Adaptive Fresnel lens: basic theory

We present a design for an adaptive Fresnel lens. The focal length can be altered by rotating two components relative to each other, each comprising modified cylindrical lenses wound into spirals. The main light-ray-direction change is due to transmission through pairs of parallel cylindrical lenses whose focal lengths have equal magnitude but opposite signs. Shifting the cylindrical lenses relative to each other perpendicular to the common cylinder-axis direction changes the directions of transmitted light rays, and by designing each component to be one long, thin, cylindrical lens whose focal length changes along its length and is bent into a suitable spiral, rotation of the components relative to each other shifts the cylindrical lenses as required. Adding the thickness profiles of the two parts of a cylindrical Alvarez-Lohmann lens to those of the two cylindrical lenses significantly improves the optical performance. We support our construction with raytracing simulations and discuss potential applications.