On the orthogonality of the phase velocity and its feasibility for plane waves

Phase and group velocities of plane waves may have different directions in a linear and homogeneous medium. This difference may lead to interesting situations such as the phase reversal propagation and the orthogonal phase velocity. Although the former has been analyzed in depth and it has found many applications, the latter has been left as a hypothesis. Actually, due to our best knowledge there is no evidence of such media where the phase velocity is orthogonal to the group velocity. In this paper we discuss this situation thoroughly. We show that a plane wave with orthogonal phase velocity cannot possess linear momentum. Moreover, we show that a plane wave without the linear momentum cannot propagate at all, so the orthogonality of the phase velocity does not take place in reality for electromagnetic plane waves.