The laser beam shaping technique is widely used in laser processing, welding, display, lighting, and other applications. The freeform surface is extensively applied in laser beam shaping because of its high design freedom, accurate control of light distribution, and high transmittance. It is known that a freeform surface with good smoothness is easier to be manufactured. Therefore, it is crucial to design a freeform lens with good smoothness. In the paper, a design method of lenses with good-smoothness freeform surfaces is presented. The optimal mesh distribution on the target plane is generated by the Poisson mesh optimization algorithm. With the optimal mesh distribution on the target plane, the normal vector at each sampling point on the free surface is calculated according to the energy mapping between input beam and output beam. The sag of the freeform surface can be obtained by solving Poisson equation established by the normal vectors and sags at sampling points. Finally, the freeform lens with good smoothness is designed. With the freeform lens, the laser beam with a circular aperture can be shaped into a rectangular spot on the target surface with uniform irradiance distribution.

Firstly, the initial meshes on the cross-section of the incident beam and the target plane are generated. The mesh distribution on the target plane is optimized by the Poisson mesh optimization algorithm, in which an error function is employed to reflect the energy distribution error between mesh on the incident section and expected energy distribution of the corresponding target surface mesh. The partial differential equation (PDE) of the error function and the pressure field is constructed by the idea of fluid mechanics. Then, the finite difference method is employed to solve the PDE so as to calculate the distribution of the pressure field. After the gradient of the pressure field is calculated, a displacement vector field can be obtained, which determines the direction and magnitude of movement of every vertex in the target plane mesh. By the method, the optimal mesh distribution on the target plane can be obtained. Given the optimal distribution, the normal vector at each sampling point on the freeform surface can be calculated according to the mapping relationship between incident and outgoing rays. The sag of the freeform surface can be obtained by the solution to Poisson equation established by the normal vectors and sags at sampling points. Finally, the assembly tolerances of the freeform lens are analyzed by a random statistical analysis method.

Two freeform lenses are designed to transform the circular laser beam with Gaussian irradiance distribution to that with uniform irradiance distribution on square and rectangular target planes with uniformity of 91% and 93% (Fig. 7), respectively. The size of the two target planes is 30 mm×30 mm and 60 mm×40 mm, respectively. To verify the smoothness of the freeform surface, a polynomial is used for fitting, which has nine terms, and the highest order is six. The RMSE after fitting is about 1.394×10^-3 (Figs. 8 and 9). The uniformity of the target plane remains almost unchanged when the freeform lens is constructed with the fitted data points. It is shown that the freeform surface designed by the method presented in the paper has good continuity and smoothness. Finally, the assembly tolerances of the freeform lens are analyzed by a random statistical analysis method. The results show that within the given tolerance range, the change in uniformity is less than 6%, and the uniformity can be maintained at about 88% for most of the samples (Figs. 10 and 11). Only about 1% of the samples report a decrease in uniformity by more than 10%.

In this paper, a freeform lens design method for laser beam shaping is proposed, which has two key steps, Poisson mesh optimization and freeform surface construction by the solution to Poisson equation. The Poisson mesh optimization algorithm is mainly used to optimize the mesh distribution on the target plane so that the light distribution on the target plane meets the expected distribution. After four iterations, the optimal mesh distribution is achieved on the target plane. Given the optimal mesh distribution on the target plane, the normal vector at each sampling point on the free surface is calculated according to the energy mapping between input beam and output beam. The sag of the freeform surface can be obtained by the solution to Poisson equation established by the normal vectors and sags at the sample points. In this way, the freeform lens is designed. To verify the feasibility of the method, this study designs two freeform lenses to transform the circular laser beam with Gaussian irradiance distribution to that with uniform irradiance distribution on square and rectangular target planes, with uniformity of 91% and 93%, respectively. A polynomial is used for fitting to verify the smoothness of the freeform surface, which has nine terms, and the highest order is six. The RMSE after fitting is about 1.394×10^-3. The uniformity of the target plane remains almost unchanged when the freeform lens is constructed with the fitted data points. Finally, the assembly tolerances of the freeform lens are analyzed by a random statistical analysis method. The results show that the uniformity of most of the samples is higher than 88% within the given tolerance range. The design is of good practical application value.