  This contribution is an update of a previous presentation of 5D matter- wave
optics and interferometry with a correction of some algebraic errors.
Electromagnetic interactions are explicitly added in the 5D metric tensor in
complete analogy with Kaluza's work. The 5D Lagrangian is rederived and an
expression for the Hamiltonian suitable for the parabolic approximation is
presented. The corresponding equations of motion are also given. The 5D action
is shown to cancel for the actual trajectory which is a null geodesics of the
5D metric. This presentation is mainly devoted to the classical aspects of the
theory and only general consequences for the quantum phase of matter-waves are
outlined. The application to Bord\'e- Ramsey interferometers is given as an
illustration.
