Angular deviations: From a cubic equation to a universal closed formula to determine the peak position of reflected and (upper) transmitted beams

TL;DR

This paper derives an analytical, universal formula for calculating the peak positions of reflected and transmitted optical beams through a dielectric prism, improving understanding of angular deviations in optics.

Contribution

It introduces a universal closed-form solution for the cubic equation governing angular deviations, applicable near Brewster angles, enhancing theoretical and experimental optical analysis.

Findings

Excellent agreement between analytic and numerical results.

Universal formula simplifies analysis near Brewster angles.

Potential applications in weak measurement experiments.

Abstract

Angular deviations and lateral displacements are optical effects widely investigated in literature. In this paper, by using the Taylor expansion of the Fresnel coefficients, we obtain an analytic expression for the beam reflected by and (upper) transmitted through a dielectric prism. These analytical approximations lead to a cubic equation which allows to determine the angular deviations of the optical beams. Near the Brewster angles, under specific conditions, we obtain a universal formulation for the cubic equation. Its explicit solution determines the peak position of the reflected and (upper) transmitted beams. The universal solution could be of great utility in future experimental implementations. The analytic results show an excellent agreement with the numerical calculation and the analytic expressions given for the reflected and (upper) transmitted beams should play an important role in the weak measurements analysis.