Freeform construction method for illumination design by using two orthogonal tangent vectors based on ray mapping.

An illumination design problem can be transformed into an optimal mass transport problem based on ray mapping. To construct a freeform surface that best fits the normal field, an efficient numerical method is put forward in this paper. In this method, the normal vectors are constructed by two adjacent orthogonal tangent vectors at each point, and then the normal vectors are substituted into Snell's law to obtain nonlinear equations describing the surface coordinates. Finally, the continuous and accurate freeform surface can be obtained by solving these nonlinear equations. The simulation results show that the proposed method not only provides lower relative standard deviation, but also significantly reduces the normal deviation more than the traditional one. It can be seen from the comparison results that different numerical integrations of a non-integrable normal field calculated by optimal mass transport can lead to different results, and the proposed method is more feasible than the traditional one, especially in the off-axis case. The simulation results of the illumination effect of some complex patterns also show that the freeform surface constructed by this method can restore the target pattern efficiently and control the normal vector error in a low range.