Teaching wave propagation in materials: Despite light propagates also in dense materials with the same wavelength as in vacuum it propagates slower- what is the real reason for this apparent contradiction?

Prof. Kurt HINGERL, head of the institute for surface and nanoanalytics at the Johannes Kepler University, Linz, Austria. 

The aim of this Tutorial is to show students how the concept of a refractive index in a solid (or liquid) arises for monochromatic plane waves without ever postulating the macroscopic Maxwell equations including the displacement (respectively magnetic) field or without using any constitutive relations for the dielectric polarization density P, the dielectric susceptibility χ, the current j or the conductivity σ. 

The underlying concept is the same as presented in the chapter on the Ewald-Oseen theorem in the book by Born and Wolf[i][i]: Matter is made up of dipoles, which start to oscillate if an external field hits them, but they also radiate, because charges are accelerated. It is then shown by comparison that he superposition results in an effective field fully equivalent to the macroscopic Maxwell equations. However, the mathematics in the mentioned reference as well as in didactical variations[ii][ii, iii, iv] is rather intricate and uses sometimes prior knowledge, e.g. the Lorentz- Lorentz relations (usually taught for longitudinal electric fields), the Fresnel relations or they use macroscopic/ mesoscopic quantities as susceptibility or refractive index. 

Contrary, here we specify a simple geometry (either a dipole sheet, or a semi-infinite half space) and use only the microscopic (thereby closed and mathematically well defined) Maxwell equations are used. Using this approach, it will get clear how an “effective” wavevector in a medium arises and that “between the atoms” light moves with the speed of light in vacuum.