The frequency crossover for the Goos–Hänchen shift

Abstract

Abstract. 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.

Maximal breaking of symmetry at critical angles and a closed-form expression for angular deviations of the Snell law

“…Shift experimentally observed in 1947 [3] and for which, one year later, Artman [4] proposed an analytical expression. The Artman formulas, valid for incidence angle greater than the critical angle, have been recently generalized for incidence at critical angle [14]. Notwithstanding the interesting nuances involved in the study of the Goos-Hanchën shift, what we aim to discuss in detail in this paper is the angular deviation from the optical path predicted by the Snell law.…”

“…The most important examples are surely represented by the Goos-Hänchen [3][4][5][6][7][8][9][10][11][12][13][14] and effects. For total internal reflection, Fresnel coefficients gain an additional phase and this phase is responsible for the transversal shift of linearly and elliptically polarized light with respect to the optical beam path predicted by the Snell law.…”

The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks

“…In section IV, we introduce the angular notation to calculate the first and second order contribution of the optical phase. As observed before, the first order contribution of the optical phase is responsible, in its geometrical part, for the optical path [14] predicted by the Snell and reflection laws and, in its Fresnel part, for the additional lateral displacement known as Goos-Hänchen shift [8,9,[25][26][27][28][29]. The second order contribution acts on the transversal symmetry and on the axial spreading of the optical beam [21].…”

Experimental confirmation of the transversal symmetry breaking in laser profiles

“…4 n cos ψ cos θ (n cos θ + cos ψ) 2 (7) are the Fresnel transmission coefficients for transverse electric (TE) and transverse magnetic (TM) waves [5]. The angle ψ is obtained from the incidence angle θ by the Snell law, sin θ = n sin ψ, and the angle ϕ (incidence at the down dielectric air interface) is given by ψ + π 4 [5].…”

“…Studies of the optical beam phase confirmed [1] or suggested deviations [2][3][4][5][6][7][8] from the law of geometrical optics [9,10]. For example, the first order Taylor expansion of the Snell phase represents an alternative method to calculate, by the stationary phase method [1], the optical path of light beams.…”