Fig. 1. Schematic diagram of a coaxial double free-form beam shaping system

Fig. 2. Grid diagram of energy mapping between the input beam and output beam

Fig. 3. (a) Schematic diagram of solving adjacent points on the free-form surface; (b) Schematic diagram of solving all discrete points of the free-form surface

Fig. 4. The difference between the virtual surface and the free-form surface in the design process of a double free-from beam shaping system

Fig. 5. Schematic diagram of the off-axis two-mirror beam shaping system

Fig. 6. (a) Influence of the beam amplification ratio on the misalignment; (b) Influence of the axial distance between two free-form surfaces on the misalignment

Fig. 7. Diagram of the virtual surface iteration process

Fig. 8. Misalignment decreases with the number of iterations

Fig. 9. (a) Ray tracing simulation diagram of a system designed by the Single Virtual Surface method; (b) Ray tracing simulation diagram of a system designed by the VSI method; (c) Simulation diagram of the irradiance of the target surface of the system designed by the Single Virtual Surface method; (d) Simulation diagram of the irradiance of the target surface of the system designed by the VSI method

Fig. 10. (a) Comparison diagram of misalignment of beam shaping system with different beam amplification ratios designed by the two methods; (b) Comparison diagram of misalignment of beam shaping system with different axial distances designed by the two methods

Fig. 11. (a) Ray tracing simulation diagram of the off-axis two-mirror beam shaping system designed by the Single Virtual Surface method; (b) Ray tracing simulation diagram of the off-axis two-mirror beam shaping system designed by the VSI method; (c) Simulation diagram of the irradiance of the target surface of the off-axis two-mirror beam shaping system designed by the Single Virtual Surface method; (d) Simulation diagram of the irradiance of the target surface of the off-axis two-mirror beam shaping system designed by the VSI method

Fig. 12. Comparison diagram of misalignment of beam shaping system with different distances between optical axes designed by the two methods

Table 1. Simulation results of beam shaping systems with different beam amplification ratios designed by the two methods

Table 2. Simulation results of beam shaping systems with different axial distances designed by the two methods