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

TL;DR

This paper investigates conditions for maximal symmetry breaking in laser beam propagation at critical angles, deriving an analytic formula for angular deviations and the Goos-Haenchen shift, with implications for optical beam control.

Contribution

It provides a new closed-form expression for angular deviations at critical angles, overcoming previous infinities, and analyzes symmetry breaking phenomena in laser beam propagation.

Findings

Maximal symmetry breaking conditions identified.

Closed-form formula for Goos-Haenchen shift derived.

Multiple peaks in angular distribution observed.

Abstract

A detailed analysis of the propagation of laser gaussian beams at critical angles shows in which conditions it is possible to maximize the breaking of symmetry in the angular distribution and for which values of the laser wavelength and beam waist is possible to find an analytic formula for angular deviations of the Snell law. For propagation throughout $N$ dielectric blocks and for a full breaking of symmetry, overcoming the well known problem of the infinity at critical angle, a closed expression for the Goos-Haenchen shift is obtained. The multiple peaks phenomenon clearly represents an additional evidence of the breaking of symmetry in the angular distribution of optical beams. Finally, laser wavelength and beam waist conditions to produce focal effects in the outgoing beam are also briefly discussed.