The optical design with arbitrary illuminance distribution is always a key concern of non-imaging optics due to its difficulty in solving the smooth mapping relationship. At present, the Monge-Ampère equation method, the supporting quadratic method (SQM), and the tailor method can solve this problem, but they all have drawbacks such as complex ideas and difficult manufacture. To seek a simple and efficient method, we start with mapping grids partitioning, and study the characteristic and optimization method of the integrability and energy matching of grids, providing references for the design of non-uniform freeform lenses.

The discussed mapping grid partitioning method is shown in Fig. 1. Equal initial quadrilateral grids are performed within the unit circle of source projection and the rectangular target plane to make the grid nodes correspond one by one. According to the relationship in Fig. 2, the energy is calculated by the grid area, and the area evaluation function is constructed based on the difference between the grid area and the ideal one. The integrability evaluation function is constructed based on the direction derivative residual of the nodes, and the weights are set according to the design. Under the guidance of the evaluation function, gradient operation is performed on the area difference to solve the area optimization vectors, and integrability optimization vectors are solved by the simplified node angle relationship in Fig. 3 and are combined based on the weights. The nodes are moved according to the comprehensive vectors and subjected to boundary conditions and topological conservation relationships. New grids are generated and a new value of the evaluation function is calculated. When the value tends towards 0, the energy matching and integrability of the mapping are improved.

To verify the effectiveness of the method, we design a freeform lens that can display a non-uniform rectangular spot with the words "ZJUT" on the target. The effective angle of the source is set to 45° and initial grids are made within the corresponding projection circle and target plane, with the final result shown in Fig. 4. According to Fig. 5, the area evaluation rapidly decreases during the iteration, and the integrability evaluation of the source also significantly reduces. However, as the grid is no longer uniform to fit the energy distribution, the integrability evaluation of the target has increased compared to the evaluation at the beginning. The trend of the comprehensive evaluation function is consistent with the area evaluation, and the overall energy distribution and integrability are ultimately balanced based on weights. The final simulation results are shown in Fig. 7, and the illumination ratio of the words to the background is 2∶1, which is in line with the design. The font's boundary is clear, and the overall rectangular boundary maintains sound. There are slight defects in the four corners, which can be attributed to the poor integrability of the four corners of the light source grid.

Based on the grid partitioning, a mapping optimization method is proposed. This method constructs a comprehensive evaluation function for energy distribution and integrability, and optimizes the mapping through iteration under the combined effect of area difference gradient and integrability optimization vectors to make the mapping have good integrability and meet the energy distribution. The lens solved via mapping has been simulated and confirmed that the spot achieves the expected shape and illumination, with slight defects on the edge. There has also been a significant improvement in design efficiency. In summary, although this method still has optimization space in boundary and non-Lambert sources, its method is simple, with sound effect and improved design efficiency. Meanwhile, it is expected to play a role in customized fields such as non-uniform spot design.