Modelling Real Valued Functions via Optical Lenses

In this study, we modeled real valued functions using freeform lenses. In our model, the bottom surface of the lens is flat whereas its top surface is determined by a function, f(x). We consider vertically coming light rays with x-coordinate x. Our aim is to find f(x) such that x is mapped to F(x), the horizontal position where the light ray leaves the bottom surface. We have found the nonlinear differential equation for a generic lens to model a given function. Namely, given F(x), the solution of the differential equation gives us the lens surface f(x). Finally, we have calculated the lens surface for four functions numerically and have provided their plots respectively.