De Broglie Wave’s Snell’s law, Fresnel equations, and Brewster’s angle

Abstract

A systematic comparative study demonstrates that the reflection and refraction of de Broglie waves at a semiconductor heterojunction are analogous to those of electromagnetic waves. We present the quantum reflection rule, Snell’s law, and Fresnel equations. We define the concept of transition energy for the de Broglie wave. Without loss of generality, we use the AlAs/GaAs heterojunction as an example. At the transition energy, electrons pass through the heterojunction without refraction or bending yet retain reflectivity. When the electron’s energy exceeds the transition energy, Brewster’s angle exists, at which the incident de Broglie wave exhibits zero reflectance, similar to the transverse magnetic (TM) mode of electromagnetic waves. However, below the transition energy, no Brewster’s angle or zero reflectance is present, similar to the transverse electric (TE) mode of electromagnetic waves. This transition energy also marks the “refractive index” inversion boundary for total internal reflection.