In numerous lighting situations, rectangular illumination is a common lighting requirement. At present, several methods can realize rectangular illumination, such as the fly-eye lens [
1,
2], micro-lens array [
3–
5], total internal reflection (TIR) lens [
6,
7], and freeform lens. The fly-eye lens is composed of a collimating system and fly-eye lenses. Each fly-eye lens has the same focal length and is embedded in a square base board. Although this method can provide rectangular illumination, the system structure is complex. The absorption from the material of multiple lenses is large, light energy is wasted, and the optical efficiency is low. The micro-lens array method uses a collimating system with a micro-lens array on top to achieve rectangular illumination. The collimating system employs a TIR collimating lens [
3] or Fresnel lens [
4] to converge the light from a LED and emit it in a small angle. The micro-lens array plays an important role in the redistribution of the beam angle and optical energy. Compared with the fly-eye lens method, the micro-lens array method provides more accurate light ray control. The system structure still contains two parts, which is relatively complex. The freeform lens technique is a common method to realize rectangular illumination directly. A system with a freeform lens has the advantages of small volume and a high degree of freedom in its design. Rectangular illumination with a TIR lens can be achieved by using composite ray mapping with a special structure of freeform lens. Traditional design methods of freeform surfaces can be classified as numerical solutions obtained by suitable group differential equations [
8–
11] and mapping between the source and the target [
12–
16]. The process of solving partial differential equations is complex and the solvability is poor, and this technique cannot solve all lens surfaces. In addition, this method cannot guarantee that every lens structure has a smooth optical surface. The mapping method needs to partition the source intensity distribution in spherical (
q,
ϕ) coordinates or (
u,
v) coordinates and partition its corresponding target irradiance distribution into a grid. The illumination quality of a lens is directly determined by dense sampling at the source and target regions. The feedback modification method is typically used together with the mapping technique. This approach requires a large number of complex iterative calculations. Hu Run et al. used the mapping method to establish the ray relationship between the source and the target plane by using vectors. The source angle
qi and the corresponding radius
ri of each ray were calculated with the energy conservation law. Although two freeform surfaces can be calculated simultaneously, the numerical solution is complex [
17,
18].