In systematical design thinking, the logic from functional design to structure design is followed. For the design tasks in homeomorphic zones, the optical behavior of the element can be considered as a local modulation of the input field, shaping the phase and redistributing the irradiance.
Usually, the homeomorphism of the irradiance redistribution problem is mathematically derived as the L2 Monge-Kantorovich problem. Numerical algorithms have been proposed to solve the problem, and mesh is generated by which the energy conservation law is satisfied locally. By connecting the corresponding mesh nodes between the two meshes, the result gives an idea of how the energy transforms locally, therefore of how the functional design could be achieved.

Pic-Design Homeomorphicsm for Light Shaping
Figure 1: Design of homeomorphisam: (a) uniform irradiance with square boundary; (b) specific irradiance distribution with circular boundary; (c), (d) corresponding mesh for both irradiance distributions.