justified by the fact that the first-order term in the Taylor expansion of this phase is responsible for the lateral shift of the optical beam which is also predicted by ray optics. From this point of view, the geometrical phase can be seen as a purely auxiliary concept. In this paper, we show how the second-order term in the Taylor expansion accounts for the symmetry breaking of the transversal spatial distribution and acts as an axial spreading modifier. These new effects clearly show the importance of the geometrical phase in describing the correct behavior of light. To test our theoretical predictions, we briefly discuss a possible experimental implementation.

Optical weak measurements without removing the Goos–Hänchen phase

Abstract Optical weak measurements are a powerful tool for measuring small shifts of optical paths. When applied to the measurement of the Goos–Hänchen shift, in particular, a special step must be added to its protocol: the removal of the relative Goos-Hänchen phase, since its presence generates a destructive influence on the measurement. There is, however, a lack of description in the literature of the precise effect of the Goos–Hänchen phase on weak measurements. In this paper we address this issue, developing an analytic study for a Gaussian beam transmitted through a dielectric structure. We obtain analytic expressions for weak measurements as a function of the relative Goos–Hänchen phase and show how to remove it without the aid of waveplates.

Oscillatory behavior of light in the composite Goos-Hänchen shift

For incidence in the critical region, the propagation of gaussian lasers through triangular dielectric blocks is characterized by the joint action of angular deviations and lateral displacements. This mixed effect, known as composite Goos-Haenchen shift, produces a lateral displacement dependent on the axial coordinate, recently confirmed by a weak measurement experiment. We discuss under which conditions this axial lateral displacement, which only exists for the composite Goos-Haenchen shift, presents an oscillatory behavior. This oscillation phenomenon shows a peculiar behavior of light for critical incidence and, if experimentally tested, could stimulate further theoretical studies and lead to interesting optical applications.

Optimizing Weak Measurements to Detect Angular Deviations

We analyze and compare the angular deviations for an optical beam reflected by and transmitted through a dielectric triangular prism. The analytic expressions derived for the angular deviations hold for arbitrary incidence angles. For incidence approaching the internal and external Brewster angles, the angular deviations transverse magnetic waves present the same behavior leading to the well-known giant Goos-Hanchen angular shift. For incidence near the critical angle a new region of large shift is seen both for transverse magnetic and transverse electric waves. While a direct measuring procedure is better in the vicinity of the Brewster region, a weak measurement breaks off the giant Goos-Hanchen effect, preserving the amplification in the critical region. We discuss under which conditions it is possible to optimize the amplification and we also determine when a weak measurement is preferred to a direct measuring procedure.