Optics that translates object to image by a constant amount

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.

• If you really just want a shift, why not use something similar to periscope? – noah Mar 21 at 16:35
• I want the optics and the translation along one line. Furthermore the optics should be shorter than the translation. – user100927 Mar 21 at 17:02
• Doesn't a rectangular glass block do this automatically? If I understand correctly in your diagram the angle of transmission is equal to the angle of incidence. So any rectangular material with a refractive index will do this. – user3502079 Mar 21 at 17:05
• the incoming and outgoing rays are parallel if we use a glass block but the translation depends on the direction of the incoming ray. A glass block does no imaging. – user100927 Mar 21 at 17:18
• Try using two identical lenses separated by 4f. – S. McGrew Mar 22 at 13:04