The Alvarez–Lohmann lens as a do-nothing machine

Abstract

A do-nothing machine uses complex methods to convert an input pattern or function into an identical output pattern or function. They are of great interest because a selective modification somewhere within that machine can often accomplish useful results. Here, we start with the simplest possible optical system—a piece of glass split between the top and bottom by some simple or complex cut and rejoined perfectly. To light passing through, the cut does nothing. But, if we displace the top and bottom parts by some small amount, the combined system can serve as a variable thickness plate, a variable deflection angle prism, a variable focal length lens, etc. as first shown independently by Alvarez and Lohmann. This appears to be the first explanation of Alvarez–Lohmann lenses in such a simple way and it leads to the possibility of new structures not contemplated by either of the inventors.