The quaternionic Goos-Haenchen shift

TL;DR

This paper explores how quaternionic potentials influence the lateral displacement of electronic waves during reflection, revealing amplified and additional shifts compared to the complex case, with potential for experimental observation.

Contribution

It introduces a theoretical framework for quaternionic potentials affecting wave reflection, providing analytical formulas to distinguish quaternionic effects from complex ones.

Findings

Quaternionic potentials amplify the Goos-Haenchen shift during total reflection.

An additional lateral shift occurs below the critical angle in the quaternionic case.

Analytical formulas enable quantitative comparison between complex and quaternionic shifts.

Abstract

We investigate the lateral displacement of electronic waves when partially and totally reflected by a quaternionic potential. Following the analogy between Quantum Mechanics and Optics we introduce a refractive index for the complex and the pure quaternionic case. For incidence greater than the critical one (total reflection) the quaternionic potentials amplify the lateral displacement found in the complex case. For incidence below the critical angle, we find, in the quaternionic case, an additional shift. The analytical formula found for the quaternionic lateral displacement gives the possibility to observe quantitative and qualitative differences between the complex and quaternionic Goos-Haenchen shift.