A remarkable focusing property of a parabolic mirror for neutrons in the gravitational field: Geometric proof

An extraordinary focusing property of a parabolic mirror for ultracold neutrons in the presence of the gravitational field was first reported by Steyerl and co-authors. It was shown that all neutrons emitted from the focus of the mirror will be reflected back upon the same focus passing a point of return in the gravitational field in between. The present note offers a complementary geometric proof of this feature and discusses its application to many-mirror systems. The results can also be applied to electrons and ions in an electric field.