The paraxial imaging geometry of any rotationallyaxially symmetric optical system can be described by three real parameters: the axial positions of the two principal planes and one focal length (i.e. $f_f=f_b$ in the diagrams below - or in the case where the input and output refractive indices are different, the focal lengths are different but their ratio is the ratio between the refractive indices). A ray entering one principal plane "teleports" to the other to emerge at the same transverse position in the other.
This is an amazingly simple (i.e. only three parameters) model for any combination of rotationallyaxially symmetric optical devices. Is there a simple proof that it always works, maybe grounded on general, Hamiltonian optics or Fermat's principle? Or, are there cases which it is not adequate for?