Analytical design of freeform optical elements generating an arbitrary-shape curve.

A method is proposed for designing refractive optical elements focusing a collimated incident beam into a curve with specified shape. A general relationship for the freeform surface of the optical element is derived as an envelope of a parametric family of hyperboloids of revolution that focuses 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. Optical elements generating line segment focus and circular arc focus are designed. The simulation results demonstrate generation of high-quality curves.