Fresnel similarity

We study Fresnel diffraction, in particular the similarity between an object, at plane z = 0, and its diffraction pattern, at z > 0. We define a degree of similarity, σ(z), which depends on the object and the distance z. Perfect self-imaging is associated with a degree of similarity equal to unity, while imperfect self-imaging is classified by a degree between zero and one. We derive the theory that allows us to find σ(z), if the object is known. And we can synthesize an object that exhibits a wanted σ(z). We describe some novel applications.