For some application I was interested in an optics s.t. for any incoming ray the outcoming ray is parallel to the incoming ray and shifted by a distance $d$ (independent of the incoming ray) along the optical axis. The system should be shorter than $d$ for the application.
Such a system would image any point to a point shifted by $d$ along the optical axis.
I am interested if there is a name for such an optical system, if it is being used somewhere how it would look like and where it is available.
I know that a grin lens with pitch 1.0 would have this property but this system would be as long as $d$ which makes it infeasible for my application.
I can theoretically construct such a system using multiple thin lenses, either four lenses with focal length f=d/8 or three lenses with focal lengths f, f/2, f in this order. Two consecutive lenses are placed 2f apart from each other in both cases (see below). However I don't know if one can make a good real design with small aberration etc. out of this.