  Systematic spectroscopic ellipsometry investigations have been performed in
order to elucidate the asymmetric insulator-to-metal transition in thin VO$_2$
films. The comprehensive analysis of the obtained macroscopic optical response
yields a hysteretic behavior, and in particular its asymmetry, when performed
in the framework of an anisotropic effective medium approximation taking into
account the volume fraction of the metal inclusions as well as their shape. We
reveal microscopic details of the percolation transition, namely that the shape
of the metal inclusions goes through several plateaus, as seen in the evolution
of the shape factor on both sides of the transition region and resulting in
different critical volume fractions at the transition for the heating and
cooling cycles.
