Analytical design of refractive optical elements generating a curve

A method for designing refractive optical elements focusing collimated incident beam into a curve with specified shape is proposed. A general relationship for the freeform surface of the optical element is derived as an envelope of a parametric family of hyperboloids of revolution which focus the incident beam into the points on the curve. Using the thin optical element approximation, the calculation of the hyperboloid parameters providing required irradiance distribution along the curve is reduced to the solution of an explicit first order differential equation. The application for the proposed method to focus into the set of points on the curve is presented. The optical element generating line segment focus and the composite optical element to focus into the curve that consists of two circular arcs are designed. The optical elements to focus into the set of points on the line-segment and on the curve that consists of two circle arcs.